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 ( α p t ) x q = 0 , < b and }^{\eta }{ \phi _{p}}\bigl[\bigl({}_{H}^{C}D_{a }^{\gamma u(t)\bigr)x(t)\bigr] v(t){\phi _{p}}\bigl(x(t)\bigr) η ϕ [ γ u ] v subject to mixed boundary conditions, respectively, where $p(t)$ , $q(t)$ $u(t)$ $v(t)$ real-valued functions $0 \beta 1 \alpha 2$ 1 2 $1 \gamma $ $\eta ${\phi _{p}}(s) |s{|^{p - 2}}s$ s | − $p > 1$ > . The value problems were firstly converted into equivalent integral with corresponding kernel functions, then derived by analytical method. Noteworthy, multi-term differential equations, creating significant challenges difficulties in investigating problems. Consequently, study provides results that can enrich existing literature on topic.