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 Da+βHC(HCDa+α+p(t))x(t)+q(t)x(t)=0,0<a<t<b,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\begin{aligned} &{}_{H}^{C}D_{a + }^{\\beta } \\bigl({}_{H}^{C}D_{a + }^{\\alpha }+ p(t)\\bigr)x(t) + q(t)x(t) = 0,\\quad 0 < a < t < b, \\end{aligned} $$\\end{document} and Da+ηHCϕp[(HCDa+γ+u(t))x(t)]+v(t)ϕp(x(t))=0,0<a<t<b,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\begin{aligned} &{}_{H}^{C}D_{a + }^{\\eta }{ \\phi _{p}}\\bigl[\\bigl({}_{H}^{C}D_{a + }^{\\gamma }+ u(t)\\bigr)x(t)\\bigr] + v(t){\\phi _{p}}\\bigl(x(t)\\bigr) = 0,\\quad 0 < a < t < b, \\end{aligned} $$\\end{document} subject to mixed boundary conditions, respectively, where p(t), q(t), u(t), v(t) are real-valued functions and 0 < beta < 1 < alpha < 2, 1 < gamma , eta < 2, {phi _{p}}(s) = |s{|^{p - 2}}s, 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.