On ϕ0 stability of a class of singular difference equations

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R E S E A R C H

Open Access

On

φ

_

-stability of a class of singular difference

equations

Peiguang Wang

1*

_{, Meng Wu}

2

_{and Yonghong Wu}

3

*_{Correspondence:}

pgwang@hbu.edu.cn

1_{College of Electronic and}

Information Engineering, Hebei University, Baoding, 071002, China Full list of author information is available at the end of the article

Abstract

This paper investigates a class of singular difference equations. Using the framework of the theory of singular difference equations and cone-valued Lyapunov functions, some necessary and sufficient conditions on the

φ

0-stability of a trivial solution of

singular diﬀerence equations are obtained. Finally, an example is provided to illustrate our results.

MSC: 39A11

Keywords: singular diﬀerence equations; cone-valued Lyapunov functions;

φ

0-stability; uniformly

φ

0-stability

1 Introduction

The singular diﬀerence equations (SDEs), which appear in the Leontiev dynamic model of multisector economy, the Leslie population growth model, singular discrete optimal control problems and so forth, have gained more and more importance in mathematical models of practical areas (see [] and references cited therein). Anh and Loi [, ] studied the solvability of initial-value problems as well as boundary-value problems for SDEs.

It is well known that stability is one of the basic problems in various dynamical systems. Many results on the stability theory of difference equations are presented, for example, by Agarwal [], Elaydi [], Halanay and Rasvan [], Martynjuk [] and Diblíket al.[]. Recently, Anh and Hoang [] obtained some necessary and sufficient conditions for the stability properties of SDEs by employing Lyapunov functions. The comparison method, which combines Lyapunov functions and inequalities, is an effective way to discuss the stability of dynamical systems. However, this approach requires that the comparison sys-tem satisfies a quasimonotone property which is too restrictive for many applications be-cause this property is not a necessary condition for a desired property like stability of the comparison system. To solve this problem, Lakshmikantham and Leela [] initiated the method of cone and cone-valued Lyapunov functions and developed the theory of differ-ential inequalities. By employing the method of cone-valued Lyapunov functions, Akpan and Akinyele [], EL-Sheikh and Soliman [], Wang and Geng [] investigated the sta-bility and theφ-stability of ordinary differential systems, functional differential equations

and diﬀerence equations, respectively.

However, to the best of our knowledge, there are few results for theφ-stability of

sin-gular difference equations. In this paper, utilizing the framework of the theory of sinsin-gular difference equations, we give some necessary and sufficient conditions for theφ-stability

of a trivial solution of singular diﬀerence equations via cone-valued Lyapunov functions.

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2 Preliminaries

The following deﬁnitions can be found in reference [].

Deﬁnition . A proper subsetKofRn_{is called a cone if}

(i) λK⊆K,λ≥; (ii) K+K⊆K; (iii) K=K; (iv) K=∅;

(v) K∩(–K) ={},

whereK andK_{denote the closure and interior of}_K_{, respectively, and}_∂_K_{denotes the}

boundary ofK. The order relation onRn_{induced by the cone}_K_{is defined as follows: Let} x,y∈K, thenx≤Kyiffy–x∈Kandx<Kyiffy–x∈K.

Deﬁnition . The setK*₌_{_φ_∈_Rn_{, (}_φ_,_x₎_≥_{, for all}_x_∈_K_}_{is said to be an adjoint cone} if it satisﬁes the properties (i)-(v).

x∈K iﬀ (φ,x) > , and

x∈∂k iﬀ (φ,x) =  for someφ∈K_*,K=K–{}.

Deﬁnition . A functiong:D→Rn_,_D_⊂_Rn_{is said to be quasimonotone relative to}_K ifx,y∈Dandy–x∈∂Kimplies that there existsφ∈K*such that

(φ,y–x) =  and

φ,g(y) –g(x)

≥.

Deﬁnition . A functiona(r) is said to belong to the classKifa∈C[R+,R+],a() = ,

anda(r) is strictly monotone increasing function inr.

Consider the following SDEs:

Anxn++Bnxn=fn(xn), n≥, (.)

whereAn,Bn∈Rm×mandfn:Rm→Rmare given. Throughout this paper, we assume that the matricesAnare singular, and the corresponding linear homogeneous equations

Anxn++Bnxn= , n≥, (.)

are of index- [–],i.e., the following hypotheses hold.

(H) rankAn=r,n≥,

(H) Sn∩kerAn–={},n≥, whereSn={ξ∈Rm:Bnξ∈imAn},n≥. For the next discussion, the following lemma from [] is needed.

Lemma . Suppose that the hypothesis(H)holds.Then the hypothesis(H)is equivalent

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(i) the matrixGn:=An+BnQn–,nis nonsingular;

(ii) Rm₌_S

n⊕kerAn–.

Let us associate SDEs (.) with the initial condition

Pn–xn=Pn–γ, n≥, (.)

whereγ is an arbitrary vector inRm_and_n

is a ﬁxed nonnegative integer.

Theorem .[] Let fn(x)be a Lipschitz continuous function with a suﬃciently small Lip-schitz coeﬃcient,i.e.,

fn(x) –fn(x)≤Lnx–x, ∀x,x∈Rm,

where

ωn:=LnQn–,nG–n, ∀n≥. Then IVP(.), (.)has a unique solution.

Setn:={x∈Rm:Qn–x=Qn–,nGn–[fn(x) –BnPn–x]}. Ifx={xn}is any solution of IVP (.), (.), then obviouslyxn∈ n(n≥n).

Deﬁnition .[] The trivial solution of (.) is said to beA-stable (P-stable) if for each

>  and anyn≥, there exists aδ=δ(,n)∈(,] such that

An–γ<δ

Pn–γ<δ

implies xn(n;γ)<,n≥n.

Deﬁnition . The trivial solution of (.) is said to be

(S) A-φ-stable (P-φ-stable) if for each> and anyn≥, there exists aδ=δ(n,)∈

(,]such that for someφ∈K*

(φ,An–γ) <δ

(φ,Pn–γ) <δ

implies φ,xn(n;γ)

<, n≥n.

(S) A-uniformlyφ-stable (P-uniformlyφ-stable) ifδin (S) is independent ofn.

(S) A-asymptoticallyφ-stable (P-asymptoticallyφ-stable) if for anyn≥there exist

positive numbersδ=δ(n)andN=N(n,)such that for someφ∈K*,

(φ,An–γ) <δ

(φ,Pn–γ) <δ

implies φ,xn(n;γ)

<, n≥n+N.

(S) A-uniformly asymptoticallyφ-stable ifδandNin (S) are independent ofn.

LetKbe a cone inRm_,_S

ρ={xn∈Rm,An–xn<ρ,ρ> }.V:Z+×Sρ→Kis continuous in the second variable, we deﬁne

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3 Main results

Lemma . The trivial solution of SDEs(.)is A-uniformlyφ-stable(P-uniformlyφ

-stable)if and only if there exists a functionψ ∈Ksuch that for any solution xnof SDEs (.)and someφ∈K*,the following inequality holds:

Conversely, suppose that the trivial solution of (.) isA-uniformlyφ-stable,i.e., for

each positive , there exists a δ =δ()∈(,] such that if xn is any solution of (.)

the proof of Lemma . is complete.

Similar to the proof of Lemma ., we have the following.

Lemma . The trivial solution of SDEs(.)is A-φ-stable(P-φ-stable)if and only if

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(iii) a[(φ,xn)]≤(φ,V(n,An–xn))for someφ∈K*anda∈K,(n,r)∈R+×Sρ. Then the trivial solution of SDEs(.)is A-φ-stable.

Proof SinceV(n, ) =  andV(n,r) is continuous inr, then givena() > ,a∈K, there

existsδsuch that

An–γ<δ implies V(n,An–γ)<a().

For someφ∈K*,

φAn–γ<φδ implies φV(n,An–γ)<φa().

Thus

(φ,An–γ) ≤ φAn–γ<φδ

implies

φ,V(n,An–γ) ≤ φV(n,An–γ)<φa().

It follows that

(φ,An–γ)≤δ implies

φ,V(n,An–γ)

≤a(),

where φδ =δ, φa() = a(), a∈K. Let xn be any solution of (.) such that (φ,An–γ) <δ. Then by (i),V is nonincreasing and so

V(n,An–xn)≤V(n,An–γ), n≥n.

Thus (φ,An–γ) <δimplies

a(φ,xn)

≤φ,V(n,An–xn)

≤φ,V(n,An–γ)

<a(),

i.e.,

(φ,An–γ) <δ implies (φ,xn) <, n≥n.

Then the trivial solution of (.) isA-φ-stable. The proof of Theorem . is complete.

Theorem . Let the hypotheses of Theorem.be satisﬁed,except the conditionV(n, An–xn)≤being replaced by

(iv) (φ,V(n,An–xn))≤–c[(φ,V(n,An–xn))],c∈K.

Then the trivial solution of SDEs(.)is A-asymptoticallyφ-stable.

Proof By Theorem ., the trivial solution of (.) isA-φ-stable. By (iv),V(n,An–xn) is a

monotone decreasing function, thus the limit

V*= lim

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exists. We prove thatV*_{= . Suppose}_V*_{= , then}_c₍_V*₎_{= ,}_c_∈_K_{. Since}_c₍_r_{) is a}

mono-tone increasing function, then

cφ,V(n,An–xn)

>cφ,V*

,

and so from (iv), we get

φ,V(n,An–xn)

≤–cφ,V*

.

Then

φ,V(n,An–xn)

≤–cφ,V*

(n–n) +

φ,V(n,An–xn)

.

Thus asn→ ∞and for someφ∈K*, we have (φ,V(n,An–xn))→–∞. This contradicts the condition (iii). It follows thatV*= . Thus

φ,V(n,An–xn)

→, n→ ∞,

and so with (iii)

(φ,xn)→, n→ ∞.

Thus for given> ,n∈R+, there existδ=δ(n) andN=N(n,) such that

(φ,An–xn) <δ implies (φ,xn) <, n≥n+N.

Then the trivial solution of (.) isA-asymptoticallyφ-stable. The proof of Theorem .

is complete.

Theorem . The trivial solution of SDEs (.)is P-φ-stable if and only if there exist

functionsψn∈Kand a Lyapunov function V∈C[Z+×Sρ,K]such that for someφ∈K*,

(i) V(n, ) = ,n≥;

(ii) (φ,y)≤(φ,V(n,Pn–y))≤ψn[(φ,Pn–y)],∀y∈ n,and someφ∈K*,

(iii) V(n,Pn–yn) :=V(n+ ,Pnyn+–V(n,Pn–yn))≤for any solutionynof(.). Proof Necessity. Suppose that the trivial solution of (.) isP-φ-stable, then, according to

Lemma ., there exist functionsψn∈K(n≥) such that for any solutionxnof (.) and for someφ∈K*,

(φ,xn)≤ψn

(φ,Pn–xn)

. (.)

Deﬁne the Lyapunov function

V(n,γ) :=sup k∈Z+

xn+k(n;γ); γ∈Rm,n∈Z+,

where xn+k:=xn+k(n;γ) is the unique solution of (.) satisfying the initial condition

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V(n, ) =  and the continuity of functionV in the second variable atγ = . For each

On the other hand, for an arbitrary solutionynof (.), by the unique solvability of the initial value problem (.) and (.), we have

V(n,Pn–yn) =sup

Suﬃciency. Assuming that the trivial solution of (.) is notP-φ-stable,i.e., there exist

a positiveand a nonnegative integernsuch that for allδ∈(,] and for someφ∈K*,

which leads to a contradiction. The proof of Theorem . is complete.

Theorem . The trivial solution of SDEs(.)is P-uniformlyφ-stable if and only if there

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(i) a[(φ,x)]≤(φ,V(n,Pn–x))≤b[(φ,Pn–x)],∀x∈ n,n≥;

(ii) V(n,Pn–xn)≤for any solutionxnof(.).

Proof The proof of the necessity part is similar to the corresponding part of Theorem .. For> , letδ=b–[a()] independent ofnfora,b∈K, andxnbe any solution of (.)

Then the trivial solution of (.) isP-uniformlyφ-stable. The proof of Theorem . is

complete.

4 Example

Consider SDEs (.) with the following data:

An= hypotheses (H), (H) are fulﬁlled, hence SDEs (.) is of index-. Clearly, the canonical

projections are

. Further, the functionfn(x) is Lipschitz continuous with the

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to Theorem ., IVP (.), (.) has a unique solution. We havex∈ nif and only if

According to Theorem ., the trivial solution of (.) isP-uniformlyφ-stable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors completed the paper together. All authors read and approved the ﬁnal manuscript.

Author details

1_{College of Electronic and Information Engineering, Hebei University, Baoding, 071002, China.}2_Fundamental

Department, Hebei Finance University, Baoding, 071051, China. 3_{Department of Mathematics and Statistics, Curtin}

University of Technology, GPO BOX U1987, Perth, WA 6845, Australia.

Acknowledgements

The authors would like to thank the reviewers for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).

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Cite this article as:Wang et al.:Onφ0-stability of a class of singular difference equations.Advances in Diﬀerence