Twin solutions to semipositone boundary value problems for fractional differential equations with coupled integral boundary conditions

This paper investigates the existence of at least two positive solutions for the following high-order fractional semipositone boundary value problem (SBVP, for short) with coupled integral boundary value conditions:  D0+u(t) + λf(t,u(t), v(t)) = 0, t ∈ (0, 1), D0+v(t) + λg(t,u(t), v(t)) = 0, t ∈ (0, 1), u(j)(0) = v(j)(0) = 0, j = 0, 1, 2, · · · ,n− 2, Dα−1 0+ u(1) = λ1 ∫η1 0 v(t)dt, Dα−1 0+ v(1) = λ2 ∫η2 0 u(t)dt, where n− 1 < α 6 n, n > 3, 0 < η1,η2 6 1, λ, λ1, λ2 are parameters and satisfy λ1λ2(η1η2) < Γ2(α+ 1), D0+ is the standard Riemann-Liouville derivative, and f,g are continuous and semipositone. By using the nonlinear alternative of Leray-Schauder type, Krasnoselskii’s fixed point theorems, and the theory of fixed point index on cone, we establish some existence results of multiple positive solutions to the considered fractional SBVP. As applications, two examples are presented to illustrate our main results. c ©2017 All rights reserved.