Global Structure Of Solutions For Some Periodic Boundary Value Problems Of Difference Equations

In this paper,we study the global structure of solutions for some periodic bound-ary value problems of difference equations by using topological degree and bifurca-tion theory.The main results are described as follows:1.We study the global structure of solutions for periodic boundary value prob-lem of second-order difference equation by using the bifurcation theorems from an interval and topological degree theory,where T>1 is an integer,T= {1,2,...,T},T = {0,1,...,T + 1};??[0,?)is a parameter;q?C(T,[0,?)),and q(t0)>0 for t0? T;??C(T×[0,?],0[0,?])and f(t,s)are not necessarily linearizable near 0 or infinity.The main results not only are the discretization of the results of Xu,Ma[Appl.Math.Comput.,2010],but also provide theory basis for numerical calculation of these class of problems.2.First of all,we show the existence of the principal eigenvalues and determine the sign of the corresponding eigenfunctions for linear periodic eigenvalue problem with sign-changing weight where q(t)>0 and q(t)(?)0 in T,and the weight function g changes its sign in T.Then,we apply our spectrum theory and the global bifurcation theorem to show the existence of positive solutions for nonlinear discrete periodic boundary value problem where f:T ×[0,?)? R is a continuous function,limf(t,s)/s =g(t),limf(t,s)/s = m(t),and m(t)>0,t ? T.The main results in this section not only generalize the corresponding results in second-order case of Brown and Lin[J.Math.Anal.Appl.,1980],but also be applied to nonlinear periodic boundary value problem of second-order difference equation,and we get the global structure of positive solutions with corresponding problem.3.We study the global structure of a class of discrete periodic boundary value problem involving ?-Laplacian operator by using the method of bifurcation from an interval,where ??[0,?)is a parameter;qt ? C(T,(0,?).qt0>0 for t0 ? T;?? C(T×[0,?),[0,?)),?(t,0)? 0.f(t,s)>0 for s>0,and f(t,s)are not,necessarily linearizable near 0.?:(-1,1)?R,?(y)= y?1-y2 is an increasing homeomorphism,?(0)= 0.The main results extend and improve the corresponding ones of Bereanu and Mawhin[J].Difference Equ.Appl.,2008].