Lyapunov-Type Inequality for a Class of Discrete Systems with Antiperiodic Boundary Conditions

Abstract

A class of higher-order 3-dimensional discrete systems with antiperiodic boundary conditions is investigated. Based on the existence of the positive solution of linear homogeneous system, several new Lyapunov-type inequalities are established.

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Academic Editor:XianHua Tang

School of Mathematics and Statistics, Central South University, Changsha 410083, China

Received 23 June 2013; Accepted 16 August 2013

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.

1. Introduction

Lyapunov-type inequalities have been proved to be very useful in oscillation theory, disconjugacy, eigenvalue problems, and numerous other applications in the theory of differential and difference equations [1-3]. In recent years, there are many literatures which improved and extended the classical Lyapunov inequality including continuous and discrete cases [4-6]. Guseinov and Kaymakçalan [7] considered the following discrete Hamiltonian system: [figure omitted; refer to PDF] where Δ denotes the forward difference operator, with the coefficients a ( t ) satisfying the condition 1 - a ( t ) ...0; 0 , t ∈ Z . They [7] presented some Lyapunov-type inequalities for discrete linear scalar Hamiltonian systems when the coefficient c ( t ) is not necessarily nonnegative value. Applying these inequalities, they [7] obtained some stability criteria for discrete Hamiltonian systems.

For simplicity, the following assumptions are introduced: [figure omitted; refer to PDF]

Recently, Zhang and Tang [8] also considered the discrete linear Hamiltonian system: [figure omitted; refer to PDF] where α ( n ) , β ( n ) , and γ ( n ) are real-valued functions defined on Z and Δ denotes the forward difference operator defined by Δ x ( n ) = x ( n + 1 ) - x ( n ) , β ( n ) ...5; 0 . They [8] obtained the following interesting Lyapunov-type inequality.

Theorem A.

Suppose that (2) holds, and let a , b ∈ Z with a < b - 1 . Assume (4) has a real solution ( x ( n ) , y ( n ) ) such that (3) holds. Then one has the following inequality: [figure omitted; refer to PDF] In 2012, the following assumptions are introduced in [9].

(H1) _{r 1} ( n ) , _{r 2} ( n ) , _{f 1} ( n ) , and _{f 2} ( n ) are real-valued functions, and _{r 1} ( n ) > 0 , and _{r 2} ( n ) > 0 .

(H2) 1 < _{p 1} , _{p 2} < ∞ , _{α 1} , _{α 2} , _{β 1} , _{β 2} > 0 satisfy _{α 1} / _{p 1} + _{α 2} / _{p 2} = 1 and _{β 1} / _{p 1} + _{β 2} / _{p 2} = 1 .

(H3) _{r i} ( n ) and _{f i} ( n ) are real-valued functions and _{r i} ( n ) > 0 for i = 1,2 , ... , m . Furthermore, 1 < _{p i} < ∞ and _{α i} ( n ) > 0 satisfy ^{∑ i = 1 m} ... ( _{α i} / _{p i} ) = 1 .

Under the boundary value conditions, Zhang and Tang [9] considered the following quasilinear difference systems with hypotheses (H1) and (H2): [figure omitted; refer to PDF] and the quasilinear difference systems involving the ( _{p 1} , _{p 2} , ... , _{p m} ) -Laplacian: [figure omitted; refer to PDF] Some Lyapunov-type inequalities are established in [9].

Recently, antiperiodic problems have received considerable attention as antiperiodic boundary conditions appear in numerous situations [10-12]. For the sake of convenience, in this paper, one will only consider the following higher-order 3-dimensional discrete system: [figure omitted; refer to PDF] where 1 < _{p k} < + ∞ for k = 1,2 , 3 ; _{q i , j} are nonnegative constants for i , j = 1,2 , 3 ; _{ψ q} ( u ) = | u ^{| q - 1} u for q > 0 with _{ψ 0} ( u ) = sign ( u ) = ± 1 for q = 0 .

Obviously, the results obtained in [9] required that _{α 1} / _{p 1} + _{α 2} / _{p 2} = 1 and _{β 1} / _{p 1} + _{β 2} / _{p 2} = 1 or ^{∑ i = 1 m} ... ( _{α i} / _{p i} ) = 1 . The order of the quasilinear difference systems considered in [9] is less than 3. In this paper, one will remove the unreasonably severe constraints _{α 1} / _{p 1} + _{α 2} / _{p 2} = 1 and _{β 1} / _{p 1} + _{β 2} / _{p 2} = 1 or ^{∑ i = 1 m} ... ( _{α i} / _{p i} ) = 1 in [9]. one will introduce the antiperiodic boundary conditions instead of boundary conditions in [9]. In this paper, one will establish some new Lyapunov-type inequalities for higher-order 3-dimensional discrete system (8) by a method different from that in [9] under the following antiperiodic boundary conditions: [figure omitted; refer to PDF] The similar results for higher-order m-dimensional discrete system are easy to obtain.

Throughout this paper, _{p i} > 1 and ^{p i [variant prime]} is a conjugate exponent; that is, 1 / _{p i} + 1 / ^{p i [variant prime]} = 1 , i = 1,2 , 3 .

2. Main Results

Theorem 1.

Let a < b , and assume that there exists a positive solution ( _{e 1} , _{e 2} , _{e 3} ) of the following linear homogeneous system: [figure omitted; refer to PDF] If ( x ( n ) , y ( n ) , z ( n ) ) is a nonzero solution of (8) satisfying the antiperiodic boundary conditions (9), then [figure omitted; refer to PDF]

Proof.

Let ( x ( n ) , y ( n ) , and z ( n ) ) be a nonzero solution of (8). By the antiperiodic boundary conditions (9), x ( a ) + x ( b ) = 0 . For n ∈ Z [ a , b ] , we have [figure omitted; refer to PDF] Using discrete Hölder inequality gives [figure omitted; refer to PDF]

Similarly, [figure omitted; refer to PDF]

Then [figure omitted; refer to PDF]

Summing (15) from a to b - 1 , we have [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF]

So [figure omitted; refer to PDF] Similarly, [figure omitted; refer to PDF]

Multiplying the first equation of (8) by ^{Δ m} x ( n ) and using inequalities (18)-(20), we have [figure omitted; refer to PDF]

Then [figure omitted; refer to PDF]

So [figure omitted; refer to PDF]

For the second and third equations of (8), we also have [figure omitted; refer to PDF]

Raising both sides of inequalities (23)-(25) to the powers _{e 1} , _{e 2} , and _{e 3} , respectively, and multiplying the resulting inequalities give [figure omitted; refer to PDF]

Since ( _{e 1} , _{e 2} , _{e 3} ) is a positive solution of the linear homogeneous system (10), then [figure omitted; refer to PDF] Summing both sides of linear homogeneous system (10) yields [figure omitted; refer to PDF]

Noting that 1 / _{p k} + 1 / ^{p k [variant prime]} = 1 , k = 1,2 , 3 , we have [figure omitted; refer to PDF]

Corollary 2.

Let a < b and assume [figure omitted; refer to PDF] If ( x ( n ) , y ( n ) , z ( n ) ) is a nonzero solution of (8) satisfying the antiperiodic boundary conditions (9), then [figure omitted; refer to PDF]

Acknowledgments

This work is partly supported by NSFC under Granst nos. 61271355 and 61070190, the ZNDXQYYJJH under Grant no. 2010QZZD015, and NFSS under Grant no. 10BJL020.

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