Positive solutions for a system of fractional integral boundary value problem

Abstract

In this paper, we shall discuss the existence of positive solutions for the system of fractional integral boundary value problem { D 0 + α u i ( t ) + f i ( t , u 1 ( t ) , u 2 ( t ) ) = 0 , 0 < t < 1 , i = 1 , 2 , u i ( 0 ) = u i ′ ( 0 ) = 0 , u i ( 1 ) = ∫ 0 1 u i ( t ) d η ( t ) , where α∈(2,3] is a real number, D 0 + α is the standard Riemann-Liouville fractional derivative of order α and f i ∈C([0,1]× R + × R + ,R), i=1,2. ∫ 0 1 u i (t)dη(t) denotes the Riemann-Stieltjes integral, i.e., η(t) has bounded variation. By virtue of some inequalities associated with Green’s function, without the assumption of the nonnegativity of f i , we utilize the fixed point index theory to establish our main results. In addition, a square function and its inverse function are used to characterize coupling behaviors of f i , so that f i are allowed to grow superlinearly and sublinearly.MSC: 34B10, 34B18, 34A34, 45G15, 45M20.