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

Abstract

We established the existence of a positive solution of nonlinear fractional differential equationsL(D)[x(t)−x(0)]=f(t,xt),t∈(0,b]with finite delayx(t)=ω(t),t∈[−τ,0], wherelimt→0f(t,xt)=+∞, that is,fis singular att=0andxt∈C([−τ,0],ℝ≥0). The operator ofL(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.