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

Peiguang Wang; Meng Wu; Yonghong Wu

doi:10.1186/1687-1847-2013-179

Research
Open Access

On $ϕ_{0}$ -stability of a class of singular difference equations

Peiguang Wang¹Email author,
Meng Wu² and
Yonghong Wu³

Advances in Difference Equations20132013:179

https://doi.org/10.1186/1687-1847-2013-179

Received: 25 February 2013
Accepted: 4 June 2013
Published: 25 June 2013

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 [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

(i)

$λ K \subseteq K$ , $λ \geq 0$ ;
(ii)

$K + K \subseteq K$ ;
(iii)

$K = \bar{K}$ ;
(iv)

$K^{0} \neq \emptyset$ ;
(v)

$K \cap (- K) = {0}$ ,

where $\bar{K}$ 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 $x, y \in K$ , then $x \leq_{K} y$ iff $y - x \in K$ and $x <_{K^{0}} y$ iff $y - x \in K^{0}$ .

Definition 2.2 The set

K^{*} = {ϕ \in R^{n}, (ϕ, x) \geq 0, for all x \in K}

is said to be an adjoint cone if it satisfies the properties (i)-(v).

x \in K^{0} iff (ϕ, x) > 0,

and

x \in \partial k iff (ϕ, x) = 0 for some ϕ \in K_{0}^{*}, 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

ϕ_{0} \in K_{0}^{*}

such that

(ϕ_{0}, y - x) = 0 and (ϕ_{0}, g (y) - g (x)) \geq 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 \geq 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 \geq 0,

(2.2)

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

(H₁) $rank A_{n} = r$ , $n \geq 0$ ,

(H₂) $S_{n} \cap ker A_{n - 1} = {0}$ , $n \geq 1$ , where $S_{n} = {ξ \in R^{m} : B_{n} ξ \in im A_{n}}$ , $n \geq 0$ .

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} γ, n \geq 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.,

∥ f_{n} (x) - f_{n} (\tilde{x}) ∥ \leq L_{n} ∥ x - \tilde{x} ∥, \forall x, \tilde{x} \in R^{m},

where

ω_{n} : = L_{n} ∥ Q_{n - 1, n} G_{n}^{- 1} ∥, \forall n \geq 0 .

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

Set $△_{n} : = {x \in R^{m} : Q_{n - 1} x = Q_{n - 1, n} G_{n}^{- 1} [f_{n} (x) - B_{n} P_{n - 1} x]}$ . If $x = {x_{n}}$ is any solution of IVP (2.1), (2.3), then obviously $x_{n} \in △_{n}$ ( $n \geq n_{0}$ ).

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} \geq 0

, there exists a

δ = δ (ϵ, n_{0}) \in (0, ϵ]

such that

∥ A_{n_{0} - 1} γ ∥ < δ (∥ P_{n_{0} - 1} γ ∥ < δ) implies ∥ x_{n} (n_{0}; γ) ∥ < ϵ, n \geq 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} \geq 0

, there exists a

δ = δ (n_{0}, ϵ) \in (0, ϵ]

such that for some

ϕ_{0} \in 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 \geq 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} \geq 0

there exist positive numbers

δ_{0} = δ_{0} (n_{0})

and

N = N (n_{0}, ϵ)

such that for some

ϕ_{0} \in 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 \geq 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} \in R^{m}, ∥ A_{n - 1} x_{n} ∥ < ρ, ρ > 0}

.

V : Z_{+} \times S_{ρ} \to K

is continuous in the second variable, we define

Δ 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

ϕ_{0}

-stable (P-uniformly

ϕ_{0}

-stable) if and only if there exists a function

ψ \in K

such that for any solution

x_{n}

of SDEs (2.1) and some

ϕ_{0} \in K_{0}^{*}

, the following inequality holds:

\begin{aligned} (ϕ_{0}, x_{n}) \leq ψ [(ϕ_{0}, A_{n_{0} - 1} x_{n_{0}})], n \geq n_{0} \\ ((ϕ_{0}, x_{n}) \leq ψ [(ϕ_{0}, P_{n_{0} - 1} x_{n_{0}})], n \geq n_{0}) . \end{aligned}

(3.1)

Proof For each positive ϵ, choose

δ = δ (ϵ) \in (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}) \leq ψ [(ϕ_{0}, A_{n_{0} - 1} x_{n_{0}})] < ψ (δ) < ϵ, n \geq 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) \in (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 \geq 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 \geq n_{0}

. Furthermore, the function

α (ϵ)

is positive and increasing, and

α (ϵ) \leq ϵ

. Considering a function

β (ϵ)

defined by

β (ϵ) : = \frac{1}{ϵ} \int_{0}^{ϵ} α (t) d t

and

β (0) : = 0

, it is easy to prove that

β \in K

and

0 < β (ϵ) < α (ϵ) \leq ϵ

. Then the inverse of β, denoted by ψ will belong to

. For some

ϕ_{0} \in 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}) = 0 \leq ψ [(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}}) \geq β (ϵ_{n})

, therefore, for some

ϕ_{0} \in K_{0}^{*}

,

(ϕ_{0}, x_{n}) = ϵ_{n} \leq 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} \in K

such that for any solution

x_{n}

of (2.1), each nonnegative integer

n_{0}

and some

ϕ_{0} \in K_{0}^{*}

, the following inequality holds:

\begin{aligned} (ϕ_{0}, x_{n}) \leq ψ_{n_{0}} [(ϕ_{0}, A_{n_{0} - 1} x_{n_{0}})], n \geq n_{0} \\ ((ϕ_{0}, x_{n}) \leq ψ_{n_{0}} [(ϕ_{0}, P_{n_{0} - 1} x_{n_{0}})], n \geq n_{0}) . \end{aligned}

(3.2)

Theorem 3.1 Assume that

(i)

$V \in C [Z_{+} \times S_{ρ}, K]$ , $V (n, 0) = 0$ , $V (n, r)$ is locally Lipschitzian in r relative to K, and for each $(n, r) \in Z_{+} \times S_{ρ}$ ,
$Δ V (n, A_{n - 1} x_{n}) \leq 0;$
(ii)

$f \in C [K, R^{m}]$ is quasimonotone in $x_{n}$ relative to K;
(iii)

$a [(ϕ_{0}, x_{n})] \leq (ϕ_{0}, V (n, A_{n - 1} x_{n}))$ for some $ϕ_{0} \in K_{0}^{*}$ and $a \in K$ , $(n, r) \in R_{+} \times S_{ρ}$ .

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} \in 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} \in 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) | \leq ∥ ϕ_{0} ∥ ∥ A_{n_{0} - 1} γ ∥ < ∥ ϕ_{0} ∥ δ_{1}

implies

| (ϕ_{0}, V (n_{0}, A_{n_{0} - 1} γ)) | \leq ∥ 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) \leq δ implies (ϕ_{0}, V (n_{0}, A_{n_{0} - 1} γ)) \leq a (ϵ),

where

∥ ϕ_{0} ∥ δ_{1} = δ

,

∥ ϕ_{0} ∥ a_{1} (ϵ) = a (ϵ)

,

a \in K

. 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}) \leq V (n_{0}, A_{n_{0} - 1} γ), n \geq n_{0} .

Thus

(ϕ_{0}, A_{n_{0} - 1} γ) < δ

implies

a [(ϕ_{0}, x_{n})] \leq (ϕ_{0}, V (n, A_{n - 1} x_{n})) \leq (ϕ_{0}, V (n_{0}, A_{n_{0} - 1} γ)) < a (ϵ),

i.e.,

(ϕ_{0}, A_{n_{0} - 1} γ) < δ implies (ϕ_{0}, x_{n}) < ϵ, n \geq 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

Δ V (n, A_{n - 1} x_{n}) \leq 0

being replaced by

(iv)

$(ϕ_{0}, Δ V (n, A_{n - 1} x_{n})) \leq - c [(ϕ_{0}, V (n, A_{n - 1} x_{n}))]$ , $c \in K$ .

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 \to \infty} V (n, A_{n - 1} x_{n})

exists. We prove that

V^{*} = 0

. Suppose

V^{*} \neq 0

, then

c (V^{*}) \neq 0

,

c \in K

. 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})) \leq - c [(ϕ_{0}, V^{*})] .

Then

(ϕ_{0}, V (n, A_{n - 1} x_{n})) \leq - c [(ϕ_{0}, V^{*})] (n - n_{0}) + (ϕ_{0}, V (n_{0}, A_{n_{0} - 1} x_{n_{0}})) .

Thus as

n \to \infty

and for some

ϕ_{0} \in K_{0}^{*}

, we have

(ϕ_{0}, V (n, A_{n - 1} x_{n})) \to - \infty

. This contradicts the condition (iii). It follows that

V^{*} = 0

. Thus

(ϕ_{0}, V (n, A_{n - 1} x_{n})) \to 0, n \to \infty,

and so with (iii)

(ϕ_{0}, x_{n}) \to 0, n \to \infty .

Thus for given

ϵ > 0

,

n_{0} \in 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 \geq 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} \in K

and a Lyapunov function

V \in C [Z_{+} \times S_{ρ}, K]

such that for some

ϕ_{0} \in K_{0}^{*}

,

(i)

$V (n, 0) = 0$ , $n \geq 0$ ;
(ii)

$(ϕ_{0}, y) \leq (ϕ_{0}, V (n, P_{n - 1} y)) \leq ψ_{n} [(ϕ_{0}, P_{n - 1} y)]$ , $\forall y \in △_{n}$ , and some $ϕ_{0} \in K_{0}^{*}$ ,
(iii)

$Δ V (n, P_{n - 1} y_{n}) : = V (n + 1, P_{n} y_{n + 1} - V (n, P_{n - 1} y_{n})) \leq 0$ 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} \in K

(

n \geq 0

) such that for any solution

x_{n}

of (2.1) and for some

ϕ_{0} \in K_{0}^{*}

,

(ϕ_{0}, x_{n}) \leq ψ_{n_{0}} [(ϕ_{0}, P_{n_{0} - 1} x_{n_{0}})] .

(3.3)

Define the Lyapunov function

V (n, γ) : = sup_{k \in Z_{+}} ∥ x_{n + k} (n; γ) ∥; γ \in R^{m}, n \in 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} \in K_{0}^{*}

,

(ϕ_{0}, V (n, γ)) \leq 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 \in △_{n}

, we have

V (n, P_{n - 1} y) : = sup_{l \in Z_{+}} ∥ x_{n + l} (n; P_{n - 1} y) ∥ \geq ∥ 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 \in △_{n}

, it follows

x_{n} (n; P_{n - 1} y) = y

, hence, for some

ϕ_{0} \in K_{0}^{*}

,

(ϕ_{0}, V (n, P_{n - 1} y)) \geq (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)) \leq 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 \in Z_{+}} ∥ x_{n + l} (n; P_{n - 1} y) ∥ = sup_{l \geq 0} ∥ y_{n + l} ∥ .

Thus

\begin{aligned} V (n + 1, P_{n} y_{n + 1}) & = sup_{l \geq 0} ∥ y_{n + l + 1} ∥ = sup_{l \geq 1} ∥ y_{n + l} ∥ \\ \leq sup_{l \geq 0} ∥ y_{n + l} ∥ = V (n, P_{n - 1} y), \end{aligned}

hence $Δ V (n, P_{n - 1} y_{n}) \leq 0$ . 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 $δ \in (0, ϵ_{0}]$ and for some $ϕ_{0} \in K_{0}^{*}$ , there exists a solution of (2.1) satisfying the inequalities $(ϕ_{0}, P_{n_{0} - 1} x_{n_{0}}) < δ$ and $(ϕ_{0}, x_{n_{1}}) \geq ϵ_{0}$ for some $n_{1} \geq n_{0}$ .

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 \in R^{m}

,

∥ ξ ∥ < δ_{0}^{'}

, we have

V (n_{0}, ξ) < ϵ_{0}

. Choosing

δ_{0} \leq 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}}) \leq δ_{0}

. However,

(ϕ_{0}, x_{n_{1}}) \geq ϵ_{0}

for some

n_{1} \geq n_{0}

. Since

(ϕ_{0}, P_{n_{0} - 1} x_{n_{0}}) < δ_{0} \leq δ_{0}^{'}

, we get

(ϕ_{0}, V (n_{0}, P_{n_{0} - 1} x_{n_{0}})) < ϵ_{0}

for some

n_{1} \geq 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}})) \geq (ϕ_{0}, V (n_{1}, P_{n_{1} - 1} x_{n_{1}})) \geq (ϕ_{0}, x_{n_{1}}) \geq 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, b \in K

and a Lyapunov function

V \in C [Z_{+} \times S_{ρ}, K]

such that for some

ϕ_{0} \in K_{0}^{*}

,

(i)

$a [(ϕ_{0}, x)] \leq (ϕ_{0}, V (n, P_{n - 1} x)) \leq b [(ϕ_{0}, P_{n - 1} x)]$ , $\forall x \in △_{n}$ , $n \geq 0$ ;
(ii)

$Δ V (n, P_{n - 1} x_{n}) \leq 0$ 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, b \in K

, 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})) \leq (ϕ_{0}, V (n_{0}, P_{n_{0} - 1} x_{n_{0}})), n \geq n_{0} .

Thus

\begin{aligned} a [(ϕ_{0}, x_{n})] & \leq (ϕ_{0}, V (n, P_{n - 1} x_{n})) \leq (ϕ_{0}, V (n_{0}, P_{n_{0} - 1} x_{n_{0}})) \\ \leq b [(ϕ_{0}, P_{n_{0} - 1} x_{n_{0}})] < b (δ) < a (ϵ), \end{aligned}

i.e.,

(ϕ_{0}, P_{n_{0} - 1} x_{n_{0}}) < δ implies (ϕ_{0}, x_{n}) < ϵ, n \geq 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} = (\begin{array}{cc} n + 3 & 0 \\ 0 & 0 \end{array}); B_{n} = (\begin{array}{cc} 1 & 0 \\ 0 & n + 2 \end{array}), n \geq - 1,

and

f_{n} (x) = \frac{1}{n + 2} {(0, 1)}^{T}; x = {(x_{1}, x_{2})}^{T}, n \geq 0 .

As

ker A_{n} = span {{(0, 1)}^{T}}

,

im A_{n} = span {{(1, 0)}^{T}}

,

n \geq - 1

and

S_{n} = span {{(1, 0)}^{T}}

,

n \geq 0

, the hypotheses (H₁), (H₂) are fulfilled, hence SDEs (2.2) is of index-1. Clearly, the canonical projections are

P_{n} = P : = (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}); Q_{n} = Q : = (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}),

therefore

Q_{n - 1, n} = (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}) = Q; G_{n} = A_{n} + B_{n} Q_{n - 1, n} = (\begin{array}{cc} n + 3 & 0 \\ 0 & n + 2 \end{array}),

hence

G_{n}^{- 1} = (\begin{array}{c} \frac{1}{n + 3} & 0 \\ 0 & \frac{1}{n + 2} \end{array})

. 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 \in △_{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

△_{n} = {x = {(x_{1}, x_{2})}^{T} : x_{2} = \frac{x_{1}}{(n + 2) (n + 3)}}, n \geq 0 .

Let

V (n, γ) : = 2 ∥ x ∥

, we get for each

x \in △_{n}

,

∥ x ∥ = {(x_{1}^{2} + \frac{x_{1}^{2}}{{(n + 2)}^{2} {(n + 3)}^{2}})}^{\frac{1}{2}} \leq 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} \in K_{0}^{*}

,

a [(ϕ_{0}, x)] \leq (ϕ_{0}, V (n, P_{n - 1} x)) \leq b [(ϕ_{0}, P_{n - 1} x)],

where

a, b \in K

and

a (r) = r

,

b (r) = 2 r

. 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{aligned} △ 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} ∥) . \end{aligned}

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} (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}) u_{n},

hence

∥ u_{n + 1} ∥ - ∥ u_{n} ∥ = - \frac{n + 2}{n + 3} ∥ u_{n} ∥ \leq \frac{1}{2} ∥ u_{n} ∥

, then

△ V (n, P_{n - 1} x_{n}) \leq - ∥ P_{n - 1} x_{n} ∥ .

According to Theorem 3.4, the trivial solution of (2.1) is P-uniformly $ϕ_{0}$ -stable.

Declarations

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).

Authors’ Affiliations

(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

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

Acknowledgements

On $ϕ_{0}$ -stability of a class of singular difference equations