Lyapunov inequality for a Caputo fractional differential equation with Riemann–Stieltjes integral boundary conditions

In this a Lyapunov‐type inequality is obtained for the fractional differential equation with Caputo derivative  CDaγx(t)+q(t)x(t)=0,a


INTRODUCTION
In 1907, Lyapunov [1] proved that if x is a nontrivial solution of boundary value problem: where q ∶ [a, b] → R is a continuous function, then the inequality b ∫ a |q(s)|ds > 4 b − a (1.2) holds, which attracted the attentions of many researchers. It has been more than hundred years, many generalizations of the Lyapunov inequalities have been obtained in the literature considering its application in eigenvalue problems, stability theory, oscillation theory, and estimation for intervals of disconjugacy. In last few decades with emergence of fractional differential equation, many improvements and generalizations of the Lyapunov inequality (1.2) have been obtained. Let us describe some previous works that exist in the literature in the direction of inequalities. In [2], Ferreira considered the Riemann-Liouville fractional boundary value problem holds. In [3], Domoshnitsky et al. studied the fractional functional differential equation with the boundary condition Cabera et al. [4] obtained some Lyapunov-type inequalities and lower bound for the eigenvalues of the fractional two-point boundary value problem: In [5], Bohner et al. generalized the boundary condition of [4] as follows: and obtained inequalities for fractional functional differential equation on the basis of Vallée-Poussin theorem. In [6], Aibout et al. studied the existence and uniqueness of solutions for a coupled system of Caputo-Hadamard fractional differential equations of the form with the multipoint boundary conditions , are given continuous function, R m for m ∈ N. In [7], Ferreria considered the problem: where q ∶ [a, b] → R is a continuous function. Ferreria [7] proved that if x is a nontrivial solution of (1.
holds. Inspired by [7], Rong and Bai [8] obtained the Lyapunov- for the boundary value problem x(a) = 0, where 1 < ≤ 1 + . Inequalities with Hadamard and Katugampola fractional derivatives can be found in [9,10]. Various inequalities involving functions, their integrals, and derivatives can be found in the book [11]. One can refer to the monograph by Agarwal et al. [12] for detailed study of Lyapunov inequalities. Motivated by above mentioned work, we consider the problem where We shall use the notations  [1]) for our use in the sequel.
This work has been divided into three section. Section 1 contains the introduction related to this present work. Section 2 deals with the necessary preliminary results that help us in deriving the Greens function for the problem (1.5)-(1.6). In Section 3, we obtain the upper estimates on the Green's function which allows us to obtain a Lyapunov-type inequality for our problem (1.5)-(1.6).

PRELIMINARIES
Let us introduce several definitions and lemmas.

Corollary 1. If x is a nontrivial solution of (3.18), and q is continuous, then
Thus, in particular, if we assume = 2, then we obtain the celebrated inequality (1.2) for the problem (1.1).