Skip to main content

On ϕ 0 -stability of a class of singular difference equations

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.

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 [1] and references cited therein). Anh and Loi [2, 3] 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 [4], Elaydi [5], Halanay and Rasvan [6], Martynjuk [7] and Diblík et al. [8]. Recently, Anh and Hoang [9] 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 system satisfies a quasimonotone property which is too restrictive for many applications because this property is not a necessary condition for a desired property like stability of the comparison system. To solve this problem, Lakshmikantham and Leela [10] initiated the method of cone and cone-valued Lyapunov functions and developed the theory of differential inequalities. By employing the method of cone-valued Lyapunov functions, Akpan and Akinyele [11], EL-Sheikh and Soliman [12], Wang and Geng [13] investigated the stability and the ϕ 0 -stability of ordinary differential systems, functional differential equations and difference equations, respectively.

However, to the best of our knowledge, there are few results for the ϕ 0 -stability of singular difference equations. In this paper, utilizing the framework of the theory of singular difference equations, we give some necessary and sufficient conditions for the ϕ 0 -stability of a trivial solution of singular difference equations via cone-valued Lyapunov functions.

2 Preliminaries

The following definitions can be found in reference [10].

Definition 2.1 A proper subset K of R n is called a cone if

  1. (i)

    λKK, λ0;

  2. (ii)

    K+KK;

  3. (iii)

    K= K ¯ ;

  4. (iv)

    K 0 ;

  5. (v)

    K(K)={0},

where K ¯ and K 0 denote the closure and interior of K, respectively, and ∂K denotes the boundary of K. The order relation on R n induced by the cone K is defined as follows: Let x,yK, then x K y iff yxK and x < K 0 y iff yx K 0 .

Definition 2.2 The set K ={ϕ R n ,(ϕ,x)0, for all xK} is said to be an adjoint cone if it satisfies the properties (i)-(v).

x K 0 iff(ϕ,x)>0,

and

xkiff(ϕ,x)=0for some ϕ K 0 , K 0 =K{0}.

Definition 2.3 A function g:D R n , D R n is said to be quasimonotone relative to K if x,yD and yxK implies that there exists ϕ 0 K 0 such that

( ϕ 0 ,yx)=0and ( ϕ 0 , g ( y ) g ( x ) ) 0.

Definition 2.4 A function a(r) is said to belong to the class if aC[ R + , R + ], a(0)=0, and a(r) is strictly monotone increasing function in r.

Consider the following SDEs:

A n x n + 1 + B n x n = f n ( x n ),n0,
(2.1)

where A n , B n R m × m and f n : R m R m are given. Throughout this paper, we assume that the matrices A n are singular, and the corresponding linear homogeneous equations

A n x n + 1 + B n x n =0,n0,
(2.2)

are of index-1 [13], i.e., the following hypotheses hold.

(H1) rank A n =r, n0,

(H2) S n ker A n 1 ={0}, n1, where S n ={ξ R m : B n ξim A n }, n0.

For the next discussion, the following lemma from [9] is needed.

Lemma 2.1 Suppose that the hypothesis (H1) holds. Then the hypothesis (H2) is equivalent to one of the following statements:

  1. (i)

    the matrix G n := A n + B n Q n 1 , n is nonsingular;

  2. (ii)

    R m = S n ker A n 1 .

Let us associate SDEs (2.1) with the initial condition

P n 0 1 x n 0 = P n 0 1 γ,n0,
(2.3)

where γ is an arbitrary vector in R m and n 0 is a fixed nonnegative integer.

Theorem 2.1 [9]

Let f n (x) be a Lipschitz continuous function with a sufficiently small Lipschitz coefficient, i.e.,

f n ( x ) f n ( x ˜ ) L n x x ˜ ,x, x ˜ R m ,

where

ω n := L n Q n 1 , n G n 1 ,n0.

Then IVP (2.1), (2.3) has a unique solution.

Set n :={x R m : Q n 1 x= Q n 1 , n G n 1 [ f n (x) B n P n 1 x]}. If x={ x n } is any solution of IVP (2.1), (2.3), then obviously x n n (n n 0 ).

Definition 2.5 [9]

The trivial solution of (2.1) is said to be A-stable (P-stable) if for each ϵ>0 and any n 0 0, there exists a δ=δ(ϵ, n 0 )(0,ϵ] such that

A n 0 1 γ<δ ( P n 0 1 γ < δ ) implies x n ( n 0 ; γ ) <ϵ,n n 0 .

Definition 2.6 The trivial solution of (2.1) is said to be

(S1) A- ϕ 0 -stable (P- ϕ 0 -stable) if for each ϵ>0 and any n 0 0, there exists a δ=δ( n 0 ,ϵ)(0,ϵ] such that for some ϕ 0 K 0

( ϕ 0 , A n 0 1 γ)<δ ( ( ϕ 0 , P n 0 1 γ ) < δ ) implies ( ϕ 0 , x n ( n 0 ; γ ) ) <ϵ,n n 0 .

(S2) A-uniformly ϕ 0 -stable (P-uniformly ϕ 0 -stable) if δ in ( S 1 ) is independent of n 0 .

(S3) A-asymptotically ϕ 0 -stable (P-asymptotically ϕ 0 -stable) if for any n 0 0 there exist positive numbers δ 0 = δ 0 ( n 0 ) and N=N( n 0 ,ϵ) such that for some ϕ 0 K 0 ,

( ϕ 0 , A n 0 1 γ)< δ 0 ( ( ϕ 0 , P n 0 1 γ ) < δ 0 ) implies ( ϕ 0 , x n ( n 0 ; γ ) ) <ϵ,n n 0 +N.

(S4) A-uniformly asymptotically ϕ 0 -stable if δ 0 and N in (S3) are independent of n 0 .

Let K be a cone in R m , S ρ ={ x n R m , A n 1 x n <ρ,ρ>0}. V: Z + × S ρ K is continuous in the second variable, we define

ΔV(n, A n 1 x n ):=V(n+1, A n x n + 1 )V(n, A n 1 x n ),

where x n is any solution of system (2.1).

3 Main results

Lemma 3.1 The trivial solution of SDEs (2.1) is A-uniformly ϕ 0 -stable (P-uniformly ϕ 0 -stable) if and only if there exists a function ψK such that for any solution x n of SDEs (2.1) and some ϕ 0 K 0 , the following inequality holds:

( ϕ 0 , x n ) ψ [ ( ϕ 0 , A n 0 1 x n 0 ) ] , n n 0 ( ( ϕ 0 , x n ) ψ [ ( ϕ 0 , P n 0 1 x n 0 ) ] , n n 0 ) .
(3.1)

Proof For each positive ϵ, choose δ=δ(ϵ)(0,ϵ] such that ψ(δ)<ϵ. If x n is an arbitrary solution of (2.1) and ( ϕ 0 , A n 0 1 x n 0 )<δ, then

( ϕ 0 , x n )ψ [ ( ϕ 0 , A n 0 1 x n 0 ) ] <ψ(δ)<ϵ,n n 0 .

Then (2.1) is A-uniformly ϕ 0 -stable.

Conversely, suppose that the trivial solution of (2.1) is A-uniformly ϕ 0 -stable, i.e., for each positive ϵ, there exists a δ=δ(ϵ)(0,ϵ] such that if x n is any solution of (2.1) which satisfies the inequality ( ϕ 0 , A n 0 1 x n 0 )<δ, then ( ϕ 0 , x n )<ϵ for all n n 0 . Denote by α(ϵ) the supremum of for the above δ(ϵ). Obviously, if ( ϕ 0 , A n 0 1 x n 0 )<α(ϵ) for some n 0 , then ( ϕ 0 , x n )<ϵ for all n n 0 . Furthermore, the function α(ϵ) is positive and increasing, and α(ϵ)ϵ. Considering a function β(ϵ) defined by β(ϵ):= 1 ϵ 0 ϵ α(t)dt and β(0):=0, it is easy to prove that βK and 0<β(ϵ)<α(ϵ)ϵ. Then the inverse of β, denoted by ψ will belong to . For some ϕ 0 K 0 , set ϵ n :=( ϕ 0 , x n ) and consider two possibilities: (i) If ( ϕ 0 , x n )=0, then ( ϕ 0 , x n )=0ψ[( ϕ 0 , A n 0 1 x n 0 )]; (ii) If for some ( ϕ 0 , A n 0 1 x n 0 )<β( ϵ n ), in which ϵ n :=( ϕ 0 , x n )>0, then ( ϕ 0 , x n )< ϵ n =( ϕ 0 , x n ), which is impossible, hence ( ϕ 0 , A n 0 1 x n 0 )β( ϵ n ), therefore, for some ϕ 0 K 0 ,

( ϕ 0 , x n )= ϵ n β 1 [ ( ϕ 0 , A n 0 1 x n 0 ) ] =ψ [ ( ϕ 0 , A n 0 1 x n 0 ) ] ,

the proof of Lemma 3.1 is complete. □

Similar to the proof of Lemma 3.1, we have the following.

Lemma 3.2 The trivial solution of SDEs (2.1) is A- ϕ 0 -stable (P- ϕ 0 -stable) if and only if there exist functions ψ n K such that for any solution x n of (2.1), each nonnegative integer n 0 and some ϕ 0 K 0 , the following inequality holds:

( ϕ 0 , x n ) ψ n 0 [ ( ϕ 0 , A n 0 1 x n 0 ) ] , n n 0 ( ( ϕ 0 , x n ) ψ n 0 [ ( ϕ 0 , P n 0 1 x n 0 ) ] , n n 0 ) .
(3.2)

Theorem 3.1 Assume that

  1. (i)

    VC[ Z + × S ρ ,K], V(n,0)=0, V(n,r) is locally Lipschitzian in r relative to K, and for each (n,r) Z + × S ρ ,

    ΔV(n, A n 1 x n )0;
  2. (ii)

    fC[K, R m ] is quasimonotone in x n relative to K;

  3. (iii)

    a[( ϕ 0 , x n )]( ϕ 0 ,V(n, A n 1 x n )) for some ϕ 0 K 0 and aK, (n,r) R + × S ρ .

Then the trivial solution of SDEs (2.1) is A- ϕ 0 -stable.

Proof Since V(n,0)=0 and V(n,r) is continuous in r, then given a 1 (ϵ)>0, a 1 K, there exists δ 1 such that

A n 0 1 γ< δ 1 implies V ( n 0 , A n 0 1 γ ) < a 1 (ϵ).

For some ϕ 0 K 0 ,

ϕ 0 A n 0 1 γ< ϕ 0 δ 1 implies ϕ 0 V ( n 0 , A n 0 1 γ ) < ϕ 0 a 1 (ϵ).

Thus

| ( ϕ 0 , A n 0 1 γ ) | ϕ 0 A n 0 1 γ< ϕ 0 δ 1

implies

| ( ϕ 0 , V ( n 0 , A n 0 1 γ ) ) | ϕ 0 V ( n 0 , A n 0 1 γ ) < ϕ 0 a 1 (ϵ).

It follows that

( ϕ 0 , A n 0 1 γ)δimplies ( ϕ 0 , V ( n 0 , A n 0 1 γ ) ) a(ϵ),

where ϕ 0 δ 1 =δ, ϕ 0 a 1 (ϵ)=a(ϵ), aK. Let x n be any solution of (2.1) such that ( ϕ 0 , A n 0 1 γ)<δ. Then by (i), V is nonincreasing and so

V(n, A n 1 x n )V( n 0 , A n 0 1 γ),n n 0 .

Thus ( ϕ 0 , A n 0 1 γ)<δ implies

a [ ( ϕ 0 , x n ) ] ( ϕ 0 , V ( n , A n 1 x n ) ) ( ϕ 0 , V ( n 0 , A n 0 1 γ ) ) <a(ϵ),

i.e.,

( ϕ 0 , A n 0 1 γ)<δimplies( ϕ 0 , x n )<ϵ,n n 0 .

Then the trivial solution of (2.1) is A- ϕ 0 -stable. The proof of Theorem 3.1 is complete. □

Theorem 3.2 Let the hypotheses of Theorem  3.1 be satisfied, except the condition ΔV(n, A n 1 x n )0 being replaced by

  1. (iv)

    ( ϕ 0 ,ΔV(n, A n 1 x n ))c[( ϕ 0 ,V(n, A n 1 x n ))], cK.

Then the trivial solution of SDEs (2.1) is A-asymptotically ϕ 0 -stable.

Proof By Theorem 3.2, the trivial solution of (2.1) is A- ϕ 0 -stable. By (iv), V(n, A n 1 x n ) is a monotone decreasing function, thus the limit

V = lim n V(n, A n 1 x n )

exists. We prove that V =0. Suppose V 0, then c( V )0, cK. Since c(r) is a monotone increasing function, then

c [ ( ϕ 0 , V ( n , A n 1 x n ) ) ] >c [ ( ϕ 0 , V ) ] ,

and so from (iv), we get

( ϕ 0 , Δ V ( n , A n 1 x n ) ) c [ ( ϕ 0 , V ) ] .

Then

( ϕ 0 , V ( n , A n 1 x n ) ) c [ ( ϕ 0 , V ) ] (n n 0 )+ ( ϕ 0 , V ( n 0 , A n 0 1 x n 0 ) ) .

Thus as n and for some ϕ 0 K 0 , we have ( ϕ 0 ,V(n, A n 1 x n )). This contradicts the condition (iii). It follows that V =0. Thus

( ϕ 0 , V ( n , A n 1 x n ) ) 0,n,

and so with (iii)

( ϕ 0 , x n )0,n.

Thus for given ϵ>0, n 0 R + , there exist δ=δ( n 0 ) and N=N( n 0 ,ϵ) such that

( ϕ 0 , A n 0 1 x n 0 )< δ 0 implies( ϕ 0 , x n )<ϵ,n n 0 +N.

Then the trivial solution of (2.1) is A-asymptotically ϕ 0 -stable. The proof of Theorem 3.2 is complete. □

Theorem 3.3 The trivial solution of SDEs (2.1) is P- ϕ 0 -stable if and only if there exist functions ψ n K and a Lyapunov function VC[ Z + × S ρ ,K] such that for some ϕ 0 K 0 ,

  1. (i)

    V(n,0)=0, n0;

  2. (ii)

    ( ϕ 0 ,y)( ϕ 0 ,V(n, P n 1 y)) ψ n [( ϕ 0 , P n 1 y)], y n , and some ϕ 0 K 0 ,

  3. (iii)

    ΔV(n, P n 1 y n ):=V(n+1, P n y n + 1 V(n, P n 1 y n ))0 for any solution y n of (2.1).

Proof Necessity. Suppose that the trivial solution of (2.1) is P- ϕ 0 -stable, then, according to Lemma 3.2, there exist functions ψ n K (n0) such that for any solution x n of (2.1) and for some ϕ 0 K 0 ,

( ϕ 0 , x n ) ψ n 0 [ ( ϕ 0 , P n 0 1 x n 0 ) ] .
(3.3)

Define the Lyapunov function

V(n,γ):= sup k Z + x n + k ( n ; γ ) ;γ R m ,n Z + ,

where x n + k := x n + k (n;γ) is the unique solution of (2.1) satisfying the initial condition P n 1 x n = P n 1 γ. Moreover, for some ϕ 0 K 0 , ( ϕ 0 ,V(n,γ)) ψ n ( ϕ 0 , P n 1 γ), which implies V(n,0)=0 and the continuity of function V in the second variable at γ=0. For each y n , we have

V(n, P n 1 y):= sup l Z + x n + l ( n ; P n 1 y ) x n ( n ; P n 1 y ) ,

where x k (n; P n 1 y) denotes the solution of (2.1) satisfying the initial condition P n 1 x n (n; P n 1 y)= P n 1 ( P n 1 y)= P n 1 y. Since x n ,y n , it follows x n (n; P n 1 y)=y, hence, for some ϕ 0 K 0 ,

( ϕ 0 , V ( n , P n 1 y ) ) ( ϕ 0 , x n ( n ; P n 1 y ) ) =( ϕ 0 ,y).

Further, the inequality (3.3) gives

( ϕ 0 , V ( n , P n 1 y ) ) ψ n ( ϕ 0 , P n 1 y).

On the other hand, for an arbitrary solution y n of (2.1), by the unique solvability of the initial value problem (2.1) and (2.3), we have

V(n, P n 1 y n )= sup l Z + x n + l ( n ; P n 1 y ) = sup l 0 y n + l .

Thus

V ( n + 1 , P n y n + 1 ) = sup l 0 y n + l + 1 = sup l 1 y n + l sup l 0 y n + l = V ( n , P n 1 y ) ,

hence ΔV(n, P n 1 y n )0. The necessity part is proved.

Sufficiency. Assuming that the trivial solution of (2.1) is not P- ϕ 0 -stable, i.e., there exist a positive ϵ 0 and a nonnegative integer n 0 such that for all δ(0, ϵ 0 ] and for some ϕ 0 K 0 , there exists a solution of (2.1) satisfying the inequalities ( ϕ 0 , P n 0 1 x n 0 )<δ and ( ϕ 0 , x n 1 ) ϵ 0 for some n 1 n 0 .

Since V( n 0 ,0)=0 and V( n 0 ,γ) is continuous at γ=0, there exists a δ 0 = δ 0 (ϵ, n 0 )>0 such that for all ξ R m , ξ< δ 0 , we have V( n 0 ,ξ)< ϵ 0 . Choosing δ 0 min{ δ 0 , ϵ 0 }, we can find a solution x n of (2.1) satisfying ( ϕ 0 , P n 0 1 x n 0 ) δ 0 . However, ( ϕ 0 , x n 1 ) ϵ 0 for some n 1 n 0 . Since ( ϕ 0 , P n 0 1 x n 0 )< δ 0 δ 0 , we get

( ϕ 0 , V ( n 0 , P n 0 1 x n 0 ) ) < ϵ 0

for some n 1 n 0 . On the other hand, using the properties (iii) of the function V, we find

( ϕ 0 , V ( n 0 , P n 0 1 x n 0 ) ) ( ϕ 0 , V ( n 1 , P n 1 1 x n 1 ) ) ( ϕ 0 , x n 1 ) ϵ 0 ,

which leads to a contradiction. The proof of Theorem 3.3 is complete. □

Theorem 3.4 The trivial solution of SDEs (2.1) is P-uniformly ϕ 0 -stable if and only if there exist functions a,bK and a Lyapunov function VC[ Z + × S ρ ,K] such that for some ϕ 0 K 0 ,

  1. (i)

    a[( ϕ 0 ,x)]( ϕ 0 ,V(n, P n 1 x))b[( ϕ 0 , P n 1 x)], x n , n0;

  2. (ii)

    ΔV(n, P n 1 x n )0 for any solution x n of (2.1).

Proof The proof of the necessity part is similar to the corresponding part of Theorem 3.3.

For ϵ>0, let δ= b 1 [a(ϵ)] independent of n 0 for a,bK, and x n be any solution of (2.1) such that ( ϕ 0 , P n 0 1 x n 0 )<δ. Then by (ii), V is nonincreasing and so

( ϕ 0 , V ( n , P n 1 x n ) ) ( ϕ 0 , V ( n 0 , P n 0 1 x n 0 ) ) ,n n 0 .

Thus

a [ ( ϕ 0 , x n ) ] ( ϕ 0 , V ( n , P n 1 x n ) ) ( ϕ 0 , V ( n 0 , P n 0 1 x n 0 ) ) b [ ( ϕ 0 , P n 0 1 x n 0 ) ] < b ( δ ) < a ( ϵ ) ,

i.e.,

( ϕ 0 , P n 0 1 x n 0 )<δimplies( ϕ 0 , x n )<ϵ,n n 0 .

Then the trivial solution of (2.1) is P-uniformly ϕ 0 -stable. The proof of Theorem 3.4 is complete. □

4 Example

Consider SDEs (2.1) with the following data:

A n = ( n + 3 0 0 0 ) ; B n = ( 1 0 0 n + 2 ) ,n1,

and

f n (x)= 1 n + 2 ( 0 , 1 ) T ;x= ( x 1 , x 2 ) T ,n0.

As ker A n =span{ ( 0 , 1 ) T }, im A n =span{ ( 1 , 0 ) T }, n1 and S n =span{ ( 1 , 0 ) T }, n0, the hypotheses (H1), (H2) are fulfilled, hence SDEs (2.2) is of index-1. Clearly, the canonical projections are

P n =P:= ( 1 0 0 0 ) ; Q n =Q:= ( 0 0 0 1 ) ,

therefore

Q n 1 , n = ( 0 0 0 1 ) =Q; G n = A n + B n Q n 1 , n = ( n + 3 0 0 n + 2 ) ,

hence G n 1 = ( 1 n + 3 0 0 1 n + 2 ) . Further, the function f n (x) is Lipschitz continuous with the Lipschitz coefficient L n = ( n + 2 ) 1 . Moreover, f n (0)=0 and ω n := L n Q n 1 , n G n 1 <1. According to Theorem 2.1, IVP (2.1), (2.3) has a unique solution. We have x n if and only if

Q n 1 x= Q n 1 , n G n 1 { f n ( x ) B n P n 1 x } ,

it leads to x 2 = x 1 ( n + 2 ) ( n + 3 ) . Thus

n = { x = ( x 1 , x 2 ) T : x 2 = x 1 ( n + 2 ) ( n + 3 ) } ,n0.

Let V(n,γ):=2x, we get for each x n ,

x= ( x 1 2 + x 1 2 ( n + 2 ) 2 ( n + 3 ) 2 ) 1 2 2| x 1 |=2 P n 1 x.

Further, V(n, P n 1 x)=2 P n 1 x=2| x 1 |. Thus, for some ϕ 0 K 0 ,

a [ ( ϕ 0 , x ) ] ( ϕ 0 , V ( n , P n 1 x ) ) b [ ( ϕ 0 , P n 1 x ) ] ,

where a,bK and a(r)=r, b(r)=2r. Suppose that x n is a solution of (2.1) and putting u n = P n 1 x n =P x n , v n = Q n 1 x n =Q x n , then we have

V ( n , P n 1 x n ) = V ( n + 1 , P n x n + 1 ) V ( n , P n 1 x ) = 2 ( P x n + 1 P x n ) = 2 ( u n + 1 u n ) .

Using equation (2.8) in [9], we find

u n + 1 = P n G n 1 B n u n + P n G n 1 f n ( x n )= 1 n + 3 ( 1 0 0 0 ) u n ,

hence u n + 1 u n = n + 2 n + 3 u n 1 2 u n , then

V(n, P n 1 x n ) P n 1 x n .

According to Theorem 3.4, the trivial solution of (2.1) is P-uniformly ϕ 0 -stable.

References

  1. 1.

    Anh PK, Du NH, Loi LC: Singular difference equations: an overview. Vietnam J. Math. 2007, 35: 339-372.

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Anh PK, Loi LC: On the solvability of initial-value problems for nonlinear implicit difference equations. Adv. Differ. Equ. 2004, 3: 195-200.

    Google Scholar 

  3. 3.

    Anh PK, Loi LC: On multipoint boundary-value problems for linear implicit non-autonomous systems of difference equations. Vietnam J. Math. 2001, 29: 281-286.

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Agarwal RP Monographs and Textbooks in Pure and Applied Mathematics 228. In Difference Equations and Inequalities, Theory, Methods and Applications. 2nd edition. Dekker, New York; 2000.

    Google Scholar 

  5. 5.

    Elaydi SN: An Introduction to Difference Equations. 3rd edition. Springer, London; 2005.

    Google Scholar 

  6. 6.

    Halanay A, Rásvan VRV: Stability and Stable Oscillations in Discrete Time Systems. Gordon & Breach Science, Taipei; 2002.

    Google Scholar 

  7. 7.

    Martynjuk DI: Lectures on the Qualitative Theory of Difference Equations. “Naukova Dumka”, Kiev; 1972.

    Google Scholar 

  8. 8.

    Diblík J, Khusainov DY, Grytsay IV, Šmarda Z: Stability of nonlinear autonomous quadratic discrete systems in the critical case. Discrete Dyn. Nat. Soc. 2010., 2010: Article ID 539087 10.1155/2010/539087

    Google Scholar 

  9. 9.

    Anh PK, Hoang DS: Stability of a class of singular difference equations. Int. J. Differ. Equ. 2006, 1: 181-193.

    MATH  Google Scholar 

  10. 10.

    Lakshmikantham V, Leela S: Cone-valued Liapunov functions. Nonlinear Anal. 1977, 1: 215-222. 10.1016/0362-546X(77)90031-1

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Akpan EP, Akineyle O:On the ϕ 0 -stability of comparison differential systems. J. Math. Anal. Appl. 1992, 164: 307-324. 10.1016/0022-247X(92)90116-U

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    EL-Sheikh MMA, Soliman AA: On stability of nonlinear systems of functional differential equations. Appl. Math. Comput. 2000, 107: 81-93. 10.1016/S0096-3003(98)10049-8

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Wang PG, Geng FJ:On ϕ 0 stability of difference equations. Appl. Math. Lett. 2005, 18: 139-147. 10.1016/j.aml.2004.05.002

    MathSciNet  Article  MATH  Google Scholar 

Download references

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).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Peiguang Wang.

Additional information

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.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Wang, P., Wu, M. & Wu, Y. On ϕ 0 -stability of a class of singular difference equations. Adv Differ Equ 2013, 179 (2013). https://doi.org/10.1186/1687-1847-2013-179

Download citation

Keywords

  • singular difference equations
  • cone-valued Lyapunov functions
  • ϕ 0 -stability
  • uniformly ϕ 0 -stability