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Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions
Advances in Difference Equations volume 2015, Article number: 82 (2015)
Abstract
In this paper, a Lyapunov-type inequality is obtained for a fractional differential equation under fractional boundary conditions. We then use this inequality to obtain an interval where a certain Mittag-Leffler function has no real zeros.
1 Introduction
The Lyapunov inequality [1] has proved to be very useful in various problems related with differential equations; for example, see [2, 3] and the references therein. The Lyapunov inequality states that a necessary condition for the boundary value problem
to have nontrivial solutions is that
where \(q:[ a , b ] \to R\) is a continuous function, and the zeros a and b of every solution \(y(t)\) are consecutive. Since then, many generalizations of the Lyapunov inequality have appeared in the literature (see [4–9] and the references therein).
Recently, the research of Lyapunov-type inequalities for fractional boundary value problem has begun. In [10], Ferreira investigated a Lyapunov-type inequality for the Caputo fractional boundary value problem
where \({}^{\mathrm{C}}_{a} D^{\alpha}\) is the Caputo fractional derivative of order α, \(1 < \alpha\le2\), the zeros a and b are consecutive, and q is a real and continuous function. It was proved that if (1.3) has a nontrivial solution, then
Obviously, if we set \(\alpha= 2\) in (1.4), one can obtain the Lyapunov classical inequality (1.2). In [11], the same author studied a differential equation that depends on the Riemann-Liouville fractional derivative and gave a Lyapunov-type inequality. In both works [10, 11], some interesting applications to the localization of real zeros of certain Mittag-Leffler functions were presented.
In [12], Jleli and Samet considered the fractional differential equation
with the mixed boundary conditions
or
For boundary conditions (1.5) and (1.6), two Lyapunov-type inequalities were established respectively as follows:
and
Motivated by the above works, we consider in this paper a Caputo fractional differential equation under boundary condition involving the Caputo fractional derivative. More precisely, we consider the boundary value problem
where \(1 < \alpha\leq2\), \(0 < \beta\le1\), and \(q:[a , b]\rightarrow R\) is a continuous function. We write (1.9) as an equivalent integral equation and then, by using some properties of its Green function, we are able to get a corresponding Lyapunov-type inequality. After that, we show that this inequality can be used to obtain a real interval where a certain Mittag-Leffler function has no real zeros. Our results generalize the main results of Jleli and Samet [12].
2 Preliminaries
In this section, we introduce the definitions of the Riemann-Liouville fractional integral and the Caputo fractional derivative and give some lemmas which are used in this article.
Definition 2.1
Let \(\alpha\geq0\) and f be a real function defined on \([a, b]\). The Riemann-Liouville fractional integral of order α is defined by \({}_{a} I^{0} f \equiv f\) and
Definition 2.2
The Caputo derivative of fractional order \(\alpha\ge0\) is defined by \({}_{a}^{\mathrm{C}} D^{0} f \equiv f\) and
where n is the smallest integer greater or equal to α.
The following results are standard within the fractional calculus theory involving the Caputo differential operator.
Lemma 2.1
([13], Chapter 2)
Let \(\gamma> \alpha> 0\), \(f\in C[a , b]\), then
Lemma 2.2
([14], Section 2)
Let \(y \in C[a , b]\) and \(1 < \alpha\le2\), then
for some real constants \(c_{0}\) and \(c_{1}\).
3 Main results
We begin by writing problem (1.9) in its equivalent integral form.
Lemma 3.1
\(y \in C[a, b]\) is a solution of the boundary value problem (1.9) if and only if y satisfies the integral equation
where
and
Proof
From (1.9) and Lemma 2.2, we obtain
where \(c_{0}\) and \(c_{1}\) are some real constants. By the boundary condition \(y(a)=0\), we can obtain that \(c_{0}=0\). Thus, we have
By (3.2), we get
Since \({}_{a}^{\mathrm{C}} D^{\beta} y(b)=0\), we have from (3.3) that
Substitute (3.4) into (3.2), we obtain
which concludes the proof. □
Lemma 3.2
If \(1 < \alpha< 2\) and \(0 < \beta < 1\), then
Proof
Consider the logarithmic derivative of the gamma function
We have by [15], p.264, that
where γ is an Euler constant. From (3.7) we obtain
Since \(\alpha< 2\), we get by (3.6) and (3.8) that \(\psi(\alpha- x) < \psi(2 - x)\), that is,
Let
Then we have by (3.9) that
Thus, \(f(0) < f(\beta) < f(1)\) (\(0 < \beta< 1\)), which implies that (3.5) holds. □
Lemma 3.3
Assume that \(0 < \beta\le1\) and \(1 < \alpha\le1+\beta\) hold. Then
Proof
Throughout the proof we consider \(\beta< 1\) since when \(\beta= 1\) our study is reduced to the case in [12]. For \(a \le t \le s \le b\), we easily know that
For convenience, let
Fixing arbitrary \(s \in[a, b)\) and differentiating \(\psi(t, s)\) with respect to t, we obtain
From (3.11) we easily know that \(\psi_{t}(t^{*}, s) = 0\) if and only if
provided \(t^{*} \le b\), i.e., as long as \(s \le b - l\), where
Hence, if \(s > b - l\), then
On the other hand, we have
Thus, we obtain
and
by \((s-a)^{\beta}+(b-s)^{\beta} \ge(b-a)^{\beta}\) and (3.5). Thus, we have by (3.13)-(3.15) that
It remains to verify the result when \(s \le b-l\), i.e., when \(t^{*} \le b\). It is easy to check that
Hence, we have
By (3.12) we have
Obviously, we have
and we obtain from Lemma 3.2 and condition \(\alpha - \beta- 1 \le0\) that
Thus, from (3.18)-(3.20) we conclude that
holds. From (3.14), (3.15), (3.21) and (3.17), we know that inequality (3.10) is true. The proof is complete. □
Theorem 3.4
Let \(0 < \beta\le1\) and \(1 < \alpha\le1+\beta\). If a nontrivial continuous solution of the fractional boundary value problem (1.9) exists, then
Proof
Let \(B = C[a, b]\) be the Banach space endowed with the norm \(\|y\|_{\infty} = \sup_{t \in[a, b]} |y(t)|\). According to Lemma 3.1, the solution of (1.9) can be written as
Now, an application of Lemma 3.3 yields
which implies that (3.22) holds. □
Remark 3.1
If \(\beta= 1\), then (3.22) reduces to the following Lyapunov-type inequality [12]:
Remark 3.2
If \(\alpha= 2\) and \(0 < \beta< 1\), then we have by Lemma 3.1 that
Similar to the proof of Theorem 3.4, it is easy to obtain that the following Lyapunov-type inequality holds:
In the following, we will use Lyapunov-type inequalities (3.22) to obtain intervals where certain Mittag-Leffler functions have no real zeros. Let \(z \in\mathbb{R}\) and consider the real zeros of the Mittag-Leffler functions \(E_{\alpha, \gamma}(z)\), where
Obviously, \(E_{\alpha, \gamma}(z)>0\) for all \(z \ge 0\). Hence, the real zeros of \(E_{\alpha, \gamma}(z)\), if they exist, must be negative real numbers.
Theorem 3.5
Assume that \(0 < \beta\le1\) and \(1 < \alpha\le1+\beta\) hold. Then the Mittag-Leffler function \(E_{\alpha, 2-\beta}(x)\) has no real zeros for
Proof
Let \(a=0\) and \(b=1\). Consider the following fractional Sturm-Liouville eigenvalue problem:
By the Laplace transform method as in [13, 16, 17], the general solution of the fractional differential equation (3.23) can be given as follows:
In the following discussion we will use the general solution (3.25) and its fractional Caputo derivative
By (3.25), (3.26) and the boundary conditions (3.24), we obtain that
Thus, the eigenvalues \(\lambda\in\mathbb{R}\) of (3.23) and (3.24) are the solutions of
and the corresponding eigenfunctions are given by
By Theorem 3.4, if a real eigenvalue λ of (3.23) and (3.24) exists, i.e., −λ is a zero of (3.27), then
that is,
which concludes the proof. □
Remark 3.3
If \(\beta= 1\), then Theorem 3.5 reduces to Theorem 3 in [12].
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Acknowledgements
The authors thank the referees for their important and valuable comments. This work is supported by the Natural Science Foundation of Jiangsu Province (BK2011407) and Natural Science Foundation of China (11271364 and 10771212).
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Rong, J., Bai, C. Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions. Adv Differ Equ 2015, 82 (2015). https://doi.org/10.1186/s13662-015-0430-x
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DOI: https://doi.org/10.1186/s13662-015-0430-x
MSC
- 34A08
- 34A40
- 26D10
- 33E12
Keywords
- Lyapunov inequality
- Caputo fractional derivative
- Mittag-Leffler function