An Enhanced Darbo-Type Fixed Point Theorems and Application to Integral Equations

This manuscript introduces a generalized operator and presents new Darbo-type fixed point theorems pivotal in the existence theory of integral and differential equations. The significance of these theorems lies in their ability to provide conditions under which solutions to complex mathematical problems can be guaranteed. We establish our results by employing the measure of noncompactness within the context of Banach spaces, a framework that allows for a comprehensive analysis of functional equations. Our findings extend existing Darbo-type fixed point theorems and offer a deeper understanding of the underlying mathematical structures. By generalizing these results, we contribute to the broader field of fixed-point theory, enhancing its applicability to various mathematical disciplines. The implications of our work are substantial, as they facilitate the development of new methods for solving integral and differential equations that arise in both theoretical and applied contexts.