New Results on Fixed Points and Coupled Fixed Points for α-Admissible Condensing Operators

This paper introduces a novel class of condensing mappings through the concept of α-admissibility. We prove several extensions of Darbo’s fixed point theorem within this framework, including a generalized contractive condition and a coupled fixed point result. Our main results extend the classical condensing condition to a flexible inequality involving α-admissibility and an auxiliary function. An example is given to demonstrate the applicability of our theorems where traditional approaches fail. Also, by leveraging the properties of α-admissible condensing operators, we proceed to establish a coupled fixed point theorem that relaxes traditional compactness conditions. At the end of the article, we apply our main theorem to prove the existence of solutions for a nonlinear integral equation.