Lyapunov-type inequalities for fractional Langevin-type equations involving Caputo-Hadamard fractional derivative

Abstract

Abstract In this study, some new Lyapunov-type inequalities are presented for Caputo-Hadamard fractional Langevin-type equations of the forms D a + β H C ( H C D a + α + p ( t ) ) x ( t ) + q ( t ) x ( t ) = 0 , 0 1 $p > 1$ . The boundary value problems of fractional Langevin-type equations were firstly converted into the equivalent integral equations with corresponding kernel functions, and then the Lyapunov-type inequalities were derived by the analytical method. Noteworthy, the Langevin-type equations are multi-term differential equations, creating significant challenges and difficulties in investigating the problems. Consequently, this study provides new results that can enrich the existing literature on the topic.

Keywords

• Lyapunov-type inequality
• Langevin-type equation
• Caputo-Hadamard fractional derivative
• Mixed boundary condition