Lyapunov-type inequality and solution for a fractional differential equation

Inequality (1) is very useful in various problems related to differential equations. Since the appearance of Lyapunov’s fundamental paper [1], many improvements and generalizations of inequality (1) for integer-order (secondand higher-order) BVPs have appeared in the literature; we refer the reader to the summary reference by Tiryaki [2]. Recently, the studies on Lyapunov’s inequality for fractional boundary value problem (FBVP) have begun, in which fractional derivatives (Riemann–Liouville derivative aD v t or


Introduction
In 1907, Lyapunov [1] stated the following outstanding result.
Inequality (1) is very useful in various problems related to differential equations. Since the appearance of Lyapunov's fundamental paper [1], many improvements and generalizations of inequality (1) for integer-order (second-and higher-order) BVPs have appeared in the literature; we refer the reader to the summary reference by Tiryaki [2].
Recently, the studies on Lyapunov's inequality for fractional boundary value problem (FBVP) have begun, in which fractional derivatives (Riemann-Liouville derivative R a D v t or Caputo derivative C a D v t ) are used instead of the classical ordinary derivative. Such a work was initiated by Ferreira [3] in 2013, who obtained a Lyapunov inequality for the following differential equation with Riemann-Liouville fractional derivative: subject to the boundary value condition y(a) = y(b) = 0.
Next, in 2014, Ferreira [4] obtained a Lyapunov inequality for the following differential equation with Caputo fractional derivative: subject to boundary value condition (3). After [3] and [4], many results appeared in the literature; we refer the reader to [5][6][7][8][9][10], where Lyapunov or Lyapunov-type inequalities are obtained for fractional differential equation subject various boundary value conditions such as Inspired by the works mentioned, in this paper, we aim to investigate the Lyapunov-type inequality for the following fractional differential equations: where δ and γ are real numbers, and q(t) ∈ L(0, 1) is not identically zero on any compact subinterval of (0, 1). Furthermore, we obtain the existence of a solution for the corresponding nonlinear problem: BVP (6) was recently studied in [11], but we should point out that only the case of δ > 1 and 0 < γ < 1 was considered in [11]. In this paper, we give a comprehensive discussion on parameters δ and γ .

Preliminaries and lemmas
For convenience, we present some definitions and lemmas from fractional calculus theory in the sense of Riemann-Liouville and Caputo.
, v > 0, be the gamma function. Then the Riemann-Liouville fractional integral of order v for y(t) is defined as By Definitions 2.1 and 2.2 we have

Lemma 2.1 A function u(t) is a solution of the boundary value problem (5) if and only if u(t) satisfies
Proof Let u(t) be a solution of (5). Then By (7) we obtain Considering u(0) = δu(1), we have considering u (0) = γ u (1), we have and thus we get Substituting (10) and (11) into (9), we obtain where G(t, s) is the Green's function:

Main result
Theorem 3.1 Suppose the boundary value problem (5) has a nonzero solution u(t).

Conclusion
In this paper, we study a linear fractional differential equation. Firstly, by obtaining the Green's function we derive a Lyapunov-type inequality for such a boundary value problem. Furthermore, we use the contraction mapping theorem to study the existence of a unique solution for the corresponding nonlinear problem.

Funding
No funding.