The Existence Of Solutions For Some Fractional Difference Equations

In this paper, by using the fxed point theorem on cones, we study the existenceof solutions for some boundary value problems of fractional diference equations.The paper is divided into three chapters.In the frst chapter, by using the Kranoselskii theorem and the Brouwer fxedpoint theorem, we study the existence of solution for a three-point boundary valueproblem of fractional diference equationwhere f:[ν-1, ν+b1]Nν-1×Râ†'R+is a continuous function, b, η∈N, ν-2<η <ν+b,1<ν <2, α>0. We establish the existence theorem of positive solutionsfor the problem under some conditions on f.In the second chapter, by using the Kranoselskii theorem, we study the existenceof positive solution for boundary value problem of fractional diference equation△νy(t)=λf(t+ν-1, y(t+ν-1)),y(ν-2)=g1(y),(P2)y(ν+b)=g2(y),where f:[ν-1, ν+b1]Nν-1×[0,∞)â†'[0,∞) is a continuous function,g1, g2∈C([ν-2, ν+b]Nν-2,[0,∞)) are given functions, and1<ν≤2. By usingthe Green function and the Kranoselskii theorem, we get the eigenvalue intervalsand some conditions which ensure the existence of solution for above problem.In the third chapter, by using the Banach contraction mapping theorem, westudy the existence of solution for boundary value problem of fractional diference equation where t∈{0,1,..., b+1}, f:{ν-1, ν,..., ν+b}×Râ†'R is continuous function,g∈C({ν-1, ν,..., ν+b}, R) is a given function, and1<ν≤2,0≤α <1, α, ν∈R.