Journal of Function Spaces
Volume 2015 (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

Mohamed Jleli,¹ Lakhdar Ragoub,² and Bessem Samet¹

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 that

This result found many practical applications in differential and difference equations (oscillation theory, disconjugacy, eigenvalue problems, etc.); see, for instance, [2–7]. 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 problem

where denotes the Riemann-Liouville fractional derivative of order . Precisely, the author proved that if (3) has a nontrivial solution, then we have

Clearly, 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 problem

where is the Caputo fractional derivative of order . It was proved that if (5) has a nontrivial solution, then we have

Similarly, 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 problem

and 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 have

where and are some real constants. Then,

Since , we obtain

From the boundary condition , we get

Using (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 have

Hence,

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 that

 On the other hand, we have

 which implies that

 From (32) and (34), we deduce

 (ii) If , in this case, we have

 Hence, there would exist such that

 As mentioned above, it is easy to verify that and . This yields

 Then,

 Observe that is equivalent to

 Therefore, we get

 Finally, using the above inequality and (39), we obtain

 which 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 function

is analytic in the whole complex plane; it will be referred to [12–14] 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 problem

The 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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