Solvability Of Some Classes Of Boundary Value Problems Of Difference Equations

In this paper, by using the Brouwer theorem, the upper and lower solutions method and the Schauder fixed-point theorem, we study the existence of solutions for two classes of boundary value problems of fractional difference equations, in addition, we establish the generalized Green’s function for two classes of boundary value problems of second order difference equation. The main works are:1. By using the Brouwer theorem and the upper and lower solutions method, we study the existence of solutions for the boundary value problem of fractional hybrid difference equation where t∈[0,b]N0, g∈C([v-1, v+b-1]Nv-1×R, R), f∈C([v-2, v+b]Nv-1×R, R\{0}) and1<v≤2is a real number. When the nonlinear term f satisfies certain conditions, we establish the existence results of solutions for above problem.In the special case f=1, the problem converts into the situation that Atici[11[studied. So our theorem extend the main result of [11].2. By using the Schauder fixed-point theorem and the upper and lower solutions method, we study the existence of positive solutions for the boundary value problem of fractional difference equation where v∈(1,2], α∈[0,1), d≥0and k∈[-1, b-1]z are real number, t∈[0, b] N0, f∈C([v-1, v+b-1]Nv-1×[0,+∞),[0,+∞)), a:[v-1,v+b-1]Nv-1�'[0,+∞) is continuous and a (?)0on any subinterval of [v-1, v+b-1]Nv-1. Under the suitable condition, we show that there exists a positive number d*such that the problem has at least one positive solution for0≤d <d*and no solution for d> d*.Obviously, in the case that d=0, a=1, the problem converts into the situation that Goodrich[12]：studied．3.When the parameter λ is an eigenvalue for the S-L problem or the periodic S-L problem by building the generalized Green’s function for the two problems，we give the：um repre-sentation of solution for七he boundary value problem Lx＝-f(t)，U1(x)＝0，U2(x)＝0and Lx=-f(t)。U3(x)=0, U4(x)=0. In other words, by building an extended Green’s function，we still can convert the originally unreversible difference operator into summa-tion form，where p：[a,b＋1]�'(0，+∞),r：[a＋1，b+1]�'(0,+∞),q(t) is defined and real valued on [a＋1,b＋1]，α12＋α22≠0，β12+β22≠0, and in the periodic S-L problem we have p(a)=p(b＋1).