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Journal of Function Spaces
Volume 2015, Article ID 468536, 5 pages
http://dx.doi.org/10.1155/2015/468536
Research Article

A Lyapunov-Type Inequality for a Fractional Differential Equation under a Robin Boundary Condition

1Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2Department of Mathematics, College of Computing and Information Systems, Al Yamamah University, Riyadh, Saudi Arabia

Received 13 July 2014; Accepted 31 August 2014

Academic Editor: Calogero Vetro

Copyright © 2015 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.

Abstract

We establish a new Lyapunov-type inequality for a class of fractional differential equations under Robin boundary conditions. The obtained inequality is used to obtain an interval where a linear combination of certain Mittag-Leffler functions has no real zeros.

1. Introduction and Preliminaries

Let the following be a continuous function. The well-known Lyapunov inequality [1] states that a necessary condition for the boundary value problem to have nontrivial solutions is thatThis result found many practical applications in differential and difference equations (oscillation theory, disconjugacy, eigenvalue problems, etc.); see, for instance, [27]. On the other hand, many improvements of (2) have been carried out, and similar inequalities have been obtained for other types of differential equations; compare to the Pachpatte monograph [8]. The search for Lyapunov-type inequalities in which the starting differential equation is constructed via fractional differential operators has begun very recently. For example, in [9], a Lyapunov-type inequality was obtained for differential equations depending on the Riemann-Liouville fractional derivative; that is, for the boundary value problemwhere denotes the Riemann-Liouville fractional derivative of order . Precisely, the author proved that if (3) has a nontrivial solution, then we haveClearly, if we let in (4), one obtains Lyapunov’s classical inequality (2).

In [10], a Lyapunov-type inequality was obtained for the Caputo fractional boundary value problemwhere is the Caputo fractional derivative of order . It was proved that if (5) has a nontrivial solution, then we haveSimilarly, if we let in (6), one obtains Lyapunov’s classical inequality (2).

Motivated by the above works, we consider in this paper a Caputo fractional differential equation under Robin boundary conditions. More precisely, we consider the boundary value problemand we get a corresponding Lyapunov-type inequality. This result is then used to obtain a real interval where a linear combination of certain Mittag-Leffler functions has no (real) zeros.

Before presenting the main results, let us start by recalling the concepts of the Riemann-Liouville fractional integral and the Caputo fractional derivative of order . For more details, we refer to [11].

Definition 1. Let and let be a real function defined on . The Riemann-Liouville fractional integral of order is defined by and

Definition 2. The Caputo fractional derivative of order is defined by and for , where is the smallest integer greater than or equal to .

The following result is standard within the fractional calculus theory involving the Caputo differential operator (see [7]).

Lemma 3. One has that is a solution to (7) if and only if where and are some real constants.

Now, we are ready to state and prove our main results.

2. Main Results

2.1. A Lyapunov-Type Inequality

At first, we consider the following notations:

Now, let us write problem (7) in its equivalent integral form.

Lemma 4. One has that is a solution to (7) if and only if satisfies the integral equation where

Proof. From Lemma 3, we havewhere and are some real constants. Then, Since , we obtainFrom the boundary condition , we getUsing (14), (16), and (17), we obtain the desired result.

Lemma 5. For all , one has

Proof. It is easy to see that, for , we have On the other hand, for , we haveHence,Now, for fixed in , we want to study the variation of the function for in . First, we have Let We distinguish two eventual cases according to the value of .

Case  1. If , in this case, we have

From (20), (21), and (24), we deduce

This yields

Observe that, in this case, we have

Using the above inequality, we obtain which implies that

From (26) and (29), we deduce

Case  2. If , in this case, we have two possibilities.(i) If , in this case, we have Therefore, we conclude thatOn the other hand, we havewhich implies thatFrom (32) and (34), we deduce(ii) If , in this case, we have Hence, there would exist such thatAs mentioned above, it is easy to verify that and . This yields Then,Observe that is equivalent to Therefore, we getFinally, using the above inequality and (39), we obtainwhich makes end to the proof.

Our first main result is as follows.

Theorem 6. If (7) admits a nontrivial continuous solution, then

Proof. Let be the Banach space endowed with norm From Lemma 4, for all , we have Now, an application of Lemma 3 yields from which the above theorem follows.

2.2. Nonexistence Result of Real Zeros for a Linear Combination of Certain Mittag-Leffler Functions

Let be fixed. The complex functionis analytic in the whole complex plane; it will be referred to [1214] as the Mittag-Leffler function with parameters .

At this stage, using the above Lyapunov-type inequality, we give an interval where a linear combination of Mittag-Leffler functions has no real zeros. More precisely, we prove

Theorem 7. Let . Then has no real zeros for

Proof. Let , and consider the fractional Sturm-Liouville eigenvalue problemThe real values of , for which there exists a non-trivial solution to (49), are called eigenvalues of (49); and the corresponding solutions are called eigenfunctions.
As established in [15], the eigenvalues of (49) must be positive; moreover, the positive number is an eigenvalue of (49) if and only if Thanks to Theorem 6, if a positive real eigenvalue of (49) exists, then Hence,which concludes the proof.

Conflict of Interests

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

Authors’ Contribution

All the authors contributed equally and significantly to writing this paper. All authors read and approved the final paper.

Acknowledgment

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project no. RG-1435-034.

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