On the Existence and Stability of Positive Solutions of Eigenvalue Problems for a Class of P-Laplacian ψ-Caputo Fractional Integro-Differential Equations

This article introduces a generalized approach for analyzing stability and establishing the existence of positive solutions in a specific type of differential equations known as p-Laplacian ψ-Caputo fractional differential equations with fractional integral boundary conditions. The study utilizes various techniques, including the analysis of Green’s function properties and the application of Guo–Krasnovelsky’s fixed point theorem on cones. By employing these methods, the research establishes novel findings concerning the existence and nonexistence of positive solutions. The investigation relies on fractional integrals, differential operators, and fundamental lemmas as fundamental tools. To assess solution stability, the Hyers–Ulam concept is employed, which extends prior research and introduces a specific definition. The article also provides numerical examples that support the obtained results, thereby demonstrating the practical applicability and accuracy of the proposed methods. Moreover, the study contributes to a deeper understanding of this subject matter and highlights real-life applications for these types of problems. Overall, this study offers a comprehensive analysis of stability and solution existence in a specific class of differential equations, with implications that extend to real-world scenarios such as engineering systems, financial modeling, population dynamics, epidemiology, and ecological studies. These types of problems arise in various fields where modeling and analyzing complex phenomena are necessary.