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

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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p(t)$\end{document}, q(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q(t)$\end{document}, u(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u(t)$\end{document}, v(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v(t)$\end{document} are real-valued functions and 0<β<1<α<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0 < \beta < 1 < \alpha < 2$\end{document}, 1<γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1 < \gamma $\end{document}, η<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\eta < 2$\end{document}, ϕp(s)=|s|p−2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\phi _{p}}(s) = |s{|^{p - 2}}s$\end{document}, p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p > 1$\end{document}. 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.


Introduction
This striking inequality is known as a Lyapunov inequality [1]. The inequality (1.1) and its generalizations have been applied in various mathematical problems, involving stability problems, oscillation theory, and eigenvalue bounds for ordinary differential equations [2][3][4]. For some improved and generalized forms, such as Lyapunov-type inequalities for higher-order differential equations, p-Laplacian differential equations, partial differential equations, difference equations, impulsive differential equations, dynamic equations on time scales, fractional differential equations, some literature studies [5][6][7][8][9][10] and the monographs [11,12] should be referred to for better comprehensive understanding. Noteworthy, a result of fractional Lyapunov-type inequality was first presented by Ferreira. In 2013, Ferreira [9] extended inequality (1.1) to the fractional case in the sense of the Riemann-Liouville fractional derivative and obtained the following classical result: (1.2) One year later, the same author obtained the analogous Lyapunov-type inequality for the fractional BVP, involving Caputo fractional derivative (see [10]). Based on the above-mentioned two studies, the subject of fractional Lyapunov-type inequalities has received significant research attention, and a variety of interesting results have been established. For some recent works on the topic, we refer the reader to the works [13][14][15][16][17][18][19][20][21][22][23][24][25][26], the survey paper [27] and the references cited therein. For example, according to the literature report [13], the authors generalized Lyapunov-type inequality (1.2) to the p-Laplacian problem:
Although the fractional Lyapunov-type inequalities have been studied by many authors, the fractional multi-term differential equations have rarely been studied to date [17,18]. Pourhadi and Mursaleen [17] analyzed a Lyapunov-type inequality for a multi-term differential equation involving Caputo fractional derivative subject to mixed boundary conditions: where C a D α denotes the Caputo fractional derivative of order α, 2 < α ≤ 3. The Lyapunovtype inequality for the BVP (1.7) is given as follows.

Theorem 1.7 Let p(t) ∈ C 1 ([a, b]) and q(t) ∈ C([a, b]). If there exists a nontrivial continuous solution of the fractional BVP
On the other hand, in 1908, Langevin proposed the following differential equation in the study of particle Brownian motion: where -ζẋ(t) represents dynamical friction experienced by the particle, x is the displacement and ζ denotes the coefficient of friction, m is the mass of particle, and F(t) is the fluctuating force. The Eq. (1.8) is called the Langevin equation, which is found to be an essential tool to describe the evolution of physical phenomena in fluctuating environments [28]. However, for systems with complex phenomena, it has been realized that the conventional integer Langevin equation does not provide an accurate description of the dynamical systems. Therefore, one way to overcome this disadvantage is to replace the integer derivative by the fractional derivative [29]. This gives rise to fractional Langevin-type equations. In recent years, fractional Langevin-type equations have been studied extensively, and further systematic explorations are still carried out [30][31][32]. For example, Ahmad et al. [31] proposed the investigation of Langevin-type equation involving two fractional orders: In the past decades, in order to meet the research needs, the p-Laplacian equation was introduced into some BVPs [32,33]. In particular, Zhou et al. [32] discussed the following fractional Langevin-type equation with the p-Laplacian operator of the form: The above-mentioned studies indicate that the Langevin-type equations are multi-term differential equations. Since there is no result available in the literature that is concerned with the Lyapunov-type inequalities for fractional Langevin-type equations, the main objective of this study is to bridge the gap and establish Lyapunov-type inequalities for the fractional Langevin-type equations involving Caputo-Hadamard fractional derivative subject to mixed boundary conditions. Precisely, the Lyapunov-type inequalities for the following problems are investigated herein: ([a, b], R). Clearly, there are two special cases of Eqs. (1.9) and (1.10), respectively; one is the p(t) ≡ 0 in Eq. (1.9) and p = 2, u(t)≡0 in Eq. (1.10), and then Eqs. (1.9) and (1.10) degenerate to the sequential fractional BVPs [24][25][26]; the other is the p(t) = u(t) = λ∈R, and then Eqs. (1.9) and (1.10) degenerate to the classical fractional Langevin-type equations (see [30][31][32]).
The remaining part of the paper is organized as follows: In Sect. 2, we recall some definitions on the fractional integral and derivative, and related basic properties which are needed later. In Sect. 3, we transform the problems (1.9) and (1.10) into equivalent integral equations with kernel functions, respectively, and give the properties of kernel functions. In Sect. 4, we present the Lyapunov-type inequalities for problem (1.9) and (1.10), respectively. Finally, we summarize our results and specify new directions for the future works in Sect. 5.

Preliminaries
In this section, we recall some definitions and lemmas about fractional integral and fractional derivative that will be used in the rest of this paper. Let x(t) be a function defined on (a, b) provided that the integral exists.

Green's functions of BVPs (1.9) and (1.10)
In this subsection, we discuss Green's functions of problems (1.9) and (1.10) and present some of their properties.

1)
where kernel functions G 1 (t, s) and G 2 (t, s) are given by Proof Applying the operator H I β a+ to both sides of Eq. (1.9) and using Lemma 2.2, we get for some c 0 ∈ R. From the boundary conditions x(a) = C H D α a+ x(a) = 0, we obtain c 0 = 0, then In view of Lemma 2.1 and Lemma 2.2, a general solution of the fractional Eq. (3.2) is given by Substituting the values c 1 and c 2 in (3.3), we have By direct computation, one can obtain the converse of the lemma. The proof is completed. (3.4) where kernel function G(t, s) and H(s, τ ) are defined by

Lemma 3.2 Let 1 p + 1 q = 1, then x(t)∈C[a, b] is a solution of the BVP (1.10) if and only if x(t) satisfies the integral equation
and . Then BVP (1.10) can be turned into the following coupled BVPs: As in the proof of Lemma 3.1, we see that BVP (3.5) has a unique solution, which is given by and BVP (3.6) has a unique solution, which is given by Substitute (3.7) into (3.8), we see that BVP (1.10) has a unique solution that is given by (3.4). Conversely, by direct computation, it can be established that (3.4) satisfies the problem (1.10). This completes the proof.
satisfies the following property:

Lemma 3.4
The function G 1 (t, s) given by Lemma 3.1 satisfies the following properties: 1 (t, s) is a nonnegative continuous function in [a, b]×[a, b]; Proof (i) Continuity is obvious. We now prove nonnegativity. To this end, we define Clearly, we have As a consequence, we get G 1 (t, s) ≥ 0. Now we show that property (ii) holds. Let v = α + β, then 2 < v < 3. In this way, the function g 11 (t, s) can be rewritten as follows: Differentiating g 11 (t, s) with respect to t for every fixed s ∈ [a, b], we obtain It follows that Easily, we can check that This implies max t∈ [s,b] g 11 (t, s) = g 11 t * s , s Differentiating g(s) on (a, b), we get .
From which we can derive that Hence, for any t, s ∈ [a, b], The lemma is proved.

Lyapunov-type inequalities for BVP (1.9) and (1.10)
In this section, we present the Lyapunov-type inequalities for problems (1.9) and (1.10), respectively. To show this, we define X = C[a, b] as the Banach space endowed with norm Proof According to Lemma 3.1 and Eq. (3.1), if x(t)∈X is a nontrivial solution of the BVP (1.9), then Hence, we derive immediately, This in combination with the Lemma 3.3 and Lemma 3.4 shows that from which the inequality (4.1) follows. Thus, Theorem 4.1 is proved.

Conclusion
In this study, Lyapunov-type inequalities were obtained for the two types of fractional Langevin-type equations in the frame of Caputo-Hadamard fractional derivative. In recent years, the fractional Langevin-type equations and Lyapunov-type inequalities are one of the research hot spots on fractional calculus theory. Therefore, this research is valuable and meaningful. Noteworthy, this is the first article to consider Lyapunov-type inequalities for fractional Langevin-type equations. However, a lot more explorations are still required in the future, such as discussing the Lyapunov-type inequalities for nonlinear fractional Langevin-type equations associated with the anti-periodic boundary conditions or other general boundary conditions.