Existence of solution to fractional differential equation with fractional integral type boundary conditions.

This paper is devoted by developing sufficient condition required for the existence of solution to a nonlinear fractional order boundary value problem Dγu(ℓ)=ψ(ℓ,u(λℓ)),ℓ∈Z=[0,1],with fractional integral boundary conditions p1u(0)+q1u(1)=1Γ(γ)∫01(1−ρ)γ−1g1(ρ,u(ρ))dρ,and p2u′(0)+q2u′(1)=1Γ(γ)∫01(1−ρ)γ−1g2(ρ,u(ρ))dρ,where γ ∈ (1, 2], 0 < λ < 1, D denotes the Caputo fractional derivative (in short CFD), ψ,g1,g2:Z×R→R are continuous functions and pi,qi(i=1,2) are positive real numbers. Using topological degree theory sufficient results are constructed for the existence of at least one and unique solution to the concerned problem. For the validity of our result, a concrete example is presented in the end. [ABSTRACT FROM AUTHOR]