Some Results for a Four $ Point Boundary Value Problems for a coupled system Involving Caputo Derivatives

Abstract

In this paper, we prove the existence and uniqueness of solutions for a system for fractional differential equations with four point boundary conditions. The results are obtained using Banach contraction principle and Krasnoselkii’s fixed point theorem          D α x ( t )+ f ( t , y ( t ) , D δ y ( t ) ) = 0, t ∈ J , D β y ( t )+ g ( t , x ( t ) , D σ x ( t ))= 0, t ∈ J , x ( 0 )= y ( 0 )= 0, x ( 1 ) − λ 1 x ( η )= 0, y ( 1 ) − λ 1 y ( η )= 0, x ′′ ( 0 )= y ′′ ( 0 )= 0, x ′′ ( 1 ) − λ 2 x ′′ ( ξ )= 0, y ′′ ( 1 ) − λ 2 y ′′ ( ξ )= 0, where 3 < α , β ≤ 4, α − 2 < σ ≤ α − 1, β − 2 < δ ≤ β − 1, 0 < ξ , η < 1, and D α , D β , D δ and D σ , are the Caputo fractional derivatives, J =[ 0, 1 ] , λ 1 , λ 2 are real constants with λ 1 η 6 = 1, λ 2 ξ 6 = 1 and f , g continuous functions on [ 0, 1 ] × R 2 .