Stability of a Class of Singular Difference Equations

Transcription

1 International Journal of Difference Equations. ISSN Volume 1 Number ), pp c Research India Publications Stability of a Class of Singular Difference Equations Pham Ky Anh and Dau Son Hoang Department of Mathematics, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam Abstract The aim of this paper is to apply Lyapunov functions to obtain some necessary and sufficient conditions for the stability of singular nonautonomous difference equations. AMS subject classification: 39A11. Keywords: Singular difference equations, initial-value problem, Lyapunov functions, stability, uniform stability, matrix pencil. 1. Introduction In practice, many problems are modeled by singular difference equations SDEs). Recently, a class of singular nonautonomous difference equations, called index-1 SDEs, has been introduced and the solvability of initial-value problems IVPs) as well as boundary-value problems BVPs) has been studied cf., [1 3, 5, 6]). Moreover, the Floquet theory has been developed for linear index-1 SDEs [4]. In this paper we apply Lyapunov functions to study stability properties of singular quasilinear difference equations. The paper is organized as follows. In Section 2 we provide basic concepts of index-1 SDEs. The unique solvability of the IVP for a class of SDEs is also established. Section 3 deals with various stability conditions for SDEs. Finally, in Section 4 some illustrative examples are considered. Received August 29, 2006; Accepted November 28, 2006

2 182 Pham Ky Anh and Dau Son Hoang 2. Basic Concepts Consider a system A n x n+1 + B n x n = f n x n ) n 0), 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, so Equation 2.1) is called an SDE. In what follows we suppose that the corresponding linear homogeneous equation A n x n+1 + B n x n = 0 n 0), 2.2) is of index-1 [1 6], i.e., the following hypotheses are assumed to be fulfilled: H1) ranka n = r n 0), H2) S n ker A n 1 = {0} n 1), where, S n = {ξ R m : B n ξ ima n }, n 0). In what follows we always assume that dim S 0 = r and let A 1 R m m be a fixed chosen matrix, such that R m = S 0 ker A 1, so the hypothesis H2 is satisfied for all n 0. Let Q n R m m be an arbitrary projection onto ker A n, n 1), i.e., Q 2 n = Q n and imq n = ker A n. Then there exists a nonsingular matrix V n R m m such that 1 Q n = V n QV n, where Q := diago r, I m r ) and O r, I m r stand for r r zero and m r) m r) identity matrices, respectively. Further we define the matrix P n := 1 I Q n and the so-called connecting operators Q n 1,n := V n 1 QV n and Q n,n 1 := 1 V n QV n 1. Obviously, the connecting operators associated with projections Q n, Q n 1 satisfy the identities Q n 1,n = Q n 1 Q n 1,n = Q n 1,n Q n, Q n 1,n Q n,n 1 = Q n 1 and Q n,n 1 Q n 1,n = Q n. For the next discussion, the following lemma from [4] is needed. Lemma 2.1. Suppose the hypothesis H1 is fulfilled. 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 ker A n 1. It is proved [4], that every index-1 SDE 2.2) can be reduced to the Kronecker normal form diagi r, O m r )y n+1 + diagw n, I m r )y n = 0. Let us associate the SDE 2.1) with the initial condition P n0 1x n0 = P n0 1γ, n 0 0, 2.3) where γ is an arbitrary vector in R m and n 0 is a fixed nonnegative integer.

3 Stability of a Class of Singular Difference Equations 183 Theorem 2.2. Let f n x) be a Lipschitz continuous function with a sufficient small Lipschitz coefficient, i.e., where f n x) f n x) L n x x, x, x R m, 2.4) Then the IVP 2.1), 2.3) has a unique solution. ω n := L n Q n 1,n G 1 n < 1, n ) Proof. The conclusion of this theorem follows from [5, Theorem 1]. However, to make our presentation self-contained we shall provide a straight-forward proof of this fact. Indeed, multiplying on both sides of Equation 2.1) from the left by P n G 1 n and Q n G 1 n, respectively and observing that G 1 n A n = P n, P n Q n = Q n P n = O we get P n x n+1 + P n G 1 n B n x n = P n G 1 n f n x n ), 2.6) Q n G 1 n B n x n = Q n Gn 1 f n x n ). 2.7) Putting u n := P n 1 x n, v n := Q n 1 x n, n 0) and noting that from 2.6) we find P n G 1 n B n Q n 1 x n = P n G 1 n B n Q n 1,n Q n,n 1 x n = P n Q n,n 1 x n = 0, u n+1 = P n G 1 n B n u n + P n G 1 n f n u n + v n ). 2.8) Now using the fact that Q n = G 1 n B n Q n 1,n we can express the left side of 2.7) as Q n G 1 n B n x n = Q n G 1 n Thus the relation 2.7) becomes B n u n + Q n G 1 B n Q n 1,n Q n,n 1 x n = Q n G 1 n B n u n + Q n,n 1 x n. Q n,n 1 x n = Q n G 1 n B n u n + Q n G 1 n f n x n ). Now acting Q n 1,n on both sides of the last relation we get n v n = Q n 1 x n = Q n 1,n G 1 n {f n u n + v n ) B n u n }. 2.9) Supposing u := u n n n 0 ) is known, where u n0 = P n0 1x n0 = P n0 1γ is given, we consider an operator T n : imq n 1 imq n 1 defined by T n v) := Q n 1,n G 1 n {f n u + v) B n u}. Since T n v) T n ṽ) L n Q n 1,n G 1 n v ṽ = ω n v ṽ,

4 184 Pham Ky Anh and Dau Son Hoang the operator T n is a contraction. Hence there exists an operator g n : imp n 1 imq n 1 giving the unique solution of 2.9) whenever u n is known. Moreover, g n is Lipschitz continuous with the Lipschitz constant β n := ω n L n + B n )L 1 n 1 ω n ) 1. Obviously, the unique solution of the IVP 2.1), 2.3) is given by x n = u n + g n u n ), 2.10) where g n u n ) is a unique solution of 2.9) with u n0 = P n0 1γ. In what follows without loss of generality we will assume that f n 0) = 0 for all n 0. Then g n 0) = 0 and Equation 2.1) always possesses a trivial solution x n 0 n 0). From 2.10) it implies that each solution x n of the IVP 2.1), 2.3) satisfies the relation x n = P n 1 x n + g n P n 1 x n ) or equivalently, Q n 1 x n = Q n 1,n G 1 n {f n x n ) B n P n 1 x n }. Set n := { x R m : Q n 1 x = Q n 1,n G 1 n [f n x) B n P n 1 x] }. If x = {x n } is any solution of the IVP 2.1), 2.3), then obviously, x n n n n 0 ). Conversely, for each α n, there exists a solution of 2.1) passing α. Indeed, let x k n; α) k n) be a solution of 2.1) satisfying the initial condition P n 1 x n = P n 1 α. Clearly, x n n; α) = P n 1 x n + g n P n 1 x n ) = P n 1 α + g n P n 1 α) = α. The following lemma shows that the set n does not depend on the choice of projections. Lemma 2.3. The following hold: i) n = Ω n := {x R m : f n x) B n x ima n } n 0). ii) Ω n ker A n 1 = {0}. Proof. i) Letting x n we have Q n 1 x = Q n 1,n G 1 n {f n x) B n P n 1 x}, hence x = P n 1 x + Q n 1 x = Q n 1,n G 1 n f n x) + I Q n 1,n G 1 n B n )P n 1 x. From the last relation we get f n x) B n x = I B n Q n 1,n G 1 n )f n x) B n I Q n 1,n G 1 n B n )P n 1 x. Observing that B n I Q n 1,n G 1 n B n )P n 1 x = I B n Q n 1,n G 1 n )B n P n 1 x, we find f n x) B n x = I B n Q n 1,n G 1 n ){f n x) B n P n 1 x}.

5 Stability of a Class of Singular Difference Equations 185 Since B n Q n 1,n G 1 n = G n A n )G 1 n = I A n G 1 n, it implies that f n x) B n x = A n G 1 n {f n x) B n P n 1 x} ima n, hence x Ω n. Conversely, let x Ω n, i.e., f n x) B n x = A n ξ, for some ξ R m. We have to prove that Q n 1 x = Q n 1,n G 1 n f n x) B n P n 1 x), or equivalently, x = Q n 1,n G 1 n [f n x) B n x] + Q n 1,n G 1 n B n Q n 1 x + P n 1 x. Denoting the right-hand side of the last relation by w n and observing that we find Q n 1,n G 1 n {f n x) B n x} = Q n 1,n G 1 n A n ξ = Q n 1,n P n ξ = 0 w n = Q n 1,n G 1 n B n Q n 1 x + P n 1 x = Q n 1,n G 1 n B n Q n 1,n )Q n,n 1 x + P n 1 x = Q n 1 x + P n 1 x = x. Thus, x n and the first part of Lemma 2.3 is proved. ii) Let x Ω n ker A n 1. Then P n 1 x = 0 and x n, hence x = P n 1 x + g n P n 1 x) = 0. The proof of Lemma 2.3 is complete. We end this section by observing that the initial condition 2.3) is equivalent to the condition A n0 1x n0 = A n0 1γ n 0 1), 2.11) which is independent of the choice of projections. Indeed, acting on both sides of 2.11) by G 1 n 0 1 and using the equality G 1 n 0 1A n0 1 = P n0 1 we get 2.3). Conversely, multiplying on both sides of 2.3) by A n0 1 and noting that A n0 1P n0 1 = A n0 1 we obtain 2.11). For the sake of convenience, we choose a matrix B 1 R m m such that the matrix pencil {A 1, B 1 } is of index-1. Then the matrix G 1 = A 1 +B 1 Q 1 is nonsingular. Moreover, A 1 = A 1 P 1 and G 1 1A 1 = P 1. Thus both initial conditions 2.3) and 2.11) are equivalent for all n 0 0. The unique solution of the IVP 2.1), 2.3) or 2.1), 2.11) will be denoted by x n n 0 ; γ). 3. Stability of Singular Difference Equations In this section the notions of stability of the trivial solution are introduced and some necessary and sufficient conditions for the stability are established. We shall restrict ourselves to the canonical projection onto kera n 1, i.e., the projection from R m m into kera n 1 along S n and will denote it again by Q n 1. Then P n 1 := I Q n 1 is the canonical projection from R m m into S n along kera n 1. Thanks to the decomposition R m = S n ker A n 1 the canonical projections are determined uniquely from the data A n, B n and A n 1. Note that if Q n 1 is an any projection

6 186 Pham Ky Anh and Dau Son Hoang onto kera n 1 n 1) and Q n 1,n is the associate connecting operator, then the canonical projection Q n can be computed as Q n = Q G 1 n 1,n n B n, where G n := A n + B n Qn 1,n. We should refer to the work [4] for details. Let R + and Z + be the set of nonnegative real numbers and integers, respectively. Definition 3.1. The trivial solution of 2.1) is said to be i) A-stable P -stable) if for each ɛ > 0 and any n 0 0 there exists a δ = δɛ, n 0 ) 0, ɛ] such that A n0 1γ < δ P n0 1γ < δ) implies x n n 0 ; γ) < ɛ for all n n 0. ii) A-uniformly P -uniformly) stable if it is A-stable P -stable) and the number δ mentioned in part i) of this definition does not depend on n 0. iii) A-asymptotically P -asymptotically) stable if for any n 0 0 there exists a δ 0 = δ 0 n 0 ) > 0 such that the inequality A n0 1γ < δ 0 P n0 1γ < δ 0 ) implies x n n 0 ; γ) 0 n 0). Remark 3.2. From the relation G 1 n A n = P n and A n P n = A n, it is easy to show that the notions A-stability and P -stability are equivalent. The same conclusion is true for the A-asymptotical stability and P -asymptotical stability. That is why in what follows we will drop the prefixes A and P when talking about the stability or asymptotical stability. Further, if the matrices A n are uniformly bounded, then A-uniform stability implies P -uniform stability. Conversely, if G 1 n have uniformly bounded inverses, then A-uniform stability follows from P -uniform stability. Denote by K the class of all continuous and strictly increasing functions ψ from [0, ) into itself, such that ψ0) = 0. Lemma 3.3. The trivial solution of 2.1) is A-uniformly P -uniformly) stable if and only if there exists a function ψ K, such that for any solution x n of 2.1) and for any nonnegative integer n 0, there holds the inequality x n ψ A n0 1x n0 ) n n 0 3.1) x n ψ P n0 1x n0 ) n n 0 ). Proof. We provide a proof of the lemma for the A-uniform stability case. The remaining P -uniform stability case can be considered similarly. Suppose first that there exists a function ψ K satisfying the condition 3.1). For each positive ɛ we choose δ = δɛ) 0, ɛ] such that ψδ) < ɛ. If x n is an arbitrary solution of 2.1) and A n0 1x n0 < δ, then x n ψ A n0 1x n0 ) < ψδ) < ɛ, n n 0. Conversely, suppose that the trivial solution of 2.1) is A-uniformly stable, i.e., for each positive ɛ there exists a δ = δɛ) 0, ɛ], such that if x n is any solution of 2.1)

7 Stability of a Class of Singular Difference Equations 187 satisfying the inequality A n0 1x n0 < δ, where n 0 is a fixed nonnegative integer, then x n < ɛ for all n n 0. Denote by αɛ) the supremum of such δɛ). Clearly, if A n0 1x n0 < αɛ) for some n 0, then x n < ɛ for all n n 0. Further, the function αɛ) is positive and increasing and moreover, αɛ) ɛ. Consider a function βɛ) defined ɛ by βɛ) := 1 αt)dt for positive ɛ and β0) := 0. It is easy to prove that β K and ɛ 0 0 < βɛ) < αɛ) ɛ. Then the inverse of β, denoted by ψ will belong to K. Let x n be a solution of 2.1) and n 0 be a fixed nonnegative integer. Set ɛ n := x n and consider two possibilities. If x n = 0, then x n = 0 ψ A n0 1x n0 ), since ψ is nonnegative. Now suppose ɛ n := x n > 0. If A n0 1x n0 < βɛ n ), then x n < ɛ n = x n, n n 0, which is impossible, hence A n0 1x n0 βɛ n ), therefore x n = ɛ n β 1 A n0 1x n0 ) = ψ A n0 1x n0 ), which was to be proved. The proof of Lemma 3.3 is complete. By arguing as in the proof of Lemma 3.3 we come to the following result. Lemma 3.4. The trivial solution of 2.1) is stable if and only if there exist functions ψ n K, n 0), such that for any solution x n of 2.1) and for each nonnegative integer n 0, there holds the inequality x n ψ n0 A n0 1x n0 ), n n 0. Theorem 3.5. The existence of the Lyapunov function V : Z + R m R + being continuous in the second variable at γ = 0 and functions ψ n K, such that i) V n, 0) = 0, n 0. ii) y V n, P n 1 y) ψ n P n 1 y ), y n, n 0. iii) V 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), is a necessary and sufficient condition for the stability of the trivial solution of the SDE 2.1). Proof. Necessity. Suppose that the trivial solution of 2.1) is stable. Then according to Lemma 3.4, there exist functions ψ n K n 0), such that for any solution x n of 2.1) x n ψ n0 A n0 1x n0 ), n n 0. Define the functions ϕ n t) = ψ n A n 1 t), t [0, ). Clearly, ϕ n K and x n ψ n0 A n0 1x n0 ) = ψ n0 A n0 1P n0 1x n0 ) ψ n0 A n0 1 P n0 1x n0 ). Thus, x n ϕ n0 P n0 1x n0 ) n n )

8 188 Pham Ky Anh and Dau Son Hoang Further, we define the Lyapunov function V n, γ) := sup k Z + x n+k n; γ) ; γ R m, n Z +, 3.3) 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. The inequality 3.2) ensures the correctness of the definition 3.3). Moreover, V n, γ) ϕ n P n 1 γ ), which implies that V n, 0) = 0 and the continuity of the function V w.r.t. 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 Further, the inequality 3.2) gives V n n, P n 1 y) x n n; P n 1 y) = y. V n, P n 1 y) = sup l Z + x n+l n; P n 1 y) ϕ n P n 1 y ). On the other hand, for an arbitrary solution y n of 2.1), due to the unique solvability of the IVP 2.1), 2.3) we have Thus V n, P n 1 y n ) = sup x n+l n; P n 1 y n ) = sup y n+l. l Z + l 0 V n + 1, P n+1 y n ) = sup l 0 y n+l+1 = sup y n+l l 1 sup y n+l = V n, P n 1 y n ), l 0 hence V n, P n 1 y n ) 0. The necessity part is proved. Sufficiency. We argue by contradiction by assuming that the trivial solution of 2.1) is not stable, i.e., there exist a positive ɛ 0 and a nonnegative integer n 0, such that for all δ 0, ɛ 0 ], there exists a solution x n of 2.1) satisfying the inequalities P n0 1x n0 < δ and x n1 ɛ 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 0ɛ, n 0 ) > 0, such that for all ξ R m, ξ < δ 0 we have V n 0, ξ) < ɛ 0. Choosing δ 0 {δ 0, ɛ 0 } we can find a solution x n of 2.1) satisfying P n0 1x n0 < δ 0, however x n1 ɛ 0 for some n 1 n 0. Since P n0 1x n0 < δ 0 δ 0, one gets V n 0, P n0 1x n0 ) < ɛ 0. On the other hand, using the properties i) and iii) of the function V, we find V n 0, P n0 1x n0 ) V n 1, P n1 1x n1 ) x n1 ɛ 0,

9 Stability of a Class of Singular Difference Equations 189 which leads to a contradiction. The proof of Theorem 3.5 is complete. A similar argument as in the proof of the sufficiency part of Theorem 3.5 leads to the next result. Theorem 3.6. Assume that there exists a function ψ K and a Lyapunov function V : Z + R m R +, which is continuous w.r.t. the second variable at γ = 0, such that i) V n, 0) = 0, n 0. ii) ψ x ) V n, A n 1 x), x Ω n, n 0. iii) V n, A n 1 x n ) := V n + 1, A n x n+1 ) V n, A n 1 x n ) 0 for any solution x n of 2.1). Then the trivial solution of 2.1) is stable. Theorem 3.7. The trivial solution of 2.1) is P -uniformly stable if and only if there exist two functions a, b K and a Lyapunov function V : Z + R m R +, such that i) a x ) V n, P n 1 x) b P n 1 x ), x n, n 0. ii) V n, P n 1 x n ) := V n + 1, P n x n+1 ) V n, P n 1 x n ) 0 for any solution x n of 2.1). Proof. The proof of the necessity part is based on Lemma 3.4 and is similar to the corresponding part of Theorem 3.5. Now suppose that the trivial solution of 2.1) is not P -uniformly stable, hence there exists ɛ 0, such that for all δ 0, ɛ 0 ], there exist a solution x n of 2.1) and two nonnegative integers n 1 n 2, such that P n1 1x n1 < δ however x n2 ɛ 0. Choose δ 0 > 0, such that δ 0 ɛ 0 and bδ 0 ) < aɛ 0 ). According to our assumption there exist a solution x n of 2.1) and two nonnegative integers n 1 n 2, such that P n1 1x n1 < δ 0 and x n2 ɛ 0. Using the first property of the Lyapunov function we have and V n 2, P n2 1x n2 ) a x n2 ) aɛ 0 ) V n 1, P n1 1x n1 ) b P n1 1x n1 ) < bδ 0 ) < aɛ 0 ). On the other hand, taking into account the second property of V, we get aɛ 0 ) > V n 1, P n1 1x n1 ) V n 2, P n2 1x n2 ) a x n2 ) aɛ 0 ), which was the desired contradiction. Thus the trivial solution of 2.1) is P -uniformly stable. Theorem 3.7 is proved. We end this section by stating a theorem on the A-uniform stability and asymptotical stability of the trivial solution of 2.1). Its proof is similar to those of Theorems 3.5, 3.7, and therefore will be omitted.

10 190 Pham Ky Anh and Dau Son Hoang Theorem 3.8. Suppose that there exist two functions a, b K and a Lyapunov function V : Z + R m R +, such that i) a x ) V n, A n 1 x) b A n 1 x ), x Ω n, n 0. ii) V n, A n 1 x n ) := V n + 1, A n x n+1 ) V n, A n 1 x n ) 0 for any solution x n of 2.1). Then the trivial solution of 2.1) is A-uniformly stable. If Condition ii) is replaced by iii) V n, A n 1 x n ) := V n + 1, A n x n+1 ) V n, A n 1 x n ) c A n 1 x n ) for any solution x n of 2.1), where c is a certain function of the class K, then the trivial solution of 2.1) is asymptotically stable. 4. Examples In this section we will use the Euclidean norms of vectors and matrices. Example 4.1. Consider the SDE 2.1) with the following data: ) ) A n = n + 2) ; B 0 0 n =, n 1, 0 n + 2 and f n x) = sin x 1 n + 2 0, 1)T ; x = x 1, x 2 ) T, n 0). As kera n = span{0, 1) T }, ima n = span{1, 0) T }, n 1 and S n = span{1, 0) T }, n 0, the hypotheses H1, H2 are fulfilled, hence ) the SDE 2.2) is of index-1. ) Clearly, the canonical projections are P n = P := ; Q 0 0 n = Q :=, therefore 0 1 Q n 1,n = Q. A simple calculation shows that G n = A n + B n Q n 1,n = n + 2)I, hence G 1 n = n+2) 1 I. 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 1 n = n + 2) 2 < 1 for all nonnegative n. According to Theorem 2.2, the IVP 2.1), 2.3) has a unique solution. From the definition of n, we have x n if and only if Q n 1 x = Q n 1,n G 1 n {f n x) B n P n 1 x}, or Qx = QG 1 n {f n x) B n P x}. The last relation leads to x 2 = n + 2) 2 sin x 1. Thus, n = Ω n = {x = x 1, x 2 ) T : x 2 = n + 2) 2 sin x 1 }, n 0. Introducing a function V n, γ) := 2 γ, we get for each x n, x = x 1 + x 2 ) 1/2 = x n + 2) 4 sin 2 x 1 ) 1/2 2 x 1 = 2 P n 1 x.

11 Stability of a Class of Singular Difference Equations 191 Further, V n, P n 1 x) = 2 P n 1 x = 2 x 1. Thus, x V n, P n 1 x) = 2 P n 1 x for all x n. Supposing 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 = Qx n we have V n, P n 1 x n ) = V n + 1, P n x n+1 ) V n, P n 1 x n ) Using Equation 2.8) we find = 2 P x n+1 P x n ) = 2 u n+1 u n ). u n+1 = P n G 1 n B n u n + P n G 1 n f n x n ) = n + 2) 1 u n, hence u n+1 u n = n + 1)n + 2) 1 u n 2 1 u n, consequently V n, P n 1 x n ) P n 1 x n. According to Theorem 3.7, the trivial solution of 2.1) is P -uniformly stable. Moreover, it is also asymptotically stable. Example 4.2. Let the data in 2.1) be as follows: ) ) n n A n = ; B n n =, n 1, n + 1 n + 1 and f n x) = sinn + 2)x 1 + x 2 ) 1, 0) T, x = x 1, x 2 ) T, n 0). 2n + 1)n + 2) In this case, kera n = span{1, n 3) T }, ima n = span{1, 1) T }, n 1 and S n = span{n, 1) T }, n 0. Clearly, S n ker A n 1 = {0}, n 0, hence Equation 2.2) is of index-1. Consider the projections Q n = ) 1 0 n 3 0 A simple calculation shows that ) 0 1 V n = 1 n 3 and P n = I Q n = ; V 1 n = ) 0 0. n ) n + 3 1, 1 0 hence Further, and Q n 1,n = V n 1 QV 1 n = ) 1 0. n 2 0 ) n G n = A n + B n Q n 1,n = n 2 n ) n = n + 1) n 2. + n 2 n + 3 G 1

12 192 Pham Ky Anh and Dau Son Hoang Observe that the function f n x) is Lipschitz continuous with a Lipschitz coefficient L n = 2n + 1)n + 2)) 1. Since ω n = L n Q n 1,n G 1 n < 1, the IVP 2.1), 2.3) has a unique solution. Denoting ξ n := n + 2)x 1 + x 2 for any x = x 1, x 2 ) T, we can rewrite P n 1 x = 0, ξ n ) T ; A n 1 x = ξ n, ξ n ) T, which leads to the relations P n 1 x = ξ n ; A n 1 x = 2 ξ n. Now note that x = x 1, x 2 ) T belongs to n if and only if Q n 1 x = Q n 1,n G 1 n {f n x) B n P n 1 x}. A direct computation shows that the last relation is equivalent to the equality { } 1 sin ξ n x 1 = ϕ n x 1, x 2 ) := n + 1) 2 2n + 1)n + 2) + nξ n. 4.1) Thus, n = {x 1, x 2 ) T : x 1 = ϕ n x 1, x 2 )}. Let us define a Lyapunov function V n, γ) := ) γ n + 1 for all nonnegative integers n and all γ R 2. For each x = x 1, x 2 ) T n we put u = P n 1 x = 0, ξ n ) T and v = Q n 1 x = x 1, n + 2)x 1 ) T. From 4.1) we have v = x n + 2) 2 x 2 1) 1/2 = [1 + n + 2) 2 ] 1/2 x 1 ) 1 + n + 2) 2 ) 1/2 n + 1) 2 1 2n + 1)n + 2) + n ξ n 2 ξ n. Since u = ξ n and taking into account the last inequality we find v 2 u, which leads to the relation x = u + v 3 u. Thus 1 3 x u = P n 1x x n. 4.2) Observing that A n 1 x = 2 ξ n for any x n we get V n, A n 1 x) = ) A n 1 x n + 1 A n 1 x = 2 ξ n = 2 u 2 3 x for all x n and n 0. Now suppose x n = x n,1, x n,2 ) T is any solution of Equation 2.1). Let u n := P n 1 x n, v n := Q n 1 x n and ξ n := n + 2)x n,1 + x n,2. Then u n is a solution of 2.8). Noting that P n G 1 n B n u n = 0, ξ n ) T and P n G 1 n f n x n ) = ) T sin ξ n 0,, we can rewrite Equation 2.8) as 2n + 1)n + 2) u n+1 = ) T sin ξ n 0, ξ n +, 2n + 1)n + 2)

13 Stability of a Class of Singular Difference Equations 193 which gives u n+1 ξ n + sin ξ n. Further computation gives 2n + 1)n + 2) V n, A n 1 x n ) = V n + 1, A n x n+1 ) V n, A n 1 x n ) = ) A n x n ) A n 1 x n n + 2 n + 1 = { ) u n ) } u n n + 2 n + 1 { ) ) sin ξ n ξ n n + 2 2n + 1)n + 2) = { } n n + 1)n + 2) sin ξ ξ n n 2 n + 1)n + 2) 2 n + 1)n + 2) sin ξ n ξ n ) 0. n + 1 Theorem 3.5 ensures the stability of the trivial solution of Equation 2.1). References ) } ξ n [1] Pham Ky Anh and Le Cong Loi. On multipoint boundary-value problems for linear implicit non-autonomous systems of difference equations. Vietnam J. Math., 293): , [2] Pham Ky Anh and Le Cong Loi. On discrete analogues of nonlinear implicit differential equations. Adv. Difference Equ., pages Art , 1 19, [3] Pham Ky Anh and Ha Thi Ngoc Yen. On the solvability of initial-value problems for nonlinear implicit difference equations. Adv. Difference Equ., 3): , [4] Pham Ky Anh and Ha Thi Ngoc Yen. Floquet theorem for linear implicit nonautonomous difference systems. J. Math. Anal. Appl., 3212): , [5] Pham Ky Anh, Ha Thi Ngoc Yen, and Tran Quoc Binh. On quasi-linear implicit difference equations. Vietnam J. Math., 321):75 85, [6] L. C. Loi, N. H. Du, and P. K. Anh. On linear implicit non-autonomous systems of difference equations. J. Difference Equ. Appl., 812): , 2002.