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 equations $𝔏\left(D\right)[x\left(t\right)-x\left(0\right)]=f(t,x_t)$, $t\in (0,b]$ with finite delay $x\left(t\right)=\omega \left(t\right)$, $t\in [-\tau ,0]$, where ${\lim }_{t\to 0}f(t,x_t)=+\infty$, that is, $f$ is singular at $t=0$ and $x_t\in C([-\tau ,0],{ℝ}^{\ge 0})$$. The operator of $𝔏\left(D\right)$ 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.