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.
Let the following
In [
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
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
Let
The Caputo fractional derivative of order
The following result is standard within the fractional calculus theory involving the Caputo differential operator (see [
One has that
Now, we are ready to state and prove our main results.
At first, we consider the following notations:
Now, let us write problem (
One has that
From Lemma
For all
It is easy to see that, for
From (
This yields
Observe that, in this case, we have
Using the above inequality, we obtain
From (
(i) If Therefore, we conclude that On the other hand, we have which implies that From ( (ii) If Hence, there would exist As mentioned above, it is easy to verify that Then, Observe that Therefore, we get Finally, using the above inequality and ( which makes end to the proof.
Our first main result is as follows.
If (
Let
Let
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
Let
Let
The real values of
As established in [
The authors declare that there is no conflict of interests regarding the publication of this paper.
All the authors contributed equally and significantly to writing this paper. All authors read and approved the final paper.
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.