Existence and Hyers–Ulam Stability Analysis of Nonlinear Multi-Term Ψ-Caputo Fractional Differential Equations Incorporating Infinite Delay

The aim of the paper is to prove the existence results and Hyers–Ulam stability to nonlinear multi-term Ψ-Caputo fractional differential equations with infinite delay. Some specified assumptions are supposed to be satisfied by the nonlinear item and the delayed term. The Leray–Schauder alternative theorem and the Banach contraction principle are utilized to analyze the existence and uniqueness of solutions for infinite delay problems. Some new inequalities are presented in this paper for delayed fractional differential equations as auxiliary results, which are convenient for analyzing Hyers–Ulam stability. Some examples are discussed to illustrate the obtained results.