The Existence of Solutions for a Nonlinear Mixed Problem of Singular Fractional Differential Equations

Abstract

By using fixed point results on cones, we study the existence of solutions for the singular nonlinear fractional boundary value problem (c)D(alpha)u(t) = f(t, u(t), u'(t), (c)D(beta)u(t)), u(0) = au(1), u'(0) = b(c)D(beta)u(1), u ''(0) = u'''(0) = u((n-1))(0) = 0, where n >= 3 is an integer, alpha is an element of (n - 1, n), 0 < beta < 1, 0 < a < 1, 0 < b < Gamma (2 - beta), f is an L-q-Caratheodory function, q > 1/alpha-1 and f(t,x,y,z) may be singular at value 0 in one dimension of its space variables x, y, z. Here, D-c stands for the Caputo fractional derivative.