Positive solutions to fractional boundary value problems with nonlinear boundary conditions

Abstract

We consider a system of boundary value problems for fractional differential equation given by D0+β φp (D 0+αu) (t) = λ1a1 (t) f1 (u (t), v (t)), t ∈ (0,1), D0+β φp (D0+αv) (t) = λ 2a2 (t) f2 (u (t), v (t)), t ∈ (0,1), where 1 < α, β ≤ 2, 2 < α + β ≤ 4, λ1,λ2 are eigenvalues, subject either to the boundary conditions D0+α u (0) = D 0+α u (1) = 0, u (0) = 0, D0+ β1 u (1) - Σi=1m-2 a1i D0+β1 u (χ1i) = 0, D0+ α v (0) = D0+α v (1) = 0, v (0) = 0, D0+β1 v (1) - Σi = 1 m-2 a2i D0+β1 v (χ2i) = 0 or D0+α u (0) = D 0+α u (1) = 0, u (0) = 0, D0+ β1 u (1) - Σi = 1m 2 a1i D0+β1 u (χ1i) = ψ1 (u), D0+α v (0) = D0+α v (1) = 0, v (0) = 0, D0+β1 v (1) - Σ i = 1 m-2 a2i D0+β1 v (χ2i) = ψ2 (v), where 0 < β1 < 1, α - β1- 1 ≥ 0 and ψ1, ψ2: C ([ 0,1 ]) → [ 0, ∞) are continuous functions. The Krasnoselskiis fixed point theorem is applied to prove the existence of at least one positive solution for both fractional boundary value problems. As an application, an example is given to demonstrate some of main results.