A Unified Approach to Solvability and Stability of Multipoint BVPs for Langevin and Sturm–Liouville Equations with CH–Fractional Derivatives and Impulses via Coincidence Theory

Abstract

The Langevin equation is a model for describing Brownian motion, while the Sturm–Liouville equation is an important mechanical model. This paper focuses on the solvability and stability of nonlinear impulsive Langevin and Sturm–Liouville equations with Caputo–Hadamard (CH) fractional derivatives and multipoint boundary value conditions. To unify the two types of equations, we investigate a general nonlinear impulsive coupled implicit system. By cleverly constructing relevant operators involving impulsive terms, we establish the coincidence degree theory and obtain the solvability. We explore the stability of solutions using nonlinear analysis and inequality techniques. As the most direct application, we naturally obtained the solvability and stability of the Langevin and Sturm–Liouville equations mentioned above. Finally, an example is provided to demonstrate the validity and availability of our major findings.

Keywords

• impulsive coupled implicit system
• CH–fractional derivative
• multipoint BVP
• solvability and stability
• coincidence degree theory