h-STABILITY AND BOUNDEDNESS IN FUNCTIONAL PERTURBED DIFFERENTIAL SYSTEMS
h-STABILITY AND BOUNDEDNESS IN FUNCTIONAL PERTURBED DIFFERENTIAL SYSTEMS
The Pure and Applied Mathematics. 2015. May, 22(2): 145-158
• Received : November 27, 2014
• Accepted : January 27, 2015
• Published : May 31, 2015
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YOON HOE, GOO

Abstract
In this paper, we investigate h -stability and boundedness for solutions of the functional perturbed differential systems using the notion of t -similarity.
Keywords
1. INTRODUCTION AND PRELIMINARIES
We consider the nonlinear nonautonomous differential system
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where f C (ℝ + × ℝ n ,ℝ n ), ℝ + = [0, ∞) and ℝ n is the Euclidean n -space. We assume that the Jacobian matrix f x = ∂f / ∂x exists and is continuous on ℝ + × ℝ n and f ( t , 0) = 0. Also, consider the functional perturbed differential systems of (1.1)
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where g C (ℝ + × ℝ n , ℝ n ), h C (ℝ + × ℝ n × ℝ n , ℝ n ), g ( t , 0) = 0, h ( t , 0, 0) = 0, and T : C (ℝ + ,ℝ n ) → C (ℝ + ,ℝ n ) is a continuous operator .
For x ∈ ℝ n , let
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For an n × n matrix A , define the norm | A | of A by | A | = sup |x|≤1 | Ax |.
Let x ( t , t 0 , x 0 ) denote the unique solution of (1.1) with x ( t 0 , t 0 , x 0 ) = x 0 , existing on [ t 0 , ∞). Then we can consider the associated variational systems around the zero solution of (1.1) and around x ( t ), respectively
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and
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The fundamental matrix Φ( t , t 0 , x 0 ) of (1.4) is given by
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and Φ( t , t 0 , 0) is the fundamental matrix of (1.3).
We recall some notions of h -stability [16] .
Definition 1.1. The system (1.1) (the zero solution x = 0 of (1.1)) is called an h-system if there exist a constant c ≥ 1, and a positive continuous function h on ℝ + such that
| x ( t )| ≤ c | x 0 | h ( t ) h ( t 0 ) −1
for t t 0 ≥ 0 and | x 0 | small enough (here
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).
Definition 1.2. The system (1.1) (the zero solution x = 0 of (1.1)) is called (hS) h-stable if there exists δ > 0 such that (1.1) is an h -system for | x 0 | ≤ δ and h is bounded.
Integral inequalities play a vital role in the study of boundedness and other qualitative properties of solutions of differential equations. In particular, Bihari’s integral inequality continuous to be an effective tool to study sophisticated problems such as stability, boundedness, and uniqueness of solutions. The behavior of solutions of a perturbed system is determined in terms of the behavior of solutions of an unperturbed system. There are three useful methods for showing the qualitative behavior of the solutions of perturbed nonlinear system : the use of integral inequalities, the method of variation of constants formula, and Lyapunov’s second method.
The notion of h -stability (hS) was introduced by Pinto [15 , 16] with the intention of obtaining results about stability for a weakly stable system (at least, weaker than those given exponential asymptotic stability) under some perturbations. That is, Pinto extended the study of exponential asymptotic stability to a variety of reasonable systems called h -systems. Choi, Ryu [2] and Choi, Koo, and Ryu [3] investigated bounds of solutions for nonlinear perturbed systems. Also, Goo [7 , 8 , 9] and Goo et al. [11] investigated boundedness of solutions for nonlinear perturbed systems.
Let M denote the set of all n × n continuous matrices A ( t ) defined on ℝ + and N be the subset of M consisting of those nonsingular matrices S ( t ) that are of class C 1 with the property that S ( t ) and S −1 ( t ) are bounded. The notion of t -similarity in M was introduced by Conti [5] .
Definition 1.3 . A matrix A ( t ) ∈ M is t - similar to a matrix B ( t ) ∈ M if there exists an n × n matrix F ( t ) absolutely integrable over ℝ + , i.e.,
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such that
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for some S ( t ) ∈ N .
The notion of t -similarity is an equivalence relation in the set of all n × n continuous matrices on ℝ + , and it preserves some stability concepts [5 , 12] .
In this paper, we investigate bounds for solutions of the nonlinear differential systems using the notion of t -similarity.
We give some related properties that we need in the sequal.
Lemma 1.4 ( [16] ). The linear system
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where A ( t ) is an n × n continuous matrix, is an h-system (respectively h-stable) if and only if there exist c ≥ 1 and a positive and continuous (respectively bounded) function h defined on + such that
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for t t 0 ≥ 0, where ϕ ( t , t 0 ) is a fundamental matrix of (1.6) .
We need Alekseev formula to compare between the solutions of (1.1) and the solutions of perturbed nonlinear system
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where g C (ℝ + × ℝ n , ℝ n ) and g ( t , 0) = 0. Let y ( t ) = y ( t , t 0 , y 0 ) denote the solution of (1.8) passing through the point ( t 0 , y 0 ) in ℝ + × ℝ n .
The following is a generalization to nonlinear system of the variation of constants formula due to Alekseev [1] .
Lemma 1.5. Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) be solutions of (1.1) and (1.8), respectively. If y 0 ∈ ℝ n , then for all t such that x ( t , t 0 , y 0 ) ∈ ℝ n ,
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Theorem 1.6 ( [2] ). If the zero solution of (1.1) is hS, then the zero solution of (1.3) is hS.
Theorem 1.7 ( [3] ). Suppose that fx ( t , 0) is t - similar to fx ( t , x ( t , t 0 , x 0 )) for t t 0 ≥ 0 and | x 0 | ≤ δ for some constant δ > 0. If the solution v = 0 of (1.3) is hS, then the solution z = 0 of (1.4) is hS .
Lemma 1.8 ( [4] ). (Bihari − type inequality) Let u , λ C (ℝ + ), w C ((0, ∞)) and w ( u ) be nondecreasing in u. Suppose that, for some c > 0
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Then
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Where
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W −1 ( u ) is the inverse of W ( u ), and
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Lemma 1.9 ( [10) . ] Let u, λ 1 , λ 2 , λ 3 , λ 4 C (ℝ + ), w C ((0, ∞)) and w ( u ) be nondecreasing in u, u w ( u ). Suppose that for some c > 0 and 0 ≤ t 0 t ,
PPT Slide
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Then
PPT Slide
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where W , W −1 are the same functions as in Lemma 1.8, and
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Lemma 1.10 ( [8] ). Let u, λ 1 , λ 2 , λ 3 , λ 4 C (ℝ + ), w C ((0, ∞)) and w ( u ) be nondecreasing in u, u w ( u ). Suppose that for some c > 0 and 0 ≤ t 0 t ,
PPT Slide
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Then
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Where W , W −1 are the same functions as in Lemma 1.8, and
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2. MAIN RESULTS
In this section, we investigate hS and boundedness for solutions of the functional perturbed differential systems via t -similarity.
Lemma 2.1. Let u , λ 1 , λ 2 , λ 3 , λ 4 , λ 5 C [ℝ + , ℝ + ] and suppose that, for some c ≥ 0 and t t 0 , we have
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Then
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Proof. Define a function v ( t ) by the right member of (2.1). Then, we have v ( t 0 ) = c and
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since v ( t ) is nondecreasing and u ( t ) ≤ v ( t ). Now, by integrating the above inequality on [ t 0 , t ] and v ( t 0 ) = c , we have
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Thus (2.3) yields the estimate (2.2). ☐
Theorem 2.2. Suppose that fx ( t , 0) is t - similar to fx ( t , x ( t , t 0 , x 0 )) for t t 0 ≥ 0 and | x 0 | ≤ δ for some constant δ > 0, the solution x = 0 of (1.1) is hS with the increasing function h, and g in (1.2) satisfies
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and
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where a, b, c, k, q C (
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),
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and
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Then, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is hS.
Proof. Using the nonlinear variation of constants formula of Alekseev [1] , any solution y ( t ) = y ( t , t 0 , y 0 ) passing through ( t 0 , y 0 ) is given by
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By Theorem 1.6, since the solution x = 0 of (1.1) is hS, the solution v = 0 of (1.3) is hS. Therefore, by Theorem 1.7, the solution z = 0 of (1.4) is hS. In view of Lemma 1.4, the hS condition of x = 0 of (1.1), (2.4),(2.5), and (2.6), we have
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Set u ( t ) = | y ( t )|| h ( t )| −1 . Now an application of Lemma 2.1 yields
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Where
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Thus, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is hS, and so the proof is complete. ☐
Theorem 2.3. Let a, b, c, k, u, w C (ℝ + ), w ( u ) be nondecreasing in u such that u w ( u ) and
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for some v > 0. Suppose that fx ( t , 0) is t - similar to fx ( t , x ( t , t 0 , x 0 )) for t t 0 ≥ 0 and | x 0 | ≤ δ for some constant δ > 0, the solution x = 0 of (1.1) is hS with the increasing function h, and g in (1.2) satisfies
PPT Slide
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and
PPT Slide
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Where
PPT Slide
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and
PPT Slide
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. Then, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is bounded on [ t 0 , ∞) and it satisfies
PPT Slide
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where W, W −1 are the same functions as in Lemma 1.8, and
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Proof. Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) be solutions of (1.1) and (1.2), respectively. By Theorem 1.6, since the solution x = 0 of (1.1) is hS, the solution v = 0 of (1.3) is hS. Therefore, by Theorem 1.7, the solution z = 0 of (1.4) is hS. Using the nonlinear variation of constants formula (2.6), the hS condition of x = 0 of (1.1), (2.7), and (2.8), we have
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Defining u ( t ) = | y ( t )|| h ( t )| −1 , then, by Lemma 1.10, we have
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Thus, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is bounded on [ t 0 , ∞). This completes the proof. ☐
Remark 2.4. Letting c ( t ) = 0 in Theorem 2.3, we obtain the same result as that of Theorem 3.2 in [7] .
Theorem 2.5. Let a, b, c, k, u, w C (ℝ + ), w ( u ) be nondecreasing in u such that u w ( u ) and
PPT Slide
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for some v > 0. Suppose that fx ( t , 0) is t - similar to fx ( t , x ( t , t 0 , x 0 )) for t t 0 ≥ 0 and | x 0 | ≤ δ for some constant δ > 0, the solution x = 0 of (1.1) is hS with the increasing function h, and g in (1.2) satisfies
PPT Slide
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and
PPT Slide
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Where
PPT Slide
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and
PPT Slide
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. Then, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is bounded on [ t 0 , ∞) and it satisfies
PPT Slide
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where W , W −1 are the same functions as in Lemma 1.8, and
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Proof . Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) be solutions of (1.1) and (1.2), respectively. By Theorem 1.6, since the solution x = 0 of (1.1) is hS, the solution v = 0 of (1.3) is hS. Therefore, by Theorem 1.7, the solution z = 0 of (1.4) is hS. Applying Lemma 1.4, the hS condition of x = 0 of (1.1), (2.6), (2.9), and (2.10), we have
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Set u ( t ) = | y ( t )| |h ( t )| −1 . Then, by Lemma 1.9, we have
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where c = c 1 | y 0 | h ( t 0 ) −1 . Thus, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is bounded on [ t 0 , ∞). Hence, the proof is complete. ☐
Remark 2.6. Letting c ( t ) = 0 and w ( u ) = u in Theorem 2.5, we obtain the same result as that of Theorem 3.1 in [6] .
Lemma 2.7. ( [8] . ] Let u, λ 1 , λ 2 , λ 3 , λ 4 C (ℝ + ), w C ((0, ∞)) and w ( u ) be nondecreasing in u, u w ( u ). Suppose that for some c > 0 and 0 ≤ t 0 t ,
PPT Slide
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Then
PPT Slide
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where W, W −1 are the same functions as in Lemma 1.8, and
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Proof. Define a function v ( t ) by the right member of (2.11). Then
PPT Slide
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which implies
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since v and w are nondecreasing, u w ( u ), and u ( t ) ≤ v ( t ) . Now, by integrating the above inequality on [ t 0 , t ] and v ( t 0 ) = c , we have
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Then, by the well-known Bihari-type inequality, (2.13) yields the estimate (2.12). ☐
Theorem 2.8. Let a, b, c, k, u, w C (ℝ + ), w ( u ) be nondecreasing in u such that u w ( u ) and
PPT Slide
Lager Image
for some v > 0. Suppose that fx ( t , 0) is t - similar to fx ( t , x ( t , t 0 , x 0 )) for t t 0 ≥ 0 and | x 0 | ≤ δ for some constant δ > 0, the solution x = 0 of (1.1) is hS with the increasing function h, and g in (1.2) satisfies
PPT Slide
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and
PPT Slide
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Where
PPT Slide
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and
PPT Slide
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. Then, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is bounded on [ t 0 , ∞) and
PPT Slide
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where W , W −1 are the same functions as in Lemma 1.8, and
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Proof. Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) be solutions of (1.1) and (1.2), respectively. By Theorem 1.6, since the solution x = 0 of (1.1) is hS, the solution v = 0 of (1.3) is hS. Therefore, by Theorem 1.7, the solution z = 0 of (1.4) is hS. Using the nonlinear variation of constants formula (2.6), the hS condition of x = 0 of (1.1), (2.14), and (2.15), we have
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Set u ( t ) = | y ( t )|| h ( t )| −1 . Then, it follows from Lemma 2.7 that we have
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where c = c 1 | y 0 | h ( t 0 ) −1 . From the above estimation, we obtain the desired result. Thus, the theorem is proved. ☐
Lemma 2.9. Let u, λ 1 , λ 2 , λ 3 , λ 4 C (ℝ + ), w C ((0, ∞)) and w ( u ) be nondecreasing in u, u w ( u ). Suppose that for some c > 0 and 0 ≤ t 0 t ,
PPT Slide
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Then
PPT Slide
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t 0 t < b 1 , where W , W −1 are the same functions as in Lemma 1.8, and
PPT Slide
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Proof . Define a function v ( t ) by the right member of (2.16) . Then
PPT Slide
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which implies
PPT Slide
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since v and w are nondecreasing, u w ( u ), and u ( t ) ≤ v ( t ) . Now, by integrating the above inequality on [ t 0 , t] and v ( t 0 ) = c , we have
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Then, by the well-known Bihari-type inequality, (2.18) yields the estimate (2.17). ☐
Theorem 2.10. Let a, b, c, k, u, w C (ℝ + ), w ( u ) be nondecreasing in u such that u w ( u ) and
PPT Slide
Lager Image
for some v > 0. Suppose that fx ( t , 0) is t - similar to fx ( t , x ( t , t 0 , x 0 )) for t t 0 ≥ 0 and | x 0 | ≤ δ for some constant δ > 0, the solution x = 0 of (1.1) is hS with the increasing function h, and g in (1.2) satisfies
PPT Slide
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and
PPT Slide
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Where
PPT Slide
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and
PPT Slide
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. Then, any solution y = 0 of (1.2) is bounded on [ t 0 , ∞) and it satisfies
PPT Slide
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t 0 t < b 1 , where c = c 1 | y 0 | h ( t 0 ) −1 , W, W −1 are the same functions as in Lemma 1.8, and
PPT Slide
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Proof. Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) be solutions of (1.1) and (1.2), respectively. By Theorem 1.6, since the solution x = 0 of (1.1) is hS, the solution v = 0 of (1.3) is hS. Therefore, by Theorem 1.7, the solution z = 0 of (1.4) is hS. Using the nonlinear variation of constants formula (2.6), the hS condition of x = 0 of (1.1), (2.19), and (2.20), we have
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Set u ( t ) = | y ( t )|| h ( t )| −1 with c = c | y 0 | h ( t 0 ) −1 . Then, an application of Lemma 2.9 yields
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where t 0 t < b 1 . Thus, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is bounded on [ t 0 , ∞), and so the proof is complete. ☐
Remark 2.11. Letting c ( t ) = 0 and b ( t ) = a ( t ) in Theorem 2.10, we obtain the similar result as that of Theorem 3.3 in [11] .
Acknowledgements
The author is very grateful for the referee’s valuable comments.
References
Alekseev V.M. 1961 An estimate for the perturbations of the solutions of ordinary differential equations Vestn. Mosk. Univ. Ser. I. Math. Mekh. (Russian) 2 28 - 36
Choi S.K. , Ryu H.S. 1993 h-stability in differential systems Bull. Inst. Math. Acad. Sinica 21 245 - 262
Choi S. K. , Koo N.J. , Ryu H.S. 1997 h-stability of differential systems via t∞-similarity Bull. Korean. Math. Soc. 34 371 - 383
Choi S.K. , Koo N.J. , Song S.M. 1999 Lipschitz stability for nonlinear functional differential systems Far East J. Math. Sci(FJMS) I (5) 689 - 708
Conti R. 1957 t∞-similitudine tra matricie l'equivalenza asintotica dei sistemi differenziali lineari Rivista di Mat. Univ. Parma 8 43 - 47
Goo Y.H. 2012 h-stability of perturbed differential systems via t∞-similarity J. Appl. Math. and Informatics 30 511 - 516
Goo Y.H. 2013 Boundedness in perturbed nonlinear differential systems J. Chungcheong Math. Soc. 26 605 - 613    DOI : 10.14403/jcms.2013.26.3.605
Goo Y.H. Boundedness in nonlinear functional perturbed differential systems submitted
Goo Y.H. 2013 Boundedness in the perturbed nonlinear differential systems Far East J. Math. Sci(FJMS) 79 205 - 217
Goo Y.H. h-stability and boundedness in the perturbed functional differential systems submitted
Goo Y.H. , Park D.G. , Ryu D.H. 2012 Boundedness in perturbed differential systems J. Appl. Math. and Informatics 30 279 - 287
Hewer G.A. 1973 Stability properties of the equation by t∞-similarity J. Math. Anal. Appl. 41 336 - 344    DOI : 10.1016/0022-247X(73)90209-6
Lakshmikantham V. , Leela S. 1969 Theory and Applications Academic Press New York and London Differential and Integral Inequalities
Pachpatte B.G. 2002 On some retarded inequalities and applications J. Ineq. Pure Appl. Math. 3 1 - 7
Pinto M. 1988 Asymptotic integration of a system resulting from the perturbation of an h-system J. Math. Anal. Appl. 131 194 - 216    DOI : 10.1016/0022-247X(88)90200-4
Pinto M. 1992 Stability of nonlinear differential systems Applicable Analysis 43 1 - 20    DOI : 10.1080/00036819208840049