Lyapunov-type inequalities for fractional Langevin-type equations involving Caputo-Hadamard fractional derivative
Tóm tắt
In this study, some new Lyapunov-type inequalities are presented for Caputo-Hadamard fractional Langevin-type equations of the forms
$$ \begin{aligned} &{}_{H}^{C}D_{a + }^{\beta } \bigl({}_{H}^{C}D_{a + }^{\alpha }+ p(t)\bigr)x(t) + q(t)x(t) = 0,\quad 0 < a < t < b, \end{aligned} $$
and
$$ \begin{aligned} &{}_{H}^{C}D_{a + }^{\eta }{ \phi _{p}}\bigl[\bigl({}_{H}^{C}D_{a + }^{\gamma }+ u(t)\bigr)x(t)\bigr] + v(t){\phi _{p}}\bigl(x(t)\bigr) = 0,\quad 0 < a < t < b, \end{aligned} $$
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.
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