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Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 595367, 18 pages
http://dx.doi.org/10.1155/2008/595367
Research Article

About the Stability and Positivity of a Class of Discrete Nonlinear Systems of Difference Equations

Department of Electricity and Electronics, Institute of Research and Development of Processes (IIDP), Faculty of Science and Technology, University of the Basque Country, P.O. Box 644, Leioa, 48080 Bilbao, Spain

Received 16 January 2008; Accepted 26 May 2008

Academic Editor: Antonia Vecchio

Copyright © 2008 M. De la Sen. 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

This paper investigates stability conditions and positivity of the solutions of a coupled set of nonlinear difference equations under very generic conditions of the nonlinear real functions which are assumed to be bounded from below and nondecreasing. Furthermore, they are assumed to be linearly upper bounded for sufficiently large values of their arguments. These hypotheses have been stated in 2007 to study the conditions permanence.

1. Introduction

There is a wide scientific literature devoted to investigate the properties of the solutions of nonlinear difference equations of several types [19]. Other equations of increasing interest are as follows:

(1)stochastic difference equations and systems (see, e.g., [10] and references therein);(2)nonstandard linear difference equations like, for instance, the case of time-varying coefficients possessing asymptotic limits and that when there are contributions of unmodeled terms to the difference equation (see, e.g., [11, 12]);(3)coupled differential and difference systems (e.g., the so-called hybrid systems of increasing interest in control theory and mathematical modeling of dynamic systems, [1316] and the study of discretized models of differential systems which are computationally easier to deal with than differential systems; see, e.g., [17, 18]). In particular, the stability, positivity, and permanence of such equations are of increasing interest. In this paper, the following system of difference equations is considered [1]:

𝑥𝑖𝑛+1=𝜆𝑖𝑥𝑖𝑛+𝑓𝑖𝛼𝑖𝑥𝑖+1𝑛𝛽𝑖𝑥𝑖+1𝑛1,𝑖𝑘=1,2,,𝑘,(1.1) with 𝑥𝑛(𝑘+1)𝑥𝑛(1), for all 𝑛𝐍; 𝜆𝑖𝐑, 𝛼𝑖𝐑, 𝛽𝑖𝐑; and 𝑓𝑖𝐑𝐑, for all 𝑖𝑘, under arbitrary initial conditions 𝑥0(𝑖),𝑥(𝑖)1, for all 𝑖𝑘. The identity 𝑥𝑛(𝑘+1)𝑥𝑛(1) allows the inclusion in a unified shortened notation via (1.1) of the dynamics:

𝑥𝑘𝑛+1=𝜆𝑖𝑥𝑘𝑛+𝑓𝑖𝛼𝑖𝑥1𝑛𝛽𝑖𝑥1𝑛1,𝑖𝑘,(1.2) as it follows by comparing (1.1) for 𝑖=𝑘 with (1.2). The solution vector sequence of (1.1) will be denoted as 𝑥𝑛=(𝑥𝑛(1),𝑥𝑛(2),,𝑥𝑛(𝑘))𝑇𝐑𝑘, for all 𝑛𝐍, under initial conditions 𝑥𝑗=(𝑥𝑗(1),𝑥𝑗(2),,𝑥𝑗(𝑘))𝑇𝐑𝑘, 𝑗=1,0. The above difference system is very useful for modeling discrete neural networks which are very useful to describe certain engineering, computation, economics, robotics, and biological processes of populations evolution or genetics [1]. The study in [1] about the permanence of the above system is performed under very generic conditions on the functions 𝑓𝑖𝐑𝐑, for all 𝑖𝑘. It is only requested that the functions be bounded from below, nondecreasing, and linearly upper bounded for large values, exceeding a prescribed threshold, of their real arguments. In this paper, general conditions for the global stability and positivity of the solutions are investigated.

1.1. Notation

𝐑+={𝑧𝐑𝑧>0}, 𝐑0+={𝑧𝐑𝑧0}, 𝐑0={𝑧𝐑𝑧0}. “” is the logic conjunction symbol. 𝐍0=𝐍{0}. If 𝑃𝐑𝑛×𝑛, then 𝑃𝑇 is the transpose of P.

𝑃>0,𝑃0,𝑃<0,𝑃0 denote, respectively, P positive definite, semidefinite positive, negative definite, and negative semidefinite. 𝑃0,𝑃>0,𝑃0 denote, respectively, P nonnegative (i.e., none of its entries is negative, also denoted as 𝑃𝐑𝑛×𝑛0+), P positive (i.e., 𝑃0 with at least one of its entries being positive), and P strictly positive (i.e., all of its entries are positive). Thus, 𝑃>0𝑃0 and 𝑃0𝑃>0𝑃0, but the converses are not generically true. The same concepts and notation of nonnegativity, positivity, and strict positivity will be used for real vectors. Then, the solution vector sequence in 𝐑𝑘 of (1.1) will be nonnegative in some interval S, denoted by 𝑥𝑛0 (identical to 𝑥𝑛𝐑𝑘0+), for all 𝑛𝑆𝐍, if all the components are nonnegative for 𝑛𝑆𝐍. If, in addition, at least one component is positive, then the solution vector is said to be positive, denoted by 𝑥𝑛>0 (implying that 𝑥𝑛𝐑𝑘0+), for all 𝑛𝑆𝐍. If all of them are positive in S, then the solution vector is said to be strictly on a discrete interval, denoted by 𝑥𝑛0 (identical to 𝑥𝑛𝐑𝑘+ and implying that 𝑥𝑛>0 and 𝑥𝑛𝐑𝑘0+), for all 𝑛𝑆𝐍.

2 and 1 are the 2 and 1 norms of vectors and induced norms of matrices, respectively. 𝐼𝑛 is the nth identity matrix.

2. Preliminaries

In order to characterize the properties of system (1.1), firstly define sets of nondecreasing and bounded-from-below functions 𝑓𝑖𝐑𝐑 in system (1.1) as follows irrespective of the initial conditions:

𝐵𝐾𝑖𝑓=𝑖𝐑𝐑𝑓𝑖𝑦𝑓𝑖𝑥𝐾𝑖,𝑥,𝑦>𝑥𝐑,𝐾𝑖𝐑,𝑖𝑘,(2.1) and sets of linearly upper bounded real functions:

𝐶𝛾𝑖,𝛿𝑖,𝑀𝑖𝑓=𝑖𝐑𝐑𝑓𝑖𝑥𝛿𝑖𝛾𝑖𝑥,𝑥>𝑀𝑖𝐑+,𝛿𝑖0,1,𝑖𝑘,(2.2) for 𝛾𝑖0 irrespective of the initial conditions as well. In a natural form, define also sets of nondecreasing, bounded-from-below, and linearly upper bounded real functions, again independent of the initial conditions, 𝐵𝐶(𝐾𝑖,𝛾𝑖,𝛿𝑖,𝑀𝑖)=𝐵(𝐾𝑖)𝐶(𝛾𝑖,𝛿𝑖,𝑀𝑖), that is,

𝐾𝐵𝐶𝑖,𝛾𝑖,𝛿𝑖,𝑀𝑖𝑓=𝑖𝐑𝐑𝑓𝑖𝑦𝑓𝑖𝑥𝐾𝑖𝑓𝑖𝑥𝛿𝑖𝛾𝑖𝑥,𝑥,𝑦>𝑥𝐑,𝐾𝑖𝐑,𝛿𝑖0,1,𝑖𝑘,(2.3) for 𝛾𝑖0. The above definitions facilitate the potential restrictions on the functions 𝑓𝑖𝐑𝐑, 𝑖𝑘, required to derive the various results of the paper. The constraints on the functions 𝑓𝑖𝐑𝐑, for all 𝑖𝑘, used in the above definitions of sets, have been proposed by Stević for 𝑓𝑖𝐵𝐶(𝐾𝑖,𝛾𝑖,𝛿𝑖,𝑀𝑖) and then used to prove the conditions of permanence of (1.1) in [1] for some 𝐾𝑖=𝐾, 𝑀𝑖=𝑀>0, and 𝛿𝑖(0,1), for all 𝑖𝑘, subject to 𝜆𝑖[0,𝛽𝑖/𝛼𝑖], 𝛼𝑖>𝛽𝑖0, for all 𝑖𝑘. The subsequent technical assumption will be then used in some of the forthcoming results.

Assumption 2.1. 𝛼𝑖>0 and 0<𝛿𝑖<min(1,𝛼𝑖1).

The following two assertions are useful for the analysis of the difference system (1.1).

Assertion 2.2. For any given 𝑖𝑘, 𝑓𝑖𝐵(𝐾𝑖)𝑓𝑖(𝛼𝑖𝑥𝑛(𝑖+1)𝛽𝑖𝑥(𝑖+1)𝑛1)𝐾𝑖, for all 𝑛𝐍{0,1}.

Assertion 2.3. (i) For any given 𝑖𝑘, 𝑓𝑖𝐶(𝛾𝑖,𝛿𝑖,𝑀𝑖)𝑓𝑖(𝛾𝑖((𝛼𝑖/𝛾𝑖)𝑥𝑛(𝑖+1)(𝛽𝑖/𝛾𝑖)𝑥(𝑖+1)𝑛1))(𝛿𝑖/𝛾𝑖)(𝛼𝑖𝑥𝑛(𝑖+1)𝛽𝑖𝑥(𝑖+1)𝑛1) if 𝑥𝑛(𝑖+1)>(𝛽𝑖/𝛼𝑖)𝑥(𝑖+1)𝑛1+(𝛾𝑖/𝛼𝑖)𝑀𝑖, for all 𝑛𝐍{0,1}, for any real constants 𝛽𝑖,𝛼𝑖>0. (ii)𝑓𝑖𝐶(𝛼𝑖,𝛿𝑖,𝑀𝑖)𝑓𝑖(𝛼𝑖(𝑥𝑛(𝑖+1)(𝛽𝑖/𝛼𝑖)𝑥(𝑖+1)𝑛1))(𝛿𝑖/𝛼𝑖)(𝛼𝑖𝑥𝑛(𝑖+1)𝛽𝑖𝑥(𝑖+1)𝑛1) if 𝑥𝑛(𝑖+1)>(𝛽𝑖/𝛼𝑖)𝑥(𝑖+1)𝑛1+𝑀𝑖, for all 𝑛𝑁{0,1}, for any real constants 𝛽𝑖,𝛼𝑖>0.(iii)𝑓𝑖𝐶(1,𝛿𝑖,𝑀𝑖)𝑓𝑖(𝛼𝑖𝑥𝑛(𝑖+1)𝛽𝑖𝑥(𝑖+1)𝑛1)𝛿𝑖(𝛼𝑖𝑥𝑛(𝑖+1)𝛽𝑖𝑥(𝑖+1)𝑛1) if 𝑥𝑛(𝑖+1)>(𝛽𝑖/𝛼𝑖)𝑥(𝑖+1)𝑛1+𝑀𝑖/𝛼𝑖, for all 𝑛𝑁{0,1}, for any real constants 𝛽𝑖,𝛼𝑖>0.(iv)𝐶(1,𝛿𝑖,𝑀𝑖)=𝐶(𝛼𝑖,𝛼𝑖𝛿𝑖,𝑀𝑖/𝛼𝑖) if Assumption 2.1 holds.

Proof. Assertion 2.3 (i)–(iii) follow directly from the definitions of 𝐵(𝐾𝑖) and 𝐶(𝛾𝑖,𝛿𝑖,𝑀𝑖), for all 𝑖𝑘.
Assertion 2.3 (iv) The proof is split into proving the two claims below.
Claim 1. 𝐶(1,𝛿𝑖,𝑀𝑖)𝐶(𝛼𝑖,𝛼𝑖𝛿𝑖,𝑀𝑖/𝛼𝑖).
Proof of Claim 1. 𝑓𝑖𝐶(1,𝛿𝑖,𝑀𝑖)𝑓𝑖(𝛼𝑖𝑥𝑛(𝑖+1)𝛽𝑖𝑥(𝑖+1)𝑛1)=𝑓𝑖(𝛼𝑖(𝑥𝑛(𝑖+1)(𝛽𝑖/𝛼𝑖)𝑥(𝑖+1)𝑛1))𝛿𝑖(𝛼𝑖𝑥𝑛(𝑖+1)𝛽𝑖𝑥(𝑖+1)𝑛1)=𝛿𝑖𝛼𝑖(𝑥𝑛(𝑖+1)(𝛽𝑖/𝛼𝑖)𝑥(𝑖+1)𝑛1) if 𝛼𝑖𝑥𝑛(𝑖+1)𝛽𝑖𝑥(𝑖+1)𝑛1>𝑀𝑖𝑓𝑖𝐶(𝛼𝑖,𝛼𝑖𝛿𝑖,𝑀𝑖/𝛼𝑖) if Assumption 2.1 holds.
Claim 2. 𝐶(𝛼𝑖,𝛼𝑖𝛿𝑖,𝑀𝑖/𝛼𝑖)𝐶(1,𝛿𝑖,𝑀𝑖).
Proof of Claim 2. 𝑓𝑖𝐶(𝛼𝑖,𝛼𝑖𝛿𝑖,𝑀𝑖/𝛼𝑖)𝑓𝑖(𝛼𝑖(𝑥𝑛(𝑖+1)(𝛽𝑖/𝛼𝑖)𝑥(𝑖+1)𝑛1))𝛼𝑖𝛿𝑖(𝑥𝑛(𝑖+1)(𝛽𝑖/𝛼𝑖)𝑥(𝑖+1)𝑛1)=𝛿𝑖(𝛼𝑖𝑥𝑛(𝑖+1)𝛽𝑖𝑥(𝑖+1)𝑛1) if 𝛼𝑖𝑥𝑛(𝑖+1)𝛽𝑖𝑥(𝑖+1)𝑛1>𝑀𝑖𝑓𝑖𝐶(1,𝛿𝑖,𝑀𝑖) if Assumption 2.1 holds.
Then, Assertion 2.3 (iv) has been proved from Claims 1-2.

The following result establishes that it is not possible to obtain equivalence classes from any collection of parts of the sets of functions in the definitions of 𝐵(𝐾𝑖), 𝐶(𝛾𝑖,𝛿𝑖,𝑀𝑖), and 𝐵𝐶(𝐾𝑖,𝛾𝑖,𝛿𝑖,𝑀𝑖).

Assertion 2.4. For any 𝑖𝑘, consider 𝐶(𝛾𝑖,𝛿𝑖,𝑀𝑖) for some given 3-tuple (𝛾𝑖,𝛿𝑖,𝑀𝑖) in 𝐑×(0,1)×𝐑, and consider any discrete collection of distinct admissible triples (𝛾𝑖𝑗𝑖𝛾,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀)𝐑×(0,1)×𝐑 (𝑗𝑖𝛾𝐽𝑖𝛾,𝑗𝑖𝛿𝐽,𝑗𝑖𝑀𝐽𝑖𝑀) subject to the constraints 𝛿𝑖𝑗𝑖𝛿𝛿𝑖 and 𝑀𝑖𝑗𝑖𝑀𝑀𝑖, for all (𝑗𝑖𝛿,𝑗𝑖𝑀)𝐽𝑖𝛿×𝐽𝑖𝑀, leading to the associated 𝐶(𝛾𝑖𝑗𝑖𝛾,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀). Define the binary relation 𝑅𝑖 in 𝐶(𝛾𝑖,𝛿𝑖,𝑀𝑖) as 𝑓𝑖𝑅𝑖𝑔𝑖𝑓𝑖,𝑔𝑖𝐶(𝛾𝑖𝑗𝑖𝛾,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀). Then, 𝑅𝑖 is not an equivalence relation so that 𝐶(𝛾𝑖𝑗𝑖𝛾,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀) are not equivalence classes in 𝐶(𝛾𝑖,𝛿𝑖,𝑀𝑖) with respect to 𝑅𝑖. Also, the sets 𝐵(𝐾𝑖𝑗𝑖𝐾) and 𝐵𝐶(𝐾𝑖𝑗𝑖𝐾,𝛾𝑖𝑗𝑖𝛾,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀) for any given respective collections 𝐾𝑖𝑗𝑖𝐾𝐾𝑖, 𝛿𝑖𝑗𝑖𝛿𝛿𝑖, 𝑀𝑖𝑗𝑖𝑀𝑀𝑖, for all (𝑗𝑖𝐾,𝑗𝑖𝛿,𝑗𝑖𝑀)𝐽𝑖𝐾×𝐽𝑖𝛿×𝐽𝑖𝑀, are not equivalence classes, respectively, in 𝐵(𝐾𝑖) and 𝐵𝐶(𝐾𝑖,𝛾𝑖,𝛿𝑖,𝑀𝑖).

Proof. In view of Assertion 2.3(iv), 𝛾𝑖𝑗𝑖𝛾 can be all set equal to unity with no loss of generality, which is done to simplify the notation in the proof. Note that 𝑓𝑖𝑅𝑖𝑔𝑖𝑓𝑖,𝑔𝑖𝐶(1,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀)𝑓𝑖,𝑔𝑖𝐶(1,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀) for some (𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀)×𝐑. Now, consider 𝐶(1,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀) with 𝛿𝑖𝑗𝑖𝛿>𝛿𝑖 such that 𝛿𝑖𝛿𝑖𝑗𝑖𝛿(>𝛿𝑖𝑗𝑖𝛿){𝛿𝑖𝑗𝑗𝐽𝑖𝛿}. Then, 𝐶(1,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀)𝐶(1,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀). Since the equivalence classes with respect to any equivalence relation are disjoint, 𝐶(1,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀) in 𝐶(1,𝛿𝑖,𝑀𝑖) with respect to 𝑅𝑖 is not an equivalence class unless 𝐶(1,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀)=𝐶(1,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝑀). Now, consider the linear function 𝑖𝐑𝐑 defined by 𝑖(𝑥)=𝛿𝑖𝑗𝑖𝛿𝑥>𝛿𝑖𝑗𝑖𝛿𝑥 so that 𝐶(1,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝛿)϶𝑖𝐶(1,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝛿). Thus, 𝐶(1,𝛿𝑖𝑗𝑖𝑖𝛿,𝑀𝑖)𝐶(1,𝛿𝑖𝑗𝑖𝛿,𝑀𝑖𝑗𝑖𝛿). Then, 𝑅𝑖 (𝑖𝑘) are not equivalence relations, and there are no equivalence classes in 𝐶(𝛾𝑖,𝛿𝑖,𝑀𝑖) (𝑖𝑘) with respect to 𝑅𝑖 (𝑖𝑘). The remaining part of the proof follows in a similar way by using the definitions of the sets 𝐵(𝐾𝑖) and 𝐵𝐶(𝐾𝑖,𝛾𝑖,𝛿𝑖,𝑀𝑖), and it is omitted.

3. Necessary Conditions for Stability and Positivity

Now, linear systems for system (1.1) with all the nonlinear functions in some specified class are investigated. Those auxiliary systems become relevant to derive necessary conditions for a given property to hold for all possible systems (1.1), whose functions are in some appropriate set 𝐵(𝐾𝑖), 𝐶(𝛾𝑖,𝛿𝑖,𝑀𝑖), or 𝐵𝐶(𝐾𝑖,𝛾𝑖,𝛿𝑖,𝑀𝑖). This allows the characterization of the above properties under few sets of conditions on the nonlinear functions in the difference system (1.1). If 𝑓𝑖𝐶(1,𝛿𝑖,𝑀𝑖), for all 𝑖𝑘, then the auxiliary linear system to (1.1) is

𝑥𝑖𝑛+1=𝜆𝑖𝑥𝑖𝑛+𝛿𝑖𝛼𝑖𝑥𝑖+1𝑛𝛽𝑖𝑥𝑖+1𝑛1,𝑖𝑘.(3.1) If 𝑓𝑖𝐶(𝛼𝑖,𝛿𝑖,𝑀𝑖), for all 𝑖𝑘, then the auxiliary linear system to (1.1) is

𝑥𝑖𝑛+1=𝜆𝑖𝑥𝑖𝑛+𝛿𝑖𝑥𝑖+1𝑛𝛽𝑖𝛼𝑖𝑥𝑖+1𝑛1,𝑖𝑘.(3.2) System (3.1) may be equivalently rewritten as follows by defining the state vector sequence 𝑥𝑛=(𝑥𝑛(1),𝑥𝑛(2),,𝑥𝑛(𝑘))𝑇𝐑𝑘, for all 𝑛𝐍, as the kth-order difference system:

𝑥𝑛+1=𝐴𝑥𝑛+𝐵𝑥𝑛1=𝑥Λ+𝐶𝑛+𝐵𝑥𝑛1=Λ𝑥𝑛+𝐵𝑥𝑛1,𝑛𝐍,(3.3) with initial conditions 𝑥𝑖=(𝑥𝑖(1),𝑥𝑖(2),,𝑥𝑖(𝑘))𝑇𝐑𝑘 for 𝑖=0,1, where 𝑥𝑛=(𝑥𝑇𝑛𝑥𝑇𝑛1)𝑇𝐑2𝑘 and

𝜆𝐴=1𝛿1𝛼1000𝜆2𝛿2𝛼200𝛿𝑘1𝛼𝑘1𝛿𝑘𝛼𝑘00𝜆𝑘𝐑𝑘×𝑘,(3.4)𝐵=0𝛿1𝛽10000𝛿2𝛽200𝛿𝑘1𝛽𝑘1𝛿𝑘𝛽𝑘000𝐑𝑘×𝑘𝜆,(3.5)Λ=Diag1,𝜆2,,𝜆𝑘,𝐶=0𝛿1𝛼10000𝛿2𝛼200𝛿𝑘1𝛼𝑘1𝛿𝑘𝛼𝑘000,(3.6)𝐵=𝐵𝐶𝐑𝑘×2𝑘.(3.7) The one-step delay may be removed by defining the following extended 2kth-order system of state vector 𝑥𝑛=(𝑥𝑇𝑛𝑥𝑇𝑛1)𝑇𝐑2𝑘 satisfying

𝑥𝑛+1=𝐴𝑥𝑛,𝑛𝐍,(3.8) with 𝑥0=(𝑥0(1),𝑥0(2),,𝑥0(𝑘),𝑥(1)1,𝑥(2)1,,𝑥(𝑘)1)𝑇𝐑2𝑘 and

𝐼𝐴=𝐴𝐵𝑘0𝐑2𝑘×2𝑘.(3.9) Note that the extended system (3.8)-(3.9) is fully equivalent to system (3.3)–(3.7) since both have identical solutions for each given common set of initial conditions. Now, let ()2 be the 2-norm of real vectors of any order and associated induced norms of matrices (i.e., spectral norms of vectors and matrices). The following definitions are useful to investigate (1.1).

Definition 3.1. System (1.1) is said to be globally Lyapunov stable (or simply globally stable) if any solution is bounded for any finite initial conditions.

Definition 3.2. System (1.1) is said to be permanent if any solution enters a compact set 𝐾 for 𝑛𝑛0 for any bounded initial conditions with 𝑛0 depending on the initial conditions.

Definition 3.3. System (1.1) is said to be positive if any solution is nonnegative for any finite nonnegative initial conditions.

The system is locally stable around an equilibrium point if any solution with initial conditions in a neighborhood of such an equilibrium point remains bounded. Local or global asymptotic stability to the equilibrium point occurs, respectively, under local or global stability around a unique equilibrium point if, furthermore, any solution tends asymptotically to such an equilibrium point as 𝑛. Definition 3.2 is the definition of permanence in the sense used in [1], which is compatible with global and local stability and with global or local asymptotic stability according to Definition 3.1 and the above comments if 0𝐾. However, it has to be pointed out that there are different definitions of permanence, like, for instance, in [2], where vanishing solutions (related to asymptotic stability to the equilibrium) or, even, negative solutions at certain intervals are not allowed for permanence. On the other hand, note that a continuous-time nonlinear differential system may be permanent without being globally stable in the case that finite escape times t of the solution exist, implying that because of unbounded discontinuities of the solution at finite time t, that solution is unbounded in [𝑡,𝑡+𝜀) for some finite 𝜀𝐑+. This phenomenon cannot occur for system (1.1) under the requirement 𝑓𝑖𝐵𝐶(𝐾𝑖,𝛾𝑖,𝛿𝑖,𝑀𝑖), for all 𝑖𝑘, which avoids the solution being infinity at finite values of the discrete index n for any finite initial conditions. The following result is concerned with necessary conditions of global Lyapunov stability of system (1.1) for all the sets of functions 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖), for all 𝑖𝑘, since the linear system defined with 𝑓𝑖(𝑥)=𝛿𝑖𝑥, for all 𝑖𝑘, in (1.1) has to be globally stable in order to keep global stability for any 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖), for all 𝑖𝑘.

Theorem 3.4. System (1.1) is globally stable and permanent for any given set of functions 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖) for any given 𝐾𝑖𝐑 and any given 𝑀𝑖𝐑, for all 𝑖𝑘, only if the subsequent properties hold.
(i) |𝜆𝑖|1, for all 𝑖𝑘.
(ii) 𝐴21, equivalently, 𝑊2=𝐴𝑇𝐴21,where𝑊=𝐴𝑇𝑊𝐴=11𝑊12𝑊𝑇12𝑊22𝐑2𝑘×2𝑘,(3.10)where 𝑊11=𝐴𝑇𝐴+𝐼𝑘=1+𝜆21+𝛿2𝑘𝛼2𝑘𝜆1𝛿1𝛼1𝜆3𝛿3𝛼3𝜆𝑘𝛿𝑘𝛼𝑘𝜆1𝛿1𝛼11+𝜆22+𝛿21𝛼21𝜆2𝛿2𝛼2𝜆𝑘1𝛿𝑘1𝛼𝑘1𝜆𝑘2𝛿𝑘2𝛼𝑘2𝜆𝑘1𝛿𝑘1𝛼𝑘1𝜆2𝑘1+𝛿2𝑘2𝛼2𝑘2𝜆2𝛿2𝛼2𝜆𝑘𝛿𝑘𝛼𝑘𝜆𝑘1𝛿𝑘1𝛼𝑘1𝜆2𝛿2𝛼21+𝜆2𝑘+𝛿2𝑘1𝛼2𝑘1,(3.11)𝑊12=𝐴𝑇𝐵, and 𝑊22=𝐵𝑇𝐵=𝐷𝑖𝑎𝑔(𝛿2𝑘𝛽𝑘,𝛿21𝛽1,,𝛿2𝑘1𝛽𝑘1), with 𝐼𝑘 being the kth identity matrix. A necessary condition is 𝑘𝑖=1(𝜆2𝑖+𝛿2𝑖(𝛼2𝑖+𝛽2𝑖))𝑘.
(iii) There exists 𝑃=𝑃𝑇=𝑃11𝑃12𝑃𝑇12𝑃220in𝐑2𝑘×2𝑘,(3.12) where 𝑃𝑖𝑗𝐑𝑘×𝑘(𝑖,𝑗=1,2), which is a solution to the matrix identity
𝐴𝑇𝑃11+𝑃𝑇12𝐴+𝐴𝑇𝑃12+𝑃22𝑃11𝐴𝑇𝑃11+𝑃𝑇12𝐵𝑃12𝐵𝑇𝑃11𝐴+𝑃12𝑃𝑇12𝐵𝑇𝑃11𝐵𝑃22=𝑄(3.13) for any given 𝑄=𝑄𝑇=𝑄11𝑄12𝑄𝑇12𝑄220𝑖𝑛𝐑2𝑘×2𝑘.(3.14)

Proof. (i) Note that the identically zero functions 𝑓𝑖𝐑0, for all 𝑖𝑘, are all in 𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖) for any 𝐾𝑖0, 𝛿𝑖(0,1), 𝑀𝑖>0, for all 𝑖𝑘. Proceed by contradiction by assuming that |𝜆𝑖|>1 and 𝑓𝑖0 for some 𝑖𝑘={1,2,,𝑘}, with the system being globally stable. Thus, |𝑥(𝑖)𝑛+1|>|𝑥𝑛(𝑖)| if 𝑥0(𝑖)0 so that |𝑥𝑛(𝑖)| as 𝑛, and then the system is unstable for some function 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖). Thus, the necessary condition for global stability has been proved, implying also the permanence of all the solutions in some compact real interval 𝐾.
(ii) Assume 𝑓𝑖(𝑥)=𝛿𝑖𝑥 with 𝛿𝑖(0,1) everywhere in R so that 𝑓𝑖𝐶(1,𝛿𝑖,𝑀𝑖), 𝑀𝑖>0. Let the spectrum of W be 𝜎(𝑊)={𝜎1,𝜎2,,𝜎𝑘}, with each eigenvalue being repeated as many times as its multiplicity. Then, 𝐴2=max1𝑖𝑘𝜎𝑖1/2. It is first proved by complete induction that if 𝑥00 is an eigenvector of 𝐴, then 𝑥𝑘 is an eigenvector of 𝐴 for any 𝑘1. Assume that 𝑥𝑘 is an eigenvector of 𝐴 for some arbitrary 𝑘1 and some eigenvalue 𝜌𝑖. Then, 𝐴𝑥𝑘+1=𝐴(𝐴𝑥𝑘)=𝐴(𝜌𝑖𝑥𝑘)=𝜌𝑖(𝐴𝑥𝑘)=𝜌𝑖𝑥𝑘+1 so that 𝑥𝑘+1 is also an eigenvector of 𝐴 for the same eigenvalue 𝜌𝑖. This property leads to
𝑥𝑘+122=𝐴𝑥𝑘22=𝑥𝑇𝑘𝐴𝑇𝐴𝑥𝑘=𝜌2𝑖𝑥𝑘22=𝜎𝑖𝑥𝑘22=𝜌𝑖2𝑘𝑥022=𝜎𝑘𝑖𝑥022.(3.15) Proceed by contradiction by assuming that system (1.1) is stable, for all 𝑓𝑖𝐶(1,𝛿𝑖,𝑀𝑖), with |𝜌𝑖|=𝜎𝑖1/2>1. From (3.15), |𝑥𝑛(𝑖)| as 𝑛, and then the system is unstable for a function 𝑓𝑖𝐶(1,𝛿𝑖,𝑀𝑖) for any real constant 𝐾𝑖 since it possesses an unbounded solution for some finite initial conditions. Now, redefine the functions 𝑓𝑖(𝑥) from the above 𝑓𝑖(𝑥), 𝑖𝑘, as follows:
𝑓𝑖𝑥=𝑓𝑖𝑥=𝛿𝑖𝑥if𝑥0,𝐑϶𝜆<(1+max1𝑖𝑘|||𝜆𝑖|||)<0if𝑥<0.(3.16) It is clear by construction that if 𝑓𝑖(𝑥)=𝑓𝑖(𝑥)=𝛿𝑖𝑥 on an interval of infinite measure and if 0>𝜆=𝑓𝑖(𝑥)𝑓𝑖(𝑥) occurs on a real interval of finite measure, then the above contradiction obtained for 𝑓𝑖𝐶(1,𝛿𝑖,𝑀𝑖) still applies for 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖) for any finite negative 𝐾𝑖<𝜆. If 𝑓𝑖(𝑥)=𝑓𝑖(𝑥) occurs on an interval of finite measure and if 𝑓𝑖(𝑥)𝑓𝑖(𝑥) occurs on an interval of infinite measure, then the linear system resulting from (1.1) with the replacement 𝑓𝑖(𝑥)𝑓𝑖(𝑥) is unstable so that any nontrivial solution is unbounded. Furthermore, since 𝑓𝑖(𝑥) as 𝑥 (function diverging to ) and 𝑓𝑖(𝑥) being unbounded on R (implying that 𝑓𝑖(𝑥𝑘) for {𝑥𝑘}0 being some monotonically increasing sequence of real numbers) are both impossible situations for some 𝑖𝑘 since 𝑓𝑖𝐑𝐑(𝑖𝑘) are all nondecreasing, it follows again that the functions are bounded from below so that 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖) for some finite 𝐾𝑖<0. If the real subintervals within which 𝑓𝑖(𝑥) equalizes 𝑓𝑖(𝑥) or differs from 𝑓𝑖(𝑥) are both of infinite measure, the result 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖) with some unbounded solution still applies trivially for some finite 𝐾𝑖<0. Thus, system (1.1) is globally stable for any given set of functions 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖) for any 𝐾𝑖𝐑 and any 𝑀𝑖𝐑, for all 𝑖𝑘, only if the subsequent equivalent properties hold: 𝐴21, 𝑊21. The necessary condition 𝑘𝑖=1(𝜆2𝑖+𝛿2𝑖(𝛼2𝑖+𝛽2𝑖))𝑘 follows by inspecting the sum of entries of the main diagonal of W which equalizes the sum of nonnegative real eigenvalues of W (which are also the squares of the modules of the eigenvalues of 𝐴, i.e., the squares of the singular values of 𝐴) which have to be all of modules not greater than unity to guarantee global stability.
(iii) The property derives directly from discrete Lyapunov global stability theorem and its associate discrete Lyapunov matrix equation 𝐴𝑇𝑃𝐴𝑃=𝑄 which has to possess a solution 𝑃0 for any given 𝑄0. This property is a necessary and sufficient condition for the global stability of the extended linear system (3.8)-(3.9), and then for that of system (3.3)–(3.7). The proof ends by noting that system (3.8)-(3.9) has to be stable in order to guarantee the global stability of system (1.1) for any set 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖), for all 𝑖𝑘, according to Property (ii).

Concerning positivity (Definition 3.3), it is well known that in the continuous-time and discrete-time linear and time-invariant cases, the positivity property may be established via a full characterization of the parameters (see, e.g., [2, 13, 17] as well as references therein). In particular, for a continuous-time linear time-invariant dynamic system to be positive, the matrix of dynamics has to be a Meztler matrix, while in a discrete-time one it has to be positive, where the control, output, and input-output interconnection matrices have to be positive in both (continuous-time and discrete-time) cases [2]. Under these conditions, each solution is always nonnegative all the time provided that all the components of the control and initial condition vectors are nonnegative [2, 13]. In general, in the nonlinear case, it is necessary to characterize the nonnegativity of the solutions over certain intervals and for certain values of initial conditions and parameters; that is, the positivity is not a general property associated with the differential system itself all the time but with some particular solutions on certain time intervals associated with certain constraints on the corresponding initial conditions. The positivity of (1.1) for linear functions 𝑓𝑖(𝑥)=𝛿𝑖𝑥 is now invoked (in terms of necessary conditions) to guarantee the positivity of all the solutions of (1.1) for any set of nonnegative initial conditions and any potential set 𝑓𝑖𝐑0+𝐑0+ with 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖) for any given 𝐾𝑖𝐑 and any given 𝑀𝑖𝐑, for all 𝑖𝑘. This is formally addressed in the subsequent result.

Theorem 3.5. System (1.1) is positive for any given set of nonnegative functions 𝑓𝑖𝐑0+𝐑0+ with 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖) for any given 𝐾𝑖𝐑 and any given 𝑀𝑖𝐑, for all 𝑖𝑘, only if 𝜆𝑖𝐑0+, 𝛼𝑖𝐑0+, 𝛽𝑖𝐑0, for all 𝑖𝑘.

Outline of proof
As argued in the proof of Theorem 3.4 for stability, the linear system has to be positive in order to guarantee that it is positive for any set 𝑓𝑖𝐑0+𝐑0+ with 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖) for any given 𝐾𝑖𝐑 and 𝑀𝑖𝐑, for all 𝑖𝑘. The linear system (3.8)-(3.9) for 𝑓𝑖(𝑥)=𝛿𝑖𝑥 is positive if and only if 𝐴𝐑𝑛×𝑛0+ [3] since, in addition, this implies 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖). The proof follows since 𝐴𝐑𝑛×𝑛0+ by direct inspection if and only if 𝜆𝑖𝐑0+, 𝛼𝑖𝐑0+, 𝛽𝑖𝐑0, for all 𝑖𝑘.

Necessary joint conditions for stability, permanence, and positivity of (1.1) for any set 𝑓𝑖𝐑0+𝐑0+ with 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖) for any given 𝐾𝑖𝐑 and 𝑀𝑖𝐑, for all 𝑖𝑘, follow directly by combining Theorems 3.4 and 3.5.

4. Main Stability Results

This section derives sufficiency-type conditions (easy to test) for global stability of the linear system (3.3)–(3.7) independently of the signs of the parameters 𝛼𝑖, 𝛽𝑖, and 𝛿𝑖, 𝑖𝑘 (which are also allowed to take values out of the interval (0,1), but on their maximum sizes). It is allowed that 𝜆𝑖 be independent of the above parameters and negative, but fulfilling that their modules are less than unity. The mechanism of proof for the linear case is then extended directly to the general nonlinear system (1.1). The 𝛼𝑖, 𝛽𝑖, and 𝜆𝑖, 𝑖𝑘, are allowed to be negative but 𝛿𝑖(0,1), 𝑖𝑘, is required to formulate an auxiliary result for the main proof.

Theorem 4.1. Assume that |𝜆𝑖|<1, for all 𝑖𝑘, and
maxmax1𝑖𝑘|||𝛼𝑖|||,max1𝑖𝑘|||𝛽𝑖|||<1max1𝑖𝑘|||𝜆𝑖|||2𝑘max1𝑖𝑘|||𝛿𝑖|||.(4.1) Then, the linear system (3.3)–(3.7), equivalently system (3.8)-(3.9), is globally Lyapunov stable for any finite arbitrary initial conditions. It is also permanent for any initial conditions:
𝑥0𝐾0𝑎1,,𝑎2𝑘,𝑏1,,𝑏2𝑘𝑥=𝑥=1,𝑥2,,𝑥2𝑘𝑇𝐑2𝑘𝑥𝑖𝑎𝑖,𝑏𝑖,>𝑏𝑖>𝑎𝑖>,𝑖2𝑘𝐑2𝑘.(4.2)

Proof. The successive use of the recursive second identity in (3.3) with initial condition 𝑥0=(𝑥𝑇0,𝑥𝑇1)𝑇 leads to
𝑥𝑛+𝑘=Λ𝑛+𝑘𝑥0+𝑛+𝑘1𝑖=0Λ𝑛+𝑘𝑖1𝐵𝑥𝑖,𝑛𝐍,𝑘𝑘,(4.3) and taking 2-norms in (4.3) with 𝜆=max1𝑖𝑘|𝜆𝑖|<1, we get
𝑥𝑛+𝑘2=Λ𝑛+𝑘2𝑥02+𝑛+𝑘1𝑖=0Λ𝑛+𝑘𝑖12𝐵2𝑥𝑖2𝜆𝑛+𝑘𝑥02+1𝜆𝑛+𝑘1𝜆𝐵2max0𝑖𝑛+𝑘1𝑥𝑖2𝜆𝑛𝑥02+1𝜆𝑛1𝜆𝐵2max0𝑖𝑛+𝑘1𝑥𝑖2𝜆𝑛𝑥02+𝛿max𝛼,𝛽1𝜆𝑛1𝜆𝑘max0𝑖𝑛+𝑘1𝑥𝑖2𝜆𝑛𝑥02+2𝛿max𝛼,𝛽1𝜆𝑛1𝜆𝑘max1𝑖𝑛+𝑘1𝑥𝑖2,𝑛𝐍,𝑘𝑘,(4.4) where 𝛿=max1𝑖𝑘(|𝛿𝑖|), 𝛼=max1𝑖𝑘(|𝛼𝑖|), and 𝛽=max1𝑖𝑘(|𝛽𝑖|) since 𝜆<1 and
𝐵2=𝜆max𝐵𝑇𝐵𝑘𝐵1𝑘𝛿max𝛼,𝛽forany𝑥𝐑𝑘,Λ𝑗22=max1𝑖𝑘|||𝜆𝑖|||2𝑗=𝜆2𝑗𝜆<1,𝑗𝐍,max0𝑖𝑛+𝑘1𝑥𝑖2=max0𝑖𝑛+𝑘1𝑥𝑇𝑖𝑥𝑖+𝑥𝑇𝑖1𝑥𝑖11/2max0𝑖𝑛+𝑘1𝑥𝑖2+𝑥𝑖122max1𝑖𝑛+𝑘1𝑥𝑖2.(4.5) Note that (4.4) is still valid if the term preceding the equality is any 𝑥𝑛+2, for all 𝐍𝑛+𝑘, since they are all upper bounded by all the right-hand side upper bounds. Then,
𝑥𝑛+2𝜆𝑛𝑥02+2𝛿max(𝛼,𝛽)1𝜆𝑛1𝜆𝑘max1𝑖𝑛+𝑘1𝑥𝑖2,(4.6) for all 𝑛𝐍, for all 𝑘𝑘, for all 𝐍𝑛+𝑘, which implies directly that
max1𝑖𝑛+𝑘1𝑥𝑛+𝑖2𝜆𝑛𝑥02+2𝛿max𝛼,𝛽1𝜆𝑘max1𝑖𝑛+𝑘1𝑥𝑖2+𝑥02+𝑥12,𝑛𝐍,𝑘𝑘.(4.7) If the condition max(𝛼,𝛽)<(1𝜆)/2𝛿𝑘 with 𝜆[0,1) holds, then the second term of the right-hand side of (4.7) may be combined with the left-hand-side term to yield
𝑥𝑛2max1𝑖𝑛+𝑘1𝑥𝑛+𝑖21𝜆1𝜆2𝑘𝛿max𝛼,𝛽1+𝜆𝑛𝑥02+𝑥121𝜆1𝜆2𝑘𝛿max𝛼,𝛽1+𝜆𝑛0𝑥02+𝑥121+𝜀1𝜆1𝜆2𝑥𝑘𝛿max𝛼,𝛽02+𝑥121+𝜀1𝜆1𝜆2(𝑘𝛿max𝛼,𝛽2𝑘𝑖=1𝑎max2𝑖,𝑏2𝑖)1/21+𝜀1𝜆1𝜆2(𝑘𝛿max𝛼,𝛽2𝑘𝑖=1|||𝑎max𝑖|||,|||𝑏𝑖|||)(4.8)31𝜆1𝜆2𝑥𝑘𝛿max𝛼,𝛽max02,𝑥12,(4.9) for all 𝜀𝐑+, for all 𝑛(𝑛0)𝐍, depending on 𝑛0, which depends on 𝜀, for any 𝐍϶𝑛0ln𝜀/ln𝜆, for all 𝑥0𝐾0. Since 𝐾0 is compact, it follows from (4.9) that any solution sequence is bounded for any 𝑛𝐍 and any finite initial conditions. Thus, the linear system (3.3)–(3.7) is globally Lyapunov stable. Also, since 𝐾0 is compact, it follows from (4.8) that any solution sequence is permanent since it enters the prefixed compact set
𝐾={𝑥𝐑𝑘|||𝑥𝑖|||1+𝜀𝑘1𝜆1𝜆2𝛿(𝑘max𝛼,𝛽2𝑘𝑖=1|||𝑎max𝑖|||,|||𝑎𝑖|||),𝑖𝑘}(4.10) for any 𝑛(𝑛0)𝐍 and any finite initial conditions (𝑥0𝑥1)𝑇 in 𝐾0. Furthermore, 𝐾 is independent of each particular set of initial conditions in 𝐾0. Thus, the linear system (3.3)–(3.7) is permanent.

The following technical result will be then useful as an auxiliary one to prove the stability of (1.1) under a set of sufficiency-type conditions based on extending the proof mechanism of Theorem 4.1 to the nonlinear case. Basically, it is proved that the functions 𝑓𝑖𝐑𝐑, 𝑖𝑘, grow at most linearly with their argument.

Lemma 4.2. 𝑓𝑖𝐶(𝛼𝑖,𝛿𝑖,𝑀𝑖)𝑓𝑖(𝑥)=𝑂(𝑥), for all 𝑖𝑘. In addition, 𝑓𝑖(𝑥) is bounded for all 𝑥𝑀𝑖. The result also holds if 𝑓𝑖𝐵𝐶(𝐾𝑖,𝛼𝑖,𝛿𝑖,𝑀𝑖), for all 𝐾𝑖𝐑, for all 𝑖𝑘.

Proof. Now, it is proved that 𝑓𝑖(𝑥)=𝑂(𝑥) (notation of “big Landau O” of x) for any 𝑓𝑖𝐶(𝛼𝑖,𝛿𝑖,𝑀𝑖), for all 𝑖𝑘. First, note that for all 𝑖𝑘 for some 𝜀𝑖[𝑀,)𝐑0+, 𝑓𝑖𝐶(𝛼𝑖,𝛿𝑖,𝑀𝑖)𝑥𝑀𝑖𝐑+𝑓𝑖(𝑥)=𝛿𝑖𝑥𝜀𝑖(𝑥)𝛿𝑖𝑥𝑓𝑖(𝑥)=𝑂(𝑥) since 𝑓𝑖(𝑥)𝛿𝑖𝑥+𝐾 for any 𝐾𝐑0+, for all 𝑥𝑀𝑖. The result also holds if 𝑓𝑖𝐵𝐶(𝐾𝑖,𝛼𝑖,𝛿𝑖,𝑀𝑖), for all 𝐾𝑖𝐑, since 𝐵𝐶(𝐾𝑖,𝛼𝑖,𝛿𝑖,𝑀𝑖)𝐶(𝐾𝑖,𝛼𝑖,𝛿𝑖,𝑀𝑖). It is now proved by a contradiction argument that if 𝑓𝑖𝐶(𝐾𝑖,𝛼𝑖,𝛿𝑖,𝑀𝑖), then it is bounded, for all 𝑥<𝑀𝑖. Assume 𝑥<𝑀𝑖𝐑+ with 𝑓𝑖(𝑥1) being arbitrarily large for some 𝑥1𝑖<𝑀𝑖. Thus, there exists 𝑀2𝑖𝐑+ being arbitrarily large so that 𝑀2𝑖𝑓𝑖(𝑥1𝑖)𝑓𝑖(𝑀𝑖)𝛿𝑖𝑀𝑖< for 𝑥1𝑖=𝛼𝑖𝑥𝑛(𝑖+1)𝛽𝑖𝑥(𝑖+1)𝑛1<𝑀𝑖 since 𝑓𝑖𝐶(𝛼𝑖,𝛿𝑖,𝑀𝑖) so that it is monotonically nondecreasing. This is a contradiction since 𝑀2𝑖 is arbitrarily large. Thus, 𝑓𝑖𝐶(𝛼𝑖,𝛿𝑖,𝑀𝑖) is bounded, for all 𝑥<𝑀𝑖. Since it is bounded, then 𝑓𝑖(𝑥)=𝑂(𝑥)|𝑓𝑖(𝑥)|𝛿|𝑥|+𝐶1 for some finite 𝐶1𝐑+ for 𝑥<𝑀𝑖 as a result, so that 𝑓𝑖(𝑥)=𝑂(𝑥) on 𝐑. Again, the result still holds if 𝑓𝑖𝐵𝐶(𝐾𝑖,𝛼𝑖,𝛿𝑖,𝑀𝑖), for all 𝐾𝑖𝐑.

Theorem 4.3. If 𝜆=max1𝑖𝑘|𝜆𝑖|<1𝛿, 𝛿=max1𝑖𝑘𝛿𝑖(0,1), 𝑓𝑖𝐵𝐶(𝐾𝑖,𝛼𝑖,𝛿𝑖,𝑀𝑖), for all 𝐾𝑖𝐑, for all 𝑖𝑘, and max(max1𝑖𝑘|𝛼𝑖|,max1𝑖𝑘|𝛽𝑖|)<(1max1𝑖𝑘|𝜆𝑖|𝛿)/4𝑘max1𝑖𝑘𝛿𝑖, then system (1.1) is globally Lyapunov stable for any finite arbitrary initial conditions. It is also permanent for any initial conditions 𝑥0𝐾0(𝑎1,,𝑎2𝑘,𝑏1,,𝑏2𝑘) with the compact set 𝐾0 defined in Theorem 4.1.

Proof. If system (1.1) is taken, then (4.4) is replaced with
𝑥𝑛+1=Λ𝑥𝑛+𝐵𝑥𝑛1+𝑓𝑥𝑛1𝐵𝑥𝑛1,𝑛,𝑗𝐍,(4.11) where
𝑓𝑥𝑛1=𝑓1𝛼2𝑥𝑛(2)𝛽𝑖+1𝑥(2)𝑛1,,𝑓𝑘𝛼1𝑥𝑛(1)𝛽𝑖+1𝑥(1)𝑛1𝑇.(4.12) The description (4.6) is similar to (1.1) via an unforced linear system (3.3)–(3.7) with a forcing sequence {(𝑓(𝑥𝑛1)𝐵𝑥𝑛1)}0 so that both solution sequences are identical under identical initial conditions. One gets directly from (4.11) that
𝑥𝑛+𝑘=Λ𝑛+𝑘𝑥0+𝑛+𝑘1𝑖=0Λ𝑛+𝑘𝑖1𝐵𝑥𝑖+𝑓𝑥𝑛1𝐵𝑥𝑛1,𝑛𝐍,𝑘𝑘,(4.13) so that
𝑥𝑛+𝑘2𝜆𝑛𝑥02+1𝜆𝑛(1𝜆𝐵2max0𝑖𝑛+𝑘1𝑥𝑖2+max0𝑖𝑛+𝑘1𝑓𝑥𝑖𝐵𝑥𝑖2)𝜆𝑛𝑥02+1𝜆𝑛21𝜆𝐵2+𝛿max0𝑖𝑛+𝑘1𝑥𝑖2+1𝜆𝑛𝐶1.1𝜆(4.14) Then by direct extension of (4.7) when using (4.14),
max1𝑖𝑛+𝑘1𝑥𝑛+𝑖2𝜆𝑛𝑥02+1(1𝜆4𝛿max𝛼,𝛽𝑘+𝛿max1𝑖𝑛+𝑘1𝑥𝑖2+𝐶1)+𝑥02+𝑥12,𝑛𝐍,𝑘𝑘,(4.15) with 𝛿(0,1) for some finite 𝐶1𝐑+ since |𝑓𝑖(𝑥)|𝛿|𝑥|+𝐶1, for all 𝑖𝑘, from Lemma 4.2. Thus, max1𝑖𝑛+𝑘1(𝑥𝑛+𝑖2) may be regrouped in the left-hand side provided that
11>1𝜆4𝛿max𝛼,𝛽<𝑘+𝛿max𝛼,𝛽1𝜆𝛿4𝛿𝑘.(4.16) Then, under similar reasoning as that used to derive (4.8)-(4.9), one gets from (4.15) that
𝑥𝑛2max1𝑖𝑛+𝑘1𝑥𝑛+𝑖211𝜆𝛿1+4𝑘max𝛼,𝛽1𝜆1+𝜆𝑛𝑥02+𝑥12+𝐶111𝜆𝛿1+4𝑘max𝛼,𝛽1𝜆1+𝜆𝑛0𝑥02+𝑥12+𝐶111+𝜀1𝜆𝛿1+4𝑥𝑘max𝛼,𝛽1𝜆02+𝑥12+𝐶111+𝜀1𝜆𝛿1+4(𝑘max𝛼,𝛽1𝜆2𝑘𝑖=1𝑎max2𝑖,𝑏2𝑖+𝐶1)1/211+𝜀1𝜆𝛿1+4𝑥𝑘max𝛼,𝛽1𝜆max02,𝑥12+𝐶1131𝜆𝛿1+4𝑥𝑘max𝛼,𝛽1𝜆max02,𝑥12+𝐶1,(4.17) for all 𝜀𝐑+, for all 𝑛(𝑛0)𝐍, depending on 𝑛0, which depends on 𝜀, for any 𝐍϶𝑛0ln𝜀/ln𝜆. The solution sequences are all bounded under any finite initial conditions and enter the compact set 𝐾 defined by
{𝑥𝐑𝑘|||𝑥𝑖|||1+𝜀𝑘(11𝜆𝛿1+4𝑘max𝛼,𝛽×(1𝜆2𝑘𝑖=1|||𝑎max𝑖|||,|||𝑎𝑖|||+𝐶1)),𝑖𝑘},(4.18) for all 𝑛(𝑛0)𝐍, for any set of initial conditions in the compact set 𝐾0. Furthermore, 𝐾 is independent of each particular set of initial conditions in 𝐾0. Then, system (1.1) is globally Lyapunov stable and permanent.

Some simple properties concerning the instability of (1.1) based on simple constraints on the nonlinear functions, such as the stated boundedness from below of the strongest one of boundedness from above and below, are now established in the subsequent result.

Theorem 4.4. The following properties hold.
(i)If |𝜆𝑖|1 and 𝑓𝑖𝐑𝐑 is bounded from above and below, then |𝑥𝑛(𝑖)| is bounded, for all 𝑛𝐍. If |𝜆𝑖|>1 and 𝑓𝑖𝐑𝐑 is bounded from above and below, then almost all solution sequences {𝑥𝑛(𝑖)}0 for sufficiently large finite absolute values of the initial conditions are unbounded. Thus, system (1.1) is unstable under sufficiently large absolute values of the initial conditions for some 𝑖𝑘.(ii)Assume that 𝑓𝑖𝐵(𝐾𝑖) and |𝜆𝑖|>1 for some 𝑖𝑘. Then |𝑥(𝑖)𝑛+1|>|𝑥𝑛(𝑖)|, for all 𝑛𝐍, and |𝑥𝑛(𝑖)| as 𝑛 if |𝑥0(𝑖)|>|𝐾𝑖|/(|𝜆𝑖|1) (|𝑥0(𝑖)||𝐾𝑖|/(|𝜆𝑖|1)if𝐾i0). Thus, system (1.1) is unstable under such sufficiently large absolute values of the initial conditions for some 𝑖𝑘.

Proof. (i) If <𝑀1𝑖𝑓𝑖(𝑥)𝑀2𝑖<, for all 𝑥𝐑, for some 𝑀𝑗𝑖, j = 1,2, and some 𝑖𝑘, then
|||𝜆𝑛𝑖|||(|||𝑥𝑖0|||+|𝑗=0𝜆𝑖𝑗1|||𝑀|max1𝑖|||,|||𝑀2𝑖|||)|||𝑥𝑖𝑛||||||𝜆𝑛𝑖|||(|||𝑥𝑖0||||𝑛1𝑗=0𝜆𝑖𝑗1|max0𝑗𝑖|||𝑓𝑖𝛼𝑖𝑥𝑖+1𝑗𝛽𝑖𝑥𝑖+1𝑗1|||)|||𝜆𝑛𝑖|||(|||𝑥𝑖0||||𝑛1𝑗=0𝜆𝑖𝑗1|||𝑀|max1𝑖|||,|||𝑀2𝑖|||)|||𝜆𝑛𝑖|||||||𝑥𝑖0||||𝑗=0𝜆𝑖𝑗1|||𝑀|max1𝑖|||,|||𝑀2𝑖||||.(4.19) If |𝜆𝑖|1, then the sequence {|𝑥𝑛(𝑖)|}0 is bounded so that the sequence {|𝑥𝑛(𝑖)|}0 may be unbounded only if |𝜆𝑖|>1. If |𝜆𝑖|>1, then 0||𝑥0(𝑖)||𝑗=0𝜆𝑖𝑗1|max(|𝑀1𝑖|,|𝑀2𝑖|)|<, and, furthermore, if |𝑥0(𝑖)|>(|𝜆𝑖|/(|𝜆𝑖|1))max(|𝑀1𝑖|,|𝑀2𝑖|)|𝑗=0𝜆𝑖𝑗1|max(|𝑀1𝑖|,|𝑀2𝑖|), then there is a strictly monotonically increasing subsequence {|𝑥𝑛(𝑖)|}𝑛𝑆 of {|𝑥𝑛(𝑖)|}0, where 𝑆={𝑛1,𝑛2,} is a countable subset of N, so that |𝑥𝑛(𝑖)𝑗+1|>|𝑥𝑛(𝑖)𝑗|, for all 𝑛𝑗𝑆, and |𝑥𝑛(𝑖)𝑗| as 𝑆϶𝑛𝑗 (i.e., it diverges).
If 𝑓𝑖𝐵𝐶(𝐾𝑖,1,𝛿𝑖,𝑀𝑖), then <|𝐾𝑖|𝑓𝑖(𝑥)𝛿𝑥, for all 𝑥(𝑀𝑖)𝐑 and all 𝐾𝑖, such that 𝐾𝑖+|𝐾𝑖|0.
(ii) From (1.1), 𝑓𝑖𝐵(𝐾𝑖), and |𝜆𝑖|>1, it follows that 𝑥𝑖𝑛+12𝑥𝑖𝑛2=𝜆2𝑖𝑥1𝑖𝑛2+𝑓2𝑖𝛼𝑖𝑥𝑖+1𝑛𝛽𝑖𝑥𝑖+1𝑛1+2𝜆𝑖𝑓𝑖𝛼𝑖𝑥𝑖+1𝑛𝛽𝑖𝑥𝑖+1𝑛1𝑥𝑖𝑛𝑔𝑖𝑛|||𝑥𝑖𝑛|||=𝐾2𝑖2𝜆𝑖|||𝐾𝑖|||𝜆2𝑖|||𝑥1𝑖𝑛|||||𝑥𝑖𝑛||>0(4.20) if |𝑥0(𝑖)|>2|𝜆𝑖||𝐾𝑖|/(𝜆2𝑖1)(|𝑥0(𝑖)||𝐾𝑖|/(|𝜆𝑖|1)if𝐾i0)|𝑥(𝑖)𝑛+1|>|𝑥𝑛(𝑖)|, for all 𝑛𝐍, so that the absolute value of the solution sequence is monotonically increasing so that it diverges. Less stringent condition for the initial conditions follows by calculating the zeros of the convex function 𝑔𝑛(𝑖)(|𝑥𝑛(𝑖)|)𝑔𝑛(𝑖)(|𝑥𝑛(𝑖)|) which are 𝑔(𝑖)2𝑛=|𝐾𝑖|/(|𝜆𝑖|1)𝑔(𝑖)1𝑛=|𝐾𝑖|/(|𝜆𝑖|+1), which implies that 𝑔𝑛(𝑖)(|𝑥𝑛(𝑖)|)0 if |𝑥𝑛(𝑖)|[𝑔(𝑖)1𝑛,𝑔(𝑖)2𝑛], and (𝑥(𝑖)𝑛+1)2(𝑥𝑛(𝑖))2𝑔𝑛(𝑖)(|𝑥𝑛(𝑖)|)>0 if |𝑥𝑛(𝑖)|(,𝑔(𝑖)2𝑛)(𝑔(𝑖)2𝑛,). This directly completes the proof.

5. Positivity Results

Some positivity properties of the solution sequences of system (1.1) are now formulated in the subsequent formal result.

Theorem 5.1. The following properties hold.
(i)Any solution vector sequence 𝑥𝑛=(𝑥𝑛(1),𝑥𝑛(2),,𝑥𝑛(𝑘))𝑇 of (1.1) is nonnegative, for all 𝑛𝐍, and any finite nonnegative 𝑥0(𝑖)0, for all 𝑖𝑘, if 𝑓𝑖(𝛼𝑖𝑥0(𝑖+1)𝛽𝑖𝑥(𝑖+1)1)𝜆𝑖𝑥0(𝑖), for all 𝑖𝑘, and𝑓𝑖𝛼𝑖𝑥𝑖+1𝑛𝛽𝑖𝑥𝑖+1𝑛1𝜆𝑖𝑥𝑖𝑛=(𝜆𝑖𝑛+1𝑥𝑖0+𝑛1𝑗=0𝜆𝑖𝑛𝑗𝑓𝑖𝛼𝑖𝑥𝑖+1𝑗𝛽𝑖𝑥𝑖+1𝑗1),(5.1) for all 𝑖𝑘, for all 𝑛𝐍. Then, system (1.1) is positive.(ii)Any solution vector sequence of (1.1) is nonnegative, for all 𝑛𝐍, and any finite nonnegative 𝑥0(𝑖)