Exploring the solutions of tempered (κ,ϖ)-Hilfer hybrid implicit boundary value problem

In this manuscript, we provide an in-depth analysis of existence and uniqueness results, along with stability assessments associated with the κ-Mittag-Leffler-Ulam-Hyers type, specifically focusing on a newly formulated category of hybrid boundary value problems (BVPs) that incorporate fractional derivatives. Our study leverages the properties of tempered (κ,ϖ)-Hilfer fractional operators to explore the mathematical underpinnings of the problem, which is characterized by implicit nonlinear fractional differential equations. To derive the results, we employ Banach's fixed point theorem, which facilitates the demonstration of the existence of solutions under certain contractive conditions. We also utilize a generalized Gronwall inequality to establish bounds and stability criteria for the solutions, thereby ensuring their robustness under perturbations. Moreover, we underscore the practical applicability of our theoretical findings by presenting several illustrative examples. These examples not only help demonstrate the effectiveness of our approach but also highlight the relevance of the results in addressing real-world scenarios where fractional dynamics are pertinent.