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 L(D)[x(t)−x(0)]=f(t,xt), t∈(0,b] with finite delay x(t)=ω(t), t∈[−τ,0], where limt→0f(t,xt)=+∞, that is, f is singular at t=0 and xt∈C([−τ,0],ℝ≥0). The operator of L(D) 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