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

Authors: M. De la Sen
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## 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.


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## Body

## 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]:(1.1)xn+1(i)=λixn(i)+fi(αixn(i+1)−βixn−1(i+1)),∀i∈k¯:={1,2,…,k}, with xn(k+1)≡xn(1), for all n∈N; λi∈R, αi∈R, βi∈R;
and fi:R→R,  for all i∈k¯,
under arbitrary initial conditions x0(i),x−1(i), for all i∈k¯.
The identity xn(k+1)≡xn(1) allows the inclusion in a unified shortened
notation via (1.1) of the dynamics:(1.2)xn+1(k)=λixn(k)+fi(αixn(1)−βixn−1(1)),∀i∈k¯, 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))T∈Rk, for all n∈N,
under initial conditions xj:=(xj(1),xj(2),…,xj(k))T∈Rk, 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 fi: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∈Rn×n,
then PT 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∈R0+n×n), 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 Rk of
(1.1) will be nonnegative in some interval S, denoted by xn≥0 (identical to xn∈R0+k), 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 xn>0 (implying that xn∈R0+k), 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 xn≫0 (identical to xn∈R+k and implying that xn>0 and xn∈R0+k), for all n∈S⊂N.∥∥2 and ∥∥1 are the ℓ2 and ℓ1 norms of vectors and induced norms of
matrices, respectively. In is the nth identity matrix.


## 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∈Rn×n,
then PT 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∈R0+n×n), 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 Rk of
(1.1) will be nonnegative in some interval S, denoted by xn≥0 (identical to xn∈R0+k), 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 xn>0 (implying that xn∈R0+k), 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 xn≫0 (identical to xn∈R+k and implying that xn>0 and xn∈R0+k), for all n∈S⊂N.∥∥2 and ∥∥1 are the ℓ2 and ℓ1 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 fi:R→R in system (1.1) as follows irrespective of the
initial conditions:(2.1)B(Ki):={fi:R⟶R:fi(y)≥fi(x)≥Ki,∀x,y(>x)∈R,Ki∈R},∀i∈k¯, and sets of
linearly upper bounded real functions:(2.2)C(γi,δi,Mi):={fi:R⟶R:fi(x)≤δiγix,∀x>Mi∈R+,δi∈(0,1)},∀i∈k¯, for γi⁢≠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(Ki,γi,δi,Mi):=B(Ki)∩C(γi,δi,Mi),
that is,(2.3)BC(Ki,γi,δi,Mi):={fi:R→R:fi(y)≥fi(x)≥Ki∧fi(x)≤δiγix,∀x,y(>x)∈R,Ki∈R,δi∈(0,1)},∀i∈k¯,
for γi⁢≠0. The
above definitions facilitate the potential restrictions on the functions fi:R→R,  i∈k¯,
required to derive the various results of the paper. The constraints on the functions fi:R→R, for all i∈k¯,
used in the above definitions of sets, have been proposed by Stević for fi∈BC(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 i∈k¯,
subject to λi∈[0,βi/αi], αi>βi≥0, for all i∈k¯.
The subsequent technical assumption will be then used in some of the
forthcoming results.Assumption 2.1.
αi>0 and 0<δi<min⁡(1,αi−1).The following
two assertions are useful for the analysis of the difference system (1.1).Assertion 2.2.
For any giveni∈k¯, fi∈B(Ki)⇒fi(αixn(i+1)−βixn−1(i+1))≥Ki, for  all n∈N∪{0,−1}.Assertion 2.3.
(i) For any giveni∈k¯, fi∈C(γi,δi,Mi)⇔fi(γi((αi/γi)xn(i+1)−(βi/γi)xn−1(i+1)))≤(δi/γi)(αixn(i+1)−βixn−1(i+1)) if xn(i+1)>(βi/αi)xn−1(i+1)+(γi/αi)Mi, for all n∈N∪{0,−1},
for any real constants βi,αi>0.
(ii)
fi∈C(αi,δi,Mi)⇔fi(αi(xn(i+1)−(βi/αi)xn−1(i+1)))≤(δi/αi)(αixn(i+1)−βixn−1(i+1)) if xn(i+1)>(βi/αi)xn−1(i+1)+Mi, for all n∈N∪{0,−1},
for any real constants βi,αi>0.(iii)
fi∈C(1,δi,Mi)⇔fi(αixn(i+1)−βixn−1(i+1))≤δi(αixn(i+1)−βixn−1(i+1)) if xn(i+1)>(βi/αi)xn−1(i+1)+Mi/αi, for all n∈N∪{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.
Assertion2.3
(i)–(iii) follow
directly from the definitions of B(Ki) and C(γi,δi,Mi), for  all i∈k¯.
Assertion2.3
(iv) The proof is split into proving the two claims
below.
Claim 1.
C(1,δi,Mi)⊂C(αi,αiδi,Mi/αi).
Proof of Claim1.
fi∈C(1,δi,Mi)⇔fi(αixn(i+1)−βixn−1(i+1))=fi(αi(xn(i+1)−(βi/αi)xn−1(i+1)))≤δi(αixn(i+1)−βixn−1(i+1))=δiαi(xn(i+1)−(βi/αi)xn−1(i+1)) if αixn(i+1)−βixn−1(i+1)>Mi⇒fi∈C(αi,αiδi,Mi/αi) if Assumption 2.1 holds.
Claim 2.
C(αi,αiδi,Mi/αi)⊂C(1,δi,Mi).
Proof of Claim2.
fi∈C(αi,αiδi,Mi/αi)⇒fi(αi(xn(i+1)−(βi/αi)xn−1(i+1)))≤αiδi(xn(i+1)−(βi/αi)xn−1(i+1))=δi(αixn(i+1)−βixn−1(i+1)) if αixn(i+1)−βixn−1(i+1)>Mi⇒fi∈C(1,δi,Mi) if Assumption 2.1 holds.
Then, Assertion2.3
(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 ofB(Ki), C(γi,δi,Mi),
and BC(Ki,γi,δi,Mi).Assertion 2.4.
For anyi∈k¯,
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 (jiγ∈J¯iγ,jiδ∈J¯,jiM∈J¯iM) subject to the constraints δijiδ≤δi and MijiM≥Mi, for all (jiδ,jiM)∈J¯iδ×J¯iM,
leading to the associated C(γijiγ,δijiδ,MijiM).
Define the binary relation Ri in C(γi,δi,Mi) as fiRigi⇔fi,gi∈C(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 KijiK≤Ki, δijiδ≤δi, MijiM≥Mi, for all (jiK,jiδ,jiM)∈J¯iK×J¯iδ×J¯iM,
are not equivalence classes, respectively,  in B(Ki) and BC(Ki,γi,δi,Mi).Proof.
In view of Assertion2.3(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 fiRigi⇔fi,gi∈C(1,δijiδ,MijiM)⇒fi,gi∈C(1,δijiδ,MijiM) for some (δijiδ,MijiM)×R.
Now, consider C(1,δijiδ′,MijiM) with δijiδ′>δi such that δi≥δijiδ′(>δijiδ)∈{δij:j∈Jiδ}. 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 hi:R→R defined by hi(x):=δijiδ′x>δijiδx so that C(1,δijiδ′,Mijiδ)϶hi⁢∉C(1,δijiδ,Mijiδ). Thus, C(1,δijiδi,Mi)⁢≠C(1,δijiδ′,Mijiδ). Then, Ri (i∈k¯) are not equivalence relations, and there are
no equivalence classes in C(γi,δi,Mi) (i∈k¯) with respect to Ri (i∈k¯). 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 fi∈C(1,δi,Mi), for all i∈k¯,
then the auxiliary linear system to (1.1) is(3.1)xn+1(i)=λixn(i)+δi(αixn(i+1)−βixn−1(i+1)),∀i∈k¯. If fi∈C(αi,δi,Mi), for all i∈k¯,
then the auxiliary linear system to (1.1) is(3.2)xn+1(i)=λixn(i)+δi(xn(i+1)−βiαixn−1(i+1)),∀i∈k¯. System (3.1)
may be equivalently rewritten as follows by defining the state vector sequence xn:=(xn(1),xn(2),…,xn(k))T∈Rk, for all n∈N,
as the kth-order difference system:(3.3)xn+1=Axn+Bxn−1=(Λ+C)xn+Bxn−1=Λxn+B¯x¯n−1,∀n∈N, with initial
conditions xi:=(xi(1),xi(2),…,xi(k))T∈Rk for i=0,−1,
where x¯n:=(xnT⁢⋮xn−1T)T∈R2k and(3.4)A=[λ1δ1α10⋯00λ2δ2α20⋯0⋮⋮⋱⋮δk−1αk−1δkαk0⋯0λk]∈Rk×k,(3.5)B=[0−δ1β10⋯000−δ2β20⋯0⋮⋮⋱⋮−δk−1βk−1−δkβk0⋯00]∈Rk×k,(3.6)Λ=Diag(λ1,λ2,…,λk),C=[0δ1α10⋯000δ2α20⋯0⋮⋮⋱⋮δk−1αk−1δkαk0⋯00],(3.7)B¯=(B⋮C)∈Rk×2k.
The  one-step
delay may be removed by defining the following extended 2kth-order
system of state vector
x¯n:=(xnT⁢⋮xn−1T)T∈R2k satisfying(3.8)x¯n+1=A¯x¯n,∀n∈N, with x¯0:=(x0(1),x0(2),…,x0(k),x−1(1),x−1(2),…,x−1(k)⁢)T∈R2k and(3.9)A¯=[A⋮B⋯Ik⋮0]∈R2k×2k. 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
asn→∞.
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 fi∈BC(Ki,γi,δi,Mi), 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 fi∈BC(Ki,1,δi,Mi), for all i∈k¯,
since the linear system defined with fi(x)=δix, for all i∈k¯,
in (1.1) has to be globally stable in order to keep global stability for any fi∈BC(Ki,1,δi,Mi), for all i∈k¯.Theorem 3.4.
System
(1.1) is globally stable and permanent for any given set of functions fi∈BC(Ki,1,δi,Mi) for any given Ki∈R and any given Mi∈R, for all i∈k¯,
only if the subsequent properties hold.
(i)|λi|≤1, for all i∈k¯.
(ii)∥A¯∥2≤1,
equivalently, (3.10)‖W‖2=‖A¯TA¯‖2≤1,whereW:=A¯TA¯=[W11⋮W12⋯W12T⋮W22]∈R2k×2k,where (3.11)W11:=ATA+Ik=[1+λ12+δk2αk2λ1δ1α1λ3δ3α3⋯λkδkαkλ1δ1α11+λ22+δ12α12λ2δ2α2⋯λk−1δk−1αk−1⋮⋮⋮λk−2δk−2αk−2λk−1δk−1αk−1⋯λk−12+δk−22αk−22λ2δ2α2λkδkαkλk−1δk−1αk−1⋯λ2δ2α21+λk2+δk−12αk−12],W12:=ATB,
and W22:=BTB=Diag(δk2βk,δ12β1,…,δk−12βk−1),
with Ik being the kth identity matrix. A
necessary condition is ∑i=1k(λi2+δi2(αi2+βi2))≤k.
(iii) There  exists(3.12)P¯=P¯T:=[P¯11⋮P¯12⋯P12T⋮P¯22]≻0in⁢R2k×2k, where P¯ij∈Rk×k⁢(i,j=1,2),
which is a solution to the matrix
identity
(3.13)[(ATP¯11+P¯12T)A+ATP¯12+P¯22−P¯11⋮(ATP¯11+P¯12T)B−P¯12⋯BT(P¯11A+P¯12)−P¯12T⋮BTP¯11B−P¯22]=−Q for any  given (3.14)Q¯=Q¯T:=[Q¯11⋮Q¯12⋯Q¯12T⋮Q¯22]≥0in⁢R2k×2k.Proof.
(i) Note that the identically zero
functionsfi:R→0, for all i∈k¯, are all in BC(Ki,1,δi,Mi) for any Ki≤0, δi∈(0,1), Mi>0, for all i∈k¯.
Proceed by contradiction by assuming that |λi|>1 and fi≡0 for some i∈k¯:={1,2,…,k},
with the system being globally stable. Thus, |xn+1(i)|>|xn(i)| if x0(i)⁢≠0 so that |xn(i)|→∞ as n→∞,
and then the system is unstable for some function fi∈BC(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) Assumefi(x)=δix with δi∈(0,1) everywhere in R so that fi∈C(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, ∥A¯∥2=max⁡1≤i≤k⁢σi1/2.  It is first proved by complete
induction that if x¯0⁢≠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 ρi.
Then, A¯x¯k+1=A¯(A¯x¯k)=A¯(ρix¯k)=ρi(A¯x¯k)=ρix¯k+1 so that x¯k+1 is also an eigenvector of A¯ for the same eigenvalue ρi.
This property leads to
(3.15)‖x¯k+1‖22=‖A¯x¯k‖22=x¯kTA¯TA¯x¯k=ρi2‖x¯k‖22=σi‖x¯k‖22=ρi2k‖x¯0‖22=σik‖x¯0‖22. Proceed by
contradiction by assuming that system (1.1) is stable, for all fi∈C(1,δi,Mi),
with |ρi|=σi1/2>1. From (3.15), |xn(i)|→∞ as n→∞,
and then the system is unstable for a function fi∈C(1,δi,Mi) for any real constant Ki since it possesses an unbounded solution for some
finite initial conditions. Now, redefine the functions f¯i(x) from the above fi(x), i∈k¯,
as follows:
(3.16)f¯i(x)={fi(x)=δixifx≥0,R϶λ<−(1+max⁡1≤i≤k(|λi|))<0ifx<0. It is clear
by construction that if f¯i(x)=fi(x)=δix on an interval of infinite measure and if 0>λ=f¯i(x)⁢≠fi(x) occurs on a real interval of finite measure,
then the above contradiction obtained for fi∈C(1,δi,Mi) still applies for f¯i∈BC(Ki,1,δi,Mi) for any finite negative Ki<−λ.
If f¯i(x)=fi(x) occurs on an interval of finite measure and if f¯i(x)⁢≠fi(x) occurs on an interval of infinite measure,
then the linear system resulting from (1.1) with the replacement fi(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(xk)→−∞ for {xk}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(Ki,1,δi,Mi) for some finite Ki<0.
If the real subintervals within which f¯i(x) equalizes fi(x) or differs from fi(x) are both of infinite measure, the result f¯i∈BC(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 fi∈BC(Ki,1,δi,Mi) for any Ki∈R and any Mi∈R, for all i∈k¯,
only if the subsequent equivalent properties hold: ∥A¯∥2≤1, ∥W∥2≤1.
The necessary condition ∑i=1k(λi2+δi2(αi2+βi2))≤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 equationA¯TP¯⁢A¯−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 fi∈BC(Ki,1,δi,Mi), for all i∈k¯,
according to Property (ii).Concerning
positivity (Definition3.3), 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 fi:R0+→R0+ with fi∈BC(Ki,1,δi,Mi) for any given Ki∈R and any given Mi∈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 fi:R0+→R0+ with fi∈BC(Ki,1,δi,Mi) for any given Ki∈R and any given Mi∈R, for all i∈k¯,
only if λi∈R0+, αi∈R0+, βi∈R0−, for all i∈k¯.Outline of proof
As argued in the proof of Theorem3.4 for
stability, the linear system has to be positive in order to guarantee that it
is positive for any set fi:R0+→R0+ with fi∈BC(Ki,1,δi,Mi) for any given Ki∈R and Mi∈R, for all i∈k¯.
The linear system (3.8)-(3.9) for fi(x)=δix is positive if and only if A¯∈R0+n×n [3] since, in addition, this implies fi∈BC(Ki,1,δi,Mi).
The proof follows since A¯∈R0+n×n by direct inspection if and only if λi∈R0+, αi∈R0+, βi∈R0−, for all i∈k¯.Necessary
joint conditions for stability, permanence, and positivity of (1.1) for any set fi:R0+→R0+ with fi∈BC(Ki,1,δi,Mi) for any given Ki∈R and Mi∈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 αi, βi,
and δi, i∈k¯ (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, i∈k¯,
are allowed to be negative but δi∈(0,1), i∈k¯,
is required to formulate an auxiliary result for the main proof.Theorem 4.1.
Assume
that|λi|<1, for all i∈k¯,
and
(4.1)max(max1≤i≤k|αi|,max1≤i≤k|βi|)<1−max1≤i≤k|λi|2kmax1≤i≤k|δi|. 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:
(4.2)x¯0∈K0(a1,…,a2k,b1,…,b2k):={x=(x1,x2,…,x2k)T∈R2k:xi∈[ai,bi],∞>bi>ai>−∞,∀i∈2k¯}⊂R2k.Proof.
The successive use of the recursive
second identity in (3.3) with initial condition x¯0=(x0T,x−1T)T leads to
(4.3)xn+k=Λn+kx0+∑i=0n+k−1Λn+k−i−1B¯x¯i,∀n∈N,∀k∈k¯, and taking ℓ2-norms in (4.3) with λ:=max⁡1≤i≤k|λi|<1,
we get
(4.4)∥xn+k∥2=∥Λn+k∥2∥x0∥2+∑i=0n+k−1‖Λn+k−i−1‖2‖B¯‖2‖x¯i‖2≤λn+k∥x0∥2+1−λn+k1−λ∥B¯∥2max⁡0≤i≤n+k−1∥x¯i∥2≤λn∥x0∥2+1−λn1−λ∥B¯∥2max⁡0≤i≤n+k−1∥x¯i∥2≤λn∥x0∥2+δmax⁡(α,β)(1−λn)1−λkmax⁡0≤i≤n+k−1∥x¯i∥2≤λn∥x0∥2+2δmax⁡(α,β)(1−λn)1−λkmax⁡−1≤i≤n+k−1∥x¯i∥2,∀n∈N,∀k∈k¯,  where δ:=max⁡1≤i≤k(|δi|), α:=max⁡1≤i≤k(|αi|),
and β:=max⁡1≤i≤k(|βi|) since λ<1 and
(4.5)∥B¯∥2=λmax(B¯TB¯)≤k∥B¯∥1≤kδmax(α,β)for anyx∈Rk,∥Λj∥22=max1≤i≤k(|λi|2j)=λ2j≤λ<1,∀j∈N,max0≤i≤n+k−1(∥x¯i∥2)=max0≤i≤n+k−1(xiTxi+xi−1Txi−1)1/2≤max0≤i≤n+k−1(∥xi∥2+∥xi−1∥2)≤2max−1≤i≤n+k−1∥xi∥2. Note that
(4.4) is still valid if the term preceding the equality is any ∥xn+ℓ∥2, for all ℓ∈N∖n+k¯,
since they are all upper bounded by all the right-hand side upper bounds. Then,
(4.6)∥xn+ℓ∥2≤λn∥x0∥2+2δ⁢max⁡(α,β)(1−λn)1−λkmax−1≤i≤n+k−1∥xi∥2, for all n∈N, for all k∈k¯, for all ℓ∈N∖n+k¯,  which implies directly that
(4.7)max−1≤i≤n+k−1(∥xn+i∥2)≤λn∥x0∥2+2δmax(α,β)1−λkmax−1≤i≤n+k−1∥xi∥2+(∥x0∥2+∥x−1∥2),∀n∈N,∀k∈k¯. If the
condition max⁡(α,β)<(1−λ)/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
(4.8)∥xn∥2≤max⁡−1≤i≤n+k−1∥xn+i∥2≤(1−λ1−λ−2kδmax⁡(α,β))((1+λn)∥x0∥2+∥x−1∥2)≤(1−λ1−λ−2kδmax⁡(α,β))((1+λn0)∥x0∥2+∥x−1∥2)≤(1+ε)(1−λ1−λ−2kδmax⁡(α,β))(∥x0∥2+∥x−1∥2)≤(1+ε)(1−λ1−λ−2kδmax⁡(α,β))(∑i=12kmax⁡(ai2,bi2))1/2≤(1+ε)(1−λ1−λ−2kδmax⁡(α,β))(∑i=12kmax⁡(|ai|,|bi|))(4.9)≤3(1−λ1−λ−2kδmax⁡(α,β))max⁡(∥x0∥2,∥x−1∥2), for all ε∈R+, for all n(≥n0)∈N,
depending on n0, which depends on ε, for any N϶n0≥ln⁡ε/ln⁡λ, for all x¯0∈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
(4.10)K:={x∈Rk:|xi|≤(1+ε)k(1−λ1−λ−2δkmax⁡(α,β))(∑i=12kmax⁡(|ai|,|ai|)),∀i∈k¯} 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 fi:R→R, i∈k¯,
grow at most linearly with their argument.Lemma 4.2.
fi∈C(αi,δi,Mi)⇒fi(x)=O(x), for all i∈k¯.
In addition, fi(x) is bounded for all x≥Mi.
The result also holds if fi∈BC(Ki,αi,δi,Mi), for all Ki∈R, for all i∈k¯.Proof.
Now, it is
proved thatfi(x)=O(x) (notation of “big Landau O” of x) for any fi∈C(αi,δi,Mi), for all i∈k¯. First, note that for all i∈k¯ for some εi:[M,∞)→R0+, fi∈C(αi,δi,Mi)∧x≥Mi∈R+⇒fi(x)=δix−εi(x)≤δix⇒fi(x)=O(x) since fi(x)≤δix+K⁢ for any K∈R0+, for all x≥Mi. The result
also holds if fi∈BC(Ki,αi,δi,Mi), for all Ki∈R,
since BC(Ki,αi,δi,Mi)⊂C(Ki,αi,δi,Mi). It is now proved by a contradiction argument
that if fi∈C(Ki,αi,δi,Mi),
then it is  bounded, for all x<Mi.
Assume x<Mi∈R+ with fi(x1) being arbitrarily large for some x1i<Mi.
Thus, there exists M2i∈R+ being arbitrarily large so that M2i≤fi(x1i)≤fi(Mi)≤δiMi<∞ for x1i=αixn(i+1)−βixn−1(i+1)<Mi since fi∈C(αi,δi,Mi) so that it is monotonically nondecreasing.
This is a contradiction since M2i is arbitrarily large. Thus, fi∈C(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 C1∈R+ for x<Mi as a result, so that fi(x)=O(x) on R.
Again, the result still holds if fi∈BC(Ki,αi,δi,Mi), for all Ki∈R.Theorem 4.3.
Ifλ:=max⁡1≤i≤k|λi|<1−δ, δ:=max⁡1≤i≤kδi∈(0,1), fi∈BC(Ki,αi,δi,Mi), for all Ki∈R, for all i∈k¯, and max⁡(max1≤i≤k|αi|,max1≤i≤k|βi|)<(1−max1≤i≤k|λi|−δ)/4k⁢max1≤i≤kδi, then system (1.1) is globally Lyapunov stable
for any finite arbitrary initial conditions. It is also permanent for any
initial conditions x¯0∈K0(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
(4.11)xn+1=Λxn+B¯x¯n−1+(f¯(x¯n−1)−B¯x¯n−1),∀n,j∈N, where
(4.12)f¯(x¯n−1)=(f1(α2xn(2)−βi+1xn−1(2)),…,fk(α1xn(1)−βi+1xn−1(1)))T. 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)−B¯x¯n−1)}0∞ so that both solution sequences are identical
under identical initial conditions. One gets directly from (4.11) that
(4.13)xn+k=Λn+kx0+∑i=0n+k−1Λn+k−i−1(B¯x¯i+(f¯(x¯n−1)−B¯x¯n−1)),∀n∈N,∀k∈k¯, so that
(4.14)∥xn+k∥2≤λn∥x0∥2+1−λn1−λ(∥B¯∥2max⁡0≤i≤n+k−1∥x¯i∥2+max⁡0≤i≤n+k−1∥f¯(x¯i)−B¯x¯i∥2)≤λn∥x0∥2+1−λn1−λ(2∥B¯∥2+δ)max⁡0≤i≤n+k−1(∥x¯i∥2)+(1−λn)C11−λ. Then by direct extension of (4.7) when using
(4.14),
(4.15)max⁡−1≤i≤n+k−1(∥xn+i∥2)≤λn∥x0∥2+11−λ((4δ⁢max⁡(α,β)k+δ)max⁡−1≤i≤n+k−1∥xi∥2+C1)+(∥x0∥2+∥x−1∥2),∀n∈N,∀k∈k¯, with δ∈(0,1) for some finite C1∈R+ since |fi(x)|≤δ|x|+C1, for all i∈k¯,
from Lemma 4.2. Thus, max⁡−1≤i≤n+k−1(∥xn+i∥2) may be regrouped in the left-hand side provided
that
(4.16)1>11−λ(4δ⁢max⁡(α,β)k+δ)⟺max⁡(α,β)<1−λ−δ4δk. Then, under
similar reasoning as that used to derive (4.8)-(4.9), one gets
from (4.15) that
(4.17)∥xn∥2≤max⁡−1≤i≤n+k−1∥xn+i∥2≤(11−λ−δ(1+4kmax⁡(α,β)))((1−λ)((1+λn)∥x0∥2+∥x−1∥2+C1))≤(11−λ−δ(1+4kmax⁡(α,β)))((1−λ)((1+λn0)∥x0∥2+∥x−1∥2+C1))≤(1+ε)(11−λ−δ(1+4kmax⁡(α,β)))((1−λ)(∥x0∥2+∥x−1∥2)+C1)≤(1+ε)(11−λ−δ(1+4kmax⁡(α,β)))((1−λ)∑i=12kmax⁡(ai2,bi2)+C1)1/2≤(1+ε)(11−λ−δ(1+4kmax⁡(α,β)))((1−λ)max⁡(∥x0∥2,∥x−1∥2)+C1)≤3(11−λ−δ(1+4kmax⁡(α,β)))((1−λ)max⁡(∥x0∥2,∥x−1∥2)+C1),   for all ε∈R+, for all n(≥n0)∈N, depending on n0,
which depends on ε,
for any N϶n0≥ln⁡ε/ln⁡λ.
The solution sequences are all bounded under any finite initial conditions and
enter the compact set K defined by
(4.18){x∈Rk:|xi|≤(1+ε)k((11−λ−δ(1+4kmax⁡(α,β)))×((1−λ)∑i=12kmax⁡(|ai|,|ai|)+C1)),∀i∈k¯}, 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 fi:R→R is bounded from above and below, then |xn(i)| is bounded, for all n∈N. If |λi|>1 and fi:R→R 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 i∈k¯.(ii)
Assume
thatfi∈B(Ki) and |λi|>1 for some i∈k¯.
Then |xn+1(i)|>|xn(i)|, for all n∈N,
and |xn(i)|→∞ as n→∞ if |x0(i)|>|Ki|/(|λi|−1) (|x0(i)|≥|Ki|/(|λi|−1)ifKi⁢≠0).
Thus, system (1.1) is unstable under such sufficiently large absolute values of
the initial conditions for some i∈k¯.Proof.
(i) If−∞<M1i≤fi(x)≤M2i<∞, for all x∈R,
for some Mji, j = 1,2, and some i∈k¯,
then
(4.19)|λin|(|x0(i)|+|∑j=0∞λi−j−1|max⁡(|M1i|,|M2i|))≥|xn(i)|≥|λin|(|x0(i)|−|∑j=0n−1λi−j−1|max⁡0≤j≤i|fi(αixj(i+1)−βixj−1(i+1))|)≥|λin|(|x0(i)|−|∑j=0n−1λi−j−1|max⁡(|M1i|,|M2i|))≥|λin|||x0(i)|−|∑j=0∞λi−j−1|max⁡(|M1i|,|M2i|)|. 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=0∞λi−j−1|max⁡(|M1i|,|M2i|)|<∞, and,
furthermore, if |x0(i)|>(|λi|/(|λi|−1))max⁡(|M1i|,|M2i|)≥|∑j=0∞λi−j−1|max⁡(|M1i|,|M2i|), then there
is a strictly monotonically increasing subsequence {|xn(i)|}n∈S of {|xn(i)|}0∞,
where S:={n1,n2,…} is a countable subset of N, so that |xnj+1(i)|>|xnj(i)|, for all nj∈S,
and |xnj(i)|→∞ as S϶nj→∞ (i.e., it diverges).
Iffi∈BC(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), fi∈B(Ki),
and |λi|>1,
it follows that   (4.20)(xn+1(i))2−(xn(i))2=(λi2−1)(xn(i))2+fi2(αixn(i+1)−βixn−1(i+1))+2λifi(αixn(i+1)−βixn−1(i+1))xn(i)≥gn(i)(|xn(i)|):=Ki2−(2λi|Ki|−(λi2−1)|xn(i)|)|xn(i)|>0 if |x0(i)|>2|λi||Ki|/(λi2−1)(|x0(i)|≥|Ki|/(|λi|−1)ifKi⁢≠0)⇒|xn+1(i)|>|xn(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 gn(i)(|xn(i)|)gn(i)(|xn(i)|) which are g2n(i)=|Ki|/(|λi|−1)≥g1n(i)=|Ki|/(|λi|+1), which implies that gn(i)(|xn(i)|)≤0 if |xn(i)|∈[g1n(i),g2n(i)],
and (xn+1(i))2−(xn(i))2≥gn(i)(|xn(i)|)>0 if |xn(i)|∈(−∞,g2n(i))∪(g2n(i),∞).
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 sequencexn:=(xn(1),xn(2),…,xn(k))T of (1.1) is nonnegative, for all n∈N,
and any finite nonnegative x0(i)≥0, for all i∈k¯,
if fi(αix0(i+1)−βix−1(i+1))≥−λix0(i), for all i∈k¯,
and
(5.1)fi(αixn(i+1)−βixn−1(i+1))≥−λixn(i)=−(λin+1x0(i)+∑j=0n−1λin−jfi(αixj(i+1)−βixj−1(i+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 x0(i)≥0, for all i∈k¯,
if λi∈R0+ and fi:R→R0+, for all i∈k¯.
Then, system (1.1) is positive.(iii)
Assume
thatλi∈R0+, for all i∈k¯,
and that there exist 2k real constants Cj(i)∈R0+, i∈k¯, j=1,2,
independent of n, such that
(5.2)−∞<−C1(i)≤fi(αix0(i+1)−βix−1(i+1))≤C2(i)<∞,∀i∈k¯,−∞<−C1(i)≤∑j=0n−1λin−j−1fi(αixj(i+1)−βixj−1(i+1))≤C2(i)<∞,∀i∈k¯,∀n∈N.
Then, the
solution vector sequence is nonnegative, for alln∈N0∖n¯0,
for some finite n0∈N0,
depending on xj(i)⁢(j=0,−1,for all⁢i∈k¯), for any
given finite x0(i)>0, for all i∈k¯.(iv)
Assume
thatfi∈B(Ki) and λi>1, for all i∈k¯.
Then, any solution vector sequence of (1.1) is nonnegative; that is, xn∈R0+n, for all n∈N,
for any given finite x−1∈Rk and some Rk϶x0≫0 of sufficiently large components (i.e., x0∈R+k and x0(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 fi∈B(Ki) with Ki∈R0+, for all i∈k¯,
then the constraints λi>1 are weakened to λi∈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 λi>1 and fi∈BC(Ki,1,δi,Mi), for all i∈k¯.
Thus, there exists x0≫0 of sufficiently large finite components so
that any solution is  strictly positive, that is, xn≫0, for all n∈N,
under initial condition
x0≫0.
The sizes are quantifiable from the knowledge of the scalars Ki,δi,Mi  (i∈k¯) and upper bounds of the nonzero entries of A
and B.Proof.
(i) The recursive use of (1.1) yields
(5.3)xn(i)=λinx0(i)+∑j=0n−1λin−j−1fi(αixj(i+1)−βixj−1(i+1)),∀i∈k¯,∀n∈N, for any given xi(i)  (i=0,−1), for all i∈k¯.
Then,
(5.4)fi(αixn(i+1)−βixn−1(i+1))≥−λixn(i)=−(λin+1x0(i)+∑j=0n−1λin−jfi(αixj(i+1)−βixj−1(i+1))),∀n∈N0⇒xn+1(i)=λixn(i)+fi(αixn(i+1)−βixn−1(i+1))≥0,∀n∈N0.
(ii)
xn(i)=λixn−1(i)+fi(αixn−1(i+1)−βixn−2(i+1))≥0, for all n∈N,
if x0(i)≥0, λi∈R0+,
and fi:R→R0+, for all i∈k¯.(iii)
xn(i)=λinx0(i)+∑j=0n−1λin−j−1fi(αixj(i+1)−βixj−1(i+1))≥λinx0(i)−C1(i)≥0, for all n≥n0:=max⁡1≤i≤k(ln⁡(C1(i)/x0(i))/ln⁡λi)−1, for all i∈k¯.
Such an n0,
being dependent on x0(i),
always exists for λi>1 since C1(i)<∞ and λinx0(i)→∞ as n→∞ for any x0(i)>0, for all i∈k¯.(iv)
Sincefi:R→R, for all i∈k¯,
are bounded from below on R, then max⁡n∈Nfi(αixn(i+1)−βixn(i+1))≥Ki>−∞, lim⁡inf⁡n→∞fi(αixn(i+1)−βixn(i+1))≥Ki>−∞ for some finite Ki∈R, for all i∈k¯.
Irrespective of the value of Ki,
since it is finite, there always exists a finite constant Ki′∈R+ fulfilling Ki≥−Ki′ such that
(5.5)max⁡n∈N⁢fi(αixn(i+1)−βixn(i+1))≥−Ki′=−|Ki′|>−∞,lim⁡inf⁡n→∞⁢fi(αixn(i+1)−βixn(i+1))≥−|Ki′|>−∞,for all i∈k¯.
Since λi>1,
the series ∑j=0∞λi−j converges so that
(5.6)∑i=0nλi−j=1−λi−(n+1)1−λi−1=λin+1−1λin(λi−1)≤∑j=0∞λi−j=11−λi−1=λiλi−1,∀i∈k¯,∀n∈N.
Then,
(5.7)xn(i)=λin(x0(i)−∑j=0n−1λi−j−1|fi(αixn(i+1)−βixn(i+1))|)≥λin(x0(i)−∑j=0n−1λi−j−1max⁡n∈N|fi(αixn(i+1)−βixn(i+1))|)≥λin(x0(i)−(λin+1−1)|Ki′|λin(λi−1))≥λin(x0(i)−λi|Ki′|λi−1),
for alli∈k¯, for all n∈N.
As a result, xn(i)∈R0+,  for all i∈k¯, for all n∈N,
if x0(i)≥λ|Ki′|/(λ−1)>0, for all i∈k¯.
Then, xn≥0, for all n∈N.
If x0(i)>λ|Ki′|/(λ−1) for at least one i∈k¯,
then xn>0, for all n∈N.
If x0(i)>λ|Ki′|/(λ−1), for all i∈k¯,
then xn≫0, for all n∈N.(v)
DefineM:=(M1,M2,…,Mk)T≫0 with the constants Mi of the sets BC(Ki,1,δi,Mi), for all i∈k¯.
Since fi(x)=δix−f˜i(x) for some f˜i:[M,∞)→R0+, for all i∈k¯, for all x≥Mi,
from the definition of the sets C(1,δi,Mi),
it follows from (3.9) that
(5.8)x¯n≥A¯nx¯0−∑i=0n−1A¯n−1−iB¯K˜′
for anyK˜′:=(K˜1′,K˜2′,…,K˜k′)T≫0 such that Ki≥−|K˜i′|, for all i∈k¯.
Since λi>1, for all i∈k¯,
then from the structure of the matrix A¯ in (3.9),
(5.9)xn(i)≥λinx0(i)+∑i=0n−1eiTΛn−1−iB¯x¯i−∑i=0n−1eiTΛn−1−iB¯K˜′≥λinx0(i)+∑i=0n−1eiTΛn−1−iB¯(A¯ix¯0−∑j=0i−1A¯i−1−jB¯K˜′)−∑i=0n−1eiTΛn−1−iB¯K˜′=λinx0(i)+∑i=0n−1eiTΛn−1−iB¯⁢A¯ix¯0−eiT(Λn−1−i+∑j=0i−1A¯i−1−j)B¯K˜′=eiT(λinIn+∑i=0n−1Λn−1−iB¯⁢A¯i)x¯0−eiT(Λn−1−i+∑j=0i−1A¯i−1−j)B¯K˜′≫Mi,
sinceλi>1, for all i∈k¯,
provided that it is sufficiently large, x0(i)≥max⁡(Mi,υi)>0 (i.e., x0≫0 has sufficiently large positive components), for all i∈k¯, for all n∈N,
where eiT is the ith unity vector in Rk of components eij=δij (the Kronecker delta), for all i,j∈k¯.Note that the
properties associated withfi∈BC(Ki,1,δi,Mi), for all i∈k¯,
have not been invoked in Theorem 5.1(i)–(iii). Theorem 5.1(ii) implicitly assumes fi∈B(Ki),
since they are assumed to be nonnegative, for all i∈k¯.



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*Source: 595367-2008-09-22.xml*