about the stability and positivity of a class of discrete nonlinear systems of difference equations

De la Sen,manuel.delasen@ehu.es Received 16 January 2008; Accepted 26 May 2008 Recommended by Antonia Vecchio This paper investigates stability conditions and positivity of the solutions

Volume 2008, Article ID 595367, 18 pages

doi:10.1155/2008/595367

Research Article

About the Stability and Positivity of

a Class of Discrete Nonlinear Systems of

Difference Equations

M De la Sen

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

Correspondence should be addressed to M De la Sen,manuel.delasen@ehu.es

Received 16 January 2008; Accepted 26 May 2008

Recommended by Antonia Vecchio

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

Copyrightq 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

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 n1 λ i x i n  f i



α i x i1 n − β i x n−1 i1

, ∀i ∈ k : {1, 2, , k}, 1.1

with x k1 n ≡ x1n , 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 x0i , x−1i , for all i ∈ k The identity x k1 n ≡ x1n allows the inclusion

in a unified shortened notation via1.1 of the dynamics:

x n1 k λ i x n k  f i



α i x n1− β i x1n−1

, ∀i ∈ k, 1.2

as it follows by comparing1.1 for i k with 1.2 The solution vector sequence of 1.1

will be denoted as x n: x1n , x n2, , x k n T ∈ Rk , for all n ∈ N, under initial conditions x j:

x1j , x2j , , x k j 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 genetics1 The study in1 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 ∈ Rn×n , then P T is the transpose of P.

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×n

0 ,

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

inRkof1.1 will be nonnegative in some interval S, denoted by x n ≥ 0 identical to x n∈ Rk

0,

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∈ Rk

0, 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∈ Rk

and

implying that x n > 0 and x n∈ Rk

0, for all n ∈ S ⊂ N.

 2 and  1 are the 2 and 1 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 system1.1, 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 γ i

i x, ∀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, BCK i , γ i , δ i , M i  : BK i  ∩ Cγ 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 γ i

i x,

∀x, y>x ∈ R, K i ∈ 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 Stevi´c

for f i ∈ BCK 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 < min1, α−1

i 

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 ∈ BK i  ⇒ f i α i x i1 n − β i x i1 n−1  ≥ K i , for all n ∈ N ∪

{0, −1}.

Assertion 2.3 i For any given i ∈ k, f i ∈ Cγ i , δ i , M i  ⇔ f i γ i α i /γ i x i1 n − β i /γ i x i1 n−1  ≤

δ i /γ i α i x i1 n − β i x i1 n−1  if x i1 n > β i /α i x i1 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 ∈ Cα i , δ i , M i  ⇔ f i α i x i1 n −β i /α i x n−1 i1  ≤ δ i /α i α i x i1 n −β i x i1 n−1  if x n i1 >

β i /α i x i1 n−1  M i , for all n ∈ N ∪ {0, −1}, for any real constants β i , α i > 0.

iii f i ∈ C1, δ i , M i  ⇔ f i α i x n i1 − β i x i1 n−1  ≤ δ i α i x i1 n − β i x n−1 i1  if x i1 n >

β i /α i x i1 n−1  M i /α i , for all n ∈ N ∪ {0, −1}, for any real constants β i , α i > 0.

iv C1, δ i , M i  Cα i , α i δ i , M i /α i ifAssumption 2.1holds

Proof. Assertion 2.3i–iii follow directly from the definitions of BK i  and Cγ i , δ i , M i, for

all i ∈ k.

Claim 1 C1, δ i , M i  ⊂ Cα i , α i δ i , M i /α i

Proof of Claim 1 f i ∈ C1, δ i , M i  ⇔ f i α i x i1 n − β i x n−1 i1  f i α i x i1 n − β i /α i x i1 n−1  ≤

δ i α i x i1 n − β i x i1 n−1  δ i α i x n i1 − β i /α i x i1 n−1  if α i x i1 n − β i x i1 n−1 > M i ⇒ f i ∈

Cα i , α i δ i , M i /α i ifAssumption 2.1holds

Claim 2 Cα i , α i δ i , M i /α i  ⊂ C1, δ i , M i.

Proof of Claim 2 f i ∈ Cα i , α i δ i , M i /α i  ⇒ f i α i x i1 n − β i /α i x n−1 i1  ≤ α i δ i x n i1 − β i /

α i x n−1 i1  δ i α i x i1 n − β i x i1 n−1  if α i x n i1 − β i x i1 n−1 > M i ⇒ f i ∈ C1, δ i , M i ifAssumption 2.1

holds

Then,Assertion 2.3iv has been proved from Claims1-2

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 BK i , Cγ i , δ i , M i,

and BCK i , γ i , δ i , M i

Assertion 2.4 For any i ∈ k, consider Cγ 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 allj iδ , j iM  ∈ J iδ × J iM , leading to the associated Cγ ijiγ , δ ijiδ , M ijiM Define

the binary relation R i in Cγ i , δ i , M i  as f i R i g i ⇔ f i , g i ∈ Cγ ijiγ , δ ijiδ , M ijiM  Then, R iis not an

equivalence relation so that Cγ ijiγ , δ ijiδ , M ijiM  are not equivalence classes in Cγ i , δ i , M i with

respect to R i Also, the sets BK ijiK  and BCK 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 allj iK , j iδ , j iM  ∈ J iK × J iδ × J iM, are not

equivalence classes, respectively, in BK i  and BCK i , γ i , δ i , M i

Proof In view of Assertion 2.3iv, 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 i R i g i ⇔ f i , g i ∈

C1, δ ijiδ , M ijiM  ⇒ f i , g i ∈ C1, δ ijiδ , M ijiM  for some δ ijiδ , M ijiM  ∈ 0, 1 × R Now,

consider C1, δ

ijiδ , M ijiM  with δ

ijiδ > δ i such that δ i ≥ δ

ijiδ > δ ijiδ  ∈ {δ ij : j ∈ J iδ}

Then, C1, δ ijiδ , M ijiM  ⊂ C1, δ

ijiδ , M ijiM Since the equivalence classes with respect to any

equivalence relation are disjoint, C1, δ ijiδ , M ijiM  in C1, δ i , M i  with respect to R i is not an

equivalence class unless C1, δ ijiδ , M ijiM  C1, δ

ijiδ , M ijiM Now, consider the linear function

h i : R → R defined by h i x : δ

ijiδ x > δ ijiδ x so that C1, δ

ijiδ , M ijiδ   h i / ∈ C1, δ ijiδ , M ijiδ

Thus, C1, δ ij iδi , M i  / C1, δ

ijiδ , M ijiδ  Then, R i i ∈ k are not equivalence relations, and there are no equivalence classes in Cγ 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 BK i and

BCK i , γ i , δ i , M i, and it is omitted

3 Necessary conditions for stability and positivity

Now, linear systems for system1.1 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 systems1.1, whose functions are in some appropriate

set BK i , Cγ i , δ i , M i , or BCK 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 ∈ C1, δ i , M i , for all i ∈ k, then the auxiliary linear system to 1.1 is

x i n1 λ i x i n  δ i



α i x i1 n − β i x i1 n−1 

, ∀i ∈ k. 3.1

If f i ∈ Cα i , δ i , M i , for all i ∈ k, then the auxiliary linear system to 1.1 is

x i n1 λ i x n i  δ i x n i1−β α i

i x i1 n−1

, ∀i ∈ k. 3.2

System3.1 may be equivalently rewritten as follows by defining the state vector sequence

x n: xn1, x2n , , x n kT∈ Rk , for all n ∈ N, as the kth-order difference system:

x n1 Ax n  Bx n−1 Λ  Cx n  Bx n−1 Λx n  Bx n−1 , ∀n ∈ N, 3.3

with initial conditions x i : x1i , x2i , , x k i T ∈ Rk for i 0, −1, where x n : xT x T

n−1T ∈

R2kand

A

⎡

⎢

⎢

. δ k−1 α k−1

⎤

⎥

B

⎡

⎢

⎢

⎣

. −δ k−1 β k−1

⎤

⎥

⎥

Λ Diagλ1, λ2, , λ k

, C

⎡

⎢

⎢

⎣

. δ k−1 α k−1

⎤

⎥

⎥

B B C

The one-step delay may be removed by defining the following extended 2kth-order system

of state vectorx n: xT x T

n−1T ∈ R2ksatisfying

x n1 Ax n , ∀n ∈ N, 3.8

with x0: x10 , x02, , x k0 , x1−1, x2−1, , x k−1T ∈ R2kand

A

⎡

⎢

⎣

A B

· · ·

I k . 0

⎤

⎥

Note that the extended system3.8-3.9 is fully equivalent to system 3.3–3.7 since both have identical solutions for each given common set of initial conditions Now, let·2be the

2-norm of real vectors of any order and associated induced norms of matricesi.e., spectral norms of vectors and matrices The following definitions are useful to investigate 1.1

Definition 3.1 System1.1 is said to be globally Lyapunov stable or simply globally stable

if any solution is bounded for any finite initial conditions

Definition 3.2 System1.1 is said to be permanent if any solution enters a compact set K for

n ≥ n0for any bounded initial conditions with n0depending on the initial conditions

Definition 3.3 System1.1 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 3.2is the definition of permanence in the sense used in1, which is compatible with global and local stability and with global or local asymptotic stability according toDefinition 3.1and the above comments if 0∈ K However,

it has to be pointed out that there are different definitions of permanence, like, for instance,

in2, 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 system1.1 under the

requirement f i ∈ BCK 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 system1.1 for all the

sets of functions f i ∈ BCK i , 1, δ i , M i , for all i ∈ k, since the linear system defined with

f i x δ i x, for all i ∈ k, in 1.1 has to be globally stable in order to keep global stability for

any f i ∈ BCK 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 ∈

BCK 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 A2≤ 1, equivalently,

W2 A T

A

2≤ 1, where W : A T A

⎡

⎢

⎣

W11 . W

12

· · ·

W T

12 . W

22

⎤

⎥

⎦ ∈ R2k×2k , 3.10

W11 : AT A  I k

⎡

⎢

⎢

⎢

⎢

1 λ2

1 δ2

k α2

k λ1δ1α1 λ3δ3α3 · · · λ k δ k α k

λ1δ1α1 1 λ2

2 δ2

1α2

1 λ2δ2α2 · · · λ k−1 δ k−1 α k−1

. .

λ k−2 δ k−2 α k−2 λ k−1 δ k−1 α k−1 · · · λ2

k−1  δ2

k−2 α2

k−2 λ2δ2α2

λ k δ k α k λ k−1 δ k−1 α k−1 · · · λ2δ2α2 1 λ2

k  δ2

k−1 α2

k−1

⎤

⎥

⎥

⎥

⎥,

3.11

W12 : AT B, and W22 : BT B Diagδ2

k β k , δ2

1β1, , δ2

k−1 β k−1 , with I k being the kth identity matrix A necessary condition isk

i 1 λ2

i  δ2

i α2

i  β2

i  ≤ k.

iii There exists

P P T :

⎡

⎢

⎣

P11 . P

12

· · ·

P T

12 . P

22

⎤

where P ij∈ Rk×k i, j 1, 2, which is a solution to the matrix identity

⎡

⎢

⎣



A T P11 P T12A  A T P12 P22− P11 . 

A T P11 P T12B − P12

· · ·

B T

P11A  P12



− P T12 . B T P11B − P22

⎤

⎥

for any given

Q Q T :

⎡

⎢

⎣

Q11 . Q

12

· · ·

Q T12 . Q

22

⎤

⎥

Proof i Note that the identically zero functions f i : R → 0, for all i ∈ k, are all in

BCK 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 n1 | > |x i n | if x i0 / 0 so that |x i n | → ∞ as n → ∞, and then the system

is unstable for some function f i ∈ BCK 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 i x with δ i ∈ 0, 1 everywhere in R so that f i ∈ C1, δ i , M i , M i >

0 Let the spectrum of W be σW : {σ1, σ2, , σ k}, with each eigenvalue being repeated

as many times as its multiplicity Then,A2 max1≤i≤kσ 1/2

i It is first proved by complete induction that ifx0/ 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 k1 AAx k  Aρ i x k  ρ i Ax k  ρ i x k1 so that x k1 is also an eigenvector of A for the same eigenvalue ρ i This property leads to

x k12

2 Ax k2

2 x T k A T A x k ρ2

i x k2

2 σ i x k2

2 ρ 2k

i x02

2 σ k

i x02

2. 3.15

Proceed by contradiction by assuming that system 1.1 is stable, for all f i ∈ C1, δ i , M i, with|ρ i | σ 1/2

i > 1 From 3.15, |x i n | → ∞ as n → ∞, and then the system is unstable for a function f i ∈ C1, δ i , M i  for any real constant K isince 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

⎧

⎪

⎪

R  λ < −1 max

1≤i≤kλ i  < 0 if x < 0. 3.16

It is clear by construction that if f i x f i x δ i x 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 ∈ C1, δ i , M i  still applies for f i ∈ BCK 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 ∈ BCK 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 ∈ BCK 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 ∈ BCK 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:

A2≤ 1, W2≤ 1 The necessary conditionk

i 1 λ2

i  δ2

i α2

i  β2

i  ≤ 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 T P A − P −Q which has to possess a

solution P

the global stability of the extended linear system3.8-3.9, 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 system1.1 for any set f i ∈ BCK i , 1, δ i , M i , for all i ∈ k, according to

Propertyii

Concerning positivityDefinition 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 parameterssee, 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 bothcontinuous-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 nonnegative2,13 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 i x 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 ∈ BCK 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 ∈ BCK 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 ofTheorem 3.4for stability, the linear system has to be positive in

order to guarantee that it is positive for any set f i :R0 → R0with f i ∈ BCK 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 i x is

positive if and only if A∈ Rn×n

0 3 since, in addition, this implies f i ∈ BCK i , 1, δ i , M i The proof follows sinceA ∈ R n×n

0 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 of1.1 for any set

f i : R0 → R0with f i ∈ BCK i , 1, δ i , M i  for any given K i ∈ R and M i ∈ R, for all i ∈ k,

follow directly by combining Theorems3.4and3.5

4 Main stability results

This section derives sufficiency-type conditions easy to test for global stability of the linear system3.3–3.7 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 interval0, 1, but on their maximum sizes It is allowed that λ ibe 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≤kα i ,max

1≤i≤kβ i  < 1 − max1≤i≤kλ i

2√

k max1≤i≤kδ i . 4.1

Then, the linear system3.3–3.7, equivalently system 3.8-3.9, is globally Lyapunov stable for

any finite arbitrary initial conditions It is also permanent for any initial conditions:

x0 ∈ K0



a1, , a 2k , b1, , b 2k

: x x1, x2, , x 2k

T

∈ R2k : x i∈a i , b i

, ∞ > b i > a i > −∞, ∀i ∈ 2k⊂ R2k 4.2

Proof The successive use of the recursive second identity in3.3 with initial condition x0

x T

0, x T

−1Tleads to

x nk Λnk x0nk−1

i 0

Λnk−i−1 Bx i , ∀n ∈ N, ∀k ∈ k, 4.3

and taking 2-norms in4.3 with λ : max1≤i≤k|λ i | < 1, we get

x nk

2 Λnk

2x0

2nk−1

i 0 Λnk−i−1

2B2x i

2

≤ λ nk x0

21− λ nk

1− λ B2 max

0≤i≤nk−1x i

2

≤ λ n x0

21− λ n

1− λ B2 max

0≤i≤nk−1x i

2

≤ λ n x0

2δ maxα, β



1− λ n

1− λ

k max

0≤i≤nk−1x i

2

≤ λ n x0

22δ maxα, β



1− λ n

1− λ

k max

−1≤i≤nk−1 x i

2, ∀n ∈ N, ∀k ∈ k,

4.4

where δ : max1≤i≤k|δ i |, α : max1≤i≤k|α i |, and β : max1≤i≤k|β i | since λ < 1 and

B2

λmax

B T B

≤kB1≤kδ maxα, β for any x ∈ R k ,

Λj2

2 max 1≤i≤kλ i2j

λ 2j ≤ λ < 1, ∀j ∈ N,

max 0≤i≤nk−1x i

2

 max 0≤i≤nk−1



x T

i x i  x T i−1 x i−11/2

≤ max 0≤i≤nk−1x i

2x i−1

2



≤ 2 max

−1≤i≤nk−1 x i

2.

4.5

Note that4.4 is still valid if the term preceding the equality is any x n2, for all  ∈ N \

n  k, since they are all upper bounded by all the right-hand side upper bounds Then,

x n

2≤ λ n x0

22δ maxα, β



1− λ n

1− λ

k max

−1≤i≤nk−1 x i

for all n ∈ N, for all k ∈ k, for all  ∈ N \ n  k, which implies directly that

max

−1≤i≤nk−1 x ni

2



≤ λ n x0

22δ maxα, β

1− λ

k max

−1≤i≤nk−1 x i

2x0

2x−1

2



,

∀n ∈ N, ∀k ∈ k.

4.7