Qualitative properties of nonlinear difference systems

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Abstract

This paper gives criteria for the existence and uniqueness of solutions to two and multipoint boundary value problems associated with a typical system of kth order nonlinear Lyapunov system. A variation of parameters formula for nonlinear kth order system is established. Using this formula, we also study various concepts of stability and establish sufficient conditions for the kth order difference system to be Lipschitz stable and uniform Lipschitz stable.

Introduction

The mathematical modeling of several important phenomenon are being presented via differential equations or difference equations. Many of the known results for the continuous system are lacking in discrete systems and many of the results in the theory of difference equations have been obtained as analogous results of the continuous system [1]. However, the situation here is different. We obtain the general solution of the kth order matrix difference system in terms of two fundamental matrix solutions of the kth order system, but since a solution is lacking in the continuous case, we consider the kth order nonlinear Lyapunov system:T(n+k)=AkT(n)Bk+F(n,T(n)),where A, B are constant square matrices of order s, whose elements are defined on Nn0+ and T(n)∈Rs×s(Cs×s). The components are functions defined on Nn0+ forNn0+={n0,n0±1,n0±2,…,n0±k,…},where kN+ and n0N,N being the set of integers.

The paper is organized as follows. In Section 2, we develop the general solution of the homogeneous matrix difference systemT(n+k)=AkT(n)Bkin terms of two fundamental matrix solutions of the system T(n+1)=AT(n) and T(n+1)=BT(n). We further develop the variation of parameters formula for the nonhomogeneous difference system in terms of the fundamental matrix solutions ϕ1(n,n0) of T(n+1)=AT(n) and ϕ2(n,n0) of T(n+1)=BT(n) and a summation involving these two fundamental matrix solutions and matrix F(n,T(n)). In Section 3, we investigate a criteria for the existence and uniqueness of solutions for two and multipoint boundary value problems using Bartles–Stewart algorithm [3] and the modified QR algorithm [4]. In Section 4, we investigate the stability properties of the homogeneous and nonhomogeneous systems and establish sufficient conditions for the Lipschitz stability and uniform Lipschitz stability of the variational system (1.1).

Section snippets

Variation of parameters formula

In this section, we first obtain the general solution of the homogeneous Lyapunov systemT(n+k)=AkT(n)Bkfor any positive integer k, in terms of the two fundamental matrix solutions of T(n+1)=AT(n) and T(n+1)=BT(n) and establish variation of parameters formula for the variational system (1.1). The proof of the following lemma is immediate.

Lemma 2.1

ϕ1(n,n0) is a fundamental matrix solution of T(n+1)=AT(n) if and only if ϕ1(n,n0) is a fundamental matrix solution of T(n+1)=AT(n).

Lemma 2.2

If ϕ1(n,n0) is a

Two (multi) point-boundary value problem

In this section, we shall be concerned with the existence and uniqueness of solutions to two and multipoint boundary value problems associated with the kth order difference systemT(n+k)=AkT(n)Bk+F(n,T(n)),L1T(n0)+L2T(n1)=W,ori=1kLiT(ni)=W,where n0,n1Nn0+ and n0<n1 (or niNn0+ and n0<n1<…<nk) L1,L2 and W are square matrices of order s×s. The general solution T(n) of the nonhomogeneous system (3.1) is given byT(n)=ϕ1(n,n0)Cn0ϕ2(n,n0)+∑j=n0n+k−1ϕ1(n,j+1)F(j,T(j))ϕ2(n,j+1).The boundary

Stability analysis

Consider the kth order difference systemT(n+k)=AkT(n)Bk+F(n,T(n)),where TRs×s, A and B are constant s×s matrices and F:Nn0×Rs×sRs×s and F(n,0)≡0. When F is small in the sense to be specified, one can consider (4.1) as a perturbation of the equationT(n+k)=AkT(n)Bkand the question that naturally arises is what are the stability properties of (4.1) that are preserved from (4.2). Such questions for the linear difference system are answered in [2]. The following theorems offer an answer to these

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