scholarly journals Estimate of Number of Periodic Solutions of Second-Order Asymptotically Linear Difference System

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Honghua Bin ◽  
Zhenkun Huang
Keyword(s):  
Perturbation Method ◽  
Periodic Solutions ◽  
Lower Bounds ◽  
Morse Theory ◽  
Discrete System ◽  
Second Order ◽  
Difference System ◽  
Asymptotically Linear ◽  
Compactness Condition ◽  
Twist Number

We investigate the number of periodic solutions of second-order asymptotically linear difference system. The main tools are Morse theory and twist number, and the discussion in this paper is divided into three cases. As the system is resonant at infinity, we use perturbation method to study the compactness condition of functional. We obtain some new results concerning the lower bounds of the nonconstant periodic solutions for discrete system.

Multiplicity results for systems of asymptotically linear second order equations

2002 ◽  
Vol 2 (4) ◽  
Author(s):  
Anna Capietto ◽  
Francesca Dalbono
Keyword(s):  
Lower Bounds ◽  
Second Order ◽  
Upper And Lower Bounds ◽  
Asymptotically Linear ◽  
Multiplicity Results ◽  
Number Of Zeros ◽  
Existence And Multiplicity ◽  
Abstract Continuation Theorem ◽  
Nodal Properties ◽  
Second Order Equations

AbstractWe prove the existence and multiplicity of solutions, with prescribed nodal properties, for a BVP associated with a system of asymptotically linear second order equations. The applicability of an abstract continuation theorem is ensured by upper and lower bounds on the number of zeros of each component of a solution.

scholarly journals Periodic solutions for a class of second-order differential delay equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xuan Wu ◽  
Huafeng Xiao
Keyword(s):  
Periodic Solutions ◽  
Index Theory ◽  
Second Order ◽  
Delay Equations ◽  
Asymptotically Linear ◽  
Differential Delay ◽  
Differential Delay Equations ◽  
Existence Of Periodic Solutions ◽  
The Difference

<p style='text-indent:20px;'>In this paper, we study the existence of periodic solutions of the following differential delay equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} z^{\prime\prime}(t) = \sum\limits_{k = 1}^{M-1}(-1)^kf(z(t-k)), \notag \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ f\in C(\mathbf{R}^N, \mathbf{R}^N) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ M,N\in \mathbf{N} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> is odd. By making use of <inline-formula><tex-math id="M4">\begin{document}$ S^1 $\end{document}</tex-math></inline-formula>-geometrical index theory, we obtain an estimation about the number of periodic solutions in term of the difference between eigenvalues of asymptotically linear matrices at the origin and at infinity.</p>

Periodic solutions of higher-dimensional discrete systems

2004 ◽  
Vol 134 (5) ◽  
pp. 1013-1022 ◽  
Author(s):  
Zhan Zhou ◽  
Jianshe Yu ◽  
Zhiming Guo
Keyword(s):  
Positive Integer ◽  
Critical Point ◽  
Periodic Solutions ◽  
Critical Point Theory ◽  
Discrete System ◽  
Discrete Systems ◽  
Point Theory ◽  
Second Order ◽  
Higher Dimensional ◽  
Image Position

Consider the second-order discrete system where f ∈ C (R × Rm, Rm), f(t + M, Z) = f(t, Z) for any (t, Z) ∈ R × Rm and M is a positive integer. By making use of critical-point theory, the existence of M-periodic solutions of (*) is obtained.

scholarly journals Nontrivial periodic solutions to delay difference equations via Morse theory

2018 ◽  
Vol 16 (1) ◽  
pp. 885-896 ◽  
Author(s):  
Yuhua Long ◽  
Haiping Shi ◽  
Xiaoqing Deng
Keyword(s):  
Periodic Solutions ◽  
Difference Equations ◽  
Morse Theory ◽  
Sufficient Conditions ◽  
Asymptotically Linear ◽  
Linear Delay ◽  
Nontrivial Periodic Solutions ◽  
Delay Difference Equations ◽  
Delay Difference

AbstractIn this paper some sufficient conditions are obtained to guarantee the existence of nontrivial 4T + 2 periodic solutions of asymptotically linear delay difference equations. The approach used is based on Morse theory.

Morse theory and multiple periodic solutions of some quasilinear difference systems with periodic nonlinearities

2017 ◽  
Vol 24 (1) ◽  
pp. 103-112 ◽  
Author(s):  
Petru Jebelean ◽  
Jean Mawhin ◽  
Călin Şerban
Keyword(s):  
Positive Integer ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Morse Theory ◽  
Difference System ◽  
System Of Difference Equations ◽  
Left Hand ◽  
Difference Systems ◽  
Multiple Periodic Solutions ◽  
The Difference

AbstractWe consider the system of difference equations$\Delta\bigg{(}\frac{\Delta u_{n-1}}{\sqrt{1-|\Delta u_{n-1}|^{2}}}\bigg{)}=% \nabla V_{n}(u_{n})+h_{n},\quad u_{n}=u_{n+T}\quad(n\in\mathbb{Z}),$with${\Delta u_{n}=u_{n+1}-u_{n}\in{\mathbb{R}}^{N}}$,${V_{n}=V_{n}(x)\in C^{2}({\mathbb{R}}^{N},\mathbb{R})}$,${V_{n+T}=V_{n}}$,${h_{n+T}=h_{n}}$for all${n\in\mathbb{Z}}$and some positive integerT,${V_{n}(x)}$is${\omega_{i}}$-periodic (${\omega_{i}>0}$) with respect to each${x_{i}}$(${i=1,\ldots,N}$) and${\sum_{j=1}^{T}h_{j}=0}$. Applying a modification argument to the corresponding problem with a left-hand member ofp-Laplacian type, and using Morse theory, we prove that if all its solutions are non-degenerate, then the difference system above has at least${2^{N}}$geometrically distinctT-periodic solutions.

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