Abstract
In this paper, we establish two Lyapunov inequalities for some half-linear higher order differential equations with anti-periodic boundary conditions. Our result improves that obtained by Wang (Appl. Math. Lett. 25:2375–2380, 2012).
Similar content being viewed by others
1 Introduction
In 1907, Lyapunov [2] proved the following remarkable result known as Lyapunov inequality. If u is a solution of
satisfying \(u(a)=u(b)=0\) (\(a< b\)) and \(u\neq 0\), then
Since then, Lyapunov inequality and many of its generalizations have gained a great deal of attention (see [3–12] and the references therein), because these results have found many applications in the study of various properties of solutions of differential and difference equations such as oscillation theory, disconjugacy, and eigenvalue problems.
In the last twenty years, a lot of efforts have been made to obtain similar results for higher order differential equations (see [1, 13–17] and the references therein), and other type integral inequalities (see [18–30] and the references therein). In particular, Çakmak [13] considered the following even higher order linear differential equation:
where \(r\in C([a,b],[0,\infty))\) and u satisfies the following boundary conditions:
and he obtained the following result. If there exists a nontrivial solution u of Eq. (1.2) satisfying (1.3), then one has
Later, Watanabe et al. [14] used a Sobolev inequality to get a new Lyapunov inequality for Eq. (1.2)
where
is the Riemann zeta function. Their result sharpened the result of Çakmak [13].
Recently, Wang [1] considered the following \((m+1)\)-order half-linear differential equation:
where \(r\in C([a,b],\mathbb{R})\), \(m\in \mathbb{N}\), \(p>1\) is a constant, and u satisfies the following anti-periodic boundary conditions:
and he obtained the following result. If there exists a nontrivial solution u of Eq. (1.5) satisfying (1.6), then
In the present paper, we shall use the Sobolev inequality established in [14] to establish two Lyapunov inequalities for Eq. (1.5) with the anti-periodic boundary conditions (1.6). Our result improves that obtained by Wang [1].
2 Main results
Lemma 2.1
([14])
For \(m\geq 1\), define the following Sobolev space:
There exists a positive constant \(C_{m}\) such that, for any \(u\in H_{m}\), the Sobolev inequality
holds, where
and the constants \(\{C_{m}\}\) are sharp,
is the Riemann zeta function.
Remark 2.1
From the definition of \(H_{m}\), we can easily get if \(u\in H_{m}\), then \(u^{(m-1)}\in H_{1}\).
Lemma 2.2
If u is a nontrivial solution of Eq. (1.5) satisfying the anti-periodic boundary conditions (1.6), then the inequalities
and
hold.
Proof
Since the nontrivial solution u of Eq. (1.5) satisfies the anti-periodic boundary conditions (1.6), then we have
So,
Similarly, we have
then (2.2) holds. It follows from (2.4) that
From (2.3) and (2.5), we obtain
i.e., (2.1) holds. □
Theorem 2.1
If u is a nontrivial solution of Eq. (1.5) satisfying the anti-periodic boundary conditions (1.6), then the inequality
holds, where \(p>2\).
Proof
Since the nontrivial solution u of Eq. (1.5) satisfies the anti-periodic boundary conditions (1.6), it is easy to see that u is an element of \(H_{m}\). Multiplying (1.5) by \(u^{(m-1)}(t)\) and integrating over \([a, b]\), yields
Using integration by parts to the first integral on the left-hand side of (2.8) and the anti-periodic boundary conditions (1.6), we have
So
By Lemma 2.1 and Remark 2.1, we obtain
and
Multiplying (2.10) and (2.11), we have
By using Hölder’s inequality
with \(f(t)=1\), \(g(t)=|u^{(m)}(t)|^{2}\), \(\alpha =\frac{p}{p-2}\), and \(\beta =\frac{p}{2}\), we obtain that
Thus
From (2.12) and (2.14), we have
Now, we claim that \(\int_{a}^{b}|u^{(m)}(t)|^{p}\,\mathrm{d}t>0\). In fact, if the above inequality is not true, we have \(\int_{a}^{b}|u^{(m)}(t)|^{p} \,\mathrm{d}t=0\), then \(u^{(m)}(t)=0\) for \(t\in [a, b]\). By the anti-periodic conditions (1.6), we obtain \(u(t)=0\) for \(t\in [a, b]\), which contradicts \(u(t)\not \equiv 0\), \(t\in [a, b]\). Thus dividing both sides of (2.16) by \(\int_{a}^{b}|u^{(m)}(t)|^{p}\,\mathrm{d}t\), we obtain the inequality
Since
we get
and
Then
So, from (2.17) and (2.18), we get (2.7) holds. Moreover, the inequality in (2.7) is strict since u is not a constant. This completes the proof of Theorem 2.1. □
Theorem 2.2
If u is a nontrivial solution of Eq. (1.5) satisfying the anti-periodic boundary conditions (1.6), then the inequality
holds, where \(1< p<2\).
Proof
As shown in the proof of Theorem 2.1, (2.9) holds, that is,
By using Hölder’s inequality (2.13) with \(f(t)=|u^{(m)}(t)|\), \(g(t)=1\), \(\alpha =p\), and \(\beta =\frac{p}{p-1}\), we obtain that
Thus
Using (2.9), (2.20), and (2.21), we get
Now, we claim that \(\int_{a}^{b}|u^{(m)}(t)|\,\mathrm{d}t>0\). In fact, if the above inequality is not true, we have \(\int_{a}^{b}|u^{(m)}(t)| \,\mathrm{d}t=0\), then \(u^{(m)}(t)=0\) for \(t\in [a, b]\). By the anti-periodic conditions (1.6), we obtain \(u(t)=0\) for \(t\in [a, b]\), which contradicts \(u(t)\not \equiv 0\), \(t\in [a, b]\). Thus, dividing both sides of (2.22) by \((\int_{a}^{b}|u^{(m)}(t)|\,\mathrm{d}t )^{p}\), we obtain the inequality
Moreover, the inequality in (2.23) is strict since u is not a constant. This completes the proof of Theorem 2.2. □
Remark 2.2
Inequality (2.7) improves inequality (1.7) significantly when \(p>2\) and \(m\geq 2\). We list the first six values of \(\zeta (2n)\), \(n = 1,2,\ldots, 6\), in Table 1.
For any \(m\in \mathbb{N}\), we have \(\zeta (2m)\leq \zeta (2)<2\), and then
Note that \(( \frac{(\frac{\pi }{2})^{m}}{2} ) ^{p-1}>1\) for \(m>\frac{\ln 2}{\ln \pi -\ln 2}\approx 1.535\) and \(p>2\). Then, using (2.24), we get
So, Theorem 2.1 improves Theorem 2.1 of [1] significantly when \(p>2\) and \(m\geq 2\).
3 Application
We give an application of the above Lyapunov inequality for an eigenvalue problem.
Example 3.1
Let λ be an eigenvalue of the following problem:
where \(r\in C([a,b],\mathbb{R})\), \(m\in \mathbb{N}\), and \(p>2\) is a constant. Then, from Theorem 2.1, we have
References
Wang, Y.Y.: Lyapunov-type inequalities for certain higher order differential equations with anti-periodic boundary conditions. Appl. Math. Lett. 25, 2375–2380 (2012)
Lyapunov, A.M.: Probleme général de la stabilité du mouvement (French translation of a Russian paper dated 1893). Ann. Fac. Sci. Univ. Toulouse 2, 27–247 (1907) Reprinted as Ann. Math. Studies, No. 17, Princeton (1947)
Cheng, S.S.: Lyapunov inequalities for differential and difference equations. Fasc. Math. 23, 25–41 (1991)
Eliason, S.B.: Lyapunov inequalities and bounds on solutions of certain second order equations. Can. Math. Bull. 17(4), 499–504 (1974)
Guseinov, G.S., Kaymakcalan, B.: Lyapunov inequalities for discrete linear Hamiltonian systems. Comput. Math. Appl. 45, 1399–1416 (2003)
Tang, X.-H., Zhang, M.: Lyapunov inequalities and stability for linear Hamiltonian systems. J. Differ. Equ. 252, 358–381 (2012)
Lee, C., Yeh, C., Hong, C., Agarwal, R.P.: Lyapunov and Wirtinger inequalities. Appl. Math. Lett. 17, 847–853 (2004)
Pachpatte, B.G.: Lyapunov type integral inequalities for certain differential equations. Georgian Math. J. 4(2), 139–148 (1997)
Tiryaki, A., Unal, M., Çakmak, D.: Lyapunov-type inequalities for nonlinear systems. J. Math. Anal. Appl. 332, 497–511 (2007)
Yang, X.J.: On inequalities of Lyapunov-type. Appl. Math. Comput. 134, 293–300 (2003)
Zhang, L.H., Zheng, Z.W.: Lyapunov type inequalities for the Riemann–Liouville fractional differential equations of higher order. Adv. Differ. Equ. 2017, 270 (2017)
Jleli, M., Kirane, M., Samet, B.: Lyapunov-type inequalities for a fractional p-Laplacian system. Fract. Calc. Appl. Anal. 20(6), 1485–1506 (2017)
Çakmak, D.: Lyapunov-type integral inequalities for certain higher order differential equations. Appl. Math. Comput. 216, 368–373 (2010)
Watanabe, K., Yamagishi, H., Kametaka, Y.: Riemann zeta function and Lyapunov-type inequalities for certain higher order differential equations. Appl. Math. Comput. 218, 3950–3953 (2011)
Yang, X.J., Lo, K.M.: Lyapunov-type inequalities for a class of higher-order linear differential equations with anti-periodic boundary conditions. Appl. Math. Lett. 34, 33–36 (2014)
Jleli, M., Nieto, J.J., Samet, B.: Lyapunov-type inequalities for a higher order fractional differential equation with fractional integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2017, 16 (2017)
Liu, H.D.: Lyapunov-type inequalities for certain higher-order difference equations with mixed non-linearities. Adv. Differ. Equ. 2018, 231 (2018)
Liu, H.D., Meng, F.W.: Nonlinear retarded integral inequalities on time scales and their applications. J. Math. Inequal. 12(1), 219–234 (2018)
Liu, H.D., Meng, F.W.: Some new generalized Volterra–Fredholm type discrete fractional sum inequalities and their applications. J. Inequal. Appl. 2016, 213 (2016)
Feng, Q.H., Meng, F.W., Fu, B.S.: Some new generalized Volterra-Fredholm type finite difference inequalities involving four iterated sums. Appl. Math. Comput. 219(15), 8247–8258 (2013)
Meng, F.W., Shao, J.: Some new Volterra–Fredholm type dynamic integral inequalities on time scales. Appl. Math. Comput. 223, 444–451 (2013)
Gu, J., Meng, F.W.: Some new nonlinear Volterra–Fredholm type dynamic integral inequalities on time scales. Appl. Math. Comput. 245, 235–242 (2014)
Liu, H.D.: A class of retarded Volterra–Fredholm type integral inequalities on time scales and their applications. J. Inequal. Appl. 2017, 293 (2017)
Cabrera, I., Sadarangani, K., Samet, B.: Hartman–Wintner-type inequalities for a class of nonlocal fractional boundary value problems. Math. Methods Appl. Sci. 40(1), 129–136 (2017)
Liu, H.D.: Some new integral inequalities with mixed nonlinearities for discontinuous functions. Adv. Differ. Equ. 2018, 22 (2018)
Zhao, D.L., Yuan, S.L., Liu, H.D.: Random periodic solution for a stochastic SIS epidemic model with constant population size. Adv. Differ. Equ. 2018, 64 (2018)
Tunç, E., Liu, H.D.: Oscillatory behavior for second-order damped differential equation with nonlinearities including Riemann–Stieltjes integrals. Electron. J. Differ. Equ. 2018, 54 (2018)
Liu, H.D., Meng, F.W.: Interval oscillation criteria for second-order nonlinear forced differential equations involving variable exponent. Adv. Differ. Equ. 2016, 291 (2016)
Wang, J.F., Meng, F.W., Gu, J.: Estimates on some power nonlinear Volterra–Fredholm type dynamic integral inequalities on time scales. Adv. Differ. Equ. 2017, 257 (2017)
Xu, R.: Some new nonlinear weakly singular integral inequalities and their applications. J. Math. Inequal. 11(4), 1007–1018 (2017)
Acknowledgements
The author is indebted to the anonymous referees for their valuable suggestions and helpful comments which helped improve the paper significantly.
Funding
This research was supported by A Project of Shandong Province Higher Educational Science and Technology Program (China) (Grant No. J14LI09), and the Natural Science Foundation of Shandong Province (China) (Grant No. ZR2018MA018), and the National Natural Science Foundation of China (Grant No. 11671227).
Author information
Authors and Affiliations
Contributions
HDL organized and wrote this paper. Further, he examined all the steps of the proofs in this research. The author read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
Liu, H. An improvement of the Lyapunov inequality for certain higher order differential equations. J Inequal Appl 2018, 215 (2018). https://doi.org/10.1186/s13660-018-1809-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-018-1809-5