Existence Of Periodic Solutions For Second-order Asymptotically Linear Difference Systems

In this paper, we study the existence and multiplicity of nontrivial periodic so-lutions for a second-order asymptotically linear difference systems by using variational method、critical point theory and Morse theory. First of all, the solution of the differ-ence systems is equivalent to the critical point of the functional. The second, through calculating the critical groups of the corresponding functional at zero and infinity and using the Morse inequality, it is proved that the difference system has at least one or two nontrivial periodic solutions when the nonlinear terms satisfy certain sufficient con-ditions. The last, we discuss the existence of nontrivial periodic solutions at the case of the double resonance.This paper consists of three chapters. The structure is as follows:In the first chapter, we introduce the background of the discussed problem of this thesis briefly, and we give the main results of this thesis.In the second chapter, we get the variational structure of discrete problem, that is, corresponding energy functional of the difference systems. Then, the solution of the difference systems is equivalent to the critical point of the functional. We give the relevant knowledge of critical group and Morse theory.In the third chapter, by using variational method、Morse theory and calculation of critical groups, we investigate existence and multiplicity of nontrivial periodic solutions for second-order asymptotically linear difference systems as follows: Where N≥3is a given positive integer. For any a, b∈Z with a≤b, Z[a,b] denotes the discrete interval {a,a+1,...,b}, Fu、Fv are the first order partial derivatives of F for their second and third variables respectively. F has continuous second order partial derivatives for their second and third variables.△is the forward difference operator, i.e.△u(i)=u(i+1)-u(i),△2u(i)=△(△u(i)).As a application, we discuss that the difference systems (P), which is resonant at both infinity and zero, has at least two nontrivial periodic solutions.