Abstract
In this paper, we establish new Lyapunov-type inequalities for a class of fractional boundary value problems. As an application, we obtain a lower bound for the eigenvalues of corresponding equations.
Similar content being viewed by others
1 Introduction
Let u be a nontrivial solution to the second order differential equation
with the Dirichlet boundary condition
where \(q:[a,b]\to\mathbb{R}\) is continuous. Then the so-called Lyapunov inequality [1]
holds, and constant 4 in (1.3) cannot be replaced by a larger number. The above inequality has several applications to various problems related to differential equations.
There are several generalizations and extensions of Lyapunov’s result. Hartman and Wintner [2] proved that if u is a nontrivial solution to (1.1)-(1.2), then
where \(q^{+}(s)\) is the positive part of q, defined as
For other generalizations and extensions of the classical Lyapunov’s inequality, we refer to [2–17] and the references therein.
Recently, some Lyapunov-type inequalities for fractional boundary value problems have been obtained. In [9], Ferreira established a Lyapunov-type inequality for a differential equation that depends on the Riemann-Liouville fractional derivative, i.e., for the boundary value problem
where he proved that if u is a nontrivial continuous solution to the above problem, then
In [8], Ferreira obtained a Lyapunov-type inequality for the Caputo fractional boundary value problem
where he established that if u is a nontrivial continuous solution to the above problem, then
Observe that if we set \(\alpha=2\) in (1.4) or (1.5), one can obtain the classical Lyapunov inequality (1.3). In [11], Jleli and Samet studied the fractional differential equation
with mixed boundary conditions
or
For boundary conditions (1.6) and (1.7), two Lyapunov-type inequalities were established respectively as follows:
and
Rong and Bai [16] established a Lyapunov-type inequality for the above fractional differential equation with the fractional boundary conditions
where \(0<\beta\leq1\) and \(1<\alpha\leq\beta+1\). They established the following result: if a nontrivial continuous solution to the above fractional boundary value problem exists, then
Observe that if \(\beta=1\), then (1.9) reduces to the Lyapunov-type inequality (1.8). For other related works, we refer to [18–21].
In all the above cited works, the fractional order α belongs to \((1.2]\). In this paper, we are concerned with the problem of finding new Lyapunov-type inequalities for the fractional boundary value problem
where \({}_{a}D^{\alpha}\) is the standard Riemann-Liouville fractional derivative of fractional order α and \(q:[a,b]\to\mathbb{R}\) is a continuous function. As an application, we obtain a lower bound for the eigenvalues of the corresponding problem.
Let f be a real function defined on \([a,b]\) (\(a< b\)).
Definition 1.1
The integral
where \(\alpha>0\), is called the Riemann-Liouville fractional integral of order α, and \(\Gamma(\alpha)\) is the Euler gamma function defined by
Definition 1.2
The expression
where \(n=[\alpha]+1\), \([\alpha]\) denotes the integer part of number α, is called the Riemann-Liouville fractional derivative of order α.
The following lemma is crucial in finding an integral representation of the fractional boundary value problem (1.10)-(1.11).
Lemma 1.3
Assume that \(f\in C(a,b)\cap L(a,b)\) with a fractional derivative of order \(\alpha>0\) that belongs to \(C(a,b)\cap L(a,b)\). Then
for some constants \(c_{i}\in\mathbb{R}\), \(i=1,\ldots,n\), \(n=[\alpha]+1\).
For more details on fractional calculus, we refer the reader to [22–24].
2 Main results
The following lemmas will be needed.
Lemma 2.1
We have that \(u\in C[a,b]\) is a solution to the boundary value problem (1.10)-(1.11) if and only if u satisfies the integral equation
where \(G(t,s)\) is the Green function of problem (1.10)-(1.11) defined as
Proof
From Lemma 1.3, \(u\in C[a,b]\) is a solution to the boundary value problem (1.10)-(1.11) if and only if
for some real constants \(c_{i}\), \(i=1,\ldots,4\). Using the boundary conditions \(u(a)=u'(a)=u''(a)=0\), we get immediately
The boundary condition \(u''(b)=0\) yields
Hence
which concludes the proof. □
Lemma 2.2
The function G defined in Lemma 2.1 satisfies the following property:
Proof
We start by fixing an arbitrary \(s\in(a,b]\). Differentiating \(G(t,s)\) with respect to t, we get
For \(a\leq t\leq s\leq b\), we have
while for \(a\leq s\leq t\leq b\), we have
Consequently, the function \(G(t,s)\) is non-decreasing with respect to t, from which it follows that
The proof is complete. □
We have the following Hartman-Wintner-type inequality.
Theorem 2.3
If a nontrivial continuous solution to the fractional boundary value problem
exists, where q is a real and continuous function in \([a,b]\), then
Proof
Let \(\mathcal{B}=C[a,b]\) be the Banach space endowed with the norm
It follows from Lemma 2.1 that a solution u to (1.10)-(1.11) satisfies the integral equation
Thus, for all \(t\in[a,b]\), we have
which yields
Since u is nontrivial, then \(\Vert u\Vert _{\infty}\neq0\), so
Now, an application of Lemma 2.2 yields
from which the inequality in (2.1) follows. □
Corollary 2.4
If a nontrivial continuous solution to the fractional boundary value problem
exists, where q is a real and continuous function in \([a,b]\), then
Proof
From Theorem 2.3, we have
Next we note
Thus we get
which gives the desired inequality (2.2). □
We have the following Lyapunov-type inequality.
Corollary 2.5
If a nontrivial continuous solution to the fractional boundary value problem
exists, where q is a real and continuous function in \([a,b]\), then
Proof
Let
Now, we differentiate \(\psi(s)\) on \((a,b)\), and we obtain after simplifications
Observe that \(\psi'(s)\) has a unique zero, attained at the point
It is easily seen that \(s^{*}\in(a,b)\), \(\psi'(s)>0\) on \((a,s^{*})\), and \(\psi'(s)<0\) on \((s^{*},b)\). We conclude that
From Corollary 2.4, we have
which yields
from which inequality (2.3) follows. □
Corollary 2.6
If a nontrivial continuous solution to the boundary value problem
exists, where q is a real and continuous function in \([a,b]\), then
Proof
Inequality (2.4) follows from Theorem 2.3 with \(\alpha=4\). □
Corollary 2.7
If a nontrivial continuous solution to the boundary value problem
exists, where q is a real and continuous function in \([a,b]\), then
Proof
Inequality (2.5) follows from Corollary 2.4 with \(\alpha=4\). □
Corollary 2.8
If a nontrivial continuous solution to the boundary value problem
exists, where q is a real and continuous function in \([a,b]\), then
Proof
Inequality (2.6) follows from Corollary 2.5 with \(\alpha=4\). □
3 Application
In this section, we give an application of the Hartman-Wintner-type inequality (2.2) for the eigenvalue problem
Theorem 3.1
If λ is an eigenvalue to the fractional boundary value problem (3.1)-(3.2), then
where B is the beta function defined by
Proof
Let λ be an eigenvalue to (3.1)-(3.2). Then there exists \(u=u_{\lambda}\), a nontrivial solution to (3.1)-(3.2). An application of Corollary 2.4 yields
Now,
from which we obtain
The proof is complete. □
References
Lyapunov, AM: Problème général de la stabilité du mouvement. Ann. of Math. Stud., vol. 17. Princeton University Press, Princeton (1949)
Hartman, P, Wintner, A: On an oscillation criterion of Liapunov. Am. J. Math. 73, 885-890 (1951)
Aktas, MF: Lyapunov-type inequalities for a certain class of n-dimensional quasilinear systems. Electron. J. Differ. Equ. 2013, 67 (2013)
Çakmak, D: On Lyapunov-type inequality for a class of nonlinear systems. Math. Inequal. Appl. 16, 101-108 (2013)
Çakmak, D: Lyapunov-type integral inequalities for certain higher order differential equations. Appl. Math. Comput. 216, 368-373 (2010)
Cheng, SS: A discrete analogue of the inequality of Lyapunov. Hokkaido Math. J. 12, 105-112 (1983)
Eliason, SB: A Lyapunov inequality for a certain nonlinear differential equation. J. Lond. Math. Soc. 2, 461-466 (1970)
Ferreira, RAC: A Lyapunov-type inequality for a fractional boundary value problem. Fract. Calc. Appl. Anal. 16(4), 978-984 (2013)
Ferreira, RAC: On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function. J. Math. Anal. Appl. 412, 1058-1063 (2014)
He, X, Tang, XH: Lyapunov-type inequalities for even order differential equations. Commun. Pure Appl. Anal. 11, 465-473 (2012)
Jleli, M, Samet, B: Lyapunov-type inequalities for a fractional differential equation with mixed boundary conditions. Math. Inequal. Appl. 18(2), 443-451 (2015)
Kwong, MK: On Lyapunov’s inequality for disfocality. J. Math. Anal. Appl. 83, 486-494 (1981)
Pachpatte, BG: On Lyapunov-type inequalities for certain higher order differential equations. J. Math. Anal. Appl. 195, 527-536 (1995)
Pachpatte, BG: Lyapunov type integral inequalities for certain differential equations. Georgian Math. J. 4, 139-148 (1997)
Parhi, N, Panigrahi, S: On Lyapunov-type inequality for third-order differential equations. J. Math. Anal. Appl. 233, 445-460 (1999)
Rong, J, Bai, C: Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions. Adv. Differ. Equ. 2015, 82 (2015)
Yang, X: On Lyapunov-type inequality for certain higher-order differential equations. Appl. Math. Comput. 134, 307-317 (2003)
Baleanu, D, Agarwal, P: Certain inequalities involving the fractional q-integral operators. Abstr. Appl. Anal. 2014, Article ID 371274 (2014)
Choi, J, Agarwal, P: Certain fractional integral inequalities involving hypergeometric operators. East Asian Math. J. 30, 283-291 (2014)
Wang, G, Agarwal, P, Baleanu, D: Certain new Gruss type inequalities involving Saigo fractional q-integral operator. J. Comput. Anal. Appl. 19(5), 862-873 (2015)
Wang, G, Agarwal, P, Chand, M: Certain Gruss type inequalities involving the generalized fractional integral operator. J. Inequal. Appl. 2014, 147 (2014)
Agarwal, RP, O’Regan, D, Stanek, S: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371(1), 57-68 (2010)
Hernández, E, O’Regan, D, Balachandran, K: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. 73(10), 3462-3471 (2010)
Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
Acknowledgements
The second author would like to extend his sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the International Research Group Project No. IRG14-04.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in drafting this manuscript and giving the main proofs. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
O’Regan, D., Samet, B. Lyapunov-type inequalities for a class of fractional differential equations. J Inequal Appl 2015, 247 (2015). https://doi.org/10.1186/s13660-015-0769-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-015-0769-2