Analysis of Caputo-Hadamard fractional neutral delay differential equations involving Hadamard integral and unbounded delays: Existence and uniqueness

In this article, we considered the nonlinear Caputo-Hadamard fractional differential equations involving Hadamard integrals and unbounded delays. We employed some standard fixed-point theorems to establish the sufficient conditions for the existence and uniqueness of solutions of the problem. The uniqueness is guaranteed by the Boyd and Wong fixed-point theorems and the Banach fixed-point theorem (BFT), while the existence result is ensured by the Leray-Schauder nonlinear alternative fixed-point theorem by utilizing a generalized Gronwall inequality (GI), which is closely related to the Hadamard derivative, the Leray-Schauder nonlinear alternative fixed-point theorem (LSFT) establishes an apriori bounds. Moreover, a new kind of continuous nondecreasing function is employed by the Boyd and Wong fixed-point theorem to transform the operator of the problem into a nonlinear contraction and produce a unique solution. Continous dependency of solutions on initial conditions (ICs) is ensured via Grownwall inequality as well. We also provide examples to support the main findings we established.