On ${\varphi }_0$-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 ${\varphi }_0$-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 ${\varphi }_0$-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 ${\varphi }_0$-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 ${\varphi }_0$-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

• (i) $\lambda K\subseteq K$, $\lambda \ge 0$;

• (ii) $K+K\subseteq K$;

• (iii) $K=\overline{K}$;

• (iv) $K^0\ne \mathrm{\varnothing }$;

• (v) $K\cap (-K)=\{0\}$,

where $\overline{K}$ and $K^0$ denote the closure and interior of K, respectively, and ∂K denotes the boundary of K. The order relation on ${\mathbb{R}}^n$ induced by the cone K is defined as follows: Let $x,y\in K$, then $x{\le }_{K}y$ iff $y-x\in K$ and $x{<}_{K^0}y$ iff $y-x\in K^0$.

Definition 2.2 The set $K^{\ast }=\{\varphi \in R^n,(\varphi ,x)\ge 0,\text{ for all }x\in K\}$ is said to be an adjoint cone if it satisfies the properties (i)-(v).

$$x\in K^0\text{iff}(\varphi ,x)>0,$$

and

$$x\in \partial k\text{iff}(\varphi ,x)=0\text{for some }\varphi \in K_0^{\ast },K_0=K-\{0\}.$$

Definition 2.3 A function $g:D\to R^n$, $D\subset R^n$ is said to be quasimonotone relative to K if $x,y\in D$ and $y-x\in \partial K$ implies that there exists ${\varphi }_0\in K_0^{\ast }$ such that

$$({\varphi }_0,y-x)=0\text{and}({\varphi }_0,g(y)-g(x))\ge 0.$$

Definition 2.4 A function $a(r)$ is said to belong to the class if $a\in C[R_{+},R_{+}]$, $a(0)=0$, and $a(r)$ is strictly monotone increasing function in r.

Consider the following SDEs:

$$A_n{x}_{n+1}+B_n{x}_n=f_n(x_n),n\ge 0,$$

(2.1)

where $A_n,B_n\in R^{m\times m}$ and $f_n:R^m\to R^m$ are given. Throughout this paper, we assume that the matrices $A_n$ are singular, and the corresponding linear homogeneous equations

$$A_n{x}_{n+1}+B_n{x}_n=0,n\ge 0,$$

(2.2)

are of index-1 [1–3], i.e., the following hypotheses hold.

(H₁) $rank{A}_n=r$, $n\ge 0$,

(H₂) $S_n\cap ker{A}_{n-1}=\{0\}$, $n\ge 1$, where $S_n=\{\xi \in R^m:B_n\xi \in im{A}_n\}$, $n\ge 0$.

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:

• (i) the matrix $G_n:=A_n+B_n{Q}_{n-1,n}$ is nonsingular;

• (ii) $R^m=S_n\oplus ker{A}_{n-1}$.

Let us associate SDEs (2.1) with the initial condition

$$P_{n_0-1}{x}_{n_0}=P_{n_0-1}\gamma ,n\ge 0,$$

(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.,

$$\parallel f_n(x)-f_n(\tilde{x})\parallel \le L_n\parallel x-\tilde{x}\parallel ,\mathrm{\forall }x,\tilde{x}\in R^m,$$

where

$${\omega }_n:=L_n\parallel Q_{n-1,n}{G}_n^{-1}\parallel ,\mathrm{\forall }n\ge 0.$$

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

Set ${\mathrm{△}}_n:=\{x\in R^m:Q_{n-1}x=Q_{n-1,n}{G}_n^{-1}[f_n(x)-B_n{P}_{n-1}x]\}$. If $x=\{x_n\}$ is any solution of IVP (2.1), (2.3), then obviously $x_n\in {\mathrm{△}}_n$ ($n\ge n_0$).

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 $n_0\ge 0$, there exists a $\delta =\delta (ϵ,n_0)\in (0,ϵ]$ such that

$$\parallel A_{n_0-1}\gamma \parallel <\delta (\parallel P_{n_0-1}\gamma \parallel <\delta )\text{implies}\parallel x_n(n_0;\gamma )\parallel <ϵ,n\ge n_0.$$

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

(S₁) A-${\varphi }_0$-stable (P-${\varphi }_0$-stable) if for each $ϵ>0$ and any $n_0\ge 0$, there exists a $\delta =\delta (n_0,ϵ)\in (0,ϵ]$ such that for some ${\varphi }_0\in K_0^{\ast }$

$$({\varphi }_0,A_{n_0-1}\gamma )<\delta (({\varphi }_0,P_{n_0-1}\gamma )<\delta )\text{implies}({\varphi }_0,x_n(n_0;\gamma ))<ϵ,n\ge n_0.$$

(S₂) A-uniformly ${\varphi }_0$-stable (P-uniformly ${\varphi }_0$-stable) if δ in ($S_1$) is independent of $n_0$.

(S₃) A-asymptotically ${\varphi }_0$-stable (P-asymptotically ${\varphi }_0$-stable) if for any $n_0\ge 0$ there exist positive numbers ${\delta }_0={\delta }_0(n_0)$ and $N=N(n_0,ϵ)$ such that for some ${\varphi }_0\in K_0^{\ast }$,

$$({\varphi }_0,A_{n_0-1}\gamma )<{\delta }_0(({\varphi }_0,P_{n_0-1}\gamma )<{\delta }_0)\text{implies}({\varphi }_0,x_n(n_0;\gamma ))<ϵ,n\ge n_0+N.$$

(S₄) A-uniformly asymptotically ${\varphi }_0$-stable if ${\delta }_0$ and N in (S₃) are independent of $n_0$.

Let K be a cone in $R^m$, $S_{\rho }=\{x_n\in R^m,\parallel A_{n-1}{x}_n\parallel <\rho ,\rho >0\}$. $V:Z_{+}\times S_{\rho }\to K$ is continuous in the second variable, we define

$$\mathrm{\Delta }V(n,A_{n-1}{x}_n):=V(n+1,A_n{x}_{n+1})-V(n,A_{n-1}{x}_n),$$

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 ${\varphi }_0$-stable (P-uniformly ${\varphi }_0$-stable) if and only if there exists a function $\psi \in \mathcal{K}$ such that for any solution $x_n$ of SDEs (2.1) and some ${\varphi }_0\in K_0^{\ast }$, the following inequality holds:

$$\begin{array}{r}({\varphi }_0,x_n)\le \psi [({\varphi }_0,A_{n_0-1}{x}_{n_0})],n\ge n_0\\ (({\varphi }_0,x_n)\le \psi [({\varphi }_0,P_{n_0-1}{x}_{n_0})],n\ge n_0).\end{array}$$

(3.1)

Proof For each positive ϵ, choose $\delta =\delta (ϵ)\in (0,ϵ]$ such that $\psi (\delta )<ϵ$. If $x_n$ is an arbitrary solution of (2.1) and $({\varphi }_0,A_{n_0-1}{x}_{n_0})<\delta$, then

$$({\varphi }_0,x_n)\le \psi [({\varphi }_0,A_{n_0-1}{x}_{n_0})]<\psi (\delta )<ϵ,n\ge n_0.$$

Then (2.1) is A-uniformly ${\varphi }_0$-stable.

Conversely, suppose that the trivial solution of (2.1) is A-uniformly ${\varphi }_0$-stable, i.e., for each positive ϵ, there exists a $\delta =\delta (ϵ)\in (0,ϵ]$ such that if $x_n$ is any solution of (2.1) which satisfies the inequality $({\varphi }_0,A_{n_0-1}{x}_{n_0})<\delta$, then $({\varphi }_0,x_n)<ϵ$ for all $n\ge n_0$. Denote by $\alpha (ϵ)$ the supremum of for the above $\delta (ϵ)$. Obviously, if $({\varphi }_0,A_{n_0-1}{x}_{n_0})<\alpha (ϵ)$ for some $n_0$, then $({\varphi }_0,x_n)<ϵ$ for all $n\ge n_0$. Furthermore, the function $\alpha (ϵ)$ is positive and increasing, and $\alpha (ϵ)\le ϵ$. Considering a function $\beta (ϵ)$ defined by $\beta (ϵ):=\frac{1}{ϵ}{\int }_0^{ϵ}\alpha (t)dt$ and $\beta (0):=0$, it is easy to prove that $\beta \in \mathcal{K}$ and $0<\beta (ϵ)<\alpha (ϵ)\le ϵ$. Then the inverse of β, denoted by ψ will belong to . For some ${\varphi }_0\in K_0^{\ast }$, set ${ϵ}_n:=({\varphi }_0,x_n)$ and consider two possibilities: (i) If $({\varphi }_0,x_n)=0$, then $({\varphi }_0,x_n)=0\le \psi [({\varphi }_0,A_{n_0-1}{x}_{n_0})]$; (ii) If for some $({\varphi }_0,A_{n_0-1}{x}_{n_0})<\beta ({ϵ}_n)$, in which ${ϵ}_n:=({\varphi }_0,x_n)>0$, then $({\varphi }_0,x_n)<{ϵ}_n=({\varphi }_0,x_n)$, which is impossible, hence $({\varphi }_0,A_{n_0-1}{x}_{n_0})\ge \beta ({ϵ}_n)$, therefore, for some ${\varphi }_0\in K_0^{\ast }$,

$$({\varphi }_0,x_n)={ϵ}_n\le {\beta }^{-1}[({\varphi }_0,A_{n_0-1}{x}_{n_0})]=\psi [({\varphi }_0,A_{n_0-1}{x}_{n_0})],$$

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-${\varphi }_0$-stable (P-${\varphi }_0$-stable) if and only if there exist functions ${\psi }_n\in \mathcal{K}$ such that for any solution $x_n$ of (2.1), each nonnegative integer $n_0$ and some ${\varphi }_0\in K_0^{\ast }$, the following inequality holds:

$$\begin{array}{r}({\varphi }_0,x_n)\le {\psi }_{n_0}[({\varphi }_0,A_{n_0-1}{x}_{n_0})],n\ge n_0\\ (({\varphi }_0,x_n)\le {\psi }_{n_0}[({\varphi }_0,P_{n_0-1}{x}_{n_0})],n\ge n_0).\end{array}$$

(3.2)

Theorem 3.1 Assume that

• (i) $V\in C[Z_{+}\times S_{\rho },K]$, $V(n,0)=0$, $V(n,r)$ is locally Lipschitzian in r relative to K, and for each $(n,r)\in Z_{+}\times S_{\rho }$,

  $$\mathrm{\Delta }V(n,A_{n-1}{x}_n)\le 0;$$

• (ii) $f\in C[K,R^m]$ is quasimonotone in $x_n$ relative to K;

• (iii) $a[({\varphi }_0,x_n)]\le ({\varphi }_0,V(n,A_{n-1}{x}_n))$ for some ${\varphi }_0\in K_0^{\ast }$ and $a\in \mathcal{K}$, $(n,r)\in R_{+}\times S_{\rho }$.

Then the trivial solution of SDEs (2.1) is A-${\varphi }_0$-stable.

Proof Since $V(n,0)=0$ and $V(n,r)$ is continuous in r, then given $a_1(ϵ)>0$, $a_1\in \mathcal{K}$, there exists ${\delta }_1$ such that

$$\parallel A_{n_0-1}\gamma \parallel <{\delta }_1\text{implies}\parallel V(n_0,A_{n_0-1}\gamma )\parallel <a_1(ϵ).$$

For some ${\varphi }_0\in K_0^{\ast }$,

$$\parallel {\varphi }_0\parallel \parallel A_{n_0-1}\gamma \parallel <\parallel {\varphi }_0\parallel {\delta }_1\text{implies}\parallel {\varphi }_0\parallel \parallel V(n_0,A_{n_0-1}\gamma )\parallel <\parallel {\varphi }_0\parallel a_1(ϵ).$$

Thus

$$|({\varphi }_0,A_{n_0-1}\gamma )|\le \parallel {\varphi }_0\parallel \parallel A_{n_0-1}\gamma \parallel <\parallel {\varphi }_0\parallel {\delta }_1$$

implies

$$\left|({\varphi }_0,V(n_0,A_{n_0-1}\gamma ))\right|\le \parallel {\varphi }_0\parallel \parallel V(n_0,A_{n_0-1}\gamma )\parallel <\parallel {\varphi }_0\parallel a_1(ϵ).$$

It follows that

$$({\varphi }_0,A_{n_0-1}\gamma )\le \delta \text{implies}({\varphi }_0,V(n_0,A_{n_0-1}\gamma ))\le a(ϵ),$$

where $\parallel {\varphi }_0\parallel {\delta }_1=\delta$, $\parallel {\varphi }_0\parallel a_1(ϵ)=a(ϵ)$, $a\in \mathcal{K}$. Let $x_n$ be any solution of (2.1) such that $({\varphi }_0,A_{n_0-1}\gamma )<\delta$. Then by (i), V is nonincreasing and so

$$V(n,A_{n-1}{x}_n)\le V(n_0,A_{n_0-1}\gamma ),n\ge n_0.$$

Thus $({\varphi }_0,A_{n_0-1}\gamma )<\delta$ implies

$$a[({\varphi }_0,x_n)]\le ({\varphi }_0,V(n,A_{n-1}{x}_n))\le ({\varphi }_0,V(n_0,A_{n_0-1}\gamma ))<a(ϵ),$$

i.e.,

$$({\varphi }_0,A_{n_0-1}\gamma )<\delta \text{implies}({\varphi }_0,x_n)<ϵ,n\ge n_0.$$

Then the trivial solution of (2.1) is A-${\varphi }_0$-stable. The proof of Theorem 3.1 is complete. □

Theorem 3.2 Let the hypotheses of Theorem  3.1 be satisfied, except the condition $\mathrm{\Delta }V(n,A_{n-1}{x}_n)\le 0$ being replaced by

• (iv) $({\varphi }_0,\mathrm{\Delta }V(n,A_{n-1}{x}_n))\le -c[({\varphi }_0,V(n,A_{n-1}{x}_n))]$, $c\in \mathcal{K}$.

Then the trivial solution of SDEs (2.1) is A-asymptotically ${\varphi }_0$-stable.

Proof By Theorem 3.2, the trivial solution of (2.1) is A-${\varphi }_0$-stable. By (iv), $V(n,A_{n-1}{x}_n)$ is a monotone decreasing function, thus the limit

$$V^{\ast }=\underset{n\to \mathrm{\infty }}{lim}V(n,A_{n-1}{x}_n)$$

exists. We prove that $V^{\ast }=0$. Suppose $V^{\ast }\ne 0$, then $c(V^{\ast })\ne 0$, $c\in \mathcal{K}$. Since $c(r)$ is a monotone increasing function, then

$$c\left[({\varphi }_0,V(n,A_{n-1}{x}_n))\right]>c\left[({\varphi }_0,V^{\ast })\right],$$

and so from (iv), we get

$$({\varphi }_0,\mathrm{\Delta }V(n,A_{n-1}{x}_n))\le -c\left[({\varphi }_0,V^{\ast })\right].$$

Then

$$({\varphi }_0,V(n,A_{n-1}{x}_n))\le -c\left[({\varphi }_0,V^{\ast })\right](n-n_0)+({\varphi }_0,V(n_0,A_{n_0-1}{x}_{n_0})).$$

Thus as $n\to \mathrm{\infty }$ and for some ${\varphi }_0\in K_0^{\ast }$, we have $({\varphi }_0,V(n,A_{n-1}{x}_n))\to -\mathrm{\infty }$. This contradicts the condition (iii). It follows that $V^{\ast }=0$. Thus

$$({\varphi }_0,V(n,A_{n-1}{x}_n))\to 0,n\to \mathrm{\infty },$$

and so with (iii)

$$({\varphi }_0,x_n)\to 0,n\to \mathrm{\infty }.$$

Thus for given $ϵ>0$, $n_0\in R_{+}$, there exist $\delta =\delta (n_0)$ and $N=N(n_0,ϵ)$ such that

$$({\varphi }_0,A_{n_0-1}{x}_{n_0})<{\delta }_0\text{implies}({\varphi }_0,x_n)<ϵ,n\ge n_0+N.$$

Then the trivial solution of (2.1) is A-asymptotically ${\varphi }_0$-stable. The proof of Theorem 3.2 is complete. □

Theorem 3.3 The trivial solution of SDEs (2.1) is P-${\varphi }_0$-stable if and only if there exist functions ${\psi }_n\in \mathcal{K}$ and a Lyapunov function $V\in C[Z_{+}\times S_{\rho },K]$ such that for some ${\varphi }_0\in K_0^{\ast }$,

• (i) $V(n,0)=0$, $n\ge 0$;

• (ii) $({\varphi }_0,y)\le ({\varphi }_0,V(n,P_{n-1}y))\le {\psi }_n[({\varphi }_0,P_{n-1}y)]$, $\mathrm{\forall }y\in {\mathrm{△}}_n$, and some ${\varphi }_0\in K_0^{\ast }$,

• (iii) $\mathrm{\Delta }V(n,P_{n-1}{y}_n):=V(n+1,P_n{y}_{n+1}-V(n,P_{n-1}{y}_n))\le 0$ for any solution $y_n$ of (2.1).

Proof Necessity. Suppose that the trivial solution of (2.1) is P-${\varphi }_0$-stable, then, according to Lemma 3.2, there exist functions ${\psi }_n\in \mathcal{K}$ ($n\ge 0$) such that for any solution $x_n$ of (2.1) and for some ${\varphi }_0\in K_0^{\ast }$,

$$({\varphi }_0,x_n)\le {\psi }_{n_0}[({\varphi }_0,P_{n_0-1}{x}_{n_0})].$$

(3.3)

Define the Lyapunov function

$$V(n,\gamma ):=\underset{k\in Z_{+}}{sup}\parallel x_{n+k}(n;\gamma )\parallel ;\gamma \in R^m,n\in Z_{+},$$

where $x_{n+k}:=x_{n+k}(n;\gamma )$ is the unique solution of (2.1) satisfying the initial condition $P_{n-1}{x}_n=P_{n-1}\gamma$. Moreover, for some ${\varphi }_0\in K_0^{\ast }$, $({\varphi }_0,V(n,\gamma ))\le {\psi }_n({\varphi }_0,P_{n-1}\gamma )$, which implies $V(n,0)=0$ and the continuity of function V in the second variable at $\gamma =0$. For each $y\in {\mathrm{△}}_n$, we have

$$V(n,P_{n-1}y):=\underset{l\in Z_{+}}{sup}\parallel x_{n+l}(n;P_{n-1}y)\parallel \ge \parallel x_n(n;P_{n-1}y)\parallel ,$$

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 $x_n,y\in {\mathrm{△}}_n$, it follows $x_n(n;P_{n-1}y)=y$, hence, for some ${\varphi }_0\in K_0^{\ast }$,

$$({\varphi }_0,V(n,P_{n-1}y))\ge ({\varphi }_0,x_n(n;P_{n-1}y))=({\varphi }_0,y).$$

Further, the inequality (3.3) gives

$$({\varphi }_0,V(n,P_{n-1}y))\le {\psi }_n({\varphi }_0,P_{n-1}y).$$

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

$$V(n,P_{n-1}{y}_n)=\underset{l\in Z_{+}}{sup}\parallel x_{n+l}(n;P_{n-1}y)\parallel =\underset{l\ge 0}{sup}\parallel y_{n+l}\parallel .$$

Thus

$$\begin{array}{rl}V(n+1,P_n{y}_{n+1})& =\underset{l\ge 0}{sup}\parallel y_{n+l+1}\parallel =\underset{l\ge 1}{sup}\parallel y_{n+l}\parallel \\ \le \underset{l\ge 0}{sup}\parallel y_{n+l}\parallel =V(n,P_{n-1}y),\end{array}$$

hence $\mathrm{\Delta }V(n,P_{n-1}{y}_n)\le 0$. The necessity part is proved.

Sufficiency. Assuming that the trivial solution of (2.1) is not P-${\varphi }_0$-stable, i.e., there exist a positive ${ϵ}_0$ and a nonnegative integer $n_0$ such that for all $\delta \in (0,{ϵ}_0]$ and for some ${\varphi }_0\in K_0^{\ast }$, there exists a solution of (2.1) satisfying the inequalities $({\varphi }_0,P_{n_0-1}{x}_{n_0})<\delta$ and $({\varphi }_0,x_{n_1})\ge {ϵ}_0$ for some $n_1\ge n_0$.

Since $V(n_0,0)=0$ and $V(n_0,\gamma )$ is continuous at $\gamma =0$, there exists a ${\delta }_0^{\prime }={\delta }_0^{\prime }(ϵ,n_0)>0$ such that for all $\xi \in R^m$, $\parallel \xi \parallel <{\delta }_0^{\prime }$, we have $V(n_0,\xi )<{ϵ}_0$. Choosing ${\delta }_0\le min\{{\delta }_0^{\prime },{ϵ}_0\}$, we can find a solution $x_n$ of (2.1) satisfying $({\varphi }_0,P_{n_0-1}{x}_{n_0})\le {\delta }_0$. However, $({\varphi }_0,x_{n_1})\ge {ϵ}_0$ for some $n_1\ge n_0$. Since $({\varphi }_0,P_{n_0-1}{x}_{n_0})<{\delta }_0\le {\delta }_0^{\prime }$, we get

$$({\varphi }_0,V(n_0,P_{n_0-1}{x}_{n_0}))<{ϵ}_0$$

for some $n_1\ge n_0$. On the other hand, using the properties (iii) of the function V, we find

$$({\varphi }_0,V(n_0,P_{n_0-1}{x}_{n_0}))\ge ({\varphi }_0,V(n_1,P_{n_1-1}{x}_{n_1}))\ge ({\varphi }_0,x_{n_1})\ge {ϵ}_0,$$

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 ${\varphi }_0$-stable if and only if there exist functions $a,b\in \mathcal{K}$ and a Lyapunov function $V\in C[Z_{+}\times S_{\rho },K]$ such that for some ${\varphi }_0\in K_0^{\ast }$,

• (i) $a[({\varphi }_0,x)]\le ({\varphi }_0,V(n,P_{n-1}x))\le b[({\varphi }_0,P_{n-1}x)]$, $\mathrm{\forall }x\in {\mathrm{△}}_n$, $n\ge 0$;

• (ii) $\mathrm{\Delta }V(n,P_{n-1}{x}_n)\le 0$ 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 $\delta =b^{-1}[a(ϵ)]$ independent of $n_0$ for $a,b\in \mathcal{K}$, and $x_n$ be any solution of (2.1) such that $({\varphi }_0,P_{n_0-1}{x}_{n_0})<\delta$. Then by (ii), V is nonincreasing and so

$$({\varphi }_0,V(n,P_{n-1}{x}_n))\le ({\varphi }_0,V(n_0,P_{n_0-1}{x}_{n_0})),n\ge n_0.$$

Thus

$$\begin{array}{rl}a[({\varphi }_0,x_n)]& \le ({\varphi }_0,V(n,P_{n-1}{x}_n))\le ({\varphi }_0,V(n_0,P_{n_0-1}{x}_{n_0}))\\ \le b[({\varphi }_0,P_{n_0-1}{x}_{n_0})]<b(\delta )<a(ϵ),\end{array}$$

i.e.,

$$({\varphi }_0,P_{n_0-1}{x}_{n_0})<\delta \text{implies}({\varphi }_0,x_n)<ϵ,n\ge n_0.$$

Then the trivial solution of (2.1) is P-uniformly ${\varphi }_0$-stable. The proof of Theorem 3.4 is complete. □

4 Example

Consider SDEs (2.1) with the following data:

$$A_n=\left(\begin{array}{cc}n+3& 0\\ 0& 0\end{array}\right);B_n=\left(\begin{array}{cc}1& 0\\ 0& n+2\end{array}\right),n\ge -1,$$

and

$$f_n(x)=\frac{1}{n+2}{(0,1)}^T;x={(x_1,x_2)}^T,n\ge 0.$$

As $ker{A}_n=span\{{(0,1)}^T\}$, $im{A}_n=span\{{(1,0)}^T\}$, $n\ge -1$ and $S_n=span\{{(1,0)}^T\}$, $n\ge 0$, the hypotheses (H₁), (H₂) are fulfilled, hence SDEs (2.2) is of index-1. Clearly, the canonical projections are

$$P_n=P:=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right);Q_n=Q:=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right),$$

therefore

$$Q_{n-1,n}=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)=Q;G_n=A_n+B_n{Q}_{n-1,n}=\left(\begin{array}{cc}n+3& 0\\ 0& n+2\end{array}\right),$$

hence $G_n^{-1}=\left(\begin{array}{cc}\frac{1}{n+3}& 0\\ 0& \frac{1}{n+2}\end{array}\right)$. Further, the function $f_n(x)$ is Lipschitz continuous with the Lipschitz coefficient $L_n={(n+2)}^{-1}$. Moreover, $f_n(0)=0$ and ${\omega }_n:=L_n\parallel Q_{n-1,n}{G}_n^{-1}\parallel <1$. According to Theorem 2.1, IVP (2.1), (2.3) has a unique solution. We have $x\in {\mathrm{△}}_n$ 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 $x_2=\frac{x_1}{(n+2)(n+3)}$. Thus

$${\mathrm{△}}_n=\{x={(x_1,x_2)}^T:x_2=\frac{x_1}{(n+2)(n+3)}\},n\ge 0.$$

Let $V(n,\gamma ):=2\parallel x\parallel$, we get for each $x\in {\mathrm{△}}_n$,

$$\parallel x\parallel ={(x_1^2+\frac{x_1^2}{{(n+2)}^2{(n+3)}^2})}^{\frac{1}{2}}\le 2|x_1|=2\parallel P_{n-1}x\parallel .$$

Further, $V(n,P_{n-1}x)=2\parallel P_{n-1}x\parallel =2|x_1|$. Thus, for some ${\varphi }_0\in K_0^{\ast }$,

$$a[({\varphi }_0,x)]\le ({\varphi }_0,V(n,P_{n-1}x))\le b[({\varphi }_0,P_{n-1}x)],$$

where $a,b\in \mathcal{K}$ 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

$$\begin{array}{rl}\mathrm{△}V(n,P_{n-1}{x}_n)& =V(n+1,P_n{x}_{n+1})-V(n,P_{n-1}x)\\ =2(\parallel P{x}_{n+1}\parallel -\parallel P{x}_n\parallel )=2(\parallel u_{n+1}\parallel -\parallel u_n\parallel ).\end{array}$$

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

$$u_{n+1}=-P_n{G}_n^{-1}{B}_n{u}_n+P_n{G}_n^{-1}{f}_n(x_n)=-\frac{1}{n+3}\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)u_n,$$

hence $\parallel u_{n+1}\parallel -\parallel u_n\parallel =-\frac{n+2}{n+3}\parallel u_n\parallel \le \frac{1}{2}\parallel u_n\parallel$, then

$$\mathrm{△}V(n,P_{n-1}{x}_n)\le -\parallel P_{n-1}{x}_n\parallel .$$

According to Theorem 3.4, the trivial solution of (2.1) is P-uniformly ${\varphi }_0$-stable.