Existence of Positive Solutions for a Class of Delay Fractional Differential Equations with Generalization to N-Term

We established the existence of a positive solution of nonlinear fractional differential equations pound (D) [x(t)-x(0)] = f(t, x(t)), t. is an element of (0, b], with finite delay x (t) = omega (t), t is an element of [-tau,0], where lim(t -> 0)f(t, x(t)) = +infinity, that is, f is singular at t = 0 and x(t) is an element of C([-tau,0], R->= 0). The operator of (D) pound involves the Riemann- Liouville fractional derivatives. In this problem, the initial conditions with fractional order and some relations among them were considered. The analysis rely on the alternative of the Leray-Schauder fixed point theorem, the Banach fixed point theorem, and the Arzela- Ascoli theorem in a cone.