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.

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: where denotes the forward difference operator, with the coefficients satisfying the condition . They [7] presented some Lyapunov-type inequalities for discrete linear scalar Hamiltonian systems when the coefficient 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:

Recently, Zhang and Tang [8] also considered the discrete linear Hamiltonian system: where , and are real-valued functions defined on and denotes the forward difference operator defined by , . They [8] obtained the following interesting Lyapunov-type inequality.

Theorem A. Suppose that (2) holds, and let with . Assume (4) has a real solution such that (3) holds. Then one has the following inequality: In 2012, the following assumptions are introduced in [9].(H1), and are real-valued functions, and , and .(H2) satisfy and .(H3) and are real-valued functions and for . Furthermore, and satisfy .
Under the boundary value conditions, Zhang and Tang [9] considered the following quasilinear difference systems with hypotheses (H1) and (H2): and the quasilinear difference systems involving the -Laplacian: 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: where for ; are nonnegative constants for ; for with for .
Obviously, the results obtained in [9] required that and or . The order of the quasilinear difference systems considered in [9] is less than 3. In this paper, one will remove the unreasonably severe constraints and or 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: The similar results for higher-order m-dimensional discrete system are easy to obtain.
Throughout this paper, and is a conjugate exponent; that is, .

2. Main Results

Theorem 1. Let , and assume that there exists a positive solution of the following linear homogeneous system: If is a nonzero solution of (8) satisfying the antiperiodic boundary conditions (9), then

Proof. Let , and be a nonzero solution of (8). By the antiperiodic boundary conditions (9), . For , we have Using discrete Hölder inequality gives
Similarly,
Then
Summing (15) from to , we have that is,
So Similarly,
Multiplying the first equation of (8) by and using inequalities (18)–(20), we have
Then
So
For the second and third equations of (8), we also have
Raising both sides of inequalities (23)–(25) to the powers , respectively, and multiplying the resulting inequalities give
Since is a positive solution of the linear homogeneous system (10), then Summing both sides of linear homogeneous system (10) yields
Noting that , we have

Corollary 2. Let and assume If is a nonzero solution of (8) satisfying the antiperiodic boundary conditions (9), then