Hartman-Wintner-Type Inequality for a Fractional Boundary Value Problem via a Fractional Derivative with respect to Another Function

Copyright © 2017 Mohamed Jleli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider a fractional boundary value problem involving a fractional derivative with respect to a certain function g. A HartmanWintner-type inequality is obtained for such problem. Next, several Lyapunov-type inequalities are deduced for different choices of the function g. Moreover, some applications to eigenvalue problems are presented.


Introduction
In this work, we are concerned with the following fractional boundary value problem: ( , ) ( ) + ( ) ( ) = 0, < < , where ( , ) ∈ R 2 , < , ∈ (1, 2), : [ , ] → R is a continuous function, and , is the fractional derivative operator of order with respect to a certain nondecreasing function ∈ 1 ([ , ]; R) with ( ) > 0, for all ∈ ( , ). A Hartman-Wintner-type inequality is derived for problem (1). As a consequence, several Lyapunov-type inequalities are deduced for different types of fractional derivatives. Next, we end the paper with some applications to eigenvalue problems. Let us start by describing some historical backgrounds about Lyapunov inequality and some related works. In the late XIX century, the mathematician A. M. Lyapunov established the following result (see [1]).
In [8], Hartman and Wintner proved that if boundary value problem (2) has a nontrivial solution, then where Using the fact that 2 Discrete Dynamics in Nature and Society Lyapunov inequality (3) follows immediately from inequality (4). Many other generalizations and extensions of inequality (3) exist in the literature; see, for instance, [7,[13][14][15][16][17][18][19][20][21][22] and references therein. Due to the positive impact of fractional calculus on several applied sciences (see, for instance, [23]), several authors investigated Lyapunov-type inequalities for various classes of fractional boundary value problems. The first work in this direction is due to Ferreira [24], where he considered the fractional boundary value problem where ( , ) ∈ R 2 , < , ∈ (1, 2), : [ , ] → R is a continuous function, and RL is the Riemann-Liouville fractional derivative operator of order . The main result obtained in [24] is the following fractional version of Theorem 1. (7) has a nontrivial solution, then

Theorem 2. If fractional boundary value problem
where Γ is the Gamma function.
In the same paper [37], the authors formulated the following question: How to get the Lyapunov inequality for the following Hadamard fractional boundary value problem: where ( , ) ∈ R 2 , 1 ≤ < , ∈ (1, 2), and : [ , ] → R is a continuous function. Note that one of our obtained results is an answer to the above question.

Preliminaries
Before stating and proving the main results in this work, some preliminaries are needed. Let = [ , ] be a certain interval in R, where ( , ) ∈ R 2 , < . We denote by AC( ; R) the space of real valued and absolutely continuous functions on . For = 1, 2, . . ., we denote by AC ( ; R) the space of real valued functions ( ) which have continuous derivatives up to order − 1 on with ( −1) ∈ AC( ; R); that is, Clearly, we have AC 1 ( ; R) = AC( ; R).
Definition 7 (see [23]). Let ∈ 1 (( , ); R). The fractional integral of order > 0 of with respect to the function is defined by Definition 8 (see [23]). Let > 0 and be the smallest integer greater than or equal to . Let : exists almost everywhere on [ , ]. In this case, the fractional derivative of order of with respect to the function is defined by for a.e. ∈ [ , ].
The following lemma is crucial for the proof of our main result.
In the sequel, we denote by Ξ ([ , ]; R) the functional space defined by We refer the reader to Ferreira [24] for the proofs of the following results.

A Hartman-Wintner-Type Inequality for Boundary Value Problem (1)
In this section, a Hartman-Wintner-type inequality is established for fractional boundary value problem (1).
We have the following result.
Therefore, V is a nontrivial solution of the Riemann-Liouville fractional boundary value problem Now, by Lemma 11, we obtain where is the Green function defined by (27). Next, let us consider the Banach space ([ , ]; R) equipped with the standard norm Clearly, since V is nontrivial, we have ‖V‖ ∞ > 0. Further, using (35) and Lemma 12, we have which yields Therefore, we obtain that is, Using the change of variable = ( ), we get ∫ ( ( ) , ( )) ( ) ( ) ≥ 1.

Lyapunov-Type Inequalities for Different Choices of the Function
In this section, using Theorem 13, several Lyapunov-type inequalities are deduced for different choices of the function .

(47)
In order to compute the value of * ( , ) for ∈ (1, 2) and > 0, we have to study the variations of the function , defined by (45). Observe that with = and = . A simple computation yields for all ∈ ( , ), where = (1 − )/ and = − 1. Next, we put We consider three cases.

Applications to Eigenvalue Problems
Now, we present an application of the Hartman-Wintner-type inequality given by Theorem 13 to eigenvalue problems.
We say that the scalar is an eigenvalue of the fractional boundary value problem Using the change of variable = ( ), we obtain which proves the desired inequality.
Taking ( ) = ln in Corollary 20, we obtain the following result.