1. Introduction

The Lyapunov inequality for linear ordinary differential equation where , gives a necessary condition for the existence of a positive solution as follows: Lyapunov [1] initiated to estimate the above inequality. Since then, there have been several results to generalize the above linear ordinary differential equation in many directions. Before stating many efforts, it is worth to mention Hartman and Pinasco's work. Hartman [2] obtained the generalized inequality by using Green's function: In fact, for by the inequality condition (1.2) is a generalization of condition (1.1).

Pinasco [3] extended linear ordinary differential equations to the following one-dimensional -Laplacian problem: where and He obtained Lyapunov inequality for () as follows: where

There have been many studies for various types of equations. Among others, one may refer to de Náploi and Pinasco [4] for the case of monotone quasilinear operators which include one-dimensional -Laplacian as a special case, Parhi and Panigrahi [5] for the case of third order differential equations, Cañada et al. [6] for the case of partial differential equations which have a weight function in , and Clark and Hinton [7] for the case of Hamiltonian systems.

Until now, the most general class of weight functions for the Lyapunov inequalities is The purpose of this paper is to get Lyapunov inequalities for single equations as well as systems of one-dimensional -Laplacian problems with singular weight functions which have a stronger singularities than those of

For this purpose, we first give three specific classes of weight functions. The first class can be given as It comes naturally from the study of the existence of positive solutions for -Laplacian problems. The second one is just the extension of Hartman's condition to -Laplacian problems given as follows: It is easy to see that and classes and are equivalent when It is also known [8] that ⫋ for and ⫋ for The third one can be given as It is obvious to see that and (see [8]).

This paper is organized as follows. In Section 2, we show Lyapunov inequality for one-dimensional -Laplacian problem when a weight function is In Section 3, we estimate Lyapunov inequality for a cycled system of one-dimensional -Laplacian problem when a weight function is Finally in Section 4, we have Lyapunov inequality for a strongly coupled system of one-dimensional -Laplacian problem when a weight function is .

2. Single Equation

Let us consider problem (). By a solution of () we mean that is absolutely continuous in any compact subinterval of , and satisfies the first equation in () in and . We assume It is known that all solutions for () are of class (see [9]).

Theorem 2.1. Assume If is a positive solution for (), then one has

Proof. By Hölder's inequality, we get For noting we have Thus, we have Similarly, by Hölder's inequality, we get For noting we get Adding (2.4) and (2.6), we have Multiplying both sides of (2.7) by and rewriting, we get Since is a solution for (), we have We note that the right-hand side makes sense because is in Integrating (2.8) on and using (2.9), we have Therefore, we get

Remark 2.2. () When the above result coincides with Hartman's estimate. But Hartman's argument does not work here by lack of Green's function for -Laplacian.
() If , for since and we have Pinasco's estimate (1.4). Thus our estimate generalizes Pinasco's.

For Pinasco [3] also estimated the lower bounds for eigenvalues of The proof mainly makes use of the nodal property of its corresponding eigenfunctions ; that is, has interior zeros in Recently, when Kajikiya et al. [9] showed the existence of eigenvalues for () and its corresponding eigenfunctions also have the nodal property. Employing Pinasco's argument ([3, Theorem ]) with (2.1), for each we have

3. Cycled System

Let us consider a cycled system: We say that is a solution of () if ,   is absolutely continuous in any compact subinterval of , each satisfies the equations in () in , and We assume that We note that all solutions for () are of class (see [10]).

Theorem 3.1. Assume If is a positive solution of (), then

Proof. We only show the case For the general case, we can prove it by repeating this procedure. As in (2.7), for we have or Multiplying the first equation of () by and integrating on , we have by (3.2) and (3.3) that Thus, we have Similarly, for the second equation in (), we have Thus, we have

Corollary 3.2. Assume for If is a positive solution of (), then one has

4. Strongly Coupled System

Let us consider a strongly coupled system: where We can give a definition for a solution of () as the definition for a solution of () and it is known that all positive solutions for () are of class (see [10]). We emphasize that it is only shown for a positive solution so far.

Theorem 4.1. Assume If is a positive solution of (), then one has

Proof. As in the proof of Theorem 3.1, we only show the case Multiplying to the first equation in () and integrating on and using (2.7), (3.2), and (3.3), we have Similarly, from the second equation of (), we have Let us denote and Then from (4.2) and (4.3), we have respectively. Equation (4.4) implies respectively. Therefore, we have Since ([11, page 38]), we get Hence, we have That is,

Corollary 4.2. Assume for If is a positive solution of (), then one has

Acknowledgment

The first author was supported by the 2009 Research Fund of the University of Ulsan.