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

Abstract

We consider a fractional boundary value problem involving a fractional derivative with respect to a certain function . A Hartman-Wintner-type inequality is obtained for such problem. Next, several Lyapunov-type inequalities are deduced for different choices of the function . Moreover, some applications to eigenvalue problems are presented.

1. Introduction

In this work, we are concerned with the following fractional boundary value problem:where , , , is a continuous function, and is the fractional derivative operator of order with respect to a certain nondecreasing function with , 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]).

Theorem 1. If the boundary value problemhas a nontrivial solution, where is a continuous function, then

Inequality (3) is known as Lyapunov inequality. It is proved to be very useful in various problems in connection with differential equations, including oscillation theory, asymptotic theory, eigenvalue problems, and disconjugacy. For more details, we refer the reader to [2–12] and references therein.

In [8], Hartman and Wintner proved that if boundary value problem (2) has a nontrivial solution, thenwhereUsing the fact thatLyapunov inequality (3) follows immediately from inequality (4). Many other generalizations and extensions of inequality (3) exist in the literature; see, for instance, [7, 13–22] 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 problemwhere , , , is a continuous function, and is the Riemann-Liouville fractional derivative operator of order . The main result obtained in [24] is the following fractional version of Theorem 1.

Theorem 2. If fractional boundary value problem (7) has a nontrivial solution, thenwhere is the Gamma function.

Observe that (3) can be deduced from Theorem 2 by passing to the limit as in (8). For other related works, we refer the reader to Ferreira [25, 26], Jleli and Samet [27, 28], Jleli et al. [29, 30], O’Regan and Samet [31], Al Arifi et al. [32], Rong and Bai [33], Chidouh and Torres [34], Agarwal and Özbekler [35], Ma [36], and the references therein.

Very recently, Ma et al. [37] investigated the fractional boundary value problemwhere , is a continuous function, and is the Hadamard fractional derivative operator of order . The main result in [37] is the following.

Theorem 3. If fractional boundary value problem (9) has a nontrivial solution, thenwhere .

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 , , , and is a continuous function. Note that one of our obtained results is an answer to the above question.

2. Preliminaries

Before stating and proving the main results in this work, some preliminaries are needed.

Let be a certain interval in , where , . We denote by the space of real valued and absolutely continuous functions on . For , we denote by the space of real valued functions which have continuous derivatives up to order on with ; that is,Clearly, we have .

Definition 4 (see [23]). Let . The Riemann-Liouville fractional integral of order of is defined by

Definition 5 (see [23]). Let and be the smallest integer greater than or equal to . Let be a function such that . Then the Riemann-Liouville fractional derivative of order of a function is defined byfor a.e. .

Let and be the smallest integer greater than or equal to . By (see [38]), one denotes the set of all functions that have the representation:where and .

Lemma 6 (see [38]). Let , be the smallest integer greater than or equal to , and . Then exists almost everywhere on if and only if ; that is, has representation (15). In such a case, one has

Let be a nondecreasing function with , for all .

Definition 7 (see [23]). Let . The fractional integral of order of with respect to the function is defined by

Definition 8 (see [23]). Let and be the smallest integer greater than or equal to . Let be a function such that exists almost everywhere on . In this case, the fractional derivative of order of with respect to the function is defined byfor a.e. .

The following lemma is crucial for the proof of our main result.

Lemma 9. Let and be the smallest integer greater than or equal to . Suppose that the function belongs to the space . Then

Proof. At first, observe that, from Lemma 6, exists for a.e. . Now, using the change of variable , , the chain rule yields Therefore, we obtain Next, using the change of variable , we obtain which proves the desired result.

In the sequel, we denote by the functional space defined by

Definition 10 (see [23]). Let and be the smallest integer greater than or equal to . Let , where and . The Hadamard fractional derivative of order of is defined by

We refer the reader to Ferreira [24] for the proofs of the following results.

Lemma 11. Let , , , and . Then is a solution of the boundary value problemif, and only if, satisfies the integral equationwhere

Lemma 12. The Green function defined by (27) satisfies the following properties:(i) for all , .(ii)For all , one has

3. 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).

Problem (1) is investigated under the following assumptions:(A1).(A2).(A3).(A4) is a nondecreasing function with , for all .

We have the following result.

Theorem 13. Under assumptions (A1)–(A4), if fractional boundary value problem (1) has a nontrivial solution , then

Proof. Suppose that is a nontrivial solution of (1). Let us define the function byUsing Lemma 9, for all , we haveOn the other hand, since is a solution of (1), we haveTherefore, is a nontrivial solution of the Riemann-Liouville fractional boundary value problemwhere , , and is the function defined byNow, by Lemma 11, we obtainwhere is the Green function defined by (27). Next, let us consider the Banach space equipped with the standard normClearly, since is nontrivial, we have . Further, using (35) and Lemma 12, we havewhich yieldsTherefore, we obtainthat is,Using the change of variable , we getNote that by (27) we haveTherefore,which is desired inequality (29).

4. 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 .

4.1. The Case ,

Taking , , in Theorem 13, we deduce the following Hartman-Wintner-type inequality.

Corollary 14. If fractional boundary value problem (1) has a nontrivial solution , where , , , then

Next, let us define the function bySince is continuous on and , there exists some such thatTherefore, from inequality (44), we obtain the following Lyapunov-type inequality.

Corollary 15. If fractional boundary value problem (1) has a nontrivial solution , where , , , then

In order to compute the value of for and , we have to study the variations of the function defined by (45). Observe thatwhere is the function defined bywith and . A simple computation yieldsfor all , where and . Next, we putWe consider three cases.

Case 1 (if ). In this case, we have and if and only if . Moreover, we have for and for . Therefore,Thus, in this case we obtainNext, using (53), we deduce from Corollary 15 the following Lyapunov-type inequality in the case .

Corollary 16 (the case ). If fractional boundary value problem (1) has a nontrivial solution , where , , , then

Case 2 (if ). In this case, we have and has two distinct zeros atwhereIt can be easily seen thatMoreover, we have for and for . Therefore,

Case 3 (if ). In this case, we have and has two distinct zeros at and . It can be easily seen thatMoreover, we have for and for . Therefore,

Observe that, for (), problem (1) is equivalent to problem (7). Moreover, in this case we haveTherefore, using (61) and Corollary 15, we obtain inequality (8), which is due to Ferreira [24].

4.2. A Lyapunov-Type Inequality via Hadamard Fractional Derivative

Taking in Theorem 13, we deduce the following Hartman-Wintner-type inequality for the following Hadamard fractional boundary value problem:where , , , and is a continuous function.

Corollary 17. If fractional boundary value problem (62) has a nontrivial solution , where , , then

Now, define the function byObserve thatwhere is the function defined bywith and . A simple computation yieldswhereObserve that has two distinct zeros atIt can be easily seen thatMoreover, we have for and for . Therefore, we deduce thatNext, combining (63) with (72), we obtain the following Lyapunov-type inequality for fractional boundary value problem (62).

Corollary 18. If fractional boundary value problem (62) has a nontrivial solution , where , , thenwhere

Observe that, in the particular case , inequality (73) reduces to inequality (10) which is due to Ma et al. [37].

Remark 19. Corollary 18 is an answer to the open problem proposed in [37].

5. 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 problemwhere , , , and with , for all , if problem (75) has at least a nontrivial solution .

We have the following result which provides a lower bound of the eigenvalues of problem (75).

Corollary 20. If is an eigenvalue of problem (75), thenwhere and .

Proof. Suppose that is an eigenvalue of problem (75). Then problem (75) admits a nontrivial solution. Applying Theorem 13 with , we obtainUsing the change of variable , we obtainwhich proves the desired inequality.

Taking , , in Corollary 20, we obtain the following result.

Corollary 21. If is an eigenvalue of problem (75) with , , , , then

Taking in Corollary 20, we obtain the following result.

Corollary 22. If is an eigenvalue of the Hadamard fractional boundary value problemwhere , , and , then

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The third author extends his appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).