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.