Existence and Hyers-Ulam stability of solutions for a delayed hyperbolic partial differential equation

Abstract

In this paper, we first prove the existence and uniqueness of the solutions for a delayed hyperbolic partial differential equation by applying the progressive contraction technique introduced by Burton (Nonlinear Dyn Syst Theory 16(4): 366-371, 2016; Fixed Point Theory 20(1): 107-113, 2019) to the corresponding fixed-point problem. Then we derive a Hyers-Ulam stability result for this differential equation by using a Wendorff-type inequality and the Abstract Gronwall Lemma.