Existence and uniqueness of mild solutions for a fractional differential equation under Sturm-Liouville boundary conditions when the data function is of Lipschitzian type

Abstract In this article, we present a sufficient condition about the length of the interval for the existence and uniqueness of mild solutions to a fractional boundary value problem with Sturm-Liouville boundary conditions when the data function is of Lipschitzian type. Moreover, we present an application of our result to the eigenvalues problem and its connection with a Lyapunov-type inequality.


Introduction
In the book [1], the authors presented the following result.
, has a unique continuous mild solution.
In [2], the author studied the same question for the following fractional boundary value problem where < ≤ α 1 2, ∈ B and + D a α denotes the fractional Riemann-Liouville derivative.
The main result appearing in [2] is the following.
: , is continuous and satisfies , then problem (2) has a unique continuous mild solution.
Recently, in [3,4], the authors considered the following fractional boundary value problem , and + D a α c denotes the Caputo fractional derivative, and they obtained a similar result to Theorem 2.
Motivated by the aforementioned papers, we study the existence and uniqueness of mild solutions for the following fractional differential equation with Sturm-Liouville-type boundary conditions , , , , , , 0 and = ( − ) + + > αγ b a αδ βγ Δ 0. Moreover, we apply our result to the eigenvalues problem and present a Lyapunov-type inequality. This kind of problem appears in a great number of papers in the literature (see [5][6][7][8][9] and references therein, among others).
The rest of the article is organized as follows. In Section 2, we recall some basic facts about fractional calculus and present an auxiliary lemma which will be used later. Section 3 contains the main result of the article, and in Section 4 we present some applications of our result.

Background
We start this section presenting some basic concepts about fractional calculus. This material can be found in [10].
, . The Riemann-Liouville fractional integral of order p of f is given by The Caputo fractional derivate of order p is defined as In order to transform problem (4) into an integral equation, we need the following result which appears in Lemma 1 of [11].
to the fractional boundary value problem is given by , as In [11], it is proved that   , , and this implies that for > ε 0 given there exists

Tx t Tx t G t s f s x s s G t s f s x s s G t s G t s f s x s s G t s G t s f s x s
for any ≥ n n 0 . Therefore, for ≥ n n 0 , we have This proves that , is said to be a mild solution to problem (4) if it is a fixed point of the operator T.
: , is continuous and there exists a constant > L 0 such that for any ∈ [ ] t a b , and ∈ x y , . If the condition holds, then the equation = Tx x, where T is the operator defined in Lemma 2, has a unique continuous solution, that is, problem (4) has a unique mild solution.
By our assumption, the Banach contraction mapping theorem says us that the equation , , that is, problem (4) has a unique mild solution. This completes the proof. □

Applications
In this section, we present an application of our result to the eigenvalues problem and a Lyapunov-type inequality. Consider the fractional Sturm-Liouville eigenvalues problem