Boundary Value Problems For Fourth-order Linear Difference Equations With Nonlinear Perturbations

This dissertation,firstly,studies the spectral structure of a linear discrete fourth-order boundary value problem by applying the method of matrix analysis.Then by using Leray-Schauder continuation theorem,the critical point theory,and mini max theorem,we obtain the existence,uniqueness of solutions for some non-linear fourth-order difference equations boundary value problem,respectively.We describe them in detail as follows.1.We study the spectral structure of linear fourth-order difference equations boundary value problem ??? by using the method of matrix analysis.where ?>0 is a parameter,T>5 is an integer,a?t?>0,t ? T₂:= ?2,3,….,T},?4y?k-2?= y?k+2?-4y?k+1?+6y?k?-4y?k-1?+ y?k-2?.we obtain the problem has exactly T-1 positive and simple eigenvalues.The result provide the basis for the study of nonlinear fourth-order difference equations.2.By using Leray-Schauder continuation theorem,we consider the existence of solutions of nonlinear fourth-order difference equation boundary value problems ??? where ?k is the k-th eigenvalue of the associated linear eigenvalue problem,h:T₂ ? R,.g:T₂ x R ? R is continuous and satisfies some resonance conditions.The results not only improve the corresponding ones of Ruyun Ma[Appl.Math.Com-put.,2010],but also provide theory basis for numerical calculation of the results of Gupta[J.Diff.Eqns.,1988].3.We concerned with the following nonlinear fouth-order difference system boundary value problems ??? where h:T₂?Rn,g:Rn?Rn is a potential function satisfying some nonres-onance conditions.By using the critical point theory and a min max theorem,we obtain the uniqueness of solutions for the problem.The results not only are the discretization of the results of Usmani[Proc.Amer.Math.Soc.,1979]and Y.S.Yang[Proc.Amer.Math.Soc.,1978],but also extend the results to the fourth-order difference system.