On A Discrete Fractional Boundary Value Problems With Nonlocal Fractional Boundary Conditions

Fractional calculus is an emerging field recently drawing attention from both theoreti-cal and applied disciplines. During the last two decades it has been successfully applied to problems in computational biology medical sciences, economics, physics and several fields in engineering. On the other hand, discrete fractional calculus is a very new area for scientists. Discrete fractional calculus has generated interest in recent years. Discrete fractional calculus includes discrete fractional initial problems and discrete fractional boundary problems. This paper will prove the solvability of discrete fractional boundary problems.This paper is devoted to introduce the discrete fractional boundary value problem with nonlocal fractional boundary conditions, for t∈[0, b+1]N0,ζ∈[0, b]N0,v∈(1,2],α,α∈[0,1], of the form-Δvy(t)=f(t+v-1,y(t+v-1), Δy(t+v-1)), subject to the boundary conditions y(v-2)=g(y),[Δαy(t)]t.=v+b-α+1=α[Δαy(t)]t=v+ζ-α. where f(t+v-1,.,.):[v-1, v+b]Nv-1×R×Râ†'[0,+∞) is continuous, g∈C([v-2,v+b+1]nv-2is a given functional. We derive the Green’s function for this problem and show that it satisfies certain properties. Some existence results are obtained by means of nonlinear alternative of Leray-Schauder type theorem and Krasnbsel-skii’s fixed point theorem.In this paper, we also investigate the discrete fractional boundary value problem with nonlocal fractional boundary conditions: Δvy(t)=f(t+v-1,y(t+v-1),Δy(t+v-2)), y(v-2)=0,[Δαy(t)]t=v+b-α+1=γ[Δαy(t)]t=v+ζ-α. We present a existence result at resonance case. The proof relies on coincidence degree theory.