]>
2018
11
3
ISSN 2008-1898
120
Ground states solutions for modified fourth-order elliptic systems with steep well potential
Ground states solutions for modified fourth-order elliptic systems with steep well potential
en
en
In this paper, we study the following modified quasilinear fourth-order
elliptic systems
\[
\left\{\begin{array}{lll}
\triangle^{2}u-\triangle u+(\lambda\alpha(x)+1)u-\frac{1}{2}\triangle(u^{2})u=\frac{p}{p+q}|u|^{p-2}|v|^{q}u,~~ \mbox{in} \;~\mathbb{R}^{N}, \\
\triangle^{2}v-\triangle v+(\lambda\beta(x)+1)v-\frac{1}{2}\triangle(v^{2})v=\frac{q}{p+q}|u|^{p}|v|^{q-2}v,~~ \mbox{in} \;~\mathbb{R}^{N},\end{array}
\right.\]
where \(\triangle^{2}=\triangle(\triangle)\) is the biharmonic operator, \(\lambda>0\), and \(2<p, 2<q,\) \(4<p+q<22^{\ast\ast}\), \(2^{\ast\ast}=\frac{2N}{N-4} \ (N\leq5)\) \((\mbox{if}~N\leq4, 2^{\ast\ast}=\infty)\) is the critical Sobolev exponent for the embedding \(W^{2,2}(\mathbb{R}^{N})\hookrightarrow L^{2^{\ast\ast}}(\mathbb{R}^{N})\). Under some appropriate assumptions on \(\alpha(x)\) and \(\beta(x)\), we obtain that the above problem has nontrivial ground state solutions via the variational methods. We also explore the phenomenon of concentration of solutions.
323
334
Liuyang
Shao
School of Mathematics and Statistics
Central South University
P. R. China
280970756@qq.com
Haibo
Chen
School of Mathematics and Statistics
Central South University
P. R. China
math_chb@163.com
Fourth-order elliptic
variational methods
ground state solutions
concentration
Article.1.pdf
[
[1]
P. Alvarez-Caudevilla, E. Coloradoa, V. A. Galaktionov , Existence of solutions for a system of coupled nonlinear stationary bi-harmonic Schrödinger equations , Nonlinear Anal., 23 (2015), 78-93
##[2]
T. Bartsch, Z. Q. Wang , Existence and multiplicity results for superlinear elliptic problems on \(R^N\) , Comm. Partial Differential Equations, 20 (1995), 1725-1741
##[3]
H. Brézis, E. Lieb , A relation between point convergence of functions and convergence of functionals , Proc. Amer. Math. Soc., 88 (1983), 486-490
##[4]
P. Candito, L. Li, R. Livrea, Infinitely many solutions for a perturbed nonlinear Navier boundary value problem involving the p-biharmonic, Nonlinear Anal., 75 (2012), 6360-6369
##[5]
S. Chen, J. Liu, X. Wu , Existence and multiplicity of nontrivial solutions for a class of modified nonlinear fourth-order elliptic equations on \(R^N\), Appl. Math. Comput., 248 (2014), 593-608
##[6]
Y. Chen, P. J. McKenna , Traveling waves in a nonlinear suspension beam: theoretical results and numerical observations, J. Differ. Equ., 136 (1997), 325-355
##[7]
Q.-H. Choi, T. Jung , Multiplicity of solutions and source terms in a fourth order nonlinear elliptic equation , Acta Math. Sci., 19 (1999), 361-374
##[8]
Z.-Y. Deng, Y.-S. Huang , Symmetric solutions for a class of singular biharmonic elliptic systems involving critical exponents, Appl. Math. Comput., 264 (2015), 323-334
##[9]
M. Hajipour, A. Malek, High accurate NRK and MWENO scheme fornonlinear degenerate parabolic PDEs, Appl. Math. Model., 36 (2012), 4439-4451
##[10]
M. Hajipour, A. Malek , High accurate modified WENO method for the solution of Black-Scholes equation, Comput. Appl. Math., 34 (2015), 125-140
##[11]
Y.-S. Huang, X.-Q. Liu , Sign-changing solutions for p-biharmonic equations with Hardy potential in the half-space , J. Math. Anal. Appl., 444 (2016), 1417-1437
##[12]
A. C. Lazer, P. J. McKenna , Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis , SIAM Rev., 32 (1990), 537-578
##[13]
A. C. Lazer, P. J. McKenna , Global bifurcation and a theorem of Tarantello , J. Math. Anal. Appl., 181 (1994), 648-655
##[14]
H. Liu, H. Chen , Least energy nodal solution for quasilinear biharmonic equations with critical exponent in \(R^N\), Appl. Math. Lett., 48 (2015), 85-90
##[15]
H. Liu, H. Chen , Ground-state solution for a class of biharmonic equations including critical exponent, Z. Angew. Math. Phys., 66 (2015), 3333-3343
##[16]
D. Lü , Multiple solutions for a class of biharmonic elliptic systems with Sobolev critical exponent,, Nonlinear Anal., 74 (2011), 6371-6382
##[17]
P. J. McKenna, W. Walter , Traveling waves in a suspension bridge , SIAM J. Appl. Math., 50 (1990), 703-715
##[18]
M. Willem , Minimax Theorems, Birkhäuser, Berlin (1996)
##[19]
M. T. O. Pimenta, S. H. M. Soares , Existence and concentration of solutions for a class of biharmonic equations, J. Math. Anal. Appl., 390 (2012), 274-289
##[20]
L.-Y. Shao, H. Chen, Multiple solutions for Schrödinger-Poisson systems with sign-changing potential and critical nonlinearity , Electron. J. Differential Equations, 2016 (2016), 1-8
##[21]
L.-Y. Shao, H. Chen , Existence and concentration result for a quasilinear Schrödinger equation with critical growth, Z. Angew. Math. Phys., 2017 (2017 ), 1-16
##[22]
H. Shi, H. Chen , Existence and multiplicity of solutions for a class of generalized quasilinear Schrödinger equations, J. Math. Anal. Appl., 452 (2017), 578-594
##[23]
J. Sun, J. Chu, T.-F. Wu , Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian , J. Differential Equations, 262 (2017), 945-977
##[24]
A. Szulkin, T. Weth , Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822
##[25]
N. S. Trudinger , On Harnack type inequalities and their application to quasilinear elliptic equations , Comm. Pure Appl. Math., 20 (1967), 721-747
##[26]
F. Wang, M. Avci, Y. An, Existence of solutions for fourth order elliptic equations of Kirchhoff type, J. Math. Anal. Appl., 409 (2014), 140-146
##[27]
Y.-J. Wang, Y.-T. Shen , Multiple and sign-changing solutions for a class of semilinear biharmonic equation, J. Differential Equations, 246 (2009), 3109-3125
##[28]
H. Xiong, Y.-T. Shen , Nonlinear biharmonic equations with critical potential , Acta Math. Sci., 21 (2005), 1285-1294
##[29]
J. Zhang, S. Li, Multiple nontrivial solutions for some fourth-order semilinear elliptic problems , Nonlinear Anal., 60 (2005), 221-230
##[30]
W. Zhang, X. Tang, J. Zhang , Infinitely many solutions for fourth-order elliptic equations with general potentials , J. Math. Anal. Appl., 407 (2013), 359-368
##[31]
J. Zhang, X. Tang, W. Zhang , Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential , J. Math. Anal. Appl., 420 (2014), 1762-1775
##[32]
W. Zhang, X. Tang, J. Zhang , Infinitely many solutions for fourth-order elliptic equations with sign-changing potential , Taiwanese J. Math., 18 (2014), 645-659
##[33]
W. Zhang, X. Tang, J. Zhang , Existence and concentration of solutions for sublinear fourth-order elliptic equations, Electron. J. Differential Equations, 2015 (2015 ), 1-9
##[34]
W. Zou, M. Schechter , Critical Point Theory and its Applications, Springer, New York (2006)
]
Some additive mappings on Banach \({\ast}\)-algebras with derivations
Some additive mappings on Banach \({\ast}\)-algebras with derivations
en
en
We take into account some additive mappings in Banach \(\ast\)-algebras with derivations.
We will first study the conditions for additive mappings with derivations on Banach \(\ast\)-algebras.
Then we prove some theorems involving linear mappings on Banach $\ast$-algebras with derivations.
So derivations on \(C^{\ast}\)-algebra are characterized.
335
341
Jae-Hyeong
Bae
Humanitas College
Kyung Hee University
Republic of Korea
jhbae@khu.ac.kr
Ick-Soon
Chang
Department of Mathematics
Chungnam National University
Republic of Korea
ischang@cnu.ac.kr
Banach \(\ast\)-algebra
\(C^{\ast}\)-algebra
additive mapping with involution
derivation
Article.2.pdf
[
[1]
R. P. Agarwal, R. Saadati, A. Salamati , Approximation of the multiplicatives on random multi-normed space , J. Inequal. Appl., 2017 (2017 ), 1-10
##[2]
T. Aoki, On the stability of the linear transformation in Banach spaces , J. Math. Soc. Japan, 2 (1950), 64-66
##[3]
Z. Baderi, R. Saadati, Generalized stability of Euler-Lagrange quadratic functional equation in random normed spaces under arbitrary t-norms , Thai J. Math., 14 (2016), 585-590
##[4]
R. Badora , On approximate ring homomorphisms , J. Math. Anal. Appl., 276 (2002), 589-597
##[5]
R. Badora , On approximate derivations, Math. Inequal. Appl., 9 (2006), 167-173
##[6]
F. F. Bonsall, J. Duncan , Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer- Verlag, New York-Heidelberg (1973)
##[7]
D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings , Duke Math. J., 16 (1949), 385-397
##[8]
M. Brešar, J. Vukman , On some additive mappings in rings with involution, Aequationes Math., 38 (1989), 178-185
##[9]
M. Brešar, J. Vukman , On left derivations and related mappings , Proc. Amer. Math. Soc., 110 (1990), 7-16
##[10]
Y. J. Cho, C.-K. Park, T. M. Rassias, R. Saadati, Stability of functional equations in Banach algebras, Springer, Cham (2015)
##[11]
M. N. Daif, M. S. Tammam El-Sayiad , On generalized derivations of semiprime rings with involution, Int. J. Algebra, 1 (2007), 551-555
##[12]
P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436
##[13]
M. E. Gordji , Nearly involutions on Banach algebras, A fixed point approach, Fixed Point Theory, 14 (2013), 117-123
##[14]
D. H. Hyers, On the stability of the linear functional equation , Proc. Nat. Acad. Sci. U. S. A., 27 (1911), 222-224
##[15]
B. E. Johnson, A. M. Sinclair , Continuity of derivations and a problem of Kaplansky , Amer. J. Math., 90 (1968), 1067-1073
##[16]
P. Kannappan, Functional equations and inequalities with applications, Springer Monographs in Mathematics, Springer, New York (2009)
##[17]
G. V. Milovanović(ed.), T. M. Rassias (ed.), Analytic number theory, approximation theory, and special functions, In honor of Hari M. Srivastava, Springer, New York (2014)
##[18]
C.-K. Park, G. A. Anastassiou, R. Saadati, S.-S. Yun , Functional inequalities in fuzzy normed spaces, J. Comput. Anal. Appl., 22 (2017), 601-612
##[19]
T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300
##[20]
P. Šemrl, The functional equation of multiplicative derivation is superstable on standard operator algebras, Integral Equations Operator Theory, 18 (1994), 118-122
##[21]
I. M. Singer, J. Wermer, Derivations on commutative normed algebras, Math. Ann., 129 (1955), 260-264
##[22]
M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math., 128 (1988), 435-460
##[23]
S. M. Ulam , A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, New York-London (1960)
]
A system of evolutionary problems driven by a system of hemivariational inequalities
A system of evolutionary problems driven by a system of hemivariational inequalities
en
en
In this paper, we introduce the differential system obtained by mixing a system of evolution equations and a system of hemivariational inequalities
((SEESHVI), for short). We prove
the superpositional measurability and upper semicontinuity for the solution set of a general system of hemivariational inequalities, and establish the
non-emptiness and compactness of the solution set of (SEESHVI).
342
357
Lu-Chuan
Ceng
Department of Mathematics
Shanghai Normal University
China
zenglc@hotmail.com
Ching-Feng
Wen
Center for Fundamental Science; and Research Center for Nonlinear Analysis and Optimization
Department of Medical Research
Kaohsiung Medical University
Kaohsiung Medical University Hospital
Taiwan
Taiwan
cfwen@kmu.edu.tw
Jen-Chih
Yao
Center for General Education
China Medical University
Taiwan
yaojc@mail.cmu.edu.tw
Yonghong
Yao
Department of Mathematics
Tianjin Polytechnic University
China
yaoyonghong@aliyun.com
Evolution equation
hemivariational inequality
\(\nu\)-condensing mapping
generalized Clarke subdifferential
Article.3.pdf
[
[1]
L.-C. Ceng, H. Gupta, C.-F. Wen, Well-posedness by perturbations of variational-hemivariational inequalities with perturbations, Filomat, 26 (2012), 881-895
##[2]
L.-C. Ceng, N.-C. Wong, J.-C. Yao , Well-posedness for a class of strongly mixed variational-hemivariational inequalities with perturbations , J. Appl. Math., 2012 (2012 ), 1-21
##[3]
X. Chen, Z. Wang, Convergence of regularized time-stepping methods for differential variational inequalities, SIAM J. Optim., 23 (2013), 1647-1671
##[4]
N. Costea, V. Rădulescu , Hartman-Stampacchia results for stably pseudomonotone operators and nonlinear hemivariational inequalities , Appl. Anal., 89 (2010), 175-188
##[5]
N. Costea, V. Rădulescu, Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term, J. Global Optim., 52 (2012), 743-756
##[6]
K. Fan , Some properties of convex sets related to fixed point theorems , Math. Ann., 266 (1984), 519-537
##[7]
A. F. Filippov, On some problems of the theory of optimal control, Vestn. Moscow Univ., 2 (1958), 25-32
##[8]
F. Giannessi, A. Khan, Regularization of non-coercive quasivariational inequalities, Control Cybern., 29 (2000), 91-110
##[9]
J. Gwinner, On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral control and given friction, Appl. Numer. Math., 59 (2009), 2774-2784
##[10]
J. Gwinner , On a new class of differential variational inequalities and a stability result , Math. Program., 139 (2013), 205-221
##[11]
L. Han, J.-S. Pang , Non-Zenoness of a class of differential quasi-variational inequalities,, Math. Program., 121 (2010), 171-199
##[12]
M. Kamemskii, V. Obukhovskii, P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Space , Walter de Gruyter, Berlin (2001)
##[13]
A. A. Khan, D. Motreanu, Existence theorems for elliptic and evolutionary variational and quasi-variational inequalities, J. Optim. Theory Appl., 167 (2015), 1136-1161
##[14]
X.-S. Li, N.-J. Huang, D. O’Regan , Differential mixed variational inequalities in finite dimensional spaces , Nonlinear Anal., 72 (2010), 3875-3886
##[15]
Z. Liu , Browder-Tikhonov regularization of non-coercive evolution hemivariational inequalities, Inverse Problems, 21 (2005), 13-20
##[16]
Z. Liu, X. Li, D. Motreanu , Approximate controllability for nonlinear evolution hemivariational inequalities in Hilbert spaces , SIAM J. Control Optim., 53 (2015), 3228-3244
##[17]
Z. Liu, N. V. Loi, V. Obukhovskii , Existence and global bifurcation of periodic solutions to a class of differential variational inequalities, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2013 (2013 ), 1-10
##[18]
Z. Liu, B. Zeng , Existence results for a class of hemivariational inequalities involving the stable (\(g, f, \alpha\))-quasimonotonicity , Topol. Methods Nonlinear Anal., 47 (2016), 195-217
##[19]
Z. Liu, S. Zeng, D. Motreanu , Evolutionary problems driven by variational inequalities , J. Differential Equations, 260 (2016), 6787-6799
##[20]
S. Migórski, A. Ochal, M. Sofonea , Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problems, Springer, New York (2012)
##[21]
D. Motreanu, P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities and Applications , Kluwer Academic, Dordrecht (1999)
##[22]
Z. Naniewicz, P. D. Panagiotopoulos , Mathematical Theory of Hemivariational Inequalities and Applications, CRC press, New York (1994)
##[23]
P. D. Panagiotopoulos , Nonconvex superpotentials in sense of F.H. Clarke and applications , Mech. Res. Comm., 8 (1981), 335-340
##[24]
P. D. Panagiotopoulos, M. Fundo, V. Rădulescu , Existence theorems of Hartman-Stampacchia type for hemivariational inequalities and applications, J. Global Optim., 15 (1999), 41-54
##[25]
J.-S. Pang, D. E. Stewart , Differential variational inequalities , Math. Program., 113 (2008), 345-424
##[26]
D. Repovš, C. Varga , A Nash type solution for hemivariational inequality systems, Nonlinear Anal., 74 (2011), 5585-5590
##[27]
X. Wang, N.-J. Huang , A class of differential vector variational inequalities in finite dimensional spaces , J. Optim. Theory Appl., 162 (2014), 633-648
##[28]
Y.-M. Wang, Y.-B. Xiao, X. Wang, Y. J. Cho , Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems, J. Nonlinear Sci. Appl., 9 (2016), 1178-1192
##[29]
Y.-B. Xiao, N.-J. Huang, Browder-Tikhonov regularization for a class of evolution second order hemivariational inequalities, J. Global Optim., 45 (2009), 371-388
##[30]
Y.-B. Xiao, N.-J. Huang , Well-posedness for a class of variational-hemivariational inequalities with perturbations, J. Optim. Theory Appl., 151 (2011), 33-51
##[31]
Y.-B. Xiao, N.-J. Huang, J. Lu, A system of time-dependent hemivariational inequalities with Volterra integral terms, J. Optim. Theory Appl., 165 (2015), 837-853
##[32]
Y.-H. Yao, N. Shahzad , An algorithmic approach to the split variational inequality and fixed point problem, J. Nonlinear Convex Anal., 18 (2017), 977-991
##[33]
H. Zegeye, N. Shahzad, Y.-H. Yao, Minimum-norm solution of variational inequality and fixed point problem in Banach spaces, Optimization, 64 (2015), 453-471
##[34]
E. Zeidler , Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York (1990)
##[35]
Y. Zhang, Y. He , On stably quasimonotone hemivariational inequalities, Nonlinear Anal., 74 (2011), 3324-3332
]
Weakly \(\mathbf{(s,r)}\)-contractive multi-valued operators on \(\mathbf{b}\)-metric space
Weakly \(\mathbf{(s,r)}\)-contractive multi-valued operators on \(\mathbf{b}\)-metric space
en
en
In this paper we introduce the notion of weakly \((s,r)\)-contractive multi-valued operator on \(b\)-metric space and establish some fixed point theorems for this operator. In addition, an application to the differential equation is given to illustrate usability of obtained results.
358
367
Lingjuan
Ye
School of Mathematics and Statistics
Beijing Institute of Technology
China
LingjuanYE@126.com
Congcong
Shen
School of Mathematics and Statistics
Beijing Institute of Technology
China
3120140519@bit.edu.cn
\(b\)-metric space
weakly \((s
r)\)-contractive multi-valued operator
fixed point theorem
Article.4.pdf
[
[1]
S. Banach, Sur les operations dans les ensembles abstraits et leures applications aux equations integrales, Fundam. Math., 3 (1922), 133-181
##[2]
M. Boriceanu, M. Bota, A. Petruşel , Multivalued fractals in b-metric spaces, Cent. Eur. J. Math., 8 (2010), 367-377
##[3]
S. Czerwik , Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5-11
##[4]
S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 263-276
##[5]
P. N. Dutta, B. S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl., 2008 (2008), 1-8
##[6]
N. Hussain, D. Đorić, Z. Kadelburg, S. Radenović, Suzuki-type fixed point results in metric type spaces, Fixed Point Theory Appl., 2012 (2012 ), 1-12
##[7]
N. Hussain, V. Parvaneh, J. R. Roshan, Z. Kadelburg , Fixed points of cyclic weakly (\(\psi,\phi, L,A, B\))-contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl., 2013 (2013 ), 1-18
##[8]
N. Hussain, N. Yasmin, N. Shafqat , Multi-valued Ćirić contractions on metric spaces with applications, Filomat, 28 (2014), 1953-1964
##[9]
H. Işık, D. Türkoğlu , Fixed point theorems for weakly contractive mappings in partially ordered metric-like spaces , Fixed Point Theory Appl., 2013 (2013 ), 1-12
##[10]
H. Işık, D. Türkoğlu, Coupled fixed point theorems for new contractive mixed monotone mappings and applications to integral equations, Filomat, 28 (2014), 1253-1264
##[11]
H. Işık, D. Türkoğlu , Generalized weakly alfa-contractive mappings and applications to ordinary differential equations , Miskolc Math. Notes, 17 (2016), 365-379
##[12]
T. Kamran, S. Hussain , Weakly (s, r)-contractive multi-valued operators, Rend. Circ. Mat. Palermo., 64 (2015), 475-482
##[13]
J. T. Markin, Continuous dependence of fixed point sets , Proc. Amer. Math. Soc., 38 (1973), 545-547
##[14]
R. Miculescu, A. Mihail, New fixed point theorems for set-valued contractions in b-metric spaces, J. Fixed Point Theory Appl., 19 (2017), 2153-2163
##[15]
S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math., 30 (1969), 475-488
##[16]
J. J. Nieto, R. Rodríguez-López , Contractive mapping theorems in partially ordered sets and applications to ordinary differential equation, Order, 22 (2005), 223-239
##[17]
O. Popescu , A new type of contractive multivalued operators , Bull. Sci. Math., 137 (2013), 30-44
##[18]
S. Reich , Fixed points of contractive functions, Boll. Un. Mat. Ital., 5 (1972), 26-42
##[19]
S. Reich, A. J. Zaslavski , Genericity in Nonlinear Analysis, Springer, New York (2014)
##[20]
B. H. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal., 47 (2001), 2683-2693
##[21]
J. R. Roshan, V. Parvaneh, Z. Kadelburg, Common fixed point theorems for weakly isotone increasing mappings in ordered b-metric spaces, J. Nonlinear Sci. Appl., 7 (2014), 229-245
##[22]
J. R. Roshan, V. Parvaneh, Z. Kadelburg, N. Hussain , New fixed point results in b-rectangular metric spaces , Nonlinear Anal. Model. Control, 21 (2016), 614-634
##[23]
I. A. Rus , Basic problems of the metric fixed point theory revisited (II) , Studia Univ. Babeş-Bolyai Math., 36 (1991), 81-89
##[24]
S. L. Singh, S. Czerwik, K. Król, A. Singh, Coincidences and fixed points of hybrid contractions, Tamsui Oxf. J. Math. Sci., 24 (2008), 401-416
##[25]
L. Wang, The fixed point method for intuitionistic fuzzy stability of a quadratic functional equation , Fixed Point Theory Appl., 2010 (2010 ), 1-7
]
Existence of solutions for a class of second-order impulsive Hamiltonian system with indefinite linear part
Existence of solutions for a class of second-order impulsive Hamiltonian system with indefinite linear part
en
en
We consider a class of second-order impulsive Hamiltonian system with indefinite linear part. By using saddle point theorem in critical point theory, an existence result is obtained, which extends and improves some existing results.
368
374
Qiongfen
Zhang
College of Science
Guilin University of Technology
P. R. China
qfzhangcsu@163.com
Impulsive Hamiltonian system
saddle point theorem
solutions
existence
Article.5.pdf
[
[1]
R. P. Agarwal, T. G. Bhaskar, K. Perea, Some results for impulsive problems via Morse theory , J. Math. Anal. Appl., 409 (2014), 752-759
##[2]
D. Baĭnov, P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific Technical, Harlow (1993)
##[3]
P. Chen, X. H. Tang , Existence of solutions for a class of p-Laplacian systems with impulsive effects , Taiwanese J. Math., 16 (2012), 803-828
##[4]
A. A. Chikrii, I. I. Matychyn, K. A. Chikrii, Differential games with impulse control , Birkhuser Boston, Boston (2007)
##[5]
E. Crück, M. Quincampoix, P. Saint-Pierre , Pursuit-evasion games with impulsive dynamics, Birkhuser Boston, Boston (2007)
##[6]
W.-Z. Gong, Q.-F. Zhang, X. H. Tang , Existence of subharmonic solutions for a class of second-order p-Laplacian systems with impulsive effects , J. Appl. Math., 2012 (2012 ), 1-18
##[7]
S. T. Kyritsi, N. S. Papageorgiou, On superquadratic periodic systems with indefinite linear part , Nonlinear Anal., 72 (2010), 946-954
##[8]
V. Lakshmikantham, D. D. Baĭnov, P. S. Simeonov , Theory of Impulsive Differential Equations, World Scientific Press, Singapore (1989)
##[9]
J. J. Nieto, D. O’Regan , Variational approach to impulsive differential equations , Nonlinear Anal. Real World Appl., 10 (2009), 680-690
##[10]
P. H. Rabinowitz , Minimax methods in critical point theory with applications to differential equations , Mathematical Society, Providence (1986)
##[11]
L. Stone, B. Shulgin, Z. Agur, Theoretical examination of the pulse vaccination policy in the SIR epidemic model, Math. Comput. Modelling, 31 (2000), 207-215
##[12]
J. Sun, H. Chen, L. Yang, Variational methods to fourth-order impulsive differential equations , J. Appl. Math. Comput., 35 (2011), 323-340
##[13]
Q. Wang, M. Wang , Existencce of solution for impulsive differential equations with indefinite linear part , Appl. Math. Lett., 51 (2016), 41-47
##[14]
Q.-F. Zhang , Existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems with impulsive effects, Abstr. Appl. Anal., 2014 (2014 ), 1-12
##[15]
X. Zhang, X. Tang, A note on the minimal periodic solutions of nonconvex super-linear Hamiltonian system, Appl. Math. Comput., 219 (2013), 7586-7590
##[16]
X. Zhang, X. Tang, Non-constant periodic solutions for second order Hamiltonian system involving the p-Laplacian, Adv. Nonlinear Stud., 13 (2013), 945-964
##[17]
X. Zhang, X. Tang , Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems, Commun. Pure Appl. Anal., 13 (2014), 75-95
]
Bifurcation and periodically semicycles for fractional difference equation of fifth order
Bifurcation and periodically semicycles for fractional difference equation of fifth order
en
en
Our paper takes into account a new bifurcation case of the cycle length and a fifth-order difference equation dynamics of
\[
y_{m+1}=\frac{y_{m} y_{m-2}^\alpha y_{m-4}^\beta+y_{m} +y_{m-2}^\alpha +y_{m-4}^\beta + \gamma }{y_{m}y_{m-2}^\alpha + y_{m-2}^\alpha y_{m-4}^\beta+y_{m} y_{m-4}^\beta+ \gamma +1} , \quad
m=0,1,2,3, \ldots,
\]
where \(\gamma \in [0, \infty )\) , \(\alpha,\beta\in \mathbb{Z^+} \), and \(y_{-4},y_{-3},y_{-1},y_{-2},y_0 \in (0, \; \infty )\) is took into consideration. The disturbance of initials lead to a distinction of cycle length principle of the non-trivial solutions of the equation. The principle of the track solutions structure for this equation is
given. The consecutive periods of negative and positive semicycles of non-trivial solutions of this equation take place periodically with only prime period fifteen and in a period with the principles represented by either \(\{3^+,1^-, 2^+, 2^-, 1^+,1^-,1^+, 4^-\}\) or \(\{3^-,1^+, 2^-, 2^+, 1^-,1^+,1^-, 4^+\}\). From this rubric we will establish that the positive fixed point has global asymptotic stability.
375
382
Tarek F.
Ibrahim
Mathematics Department, College of Sciences and Arts for Girls in sarat Abida
Mathematics Department, Faculty of Science
King Khalid University
Mansoura University
Saudi Arabia
Egypt
tfibrahem@mans.edu.eg
Semicycles
solutions
difference equations
oscillatory solution
global stability
Article.6.pdf
[
[1]
A. M. Ahmed , On the dynamics of a higher-order rational difference equation, Discrete Dyn. Nat. Soc., 2011 (2011), 1-8
##[2]
Q. Din , Dynamics of a discrete Lotka-Volterra model , Adv. Difference Equ., 2013 (2013), 1-13
##[3]
E. M. Elabbasy, E. M. Elsayed, On the global attractivity of difference equation of higher order, Carpathian J. Math., 24 (2008), 45-53
##[4]
H. El-Metwally, E. A. Grove, G. Ladas, H. D. Voulov , On the global attractivity and the periodic character of some difference equations, J. Differ. Equations Appl., 7 (2001), 837-850
##[5]
E. M. Elsayed, T. F. Ibrahim , Periodicity and Solutions for some systems of nonlinear rational difference equations, Hacet. J. Math. Stat., 44 (2015), 1361-1390
##[6]
E. M. Elsayed, T. F. Ibrahim , Solutions and Periodicity of a Rational Recursive Sequences of order Five, Bull. Malays. Math. Sci. Soc., 38 (2015), 95-112
##[7]
T. F. Ibrahim , Periodicity and Global Attractivity of Difference Equation of Higher Order, J. Comput. Anal. Appl., 16 (2014), 552-564
##[8]
T. F. Ibrahim, M. A. El-Moneam, Global stability of a higher-order difference equation, Iran. J. Sci. Technol. Trans. A Sci., 41 (2017), 51-58
##[9]
T. F. Ibrahim, N. Touafek , Max-Type System Of Difference Equations With Positive Two-Periodic Sequences, Math. Methods Appl. Sci., 37 (2014), 2541-2553
##[10]
V. L. Kocić, G. Ladas, Global behavior of Nonlinear Difference Equations of Higher Order with Applications, Academic Publishers Group, Dordrecht (1993)
##[11]
G. Ladas, Open problems and conjectures, J. Difference Equa. Appl., 10 (2004), 1119-1127
##[12]
X. Li , Global behavior for a fourth-order rational difference equation, J. Math. Anal. Appl., 312 (2005), 555-563
##[13]
X. Li , The rule of trajectory structure and global asymptotic stability for a nonlinear difference equation, Appl. Math. Lett., 19 (2006), 1152-1158
##[14]
N. Touafek, On a second order rational difference equation , Hacet. J. Math. Stat., 41 (2012), 867-874
##[15]
I. Yalinkaya, C. Cinar, On the positive solutions of the difference equation system \(x_{n+1} = \frac{1}{ z_n} , y_{n+1} = \frac{y_n}{ x_n+y_{n-1}} , z_{n+1} = \frac{1}{ x_{n-1}}\) , J. Inst. Math. Comp. Sci., 18 (2005), 135-136
##[16]
E. M. E. Zayed, M. A. El-Moneam, On the global attractivity of two nonlinear difference equations, J. Math. Sci., 177 (2011), 487-499
]
Some identities of degenerate Fubini polynomials arising from differential equations
Some identities of degenerate Fubini polynomials arising from differential equations
en
en
Recently, Kim et al. have studied degenerate Fubini polynomials
in [T. Kim, D. V. Dolgy, D. S. Kim, J. J. Seo, J. Nonlinear Sci. Appl., \({\bf 9}\) (2016), 2857--2864]. Jang and Kim presented some identities of Fubini polynomials
arising from differential equations in [G.-W. Jang, T. Kim, Adv. Studies Contem. Math.,
\({\bf 28}\) (2018), to appear]. In this paper,
we drive differential equations from the generating function of the
degenerate Fubini polynomials. In addition, we obtain some
identities from those differential equations.
383
393
Sung-Soo
Pyo
Department of Mathematics Education
Silla University, Busan, Republic of Korea
Republic of Korea
ssoopyo@silla.ac.kr
Differential equations
Fubini polynomials
degenerate Fubini polynomials
Article.7.pdf
[
[1]
L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math., 15 (1979), 51-88
##[2]
J.-M. De Koninck, Those Fascinating Numbers, American Mathematical Society, Providence (2009)
##[3]
G.-W. Jang, T. Kim, Some identities of Fubini polynomials arising from differential equations, Adv. Studies Contem. Math., 28 (2018), -
##[4]
T. Kim, Identities involving Frobenius-Euler polynomials arising from non-linear differential equations, J. Number Theory, 132 (2012), 2854-2865
##[5]
T. Kim, λ-analogue of Stirling numbers of the first kind, Adv. Stud. Contemp. Math., Kyungshang, 27 (2017), 423-429
##[6]
T. Kim, D. V. Dolgy, D. S. Kim, J. J. Seo, Differential equations for Changhee polynomials and their applications, J. Nonlinear Sci. Appl., 9 (2016), 2857-2864
##[7]
D. S. Kim, T. Kim, Some identities for Bernoulli numbers of the second kind arising from a nonlinear differential equation, Bull. Korean Math. Soc., 52 (2015), 2001-2010
##[8]
T. Kim, D. S. Kim, A note on nonlinear Changhee differential equations, Russ. J. Math. Phys., 23 (2016), 88-92
##[9]
T. Kim, D. S. Kim, G.-W. Jang, A note on degenerate Fubini polynomials, Proc. Jangjeon. Math. Soc., 20 (2017), 521-531
##[10]
T. Kim, D. S. Kim, L. C. Jang, H. I. Kwon, Differential equations associated with Mittag-Leffer polynomials, Glob. J. Pure Appl. Math., 12 (2016), 2839-2847
##[11]
S. Kim, B. M. Kim, J. Kwon, Differential equations associated with Genocchi polynomials, Glob. J. Pure Appl. Math., 12 (2016), 4579-4585
##[12]
T. Kim, D. S. Kim, J. J. Seo, Differential equations associated with degenerate Bell polynomials, Inter. J. Pure Appl. Math., 108 (2016), 551-559
##[13]
T. Kim, J. J. Seo, Revisit nonlinear differential equations arising from the generating functions of degenerate Bernoulli numbers, Adv. Stud. Contemp. Math., 2016 (26), 401-406
##[14]
H. I. Kwon, T. Kim, J. J. Seo, A note on Daehee numbers arising from differential equations, Glob. J. Pure Appl., 12 (2016), 2349-2354
##[15]
D. Lim, Some identities of degenerate Genocchi polynomials, Bull. Korean Math. Soc., 53 (2016), 569-579
##[16]
M. Muresan, G. Toader, A generalization of Fubini’s number, Studia Univ. Babeş-Bolyai Math. , 31 (1986), 60-65
##[17]
N. Pippenger, The hypercube of resistors, asymptotic expansions, and preferential arrangements, Math. Mag., 83 (2010), 331-346
##[18]
S.-S. Pyo, T. Kim, S.-H. Rim, Identities of the degenerate Daehee numbers with the Bernoulli numbers of the second kind arising from nonlinear Differential equation, J. Nonlinear Sci. Appl., 10 (2017), 6219-6228
##[19]
S.-S. Pyo, Degenerate Cauchy numbers and polynomials of the fourth kind, Adv. Studies. Contemp. Math., 2018 (28), -
##[20]
S.-S. Pyo, T. Kim, S.-H. Rim, Degenerate Cauchy numbers of the third kind, preprint, (2018), -
##[21]
D. J. Velleman, G. S. Call, Permutations and combination locks, Math. Mag., 68 (1995), 243-253
]
Simultaneous iteration for variational inequalities over common solutions for finite families of nonlinear problems
Simultaneous iteration for variational inequalities over common solutions for finite families of nonlinear problems
en
en
In this paper, we apply Theorem 3.2 of [G. M. Lee, L.-J. Lin, J. Nonlinear Convex Anal., \({\bf 18}\) (2017), 1781--1800] to study
the variational inequality over split equality fixed point problems
for three finite families of strongly quasi-nonexpansive mappings.
Then we use this result to study variational inequalities over split
equality for three various finite families of nonlinear mappings. We
give a unified method to study split equality for three various
finite families of nonlinear problems. Our results contain many
results on split equality fixed point problems and multiple sets
split feasibility problems as special cases. Our results can treat
large scale of nonlinear problems by group these problems into
finite families of nonlinear problems, then we use simultaneous
iteration to find the solutions of these problems. Our results will
give a simple and quick method to study large scale of nonlinear
problems and will have many applications to study large scale of
nonlinear problems.
394
416
Lai-Jiu
Lin
Department of Mathematics
National Changhua University of Education
Taiwan
maljlin@cc.ncue.edu.tw
Split equality fixed point problem
split fixed point problem
quasi-pseudocontractive mapping
demicontractive mapping
pseudo-contractive mapping
Article.8.pdf
[
[1]
H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, Springer, , New York (2011)
##[2]
E. Blum, W. Oettli, From optimization and variational inequalities, Math. Student,, 63 (1994), 123-146
##[3]
F. E. Browder, Fixed point theorems for noncompact mappings in Hilbert spaces, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1272-1276
##[4]
G. Cai, Y. Shehu, An iteration for fixed point problem and convex minimization problems with applications, Fixed Point Theory Appl., 2015 (2015), 1-17
##[5]
A. Cegielski, General methods for solving the split common fixed point problem, J. Optim. Theory Appl., 165 (2015), 385-404
##[6]
Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projection in a product space, Numer. Algorithms,, 8 (1994), 221-239
##[7]
Y. Censor, A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587-600
##[8]
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 problem, Fixed Point Theory Appl., 2014 (2014), 1-17
##[9]
S.-S. Chang, L.Wang, L.-J. Qin, Split equality fixed point problem for quasi-pseudo-contractive mappings with applications, Fixed Point Theory Appl., 2015 (2015), 1-12
##[10]
H. Che, M. Li, A simultaneous iteration methods for split equality problems of two finite families of strictly pseudononspreading mappings without prior knowledge of operator norms, Fixed Point Theory Appl., 2015 (2015), 1-14
##[11]
C.-S. Chuang, L.-J. Lin, Z.-T. Yu, Mathematical programming over the solution set of the minimization problem for the sum of two convex functions, J. Nonlinear Convex Anal., 17 (2016), 2105-2118
##[12]
P. L. Combettes, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136
##[13]
M. Eslamian, P. Eslamian, Strong convergence of split common fixed point problem, Numer. Funct. Anal. Optim., 37 (2016), 1248-1266
##[14]
14] G. M. Lee, L.-J. Lin, Variational inequalities over split equality fixed point sets of strongly quasi-nonexpansive mappings, J. Nonlinear Convex Anal., 18 (2017), 1781-1800
##[15]
P.-E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912
##[16]
G. Marino, H.-K. Xu, Weak and strong convergence theorems for strict pseudo-contraction in Hilbert spaces, J. Math. Anal. Appl., 329 (2007), 336-346
##[17]
A. Moudafi, A note on the split common fixed point problem for quasi-nonexpansive operators, Nonlinear Anal., 74 (2011), 4083-4087
##[18]
A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Anal., 79 (2013), 117-121
##[19]
A. Moudafi, E. AI-Shemas, Simultaneous iterative methodsfor split equality problems, Trans. Math. Program Appl., 2013 (2013), 1-10
##[20]
M. O. Osilike, F. O. Isiogugu, Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces, Nonlinear Anal., 74 (2011), 1814-1822
##[21]
W. Takahashi, H.-K. Xu, J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. Anal., 23 (2015), 205-221
##[22]
Y. Wang, X. Fang, Viscosity approximation for the multiple -set split equality fixed point problem of demicontractive mappings, J. Nonlinear Sci. Appl., 10 (2017), 4254-4268
##[23]
Y.Wang, T. H. Kim, Simultaneous iterative algorithm for the split equality fixed point problem of demicontractive mappings, J. Nonlinear Sci. Appl., 10 (2017), 154-165
##[24]
Y. Wang, T.-H. Kim, X. Fang, H. He, The split common fixed point for demicontractive mappings and quasi-nonexpansive mappings, J. Nonlinear Sci. Appl., 10 (2017), 2976-2985
##[25]
Z.-T. Yu, L.-J. Lin, C.-S. Chuang, Mathematical programing with multiple sets split monotone variational inclusion constraints, Fixed Point Theory Appl., 2014 (2014), 1-27
##[26]
J. Zhao, S. N. He, Simultaneous iterative algorithm for the split common fixed point problem governed by quasinonexpansive mappings, J. Nonlinear and Convex Anal., (accepted), -
##[27]
J. Zhao, S. Wang, Viscosity approximate methods for trhe split equality quasi-nonexpansive operators, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 1474-1486
]
Fixed point belonging to the zero-set of a given function
Fixed point belonging to the zero-set of a given function
en
en
We prove the existence and uniqueness of fixed point belonging to the zero-set of a given function. The results are established in the setting of metric spaces and partial metric spaces. Our approach combines the recent notions of \((F,\varphi)\)-contraction and \(\mathcal{Z}\)-contraction. The main result allows to deduce, as a particular case, some of the most known results in the literature. An example supports the theory.
417
424
Francesca
Vetro
Nonlinear Analysis Research Group
Faculty of Mathematics and Statistics
Ton Duc Thang University
Ton Duc Thang University
Vietnam
Vietnam
francescavetro@tdt.edu.vn
Fixed point
metric space
partial metric space
nonlinear contraction
simulation function
Article.9.pdf
[
[1]
S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133-181
##[2]
D. W. Boyd, J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458-464
##[3]
M. Jleli, B. Samet, C. Vetro, Fixed point theory in partial metric spaces via \(\varphi\)-fixed point’s concept in metric spaces, J. Inequal. Appl., 2014 (2014), 1-9
##[4]
F. Khojasteh, S. Shukla, S. Radenović, A new approach to the study of fixed point theorems via simulation functions, Filomat, 29 (2015), 1189-1194
##[5]
S. G. Matthews, Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Applications, Ann. New York Acad. Sci., 728 (1994), 183-197
##[6]
S. J. O’Neill, Partial metrics, valuations and domain theory, in: Proc. 11th Summer Conference on General Topology and Applications, Ann. New York Acad. Sci. , 806 (1996), 304-315
##[7]
D. Paesano, P. Vetro, Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces, Topology Appl., 159 (2012), 911-920
##[8]
S. Reich, Fixed points of contractive functions, Boll. Un. Mat. Ital., 5 (1972), 26-42
##[9]
B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal., 47 (2001), 2683-2693
##[10]
B. Samet, C. Vetro, F. Vetro, From metric spaces to partial metric spaces, Fixed Point Theory Appl., 2013 (2013 ), 1-11
##[11]
C. Vetro, F. Vetro, Metric or partial metric spaces endowed with a finite number of graphs: a tool to obtain fixed point results, Topology Appl., 164 (2014), 125-137
]
Erratum to "On some fixed points of \(\alpha-\psi\) contractive mappings with rational expressions, J. Nonlinear Sci. Appl., 10 (2017), 1569 - 1581"
Erratum to "On some fixed points of \(\alpha-\psi\) contractive mappings with rational expressions, J. Nonlinear Sci. Appl., 10 (2017), 1569 - 1581"
en
en
The aim of this note is to correct the affiliation of the authors in [E. Karapinar, A. Dehici, N. Redjel, J. Nonlinear Sci. Appl., \({\bf10}\) (2017), 1569--1581]. We shall also extend the main result of this paper further.
425
427
Erdal
Karapinar
Department of Mathematics
Atilim University 06836
Turkey
erdal.karapinar@atilim.edu.tr
Complete metric space
\((c)\)-comparison function
fixed point
\(\alpha\)-admissible mapping
cyclic mapping
Article.10.pdf
[
[1]
H. H. Alsulami, E. Karapınar, F. Khojasteh, A.-F. Roldán-López-de-Hierro , A proposal to the study of contractions in quasi-metric spaces, Discrete Dyn. Nat. Soc., 2014 (2014), 1-10
##[2]
E. Karapınar, A. Dehici, N. Redjel, On some fixed points of \(\alpha-\psi\) contractive mappings with rational expressions, J. Nonlinear Sci. Appl., 10 (2017), 1569-1581
##[3]
F. Khojasteh, S. Shukla, S. Radenović, A new approach to the study of fixed point theorems via simulation functions , Filomat, 29 (2015), 1189-1194
##[4]
A.-F. Roldán-López-de-Hierro, E. Karapınar, C. Roldán-López-de-Hierro, J. Martínez-Moreno , Coincidence point theorems on metric spaces via simulation functions, J. Comput. Appl. Math., 275 (2015), 345-355
]
The uniqueness of solution for initial value problems for fractional differential equation involving the Caputo-Fabrizio derivative
The uniqueness of solution for initial value problems for fractional differential equation involving the Caputo-Fabrizio derivative
en
en
In this paper, we study some results about the expression of solutions to some linear differential equations for the Caputo-Fabrizio fractional derivative. Furthermore, by the Banach contraction principle, the unique existence of the solution to an initial value problem for nonlinear differential equation involving the Caputo-Fabrizio fractional derivative is obtained.
428
436
Shuqin
Zhang
School of Science,
China University of Mining and Technology (Beijing)
P. R. China
zsqjk@163.com
Lei
Hu
School of Science
Shandong Jiaotong University
P. R. China
huleimath@163.com
Sujing
Sun
College of Mathematics and System Science
Shandong University of Science and Technology
P. R. China
kdssj@163.com
The Caputo-Fabrizio fractional derivative
initial value problem
fractional differential equations
Banach contraction principle
uniqueness
Article.11.pdf
[
[1]
T. Abdeljawad, D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11-27
##[2]
M. Abdulhameed, D. Vieru, R. Roslan, Modeling electro-magneto-hydrodynamic thermo-fluidic transpot of biofluids with new trend of fractional derivative without singular kernel, Phys. A, 484 (2017), 233-252
##[3]
N. Al-Salti, E. Karimov, K. Sadarangani, On a differential Equation with Caputo-Fabrizio fractional derivative of order 1 < \beta\leq 2 and applications to Mass-Spring-Damper system, Progr. Fract. Differ. Appl., 2 (2016), 257-263
##[4]
A. Atangana, On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation, Appl. Math. Comput., 273 (2016), 948-956
##[5]
S. Aydogan, D. Baleanu, A. Mousalou, S. Rezapour, On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equtions, Adv. Difference Equ., 2017 (2017), 1-11
##[6]
R. L. Bagley, P. J. Torvik, On the fractional calculus model of viscoelastic behavior, J. Rheol., 30 (1986), 133-155
##[7]
Z. Bai, X. Dong, C. Yin, Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions, Bound. Value Probl., 2016 (2016), 1-11
##[8]
Z. Bai, T. Qiu, Existence of positive solution for singular fractional differential equation, Appl. Math. Comput., 215 (2009), 2761-2767
##[9]
Z. Bai, Y. Zhang, The existence of solutions for a fractional multi-point boundary value problem, Comput. Math. Appl., 60 (2010), 2364-2372
##[10]
Z. Bai, S. Zhang, S. Sun, C. Yin, Monotone iterative method for a class of fractional differential equations, Electron. J. Differential Equations, 2016 (2016), 1-8
##[11]
D. Baleanu, A. Mousalou, S. Rezapour, On the existence of solutions for some infinite coefficient-symmetric Caputo- Fabrizio fractional integro-differential equations, Bound. Value Probl., 2017 (2017), 1-9
##[12]
D. Baleanu, A. Mousalou, S. Rezapour, A new method for investigationg approximate solutions of some fractional integrodifferential equations involving the Caputo-Fabrizio derivative, Adv. Difference Equ., 2017 (2017), 1-12
##[13]
A. R. Butt, M. Abdullah, N. Raza, M. A. Imran, Influence of non-integer order parameter and Hartmann number on the heat and mass transfer flow of a Jeffery fluid over an oscillationg vertical plate via Caputo-Fabrizio tiem fractional derivatives, Eur. Phys. J. plus, 2017 (2017), 1-16
##[14]
M. Caputo, M. Fabrizio, A New Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85
##[15]
M. Caputo, M. Fabrizio, Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11
##[16]
Y. Cui, Uniqueness of solution for boundary value problems for fractional differential equations, Appl. Math. Lett., 51 (2016), 48-54
##[17]
J. F. GÓmez-Aguilar, Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel, Phys. A, 465 (2017), 562-572
##[18]
R. Hilfer, Applications of Fractional calculus in Physics, World Scientific, Singapore (2000)
##[19]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo , Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam (2006)
##[20]
C. Kou, H. Zhou, Y. Yan, Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis, Nonlinear Anal., 74 (2011), 5975-5986
##[21]
D. Kumar, J. Singh, M. Al Qurashi, D. Baleanu, Analysis of logistic equation pertaining to a new fractional derivative with non-singular kernel, Adv. Mech. Eng., 2017 (2017), 1-8
##[22]
J. Losada, J. J. Nieto, Properties of a New Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1 (2015), 87-92
##[23]
I. A. Mirza, D. Vieru, Fundamental solutions to advection-diffusion equation with time-fractional Caputo-Fabrizio derivative, Comput. Math. Appl., 73 (2017), 1-10
##[24]
J. Singh, D. Kumar, Z. Hammouch, A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504-515
]
A classification of minimal translation surfaces in Minkowski space
A classification of minimal translation surfaces in Minkowski space
en
en
Minimal surfaces are well known as a class of surfaces with
vanishing mean curvature which minimize area within a given boundary
configuration since 19th century. This fact was implicitly proved by
Lagrange for nonparametric surfaces in 1760, and then by Meusnier in
1776 who used the analytic expression for the mean curvature. Mathematically, a minimal surface corresponds to the solution of a
nonlinear partial differential equation. By solving some
differential equations, in this paper we give a complete and
explicit classification of minimal translation surfaces in an
\(n\)-dimensional Minkowski space.
437
443
Dan
Yang
School of Mathematics
Liaoning University
P. R. China
dlutyangdan@126.com
Wei
Dan
School of Mathematics and Statistic
Faculty of Arts and Sciences
Guangdong University of Finance and Economics
Shenzhen Technology University
P. R. China
P. R. China
wdan@gdufe.edu.cn
Yu
Fu
School of Mathematics
Dongbei University of Finance and Economics
P. R. China
fuyumath@163.com
Minimal surfaces
translation surfaces
Minkowski space
Article.12.pdf
[
[1]
A. Bueno, R. López, Translation surfaces of linear Weingarten type, arXive, 2014 (2014), 1-7
##[2]
B.-Y. Chen, Geometry of Submanifolds, Marcel Dekker, New York (1973)
##[3]
B.-Y. Chen, Pseudo-Riemannian Geometry, \(\delta\) -invariants and Applications, With a foreword by Leopold Verstraelen, World Scientific Publishing Co., Hackensack (2011)
##[4]
F. Dillen, L. Verstraelen, G. Zafindratafa, A generalization of the translation surfaces of Scherk, Diff. Geom. in honor of Radu Rosca (KUL), (1991), 107-109
##[5]
W. Goemans, Surfaces in three-dimensional Euclidean and Minkowski space, in particular a study of Weingarten surfaces, PhD Thesis, Katholieke Univ. Leuven, Leuven (2010)
##[6]
B. P. Lima, N. L. Santos, P. A. Sousa, Translation hypersurfaces with constant scalar curvature into the Euclidean space, Israel J. Math., 201 (2014), 797-811
##[7]
H. Liu, Translation surfaces with constant mean curvature in 3-dimensional spaces, J. Geom., 64 (1999), 141-149
##[8]
M. Marin, On weak solutions in elasticity of dipolar bodies with voids, J. Comp. Appl. Math., 82 (1997), 291-297
##[9]
M. Marin, Harmonic vibrations in thermoelasticity of microstretch materials, J. Vib. Acoust. ASME, 2010 (2010), 6 pages, 2010 (2010), 1-6
##[10]
M. I. Munteanu, A. I. Nistor, Polynomial Translation Weingarten Surfaces in 3-dimensional Euclidean space, Differential geometry, 316–320, World Sci. Publ., Hackensack, NJ (2009)
##[11]
K. Seo, Translation hypersurfaces with constant curvature in space forms, Osaka J. Math., 50 (2013), 631-641
##[12]
K. Sharma, M. Marin, Effect of distinct conductive and thermodynamic temperatures on the reflection of plane waves in micropolar elastic half-space, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 75 (2013), 121-132
##[13]
L. Verstraelen, J. Walrave, S. Yaprak, The minimal translation surfaces in Euclidean space, Soochow J. Math., 20 (1994), 77-82
]