]>
2016
9
4
ISSN 2008-1898
541
Construction of Tri-parametric derivative free fourth order with and without memory iterative method
Construction of Tri-parametric derivative free fourth order with and without memory iterative method
en
en
We have given two general methods of converting with derivative two-step methods to without derivative two-step methods. It can also be observed that this conversion not only retain the optimal order
of convergence of the two-step methods but the resulting derivative free families of iterative methods are
also extendable to with memory class. The with-memory methods show greater acceleration in the order
of convergence. In this way, order of convergence is accelerated from 4 to 7.53 at the most. An extensive
comparison of our methods is done with the recent methods of respective domain.
1410
1423
F.
Zafar
Centre for Advanced Studies in Pure and Applied Mathematics(CASPAM)
Bahauddin Zakariya University
Pakistan
fizazafar@gmail.com
N.
Yasmin
Centre for Advanced Studies in Pure and Applied Mathematics(CASPAM)
Bahauddin Zakariya University
Pakistan
nusyasmin@yahoo.com
M. A.
Kutbi
Department of Mathematics
King Abdulaziz University
Saudi Arabia
mkutbi@yahoo.com
M.
Zeshan
Centre for Advanced Studies in Pure and Applied Mathematics(CASPAM)
Bahauddin Zakariya University
Pakistan
zeeshan.zeshan@hotmail.com
With and without memory methods
derivative free
self accelerating parameters
accelerated order of convergence.
Article.1.pdf
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[1]
A. Cordero, J. R. Torregrosa, Low complexity root finding iteration functions with no derivatives of any order of convergence, J. Comput. Appl. Math., 275 (2015), 502-515
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A. R. Khan, V. Kumar, N. Hussain, Analytical and numerical treatment of Jungck-Type iterative schemes, Appl. Math. Comput., 231 (2014), 521-535
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T. Lotfi, F. Soleymani, M. Ghorbanzadeh, P. Assari, On the construction of some tri-parametric iterative methods with memory , Numer. Algor., 2015 (2015), 1-11
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J. M. Ortega, W. G. Rheinboldt, Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York (1970)
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F. Soleymani, T. Lotfi, E. Tanakli, F. K. Haghani , Several iterative methods with memory using self accelerator, Appl. Math. Comput., 254 (2015), 452-458
##[8]
J. F. Traub , Iterative Methods for the Solution of Equations, Prentice Hall, New York, USA (1964)
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X. Wang, T. Zhang, Y. Qin, Efficient two-step derivative-free iterative methods with memory and their dynamics, Int. J. Comp. Math., 2015 (2015), 1-24
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F. Zafar, N. Hussain, Z. Fatimah, A. Kharal, Optimal sixteenth order convergent method based on Quasi-Hermite interpolation for computing roots, Sci. World J., 2014 (2014), 1-18
]
Iterative algorithm for strongly continuous semigroup of Lipschitz pseudocontraction mappings
Iterative algorithm for strongly continuous semigroup of Lipschitz pseudocontraction mappings
en
en
In this paper, an implicit iterative process is considered for strongly continuous semigroup of Lipschitz
pseudocontraction mappings. Weak and strong convergence theorems for common fixed points of strongly
continuous semigroup of Lipschitz pseudocontraction mappings are established in a real Banach space.
1424
1431
Liping
Yang
School of Applied Mathematics
Guangdong University of Technology
China
yanglping2003@126.com
Semigroup of pseudocontraction mappings
uniformly convex Banach spaces
Opial's condition
variational inequality.
Article.2.pdf
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C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120
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R. Dewangan, B. S. Thakur, M. Postolache , Strong convergence of asymptotically pseudocontractive semigroup by viscosity iteration, Appl. Math. Comput., 248 (2014), 160-168
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C. H. Morales, J. S. Jung, Convergence of paths for pseudocontractive mappings in Banach spaces , Proc. Amer. Math. Soc., 128 (2000), 3411-3419
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W. Takahashi , Nonlinear Functional Analysis-Fixed Point Theory and Its Applications, Yokohama Publishers Inc., Yokohama (2000)
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B. B. Thakur, R. Dewangan, M. Postolache, Strong convergence of new iteration process for a strongly continuous semigroup of asymptotically pseudocontractive mappings, Numer. Funct. Anal. Optim., 34 (2013), 1418-1431
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B. S. Thakur, R. Dewangan, M. Postolache , General composite implicit iteration process for a finite family of asymptotically pseudocontractive mappings, Fixed Point Theory Appl., 2014 (2014), 1-15
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H. K. Xu, R. G. Ori , An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim., 22 (2001), 767-773
##[20]
Y. Yao, M. Postolache, S. M. Kang , Strong convergence of approximated iterations for asymptotically pseudocon-tractive mappings, Fixed Point Theory Appl., 2014 (2014), 1-13
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D. Youla , On deterministic convergence of iterations of related projection operators, J. Vis. Commun. Image Represent., 1 (1990), 12-20
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S. S. Zhang, Convergence theorem of common fixed points for Lipschitzian pseudo-contraction semigroups in Banach spaces, Appl. Math. Mech., 30 (2009), 145-152
]
Ground state solutions for an asymptotically periodic and superlinear Schrodinger equation
Ground state solutions for an asymptotically periodic and superlinear Schrodinger equation
en
en
We consider the semilinear Schrödinger equation
\[
\begin{cases}
-\Delta u + V(x)u= f(x,u) ,\,\,\,\,\, x\in R^N,\\
u\in H^1(R^N),
\end{cases}
\]
where V (x) is asymptotically periodic and sign-changing, f(x; u) is a superlinear, subcritical nonlinearity.
Under asymptotically periodic V (x) and a super-quadratic condition about f(x; u). We prove that the
above problem has a ground state solution which minimizes the corresponding energy among all nontrivial
solutions.
1432
1439
Huxiao
Luo
School of Mathematics and Statistics
Central South University
P. R. China
wshrm7@126.com
Schrödinger equation
ground state solutions
asymptotically periodic
sign-changing
super-quadratic condition.
Article.3.pdf
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M. Yang, Ground state solutions for a periodic periodic Schrödinger equation with superlinear nonlinearities, Nonlinear. Anal., 72 (2010), 2620-2627
]
Solution to an ice melting cylindrical problem
Solution to an ice melting cylindrical problem
en
en
We give a solution to an ice melting cylindrical problem using the ''modified variable time step method'',
earlier suggested by the author. New numerical techniques are proposed for the one-dimensional melting
problem. The numerical results are obtained for the position of the moving boundary, time and temperatures.
1440
1452
Abdellatif
Boureghda
Department of Mathematics
Ferhat Abbas University
Algeria
abdellatif.boureghda@uha.fr
Stefan problems
phase changes
moving boundary problems
partial differential equations
finite difference methods
heat equation.
Article.4.pdf
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N. S. Asaithambi, A Galerkin method for Stefan problems, Appl. Math. Comput., 52 (1992), 239-250
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A. Boureghda, Moving boundary value problems, Doctorat en Sciences Mathématiques (Ph.D thesis), between Department of Mathematics, Ferhat Abbas University, Sétif Algeria and LMIA Haute Alsace University, Mulhouse, France (2008)
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A. Boureghda, A modified variable time step method for solving ice melting problem, J. Difference Equ. Appl., 18 (2012), 1443-1455
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J. Caldwell, Y. Y. Kwan, Starting solutions for the boundary immobilization method, Comm. Numer. Methods Engrg., 21 (2005), 289-295
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]
A monotone projection algorithm for solving fixed points of nonlinear mappings and equilibrium problems
A monotone projection algorithm for solving fixed points of nonlinear mappings and equilibrium problems
en
en
In this paper, fixed points of asymptotically quasi-\(\phi\)-nonexpansive mappings in the intermediate sense
and equilibrium problems are investigated based on a monotone projection algorithm. Strong convergence
theorems are established in the framework of reflexive Banach spaces.
1453
1462
Mingliang
Zhang
School of Mathematics and Statistics
Henan University
China
hdzhangml@yeah.net
Sun Young
Cho
Department of Mathematics
Gyeongsang National University
Korea
ooly61@hotmail.com
Asymptotically quasi-\(\phi\)-nonexpansive mapping
equilibrium problem
fixed point
generalized projection.
Article.5.pdf
[
[1]
R. P. Agarwal, Y. J. Cho, X. Qin, Generalized projection algorithms for nonlinear operators, Numer. Funct. Anal. Optim., 28 (2007), 1197-1215
##[2]
Y. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, in: A. G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Marcel Dekker, New York (1996)
##[3]
B. A. Bin Dehaish, X. Qin, A. Latif, H. Bakodah, Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., 16 (2015), 1321-1336
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D. Butnariu, S. Reich, A. J. Zaslavski, Weak convergence of orbits of nonlinear operators in reflexive Banach spaces, Numer. Funct. Anal. Optim., 24 (2003), 489-508
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S. Y. Cho, X. Qin, On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems, Appl. Math. Comput., 235 (2014), 430-438
##[8]
S. Y. Cho, X. Qin, S. M. Kang , Iterative processes for common fixed points of two different families of mappings with applications, J. Global Optim., 57 (2013), 1429-1446
##[9]
S. Y. Cho, X. Qin, L. Wang , Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., 2014 (2014), 1-15
##[10]
B. S. Choudhury, S. Kundu, A viscosity type iteration by weak contraction for approximating solutions of generalized equilibrium problem, J. Nonlinear Sci. Appl., 5 (2012), 243-251
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J. Gwinner, F. Raciti, Random equilibrium problems on networks, Math. Comput. Modelling, 43 (2006), 880-891
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R. H. He, Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FC-spaces, Adv. Fixed Point Theory, 2 (2012), 47-57
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J. S. Jung, Strong convergence of composite iterative methods for equilibrium problems and fixed point problems, Appl. Math. Comput., 213 (2009), 498-505
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J. K. Kim , Strong convergence theorems byhybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-\(\phi\)-nonexpansive mappings, Fixed Point Theory Appl., 2011 (2011), 1-15
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J. K. Kim, P. N. Anh, Y. M. Nam, Strong convergence of an extended extragradient method for equilibrium problems and fixed point problems, J. Korean Math. Soc., 49 (2012), 187-200
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B. Liu, C. Zhang, Strong convergence theorems for equilibrium problems and quasi-\(\phi\)-nonexpansive mappings , Nonlinear Funct. Anal. Appl., 16 (2011), 365-385
##[18]
X. Qin, Y. J. Cho, S. M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math., 225 (2009), 20-30
##[19]
X. Qin, Y. J. Cho, S. M. Kang, On hybrid projection methods for asymptotically quasi-\(\phi\)-nonexpansive mappings, Appl. Math. Comput., 215 (2010), 3874-3883
##[20]
X. Qin, L. Wang, On asymptotically quasi-\(\phi\)-nonexpansive mappings in the intermediate sense, Abst. Appl. Anal., 2012 (2012), 1-13
##[21]
S. Reich, A weak convergence theorem for the alternating method with Bregman distance, Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Appl. Math., Dekker, New York, 178 (1996), 313-318
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W. Takahashi, K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., 70 (2009), 45-57
##[23]
Z. M. Wang, X. Zhang, Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems, J. Nonlinear Funct. Anal., 2014 (2014), 1-25
##[24]
Y. Yao, Y. J. Cho, Y. C. Liou, Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems, European J. Oper. Res., 212 (2011), 242-250
##[25]
Q. Yu, D. Fang, W. Du , Solving the logit-based stochastic user equilibrium problem with elastic demand based on the extended traffic network model, European J. Oper. Res., 239 (2014), 112-118
##[26]
L. Zhang, H. Tong, An iterative method for nonexpansive semigroups, variational inclusions and generalized equilibrium problems, Adv. Fixed Point Theory, 4 (2014), 325-343
##[27]
J. Zhao, Strong convergence theorems for equilibrium problems, fixed point problems of asymptotically nonexpansive mappings and a general system of variational inequalities, Nonlinear Funct. Anal. Appl., 16 (2011), 447-464
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L. C. Zhao, S. S. Chang , Strong convergence theorems for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Nonlinear Sci. Appl., 2 (2009), 78-91
]
Dynamics and behavior of a higher order rational difference equation
Dynamics and behavior of a higher order rational difference equation
en
en
We study the global result, boundedness, and periodicity of solutions of the difference equation
\[x_{n+1} = a +\frac{bx_{n-l} + cx_{n-k}}{dx_{n-l} + ex_{n-k}};\qquad
n = 0; 1; ... ;\]
where the parameters a; b; c; d, and e are positive real numbers and the initial conditions \(x_{-t}; x_{-t+1}; ...; x_{-1}\)
and \(x_0\) are positive real numbers where \(t = \max\{l; k\}; l \neq k\).
1463
1474
E. M.
Elsayed
Mathematics Department, Faculty of Science
King AbdulAziz University
Saudi Arabia
emmelsayed@yahoo.com;emelsayed@mans.edu.eg
Rational difference equations
rational systems
periodicity.
Article.6.pdf
[
[1]
R. P. Agarwal, E. M. Elsayed, Periodicity and stability of solutions of higher order rational difference equation, Adv. Stud. Contemp. Math., 17 (2008), 181-201
##[2]
M. Aloqeili , Dynamics of a rational difference equation, Appl. Math. Comput., 176 (2006), 768-774
##[3]
C. Çinar , On the positive solutions of the difference equation \(x_{n+1} =\frac{ ax_{n-1}}{ 1 + bx_nx_{n-1}}\), Appl. Math. Comput., 156 (2004), 587-590
##[4]
M. Dehghan, R. Mazrooei-sebdani, Dynamics of \(x_{n+1} =\frac{ x_{n-2k+1}}{ x_{n-2k+1} + \alpha x_{n-2l}}\) , Appl. Math. Comput., 185 (2007), 464-472
##[5]
Q. Din, E. M. Elsayed, Stability analysis of a discrete ecological model, Comput. Ecol. Software, 4 (2014), 89-103
##[6]
E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, On the difference equation \(x_{n+1} = ax_n - \frac{ bx_n}{ cx_{n }- dx_{n-1}}\), Adv. Differ. Equ., 2006 (2006), 1-10
##[7]
E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, Global attractivity and periodic character of a fractional difference equation of order three , Yokohama Math. J., 53 (2007), 89-100
##[8]
E. M. Elabbasy, H. El-Metwally, E. M. Elsayed , On the difference equations\( x_{n+1} = \frac{\alpha x_{n-k}}{ \beta + \gamma\Pi^k_{ i=0} x_{n-i}}\), J. Concr. Appl. Math., 5 (2007), 101-113
##[9]
E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, Qualitative behavior of higher order difference equation, Soochow J. Math., 33 (2007), 861-873
##[10]
E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, Some Properties and Expressions of Solutions for a Class of Nonlinear Difference Equation, Util. Math., 87 (2012), 93-110
##[11]
H. El-Metwally, M. M. El-Afifi, On the behavior of some extension forms of some population models, Chaos Solitons Fractals, 36 (2008), 104-114
##[12]
H. El-Metwally, E. M. Elsayed, Solution and behavior of a third rational difference equation, Util. Math., 88 (2012), 27-42
##[13]
H. El-Metwally, E. A. Grove, G. Ladas , On the difference equation \(y_{n+1} = \frac{y_{n-(2k+1)} + p}{ y_{n-(2k+1) }+ qy_{n-2l}}\) , Proceedings of the 6th ICDE, Taylor and Francis, London (2004)
##[14]
M. A. El-Moneam, On the dynamics of the higher order nonlinear rational difference equation, Math. Sci. Lett., 3 (2014), 121-129
##[15]
M. A. El-Moneam, On the dynamics of the solutions of the rational recursive sequences , Br. J. Math. Comput. Sci., 5 (2015), 654-665
##[16]
E. M. Elsayed, On the solution of recursive sequence of order two, Fasc. Math., 40 (2008), 5-13
##[17]
E. M. Elsayed, A solution form of a class of rational difference equations, Int. J. Nonlinear Sci., 8 (2009), 402-411
##[18]
E. M. Elsayed , Dynamics of recursive sequence of order two, Kyungpook Math. J., 50 (2010), 483-497
##[19]
E. M. Elsayed, Solutions of rational difference system of order two, Math. Comput. Modelling, 55 (2012), 378-384
##[20]
E. M. Elsayed, Behavior and expression of the solutions of some rational difference equations, J. Comput. Anal. Appl., 15 (2013), 73-81
##[21]
E. M. Elsayed, Solution for systems of difference equations of rational form of order two, Comput. Appl. Math., 33 (2014), 751-765
##[22]
E. M. Elsayed, On the solutions and periodic nature of some systems of difference equations, Int. J. Biomath., 7 (2014), 1-26
##[23]
E. M. Elsayed, New method to obtain periodic solutions of period two and three of a rational difference equation, Nonlinear Dyn., 79 (2015), 241-250
##[24]
E. M. Elsayed, The expressions of solutions and periodicity for some nonlinear systems of rational difference equations, Adv. Stud. Contemp. Math., 25 (2015), 341-367
##[25]
E. M. Elsayed, M. M. El-Dessoky, Dynamics and global behavior for a fourth-order rational difference equation, Hacet. J. Math. Stat., 42 (2013), 479-494
##[26]
E. M. Elsayed, H. El-Metwally, Stability and solutions for rational recursive sequence of order three, J. Comput. Anal. Appl., 17 (2014), 305-315
##[27]
E. M. Elsayed, H. El-Metwally, Global behavior and periodicity of some difference equations, J. Comput. Anal. Appl., 19 (2015), 298-309
##[28]
A. E. Hamza, S. G. Barbary, Attractivity of the recursive sequence \(x_{n+1} = (\alpha-\beta x_n)F(x_{n-1};... ; x_{n-k})\), Math. Comput. Modelling, 48 (2008), 1744-1749
##[29]
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]
A viscosity of Cesaro mean approximation method for split generalized equilibrium, variational inequality and fixed point problems
A viscosity of Cesaro mean approximation method for split generalized equilibrium, variational inequality and fixed point problems
en
en
In this paper, we introduce and study an iterative viscosity approximation method by modified Cesàro
mean approximation for finding a common solution of split generalized equilibrium, variational inequality and
fixed point problems. Under suitable conditions, we prove a strong convergence theorem for the sequences
generated by the proposed iterative scheme. The results presented in this paper generalize, extend and
improve the corresponding results of Shimizu and Takahashi [K. Shimoji, W. Takahashi, Taiwanese J.
Math., 5 (2001), 387-404].
1475
1496
Jitsupa
Deepho
Department of Mathematics, Faculty of Science
Department of Mathematics, Faculty of Science
King Mongkut's University of Technology Thonburi (KMUTT)
University of Jaén, Campus Las Lagunillas
Thailand
Spain
jitsupa.deepho@mail.kmutt.ac.th
Juan
Martínez-Moreno
Department of Mathematics, Faculty of Science
University of Jaén, Campus Las Lagunillas
Spain
jmmoreno@ujaen.es
Poom
Kumam
Department of Mathematics, Faculty of Science
Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science
King Mongkut's University of Technology Thonburi (KMUTT)
King Mongkuts University of Technology Thonburi (KMUTT)
Thailand
Thailand
splernn@gmail.ac.th
Fixed point
variational inequality
viscosity approximation
nonexpansive mapping
Hilbert space
split generalized equilibrium problem
Cesàro mean approximation method.
Article.7.pdf
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]
q-Durrmeyer operators based on Pólya distribution
q-Durrmeyer operators based on Pólya distribution
en
en
We introduce a q analogue of Durrmeyer type modification of Bernstein operators based on Pólya distributions. We study the ordinary approximation properties of operators using modulus of continuity and
Peetre K-functional of second order. Further, we establish the weighted approximation properties for these
operators.
1497
1504
Vijay
Gupta
Department of Mathematics
Netaji Subhas Institute of Technology Sector 3 Dwarka
India
vijaygupta2001@hotmail.com
Themistocles M.
Rassias
Department of Mathematics
National Technical University of Athens, Zografou Campus
Greece
trassias@math.ntua.gr
Honey
Sharma
Department of Applied Sciences
Gulzar Group of Institutes
India
pro.sharma.h@gmail.com
Pólya distribution
q-integers
q-Bernstein operators
modulus of continuity
Peetre K-functional
weighted modulus of continuity.
Article.8.pdf
[
[1]
A. Aral, V. Gupta, R. P. Agarwal , Applications of q Calculus in Operator Theory, Springer, New York (2013)
##[2]
I. Büyükyazıcı, H. Sharma, Approximation properties of two-dimensional q-Bernstein-Chlodowsky-Durrmeyer operators, Numer. Funct. Anal. Optim., 33 (2012), 1351-1371
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R. A. De Vore, G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin (1993)
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Z. Ditzian, V. Totik, Moduli of Smoothness, Springer-Verlag, New York (1987)
##[5]
Z. Finta, V. Gupta, Approximation by q-Durrmeyer operators, J. Appl. Math. Comput., 29 (2009), 401-415
##[6]
V. Gupta, Some approximation properties of q-Durrmeyer operators, Appl. Math. Comput., 197 (2008), 172-178
##[7]
V. Gupta, Z. Finta, On certain q-Durrmeyer type operators, Appl. Math. Comput., 209 (2009), 415-420
##[8]
V. Gupta, W. Heping, The rate of convergence of q-Durrmeyer operators for 0 < q < 1, Math. Methods Appl. Sci., 31 (2008), 1946-1955
##[9]
V. Gupta, T. M. Rassias, Lupaş-Durrmeyer operators based on Pólya distribution, Banach J. Math. Anal., 8 (2014), 146-155
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V. Gupta, H. Sharma, Recurrence formula and better approximation for q-Durrmeyer Operators, Lobachevskii J. Math., 32 (2011), 140-145
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V. Gupta, H. Sharma, T. Kim, S. Lee, Properties of q-analogue of Beta operator, Adv. Difference Equ., 2012 (2012), 1-16
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G. Nowak, Approximation properties for generalized q-Bernstein polynomials, J. Math. Anal. Appl., 350 (2009), 50-55
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]
Existence of periodic solutions for second-order nonlinear difference equations
Existence of periodic solutions for second-order nonlinear difference equations
en
en
By using the critical point method, the existence of periodic solutions for second-order nonlinear difference equations is obtained. The proof is based on the Saddle Point Theorem in combination with variational
technique. The problem is to solve the existence of periodic solutions of second-order nonlinear difference
equations. One of our results obtained complements the result in the literature.
1505
1514
Zhiguo
Ren
Department of Information Engineering
Jieyang Vocational and Technical College
China
rzhg-1225@163.com
Jie
Li
Quality Control Office
Zhongshan Torch College
China
53457698@qq.com
Haiping
Shi
Modern Business and Management Department
Guangdong Construction Vocational Technology Institute
China
shp7971@163.com
Existence
periodic solutions
second-order
nonlinear difference equations
discrete variational theory.
Article.9.pdf
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[1]
R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker, New York (2000)
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]
Feasible iterative algorithms and strong convergence theorems for bi-level fixed point problems
Feasible iterative algorithms and strong convergence theorems for bi-level fixed point problems
en
en
The purpose of this paper is to introduce and study the bi-level split fixed point problems in the setting
of infinite-dimensional Hilbert spaces. For solving this kind problems, some new simultaneous iterative
algorithms are proposed. Under suitable conditions, some strong convergence theorems for the sequences
generated by the proposed algorithm are proved. As applications, we shall utilize the results presented in
the paper to study bi-level split equilibrium problem, bi-level split optimization problems and the bi-level
split variational inequality problems. The results presented in the paper are new which also extend and
improve many recent results.
1515
1528
Shih-Sen
Chang
Center for General Education
China Medical University
Taiwan
changss2013@163.com
Jing
Quan
Department of Mathematics
Yibin University
China
quanjingcq@163.com
Jingai
Liu
Department of Mathematics and Physics
Beijing Institute of Petro-Chemical Technology
China
liujingai@bipt.edu.cn
Bi-level split fixed point problem
bi-level split equilibrium problem
bi-evel split optimization problem
bi-level split variational inequality problem
split feasibility problem.
Article.10.pdf
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[1]
Q. H. Ansari, A. Rehan, Split feasibility and fixed point problems, Nonlinear anal., 2014 (2014), 282-322
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E. Blum, W. Oettli , From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145
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C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Problems, 18 (2002), 441-453
##[4]
Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for inversion problem in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365
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Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239
##[6]
Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084
##[7]
Y. Censor, A. Motova, A. Segal , Perturbed projections ans subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl, 327 (2007), 1244-1256
##[8]
Y. Censor, A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587-600
##[9]
, , , (), -
##[10]
S. S. Chang, R. P. Agarwal , Strong convergence theorems of general split equality problems for quasi-nonexpansive mappings, J. Inequal. Appl., 2014 (2014), 1-14
##[11]
S. S. Chang, Y. J. Cho, J. K. Kim, W. B. Zhang, L. Yang, Multiple-set split feasibility problems for asymptotically strict pseudocontractions, Abstr. Appl. Anal., 2012 (2012), 1-12
##[12]
S. S. Chang, H. W. J. Lee, C. K. Chan, W. B. Zhang, A modified Halpern-type iterative algorithm for totally quasi-\(\phi\)-asymptotically nonexpansive mappings with applications, Appl. Math. Comput., 218 (2012), 6489-6497
##[13]
S. S. Chang, L. Wang, Strong convergence theorems for the general split variational inclusion problem in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), 1-14
##[14]
S. S. Chang, L. Wang, Y. K. Tang, G. Wang, Moudafi's open question and simultaneous iterative algorithm for general split equality variational inclusion problems and general split equality optimization problems, Fixed Point Theory Appl., 2014 (2014), 1-17
##[15]
S. S. Chang, L. Wang, Y. K. Tang, L. Yang, The split common fixed point problem for total asymptotically strictly pseudocontractive mappings, J. Appl. Math., 2012 (2012), 1-13
##[16]
S. S. Chang, L. Wang, X. R. Wang, G. Wang, General Split Equality Equilibrium Problems with Application to Split Optimization Problems, J. Optim Theory Appl., 166 (2015), 377-390
##[17]
R. Chen, J. Wang, H. Zhang, General split equality problems in Hilbert spaces , Fixed Point Theory Appl., 2014 (2014), 1-8
##[18]
C. S. Chuang , Strong convergence theorems for the split variational inclusion problem in Hilbert spaces, Fixed Point Theory Appl., 2013 (2013), 1-20
##[19]
M. Eslamian, A. Latif, General split feasibility problems in Hilbert spaces, Abstr. Appl. Anal., 2013 (2013), 1-6
##[20]
K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge (1990)
##[21]
Z. He, W. S. Du, New feasible iterative algorithms and strong convergence theorems for bilevel split equilibrium problems, Fixed Point Theory Appl., 2014 (2014), 1-17
##[22]
P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912
##[23]
A. Moudafi, Split monotone variational inclusions , J. Optim. Theory Appl., 150 (2011), 275-283
##[24]
A. Moudafi , A relaxed alternating CQ algorithm for convex feasibility problems, Nonlinear Anal., 79 (2013), 117-121
##[25]
A. Moudafi, A. S. Eman, Simultaneous iterative methods for split equality problem , Trans. Math. Program. Appl., 1 (2013), 1-11
##[26]
E. Naraghirad , On an open question of Moudafi for convex feasibility problem in Hilbert spaces, Taiwanese J. Math., 18 (2014), 371-408
##[27]
H. K. Xu, A variable Krasnosel'skii-Mann algorithm and the multiple-sets split feasibility problem , Inverse Problems, 22 (2006), 2021-2034
##[28]
Q. Yang, The relaxed CQ algorithm for solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266
##[29]
J. Zhao, Q. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791-1799
]
Refinements of bounds for Neuman means with applications
Refinements of bounds for Neuman means with applications
en
en
In this article, we present the sharp bounds for the Neuman means derived from the Schwab-Borchardt,
geometric, arithmetic and quadratic means in terms of the arithmetic and geometric combinations of harmonic, arithmetic and contra-harmonic means.
1529
1540
Yue-Ying
Yang
School of Mechanical and Electrical Engineering
Huzhou Vocational & Technical College
China
xiafangli2005@126.com
Wei-Mao
Qian
School of Distance Education
Huzhou Broadcast and TV University
China
qwm661977@126.com
Yu-Ming
Chu
Department of Mathematics
Huzhou University
China
chuyuming2005@126.com
Neuman mean
Schwab-Borchardt mean
harmonic mean
geometric mean
quadratic mean
contra-harmonic mean
arithmetic mean.
Article.11.pdf
[
[1]
E. Neuman , On a new bivariate mean, Aequationes Math., 88 (2014), 277-289
##[2]
E. Neuman, J. Sándor, On the Schwab-Borchardt mean, Math. Pannon., 14 (2003), 253-266
##[3]
E. Neuman, J. Sándor , On the Schwab-Borchardt mean II, Math. Pannon., 17 (2006), 49-59
##[4]
W. M. Qian, Z. H. Shao, Y. M. Chu, Sharp inequalities involving Neuman means of the second kind, J. Math. Inequal., 9 (2015), 531-540
##[5]
Y. Y. Yang, W. M. Qian, The optimal convex combination bounds of harmonic, arithmetic and contraharmonic means for the Neuman means, Int. Math. Fourm, 9 (2014), 1295-1307
##[6]
L. Yang, Y. Y. Yang, Q. Wang, W.-M. Qian, The optimal geometric combination bounds for Neuman means of harmonic, arithmetic and contra-harmonic means, Pac. J. Appl. Math., 6 (2014), 283-292
##[7]
Y. Zhang, Y. M. Chu, Y.-L. Jiang, Sharp geometric mean bounds for Neuman means, Abstr. Appl. Anal., 2014 (2014), 1-6
]
Some results on fixed points of nonlinear operators and solutions of equilibrium problems
Some results on fixed points of nonlinear operators and solutions of equilibrium problems
en
en
The purpose of this paper is to investigate fjxed points of an asymptotically quasi-\(\phi\)-nonexpansive mapping in the intermediate sense and a bifunction equilibrium problem. We obtain a strong convergence
theorem of solutions in the framework of Banach spaces.
1541
1548
Peng
Cheng
School of Mathematics and Information Science
North China University of Water Resources and Electric Power
China
hschengp@yeah.net
Zhaocui
Min
School of Science
Hebei University of Engineering
China
Asymptotically quasi-\(\phi\)-nonexpansive mapping
equilibrium problem
fixed point
variational inequality
iterative process.
Article.12.pdf
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]
Analysis of a TB model with treatment interruptions
Analysis of a TB model with treatment interruptions
en
en
In this article, a TB transmission model with treatment interruptions is established. The control reproduction numbers which completely determine the long behaviors of the TB model are explicitly given.
By applying the comparison principle and constructing proper Lyapunov functions, the global asymptotic
stability of equilibria is analyzed. The numerical simulations show that the treatment of active TB cases has
always a positive effect on controlling the TB epidemic; while treatment interruptions may have a negative,
positive or no effect on combating the TB epidemic.
1549
1563
Luju
Liu
School of Mathematics and Statistics
Henan University of Science and Technology
China
lujuliu@126.com
Yan
Wang
College of Science
China University of Petroleum
China
wangyan@upc.edu.cn
TB transmission model
treatment interruptions
the control reproduction number
Lyapunov function
globally asymptotically stable.
Article.13.pdf
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]
Some new fixed point results in partial ordered metric spaces via admissible mappings and two new functions
Some new fixed point results in partial ordered metric spaces via admissible mappings and two new functions
en
en
The purpose of this paper is to discuss the existence of fixed points for new classes of mappings defined
on an ordered metric space. The obtained results generalize and improve some fixed point results in the
literature. Some examples show the usefulness of our results.
1564
1580
Xiao-lan
Liu
School of Science
Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing
Sichuan University of Science and Engineering
Sichuan Province University
China
China
stellalwp@163.com
Arslan Hojat
Ansari
Department of Mathematics
Department of Mathematics
Payame Noor University
Karaj Branch, Islamic Azad University
Iran
Iran
mathanalsisamir4@gmail.com
Sumit
Chandok
Department of Applied Science, Khalsa College of Engineering and Technology
Punjab Technical University
India
chansok.s@gmail.com
Choonkil
Park
Research Institute for Natural Sciences
Hanyang University
Republic of Korea
baak@hanyang.ac.kr
Common fixed point
generalized weakly contraction
generalized metric spaces
upper class
C-class function.
Article.14.pdf
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[1]
M. Abbas, D. Dorić , Common fixed point theorem for four mappings satisfying generalized weak contractive condition, Filomat, 24 (2010), 1-10
##[2]
H. Alikhani, S. Rezapour, N. Shahzad, Fixed points of a new type of contractive mappings and multifunctions, Filomat, 27 (2013), 1315-1319
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A. H. Ansari , Note on ''\(\varphi,\psi\)-contractive type mappings and related fixed point'', The 2nd Regional Conference on Mathematics and Applications, Payame Noor University, (2014), 377-380
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A. H. Ansari , Note on ''\(\alpha\)-admissible mappings and related fixed point theorems'', The 2nd Regional Conference on Mathematics and Applications, Payame Noor University, (2014), 373-376
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A. H. Ansari, S. Chandok, C. Ionescu, Fixed point theorems on b-metric spaces for weak contractions with auxiliary functions, J. Inequal. Appl., 2014 (2014), 1-17
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A. H. Ansari, S. Shukla, Some fixed point theorems for ordered F-(F; h)-contraction and subcontractions in 0-f- orbitally complete partial metric spaces, J. Adv. Math. Stud., 9 (2016), 37-53
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J. H. Asl, S. Rezapour, N. Shahzad , On fixed points of \(\alpha-\psi\)-contractive multifunctions, Fixed Point Theory Appl., 2012 (2012), 1-6
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H. Aydi, E. Karapinar, B. Samet , Remarks on some recent fixed point theorems, Fixed Point Theory Appl., 2012 (2012), 1-6
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N. Hussain, E. Karapinar, P. Salimi, F. Akbar, \(\alpha\)-admissible mappings and related fixed point theorems, J. Inequal. Appl., 2013 (2013), 1-11
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N. Hussain, E. Karapinar, P. Salimi, P. Vetro, Fixed point results for \(G^m\)-Meir-Keeler contractive and \(G-\alpha-\psi\)-Meir-Keeler contractive mappings, Fixed Point Theory Appl., 2013 (2013), 1-14
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N. Hussain, P. Salimi, A. Latif , Fixed point results for single and set-valued \(\alpha-\eta-\psi\)-contractive mappings, Fixed Point Theory Appl., 2013 (2013), 1-23
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E. Karapinar, P. Kumam, P. Salimi, On \(\alpha-\psi\)-Meir-Keeler contractive mappings, Fixed Point Theory Appl., 2013 (2013), 1-12
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E. Karapinar, P. Salimi, Fixed point theorems via auxiliary functions, J. Appl. Math., 2012 (2012), 1-9
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A. Latif, H. Isikb, A. H. Ansari, Fixed points and functional equation problems via cyclic admissible generalized contractive type mappings, J. Nonlinear Sci. Appl., 9 (2016), 1129-1142
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W. Long, S. Khaleghizadeh, P. Salimi, S. Radenović, S. Shukla , Some new fixed point results in partial ordered metric spaces via admissible mappings, Fixed Point Theory Appl., 2014 (2014), 1-18
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]
Fixed point and common fixed point theorems on ordered cone metric spaces over Banach algebras
Fixed point and common fixed point theorems on ordered cone metric spaces over Banach algebras
en
en
The purpose of this paper is to obtain some fixed point and common fixed point results of comparable
maps satisfying certain contractive conditions on partially ordered cone metric spaces over Banach algebras.
Moreover, an example is given, which shows that our main results are more useful than the presented results
in some recent literatures.
1581
1589
Qi
Yan
Department of Mathematics
Nanchang University
P. R. China
qiyanmath@163.com
Jiandong
Yin
Department of Mathematics
Nanchang University
P. R. China
yjdaxf@163.com
Tao
Wang
Department of Mathematics
Nanchang University
P. R. China
taowangmath@163.com
Fixed points
cone metric spaces over Banach algebras
ordered sets.
Article.15.pdf
[
[1]
M. Abbas, G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl., 341 (2008), 416-420
##[2]
M. Abbas, B. Rhoades, T. Nazir, Common fixed points for four maps in cone metric spaces, Appl. Math. Comput., 216 (2010), 80-86
##[3]
I. Altun, B. Damjanović, D. Djorić, Fixed point and common fixed point theorems on ordered cone metric spaces , Appl. Math. Lett., 23 (2010), 310-316
##[4]
I. Altun, G. Durmaz, Some fixed point theorems on ordered cone metric spaces, Rend. Circ. Mate. Palermo, 58 (2009), 319-325
##[5]
L. G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468-1476
##[6]
D. Ilić, V. Rakočević, Common fixed points for maps on cone metric space, J. Math. Anal. Appl., 341 (2008), 876-882
##[7]
D. Ilić, V. Rakočević , Quasi-contraction on a cone metric space, Appl. Math. Lett., 22 (2009), 728-731
##[8]
S. Janković, Z. Kaselburg, S. Radenović , On the cone metric space: a survey, Nonliner Anal., 74 (2011), 2591-2601
##[9]
Z. Kadelburg, S. Radenović, V. Rakočević , A note on the equivalence of some metric and cone metric fixed point results, Appl. Math. Lett., 24 (2011), 370-374
##[10]
H. Liu, S. Xu , Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz maps, Fixed Point Theory Appl., 2013 (2013), 1-10
##[11]
S. Radenović , Common fixed points under contractive conditions in cone metric spaces, Comput. Math. Appl., 58 (2009), 1273-1278
##[12]
S. Radenović, B. E. Rhoades, Fixed point theorem for two non-self maps in cone metric spaces , Comput. Math. Appl., 57 (2009), 1701-1707
##[13]
S. Rezapour, R. Hamlbarani , Some notes on the paper Cone metric spaces and fixed point theorems of contractive mapping', J. Math. Anal. Appl., 345 (2008), 719-724
##[14]
W. Rudin , Functional Analysis, 2nd edn, McGraw-Hill , New York (1991)
##[15]
S. Xu, S. Radenović, Fixed point theorems of generalized Lipschitz maps on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory Appl., 2014 (2014), 1-12
]
Tripled coincidence points for mixed comparable mappings in partially ordered cone metric spaces over Banach algebras
Tripled coincidence points for mixed comparable mappings in partially ordered cone metric spaces over Banach algebras
en
en
Let \((X; d)\) be a complete partially ordered cone metric space, \(g : X \rightarrow X \)and \(F : X \times X \times X \rightarrow X\)
be two mappings. In this paper, a new concept of F having the mixed comparable property with respect
to g is introduced and some tripled coincidence point results of F and g are obtained if F has the mixed
comparable property with respect to g and some other natural conditions are satisfied. Moreover, a support
example of one of our results is given.
1590
1599
Jiandong
Yin
Department of Mathematics
Nanchang University
P. R. China
yjdaxf@163.com
Qi
Yan
Department of Mathematics
Nanchang University
P. R. China
qiyanmath@163.com
Tao
Wang
Department of Mathematics
Nanchang University
P. R. China
taowangmath@163.com
Ling
Liu
Department of Mathematics
Nanchang University
P. R. China
lliumath@163.com
Cone metric spaces over Banach algebras
mixed comparable properties
tripled coincidence points
spectral radius.
Article.16.pdf
[
[1]
V. Berinde, M. Borcut, Tripled coincidence point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal., 74 (2011), 4889-4897
##[2]
A. G. Bin Ahmad, Z. M. Fadail, M. Abbas, Z. Kadelburg, S. Radenović , Some fixed and periodic points in abstract metric spaces, Abstr. Appl. Anal., 2012 (2012), 1-15
##[3]
H. S. Ding, Z. Kadelburg, E. Karapinar, S. Radenović, Common fixed points of weak contractions in cone metric spaces, Abstr. Appl. Anal., 2012 (2012), 1-18
##[4]
W. S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal., 72 (2010), 2259-2261
##[5]
L. Gajić, V. Rakočević, Quasi-contractions on a nonnormal cone metric space, Funct. Anal. Appl., 46 (2012), 62-65
##[6]
T. Gnana Bhaskar, T. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 265 (2006), 1379-1393
##[7]
L. G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468-1476
##[8]
D. Ilić, V. Rakočević, Quasi-contraction on a cone metric space, Appl. Math. Lett., 22 (2009), 728-731
##[9]
S. Janković, Z. Kadelburg, S. Radenović, On the cone metric space: a survey, Nonlinear Anal., 74 (2011), 2591-2601
##[10]
Z. Kadelburg, S. Radenović, Generalized quasicontractions in orbitally complete abstract metric spaces, Fixed Point Theory, 13 (2012), 527-536
##[11]
Z. Kadelburg, S. Radenović, A note on various types of cones and fixed point results in cone metric spaces, Asian J. Math. Appl., 2013 (2013), 1-7
##[12]
Z. Kadelburg, S. Radenović, V. Rakočević, Topological vector space-valued cone metric spaces and fixed point theorems, Fixed Point Theory Appl., 2010 (2010), 1-17
##[13]
Z. Kadelburg, S. Radenović, V. Rakočević, A note on the equivalence of some metric and cone metric fixed point results, Appl. Math. Lett., 24 (2011), 370-374
##[14]
H. Liu, S. Xu , Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings , Fixed Point Theory Appl., 2013 (2013), 1-10
##[15]
H. Liu, S. Xu, Fixed point theorem of quasi-contractions on cone metric spaces with Banach algebras, Abstr. Appl. Anal., 2013 (2013), 1-5
##[16]
S. Radenović, Z, Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, Banach J. Math. Anal., 5 (2011), 38-50
##[17]
S. Radenović, B. E. Rhoades, Fixed point theorems for two non-self mappings in cone metric spaces, Comput. Math. Appl., 57 (2009), 1701-1707
##[18]
S. Rezapour, R. Hamlbarani, Some notes on the paper: ''Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 345 (2008), 719-724
##[19]
F. Sabetghadam, H. P. Masiha, A. H. Sanatpour, Some coupled fixed point theorems in cone metric spaces , Fixed Point Theory Appl., 2009 (2009), 1-8
##[20]
S. Xu, S. Radenović, Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory Appl., 2014 (2014), 1-12
##[21]
J. Yin, T.Wang, Q. Yan , Fixed point theorems of ordered contractive mappings on cone metric spaces over Banach algebras , Fixed Point Theory Appl., 2015 (2015), 1-13
]
Oscillation properties for solutions of a kind of partial fractional differential equations with damping term
Oscillation properties for solutions of a kind of partial fractional differential equations with damping term
en
en
The aim of the present paper is to obtain sufficient conditions for oscillation of solutions of partial
fractional differential equations with the damping term of the form
\[D^{1+\alpha}_{+;t} u(x; t) + p(t)D^\alpha _{+;t} u(x; t) = a(t)\Delta u(x; t) + \Sigma^m_{i=1}
a_i(t)\Delta u(x; t - \tau_i)
- q(x; t)
\int^t_0
(t - \xi)^{-\alpha} u(x; \xi)d\xi; \quad (x; t) \in
\Omega\times \mathbb{R}_+ \equiv G.\]
Two examples are given to illustrate the main results.
1600
1608
Wei Nian
Li
Department of Mathematics
Binzhou University
P. R. China
wnli@263.net
Weihong
Sheng
Department of Mathematics
Binzhou University
P. R. China
wh-sheng@163.com
Oscillation
fractional partial differential equation
Riemann-Liouville derivative
damping term.
Article.17.pdf
[
[1]
S. Abbas, M. Benchohra, G. M. N'Guérékata, Topics in Fractional Differential Equations, Springer, New York (2012)
##[2]
C. Chen, Y. L. Jiang, Lie group analysis method for two classes of fractional partial differential equations, Commun. Nonlinear Sci. Numer. Simul., 26 (2015), 24-35
##[3]
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]
A topological analysis on patches of optical flow
A topological analysis on patches of optical flow
en
en
The research of optical
flow is vitally important topic in computer vision. In this paper we research a
topological analysis of space of optical
flow locally. We use the methods of computing topology to the spaces
of \(4 \times 4\) and \(6 \times 6\) high contrast optical
flow patches. We experimentally prove that in both cases there
exist subspaces of the spaces of all high contrast optical
flow patches that is topologically equivalent to a
circle, which states that some results on the topological analysis of natural images and range images can be
extended to the scope of image motion.
1609
1618
Shengxiang
Xia
College of Science
Shandong Jianzhu University
P. R. China
xias@sdjzu.edu.cn
Optical flow
high contrast patches
persistent homology
plex barcodes
Klein bottle.
Article.18.pdf
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[1]
H. Adams, A. Atanasov, G. Carlsson, Nudged elastic band in topological data analysis, Topol. Methods Nonlinear Anal., 45 (2015), 247-272
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H. Adams, G. Carlsson, On the nonlinear statistics of range image patches, SIAM J. Image Sci., 2 (2009), 110-117
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K. Jia, X. Wang, X. Tang, Optical flow estimation using learned sparse model, IEEE International Conference on Computer Vision, (2011), 2391-2398
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A. B. Lee, K. S. Pedersen, D. Mumford , The nonlinear statistics of high-contrast patches in natural images, Int. J. Comput. Vision, 54 (2003), 83-103
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S. Roth, M. J. Black, On the spatial statistics of optical flow, Int. J. Comput. Vision, 74 (2007), 33-50
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V. d. Silva, G. Carlsson, Topological estimation using witness complexes, Proc. Sympos. Point-Based Graphics, (2004), 157-166
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D. Sun, S. Roth, M. J. Black, A quantitative analysis of current practices in optical flow estimation and the principles behind them, Int. J. Comput. Vision, 106 (2014), 115-137
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S. Xia, On the local behavior of spaces of range image patches, , (To appear), -
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S. Xia , Y. Yin, On the nonlinear analysis of optical flow, , (Submitted), -
##[18]
Q. Yin, W. Wang, An analysis of spaces of range image small patches, Open Cybern Syst. J., 9 (2015), 275-279
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A. Zomorodian, G. Carlsson, Computing persistent homology, Discrete Comput. Geom., 33 (2005), 249-274
]
Almost strongly \(\theta\)-e-continuous functions
Almost strongly \(\theta\)-e-continuous functions
en
en
We introduce and investigate a new class of functions called almost strongly \(\theta\)-e-continuous functions,
containing the classes of almost strongly \(\theta\)-precontinuous [J. H. Park, S. W. Bae, Y. B. Park, Chaos Solitons
Fractals, 28 (2006), 32-41], almost strongly \(\theta\)-semicontinuous [Y. Beceren, S. Yüksel, E. Hatir, Bull.
Calcutta Math. Soc., 87 (1995), 329-334] and strongly \(\theta\)-e-continuous functions [M. Özkoç, G. Aslım,
Bull. Korean Math. Soc., 47 (2010), 1025-1036]. Several characterizations concerning almost strongly
\(\theta\)-e-continuous functions are obtained. Also we investigate the relationships between almost strongly \(\theta\)-e-
continuous functions and separation axioms and almost strongly e-closedness of graphs of functions.
1619
1635
Murad
Özkoç
Department of Mathematics, Faculty of Science
Muğla Sıtkı Koçman University
Turkey
murad.ozkoc@mu.edu.tr
Burcu Sünbül
Ayhan
Department of Mathematics, Faculty of Science
Muğla Sıtkı Koçman University
Turkey
brcyhn@gmail.com
Almost strong \(\theta\)-e-continuity
e-open
e-\(\theta\)-open
almost e-regular
almost strongly e-closed.
Article.19.pdf
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M. Özkoç, N. Elçi, On e-locally closed sets and related topics, , (Submitted), -
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J. H. Park, S. W. Bae, Y. B. Park, Almost strongly \(\theta\)-precontinuous functions, Chaos Solitons Fractals, 28 (2006), 32-41
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]
Existence results to certain functional equations in probabilistic Banach spaces with an application to integral equations
Existence results to certain functional equations in probabilistic Banach spaces with an application to integral equations
en
en
We consider some classes of functional equations posed in PB-spaces, for which we establish existence
and uniqueness of solutions that belong to a cone. An application to integral equations is presented.
1636
1644
Mohamed
Jleli
Department of Mathematics, College of Science
King Saud University
Saudi Arabia
jleli@ksu.edu.sa
Bessem
Samet
Department of Mathematics, College of Science
King Saud University
Saudi Arabia
bsamet@ksu.edu.sa
Functional equation
PB-space
normal cone
partial order
integral equation.
Article.20.pdf
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H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709
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M. Jleli, B. Samet , Positive fixed points for convex and decreasing operators in probabilistic Banach spaces with an application to a two-point boundary value problem, Fixed Point Theory Appl., 2015 (2015), 1-19
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]
Construction of a common solution of a finite family of variational inequality problems for monotone mappings
Construction of a common solution of a finite family of variational inequality problems for monotone mappings
en
en
Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let \(A_i : C \rightarrow H\), for \(i = 1; 2\);
be two \(L_i\)-Lipschitz monotone mappings and let \(f : C \rightarrow C\) be a contraction mapping. It is our purpose in
this paper to introduce an iterative process for finding a point in \(V I(C;A_1) \cap V I(C;A_2) \)under appropriate
conditions. As a consequence, we obtain a convergence theorem for approximating a common solution of
a finite family of variational inequality problems for Lipschitz monotone mappings. Our theorems improve
and unify most of the results that have been proved for this important class of nonlinear operators.
1645
1657
Mohammed Ali
Alghamdi
Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science
King Abdulaziz University
Saudi Arabia
Proff-malghamdi@hotmail.com
Naseer
Shahzad
Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science
King Abdulaziz University
Saudi Arabia
nshahzad@kau.edu.sa
Habtu
Zegeye
Department of Mathematics
University of Botswana
Botswana
habtuzh@yahoo.com
Fixed points of a mapping
monotone mapping
strong convergence
variational inequality.
Article.21.pdf
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[1]
Y. I. Alber, A. N. Iusem, Extension of subgradient techniques for nonsmooth optimization in Banach spaces, Set-Valued Anal., 9 (2001), 315-335
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J. Y. Bello Cruz, A. N. Iusem, A strongly convergent direct method for monotone variational inequalities in Hilbert spaces, Numer. Funct. Anal. Optim., 30 (2009), 23-36
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G. Cai, S. Bu, An iterative algorithm for a general system of variational inequalities and fixed point problems in q-uniformly smooth Banach spaces, Optim. Lett., 7 (2013), 267-287
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Y. Censor, A. Gibali, S. Reich , The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318-335
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Y. Censor, A. Gibali, S. Reich , Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119-1132
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N. Shahzad, H. Zegeye, Approximation methods for a common minimum-norm point of a solution of variational inequality and fixed point problems in Banach spaces, Bull. Korean Math. Soc., 51 (2014), 773-788
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W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Japan (2000)
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W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428
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I. Yamada, The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings, Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Edited by D. Butnariu, Y. Censor, and S. Reich, North-Holland, Amsterdam, (2001), 473-504
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Y. Yao, Y. C. Liou, S. M. Kang, Algorithms construction for variational inequalities, Fixed Point Theory Appl., 2011 (2011), 1-12
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Y. Yao, G. Marino, L. Muglia, A modified Korpelevich's method convergent to the minimum-norm solution of a variational inequality, Optimization, 63 (2014), 559-569
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Y. Yao, M. Postolache, Y. C. Liou, Variant extragradient-type method for monotone variational inequalities, Fixed Point Theory Appl., 2013 (2013), 1-15
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Y. Yao, H. K. Xu, Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications, Optimization, 60 (2011), 645-658
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H. Zegeye, E. U. Ofoedu, N. Shahzad, Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings, Appl. Math. Comput., 216 (2010), 3439-3449
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H. Zegeye, N. Shahzad, Strong convergence theorems for a common zero of a countably infinite family of \(\alpha\)-inverse strongly accretive mappings, Nonlinear Anal., 71 (2009), 531-538
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H. Zegeye, N. Shahzad, A hybrid approximation method for equilibrium, variational inequality and fixed point problems, Nonlinear Anal. Hybrid Syst., 4 (2010), 619-630
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H. Zegeye, N. Shahzad, A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems, Nonlinear Anal., 74 (2011), 263-272
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H. Zegeye, N. Shahzad, Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces, Optim. Lett., 5 (2011), 691-704
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H. Zegeye, N. Shahzad, Convergence of Mann's type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl., 62 (2011), 4007-4014
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H. Zegeye, N. Shahzad, Extragradient method for solutions of variational inequality problems in Banach spaces, Abstr. Appl. Anal., 2013 (2013), 1-8
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H. Zegeye, N. Shahzad, Solutions of variational inequality problems in the set of fixed points of pseudocontractive mappings, Carpathian J. Math., 30 (2014), 257-265
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H. Zegeye, N. Shahzad, Algorithms for solutions of variational inequalities in the set of common fixed points of finite family of \(\lambda\)-strictly pseudocontractive mappings, Numer. Funct. Anal. Optim., 36 (2015), 799-816
]
Some fixed point theorems in generalized quasi- partial metric spaces
Some fixed point theorems in generalized quasi- partial metric spaces
en
en
In this paper, a new concept of generalized quasi-partial metric spaces is presented. Some fixed point
results due to Karapinar et. al., [E. Karapinar, I. M. Erhan, A. Öztürk, Math. Comput. Modelling, 57
(2013), 2442-2448] are extended in the setting of the generalized quasi-partial metric spaces.
1658
1674
Xiaoming
Fan
School of Mathematical Sciences
Harbin Normal University
P. R. China
fanxm093@163.com
Zhigang
Wang
School of Mathematical Sciences
Harbin Normal University
P. R. China
wangzg2003205@163.com
Generalized quasi-partial metric space
fixed point theorems
quasi-partial metric space
generalized dislocated quasi-metric.
Article.22.pdf
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A. Ait Taleb, E. Hanebaly, A fixed point theorem and its application to integral equations in modular function spaces, Proc. Amer. Math. Soc., 128 (2000), 419-426
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M. Arshad, A. Shoaib, I. Beg, Fixed point of a pair of contractive dominated mappings on a closed ball in an ordered dislocated metric space, Fixed Point Theory Appl., 2013 (2013), 1-15
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P. Chaipunya, Y. J. Cho, P. Kumam, Geraghty-type theorems in modular metric spaces with an application to partial differential equation, Adv. Difference Equ., 2012 (2012), 1-12
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P. Hitzler, A. K. Seda, Dislocated topologies, J. Electr. Eng., 51 (2000), 3-7
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N. Hussain, D. Dorić, Z. Kadelburg, S. Radenović, Suzuki-type fixed point results in metric type spaces, Fixed Point Theory Appl., 2012 (2012), 1-12
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E. Karapinar, I. M. Erhan, A. Öztürk, Fixed point theorems on quasi-partial metric spaces, Math. Comput. Modelling, 57 (2013), 2442-2448
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E. Karapinar, P. Salimi , Dislocated metric space to metric spaces with some fixed point theorems, Fixed Point Theory Appl., 2013 (2013), 1-19
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K. Kuaketa, P. Kumam, Fixed points of asymptotic pointwise contractions in modular spaces, Appl. Math. Lett., 24 (2011), 1795-1798
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F. M. Zeyada, G. H. Hassan, M. A. Ahmed, A generalization of a fixed point theorem due to Hitzler and Seda in dislocated quasi-metric spaces, Arab. J. Sci. Eng. Sect. A Sci., 31 (2006), 111-114
]
Some results on asymptotically quasi-phi-nonexpansive mappings in the intermediate sense and Ky Fan inequalities
Some results on asymptotically quasi-phi-nonexpansive mappings in the intermediate sense and Ky Fan inequalities
en
en
In this paper, we study asymptotically quasi-\(\phi\)- nonexpansive mappings in the intermediate sense and Ky
Fan inequalities. A convergence theorem is established in a strictly convex and uniformly smooth Banach
space. The results presented in the paper improve and extend some recent results.
1675
1684
Hongwei
Liang
School of Mathematics and Statistics
Henan University
China
hdlianghw@yeah.net
Mingliang
Zhang
School of Mathematics and Statistics
Henan University
China
hdzhangml@yeah.net
Asymptotically nonexpansive mapping
quasi-\(\phi\)-nonexpansive mapping
fixed point
convergence theorem.
Article.23.pdf
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R. P. Agarwal, Y. J. Cho, X. Qin , Generalized projection algorithms for nonlinear operators, Numer. Funct. Anal. Optim., 28 (2007), 1197-1215
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B. A. Bin Dehaish, X. Qin, A. Latif, H. Bakodah, Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., 16 (2015), 1321-1336
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C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103-120
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]
Blow-up of solutions for the heat equations with variable source on graphs
Blow-up of solutions for the heat equations with variable source on graphs
en
en
In this paper, we mainly consider the blow-up problem for the discrete heat equations with variable
source on finite graphs
\[u_t = \Delta_\omega u + u^{p(x)}\]
with homogeneous Dirichlet boundary conditions and positive initial energy. We prove that the corresponding solutions blow up at a finite time with large enough initial data. Moreover, the blow-up rate is also
considered.
1685
1692
Qiao
Xin
College of Mathematics and Statistics
Yili Normal University
P. R. China
xinqiaoylsy@163.com
Dengming
Liu
School of Mathematics and Computational Science
Hunan University of Science and Technology
P. R. China
liudengming08@163.com
Blow-up
discrete heat equation
variable reaction
finite graphs.
Article.24.pdf
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[1]
F. R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence (1997)
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S. Y. Chung , Critical blow-up and global existence for discrete nonlinear p-laplacian parabolic equations, Discrete Dyn. Nat. Soc., 2014 (2014), 1-10
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S. Y. Chung, C. A. Berenstein, \(\omega\)-harmonic functions and inverse conductivity problems on networks, SIAM J. Appl. Math., 65 (2005), 1200-1226
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Y. S. Chung, Y. S. Lee, S. Y. Chung, Extinction and positivity of the solutions of the heat eqautions with absorption on networks, J. Math. Anal. Appl., 380 (2011), 642-652
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J. P. Pinasco, Blow-up for parabolic and hyperbolic problems with variable exponents, Nonlinear Anal., 71 (2009), 1094-1099
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X. Wu, B. Guo, W. Gao, Blow-up of solutions for a semilinear parabolic equation involving variable source and positive initial energy, Appl. Math. Lett., 26 (2013), 539-543
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Q. Xin, L. Xu, C. Mu , Blow-up for the \(\omega\)-heat equation with dirichlet boundary conditions and a reaction term on graphs, Appl. Anal., 93 (2014), 1691-1701
##[12]
W. Zhou, M. Chen, W. Liu, Critical exponent and blow-up rate for the \(\omega\)-diffusion equations on graphs with dirichlet boundary conditions, Electron. J. Differential Equations, 2014 (2014), 1-13
]
Stability of cubic and quartic rho-functional inequalities in fuzzy normed spaces
Stability of cubic and quartic rho-functional inequalities in fuzzy normed spaces
en
en
In this paper, we solve the following cubic \(\rho\)-functional inequality
\[N(f(2x + y) + f(2x - y) - 2f(x + y) - 2f(x - y) - 12f(x) - \rho
(4f
(
x +\frac{y}{2}
)
+ 4f
(
x - \frac{y}{2})
- f(x + y) - f(x - y) - 6f(x)); t) \geq \frac
{t}
{t + \varphi(x; y)}\quad (1)\]
and the following quartic \(\rho\)-functional inequality
\[N(f(2x + y) + f(2x - y) - 4f(x + y) - 4f(x - y) - 24f(x) + 6f(y)- \rho
(8f
(
x +\frac{y}{2}
)
+ 8f
(
x - \frac{y}{2})
- 2f(x + y) - 2f(x - y) - 12f(x)+ 3 f(y)); t) \geq \frac
{t}
{t + \varphi(x; y)}\quad (2)\]
in fuzzy normed spaces, where \(\rho\) is a fixed real number with \(\rho\neq 2\).
Using the direct method, we prove the Hyers-Ulam stability of the cubic \(\rho\)-functional inequality (1) and
the quartic \(\rho\)-functional inequality (2) in fuzzy Banach spaces.
1693
1701
Choonkill
Park
Research Institute for Natural Sciences
Hanyang University
Korea
baak@hanyang.ac.kr
Sungsik
Yun
Department of Financial Mathematics
Hanshin University
Korea
ssyun@hs.ac.kr
fuzzy Banach space
cubic \(\rho\) -functional inequality
quartic \(\rho\) -functional inequality
Hyers-Ulam stability.
Article.25.pdf
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##[32]
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J. Z. Xiao, X. H. Zhu, Fuzzy normed spaces of operators and its completeness , Fuzzy Sets and Systems, 133 (2003), 389-399
]
Generalized dynamic process for generalized (f,L)-almost F-contraction with applications
Generalized dynamic process for generalized (f,L)-almost F-contraction with applications
en
en
Recently Abbas [M. Abbas, Fixed Point Theory, 13 (2012), 3-10] introduced the concept of f-almost
contraction which in turn extended the class of multivalued almost contraction mapping and obtained coincidence point results for this new class of mappings. The aim of this paper is to introduce the notion
of dynamic process for generalized (f;L)-almost F-contraction mappings and to obtain coincidence and
common fixed point results for such process. It is worth mentioning that our results do not rely on the commonly used range inclusion condition. We provide some examples to support our results. As an application
of our results, we obtain the existence and uniqueness of solutions of dynamic programming and integral
equations. Our results provide extension as well as substantial generalizations and improvements of several
well known results in the existing comparable literature.
1702
1715
Nawab
Hussain
Department of Mathematics
King Abdulaziz University
Saudi Arabia
nhusain@kau.edu.sa
Muhammad
Arshad
Department of Mathematics
International Islamic University
Pakistan
marshadzia@iiu.edu.pk
Mujahid
Abbas
Department of Mathematics
Department of Mathematics and Applied Mathematics
King Abdulaziz University
University of Pretoria
Saudi Arabia
South Africa
mujahid.abbas@up.ac.za
Aftab
Hussain
Department of Mathematics
International Islamic University
Pakistan
aftabshh@gmail.com
Coincidence point
generalized dynamic process
integral equations
(f،L)-almost F-contraction
dynamic programming.
Article.26.pdf
[
[1]
M. Abbas, Coincidence points of multivalued f-almost nonexpansive mappings, Fixed Point Theory, 13 (2012), 3-10
##[2]
M. Abbas, B. Ali, S. Romaguera , Fixed and periodic points of generalized contractions in metric spaces, Fixed Point Theory Appl., 2013 (2013), 1-11
##[3]
Ö. Acar, G. Durmaz, G. Minak, Generalized multivalued F-contraction on complete metric spaces, Bull. Iranian Math. Soc., 40 (2014), 1469-1478
##[4]
R. P. Agarwal, N. Hussain, M. A. Taoudi , Fixed point theorems in ordered Banach spaces and applications to non linear integral equations, Abstr. Appl. Anal., 2012 (2012), 1-15
##[5]
J. Ahmad, A. Al-Rawashdeh, A. Azam , New fixed point theorems for generalized F-contractions in complete metric spaces, Fixed Point Theory Appl., 2015 (2015), 1-18
##[6]
J. Ahmad, N. Hussain, A. R. Khan, A. Azam, Fixed point results for generalized multi-valued contractions, J. Nonlinear Sci. Appl., 8 (2015), 909-918
##[7]
I. Altun, G. Minak, H. Dağ, Multivalued F-contraction on complete metric spaces, J. Nonlinear Convex Anal., 16 (2015), 659-666
##[8]
M. Arshad, E. Ameer, A. Hussain, Hardy-Rogers type fixed point theorems for \(\alpha\)-GF-contractions, Arch. Math., 51 (2015), 129-141
##[9]
R. Baskaran, P. V. Subrahmanyam, A note on the solution of a class of functional equations, Appl. Anal., 22 (1986), 235-241
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##[12]
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##[13]
V. Berinde, Some remarks on a fixed point theorem for Ciric-type almost contractions, Carpathian J. Math., 25 (2009), 157-162
##[14]
M. Berinde, V. Berinde, On general class of multivalued weakly Picard mappings, J. Math. Anal. Appl., 326 (2007), 772-782
##[15]
V. Berinde, M. Păcurar, A note on the paper ''Remarks on fixed point theorems of Berinde'', Nonlinear Anal. Forum, 14 (2009), 119-124
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##[19]
N. Hussain, M. A. Taoudi, Krasnosel'skii-type fixed point theorems with applications to Volterra integralequations, Fixed Point Theory Appl, 2013 (2013), 1-16
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T. Kamran, Multivalued f-weakly Picard mappings, Nonlinear Anal., 67 (2007), 2289-2296
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D. Klim, D. Wardowski, Fixed points of dynamic processes of set-valued F-contractions and application to functional equations, Fixed Point Theory Appl., 2015 (2015), 1-9
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M. A. Kutbi, M. Arshad, A. Hussain, Fixed point results for Ćirić type \(\alpha-\eta\)-GF-contractions, J. Comput. Anal. Appl., 21 (2016), 466-481
##[23]
G. Minak, A. Halvaci, I. Altun, Ćirić type generalized F-contractions on complete metric spaces and fixed point results, Filomat, 28 (2014), 1143-1151
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M. Sgroi, C. Vetro , Multi-valued F-contractions and the solution of certain Functional and integral Equations, Filomat, 27 (2013), 1259-1268
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D. Wardowski, Fixed points of new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 1-6
]
Periodic orbits around the collinear libration points
Periodic orbits around the collinear libration points
en
en
The locations for the collinear libration points in the framework of the restricted three-body problem
are determined when the bigger primary is a triaxial rigid body. The analysis of the periodic motion around
these points is performed and given up to second order in the case that the initial state of the motion gives
rise to periodic orbits. Moreover, some numerical results for the locations of collinear points are provided
and the graphical investigations for the periodic motion are plotted, as well. It is worth mentioning that the
collinear libration points and associated periodic orbits are considered the optimal placement to transfer a
spacecraft to the nominal periodic orbits or to an associated stable manifold.
1716
1727
E. I.
Abouelmagd
Celestial Mechanics Unit, Astronomy Department
Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics
National Research Institute of Astronomy and Geophysics (NRIAG)
King Abdulaziz University
Egypt
Saudi Arabia
eabouelmagd@gmail.com;eabouelmagd@nriag.sci.eg;eabouelmagd@kau.edu.s
F.
Alzahrani
Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics
King Abdulaziz University
Saudi Arabia
A.
Hobiny
Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics
King Abdulaziz University
Saudi Arabia
J. L. G.
Guirao
Departamento de Matemática Aplicada y Estadística
Universidad Politécnica de Cartagena
Spain
juan.garcia@upct.es
M.
Alhothuali
Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics
King Abdulaziz University
Saudi Arabia
Restricted three-body problem
collinear points
periodic orbits.
Article.27.pdf
[
[1]
F. A. Abd El-Salam, Stability of triangular equilibrium points in the elliptic restricted three body problem with oblate and triaxial primaries, Astrophysics Space Sci., 357 (2015), 1-9
##[2]
E. I. Abouelmagd, Existence and stability of triangular points in the restricted three-body problem with numerical applications, Astrophysics Space Sci., 342 (2012), 45-53
##[3]
E. I. Abouelmagd, Stability of the triangular points under combined effects of radiation and oblateness in the restricted three-body problem, Earth Moon Planets, 110 (2013), 143-155
##[4]
E. I. Abouelmagd , The effect of photogravitational force and oblateness in the perturbed restricted three-body problem, Astrophysics Space Sci., 346 (2013), 51-69
##[5]
E. I. Abouelmagd, M. S. Alhothuali, J. L. G. Guirao, H. M. Malaikah, On the periodic structure in the planar photogravitational Hill problem, Appl. Math. Inf. Sci., 9 (2015), 2409-2416
##[6]
E. I. Abouelmagd, M. S. Alhothuali, J. L. G. Guirao, H. M. Malaikah, Periodic and secular solutions in the restricted threebody problem under the effect of zonal harmonic parameters, Appl. Math. Inf. Sci., 2015 (9), 1659-1669
##[7]
E. I. Abouelmagd, M. S. Alhothuali, J. L. G. Guirao, H. M. Malaikah, The effect of zonal harmonic coefficients in the framework of the restricted three-body problem, Adv. Space Res., 55 (2015), 1660-1672
##[8]
E. I. Abouelmagd, H. M. Asiri, M. A. Sharaf, The effect of oblateness in the perturbed restricted three-body problem, Meccanica, 48 (2013), 2479-2490
##[9]
E. I. Abouelmagd, M. E. Awad, E. M. A. Elzayat, I. A. Abbas, Reduction the secular solution to periodic solution in the generalized restricted three-body problem , Astrophysics Space Sci., 350 (2014), 495-505
##[10]
E. I. Abouelmagd, S. M. El-Shaboury, Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodies, Astrophysics Space Sci., 341 (2012), 331-341
##[11]
E. I. Abouelmagd, J. L. G. Guirao, A. Mostafa , Numerical integration of the restricted thee-body problem with Lie series , Astrophysics Space Sci., 354 (2014), 369-378
##[12]
E. I. Abouelmagd, A. Mostafa, Out of plane equilibrium points locations and the forbidden movement regions in the restricted three-body problem with variable mass, Astrophysics Space Sci., 357 (2015), 1-10
##[13]
E. I. Abouelmagd, M. A. Sharaf, The motion around the libration points in the restricted three-body problem with the effect of radiation and oblateness, Astrophysics Space Sci., 344 (2013), 321-332
##[14]
S. M. El-Shaboury, A. Mostafa, The singly averaged elliptical restricted three-body problem, Astrophys. Space Sci., 348 (2013), 385-391
##[15]
G. Gómez, J. M. Mondelo, The dynamics around the collinear equilibrium points of the RTBP , Phys. D, 157 (2001), 283-321
##[16]
V. S. Kalantonis, C. N. Douskos, E. A. Perdios, Numerical determination of homoclinic and heteroclinic orbits at collinear equilibria in the restricted Three-Body Problem with Oblateness, Celestial Mech. Dynam. Astronom., 94 (2006), 135-153
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A. Mittal, I. Ahmad, K. B. Bhatnagar, Periodic orbits in the photogravitational restricted problem with the smaller primary an oblate body, Astrophysics Space Sci., 323 (2009), 65-73
##[19]
A. Narayan, T. Usha, Stability of triangular equilibrium points in the elliptic restricted problem of three bodies with radiating and triaxial primaries, Astrophysics Space Sci., 351 (2014), 135-142
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##[23]
J. Singh, J. J. Taura, Stability of Triangular Equilibrium Points in the Photo gravitational Restricted Three-Body Problem with Oblateness and Potential from a Belt, Astrophysics Space Sci., 35 (2014), 107-119
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##[25]
G. A. Tsirogiannis, C. N. Douskos, E. A. Perdios, Computation of the Liapunov orbits in the photo gravitational RTBP with oblateness, Astrophysics Space Sci., 305 (2006), 389-398
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T. Usha, A. Narayan, B. Ishwar , Effects of radiation and triaxiality of primaries on triangular equilibrium points in elliptic restricted three body problem , Astrophysics Space Sci., 349 (2014), 151-164
]
\(\beta_1\)-paracompact spaces
\(\beta_1\)-paracompact spaces
en
en
We introduce the class of \(\beta_1\)-paracompact spaces in topological spaces and give characterizations of such
spaces. We study subsets and subspaces of \(\beta_1\)-paracompact spaces and discuss their properties. Also, we
investigate the invariants of \(\beta_1\)-paracompact spaces by functions.
1728
1734
Heyam Hussain
Aljarrah
Department of Mathematics, Faculty of Science
Yarmouk University
Jordan
hiamaljarah@yahoo.com
Paracompact
\(\beta_1\)-paracompact
\(\beta\)-open set
locally finite collection.
Article.28.pdf
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A research on the some properties and distribution of zeros for Stirling polynomials
A research on the some properties and distribution of zeros for Stirling polynomials
en
en
We find some identities of the Stirling polynomials and relations between these polynomials and other
numbers and polynomials such as generalized Bernoulli numbers. We also display some properties and
figures that are related to the distribution of fixed points in the Stirling polynomials from the Newton
dynamical system containing iterated mapping.
1735
1747
Jung Yoog
Kang
Department of Mathematics
Hannam University
Korea
rkdwjddnr2002@yahoo.co.kr
Cheon Seoung
Ryoo
Department of Mathematics
Hannam University
Korea
ryoocs@hnu.kr
Stirling polynomials
generalized Bernoulli numbers and polynomials
Euler polynomials of the second kind
Newton dynamical system
fixed point.
Article.29.pdf
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]
On the dynamics of positive solutions for the difference equation in a new population model
On the dynamics of positive solutions for the difference equation in a new population model
en
en
In this paper we study the boundedness and the asymptotic behavior of positive solutions for the difference equation
\[x_{n+1} = a + bx_ne^{-x_{n-1}}\] ;
where \(a; b\) are positive constants, and the initial values \(x_{-1}; x_0\) are nonnegative numbers.
1748
1754
Wenjie
Wang
Department of Mathematics
Lanzhou Jiaotong University
China
wangwenjie@mail.lzjtu.cn
Hui
Feng
Department of Mathematics
Northwest Normal University
China
fh_9237@163.com
Difference equations
boundedness
asymptotic behavior.
Article.30.pdf
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J. R. Beddington, C. A. Free, J. H. Lawton, Dynamic complexity in predator prey models framed in difference equations, Nature, 255 (1975), 58-60
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R. Devault, W. Kosmala, G. Ladas, S. W. Schultz, Global Behavior of \(y_{n+1} = \frac{p+y_{n-k}}{ qy_n+y_{n-k}}\), Nonlinear Anal., 47 (2001), 4743-4751
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E. El-Metwally, E. A. Grove, G. Ladas, R. Levins, M. Radin, On the difference equation \(x_{n+1 }= \alpha + \beta x_{n-1}e^{-x_n}\), Nonlinear Anal., 47 (2001), 4623-4634
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N. Fotiades, G. Papaschinopoulos, Existence, uniqueness and attractivity of prime period two solution for a difference equation of exponential form, Appl. Math. Comput. Model., 218 (2012), 11648-11653
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Several improvements of Mitrinovic-Adamovic and Lazarevics inequalities with applications to the sharpening of Wilker-type inequalities
Several improvements of Mitrinovic-Adamovic and Lazarevics inequalities with applications to the sharpening of Wilker-type inequalities
en
en
In this paper, we give several improvements of Mitrinović-Adamović's inequality and Lazarević's inequality. Our results show some interesting relationships between Mitrinović-Adamović's inequality and
Lazarević's inequality. At the end of the paper, the improved Lazarević's inequality is applied to the sharpening of Wilker-type inequalities for hyperbolic functions.
1755
1765
Shan-He
Wu
Department of Mathematics
Longyan University
P. R. China
shanhewu@gmail.com
Huan-Peng
Yue
Department of Mathematics
Longyan University
P. R. China
huanpengyue@aliyun.com
Yong-Ping
Deng
Department of Mathematics
Longyan University
P. R. China
yongpingdengfj@163.com
Yu-Ming
Chu
School of Mathematics and Computation Science
Hunan City University
P. R. China
chuyuming2005@126.com
Wilker-type inequalities
hyperbolic functions
Mitrinović-Adamović's inequality
Lazarević's inequality
improvement.
Article.31.pdf
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L. Yin, L. Huang, F. Qi, Some inequalities for the generalized trigonometric and hyperbolic functions, Turkish J. Anal. Number theor., 2 (2014), 96-101
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]
Calculation of generalized period constants via time-angle difference for complex analytic systems with resonant ratio
Calculation of generalized period constants via time-angle difference for complex analytic systems with resonant ratio
en
en
In the case of a critical point being a center, the isochronicity problem (or linearizability problem) is far
to be solved in general. A progressive way to find necessary conditions for isochronicity is to compute period
constants. In this paper, we establish a new recursive algorithm of calculation of the so-called generalized
period constants. Furthermore, we verify the new algorithm by the existing results for the Lotka-Volterra
system with 3 : -2 resonance. Finally, the algorithm is applied to solve the linearizability problem for the
Lotka-Volterra system in the ratio 4 : -5.
1766
1775
Yusen
Wu
School of Mathematics and Statistics
Henan University of Science and Technology
P. R. China
wuyusen621@126.com
Generalized period constant
recursive algorithm
linearizability
Lotka-Volterra system.
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Strong convergence of hybrid Halpern processes in a Banach space
Strong convergence of hybrid Halpern processes in a Banach space
en
en
The purpose of this paper is to investigate convergence of a hybrid Halpern process for fixed point and
equilibrium problems. Strong convergence theorems of common solutions are established in a strictly convex
and uniformly smooth Banach space which also has the Kadec-Klee property.
1776
1786
Yuan
Hecai
School of Mathematics and Information Science
North China University of Water Resources and Electric Power
China
hsyuanhc@yeah.net
Zhaocui
Min
School of Science
Hebei University of Engineering
China
Asymptotically nonexpansive mapping
fixed point
quasi-\(\phi\)-nonexpansive mapping
equilibrium problem
generalized projection.
Article.33.pdf
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R. P. Agarwal, Y. J. Cho, X. Qin, Generalized projection algorithms for nonlinear operators, Numer. Funct. Anal. Optim., 28 (2007), 1197-1215
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Y. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, Lecture Notes in Pure and Appl. Math., Dekker, New York, 178 (1996), 15-50
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B. A. Bin Dehaish, X. Qin, A. Latif, H. O. Bakodah , Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., 16 (2015), 1321-1336
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S. Y. Cho, X. Qin, S. M. Kang, Iterative processes for common fixed points of two different families of mappings with applications, J. Global Optim., 57 (2013), 1429-1446
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S. Y. Cho, X. Qin, L. Wang, Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., 2014 (2014), 1-15
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W. Cholamjiak, P. Cholamjiak, S. Suantai, Convergence of iterative schemes for solving fixed point problems for multi-valued nonself mappings and equilibrium problems, J. Nonlinear Sci. Appl., 8 (2015), 1245-1256
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Y. Hao, Some results on a modified Mann iterative scheme in a reFLexive Banach space, Fixed Point Theory Appl., 2013 (2013), 1-14
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J. K. Kim, Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-\(\phi\)-nonexpansive mappings, Fixed Point Theory Appl., 2011 (2011), 1-15
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J. K. Kim, Convergence theorems of iterative sequences for generalized equilibrium problems involving strictly pseudocontractive mappings in Hilbert spaces, J. Comput. Anal. Appl., 18 (2015), 454-471
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Y. Liu, Convergence theorems for a generalized equilibrium problem and two asymptotically nonexpansive mappings in Hilbert spaces , Nonlinear Funct. Anal. Appl., 19 (2014), 317-328
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M. A. Noor, K. I. Noor, M. Waseem, Decomposition method for solving system of linear equations, Eng. Math. Lett., 2 (2013), 34-41
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S. Park, A review of the KKM theory on \(\phi_A\)-space or GFC-spaces , Adv. Fixed Point Theory, 3 (2013), 355-382
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X. Qin, S. Y. Cho, S. M. Kang, On hybrid projection methods for asymptotically quasi-\(\phi\)-nonexpansive mappings, Appl. Math. Comput., 215 (2010), 3874-3883
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X. Qin, L. Wang, On asymptotically quasi-\(\phi\)-nonexpansive mappings in the intermediate sense, Abstr. Appl. Anal., 2012 (2012), 1-13
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J. Song, M. Chen, On generalized asymptotically quasi-\(\phi\)-nonexpansive mappings and a Ky Fan inequality, Fixed Point Theory Appl., 2013 (2013), 1-15
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T. V. Su, Second-order optimality conditions for vector equilibrium problems, J. Nonlinear Funct. Anal., 2015 (2015), 1-31
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Y. Su, X. Qin, Strong convergence of modified Ishikawa iterations for nonlinear mappings, Proc. Indian Academy Sci., 117 (2007), 97-107
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W. Wang, J. Song, Hybrid projection methods for a bifunction and relatively asymptotically nonexpansive mappings, Fixed Point Theory Appl., 2013 (2013), 1-10
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Z. M. Wang, X. Zhang , Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems, J. Nonlinear Funct. Anal., 2014 (2014), 1-25
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J. Ye, J. Huang , An iterative method for mixed equilibrium problems, fixed point problems of strictly pseudo-contractive mappings and nonexpansive semi-groups, Nonlinear Funct. Anal. Appl., 18 (2013), 307-325
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H. Zegeye, N. Shahzad, Strong convergence theorem for a common point of solution of variational inequality and fixed point problem, Adv. Fixed Point Theory, 2 (2012), 374-397
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J. Zhao, Strong convergence theorems for equilibrium problems, fixed point problems of asymptotically nonexpansive mappings and a general system of variational inequalities, Nonlinear Funct. Anal. Appl., 16 (2011), 447-464
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L. Zhang, H. Tong , An iterative method for nonexpansive semigroups, variational inclusions and generalized equilibrium problems, Adv. Fixed Point Theory, 4 (2014), 325-343
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L. Zhang, Y. Hao , Fixed point methods for solving solutions of a generalized equilibrium problem, J. Nonlinear Sci. Appl., 9 (2016), 149-159
]
A fixed point approach to the stability of an AQCQ-functional equation in RN-spaces
A fixed point approach to the stability of an AQCQ-functional equation in RN-spaces
en
en
Using the fixed point method, we prove the Hyers-Ulam stability of the following additive-quadratic-
cubic-quartic functional equation
\[f(x + 2y) + f(x - 2y) = 4f(x + y) + 4f(x - y) - 6f(x) + f(2y) + f(-2y) - 4f(y) - 4f(-y)\]
in random normed spaces.
1787
1806
Choonkil
Park
Research Institute for Natural Sciences
Hanyang University
Republic of Korea
baak@hanyang.ac.kr
Dong Yun
Shin
Department of Mathematics
University of Seoul
Republic of Korea
dyshin@uos.ac.kr
Sungjin
Lee
Department of Mathematics
Daejin University
Republic of Korea
hyper@daejin.ac.kr
Random normed space
fixed point
Hyers-Ulam stability
additive-quadratic-cubic-quartic functional equation.
Article.34.pdf
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J. Aczel, J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge (1989)
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]
Multi-soliton solutions of the BBM equation arisen in shallow water
Multi-soliton solutions of the BBM equation arisen in shallow water
en
en
In this work, multiple soliton solutions and multiple singular soliton solutions are formally derived for
the BBM equation. A novel transformation method combined with the Hirota's bilinear method are used
to determine the two sets of solutions, where each set has a distinct structure. The resonance phenomenon
does not exist for the model under the study.
1807
1814
O.
Alsayyed
Department of Mathematics
Hashemite University
Jordan
o.alsayyed@yahoo.com
H. M.
Jaradat
Department of Mathematics
Department of Mathematics and Applied Sciences
Al al-Bayt University
Dhofar University
Jordan
Oman
husseinjaradat@yahoo.com
M. M. M.
Jaradat
Department of Mathematics, Statistics and Physics
Qatar University
Qatar
mmjst4@qu.edu.qa
Z.
Mustafa
Department of Mathematics, Statistics and Physics
Qatar University
Qatar
zead@qu.edu.qa
F.
Shatat
General Courses Department
Emirates College of Technology
Emirates
feras.shatat@ect.ac.ae
N-soliton solutions
BBM equation
shallow water waves.
Article.35.pdf
[
[1]
A. S. Abdel Rady, E. S. Osman, M. Khalfallah, The homogeneous balance method and its application to the Benjamin-Bona-Mahoney (BBM) equation, Appl. Math. Comput., 217 (2010), 1385-1390
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]
Reduced-order synchronization of fractional order chaotic systems with fully unknown parameters using modified adaptive control
Reduced-order synchronization of fractional order chaotic systems with fully unknown parameters using modified adaptive control
en
en
A novel reduced-order adaptive controller is extended and developed to synchronize two different fractional
order chaotic systems with different dimensions. Based upon the parameters modulation and the
adaptive control techniques, we show that dynamical evolution of third{order fractional order chaotic system
can be synchronized with the projection of a fourth{order fractional order chaotic system even though
their parameters are unknown. The techniques are successfully applied to fractional order hyperchaotic
Chen (4th-order) and fractional order chaotic Liu (3rd-order) systems. Theoretical analysis and numerical
simulations are shown to verify the results.
1815
1825
M. M.
Al-sawalha
Mathematics Department, Faculty of Science
University of Hail
Kingdom of Saudi Arabia
sawalha_moh@yahoo.com
M.
Shoaib
Abu Dhabi Men's College
Higher Colleges of Technology
United Arab Emirates
safridi@gmail.com
Reduced-order
synchronization
adaptive control
fractional order chaos.
Article.36.pdf
[
[1]
S. K. Agrawal, S. Das, A modified adaptive control method for synchronization of some fractional chaotic systems with unknown parameters, Nonlinear. Dynam., 73 (2013), 907-919
##[2]
S. K. Agrawal, S. Das , Function projective synchronization between four dimensional chaotic systems with uncertain parameters using modified adaptive control method , J. Process Control, 24 (2014), 517-530
##[3]
S. K. Agrawal, M. Srivastava, S. Das , Synchronization of fractional order chaotic systems using active control method, Chaos Solitons Fractals, 45 (2012), 737-752
##[4]
M. M. Al-Sawalha, M. S. M. Noorani , On anti-synchronization of chaotic systems via nonlinear control, Chaos Solitons Fractals, 42 (2009), 170-179
##[5]
M. M. Al-Sawalha, M. S. M. Noorani , Anti-synchronization of two hyperchaotic systems via nonlinear control , Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 3402-3411
##[6]
M. M. Al-Sawalha, M. S. M. Noorani, Anti-synchronization between two Different hyperchaotic systems, J. Uncertain Sys., 3 (2009), 192-200
##[7]
M. M. Al-Sawalha, M. S. M. Noorani, Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters , Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 1036-1047
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M. M. Al-Sawalha, M. S. M. Noorani , Adaptive reduced-order anti-synchronization of chaotic systems with fully unknown parameters, Commun. Nonlinear. Sci. Numer. Simul., 15 (2010), 3022-3034
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M. M. Al-Sawalha, M. S. M. Noorani , Chaos reduced-order anti-synchronization of chaotic systems with fully unknown parameters, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1908-1920
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S. Bhalekar, V. D. Gejji , Synchronization of different fractional order chaotic systems using active control , Commun. Nonlinear. Sci. Numer. Simul., 15 (2010), 3536-3546
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H. Dua, Q. Zeng, C. Wang, M. Ling , Function projective synchronization in coupled chaotic systems, Nonlinear Anal. Real World Appl., 11 (2010), 705-712
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Z. Gao, X. Liao, Integral sliding mode control for fractional-order systems with mismatched uncertainties, Nonlinear Dynam., 72 (2013), 27-35
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R. Hilfer, Applications of Fractional Calculus in Physics World Scientific, New Jersey, (2001)
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W. Jawaada, M. S. M. Noorani, M. M. Al-sawalha, Robust active sliding mode anti-synchronization of hyperchaotic systems with uncertainties and external disturbances, Nonlinear Anal. Real World Appl., 13 (2012), 2403-2413
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J. A. Laoye, U. E. Vincent, O. O. Akigbogun, Chaos Control and Reduced-order Synchronization of the Rigid Body, Int. J. Nonlinear Sci., 6 (2008), 106-113
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Z. Li, X. Zhao , Generalized function projective synchronization of two different hyperchaotic systems with unknown parameters, Nonlinear Anal. Real World Appl., 12 (2011), 2607-2615
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C. Liu, L. Liu, T. Liu, A novel three-dimensional autonomous chaos system , Chaos Solitons Fractals, 39 (2009), 1950-1958
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##[23]
L. Pana, W. Zhoua, J. Fanga, D. Lic, Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems via novel active pinning control , Commun. Nonlinear. Sci. Numer. Simul., 15 (2010), 3754-3762
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J. Park , Adaptive synchronization of hyperchaotic Chen system with uncertain parameters , Chaos Solitons Fractals, 26 (2005), 959-964
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I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999)
##[26]
C. Shihua, J. Lü , Parameters identification and synchronization of chaotic systems based upon adaptive control , Phys. Lett. A, 299 (2002), 353-358
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M. Zribi, N. Smaoui, H. Salim , Synchronization of the unified chaotic systems using a sliding mode controller , Chaos Solitons Fractals, 42 (2009), 3197-3209
]
Multistage Optimal Homotopy Asymptotic Method for Solving Initial-Value Problems
Multistage Optimal Homotopy Asymptotic Method for Solving Initial-Value Problems
en
en
In this paper, a new approximate analytical algorithm namely multistage optimal homotopy asymptotic
method (MOHAM) is presented for the first time to obtain approximate analytical solutions for linear,
nonlinear and system of initial value problems (IVPs). This algorithm depends on the standard optimal
homotopy asymptotic method (OHAM), in which it is treated as an algorithm in a sequence of subinterval.
The main advantage of this study is to obtain continuous approximate analytical solutions for a long time
span. Numerical examples are tested to highlight the important features of the new algorithm. Comparison
of the MOHAM results, standard OHAM, available exact solution and the fourth-order Runge Kutta (RK4)
reveale that this algorithm is effective, simple and more impressive than the standard OHAM for solving
IVPs.
1826
1843
N. R.
Anakira
Department of Mathematics, Faculty of Science and Technology
Irbid National University
Jordan
alanaghreh_nedal@yahoo.com
A. k.
Alomari
Department of Mathematics, Faculty of Science
Yarmouk University
Jordan
abdomari2008@yahoo.com
A. F.
Jameel
School of Quantitative Sciences
Universiti Utara Malaysia (UUM)
Malaysia
kakarotte79@gmail.com
I.
Hashim
School of Mathematical Sciences
Universiti Kebangsaan Malaysia
Malaysia
ishak_h@ukm.edu.my
Optimal homotopy asymptotic method (OHAM)
multistage optimal homotopy asymptotic method (MOHAM)
initial value problems
series solution
Mathematica 9.
Article.37.pdf
[
[1]
J. Ali, S. Islam, S. Islam, G. Zaman, The solution of multipoint boundary value problems by the optimal homotopy asymptotic method, Comput. Math. Appl., 59 (2010), 2000-2006
##[2]
A. K. Alomari, A novel solution for fractional chaotic Chen system, J. Nonlinear Sci. Appl., 8 (2015), 478-488
##[3]
A. K. Alomari, F. Awawdeh, N. Tahat, F. Bani Ahmad, W. Shatanawi, Multiple solutions for fractional differential equations: Analytic approach, Appl. Math. Comput., 219 (2013), 8893-8903
##[4]
A. K. Alomari, N. Ratib Anakira, A. S. Bataineh, I. Hashim, Approximate solution of nonlinear system of BVP arising in fluid flow problem, Math. Probl. Eng., 2013 (2013), 1-7
##[5]
A. K. Alomari, N. Ratib Anakira, I. Hashim, Multiple solutions of problems in fluid mechanics by predictor optimal homotopy asymptotic method , Adv. Mech. Eng., 6 (2014), 1-20
##[6]
J. D. Faires, R. Burden, Numerical Methods, Brooks/Cole Publishing Co, Pacific Grove (1993)
##[7]
M. Ghoreishi, A. Ismail, A. K. Alomari , Application of the homotopy analysis method for solving a model for HIV infection of CD4(+) T-cells, Math. Comput. Model, 54 (2013), 3007-3015
##[8]
M. Ghoreishi, A. Ismail, A. K. Alomari, A. S. Bataineh, The comparison between homotopy analysis method and optimal homotopy asymptotic method for nonlinear age-structured population models, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1163-1177
##[9]
S. Ghosh, A. Roy, D. Roy, An adaptation of adomian decomposition for numericanalytic integration of strongly nonlinear and chaotic oscillators , Comput. Methods Appl. Mech. Eng., 196 (2007), 1133-1153
##[10]
S. Haq, M. Ishaq, Solution of coupled Whitham-Broer-Kaup equations using optimal homotopy asymptotic method, Ocean Eng., 84 (2014), 81-88
##[11]
I. Hashim, M. S. H. Chowdhury , Adaptation of homotopy-perturbation method for numeric-analytic solution of system of ODEs , Phys. Lett. A, 372 (2008), 470-481
##[12]
M. S. Hashim, N. Khan, S. Iqbal, Optimal homotopy asymptotic method for solving nonlinear Fredholm integral equations of second kind , Appl. Math. Comput., 218 (2012), 10982-10989
##[13]
I. Hashim, M. S. M. Noorani, R. Ahmad, S. A. Bakar, E. S. Ismail, A. M. Zakaria, Accuracy of the Adomian decomposition method applied to the Lorenz system, Chaos Solitons Fractals, 28 (2006), 1149-1158
##[14]
N. Herişanu, V. Marinca , Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method, Comput. Math. Appl., 60 (2010), 1607-1615
##[15]
N. Herişanu, V. Marinca , Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia, Meccanica, 45 (2010), 847-855
##[16]
S. Iqbal, A. Javed , Application of optimal homotopy asymptotic method for the analytic solution of singular Lane-Emden type equation , Appl. Math. Comput., 217 (2011), 7753-7761
##[17]
M. J. Jang, C. L. Chen, Y. C. Liy, On solving the initial-value problems using the differential transformation method, Appl. Math. Comput., 115 (2000), 145-160
##[18]
V. Marinca, R. D. Ene, Dual approximate solutions of the unsteady viscous flow over a shrinking cylinder with optimal homotopy asymptotic method, Adv. Math. Phys., 2014 (2014), 1-11
##[19]
V. Marinca, R. D. Ene, B. Marinca, Analytic approximate solution for Falkner-Skan Equation , Sci. World J., 2014 (2014), 1-22
##[20]
V. Marinca, N. Herişanu , Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer , Int. Commun. Heat Mass, 35 (2008), 710-715
##[21]
V. Marinca, N. Herişanu, The optimal homotopy asymptotic method for solving Blasius equation, Appl. Math. Comput., 231 (2014), 134-139
##[22]
V. Marinca, N. Herişanu, C. Bota, B. Marinca, An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate, Appl. Math. Lett., 22 (2009), 245-251
##[23]
V. Marinca, N. Herişanu, I. Nemeş , Optimal Homotopy Asymptotic Method with application to thin film flow, Cent. Eur. J. Phys., 6 (2008), 648-653
##[24]
N. Ratib Anakira, A. K. Alomari, I. Hashim , Numerical scheme for solving singular two-point boundary value problems, J. Appl. Math., 2013 (2013), 1-8
##[25]
N. Ratib Anakira, A. K. Alomari, I. Hashim, Optimal homotopy asymptotic method for solving delay differential equations, Math. Probl. Eng., 2013 (2013), 1-11
##[26]
N. Shawagfeh, D. Kaya, Comparing numerical method for the solutions of systems of ordinary differenial equations , Appl. Math. Lett., 17 (2004), 323-328
##[27]
P. Vadasza, S. Olekb, Convergence and accuracy of Adomian's decomposition method for the solution of Lorenz equations, Int. J. Heat Mass Transfer, 43 (2000), 1715-1734
]
Derivation and applications of inequalities of Ostrowski type for n-times differentiable mappings for cumulative distribution function and some quadrature rules
Derivation and applications of inequalities of Ostrowski type for n-times differentiable mappings for cumulative distribution function and some quadrature rules
en
en
In this paper new integral inequalities of Ostrowski type are developed for n-times differentiable mappings.
Some well known inequalities become special cases of the inequalities obtained in this paper. With
the help of obtained inequalities, we will derive new and efficient quadrature rules which are analyzed with
the help of specific examples. We also give applications for cumulative distribution function.
1844
1857
Ather
Qayyum
Department of Fundamental and Applied Sciences
Universiti Teknologi PETRONAS
Malaysia
atherqayyum@gmail.com
Abdul Rehman
Kashif
Department of Mathematics
University of Hail
Saudi Arabia
kashmology@gmail.com
Muhammad
Shoaib
Abu Dhabi Mens College
Higher Colleges of Technology
United Arab Emirates
safridi@gmail.com
Ibrahima
Faye
Department of Fundamental and Applied Sciences
Universiti Teknologi PETRONAS
Malaysia
ibrahima faye@petronas.com.my
Ostrowski inequality
numerical integration
composite quadrature rule
cumulative distributive function.
Article.38.pdf
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[1]
R. P. Agarwal, V. Čuljak, J. Pečarić, Some integral inequalities involving bounded higher order derivatives, Math. Comput. Modelling, 28 (1998), 51-57
##[2]
M. Alomari, M. Darus, Some Ostrowski type inequalities for convex functions with applications to special means, RGMIA Res. Rep. Coll., (2010)
##[3]
P. L. Čebyšev , Sur less expressions approximatives des integrales definies par les autres prises entre les memes limites, Proc. Math. Soc. Charkov, 2 (1882), 93-98
##[4]
P. Cerone, A new Ostrowski Type Inequality Involving Integral Means Over End Intervals, Tamkang J. Math., 33 (2002), 109-118
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]
Fixed points and quadratic rho-functional equations
Fixed points and quadratic rho-functional equations
en
en
In this paper, we solve the quadratic \(\rho\)-functional equations
\[f(x + y) + f(x - y) - 2f(x) - 2f(y) = \rho
\left(
2f
(\frac{x + y}{2})
+ 2f
(\frac{x - y}{2})
- f(x) - f(y)\right), \qquad (1)\]
where \(\rho\) is a fixed non-Archimedean number or a fixed real or complex number with \(\rho\neq 1;2\), and
\[2f
(\frac{x + y}{2})
+ 2f
(\frac{x - y}{2})
- f(x) - f(y) = \rho
\left(f(x + y) + f(x - y) - 2f(x) - 2f(y)\right); \qquad (2)\]
where \(\rho\) is a fixed non-Archimedean number or a fixed real or complex number with \(\rho\neq 1; \frac{-1}{2}\).
Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic \(\rho\)-functional equations
(1) and (2) in non-Archimedean Banach spaces and in Banach spaces.
1858
1871
Choonkil
Park
Research Institute for Natural Sciences
Hanyang University
Korea
baak@hanyang.ac.kr
Sang Og
Kim
Department of Mathematics
Hallym University
Korea
sokim@hallym.ac.kr
Hyers-Ulam stability
non-Archimedean normed space
fixed point
quadratic \(\rho\)-functional equation.
Article.39.pdf
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]
On the Appell type \(\lambda\)-Changhee polynomials
On the Appell type \(\lambda\)-Changhee polynomials
en
en
In the paper, by virtue of the p-adic fermionic integral on \(\mathbb{Z}_p\), the authors consider a \(\lambda\)-analogue of the
Changhee polynomials and present some properties and identities of these polynomials.
1872
1876
Dongkyu
Lim
School of Mathematical Sciences
Nankai University
China
dgrim84@gmail.com
Feng
Qi
Department of Mathematics, College of Science
Institute of Mathematics
Tianjin Polytechnic University
Henan Polytechnic University
China
China
qifeng618@gmail.com;qifeng618@hotmail.com
Identity
property
Appell type \(\lambda\)-Changhee polynomial
Changhee polynomial
degenerate Changhee polynomial.
Article.40.pdf
[
[1]
E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277
##[2]
D. V. Dolgy, T. Kim, S.-H. Rim, J.-J. Seo, A note on Changhee polynomials and numbers with q-parameter, Int. J. Math. Anal., 8 (2014), 1255-1264
##[3]
T. Kim, A note on p-adic q-integral on \(\mathbb{Z}_p\) associated with q-Euler numbers, Adv. Stud. Contemp. Math. (Kyung- shang), 15 (2007), 133-138
##[4]
D. S. Kim, T. Kim, J.-J. Seo, A note on Changhee numbers and polynomials, Adv. Stud. Theor. Phys., 7 (2013), 993-1003
##[5]
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##[6]
H. I. Kwon, T. Kim, J.-J. Seo, A note on degenerate Changhee numbers and polynomials, Proc. Jangjeon Math. Soc., 18 (2015), 295-305
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]
Chaos analysis of the nonlinear duffing oscillators based on the new Adomian polynomials
Chaos analysis of the nonlinear duffing oscillators based on the new Adomian polynomials
en
en
Numerical recurrence formulae are given to investigate the chaotic motion of the famous Duffing system.
The new Adomian polynomial is adopted to treat the cubic nonlinear term. With the numerical simulation
of the phase portraits and the Poincare sections, the chaotic behaviors are discussed for varied frequencies,
damping coefficients and forces. The results show that the numerical method is reliable to investigate chaotic
systems.
1877
1881
L. L.
Huang
Institute of Applied Nonlinear Science, College of Mathematics and Information Science
Neijiang Normal University
China
G. C.
Wu
Institute of Applied Nonlinear Science, College of Mathematics and Information Science
Neijiang Normal University
China
wuguocheng@gmail.com
M. M.
Rashidi
Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems
ENN-Tongji Clean Energy Institute of Advanced Studies
Tongji University
China
China
mm_rashidi@tongji.edu.cn
W. H.
Luo
Institute of Applied Nonlinear Science, College of Mathematics and Information Science
Neijiang Normal University
China
New Adomian polynomial
duffing systems
chaos.
Article.41.pdf
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]
Iterative common solutions of fixed point and variational inequality problems
Iterative common solutions of fixed point and variational inequality problems
en
en
In this paper, fixed point and variational inequality problems are investigated based on a viscosity
approximation method. Strong convergence theorems are established in the framework of Hilbert spaces.
1882
1890
Yunpeng
Zhang
College of Electric Power
North China University of Water Resources and Electric Power
China
zhangypliyl@yeah.net
Qing
Yuan
Department of Mathematics
Linyi University
China
zjyuanq@yeah.net
Inverse-strongly monotone operator
nonexpansive mapping
variational inequality
fixed point.
Article.42.pdf
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[1]
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Z. M. Wang, X. Zhang, Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems , J. Nonlinear Funct. Anal., 2014 (2014), 1-25
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]
A new convergence theorem in a reflexive Banach space
A new convergence theorem in a reflexive Banach space
en
en
In this paper, fixed points of an asymptotically quasi-\(\phi\)-nonexpansive mapping in the intermediate sense
and a bifunction are investigated based on a monotone projection algorithm. A strong convergence theorem
is established in a reflexive Banach space.
1891
1901
Xiaoying
Gong
Department of Mathematics and Sciences
Shijiazhuang University of Economics
China
hbgongxy@sohu.com
Wenxin
Wang
Department of Applied Mathematics and Physics
North China Electric Power University
China
hdwangwx@sohu.com
Equilibrium problem point
quasi-\(\phi\)-nonexpansive mapping
fixed point
projection
variational inequality.
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Y. J. Cho, S. M. Kang, X. Qin, On systems of generalized nonlinear variational inequalities in Banach spaces, Appl. Math. Comput., 206 (2008), 214-220
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Y. J. Cho, X. Qin, S. M. Kang, Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems, Nonlinear Anal., 71 (2009), 4203-4214
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S. Y. Cho, X. Qin, L. Wang, Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., 2014 (2014), 1-15
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A. Genel, J. Lindenstruss, An example concerning fixed points, Israel J. Math., 22 (1975), 81-86
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Y. Hao, Some results on a modified Mann iterative scheme in a reflexive Banach space, Fixed Point Theory Appl., 2013 (2013), 1-14
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R. H. He, Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequal- ities in FC-spaces, Adv. Fixed Point Theory, 2 (2012), 47-57
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M. A. Khan, N. C. Yannelis, Equilibrium Theory in Infinite dimensional Spaces, Springer-Verlage, New York (1991)
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J. K. Kim, Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-\(\phi\)-nonexpansive mappings, Fixed Point Theory Appl., 2011 (2011), 1-15
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B. Liu, C. Zhang, Strong convergence theorems for equilibrium problems and quasi-\(\phi\)-nonexpansive mappings, Nonlinear Funct. Anal. Appl., 16 (2011), 365-385
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Z. M. Wang, X. Zhang, Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems , J. Nonlinear Funct. Anal., 2014 (2014), 1-25
##[29]
C. Wu, Strong convergence theorems for common solutions of variational inequality and fixed point problems, Adv. Fixed Point Theory, 4 (2014), 229-244
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Y. Yao, M. Aslam Noor, Y. C. Liou, Strong convergence of the modified hybrid steepest-descent methods for general variational inequalities, J. Appl. Math. Comput., 24 (2007), 179-190
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Q. N. Zhang, H. Wu, Hybrid algorithms for equilibrium and common fixed point problems with applications, J. Inequal. Appl., 2014 (2014), 1-13
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J. Zhao , Approximation of solutions to an equilibrium problem in a non-uniformly smooth Banach space, J. Inequal. Appl., 2013 (2013), 1-10
]
Finite time blow up of solutions to an inverse problem for a quasilinear parabolic equation with power nonlinearity
Finite time blow up of solutions to an inverse problem for a quasilinear parabolic equation with power nonlinearity
en
en
We consider an inverse problem for quasilinear parabolic equations with type power nonlinearity.
Sufficient conditions on initial data for blow up result are obtained with positive initial energy. Overdetermination
condition is given as an integral form. To get the blow up result for considered nonlinear
inverse parabolic equation, we use the concavity of a special positive function. The life span of the solution
is also computed.
1902
1910
Şevket
Gür
Department of Mathematics
Sakarya University
Turkey
sgur@sakarya.edu.tr
Metin
Yaman
Department of Mathematics
Sakarya University
Turkey
myaman@sakarya.edu.tr
Yalçın
Yılmaz
Department of Mathematics
Sakarya University
Turkey
yalciny@sakarya.edu.tr
Blow-up
inverse problem
quasilinear parabolic equation.
Article.44.pdf
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[1]
B. A. Bilgin, V. K. Kalantarov, Blow up of solutions to the initial boundary value problem for quasilinear strongly damped wave equations, J. Math. Anal. Appl., 403 (2013), 89-94
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]
Best proximity point theorems for multivalued mappings on partially ordered metric spaces
Best proximity point theorems for multivalued mappings on partially ordered metric spaces
en
en
In this paper, we prove some best proximity point theorems for multivalued mappings in the setting of
complete partially ordered metric spaces. As an application, we infer best proximity point and fixed point
results for single valued mappings in partially ordered metric spaces. The results presented generalize and
improve various known results from best proximity and fixed point theory.
1911
1921
V.
Pragadeeswarar
Department of Mathematics
School of Engineering
India
pragadeeswarar@gmail.com
M.
Marudai
Department of Mathematics
Bharathidasan University
India
marudaim@hotmail.com
P.
Kumam
Department of Mathematics and Theoretical and Computational Science (TaCS) Center, Faculty of Science
China Medical University
King Mongkut's University of Technology Thonburi (KMUTT)
Taiwan
Thailand
poom.kum@kmutt.ac.th; poom.kum@mail.cmu.edu.tw
Partially ordered set
optimal approximate solution
proximally increasing mapping
fixed point
best proximity point.
Article.45.pdf
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[1]
A. Abkar, M. Gabeleh, Best proximity points for cyclic mappings in ordered metric spaces, J. Optim. Theory Appl., 150 (2011), 188-193
##[2]
A. Abkar, M. Gabeleh, Generalized cyclic contractions in partially ordered metric spaces , Optim. Lett., 6 (2012), 1819-1830
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A. Abkar, M. Gabeleh, The existence of best proximity points for multivalued non-self-mappings, Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM, 107 (2013), 319-325
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M. A. Al-Thagafi, N. Shahzad , Convergence and existence results for best proximity points, Nonlinear Anal., 70 (2009), 3665-3671
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I. Beg, A. R. Butt, Common fixed point for generalized set valued contractions satisfying an implicit relation in partially ordered metric spaces, Math. Commun., 15 (2010), 65-76
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A. Eldred, P. Veeramani , Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001-1006
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K. Fan, Extensions of two fixed point theorems of F.E. Browder, Math. Z., 122 (1969), 234-240
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M. Gabeleh, Best proximity points: global minimization of multivalued non-self mappings , Optim. Lett., 8 (2014), 1101-1112
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W. K. Kim, K. H. Lee, Existence of best proximity pairs and equilibrium pairs, J. Math. Anal. Appl., 316 (2006), 433-446
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W. A. Kirk, S. Reich, P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim., 24 (2003), 851-862
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V. Pragadeeswarar, M. Marudai, Best proximity points for generalized proximal weak contractions in partially ordered metric spaces, Optim. Lett., 9 (2015), 105-118
##[15]
V. Pragadeeswarar, M. Marudai, P. Kumam, K. Sitthithakerngkiet , The existence and uniqueness of coupled best proximity point for proximally coupled contraction in a complete ordered metric space, Abstr. Appl. Anal., 2014 (2014), 1-7
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V. S. Raj , A best proximity point theorem for weakly contractive non-self-mappings, Nonlinear Anal., 74 (2011), 4804-4808
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S. Sadiq Basha , Best proximity point theorems on partially ordered sets, Optim. Lett., 7 (2013), 1035-1043
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##[20]
P. S. Srinivasan, P. Veeramani, On existence of equilibrium pair for constrained generalized games, Fixed Point Theory Appl., 1 (2004), 21-29
]
Sharp upper bound involving circuit layout system
Sharp upper bound involving circuit layout system
en
en
In this paper, the circuit layout system in a Euclidean space is defined. By means of the algebraic,
analytic, geometry and inequality theories, a sharp upper bound involving circuit layout system is obtained
as follows:
\[\frac{1}{2}\sum_{1\leq j-i\leq N-1, 1\leq i\leq N}\| A_j^*-A^*_i\|\leq\frac{1}{4}\sqrt{N(1+\max|\cos\angle A_i|)}\csc^2\frac{\pi}{2N}\sqrt{\sum^n_{i=1}\| A_{i+1}-A_i\|^2}.\]
1922
1935
Tianyong
Han
College of Mathematics and Computer Science
Chengdu University
P. R. China
hantian123_123@163.com
Shanhe
Wu
Department of Mathematics
Longyan University
P. R. China
shanhewu@163.com
Jiajin
Wen
College of Mathematics and Computer Science
Chengdu University
P. R. China
wenjiajin623@163.com
Circuit layout system
Euclidean space
power mean
Jensen's inequality.
Article.46.pdf
[
[1]
C. B. Gao, J. J. Wen, Theory of surround system and associated inequalities, Comput. Math. Appl., 63 (2012), 1621-1640
##[2]
J. J. Wen, T. Y. Han, S. S. Cheng, Inequalities involving Dresher variance mean, J. Inequal. Appl., 2013 (2013), 1-29
##[3]
J. J. Wen, Y. Huang, S. S. Cheng, Theory of \(\phi\)-Jensen variance and its applications in higher education, J. Inequal. Appl., 2015 (2015), 1-40
##[4]
J. J. Wen, W. L. Wang, The optimization for the inequalities of power means, J. Inequal. Appl., 2006 (2006), 1-25
##[5]
J. J. Wen, W. L. Wang, Chebyshev type inequalities involving permanents and their applications, Linear Algebra Appl., 422 (2007), 295-303
##[6]
J. J. Wen, S. H. Wu, C. B. Gao, Sharp lower bounds involving circuit layout system, J. Inequal. Appl., 2013 (2013), 1-22
##[7]
J. J. Wen, Z. H. Zhang, Jensen type inequalities involving homogeneous polynomials, J. Inequal. Appl., 2010 (2010), 1-21
]
Stationary distribution and pathwise estimation of n-species mutualism system with stochastic perturbation
Stationary distribution and pathwise estimation of n-species mutualism system with stochastic perturbation
en
en
In this paper, we develop a new stochastic mutualism population model
\[dx_i(t)=x_i(t)\left[\left(r_i+ \sum^n_{j=1}a_{ij}x_j(t)\right)dt + \sigma_i\sigma x_i(t)dB_i(t)\right], \qquad i=1,2,...,n.\]
By constructing suitable Lyapunov functions, we show the system has a stationary distribution. We also
discuss the pathwise behaviour of the solution. The conclusions of this paper is very powerful since they
are independent both of the system parameters and of the initial value. It is also independent of the noise
intensity as long as the noise intensity \(\sigma_i^2 > 0\). Computer simulations are used to illustrated our results.
1936
1943
Weiwei
Fang
School of Mathematical Sciences
Harbin Normal University
P. R. China
fangww11@163.com
Qixing
Han
School of Mathematics
Changchun Normal University
P. R. China
hanqixing123@163.com
Xiangdan
Wen
Department of Mathematics
Yanbian University
P. R. China
xdwen0502@yeah.net
Qiuyue
Li
Department of Foundation Courses
Aviation University of Airforce
P. R. China
qiuyueli:410881136@qq.com
Mutualism model
Itô's formula
stationary distribution
ergodicity
pathwise estimation.
Article.47.pdf
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[1]
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Y. Li, H. Zhang, Existence of periodic solutions for a periodic mutualism model on time scales, J. Math. Anal. Appl., 343 (2008), 818-825
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M. Liu, K. Wang, Analysis of a stochastic autonomous mutualism model , J. Math. Anal. Appl., 402 (2013), 392-403
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Q. Luo, X. Mao , Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84
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X. Mao , Stochastic Differential Equations and Applications, Horwood Publishing, Chichester (1997)
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X. Mao, Delay population dynamics and environmental noise, Stoch. Dyn., 5 (2005), 149-162
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X. Mao, G. Marion, E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110
##[16]
C. Y. Wang, S. Wang, F. P. Yang, L. R. Li , Global asymptotic stability of positive equilibrium of three-species Lotka-Volterra mutualism models with diffusion and delay effects, Appl. Math. Model., 34 (2010), 4278-4288
##[17]
C. Zhu, G. Yin , Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179
]
Solutions of fractional differential equations by Sumudu transform and variational iteration method
Solutions of fractional differential equations by Sumudu transform and variational iteration method
en
en
With the help of the Sumudu transform and the variational iteration method, we solve differential
equations and fractional differential equations related to entropy, wavelets etc. The methods which produce
solutions in terms of convergent series is explained. Some examples are provided to show the accuracy and
easy implementation and to show the methodology.
1944
1951
Pranay
Goswami
School of Liberal Studies
Ambedkar University Delhi
India
pranaygoswami83@gmail.com
Rubayyi T
Alqahtani
Department of Mathematics and Statistics, College of Science
Al-Imam Mohammad Ibn Saud Islamic University (IMSIU)
Saudi Arabia
rr-gahtani@hotmail.com
Variational iteration method
Sumudu transform
fractional differential equation.
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T. Allahviranloo, S. Abbasbandy, H. Rouhparvar, The exact solutions of fuzzy wave-like equations with variable coefficients by a variational iteration method, Appl. Soft Comput., 11 (2011), 2186-2192
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A. Atangana, D. Boleanu, Nonlinear fractional Jaulent-Miodek and Whitham-Broer-Kaup equations within Sumudu transform, Abstr. Appl. Anal., 2013 (2013), 1-8
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A. Atangana, A. Kilicman , The use of Sumudu transform for solving certain nonlinear fractional heat-like equations, Abstr. Appl. Anal., 2013 (2013), 1-12
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S. Das, Analytical Solution of a Fractional Diffusion Equation by Variational Iteration Method, Comput. Math. Appl., 57 (2009), 483-487
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A. Kadem, D. Baleanu , Homotopy perturbation method for the coupled fractional Lotka-Volterra equations, Romanian J. Phys., 56 (2011), 332-338
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]