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On {\varphi}_{0}stability of a class of singular difference equations
Advances in Difference Equations volume 2013, Article number: 179 (2013)
Abstract
This paper investigates a class of singular difference equations. Using the framework of the theory of singular difference equations and conevalued 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 initialvalue problems as well as boundaryvalue 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 conevalued Lyapunov functions and developed the theory of differential inequalities. By employing the method of conevalued Lyapunov functions, Akpan and Akinyele [11], ELSheikh 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 conevalued 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 yx\in K and x{<}_{{K}^{0}}y iff yx\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).
and
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 yx\in \partial K implies that there exists {\varphi}_{0}\in {K}_{0}^{\ast} such that
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:
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
are of index1 [1–3], i.e., the following hypotheses hold.
(H_{1}) rank{A}_{n}=r, n\ge 0,
(H_{2}) {S}_{n}\cap ker{A}_{n1}=\{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 (H_{1}) holds. Then the hypothesis (H_{2}) is equivalent to one of the following statements:

(i)
the matrix {G}_{n}:={A}_{n}+{B}_{n}{Q}_{n1,n} is nonsingular;

(ii)
{R}^{m}={S}_{n}\oplus ker{A}_{n1}.
Let us associate SDEs (2.1) with the initial condition
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.,
where
Then IVP (2.1), (2.3) has a unique solution.
Set {\mathrm{\u25b3}}_{n}:=\{x\in {R}^{m}:{Q}_{n1}x={Q}_{n1,n}{G}_{n}^{1}[{f}_{n}(x){B}_{n}{P}_{n1}x]\}. If x=\{{x}_{n}\} is any solution of IVP (2.1), (2.3), then obviously {x}_{n}\in {\mathrm{\u25b3}}_{n} (n\ge {n}_{0}).
Definition 2.5 [9]
The trivial solution of (2.1) is said to be Astable (Pstable) if for each \u03f5>0 and any {n}_{0}\ge 0, there exists a \delta =\delta (\u03f5,{n}_{0})\in (0,\u03f5] such that
Definition 2.6 The trivial solution of (2.1) is said to be
(S_{1}) A{\varphi}_{0}stable (P{\varphi}_{0}stable) if for each \u03f5>0 and any {n}_{0}\ge 0, there exists a \delta =\delta ({n}_{0},\u03f5)\in (0,\u03f5] such that for some {\varphi}_{0}\in {K}_{0}^{\ast}
(S_{2}) Auniformly {\varphi}_{0}stable (Puniformly {\varphi}_{0}stable) if δ in ({S}_{1}) is independent of {n}_{0}.
(S_{3}) Aasymptotically {\varphi}_{0}stable (Pasymptotically {\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},\u03f5) such that for some {\varphi}_{0}\in {K}_{0}^{\ast},
(S_{4}) Auniformly asymptotically {\varphi}_{0}stable if {\delta}_{0} and N in (S_{3}) are independent of {n}_{0}.
Let K be a cone in {R}^{m}, {S}_{\rho}=\{{x}_{n}\in {R}^{m},\parallel {A}_{n1}{x}_{n}\parallel <\rho ,\rho >0\}. V:{Z}_{+}\times {S}_{\rho}\to K is continuous in the second variable, we define
where {x}_{n} is any solution of system (2.1).
3 Main results
Lemma 3.1 The trivial solution of SDEs (2.1) is Auniformly {\varphi}_{0}stable (Puniformly {\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:
Proof For each positive ϵ, choose \delta =\delta (\u03f5)\in (0,\u03f5] such that \psi (\delta )<\u03f5. If {x}_{n} is an arbitrary solution of (2.1) and ({\varphi}_{0},{A}_{{n}_{0}1}{x}_{{n}_{0}})<\delta, then
Then (2.1) is Auniformly {\varphi}_{0}stable.
Conversely, suppose that the trivial solution of (2.1) is Auniformly {\varphi}_{0}stable, i.e., for each positive ϵ, there exists a \delta =\delta (\u03f5)\in (0,\u03f5] 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})<\u03f5 for all n\ge {n}_{0}. Denote by \alpha (\u03f5) the supremum of for the above \delta (\u03f5). Obviously, if ({\varphi}_{0},{A}_{{n}_{0}1}{x}_{{n}_{0}})<\alpha (\u03f5) for some {n}_{0}, then ({\varphi}_{0},{x}_{n})<\u03f5 for all n\ge {n}_{0}. Furthermore, the function \alpha (\u03f5) is positive and increasing, and \alpha (\u03f5)\le \u03f5. Considering a function \beta (\u03f5) defined by \beta (\u03f5):=\frac{1}{\u03f5}{\int}_{0}^{\u03f5}\alpha (t)\phantom{\rule{0.2em}{0ex}}dt and \beta (0):=0, it is easy to prove that \beta \in \mathcal{K} and 0<\beta (\u03f5)<\alpha (\u03f5)\le \u03f5. Then the inverse of β, denoted by ψ will belong to . For some {\varphi}_{0}\in {K}_{0}^{\ast}, set {\u03f5}_{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 ({\u03f5}_{n}), in which {\u03f5}_{n}:=({\varphi}_{0},{x}_{n})>0, then ({\varphi}_{0},{x}_{n})<{\u03f5}_{n}=({\varphi}_{0},{x}_{n}), which is impossible, hence ({\varphi}_{0},{A}_{{n}_{0}1}{x}_{{n}_{0}})\ge \beta ({\u03f5}_{n}), therefore, for some {\varphi}_{0}\in {K}_{0}^{\ast},
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:
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}_{n1}{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}_{n1}{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}(\u03f5)>0, {a}_{1}\in \mathcal{K}, there exists {\delta}_{1} such that
For some {\varphi}_{0}\in {K}_{0}^{\ast},
Thus
implies
It follows that
where \parallel {\varphi}_{0}\parallel {\delta}_{1}=\delta, \parallel {\varphi}_{0}\parallel {a}_{1}(\u03f5)=a(\u03f5), 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
Thus ({\varphi}_{0},{A}_{{n}_{0}1}\gamma )<\delta implies
i.e.,
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}_{n1}{x}_{n})\le 0 being replaced by

(iv)
({\varphi}_{0},\mathrm{\Delta}V(n,{A}_{n1}{x}_{n}))\le c[({\varphi}_{0},V(n,{A}_{n1}{x}_{n}))], c\in \mathcal{K}.
Then the trivial solution of SDEs (2.1) is Aasymptotically {\varphi}_{0}stable.
Proof By Theorem 3.2, the trivial solution of (2.1) is A{\varphi}_{0}stable. By (iv), V(n,{A}_{n1}{x}_{n}) is a monotone decreasing function, thus the limit
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
and so from (iv), we get
Then
Thus as n\to \mathrm{\infty} and for some {\varphi}_{0}\in {K}_{0}^{\ast}, we have ({\varphi}_{0},V(n,{A}_{n1}{x}_{n}))\to \mathrm{\infty}. This contradicts the condition (iii). It follows that {V}^{\ast}=0. Thus
and so with (iii)
Thus for given \u03f5>0, {n}_{0}\in {R}_{+}, there exist \delta =\delta ({n}_{0}) and N=N({n}_{0},\u03f5) such that
Then the trivial solution of (2.1) is Aasymptotically {\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}_{n1}y))\le {\psi}_{n}[({\varphi}_{0},{P}_{n1}y)], \mathrm{\forall}y\in {\mathrm{\u25b3}}_{n}, and some {\varphi}_{0}\in {K}_{0}^{\ast},

(iii)
\mathrm{\Delta}V(n,{P}_{n1}{y}_{n}):=V(n+1,{P}_{n}{y}_{n+1}V(n,{P}_{n1}{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},
Define the Lyapunov function
where {x}_{n+k}:={x}_{n+k}(n;\gamma ) is the unique solution of (2.1) satisfying the initial condition {P}_{n1}{x}_{n}={P}_{n1}\gamma. Moreover, for some {\varphi}_{0}\in {K}_{0}^{\ast}, ({\varphi}_{0},V(n,\gamma ))\le {\psi}_{n}({\varphi}_{0},{P}_{n1}\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{\u25b3}}_{n}, we have
where {x}_{k}(n;{P}_{n1}y) denotes the solution of (2.1) satisfying the initial condition {P}_{n1}{x}_{n}(n;{P}_{n1}y)={P}_{n1}({P}_{n1}y)={P}_{n1}y. Since {x}_{n},y\in {\mathrm{\u25b3}}_{n}, it follows {x}_{n}(n;{P}_{n1}y)=y, hence, for some {\varphi}_{0}\in {K}_{0}^{\ast},
Further, the inequality (3.3) gives
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
Thus
hence \mathrm{\Delta}V(n,{P}_{n1}{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 {\u03f5}_{0} and a nonnegative integer {n}_{0} such that for all \delta \in (0,{\u03f5}_{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 {\u03f5}_{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}(\u03f5,{n}_{0})>0 such that for all \xi \in {R}^{m}, \parallel \xi \parallel <{\delta}_{0}^{\prime}, we have V({n}_{0},\xi )<{\u03f5}_{0}. Choosing {\delta}_{0}\le min\{{\delta}_{0}^{\prime},{\u03f5}_{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 {\u03f5}_{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
for some {n}_{1}\ge {n}_{0}. On the other hand, using the properties (iii) of the function V, we find
which leads to a contradiction. The proof of Theorem 3.3 is complete. □
Theorem 3.4 The trivial solution of SDEs (2.1) is Puniformly {\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}_{n1}x))\le b[({\varphi}_{0},{P}_{n1}x)], \mathrm{\forall}x\in {\mathrm{\u25b3}}_{n}, n\ge 0;

(ii)
\mathrm{\Delta}V(n,{P}_{n1}{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 \u03f5>0, let \delta ={b}^{1}[a(\u03f5)] 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
Thus
i.e.,
Then the trivial solution of (2.1) is Puniformly {\varphi}_{0}stable. The proof of Theorem 3.4 is complete. □
4 Example
Consider SDEs (2.1) with the following data:
and
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 (H_{1}), (H_{2}) are fulfilled, hence SDEs (2.2) is of index1. Clearly, the canonical projections are
therefore
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}_{n1,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{\u25b3}}_{n} if and only if
it leads to {x}_{2}=\frac{{x}_{1}}{(n+2)(n+3)}. Thus
Let V(n,\gamma ):=2\parallel x\parallel, we get for each x\in {\mathrm{\u25b3}}_{n},
Further, V(n,{P}_{n1}x)=2\parallel {P}_{n1}x\parallel =2{x}_{1}. Thus, for some {\varphi}_{0}\in {K}_{0}^{\ast},
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}_{n1}{x}_{n}=P{x}_{n}, {v}_{n}={Q}_{n1}{x}_{n}=Q{x}_{n}, then we have
Using equation (2.8) in [9], we find
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
According to Theorem 3.4, the trivial solution of (2.1) is Puniformly {\varphi}_{0}stable.
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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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Wang, P., Wu, M. & Wu, Y. On {\varphi}_{0}stability of a class of singular difference equations. Adv Differ Equ 2013, 179 (2013). https://doi.org/10.1186/168718472013179
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DOI: https://doi.org/10.1186/168718472013179