Existence and Stability of Ulam–Hyers and Generalized Ulam–Hyers for the Generalized Langevin–Sturm–Liouville Equation Involving Generalized Liouville–Caputo Type

This paper focuses on investigating the existence, uniqueness, and stability of Ulam–Hyers (U-H) and generalized Ulam–Hyers (G-U-H) solutions for the generalized Langevin–Sturm–Liouville equation, which involves generalized Liouville–Caputo derivatives and antiperiodic boundary conditions. We can divide this manuscript into six parts. The first section employs the Leray–Schauder alternative to prove the problem’s solution. In the second portion, the analysis of uniqueness is discussed with the help of Banach’s fixed-point theorem. The third section covers both the G-U-H stability and the U-H stability to solve the given generalized Langevin–Sturm–Liouville boundary value problem. The fourth section deals with the variations of the stated problem, the generalized Sturm–Liouville equation with generalized Liouville–Caputo derivatives. The fifth section deals with the variations of the stated problem, the generalized Langevin with generalized Liouville–Caputo derivatives. Lastly, three examples, which serve as applications, are included to demonstrate the highlights of our results.