Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2011 (2011), Article ID 418136, 10 pages
http://dx.doi.org/10.1155/2011/418136
Research Article

Lyapunov-Type Inequalities for Some Quasilinear Dynamic System Involving the -Laplacian on Time Scales

1College of Mathematics and Computer Science, Jishou University, Jishou 416000, Hunan, China
2College of Science, Hunan University of Technology, Zhuzhou 412007, Hunan, China

Received 30 June 2011; Accepted 25 August 2011

Academic Editor: Yansheng Liu

Copyright © 2011 Xiaofei He and Qi-Ming Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We establish several new Lyapunov-type inequalities for some quasilinear dynamic system involving the -Laplacian on an arbitrary time scale , which generalize and improve some related existing results including the continuous and discrete cases.

1. Introduction

In recent years, the theory of time scales (or measure chains) has been developed by several authors with one goal being the unified treatment of differential equations (the continuous case) and difference equations (the discrete case). A time scale is an arbitrary nonempty closed subset of the real numbers . We assume that is a time scale and has the topology that it inherits from the standard topology on the real numbers . The two most popular examples are and . In Section 2, we will briefly introduce the time scale calculus and some related basic concepts of Hilger [13]. For further details, we refer the reader to the books independently by Kaymakcalan et al. [4] and by Bohner and Peterson [5, 6].

Consider the following quasilinear dynamic system involving the -Laplaci-an on an arbitrary time scale :

It is obvious that system (1.1) covers the continuous quasilinear system and the corresponding discrete case, respectively, when and ; that is,

In 1907, Lyapunov [7] established the first so-called Lyapunov inequality if the Hill equation has a real solution such that Moreover the constant 4 in (1.3) cannot be replaced by a larger number, where is a piece-wise continuous and nonnegative function defined on .

It is a classical topic for us to study Lyapunov-type inequalities which have proved to be very useful in oscillation theory, disconjugacy, eigenvalue problems, and numerous other applications in the theory of differential and difference equations. So far, there are many literatures which improved and extended the classical Lyapunov including continuous and discrete cases. For example, inequality (1.3) has been generalized to discrete linear Hamiltonian system by Zhang and Tang [8], to second-order nonlinear differential equations by Eliason [9] and by Pachpatte [10], to second-order nonlinear difference system by He and Zhang [11], to the second-order delay differential equations by Eliason [12] and by Dahiya and Singh [13], to higher-order differential equations by Pachpatte [14], Yang [15, 16], Yang and Lo [17] and Cakmak and Tiryaki [18, 19]. Lyapunov-type inequalities for the Emden-Fowler-type equations can be found in Pachpatte [10], and for the half-linear equations can be found in Lee et al. [20] and Pinasco [21]. Recently, there has been much attention paid to Lyapunov-type inequalities for dynamic systems on time scales and some authors including Agarwal et al. [22], Jiang and Zhou [23], He [24], He et al. [25], Saker [26], Bohner et al. [27], and Ünal and Cakmak [28] have contributed the above results.

In this paper, we use the methods in [29] to establish some Lyapunov-type inequalities for system (1.1) on an arbitrary time scale .

2. Preliminaries about the Time Scales Calculus

We introduce some basic notions connected with time scales.

Definition 2.1 (see [6]). Let . We define the forward jump operator by while the backward jump operator by In this definition, we put (i.e., if has a maximum ) and (i.e., if has a minimum ), where denotes the empty set. If , we say that is right-scattered, while if , we say that is left-scattered. Also, if and , then is called right-dense, and if and , then is called left-dense. Points that are right-scattered and left-scattered at the same time are called isolated. Points that are right-dense and left-dense at the same time are called dense. If has a left-scattered maximum , then we define otherwise; . The graininess function is defined by We consider a function and define so-called delta (or Hilger) derivative of at a point .

Definition 2.2 (see [6]). Assume that is a function, and let . Then, we define to be the number (provided it exists) with the property that given any , there is a neighborhood of (i.e., for some ) such that We call the delta (or Hilger) derivative of at .

Lemma 2.3 (see [6]). Assume that are differential at , then, (i)for any constant and , the sum is differential at with (ii)if exists, then is continuous at ,(iii)if exists, then ,(iv)the product is differential at with (v)if , then is differential at and

Definition 2.4 (see [6]). A function is called rd-continuous, provided it is continuous at right-dense points in and left-sided limits exist (finite) at left-dense points in and denotes by .

Definition 2.5 (see [6]). A function is called an antiderivative of , provided holds for all . We define the Cauchy integral by

The following lemma gives several elementary properties of the delta integral.

Lemma 2.6 (see [6]). If and , then (i), (ii), (iii), (iv), (v), (vi)if , then

The notation and will denote time scales intervals. For example, . To prove our results, we present the following lemma.

Lemma 2.7 (see [6]). Let and with . For , one has

Lemma 2.8 (see [6]). Let and with for . For , one has

3. Lyapunov-Type Inequalities

Denote

First, we give the following hypothesis.(H1) and are rd-continuous real functions and for and . Furthermore, and satisfy for .

Theorem 3.1. Let with . Suppose that hypothesis (H1) is satisfied. If (1.1) has a real solution satisfying the boundary value conditions then one has where and in what follows for .

Proof. By (1.1) and Lemma 2.3(iv), we obtain where . From Definition 2.5, integrating (3.5) from to , together with (3.3), we get It follows from (3.1), (3.3), and Lemma 2.7 that Similarly, it follows from (3.2), (3.3), and Lemma 2.7 that From (3.7) and (3.8), we have So, from (3.3), (3.6), (3.9), (H1), and Lemma 2.8, we have where
Next, we prove that If (3.12) is not true, there exist such that From (3.6), (3.13), and Lemma 2.8, we have It follows from the fact that that Combining (3.7) with (3.15), we obtain that for , which contradicts (3.3). Therefore, (3.12) holds. From (3.10), (3.12), and (H1), we have It follows from (3.11) and (3.16) that (3.4) holds.

Corollary 3.2. Let with . Suppose that hypothesis (H1) is satisfied. If (1.1) has a real solution satisfying the boundary value conditions (3.3), then one has

Proof. Since it follows from (3.4) and (H1) that (3.17) holds.

Corollary 3.3. Let with . Suppose that hypothesis (H1) is satisfied. If (1.1) has a real solution satisfying the boundary value conditions (3.3), then one has where .

Proof. Since it follows from (3.20) and (H1) that (3.19) holds.

When , , and , system (1.1) reduces to a second-order half-linear dynamic equation, and denote by

We can easily derive the following corollary for (3.21).

Corollary 3.4. Let with . If (3.21) has a solution satisfying then

Especially, while , , and , system (1.1) reduces to a second-order linear dynamic equation and denote by

Obviously, (3.24) is a special case of (3.21). One can also obtain a corollary immediately.

Corollary 3.5. Let with . If (3.24) has a solution satisfying then

Acknowledgments

This work is partially supported by the NNSF (no. 11171351) of China and by Scientific Research Fund of Hunan Provincial Education Department (no. 10C0655 and no. 11A095).

References

  1. S. Hilger, Einßmakettenkalküul mit anwendung auf zentrumsmannigfaltigkeiten, Ph.D. thesis, Universitäat Wüurzburg, 1988.
  2. S. Hilger, “Analysis on measure chains-a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990. View at Google Scholar · View at Zentralblatt MATH
  3. S. Hilger, “Differential and difference calculus-unified!,” vol. 30, no. 5, pp. 2683–2694. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. B. Kaymakcalan, V. Lakshmikantham, and S. Sivasundaram, Dynamic Systems on Measure Chains, vol. 370 of Mathematics and its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1996.
  5. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, Boston, Mass, USA, 2003. View at Zentralblatt MATH
  6. C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian systems: Difference Equations, Continued Fractions, and Riccati Equations, vol. 16 of Kluwer Texts in the Mathematical Sciences, Kluwer Academic, Boston, Mass, USA, 1996.
  7. A. M. Lyapunov, “Problème général de la stabilité du mouvement,” Annde la Faculté, vol. 2, no. 9, pp. 203–474, 1907. View at Google Scholar
  8. Q. Zhang and X. H. Tang, “Lyapunov inequalities and stability for discrete linear Hamiltonian system,” Applied Mathematics and Computation, vol. 218, pp. 574–582, 2011. View at Publisher · View at Google Scholar
  9. S. B. Eliason, “A Lyapunov inequality for a certain second order nonlinear differential equation,” Journal of the London Mathematical Society, vol. 2, pp. 461–466, 1970. View at Google Scholar
  10. B. G. Pachpatte, “Inequalities related to the zeros of solutions of certain second order differential equations,” Facta Universitatis. Series: Mathematics and Informatics, no. 16, pp. 35–44, 2001. View at Google Scholar · View at Zentralblatt MATH
  11. X. He and Q. Zhang, “A discrete analogue of Lyapunov-type inequalities for nonlinear difference systems,” Computers & Mathematics with Applications, vol. 62, pp. 677–684, 2011. View at Publisher · View at Google Scholar
  12. S. B. Eliason, “Lyapunov type inequalities for certain second order functional differential equations,” SIAM Journal on Applied Mathematics, vol. 27, pp. 180–199, 1974. View at Google Scholar · View at Zentralblatt MATH
  13. R. S. Dahiya and B. Singh, “A Lyapunov inequality and nonoscillation theorem for a second order non-linear differential-difference equation,” vol. 7, pp. 163–170, 1973. View at Google Scholar
  14. B. G. Pachpatte, “On Lyapunov-type inequalities for certain higher order differential equations,” Journal of Mathematical Analysis and Applications, vol. 195, no. 2, pp. 527–536, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. X. Yang, “On Liapunov-type inequality for certain higher-order differential equations,” Applied Mathematics and Computation, vol. 134, no. 2-3, pp. 307–317, 2003. View at Publisher · View at Google Scholar
  16. X. Yang, “On inequalities of Lyapunov type,” Applied Mathematics and Computation, vol. 134, no. 2-3, pp. 293–300, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. X. Yang and K. Lo, “Lyapunov-type inequality for a class of even-order differential equations,” Applied Mathematics and Computation, vol. 215, no. 11, pp. 3884–3890, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. D. Cakmak and A. Tiryaki, “On Lyapunov-type inequality for quasilinear systems,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3584–3591, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. D. Cakmak and A. Tiryaki, “Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the p1,p2,,pn-Laplacian,” Journal of Mathematical Analysis and Applications, vol. 369, no. 1, pp. 76–81, 2010. View at Publisher · View at Google Scholar
  20. C. Lee, C. Yeh, C. Hong, and R. P. Agarwal, “Lyapunov and Wirtinger inequalities,” Applied Mathematics Letters, vol. 17, no. 7, pp. 847–853, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. J. P. Pinasco, “Lower bounds for eigenvalues of the one-dimensional p-Laplacian,” Abstract and Applied Analysis, vol. 2004, no. 2, pp. 147–153, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. R. P. Agarwal, M. Bohner, and P. Rehak, “Half-linear dynamic equations,” Nonlinear Analysis and Applications, vol. 1, pp. 1–56, 2003. View at Google Scholar · View at Zentralblatt MATH
  23. L. Q. Jiang and Z. Zhou, “Lyapunov inequality for linear Hamiltonian systems on time scales,” Journal of Mathematical Analysis and Applications, vol. 310, no. 2, pp. 579–593, 2005. View at Google Scholar · View at Zentralblatt MATH
  24. Z. He, “Existence of two solutions of m-point boundary value problem for second order dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 296, no. 1, pp. 97–109, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. X. He, Q. Zhang, and X. H. Tang, “On inequalities of Lyapunov for linear Hamiltonian systems on time scales,” Journal of Mathematical Analysis and Applications, vol. 381, pp. 695–705, 2011. View at Publisher · View at Google Scholar
  26. S. H. Saker, “Oscillation of nonlinear dynamic equations on time scales,” Applied Mathematics and Computation, vol. 148, no. 1, pp. 81–91, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. M. Bohner, S. Clark, and J. Ridenhour, “Lyapunov inequalities for time scales,” Journal of Inequalities and Applications, vol. 7, no. 1, pp. 61–77, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. M. Ünal and D. Cakmak, “Lyapunov-type inequalities for certain nonlinear systems on time scales,” Turkish Journal of Mathematics, vol. 32, no. 3, pp. 255–275, 2008. View at Google Scholar · View at Zentralblatt MATH
  29. X. H. Tang and X. He, “Lower bounds for generalized eigenvalues of the quasilinear systems,” Journal of Mathematical Analysis and Applications, vol. 385, pp. 72–85, 2012. View at Publisher · View at Google Scholar