Two positive solutions for ( n − 1, 1)-type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations

Abstract

In this paper, we consider the ( n − 1, 1)-type integral boundary value problem of nonlinear fractional differential equation D 0 + α u ( t ) + λ f ( t , v ( t ) ) = 0 , 0 < t < 1 D 0 + α v ( t ) + λ g ( t , u ( t ) ) = 0 , u ( j ) ( 0 ) = v ( j ) ( 0 ) = 0 , 0 ⩽ j ⩽ n - 2 , u ( 1 ) = μ ∫ 0 1 u ( s ) ds , v ( 1 ) = μ ∫ 0 1 v ( s ) ds , where λ , μ are parameter and 0 < μ < α , α ∈ ( n - 1 , n ] is a real number and n ⩾ 3 , D 0 + α is the Riemann–Liouville’s fractional derivative, f , g are continuous and semipositone. We gave the corresponding Green’s function for the boundary value problem and its some properties. Moreover, we derive an interval of λ such that any λ lying in this interval, the semipositone boundary value problem has multiple positive solutions.