This article focuses on the study of the existence of solution to Hadamard–Caputo fractional nonlinear differential equation with time delay. For time delay, two different cases of finite delay and infinite delay are considered. Based on the weighted function solution spaces, the existence and uniqueness of solution to the differential equations are verified by Leray–Schauder selection theorem and Banach fixed point theorem respectively. Finally, examples are given to illustrate the abstract results of this article.
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