]>
2019
12
2
ISSN 2008-1898
69
Stability of a fractional difference equation of high order
Stability of a fractional difference equation of high order
en
en
In this paper we investigate the local stability, global stability, and
boundedness of solutions of the recursive sequence%
\[
x_{n+1}=x_{n-p}\ \left( \frac{2\ x_{n-q}\ +a\ x_{n-r}}{x_{n-q}\ +a\ x_{n-r}}%
\right),
\]
where \(x_{-q+k}\ \neq -a\ x_{-r+k} \) for \( k=0,1,\dots,\min (q,r) , a\in \mathbb{R},\ p ,q, r \geq 0\) with the initial condition \(x_{-p},x_{-p+1} ,\dots, x_{-q},\) \(x_{-q+1} ,\dots, x_{-r},x_{-r+1} ,\dots, x_{-1}\)
and \(x_{0}\in (0,\infty )\). Some numerical examples will be given to
illustrate our results.
65
74
M. A.
El-Moneam
Mathematics Department, Faculty of Science
Jazan University
Saudi Arabia
mabdelmeneam2014@yahoo.com
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
S.
Elamody
Mathematics Department, Faculty of Science
Jazan University
Saudi Arabia
Difference equations
prime period two solution
boundedness character
locally asymptotically stable
global attractor
global stability
high orders
Article.1.pdf
[
[1]
R. P. Agarwal, E. M. Elsayed, On the solution of fourth-order rational recursive sequence , Adv. Stud. Contemp. Math. (Kyungshang), 20 (2010), 525-545
##[2]
R. Devault, W. Kosmala, G. Ladas, S. W. Schaultz, Global Behavior of \(y_{n+1} = \frac{p+y_{n-k}}{ qy_n+y_{n-k}}\), Nonlinear Anal., 47 (2001), 4743-4751
##[3]
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. Difference Equ., 2006 (2006), 1-10
##[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]
M. A. El-Moneam , On the dynamics of the higher order nonlinear rational difference equation, Math. Sci. Lett., 3 (2014), 121-129
##[6]
M. A. El-Moneam, On the dynamics of the solutions of the rational recursive sequences, British J. Math. Computer Sci., 5 (2015), 654-665
##[7]
M. A. El-Moneam, S. O. Alamoudy, On study of the asymptotic behavior of some rational difference equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 21 (2014), 89-109
##[8]
M. A. El-Moneam, E. M. E. Zayed, Dynamics of the rational difference equation, Inf. Sci. Lett., 3 (2014), 1-9
##[9]
E. M. Elsayed, Solution and Attractivity for a Rational Recursive Sequence, Discrete Dyn. Nat. Soc., 2011 (2011), 1-17
##[10]
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
##[11]
E. A. Grove, G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman & Hall/CRC , Boca Raton (2005)
##[12]
T. F. Ibrahim , Boundedness and stability of a rational difference equation with delay, Rev. Roum. Math. Pures Appl., 57 (2012), 215-224
##[13]
T. F. Ibrahim, Periodicity and global attractivity of difference equation of higher order, J. Comput. Anal. Appl., 16 (2014), 552-564
##[14]
T. F. Ibrahim, Three-dimensional max-type cyclic system of difference equations , Int. J. Phys. Sci., 8 (2013), 629-634
##[15]
T. F. Ibrahim, N. Touafek, On a third-order rational difference equation with variable coefficients , Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 20 (2013), 251-264
##[16]
V. L. Kocic, G. Ladas , Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht (1993)
##[17]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive two sequences \(x_{n+1} = ax_{n-k} +bx_{n-k}/ (cx_n + \delta dx_{n-k})\), Acta Math. Vietnam., 35 (2010), 355-369
##[18]
E. M. E. Zayed, M. A. El-Moneam, On the global attractivity of two nonlinear difference equations, J. Math. Sci., 177 (2011), 487-499
]
A motion of complex curves in \(\mathbb C^3\) and the nonlocal nonlinear Schrödinger equation
A motion of complex curves in \(\mathbb C^3\) and the nonlocal nonlinear Schrödinger equation
en
en
This paper shows that soliton solutions to the nonlocal nonlinear Schrödinger equation (NNLS) proposed recently by Ablowitz and Musslimani [M. J. Ablowitz, Z. H. Musslimani, Phys. Rev. Lett., \(\bf 110\) (2013), 5 pages] describe a motion of three distinct complex curves in \(\mathbb C^3\) with initial data being suitably restricted. This gives a geometric interpretation of NNLS.
75
85
Shiping
Zhong
School of Mathematics and Computer Sciences
Gannan Normal University
P. R. China
spzhong15@fudan.edu.cn
Complex moving curve
geometric interpretation
uniqueness
Article.2.pdf
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]
Viscosity iterative method for split common null point problems and fixed point problems
Viscosity iterative method for split common null point problems and fixed point problems
en
en
In this paper, we introduce and study
an Ishikawa-like iterative algorithm to approximate a common solution of a split common null point problem and a fixed point problem of asymptotically
pseudo-contractive mappings in the intermediate sense on unbounded domains.
We prove that the sequence generated by the iterative scheme strongly converges to
a common solution of the above-said problems. The method in this paper is novel and different from those given
in many other papers. The results are the extension and improvement
of the recent results in the literature.
86
101
Yanlai
Song
College of Science
Zhongyuan University of Technology
China
songyanlai2009@163.com
Xinhong
Chen
College of Science
Zhongyuan University of Technology
China
chenxh6@163.com
Banach space
split common null point problem
fixed point
metric resolvent
asymptotically pseudocontractive mapping in the intermediate sense
Article.3.pdf
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Y. I. Alber, Metric and generalized projections operators in Banach spaces: properties and applications, Dekker, New York (1996)
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S. Y. Cho, Strong convergence analysis of a hybrid algorithm for nonlinear operators in a Banach space, J. Appl. Anal. Comput., 8 (2018), 19-31
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C.-S. Ge, A hybrid algorithm with variable coefficients for asymptotically pseudocontractive mappings in the intermediate sense on unbounded domains, Nonlinear Anal., 75 (2012), 2859-2866
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X. Qin, S. Y. Cho , Convergence analysis of a monotone projection algorithm in reflexive Banach spaces, Acta Math. Sci. Ser. B (Engl. Ed.), 37 (2017), 488-502
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X. Qin, S. Y. Cho, J. K. Kim , Convergence theorems on asymptotically pseudocontractive mappings in the intermediate sense, Fixed Point Theory Appl., 2010 (2010), 1-14
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X. Qin, J.-C. Yao, Projection splitting algorithms for nonself operators, J. Nonlinear Convex Anal., 18 (2017), 925-935
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W. Takahashi , Convex Analysis and Approximation of Fixed Points, Yokohama Publ., Yokohama (2000)
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W. Takahashi, Nonlinear Functional Analysis, Yokohama Publ., Yokohama (2000)
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W. Takahashi, J.-C. Yao, Strong convergence theorems by hybrid methods for the split common null point problem in Banach spaces , Fixed Point Theory Appl., 2015 (2015), 1-13
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H. Zhou, Demiclosedness principle with applications for asymptotically pseudo-contraction in Hilbert spaces, Nonlinear Anal., 70 (2009), 3140-3145
]
On the rational difference equation \(y_{{n+1}}={\frac{\alpha _{{0}}y_{{n}}+\alpha _{{1}}y_{{n-p}}+\alpha _{{2}}y_{{n-q}}+\alpha _{{3}}y_{{n-r}}+\alpha _{{4}}y_{{n-s}}+\alpha _{{5}}y_{{n-t}}}{\beta _{{0}}y_{{n}}+\beta _{{1}}y_{{n-p}}+\beta _{{2}}y_{{n-q}}+\beta _{{3}}y_{{n-r}}+\beta _{{4}}y_{{n-s}}+\beta _{{5}}y_{{n-t}}}}\)
On the rational difference equation \(y_{{n+1}}={\frac{\alpha _{{0}}y_{{n}}+\alpha _{{1}}y_{{n-p}}+\alpha _{{2}}y_{{n-q}}+\alpha _{{3}}y_{{n-r}}+\alpha _{{4}}y_{{n-s}}+\alpha _{{5}}y_{{n-t}}}{\beta _{{0}}y_{{n}}+\beta _{{1}}y_{{n-p}}+\beta _{{2}}y_{{n-q}}+\beta _{{3}}y_{{n-r}}+\beta _{{4}}y_{{n-s}}+\beta _{{5}}y_{{n-t}}}}\)
en
en
In this paper, we examine and explore the boundedness, periodicity, and
global stability of the positive solutions of the rational difference
equation
\[
y_{{n+1}}={\frac{\alpha _{{0}}y_{{n}}+\alpha _{{1}}y_{{n-p}}+\alpha _{{2}}y_{%
{n-q}}+\alpha _{{3}}y_{{n-r}}+\alpha _{{4}}y_{{n-s}}+\alpha _{{5}}y_{{n-t}}}{%
\beta _{{0}}y_{{n}}+\beta _{{1}}y_{{n-p}}+\beta _{{2}}y_{{n-q}}+\beta _{{3}%
}y_{{n-r}}+\beta _{{4}}y_{{n-s}}+\beta _{{5}}y_{{n-t}}}},
\]
where the coefficients \({ \alpha _{i},\beta _{i}\in (0,\infty
),\ i=0,1,2,3,4,5},\) and \(p,q,r,s,\) and \(t\) are positive integers. The
initial conditions \(y_{-t} ,\) \(\ldots, y_{-s} ,\ldots, y_{-r} ,\ldots, y_{-q} ,\ldots, y_{{%
-p}} ,\ldots, y_{-1} , y_{0}\) are arbitrary positive real numbers such that \(%
p<q<r<s<t\). Some numerical examples will be given to illustrate our result.
102
119
M. A.
El-Moneam
Mathematics Department, Faculty of Science
Jazan University
Kingdom of Saudi Arabia
maliahmedibrahim@jazanu.edu.sa
E. S.
Aly
Mathematics Department, Faculty of Science
Jazan University
Kingdom of Saudi Arabia
elkhateeb@jazanu.edu.sa
M. A.
Aiyashi
Mathematics Department, Faculty of Science
Jazan University
Kingdom of Saudi Arabia
maiyashi@jazanu.edu.sa
Difference equation
boundedness
prime
period two solution
stability
Article.4.pdf
[
[1]
E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, On the difference equation \(x_{n+1} = ax_n - bx_n/ (cx_n - dx_{n-1})\), Adv. Difference Equ., 2006 (2006), 1-10
##[2]
E. M. Elabbasy, H. El-Metwally, E. M. Elsayed , On the difference equation \(x_{n+1} = (\alpha x_{n-l} + \beta x_{n-k}) / (Ax_{n-l} + Bx_{n-k})\), Acta Math. Vietnam., 33 (2008), 85-94
##[3]
M. A. El-Moneam, S. O. Alamoudy , On study of the asymptotic behavior of some rational difference equations , Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 21 (2014), 89-109
##[4]
E. M. Elsayed , On the global attractivity and periodic character of a recursive sequence, Opuscula Math., 30 (2010), 431-446
##[5]
E. A. Grove, G. Ladas, Periodicities in nonlinear difference equations, Chapman & Hall/CRC, Boca Raton (2005)
##[6]
W. T. Li, H. R. Sun , Dynamics of a rational difference equation, Appl. Math. Comput., 163 (2005), 577-591
##[7]
M. A. Obaid, E. M. Elsayed, M. M. El-Dessoky, Global attractivity and periodic character of difference equation of order four, Discrete Dyn. Nat. Soc., 2012 (2012), 1-20
##[8]
M. Saleh, S. Abu-Baha, Dynamics of a higher order rational difference equation, Appl. Math. Comput., 181 (2006), 84-102
##[9]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1} = (D+ \alpha x_n + \beta x_{n-1} + \gamma x_{n-2})/(Ax_n + Bxn-1 + Cx_{n-2})\) , Comm. Appl. Nonlinear Anal., 12 (2005), 15-28
##[10]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1} = (\alpha x_n + \beta x_{n-1} + \gamma x_{n-2} + \delta x_{n-3})/(Ax_n + Bx_{n-1} + Cx_{n-2} + Dx_{n-3})\), J. Appl. Math. Comput., 22 (2006), 247-262
##[11]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1} = \left( A + \Sigma_{i=0}^k \alpha_ix_{n-i} \right)/ \Sigma_{i=0}^k \beta_ix_{n-i}\), Math. Bohem., 133 (2008), 225-239
##[12]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1} = \left( A + \Sigma_{i=0}^k \alpha_ix_{n-i} \right) /\left( \Sigma_{i=0}^k \beta_ix_{n-i}\right)\) , Int. J. Math. Math. Sci., 2007 (2007), 1-12
##[13]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1} = ax_n - bx_n/ (cx_n - dx_{n-k})\), Comm. Appl. Nonlinear Anal., 15 (2008), 47-57
##[14]
E. M. E. Zayed, M. A. El-Moneam, On the Rational Recursive Sequence \(x_{n+1} = (\alpha + \beta x_{n-k})/ (\gamma - x_n)\), J. Appl. Math. Comput., 31 (2009), 229-237
##[15]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1} = Ax_n +(\beta x_n + \gamma x_{n-k}) / (Cx_n + Dx_{n-k})\), Comm. Appl. Nonlinear Anal., 16 (2009), 91-106
##[16]
E. M. E. Zayed, M. A. El-Moneam, On the Rational Recursive Sequence \(x_{n+1} = x_{n-k} + (ax_n + bx_{n-k}) / (cx_n - dx_{n-k})\), Bull. Iranian Math. Soc., 36 (2010), 103-115
##[17]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1} = (\alpha_0x_n + \alpha_1x_{n-l} + \alpha_2x_{n-k}) / (\beta_0x_n + \beta_1x_{n-l} + \beta_2x_{n-k})\) , Math. Bohem., 135 (2010), 319-336
##[18]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1} = Ax_n + Bx_{n-k} + (\beta x_n + \gamma x_{n-k}) / (Cx_n + Dx_{n-k})\), Acta Appl. Math., 111 (2010), 287-301
##[19]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive two sequences \(x_{n+1} = ax_{n-k} + bx_{n-k}/ (cx_n + \delta dx_{n-k})\), Acta Math. Vietnam., 35 (2010), 355-369
##[20]
E. M. E. Zayed, M. A. El-Moneam , On the global attractivity of two nonlinear difference equations, J. Math. Sci. (N.Y.), 177 (2011), 487-499
##[21]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1} = (A + \alpha_0x_n + \alpha_1x_{n-\sigma}) / (B + \beta_0x_n + \beta_1x_{n-\tau})\), Acta Math. Vietnam, 36 (2011), 73-87
##[22]
E. M. E. Zayed, M. A. El-Moneam, On the global asymptotic stability for a rational recursive sequence, Iran. J. Sci. Technol. Trans. A Sci., 35 (2011), 333-339
##[23]
E. M. E. Zayed, M. A. El-Moneam, On the global stability of the nonlinear difference equation \(x_{n+1} = \frac{\alpha_0x_n+\alpha_1x_{n-l}+\alpha_2x_{n-m}+\alpha_3x_{n-k}}{ \beta_0x_n+\beta_1x_{n-l}+\beta_2x_{n-m}+\beta_3x_{n-k}}\), WSEAS Transactions on Mathematics, 11 (2012), 357-366
##[24]
E. M. E. Zayed, M. A. El-Moneam, On the qualitative study of the nonlinear difference equation \(x_{n+1} =\frac{ \alpha x_{n-\sigma}}{ \beta+\gamma x^p _{n-\tau}}\), Fasc. Math., 50 (2013), 137-147
]
Weak mixing in general semiflows implies multi-sensitivity, but not thick sensitivity
Weak mixing in general semiflows implies multi-sensitivity, but not thick sensitivity
en
en
It was proved by Wang et al. [Wang, J. Yin, Q. Yan, J. Nonlinear Sci. Appl., \({\bf 9}\) (2016), 989--997] that any weakly mixing semiflow on a compact metric space, whose all transition maps are surjective, is thickly sensitive. We consider what happens if we do not have the assumptions of compactness and surjectivity. We prove that even in that case any weakly mixing semiflow is multi-sensitive, and that, however, it does not have to be thickly sensitive.
120
123
Alica
Miller
Department of Mathematics
University of Louisville
USA
alica.miller@louisville.edu
Weak mixing
sensitivity
multi-sensitivity
thick sensitivity
semi-flow
Article.5.pdf
[
[1]
G. Cairns, A. Kolganova, A. Nielsen, Topological transitivity and mixing notions for group actions, Rocky Mount. J. Math., 37 (2007), 371-397
##[2]
L. He, X. Yan, L. Wang, Weak-mixing implies sensitive dependence, J. Math. Anal. Appl., 299 (2004), 300-304
##[3]
E. Kontorovich, M. Megrelishvili, A note on sensitivity of semigroup actions, Semigroup Forum, 76 (2008), 133-141
##[4]
S. Lardjane, On some stochastic properties in Devaney’s chaos, Chaos. Solitons Fractals, 28 (2006), 668-672
##[5]
R. Li, T. Lu, Y. Zhao, H. Wang, H. Liang, Multi-sensitivity, syndetical sensitivity and the asymptotic average-shadowing property for continuous semi-flows, J. Nonlinear Sci. Appl., 10 (2017), 4940-4953
##[6]
A. Miller, Envelopes of syndetic subsemigroups of the acting topological semigroup in a semiflow, Topology Appl., 158 (2011), 291-297
##[7]
T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20 (2007), 2115-2126
##[8]
T. Wang, J. Yin, Q. Yan, The sufficient conditions for dynamical systems of semigroup actions to have some stronger forms of sensitivities, J. Nonlinear Sci. Appl., 9 (2016), 989-997
]
Implicit hybrid methods for solving fractional Riccati equation
Implicit hybrid methods for solving fractional Riccati equation
en
en
In this paper, we modify the implicit hybrid methods for solving fractional Riccati equation. Similar methods are implemented for the ordinary derivative and we are the first who implement it for fractional derivative case. This approach is of higher order comparing with the existing methods in the literature. We study the convergence, zero stability, consistency, and region of absolute stability. Numerical results are presented to show the efficiency of the proposed method.
124
134
Muhammed I.
Syam
Department of Mathematical Sciences
United Arab Emirates University
UAE
m.syam@uaeu.ac.ae
Azza
Alsuwaidi
Department of Mathematical Sciences
United Arab Emirates University
UAE
Asia
Alneyadi
Department of Mathematical Sciences
United Arab Emirates University
UAE
Safa
Al Refai
Department of Mathematical Sciences
United Arab Emirates University
UAE
Sondos
Al Khaldi
Department of Mathematical Sciences
United Arab Emirates University
UAE
Fractional Riccati equation
implicit hybrid methods
convergence
Article.6.pdf
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