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

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 [1–9]. 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, [13–16] 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]:

𝑥^𝑖_𝑛+₁=𝜆_𝑖𝑥^𝑖_𝑛+𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑛−𝛽_𝑖𝑥^𝑖+1_𝑛−1,∀𝑖∈𝑘∶=1,2,…,𝑘,(1.1) with 𝑥^𝑛(𝑘+1)≡𝑥^𝑛(1), for all 𝑛∈𝐍; 𝜆_𝑖∈𝐑, 𝛼_𝑖∈𝐑, 𝛽_𝑖∈𝐑; and 𝑓_𝑖∶𝐑→𝐑, for all 𝑖∈𝑘, under arbitrary initial conditions 𝑥^0(𝑖),𝑥^(𝑖)−1, for all 𝑖∈𝑘. The identity 𝑥^𝑛(𝑘+1)≡𝑥^𝑛(1) allows the inclusion in a unified shortened notation via (1.1) of the dynamics:

𝑥^𝑘_𝑛+1=𝜆_𝑖𝑥^𝑘_𝑛+𝑓_𝑖𝛼^𝑖𝑥¹_𝑛−𝛽_𝑖𝑥¹_𝑛−₁,∀𝑖∈𝑘,(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 𝑥_𝑛∶=(𝑥^𝑛(1),𝑥^𝑛(2),…,𝑥^𝑛(𝑘))^𝑇∈𝐑^𝑘, for all 𝑛∈𝐍, under initial conditions 𝑥_𝑗∶=(𝑥^𝑗(1),𝑥^𝑗(2),…,𝑥^𝑗(𝑘))^𝑇∈𝐑^𝑘, 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 𝑓_𝑖∶𝐑→𝐑, for all 𝑖∈𝑘. 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₊∶={𝑧∈𝐑∶𝑧>0}, 𝐑₀₊∶={𝑧∈𝐑∶𝑧≥0}, 𝐑₀₋∶={𝑧∈𝐑∶𝑧≤0}. “∧” is the logic conjunction symbol. 𝐍₀∶=𝐍∪{0}. If 𝑃∈𝐑^𝑛×𝑛, then 𝑃^𝑇 is the transpose of P.

𝑃>0,𝑃≥0,𝑃<0,𝑃≤0 denote, respectively, P positive definite, semidefinite positive, negative definite, and negative semidefinite. 𝑃≥0,𝑃>0,𝑃≫0 denote, respectively, P nonnegative (i.e., none of its entries is negative, also denoted as 𝑃∈𝐑^𝑛×𝑛0+), P positive (i.e., 𝑃≥0 with at least one of its entries being positive), and P strictly positive (i.e., all of its entries are positive). Thus, 𝑃>0⇒𝑃≥0 and 𝑃≫0⇒𝑃>0⇒𝑃≥0, 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 𝐑^𝑘 of (1.1) will be nonnegative in some interval S, denoted by 𝑥_𝑛≥0 (identical to 𝑥_𝑛∈𝐑^𝑘0+), for all 𝑛∈𝑆⊂𝐍, if all the components are nonnegative for 𝑛∈𝑆⊂𝐍. If, in addition, at least one component is positive, then the solution vector is said to be positive, denoted by 𝑥_𝑛>0 (implying that 𝑥_𝑛∈𝐑^𝑘0+), for all 𝑛∈𝑆⊂𝐍. If all of them are positive in S, then the solution vector is said to be strictly on a discrete interval, denoted by 𝑥_𝑛≫0 (identical to 𝑥_𝑛∈𝐑^𝑘+ and implying that 𝑥_𝑛>0 and 𝑥_𝑛∈𝐑^𝑘0+), for all 𝑛∈𝑆⊂𝐍.

‖‖₂ and ‖‖₁ are the ℓ₂ and ℓ₁ norms of vectors and induced norms of matrices, respectively. 𝐼_𝑛 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 𝑓_𝑖∶𝐑→𝐑 in system (1.1) as follows irrespective of the initial conditions:

𝐵𝐾_𝑖∶=𝑓^𝑖∶𝐑⟶𝐑∶𝑓_𝑖𝑦≥𝑓_𝑖𝑥≥𝐾_𝑖,∀𝑥,𝑦>𝑥∈𝐑,𝐾_𝑖∈𝐑,∀𝑖∈𝑘,(2.1) and sets of linearly upper bounded real functions:

𝐶𝛾_𝑖,𝛿_𝑖,𝑀_𝑖∶=𝑓^𝑖∶𝐑⟶𝐑∶𝑓_𝑖𝑥≤𝛿^𝑖𝛾_𝑖𝑥,∀𝑥>𝑀_𝑖∈𝐑₊,𝛿_𝑖∈0,1,∀𝑖∈𝑘,(2.2) for 𝛾_𝑖≠0 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(𝐾_𝑖,𝛾_𝑖,𝛿_𝑖,𝑀_𝑖)∶=𝐵(𝐾_𝑖)∩𝐶(𝛾_𝑖,𝛿_𝑖,𝑀_𝑖), that is,

𝐵𝐶𝐾_𝑖,𝛾_𝑖,𝛿_𝑖,𝑀_𝑖∶=𝑓^𝑖∶𝐑→𝐑∶𝑓_𝑖𝑦≥𝑓_𝑖𝑥≥𝐾_𝑖∧𝑓_𝑖𝑥≤𝛿^𝑖𝛾_𝑖∀𝑥,𝑦>𝑥∈𝐑,𝐾_𝑖𝑥,∈𝐑,𝛿_𝑖∈0,1,∀𝑖∈𝑘,(2.3) for 𝛾_𝑖≠0. The above definitions facilitate the potential restrictions on the functions 𝑓_𝑖∶𝐑→𝐑, 𝑖∈𝑘, required to derive the various results of the paper. The constraints on the functions 𝑓_𝑖∶𝐑→𝐑, for all 𝑖∈𝑘, used in the above definitions of sets, have been proposed by Stević for 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,𝛾_𝑖,𝛿_𝑖,𝑀_𝑖) and then used to prove the conditions of permanence of (1.1) in [1] for some 𝐾_𝑖=𝐾, 𝑀_𝑖=𝑀>0, and 𝛿_𝑖∈(0,1), for all 𝑖∈𝑘, subject to 𝜆_𝑖∈[0,𝛽_𝑖/𝛼_𝑖], 𝛼_𝑖>𝛽_𝑖≥0, for all 𝑖∈𝑘. The subsequent technical assumption will be then used in some of the forthcoming results.

Assumption 2.1. 𝛼_𝑖>0 and 0<𝛿_𝑖<min(1,𝛼^𝑖−1).

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

Assertion 2.2. For any given 𝑖∈𝑘, 𝑓_𝑖∈𝐵(𝐾_𝑖)⇒𝑓_𝑖(𝛼_𝑖𝑥^𝑛(𝑖+1)−𝛽_𝑖𝑥^{(𝑖+1)𝑛−1})≥𝐾_𝑖, for all 𝑛∈𝐍∪{0,−1}.

Assertion 2.3. (i) For any given 𝑖∈𝑘, 𝑓_𝑖∈𝐶(𝛾_𝑖,𝛿_𝑖,𝑀_𝑖)⇔𝑓_𝑖(𝛾_𝑖((𝛼_𝑖/𝛾_𝑖)𝑥^𝑛(𝑖+1)−(𝛽_𝑖/𝛾_𝑖)𝑥^{(𝑖+1)𝑛−1}))≤(𝛿_𝑖/𝛾_𝑖)(𝛼_𝑖𝑥^𝑛(𝑖+1)−𝛽_𝑖𝑥^{(𝑖+1)𝑛−1}) if 𝑥^𝑛(𝑖+1)>(𝛽_𝑖/𝛼_𝑖)𝑥^{(𝑖+1)𝑛−1}+(𝛾_𝑖/𝛼_𝑖)𝑀_𝑖, for all 𝑛∈𝐍∪{0,−1}, for any real constants 𝛽_𝑖,𝛼_𝑖>0. (ii)𝑓_𝑖∈𝐶(𝛼_𝑖,𝛿_𝑖,𝑀_𝑖)⇔𝑓_𝑖(𝛼_𝑖(𝑥^𝑛(𝑖+1)−(𝛽_𝑖/𝛼_𝑖)𝑥^{(𝑖+1)𝑛−1}))≤(𝛿_𝑖/𝛼_𝑖)(𝛼_𝑖𝑥^𝑛(𝑖+1)−𝛽_𝑖𝑥^{(𝑖+1)𝑛−1}) if 𝑥^𝑛(𝑖+1)>(𝛽_𝑖/𝛼_𝑖)𝑥^{(𝑖+1)𝑛−1}+𝑀_𝑖, for all 𝑛∈𝑁∪{0,−1}, for any real constants 𝛽_𝑖,𝛼_𝑖>0.(iii)𝑓_𝑖∈𝐶(1,𝛿_𝑖,𝑀_𝑖)⇔𝑓_𝑖(𝛼_𝑖𝑥^𝑛(𝑖+1)−𝛽_𝑖𝑥^{(𝑖+1)𝑛−1})≤𝛿_𝑖(𝛼_𝑖𝑥^𝑛(𝑖+1)−𝛽_𝑖𝑥^{(𝑖+1)𝑛−1}) if 𝑥^𝑛(𝑖+1)>(𝛽_𝑖/𝛼_𝑖)𝑥^{(𝑖+1)𝑛−1}+𝑀_𝑖/𝛼_𝑖, for all 𝑛∈𝑁∪{0,−1}, for any real constants 𝛽_𝑖,𝛼_𝑖>0.(iv)𝐶(1,𝛿_𝑖,𝑀_𝑖)=𝐶(𝛼_𝑖,𝛼_𝑖𝛿_𝑖,𝑀_𝑖/𝛼_𝑖) if Assumption 2.1 holds.

Proof. Assertion 2.3 (i)–(iii) follow directly from the definitions of 𝐵(𝐾_𝑖) and 𝐶(𝛾_𝑖,𝛿_𝑖,𝑀_𝑖), for all 𝑖∈𝑘.
Assertion 2.3 (iv) The proof is split into proving the two claims below.
Claim 1. 𝐶(1,𝛿_𝑖,𝑀_𝑖)⊂𝐶(𝛼_𝑖,𝛼_𝑖𝛿_𝑖,𝑀_𝑖/𝛼_𝑖).
Proof of Claim 1. 𝑓_𝑖∈𝐶(1,𝛿_𝑖,𝑀_𝑖)⇔𝑓_𝑖(𝛼_𝑖𝑥^𝑛(𝑖+1)−𝛽_𝑖𝑥^{(𝑖+1)𝑛−1})=𝑓_𝑖(𝛼_𝑖(𝑥^𝑛(𝑖+1)−(𝛽_𝑖/𝛼_𝑖)𝑥^{(𝑖+1)𝑛−1}))≤𝛿_𝑖(𝛼_𝑖𝑥^𝑛(𝑖+1)−𝛽_𝑖𝑥^{(𝑖+1)𝑛−1})=𝛿_𝑖𝛼_𝑖(𝑥^𝑛(𝑖+1)−(𝛽_𝑖/𝛼_𝑖)𝑥^{(𝑖+1)𝑛−1}) if 𝛼_𝑖𝑥^𝑛(𝑖+1)−𝛽_𝑖𝑥^{(𝑖+1)𝑛−1}>𝑀_𝑖⇒𝑓_𝑖∈𝐶(𝛼_𝑖,𝛼_𝑖𝛿_𝑖,𝑀_𝑖/𝛼_𝑖) if Assumption 2.1 holds.
Claim 2. 𝐶(𝛼_𝑖,𝛼_𝑖𝛿_𝑖,𝑀_𝑖/𝛼_𝑖)⊂𝐶(1,𝛿_𝑖,𝑀_𝑖).
Proof of Claim 2. 𝑓_𝑖∈𝐶(𝛼_𝑖,𝛼_𝑖𝛿_𝑖,𝑀_𝑖/𝛼_𝑖)⇒𝑓_𝑖(𝛼_𝑖(𝑥^𝑛(𝑖+1)−(𝛽_𝑖/𝛼_𝑖)𝑥^{(𝑖+1)𝑛−1}))≤𝛼_𝑖𝛿_𝑖(𝑥^𝑛(𝑖+1)−(𝛽_𝑖/𝛼_𝑖)𝑥^{(𝑖+1)𝑛−1})=𝛿_𝑖(𝛼_𝑖𝑥^𝑛(𝑖+1)−𝛽_𝑖𝑥^{(𝑖+1)𝑛−1}) if 𝛼_𝑖𝑥^𝑛(𝑖+1)−𝛽_𝑖𝑥^{(𝑖+1)𝑛−1}>𝑀_𝑖⇒𝑓_𝑖∈𝐶(1,𝛿_𝑖,𝑀_𝑖) 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(𝐾_𝑖), 𝐶(𝛾_𝑖,𝛿_𝑖,𝑀_𝑖), and 𝐵𝐶(𝐾_𝑖,𝛾_𝑖,𝛿_𝑖,𝑀_𝑖).

Assertion 2.4. For any 𝑖∈𝑘, consider 𝐶(𝛾_𝑖,𝛿_𝑖,𝑀_𝑖) for some given 3-tuple (𝛾_𝑖,𝛿_𝑖,𝑀_𝑖) in 𝐑×(0,1)×𝐑, and consider any discrete collection of distinct admissible triples (𝛾_{𝑖𝑗𝑖𝛾},𝛿_{𝑖𝑗𝑖𝛿},𝑀_{𝑖𝑗𝑖𝑀})∈𝐑×(0,1)×𝐑 (𝑗_𝑖𝛾∈𝐽_𝑖𝛾,𝑗_𝑖𝛿∈𝐽,𝑗_𝑖𝑀∈𝐽_𝑖𝑀) subject to the constraints 𝛿_{𝑖𝑗𝑖𝛿}≤𝛿_𝑖 and 𝑀_{𝑖𝑗𝑖𝑀}≥𝑀_𝑖, for all (𝑗_𝑖𝛿,𝑗_𝑖𝑀)∈𝐽_𝑖𝛿×𝐽_𝑖𝑀, leading to the associated 𝐶(𝛾_{𝑖𝑗𝑖𝛾},𝛿_{𝑖𝑗𝑖𝛿},𝑀_{𝑖𝑗𝑖𝑀}). Define the binary relation 𝑅_𝑖 in 𝐶(𝛾_𝑖,𝛿_𝑖,𝑀_𝑖) as 𝑓_𝑖𝑅_𝑖𝑔_𝑖⇔𝑓_𝑖,𝑔_𝑖∈𝐶(𝛾_{𝑖𝑗𝑖𝛾},𝛿_{𝑖𝑗𝑖𝛿},𝑀_{𝑖𝑗𝑖𝑀}). Then, 𝑅_𝑖 is not an equivalence relation so that 𝐶(𝛾_{𝑖𝑗𝑖𝛾},𝛿_{𝑖𝑗𝑖𝛿},𝑀_{𝑖𝑗𝑖𝑀}) are not equivalence classes in 𝐶(𝛾_𝑖,𝛿_𝑖,𝑀_𝑖) with respect to 𝑅_𝑖. Also, the sets 𝐵(𝐾_{𝑖𝑗𝑖𝐾}) and 𝐵𝐶(𝐾_{𝑖𝑗𝑖𝐾},𝛾_{𝑖𝑗𝑖𝛾},𝛿_{𝑖𝑗𝑖𝛿},𝑀_{𝑖𝑗𝑖𝑀}) for any given respective collections 𝐾_{𝑖𝑗𝑖𝐾}≤𝐾_𝑖, 𝛿_{𝑖𝑗𝑖𝛿}≤𝛿_𝑖, 𝑀_{𝑖𝑗𝑖𝑀}≥𝑀_𝑖, for all (𝑗_𝑖𝐾,𝑗_𝑖𝛿,𝑗_𝑖𝑀)∈𝐽_𝑖𝐾×𝐽_𝑖𝛿×𝐽_𝑖𝑀, are not equivalence classes, respectively, in 𝐵(𝐾_𝑖) and 𝐵𝐶(𝐾_𝑖,𝛾_𝑖,𝛿_𝑖,𝑀_𝑖).

Proof. In view of Assertion 2.3(iv), 𝛾_{𝑖𝑗𝑖𝛾} can be all set equal to unity with no loss of generality, which is done to simplify the notation in the proof. Note that 𝑓_𝑖𝑅_𝑖𝑔_𝑖⇔𝑓_𝑖,𝑔_𝑖∈𝐶(1,𝛿_{𝑖𝑗𝑖𝛿},𝑀_{𝑖𝑗𝑖𝑀})⇒𝑓_𝑖,𝑔_𝑖∈𝐶(1,𝛿_{𝑖𝑗𝑖𝛿},𝑀_{𝑖𝑗𝑖𝑀}) for some (𝛿_{𝑖𝑗𝑖𝛿},𝑀_{𝑖𝑗𝑖𝑀})×𝐑. Now, consider 𝐶(1,𝛿^𝑖𝑗_𝑖𝛿,𝑀_{𝑖𝑗𝑖𝑀}) with 𝛿^𝑖𝑗_𝑖𝛿>𝛿_𝑖 such that 𝛿_𝑖≥𝛿^𝑖𝑗_𝑖𝛿(>𝛿_{𝑖𝑗𝑖𝛿})∈{𝛿_𝑖𝑗∶𝑗∈𝐽_𝑖𝛿}. Then, 𝐶(1,𝛿_{𝑖𝑗𝑖𝛿},𝑀_{𝑖𝑗𝑖𝑀})⊂𝐶(1,𝛿^𝑖𝑗_𝑖𝛿,𝑀_{𝑖𝑗𝑖𝑀}). Since the equivalence classes with respect to any equivalence relation are disjoint, 𝐶(1,𝛿_{𝑖𝑗𝑖𝛿},𝑀_{𝑖𝑗𝑖𝑀}) in 𝐶(1,𝛿_𝑖,𝑀_𝑖) with respect to 𝑅_𝑖 is not an equivalence class unless 𝐶(1,𝛿_{𝑖𝑗𝑖𝛿},𝑀_{𝑖𝑗𝑖𝑀})=𝐶(1,𝛿^𝑖𝑗_𝑖𝛿,𝑀_{𝑖𝑗𝑖𝑀}). Now, consider the linear function ℎ_𝑖∶𝐑→𝐑 defined by ℎ_𝑖(𝑥)∶=𝛿^𝑖𝑗_𝑖𝛿𝑥>𝛿_{𝑖𝑗𝑖𝛿}𝑥 so that 𝐶(1,𝛿^𝑖𝑗_𝑖𝛿,𝑀_{𝑖𝑗𝑖𝛿})϶ℎ_𝑖∉𝐶(1,𝛿_{𝑖𝑗𝑖𝛿},𝑀_{𝑖𝑗𝑖𝛿}). Thus, 𝐶(1,𝛿_{𝑖𝑗𝑖𝑖𝛿},𝑀_𝑖)≠𝐶(1,𝛿^𝑖𝑗_𝑖𝛿,𝑀_{𝑖𝑗𝑖𝛿}). Then, 𝑅_𝑖 (𝑖∈𝑘) are not equivalence relations, and there are no equivalence classes in 𝐶(𝛾_𝑖,𝛿_𝑖,𝑀_𝑖) (𝑖∈𝑘) with respect to 𝑅_𝑖 (𝑖∈𝑘). The remaining part of the proof follows in a similar way by using the definitions of the sets 𝐵(𝐾_𝑖) and 𝐵𝐶(𝐾_𝑖,𝛾_𝑖,𝛿_𝑖,𝑀_𝑖), 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(𝐾_𝑖), 𝐶(𝛾_𝑖,𝛿_𝑖,𝑀_𝑖), or 𝐵𝐶(𝐾_𝑖,𝛾_𝑖,𝛿_𝑖,𝑀_𝑖). This allows the characterization of the above properties under few sets of conditions on the nonlinear functions in the difference system (1.1). If 𝑓_𝑖∈𝐶(1,𝛿_𝑖,𝑀_𝑖), for all 𝑖∈𝑘, then the auxiliary linear system to (1.1) is

𝑥^𝑖_𝑛+₁=𝜆_𝑖𝑥^𝑖_𝑛+𝛿_𝑖𝛼^𝑖𝑥^𝑖+1_𝑛−𝛽_𝑖𝑥^𝑖+1_𝑛−1,∀𝑖∈𝑘.(3.1) If 𝑓_𝑖∈𝐶(𝛼_𝑖,𝛿_𝑖,𝑀_𝑖), for all 𝑖∈𝑘, then the auxiliary linear system to (1.1) is

𝑥^𝑖_𝑛+₁=𝜆_𝑖𝑥^𝑖_𝑛+𝛿_𝑖𝑥^𝑖+1_𝑛−𝛽^𝑖𝛼_𝑖𝑥^𝑖+1_𝑛−1,∀𝑖∈𝑘.(3.2) System (3.1) may be equivalently rewritten as follows by defining the state vector sequence 𝑥_𝑛∶=(𝑥^𝑛(1),𝑥^𝑛(2),…,𝑥^𝑛(𝑘))^𝑇∈𝐑^𝑘, for all 𝑛∈𝐍, as the kth-order difference system:

𝑥_𝑛+1=𝐴𝑥_𝑛+𝐵𝑥_𝑛−1=Λ+𝐶𝑥_𝑛+𝐵𝑥_𝑛−1=Λ𝑥_𝑛+𝐵𝑥_𝑛−1,∀𝑛∈𝐍,(3.3) with initial conditions 𝑥_𝑖∶=(𝑥^𝑖(1),𝑥^𝑖(2),…,𝑥^𝑖(𝑘))^𝑇∈𝐑^𝑘 for 𝑖=0,−1, where 𝑥_𝑛∶=(𝑥^𝑇𝑛⋮𝑥^𝑇𝑛−1)^𝑇∈𝐑^2𝑘 and

𝐴=⎡⎢⎢⎢⎢⎢⎣𝜆¹0𝛿𝜆₁𝛼₁₂0⋯0⋮⋮𝛿⋱₂𝛼₂⋮0𝛿⋯_𝑘−10𝛼_𝑘−1𝛿_𝑘𝛼_𝑘0⋯0𝜆_𝑘⎤⎥⎥⎥⎥⎥⎦∈𝐑_𝑘×𝑘𝐵=⎡⎢⎢⎢⎢⎢⎣0−𝛿₁,(3.4)00𝛽₁−𝛿0₂⋯0⋮⋮⋱𝛽₂⋮0−⋯𝛿_𝑘−10𝛽_𝑘−1−𝛿_𝑘𝛽_𝑘0⋯00⎤⎥⎥⎥⎥⎥⎦∈𝐑^𝑘×𝑘Λ=Diag𝜆₁,(3.5),𝜆₂,…,𝜆_𝑘,𝐶=⎡⎢⎢⎢⎢⎢⎣0𝛿¹00𝛼₁𝛿₂0⋯0⋮⋮⋱𝛼₂⋮0𝛿⋯_𝑘−10𝛼_𝑘−1𝛿_𝑘𝛼_𝑘0⋯00⎤⎥⎥⎥⎥⎥⎦,(3.6)𝐵=𝐵⋮𝐶∈𝐑^𝑘×2𝑘.(3.7) The one-step delay may be removed by defining the following extended 2kth-order system of state vector 𝑥_𝑛∶=(𝑥^𝑇𝑛⋮𝑥^𝑇𝑛−1)^𝑇∈𝐑^2𝑘 satisfying

𝑥_𝑛+1=𝐴𝑥_𝑛,∀𝑛∈𝐍,(3.8) with 𝑥₀∶=(𝑥⁰⁽¹⁾,𝑥⁰⁽²⁾,…,𝑥^0(𝑘),𝑥⁽¹⁾⁻¹,𝑥⁽²⁾⁻¹,…,𝑥^(𝑘)−1)^𝑇∈𝐑^2𝑘 and

𝐴=⎡⎢⎢⎢⎣𝐼𝐴⋯⋮𝐵_𝑘⋮0⎤⎥⎥⎥⎦∈𝐑^2𝑘×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 ℓ₂-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 𝑛≥𝑛₀ 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 0∈𝐾. 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 𝜀∈𝐑₊. This phenomenon cannot occur for system (1.1) under the requirement 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,𝛾_𝑖,𝛿_𝑖,𝑀_𝑖), for all 𝑖∈𝑘, 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 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖), for all 𝑖∈𝑘, since the linear system defined with 𝑓_𝑖(𝑥)=𝛿_𝑖𝑥, for all 𝑖∈𝑘, in (1.1) has to be globally stable in order to keep global stability for any 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖), for all 𝑖∈𝑘.

Theorem 3.4. System (1.1) is globally stable and permanent for any given set of functions 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖) for any given 𝐾_𝑖∈𝐑 and any given 𝑀_𝑖∈𝐑, for all 𝑖∈𝑘, only if the subsequent properties hold.
(i) |𝜆_𝑖|≤1, for all 𝑖∈𝑘.
(ii) ‖𝐴‖₂≤1, equivalently, ‖𝑊‖₂=‖𝐴^𝑇𝐴‖₂≤1,where𝑊∶=𝐴^𝑇𝐴=⎡⎢⎢⎢⎣𝑊¹¹⋮𝑊₁₂𝑊_𝑇12⋯⋮𝑊₂₂⎤⎥⎥⎥⎦∈𝐑_2𝑘×2𝑘,(3.10)where 𝑊₁₁∶=𝐴^𝑇𝐴+𝐼_𝑘=⎡⎢⎢⎢⎢⎢⎢⎢⎣1+𝜆²¹+𝛿^2𝑘𝛼^2𝑘𝜆₁𝛿₁𝛼₁𝜆₃𝛿₃𝛼₃⋯𝜆_𝑘𝛿_𝑘𝛼_𝑘𝜆₁𝛿₁𝛼₁1+𝜆²²+𝛿²¹𝛼²¹𝜆₂𝛿₂𝛼₂⋯𝜆_𝑘−1𝛿_𝑘−1𝛼_𝑘−1𝜆⋮⋮⋮_𝑘−2𝛿_𝑘−2𝛼_𝑘−2𝜆_𝑘−1𝛿_𝑘−1𝛼_𝑘−1⋯𝜆^2𝑘−1+𝛿^2𝑘−2𝛼^2𝑘−2𝜆₂𝛿₂𝛼₂𝜆_𝑘𝛿_𝑘𝛼_𝑘𝜆_𝑘−1𝛿_𝑘−1𝛼_𝑘−1⋯𝜆₂𝛿₂𝛼₂1+𝜆^2𝑘+𝛿^2𝑘−1𝛼^2𝑘−1⎤⎥⎥⎥⎥⎥⎥⎥⎦,(3.11)𝑊₁₂∶=𝐴^𝑇𝐵, and 𝑊₂₂∶=𝐵^𝑇𝐵=𝐷𝑖𝑎𝑔(𝛿^2𝑘𝛽_𝑘,𝛿²¹𝛽₁,…,𝛿^2𝑘−1𝛽_𝑘−1), with 𝐼_𝑘 being the kth identity matrix. A necessary condition is ∑^𝑘𝑖=1(𝜆^2𝑖+𝛿^2𝑖(𝛼^2𝑖+𝛽^2𝑖))≤𝑘.
(iii) There exists 𝑃=𝑃^𝑇∶=⎡⎢⎢⎢⎣𝑃₁₁⋮𝑃₁₂𝑃_𝑇12⋯⋮𝑃₂₂⎤⎥⎥⎥⎦≻0in𝐑_2𝑘×2𝑘,(3.12) where 𝑃_𝑖𝑗∈𝐑^𝑘×𝑘(𝑖,𝑗=1,2), which is a solution to the matrix identity
⎡⎢⎢⎢⎢⎣𝐴^𝑇𝑃₁₁+𝑃^𝑇12𝐴+𝐴^𝑇𝑃₁₂+𝑃₂₂−𝑃₁₁⋮𝐴^𝑇𝑃₁₁+𝑃^𝑇12𝐵−𝑃₁₂𝐵_𝑇⋯𝑃₁₁𝐴+𝑃₁₂−𝑃^𝑇12⋮𝐵^𝑇𝑃₁₁𝐵−𝑃₂₂⎤⎥⎥⎥⎥⎦=−𝑄(3.13) for any given 𝑄=𝑄^𝑇∶=⎡⎢⎢⎢⎢⎣𝑄₁₁⋮𝑄₁₂⋯𝑄^𝑇12⋮𝑄₂₂⎤⎥⎥⎥⎥⎦≥0𝑖𝑛𝐑_2𝑘×2𝑘.(3.14)

Proof. (i) Note that the identically zero functions 𝑓_𝑖∶𝐑→0, for all 𝑖∈𝑘, are all in 𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖) for any 𝐾_𝑖≤0, 𝛿_𝑖∈(0,1), 𝑀_𝑖>0, for all 𝑖∈𝑘. Proceed by contradiction by assuming that |𝜆_𝑖|>1 and 𝑓_𝑖≡0 for some 𝑖∈𝑘∶={1,2,…,𝑘}, with the system being globally stable. Thus, |𝑥^(𝑖)𝑛+1|>|𝑥^𝑛(𝑖)| if 𝑥^0(𝑖)≠0 so that |𝑥^𝑛(𝑖)|→∞ as 𝑛→∞, and then the system is unstable for some function 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖). 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 𝑓_𝑖(𝑥)=𝛿_𝑖𝑥 with 𝛿_𝑖∈(0,1) everywhere in R so that 𝑓_𝑖∈𝐶(1,𝛿_𝑖,𝑀_𝑖), 𝑀_𝑖>0. Let the spectrum of W be 𝜎(𝑊)∶={𝜎₁,𝜎₂,…,𝜎_𝑘}, with each eigenvalue being repeated as many times as its multiplicity. Then, ‖𝐴‖₂=max_{1≤𝑖≤𝑘}𝜎^𝑖1/2. It is first proved by complete induction that if 𝑥₀≠0 is an eigenvector of 𝐴, then 𝑥_𝑘 is an eigenvector of 𝐴 for any 𝑘≥1. Assume that 𝑥_𝑘 is an eigenvector of 𝐴 for some arbitrary 𝑘≥1 and some eigenvalue 𝜌_𝑖. Then, 𝐴𝑥_𝑘+1=𝐴(𝐴𝑥_𝑘)=𝐴(𝜌_𝑖𝑥_𝑘)=𝜌_𝑖(𝐴𝑥_𝑘)=𝜌_𝑖𝑥_𝑘+1 so that 𝑥_𝑘+1 is also an eigenvector of 𝐴 for the same eigenvalue 𝜌_𝑖. This property leads to
‖𝑥_𝑘+1‖²²=‖𝐴𝑥_𝑘‖²²=𝑥^𝑇𝑘𝐴^𝑇𝐴𝑥_𝑘=𝜌^2𝑖‖𝑥_𝑘‖²²=𝜎_𝑖‖𝑥_𝑘‖²²=𝜌^𝑖2𝑘‖𝑥₀‖²²=𝜎^𝑘𝑖‖𝑥₀‖²².(3.15) Proceed by contradiction by assuming that system (1.1) is stable, for all 𝑓_𝑖∈𝐶(1,𝛿_𝑖,𝑀_𝑖), with |𝜌_𝑖|=𝜎^𝑖1/2>1. From (3.15), |𝑥^𝑛(𝑖)|→∞ as 𝑛→∞, and then the system is unstable for a function 𝑓_𝑖∈𝐶(1,𝛿_𝑖,𝑀_𝑖) for any real constant 𝐾_𝑖 since it possesses an unbounded solution for some finite initial conditions. Now, redefine the functions 𝑓_𝑖(𝑥) from the above 𝑓_𝑖(𝑥), 𝑖∈𝑘, as follows:
𝑓_𝑖𝑥=⎧⎪⎨⎪⎩𝑓^𝑖𝐑϶𝜆𝑥<=−𝛿(1_𝑖𝑥+max_{1≤𝑖≤𝑘}if𝑥≥0,|||𝜆^𝑖|||)<0if𝑥<0.(3.16) It is clear by construction that if 𝑓_𝑖(𝑥)=𝑓_𝑖(𝑥)=𝛿_𝑖𝑥 on an interval of infinite measure and if 0>𝜆=𝑓_𝑖(𝑥)≠𝑓_𝑖(𝑥) occurs on a real interval of finite measure, then the above contradiction obtained for 𝑓_𝑖∈𝐶(1,𝛿_𝑖,𝑀_𝑖) still applies for 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖) for any finite negative 𝐾_𝑖<−𝜆. If 𝑓_𝑖(𝑥)=𝑓_𝑖(𝑥) occurs on an interval of finite measure and if 𝑓_𝑖(𝑥)≠𝑓_𝑖(𝑥) occurs on an interval of infinite measure, then the linear system resulting from (1.1) with the replacement 𝑓_𝑖(𝑥)→𝑓_𝑖(𝑥) is unstable so that any nontrivial solution is unbounded. Furthermore, since 𝑓_𝑖(𝑥)→−∞ as 𝑥→∞ (function diverging to −∞) and 𝑓_𝑖(𝑥) being unbounded on R (implying that 𝑓_𝑖(𝑥_𝑘)→−∞ for {𝑥_𝑘}^∞0 being some monotonically increasing sequence of real numbers) are both impossible situations for some 𝑖∈𝑘 since 𝑓_𝑖∶𝐑→𝐑(𝑖∈𝑘) are all nondecreasing, it follows again that the functions are bounded from below so that 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖) for some finite 𝐾_𝑖<0. If the real subintervals within which 𝑓_𝑖(𝑥) equalizes 𝑓_𝑖(𝑥) or differs from 𝑓_𝑖(𝑥) are both of infinite measure, the result 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖) with some unbounded solution still applies trivially for some finite 𝐾_𝑖<0. Thus, system (1.1) is globally stable for any given set of functions 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖) for any 𝐾_𝑖∈𝐑 and any 𝑀_𝑖∈𝐑, for all 𝑖∈𝑘, only if the subsequent equivalent properties hold: ‖𝐴‖₂≤1, ‖𝑊‖₂≤1. The necessary condition ∑^𝑘𝑖=1(𝜆^2𝑖+𝛿^2𝑖(𝛼^2𝑖+𝛽^2𝑖))≤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 𝐴^𝑇𝑃𝐴−𝑃=−𝑄 which has to possess a solution 𝑃≻0 for any given 𝑄⪰0. 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 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖), for all 𝑖∈𝑘, 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 𝑓_𝑖(𝑥)=𝛿_𝑖𝑥 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 𝑓_𝑖∶𝐑₀₊→𝐑₀₊ with 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖) for any given 𝐾_𝑖∈𝐑 and any given 𝑀_𝑖∈𝐑, for all 𝑖∈𝑘. This is formally addressed in the subsequent result.

Theorem 3.5. System (1.1) is positive for any given set of nonnegative functions 𝑓_𝑖∶𝐑₀₊→𝐑₀₊ with 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖) for any given 𝐾_𝑖∈𝐑 and any given 𝑀_𝑖∈𝐑, for all 𝑖∈𝑘, only if 𝜆_𝑖∈𝐑₀₊, 𝛼_𝑖∈𝐑₀₊, 𝛽_𝑖∈𝐑₀₋, for all 𝑖∈𝑘.

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 𝑓_𝑖∶𝐑₀₊→𝐑₀₊ with 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖) for any given 𝐾_𝑖∈𝐑 and 𝑀_𝑖∈𝐑, for all 𝑖∈𝑘. The linear system (3.8)-(3.9) for 𝑓_𝑖(𝑥)=𝛿_𝑖𝑥 is positive if and only if 𝐴∈𝐑^𝑛×𝑛0+ [3] since, in addition, this implies 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖). The proof follows since 𝐴∈𝐑^𝑛×𝑛0+ by direct inspection if and only if 𝜆_𝑖∈𝐑₀₊, 𝛼_𝑖∈𝐑₀₊, 𝛽_𝑖∈𝐑₀₋, for all 𝑖∈𝑘.

Necessary joint conditions for stability, permanence, and positivity of (1.1) for any set 𝑓_𝑖∶𝐑₀₊→𝐑₀₊ with 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖) for any given 𝐾_𝑖∈𝐑 and 𝑀_𝑖∈𝐑, for all 𝑖∈𝑘, 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 𝛼_𝑖, 𝛽_𝑖, and 𝛿_𝑖, 𝑖∈𝑘 (which are also allowed to take values out of the interval (0,1), but on their maximum sizes). It is allowed that 𝜆_𝑖 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 𝛼_𝑖, 𝛽_𝑖, and 𝜆_𝑖, 𝑖∈𝑘, are allowed to be negative but 𝛿_𝑖∈(0,1), 𝑖∈𝑘, is required to formulate an auxiliary result for the main proof.

Theorem 4.1. Assume that |𝜆_𝑖|<1, for all 𝑖∈𝑘, and
maxmax_{1≤𝑖≤𝑘}|||𝛼_𝑖|||,max_{1≤𝑖≤𝑘}|||𝛽_𝑖|||<1−max^{1≤𝑖≤𝑘}|||𝜆_𝑖|||2√𝑘max_{1≤𝑖≤𝑘}|||𝛿_𝑖|||.(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:
∶=𝑥=𝑥₁𝑥₀∈𝐾₀𝑎¹,…,𝑎_2𝑘,𝑏₁,…,𝑏_2𝑘,𝑥₂,…,𝑥_2𝑘^𝑇∈𝐑^2𝑘∶𝑥_𝑖∈𝑎_𝑖,𝑏_𝑖,∞>𝑏^𝑖>𝑎_𝑖>−∞,∀𝑖∈2𝑘⊂𝐑^2𝑘.(4.2)

Proof. The successive use of the recursive second identity in (3.3) with initial condition 𝑥₀=(𝑥^𝑇0,𝑥^𝑇−1)^𝑇 leads to
𝑥_𝑛+𝑘=Λ^𝑛+𝑘𝑥₀+^{𝑛+𝑘−1}_𝑖=0Λ^{𝑛+𝑘−𝑖−1}𝐵𝑥_𝑖,∀𝑛∈𝐍,∀𝑘∈𝑘,(4.3) and taking ℓ₂-norms in (4.3) with 𝜆∶=max_{1≤𝑖≤𝑘}|𝜆_𝑖|<1, we get
‖‖𝑥_𝑛+𝑘‖‖₂=‖‖Λ^𝑛+𝑘‖‖₂‖‖𝑥₀‖‖₂+^{𝑛+𝑘−1}_𝑖=0‖Λ^{𝑛+𝑘−𝑖−1}‖₂‖𝐵‖₂‖𝑥_𝑖‖₂≤𝜆^𝑛+𝑘‖‖𝑥₀‖‖₂+1−𝜆^𝑛+𝑘1−𝜆‖‖𝐵‖‖₂max_{0≤𝑖≤𝑛+𝑘−1}‖‖𝑥_𝑖‖‖₂≤𝜆^𝑛‖‖𝑥₀‖‖₂+1−𝜆^𝑛1−𝜆‖‖𝐵‖‖₂max_{0≤𝑖≤𝑛+𝑘−1}‖‖𝑥_𝑖‖‖₂≤𝜆^𝑛‖‖𝑥₀‖‖₂+𝛿max𝛼,𝛽1−𝜆^𝑛1−𝜆√𝑘max_{0≤𝑖≤𝑛+𝑘−1}‖‖𝑥_𝑖‖‖₂≤𝜆^𝑛‖‖𝑥₀‖‖₂+2𝛿max𝛼,𝛽1−𝜆^𝑛1−𝜆√𝑘max_{−1≤𝑖≤𝑛+𝑘−1}‖‖𝑥_𝑖‖‖₂,∀𝑛∈𝐍,∀𝑘∈𝑘,(4.4) where 𝛿∶=max_{1≤𝑖≤𝑘}(|𝛿_𝑖|), 𝛼∶=max_{1≤𝑖≤𝑘}(|𝛼_𝑖|), and 𝛽∶=max_{1≤𝑖≤𝑘}(|𝛽_𝑖|) since 𝜆<1 and
‖‖𝐵‖‖₂=𝜆‖‖_maxΛ_𝑗𝐵^𝑇𝐵≤√𝑘‖‖𝐵‖‖₁≤√𝑘𝛿max𝛼,𝛽forany𝑥∈𝐑^𝑘,‖‖²²=max_{1≤𝑖≤𝑘}|||𝜆^𝑖|||^2𝑗max_{0≤𝑖≤𝑛+𝑘−1}=𝜆^2𝑗≤𝜆<1,∀𝑗∈𝐍,‖‖𝑥_𝑖‖‖₂=max_{0≤𝑖≤𝑛+𝑘−1}𝑥^𝑇𝑖𝑥_𝑖+𝑥^𝑇𝑖−1𝑥_𝑖−1^1/2≤≤2mmaxax_{0≤𝑖≤𝑛+𝑘−1}‖‖_{−1≤𝑖≤𝑛+𝑘−1}𝑥^𝑖‖‖₂+‖‖𝑥_𝑖−1‖‖₂‖‖𝑥_𝑖‖‖₂.(4.5) Note that (4.4) is still valid if the term preceding the equality is any ‖𝑥_𝑛+ℓ‖₂, for all ℓ∈𝐍⧵𝑛+𝑘, since they are all upper bounded by all the right-hand side upper bounds. Then,
‖‖𝑥_𝑛+ℓ‖‖₂≤𝜆^𝑛‖‖𝑥₀‖‖₂+2𝛿max(𝛼,𝛽)1−𝜆^𝑛1−𝜆√𝑘max_{−1≤𝑖≤𝑛+𝑘−1}‖‖𝑥_𝑖‖‖₂,(4.6) for all 𝑛∈𝐍, for all 𝑘∈𝑘, for all ℓ∈𝐍⧵𝑛+𝑘, which implies directly that
≤ma𝜆x_{−1≤𝑖≤𝑛+𝑘−1𝑛}‖‖𝑥^𝑛+𝑖‖‖₂‖‖𝑥₀‖‖₂+2𝛿max𝛼,𝛽1−𝜆√𝑘max_{−1≤𝑖≤𝑛+𝑘−1}∀‖‖𝑛𝑥_𝑖∈‖‖₂𝐍+,∀‖‖𝑘𝑥∈⁰‖‖₂+‖‖𝑥₋₁‖‖₂,𝑘.(4.7) If the condition max(𝛼,𝛽)<(1−𝜆)/2𝛿√𝑘 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
‖‖𝑥_𝑛‖‖₂≤max_{−1≤𝑖≤𝑛+𝑘−1}‖‖𝑥_𝑛+𝑖‖‖₂≤1−𝜆1−𝜆−2√𝑘𝛿max𝛼,𝛽1+𝜆^𝑛‖‖𝑥⁰‖‖₂+‖‖𝑥₋₁‖‖₂≤1−𝜆1−𝜆−2√𝑘𝛿max𝛼,𝛽1+𝜆^𝑛0‖‖𝑥⁰‖‖₂+‖‖𝑥₋₁‖‖₂≤1+𝜀1−𝜆1−𝜆−2√𝑘𝛿max𝛼,𝛽‖‖𝑥⁰‖‖₂+‖‖𝑥₋₁‖‖₂≤1+𝜀1−𝜆1−𝜆−2√𝑘𝛿max𝛼,𝛽(^2𝑘_𝑖=1max𝑎^2𝑖,𝑏^2𝑖)^1/2≤1+𝜀1−𝜆1−𝜆−2√𝑘𝛿max𝛼,𝛽(^2𝑘_𝑖=1max|||𝑎_𝑖≤31−𝜆|||,|||𝑏_𝑖|||)(4.8)1−𝜆−2√𝑘𝛿max𝛼,𝛽max‖‖𝑥⁰‖‖₂,‖‖𝑥₋₁‖‖₂,(4.9) for all 𝜀∈𝐑₊, for all 𝑛(≥𝑛₀)∈𝐍, depending on 𝑛₀, which depends on 𝜀, for any 𝐍϶𝑛₀≥ln𝜀/ln𝜆, for all 𝑥₀∈𝐾₀. Since 𝐾₀ is compact, it follows from (4.9) that any solution sequence is bounded for any 𝑛∈𝐍 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∶={𝑥∈𝐑^𝑘∶|||𝑥_𝑖|||≤1+𝜀𝑘1−𝜆−2𝛿1√−𝑘𝜆max𝛼,𝛽(^2𝑘_𝑖=1max|||𝑎_𝑖|||,|||𝑎_𝑖|||),∀𝑖∈𝑘}(4.10) for any 𝑛(≥𝑛₀)∈𝐍 and any finite initial conditions (𝑥₀⋮𝑥₋₁)^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 𝑓_𝑖∶𝐑→𝐑, 𝑖∈𝑘, grow at most linearly with their argument.

Lemma 4.2. 𝑓_𝑖∈𝐶(𝛼_𝑖,𝛿_𝑖,𝑀_𝑖)⇒𝑓_𝑖(𝑥)=𝑂(𝑥), for all 𝑖∈𝑘. In addition, 𝑓_𝑖(𝑥) is bounded for all 𝑥≥𝑀_𝑖. The result also holds if 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,𝛼_𝑖,𝛿_𝑖,𝑀_𝑖), for all 𝐾_𝑖∈𝐑, for all 𝑖∈𝑘.

Proof. Now, it is proved that 𝑓_𝑖(𝑥)=𝑂(𝑥) (notation of “big Landau O” of x) for any 𝑓_𝑖∈𝐶(𝛼_𝑖,𝛿_𝑖,𝑀_𝑖), for all 𝑖∈𝑘. First, note that for all 𝑖∈𝑘 for some 𝜀_𝑖∶[𝑀,∞)→𝐑₀₊, 𝑓_𝑖∈𝐶(𝛼_𝑖,𝛿_𝑖,𝑀_𝑖)∧𝑥≥𝑀_𝑖∈𝐑₊⇒𝑓_𝑖(𝑥)=𝛿_𝑖𝑥−𝜀_𝑖(𝑥)≤𝛿_𝑖𝑥⇒𝑓_𝑖(𝑥)=𝑂(𝑥) since 𝑓_𝑖(𝑥)≤𝛿_𝑖𝑥+𝐾 for any 𝐾∈𝐑₀₊, for all 𝑥≥𝑀_𝑖. The result also holds if 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,𝛼_𝑖,𝛿_𝑖,𝑀_𝑖), for all 𝐾_𝑖∈𝐑, since 𝐵𝐶(𝐾_𝑖,𝛼_𝑖,𝛿_𝑖,𝑀_𝑖)⊂𝐶(𝐾_𝑖,𝛼_𝑖,𝛿_𝑖,𝑀_𝑖). It is now proved by a contradiction argument that if 𝑓_𝑖∈𝐶(𝐾_𝑖,𝛼_𝑖,𝛿_𝑖,𝑀_𝑖), then it is bounded, for all 𝑥<𝑀_𝑖. Assume 𝑥<𝑀_𝑖∈𝐑₊ with 𝑓_𝑖(𝑥₁) being arbitrarily large for some 𝑥_1𝑖<𝑀_𝑖. Thus, there exists 𝑀_2𝑖∈𝐑₊ being arbitrarily large so that 𝑀_2𝑖≤𝑓_𝑖(𝑥_1𝑖)≤𝑓_𝑖(𝑀_𝑖)≤𝛿_𝑖𝑀_𝑖<∞ for 𝑥_1𝑖=𝛼_𝑖𝑥^𝑛(𝑖+1)−𝛽_𝑖𝑥^{(𝑖+1)𝑛−1}<𝑀_𝑖 since 𝑓_𝑖∈𝐶(𝛼_𝑖,𝛿_𝑖,𝑀_𝑖) so that it is monotonically nondecreasing. This is a contradiction since 𝑀_2𝑖 is arbitrarily large. Thus, 𝑓_𝑖∈𝐶(𝛼_𝑖,𝛿_𝑖,𝑀_𝑖) is bounded, for all 𝑥<𝑀_𝑖. Since it is bounded, then 𝑓_𝑖(𝑥)=𝑂(𝑥)≤|𝑓_𝑖(𝑥)|≤𝛿|𝑥|+𝐶₁ for some finite 𝐶₁∈𝐑₊ for 𝑥<𝑀_𝑖 as a result, so that 𝑓_𝑖(𝑥)=𝑂(𝑥) on 𝐑. Again, the result still holds if 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,𝛼_𝑖,𝛿_𝑖,𝑀_𝑖), for all 𝐾_𝑖∈𝐑.

Theorem 4.3. If 𝜆∶=max_{1≤𝑖≤𝑘}|𝜆_𝑖|<1−𝛿, 𝛿∶=max_{1≤𝑖≤𝑘}𝛿_𝑖∈(0,1), 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,𝛼_𝑖,𝛿_𝑖,𝑀_𝑖), for all 𝐾_𝑖∈𝐑, for all 𝑖∈𝑘, and max(max_{1≤𝑖≤𝑘}|𝛼_𝑖|,max_{1≤𝑖≤𝑘}|𝛽_𝑖|)<(1−max_{1≤𝑖≤𝑘}|𝜆_𝑖|−𝛿)/4√𝑘max_{1≤𝑖≤𝑘}𝛿_𝑖, then system (1.1) is globally Lyapunov stable for any finite arbitrary initial conditions. It is also permanent for any initial conditions 𝑥₀∈𝐾₀(𝑎₁,…,𝑎_2𝑘,𝑏₁,…,𝑏_2𝑘) with the compact set 𝐾₀ defined in Theorem 4.1.

Proof. If system (1.1) is taken, then (4.4) is replaced with
𝑥_𝑛+1=Λ𝑥_𝑛+𝐵𝑥_𝑛−1+𝑓𝑥_𝑛−1−𝐵𝑥_𝑛−1,∀𝑛,𝑗∈𝐍,(4.11) where
𝑓𝑥_𝑛−1=𝑓¹𝛼²𝑥^𝑛(2)−𝛽_𝑖+1𝑥^(2)𝑛−1,…,𝑓^𝑘𝛼¹𝑥^𝑛(1)−𝛽_𝑖+1𝑥^(1)𝑛−1^𝑇.(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(𝑥_𝑛−1)−𝐵𝑥_𝑛−1)}^∞0 so that both solution sequences are identical under identical initial conditions. One gets directly from (4.11) that
𝑥_𝑛+𝑘=Λ^𝑛+𝑘𝑥₀+^{𝑛+𝑘−1}_𝑖=0Λ^{𝑛+𝑘−𝑖−1}𝐵𝑥_𝑖+𝑓𝑥_𝑛−1−𝐵𝑥_𝑛−1,∀𝑛∈𝐍,∀𝑘∈𝑘,(4.13) so that
‖‖𝑥_𝑛+𝑘‖‖₂≤𝜆^𝑛‖‖𝑥₀‖‖₂+1−𝜆^𝑛1−𝜆(‖‖≤𝜆_𝑛𝐵‖‖₂max_{0≤𝑖≤𝑛+𝑘−1}‖‖𝑥_𝑖‖‖₂+max_{0≤𝑖≤𝑛+𝑘−1}‖‖𝑓𝑥_𝑖−𝐵𝑥_𝑖‖‖₂)‖‖𝑥₀‖‖₂+1−𝜆^𝑛1−𝜆2‖‖𝐵‖‖₂+𝛿max_{0≤𝑖≤𝑛+𝑘−1}‖‖𝑥_𝑖‖‖₂+1−𝜆^𝑛𝐶¹1−𝜆.(4.14) Then by direct extension of (4.7) when using (4.14),
max_{−1≤𝑖≤𝑛+𝑘−1}‖‖𝑥^𝑛+𝑖‖‖₂≤𝜆^𝑛‖‖𝑥₀‖‖₂+11−𝜆(4𝛿max𝛼,𝛽√+‖‖𝑥₀𝑘+𝛿max_{−1≤𝑖≤𝑛+𝑘−1}‖‖𝑥_𝑖‖‖₂+𝐶₁)‖‖₂+‖‖𝑥₋₁‖‖₂,∀𝑛∈𝐍,∀𝑘∈𝑘,(4.15) with 𝛿∈(0,1) for some finite 𝐶₁∈𝐑₊ since |𝑓_𝑖(𝑥)|≤𝛿|𝑥|+𝐶₁, for all 𝑖∈𝑘, from Lemma 4.2. Thus, max_{−1≤𝑖≤𝑛+𝑘−1}(‖𝑥_𝑛+𝑖‖₂) may be regrouped in the left-hand side provided that
1>11−𝜆4𝛿max𝛼,𝛽√𝑘+𝛿⟺max𝛼,𝛽<1−𝜆−𝛿4𝛿√𝑘.(4.16) Then, under similar reasoning as that used to derive (4.8)-(4.9), one gets from (4.15) that
‖‖𝑥_𝑛‖‖₂≤max_{−1≤𝑖≤𝑛+𝑘−1}‖‖𝑥_𝑛+𝑖‖‖₂≤11−𝜆−𝛿1+4√𝑘max𝛼,𝛽1−𝜆1+𝜆^𝑛‖‖𝑥⁰‖‖₂+‖‖𝑥₋₁‖‖₂+𝐶₁≤11−𝜆−𝛿1+4√𝑘max𝛼,𝛽1−𝜆1+𝜆^𝑛0‖‖𝑥⁰‖‖₂+‖‖𝑥₋₁‖‖₂+𝐶₁≤1+𝜀11−𝜆−𝛿1+4√𝑘max𝛼,𝛽1−𝜆‖‖𝑥⁰‖‖₂+‖‖𝑥₋₁‖‖₂≤1+𝜀1+𝐶¹1−𝜆−𝛿1+4√𝑘max𝛼,𝛽(1−𝜆^2𝑘_𝑖=1max𝑎^2𝑖,𝑏^2𝑖+𝐶¹)^1/2≤1+𝜀11−𝜆−𝛿1+4√𝑘max𝛼,𝛽1−𝜆max‖‖𝑥⁰‖‖₂,‖‖𝑥₋₁‖‖₂≤31+𝐶¹1−𝜆−𝛿1+4√𝑘max𝛼,𝛽1−𝜆max‖‖𝑥⁰‖‖₂,‖‖𝑥₋₁‖‖₂+𝐶¹,(4.17) for all 𝜀∈𝐑₊, for all 𝑛(≥𝑛₀)∈𝐍, depending on 𝑛₀, which depends on 𝜀, for any 𝐍϶𝑛₀≥ln𝜀/ln𝜆. The solution sequences are all bounded under any finite initial conditions and enter the compact set 𝐾 defined by
{𝑥∈𝐑^𝑘∶|||𝑥_𝑖|||≤1+𝜀𝑘×((11−−𝜆𝜆−_2𝑘𝛿1+41√𝑘max𝛼,𝛽_𝑖∑₌₁max|||𝑎_𝑖|||,|||𝑎_𝑖|||+𝐶₁)),∀𝑖∈𝑘},(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 |𝜆_𝑖|≤1 and 𝑓_𝑖∶𝐑→𝐑 is bounded from above and below, then |𝑥^𝑛(𝑖)| is bounded, for all 𝑛∈𝐍. If |𝜆_𝑖|>1 and 𝑓_𝑖∶𝐑→𝐑 is bounded from above and below, then almost all solution sequences {𝑥^𝑛(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 𝑖∈𝑘.(ii)Assume that 𝑓_𝑖∈𝐵(𝐾_𝑖) and |𝜆_𝑖|>1 for some 𝑖∈𝑘. Then |𝑥^(𝑖)𝑛+1|>|𝑥^𝑛(𝑖)|, for all 𝑛∈𝐍, and |𝑥^𝑛(𝑖)|→∞ as 𝑛→∞ if |𝑥^0(𝑖)|>|𝐾_𝑖|/(|𝜆_𝑖|−1) (|𝑥^0(𝑖)|≥|𝐾_𝑖|/(|𝜆_𝑖|−1)if𝐾_i≠0). Thus, system (1.1) is unstable under such sufficiently large absolute values of the initial conditions for some 𝑖∈𝑘.

Proof. (i) If −∞<𝑀_1𝑖≤𝑓_𝑖(𝑥)≤𝑀_2𝑖<∞, for all 𝑥∈𝐑, for some 𝑀_𝑗𝑖, j = 1,2, and some 𝑖∈𝑘, then
|||𝜆_𝑛𝑖|||(|||𝑥^𝑖₀|||+|^∞_𝑗=0𝜆^{𝑖−𝑗−1}|max|||𝑀_1𝑖≥|||𝑥_𝑖|||,|||𝑀_2𝑖|||)_𝑛|||≥|||𝜆_𝑛𝑖|||(|||𝑥^𝑖₀|||−|^𝑛−1_𝑗=0𝜆^{𝑖−𝑗−1}|max_{0≤𝑗≤𝑖}|||𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑗−𝛽_𝑖𝑥^𝑖+1_𝑗−1≥|||𝜆_𝑛𝑖|||)|||(|||𝑥^𝑖₀|||−|^𝑛−1_𝑗=0𝜆^{𝑖−𝑗−1}|max|||𝑀_1𝑖≥|||𝜆_𝑛𝑖|||,|||𝑀_2𝑖|||)|||||||𝑥^𝑖₀|||−|^∞_𝑗=0𝜆^{𝑖−𝑗−1}|max|||𝑀_1𝑖|||,|||𝑀_2𝑖||||.(4.19) If |𝜆_𝑖|≤1, then the sequence {|𝑥^𝑛(𝑖)|}^∞0 is bounded so that the sequence {|𝑥^𝑛(𝑖)|}^∞0 may be unbounded only if |𝜆_𝑖|>1. If |𝜆_𝑖|>1, then 0≤||𝑥^0(𝑖)|−|∑^∞𝑗=0𝜆^{𝑖−𝑗−1}|max(|𝑀_1𝑖|,|𝑀_2𝑖|)|<∞, and, furthermore, if |𝑥^0(𝑖)|>(|𝜆_𝑖|/(|𝜆_𝑖|−1))max(|𝑀_1𝑖|,|𝑀_2𝑖|)≥|∑^∞𝑗=0𝜆^{𝑖−𝑗−1}|max(|𝑀_1𝑖|,|𝑀_2𝑖|), then there is a strictly monotonically increasing subsequence {|𝑥^𝑛(𝑖)|}_𝑛∈𝑆 of {|𝑥^𝑛(𝑖)|}^∞0, where 𝑆∶={𝑛₁,𝑛₂,…} is a countable subset of N, so that |𝑥^𝑛(𝑖)_𝑗+1|>|𝑥^𝑛(𝑖)_𝑗|, for all 𝑛_𝑗∈𝑆, and |𝑥^𝑛(𝑖)_𝑗|→∞ as 𝑆϶𝑛_𝑗→∞ (i.e., it diverges).
If 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖), then −∞<−|𝐾^𝑖|≤𝑓_𝑖(𝑥)≤𝛿𝑥, for all 𝑥(≥𝑀_𝑖)∈𝐑 and all 𝐾^𝑖, such that 𝐾_𝑖+|𝐾^𝑖|≥0.
(ii) From (1.1), 𝑓_𝑖∈𝐵(𝐾_𝑖), and |𝜆_𝑖|>1, it follows that 𝑥^𝑖_𝑛+₁²−𝑥^𝑖_𝑛²=𝜆^2𝑖−1𝑥^𝑖_𝑛²+𝑓^2𝑖𝛼^𝑖𝑥^𝑖+1_𝑛−𝛽_𝑖𝑥^𝑖+1_𝑛−1+2𝜆^𝑖𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑛−𝛽_𝑖𝑥^𝑖+1_𝑛−1𝑥^𝑖_𝑛≥𝑔^𝑖_𝑛|||𝑥^𝑖_𝑛|||∶=𝐾_2𝑖−2𝜆_𝑖|||𝐾_𝑖|||−𝜆_2𝑖−1|||𝑥^𝑖_𝑛|||||𝑥^𝑖_𝑛||>0(4.20) if |𝑥^0(𝑖)|>2|𝜆_𝑖||𝐾_𝑖|/(𝜆^2𝑖−1)(|𝑥^0(𝑖)|≥|𝐾_𝑖|/(|𝜆_𝑖|−1)if𝐾_i≠0)⇒|𝑥^(𝑖)𝑛+1|>|𝑥^𝑛(𝑖)|, for all 𝑛∈𝐍, 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 𝑔^𝑛(𝑖)(|𝑥^𝑛(𝑖)|)𝑔^𝑛(𝑖)(|𝑥^𝑛(𝑖)|) which are 𝑔^(𝑖)2𝑛=|𝐾_𝑖|/(|𝜆_𝑖|−1)≥𝑔^(𝑖)1𝑛=|𝐾_𝑖|/(|𝜆_𝑖|+1), which implies that 𝑔^𝑛(𝑖)(|𝑥^𝑛(𝑖)|)≤0 if |𝑥^𝑛(𝑖)|∈[𝑔^(𝑖)1𝑛,𝑔^(𝑖)2𝑛], and (𝑥^(𝑖)𝑛+1)²−(𝑥^𝑛(𝑖))²≥𝑔^𝑛(𝑖)(|𝑥^𝑛(𝑖)|)>0 if |𝑥^𝑛(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 𝑥_𝑛∶=(𝑥^𝑛(1),𝑥^𝑛(2),…,𝑥^𝑛(𝑘))^𝑇 of (1.1) is nonnegative, for all 𝑛∈𝐍, and any finite nonnegative 𝑥^0(𝑖)≥0, for all 𝑖∈𝑘, if 𝑓_𝑖(𝛼_𝑖𝑥^0(𝑖+1)−𝛽_𝑖𝑥^(𝑖+1)−1)≥−𝜆_𝑖𝑥^0(𝑖), for all 𝑖∈𝑘, and𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑛−𝛽_𝑖𝑥^𝑖+1_𝑛−1≥−𝜆^𝑖𝑥^𝑖_𝑛=−(𝜆^𝑖𝑛+1𝑥^𝑖₀+^𝑛−1_𝑗=0𝜆^{𝑖𝑛−𝑗}𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑗−𝛽_𝑖𝑥^𝑖+1_𝑗−1),(5.1) for all 𝑖∈𝑘, for all 𝑛∈𝐍. Then, system (1.1) is positive.(ii)Any solution vector sequence of (1.1) is nonnegative, for all 𝑛∈𝐍, and any finite nonnegative 𝑥^0(𝑖)≥0, for all 𝑖∈𝑘, if 𝜆_𝑖∈𝐑₀₊ and 𝑓_𝑖∶𝐑→𝐑₀₊, for all 𝑖∈𝑘. Then, system (1.1) is positive.(iii)Assume that 𝜆_𝑖∈𝐑₀₊, for all 𝑖∈𝑘, and that there exist 2𝑘 real constants 𝐶^𝑗(𝑖)∈𝐑⁺⁰, 𝑖∈𝑘, 𝑗=1,2, independent of n, such that−∞<−𝐶^𝑖₁≤𝑓_𝑖𝛼^𝑖𝑥^𝑖+1₀−𝛽_𝑖𝑥^𝑖+1₋₁≤𝐶^𝑖₂−∞<−𝐶_𝑖<∞,∀𝑖∈𝑘,₁≤∑^𝑛−1_𝑗=0𝜆^{𝑖𝑛−𝑗−1}𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑗−𝛽_𝑖𝑥^𝑖+1_𝑗−1≤𝐶^𝑖₂<∞,∀𝑖∈𝑘,∀𝑛∈𝐍.(5.2)Then, the solution vector sequence is nonnegative, for all 𝑛∈𝑁₀⧵𝑛₀, for some finite 𝑛₀∈𝐍₀, depending on 𝑥^𝑗(𝑖)(𝑗=0,−1,forall𝑖∈𝑘), for any given finite 𝑥^0(𝑖)>0, for all 𝑖∈𝑘.(iv)Assume that 𝑓_𝑖∈𝐵(𝐾_𝑖) and 𝜆_𝑖>1, for all 𝑖∈𝑘. Then, any solution vector sequence of (1.1) is nonnegative; that is, 𝑥_𝑛∈𝐑^𝑛0+, for all 𝑛∈𝐍, for any given finite 𝑥₋₁∈𝐑^𝑘 and some 𝐑^𝑘϶𝑥₀≫0 of sufficiently large components (i.e., 𝑥₀∈𝐑^𝑘+ and 𝑥^0(𝑖)≥𝜐^(𝑖)>0, for some positive lower bound, with 𝜐^(𝑖) being sufficiently large, for all 𝑖∈𝑘). The solution vector sequence is positive by increasing the size of the initial condition of at least one component, and strictly positive by increasing simultaneously the sizes of the initial conditions of all the components. If 𝑓_𝑖∈𝐵(𝐾_𝑖) with 𝐾_𝑖∈𝐑₀₊, for all 𝑖∈𝑘, then the constraints 𝜆_𝑖>1 are weakened to 𝜆_𝑖∈𝐑₀₊, for all 𝑖∈𝑘 (Property (ii)).(v)Assume that [𝐴⋮𝐵]>0 with at least a positive entry per row, with the matrices A and B defined in (3.4), and that 𝜆_𝑖>1 and 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖), for all 𝑖∈𝑘. Thus, there exists 𝑥₀≫0 of sufficiently large finite components so that any solution is strictly positive, that is, 𝑥_𝑛≫0, for all 𝑛∈𝐍, under initial condition 𝑥₀≫0. The sizes are quantifiable from the knowledge of the scalars 𝐾_𝑖,𝛿_𝑖,𝑀_𝑖 (𝑖∈𝑘) and upper bounds of the nonzero entries of A and B.

Proof. (i) The recursive use of (1.1) yields
𝑥^𝑖_𝑛=𝜆^𝑛𝑖𝑥^𝑖₀+^𝑛−1_𝑗=0𝜆^{𝑖𝑛−𝑗−1}𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑗−𝛽_𝑖𝑥^𝑖+1_𝑗−1,∀𝑖∈𝑘,∀𝑛∈𝐍,(5.3) for any given 𝑥^𝑖(𝑖) (𝑖=0,−1), for all 𝑖∈𝑘. Then,
𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑛−𝛽_𝑖𝑥^𝑖+1_𝑛−1≥−𝜆^𝑖𝑥^𝑖_𝑛=−(𝜆^𝑖𝑛+1𝑥^𝑖₀+^𝑛−1_𝑗=0𝜆^{𝑖𝑛−𝑗}𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑗−𝛽_𝑖𝑥^𝑖+1_𝑗−1),∀𝑛∈𝐍⁰⇒𝑥^𝑖_𝑛+₁=𝜆_𝑖𝑥^𝑖_𝑛+𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑛−𝛽_𝑖𝑥^𝑖+1_𝑛−1≥0,∀𝑛∈𝐍⁰.(5.4)
(ii)𝑥^𝑛(𝑖)=𝜆_𝑖𝑥^{(𝑖)𝑛−1}+𝑓_𝑖(𝛼_𝑖𝑥^{(𝑖+1)𝑛−1}−𝛽_𝑖𝑥^{(𝑖+1)𝑛−2})≥0, for all 𝑛∈𝐍, if 𝑥^0(𝑖)≥0, 𝜆_𝑖∈𝐑₀₊, and 𝑓_𝑖∶𝐑→𝐑₀₊, for all 𝑖∈𝑘.(iii)𝑥^𝑛(𝑖)=𝜆^𝑛𝑖𝑥^0(𝑖)+∑^{𝑛−1𝑗=0}𝜆^{𝑖𝑛−𝑗−1}𝑓_𝑖(𝛼_𝑖𝑥^𝑗(𝑖+1)−𝛽_𝑖𝑥^{(𝑖+1)𝑗−1})≥𝜆^𝑛𝑖𝑥^0(𝑖)−𝐶^1(𝑖)≥0, for all 𝑛≥𝑛₀∶=max_{1≤𝑖≤𝑘}(ln(𝐶^1(𝑖)/𝑥^0(𝑖))/ln𝜆_𝑖)−1, for all 𝑖∈𝑘. Such an 𝑛₀, being dependent on 𝑥^0(𝑖), always exists for 𝜆_𝑖>1 since 𝐶^1(𝑖)<∞ and 𝜆^𝑛𝑖𝑥^0(𝑖)→∞ as 𝑛→∞ for any 𝑥^0(𝑖)>0, for all 𝑖∈𝑘.(iv)Since 𝑓_𝑖∶𝐑→𝐑, for all 𝑖∈𝑘, are bounded from below on R, then max_𝑛∈𝐍𝑓_𝑖(𝛼_𝑖𝑥^𝑛(𝑖+1)−𝛽_𝑖𝑥^𝑛(𝑖+1))≥𝐾_𝑖>−∞, liminf_𝑛→∞𝑓_𝑖(𝛼_𝑖𝑥^𝑛(𝑖+1)−𝛽_𝑖𝑥^𝑛(𝑖+1))≥𝐾_𝑖>−∞ for some finite 𝐾_𝑖∈𝐑, for all 𝑖∈𝑘. Irrespective of the value of 𝐾_𝑖, since it is finite, there always exists a finite constant 𝐾^𝑖∈𝐑₊ fulfilling 𝐾_𝑖≥−𝐾^𝑖 such thatmax_𝑛∈𝐍𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑛−𝛽_𝑖𝑥^𝑖+1_𝑛≥−𝐾^𝑖=−|||𝐾^𝑖liminf_𝑛→∞|||>−∞,𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑛−𝛽_𝑖𝑥^𝑖+1_𝑛≥−|||𝐾^𝑖|||>−∞,(5.5)for all 𝑖∈𝑘. Since 𝜆_𝑖>1, the series ∑^∞𝑗=0𝜆^𝑖−𝑗 converges so that^𝑛_𝑖=0𝜆^𝑖−𝑗=1−𝜆⁻^𝑛+1_𝑖1−𝜆^𝑖−1=𝜆^𝑖𝑛+1−1𝜆^𝑛𝑖𝜆^𝑖−1≤^∞_𝑗=0𝜆^𝑖−𝑗=11−𝜆^𝑖−1=𝜆^𝑖𝜆_𝑖−1,∀𝑖∈𝑘,∀𝑛∈𝐍.(5.6)Then,𝑥^𝑖_𝑛=𝜆^𝑛𝑖(𝑥^𝑖₀−^𝑛−1_𝑗=0𝜆^{𝑖−𝑗−1}|||𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑛−𝛽_𝑖𝑥^𝑖+1_𝑛≥𝜆_𝑛𝑖|||)(𝑥^𝑖₀−^𝑛−1_𝑗=0𝜆^{𝑖−𝑗−1}max_𝑛∈𝐍|||𝑓_𝑖𝛼^𝑖𝑥^𝑖+1_𝑛−𝛽_𝑖𝑥^𝑖+1_𝑛≥𝜆_𝑛𝑖|||)𝑥^𝑖₀−𝜆^𝑖𝑛+1−1|||𝐾^𝑖|||≥𝜆_𝑛𝑖𝜆^𝑛𝑖𝜆^𝑖−1𝑥^𝑖₀−𝜆^𝑖|||𝐾_𝑖|||𝜆_𝑖−1,(5.7) for all 𝑖∈𝑘, for all 𝑛∈𝐍. As a result, 𝑥^𝑛(𝑖)∈𝐑₀₊, for all 𝑖∈𝑘, for all 𝑛∈𝐍, if 𝑥^0(𝑖)≥𝜆|𝐾^𝑖|/(𝜆−1)>0, for all 𝑖∈𝑘. Then, 𝑥_𝑛≥0, for all 𝑛∈𝐍. If 𝑥^0(𝑖)>𝜆|𝐾^𝑖|/(𝜆−1) for at least one 𝑖∈𝑘, then 𝑥_𝑛>0, for all 𝑛∈𝐍. If 𝑥^0(𝑖)>𝜆|𝐾^𝑖|/(𝜆−1), for all 𝑖∈𝑘, then 𝑥_𝑛≫0, for all 𝑛∈𝐍.(v)Define 𝑀∶=(𝑀₁,𝑀₂,…,𝑀_𝑘)^𝑇≫0 with the constants 𝑀_𝑖 of the sets 𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖), for all 𝑖∈𝑘. Since 𝑓_𝑖(𝑥)=𝛿_𝑖𝑥−𝑓_𝑖(𝑥) for some 𝑓_𝑖∶[𝑀,∞)→𝐑₀₊, for all 𝑖∈𝑘, for all 𝑥≥𝑀_𝑖, from the definition of the sets 𝐶(1,𝛿_𝑖,𝑀_𝑖), it follows from (3.9) that𝑥_𝑛≥𝐴^𝑛𝑥₀−^𝑛−1_𝑖=0𝐴^{𝑛−1−𝑖}𝐵𝐾^(5.8)for any 𝐾^∶=(𝐾^1,𝐾^2,…,𝐾^𝑘)^𝑇≫0 such that 𝐾_𝑖≥−|𝐾^𝑖|, for all 𝑖∈𝑘. Since 𝜆_𝑖>1, for all 𝑖∈𝑘, then from the structure of the matrix 𝐴 in (3.9),𝑥^𝑛(𝑖)≥𝜆^𝑛𝑖𝑥^0(𝑖)+^𝑛−1_𝑖=0𝑒^𝑇𝑖Λ^{𝑛−1−𝑖}𝐵𝑥_𝑖−^𝑛−1_𝑖=0𝑒^𝑇𝑖Λ^{𝑛−1−𝑖}𝐵𝐾^≥𝜆^𝑛𝑖𝑥^0(𝑖)+^𝑛−1_𝑖=0𝑒^𝑇𝑖Λ^{𝑛−1−𝑖}𝐵(𝐴^̄‌𝑖𝑥₀−^𝑖−1_𝑗=0𝐴^{𝑖−1−𝑗}𝐵𝐾^)−^𝑛−1_𝑖=0𝑒^𝑇𝑖Λ^{𝑛−1−𝑖}𝐵𝐾^=𝜆^𝑛𝑖𝑥^0(𝑖)+^𝑛−1_𝑖=0𝑒^𝑇𝑖Λ^{𝑛−1−𝑖}𝐵𝐴^𝑖𝑥₀−𝑒^𝑇𝑖(Λ^{𝑛−1−𝑖}+^𝑖−1_̄𝑗=0𝐴^{𝑖−1−𝑗})𝐵𝐾^=𝑒^𝑇𝑖(𝜆^𝑛𝑖𝐼_𝑛+^𝑛−1_𝑖=0Λ^{𝑛−1−𝑖}𝐵𝐴^̄‌𝑖)𝑥₀−𝑒^𝑇𝑖(Λ^{𝑛−1−𝑖}+^𝑖−1_𝑗=0𝐴^{𝑖−1−𝑗})𝐵𝐾^≫𝑀_𝑖,(5.9)since 𝜆_𝑖>1, for all 𝑖∈𝑘, provided that it is sufficiently large, 𝑥^0(𝑖)≥max(𝑀_𝑖,𝜐_𝑖)>0 (i.e., 𝑥₀≫0 has sufficiently large positive components), for all 𝑖∈𝑘, for all 𝑛∈𝐍, where 𝑒^𝑇𝑖 is the ith unity vector in 𝐑^𝑘 of components 𝑒_𝑖𝑗=𝛿_𝑖𝑗 (the Kronecker delta), for all 𝑖,𝑗∈𝑘.

Note that the properties associated with 𝑓_𝑖∈𝐵𝐶(𝐾_𝑖,1,𝛿_𝑖,𝑀_𝑖), for all 𝑖∈𝑘, have not been invoked in Theorem 5.1(i)–(iii). Theorem 5.1(ii) implicitly assumes 𝑓_𝑖∈𝐵(𝐾_𝑖), since they are assumed to be nonnegative, for all 𝑖∈𝑘.

Acknowledgments

The author is very grateful to MCYT due to the partial support of this work through Grant no. DPI2006-00714, and to the Basque Government due to its support of this work via Research Grants Research Groups no. IT-269-07. The author is also grateful to the reviewers for their useful comments and corrections which helped him to improve the original manuscript.