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

Abstract

In this study, some new Lyapunov-type inequalities are presented for Caputo-Hadamard fractional Langevin-type equations of the forms $$ \begin{aligned} &{}_{H}^{C}D_{a + }^{\beta } \bigl({}_{H}^{C}D_{a + }^{\alpha }+ p(t)\bigr)x(t) + q(t)x(t) = 0,\quad 0 < a < t < b, \end{aligned} $$ D a + β H C ( H C D a + α + p ( t ) ) x ( t ) + q ( t ) x ( t ) = 0 , 0 < a < t < b , and $$ \begin{aligned} &{}_{H}^{C}D_{a + }^{\eta }{ \phi _{p}}\bigl[\bigl({}_{H}^{C}D_{a + }^{\gamma }+ u(t)\bigr)x(t)\bigr] + v(t){\phi _{p}}\bigl(x(t)\bigr) = 0,\quad 0 < a < t < b, \end{aligned} $$ D a + η H C ϕ p [ ( H C D a + γ + u ( t ) ) x ( t ) ] + v ( t ) ϕ p ( x ( t ) ) = 0 , 0 < a < t < b , subject to mixed boundary conditions, respectively, where $p(t)$ p ( t ) , $q(t)$ q ( t ) , $u(t)$ u ( t ) , $v(t)$ v ( t ) are real-valued functions and $0 < \beta < 1 < \alpha < 2$ 0 < β < 1 < α < 2 , $1 < \gamma $ 1 < γ , $\eta < 2$ η < 2 , ${\phi _{p}}(s) = |s{|^{p - 2}}s$ ϕ p ( s ) = | s | p − 2 s , $p > 1$ 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.