Existence of solutions for fractional differential equations with integral boundary conditions

*Correspondence: [email protected] School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, PR China Abstract In this paper, we study boundary-value problems for the following nonlinear fractional differential equations involving the Caputo fractional derivative: D0+x(t) = f (t, x(t), Dβ0+x(t)), t ∈ [0, 1], x(0) + x′(0) = y(x), ∫ 1 0 x(t)dt =m, x′′(0) = x′′′(0) = · · · = x(n–1)(0) = 0, where D0+, Dβ0+ are the Caputo fractional derivatives, f : [0, 1]×R×R→R is a continuous function, y : C([0, 1],R)→R is a continuous function andm ∈R, n – 1 < α < n (n≥ 2), 0 < β < 1 is a real number. By means of the Banach fixed-point theorem and the Schauder fixed-point theorem, some solutions are obtained, respectively. As applications, some examples are presented to illustrate our main results. MSC: 34A08; 34B10