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, [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 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
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 functionis 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 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.