On Nonlocal Boundary Value Problems for Nonlinear Integro-differential Equations of Arbitrary Fractional Order

In this paper, we prove the existence of solutions of a nonlocal boundary value problem for nonlinear integro-differential equations of fractional order given by (c)D(q)x(t) = f(t,x(t), (phi x)(t),(psi x)(t)), 0 < t < 1, x(0) = beta x(eta),x'(0) = 0, x ''(0) = =, ... , x((m - 2))(0) 0 0,x(1) = alpha x(eta), where q is an element of (m - 1,m], m is an element of N, m >= 2, 0 < eta < 1, and phi x and psi x are integral operators. The existence results are established by means of the contraction mapping principle and Krasnoselskii's fixed point theorem. An illustrative example is also presented.