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h-STABILITY AND BOUNDEDNESS IN FUNCTIONAL PERTURBED DIFFERENTIAL SYSTEMS
h-STABILITY AND BOUNDEDNESS IN FUNCTIONAL PERTURBED DIFFERENTIAL SYSTEMS
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2015. May, 22(2): 145-158
Copyright © 2015, Korean Society of Mathematical Education
  • 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
PPT Slide
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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)
PPT Slide
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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
PPT Slide
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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
PPT Slide
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and
PPT Slide
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The fundamental matrix Φ( t , t 0 , x 0 ) of (1.4) is given by
PPT Slide
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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
PPT Slide
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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.,
PPT Slide
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such that
PPT Slide
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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
PPT Slide
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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
PPT Slide
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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
PPT Slide
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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 ,
PPT Slide
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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
PPT Slide
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Then
PPT Slide
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Where
PPT Slide
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W −1 ( u ) is the inverse of W ( u ), and
PPT Slide
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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
PPT Slide
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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
PPT Slide
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Where W , W −1 are the same functions as in Lemma 1.8, and
PPT Slide
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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
PPT Slide
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Then
PPT Slide
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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
PPT Slide
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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
PPT Slide
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and
PPT Slide
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where a, b, c, k, q C (
PPT Slide
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),
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 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
PPT Slide
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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
PPT Slide
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Set u ( t ) = | y ( t )|| h ( t )| −1 . Now an application of Lemma 2.1 yields
PPT Slide
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Where
PPT Slide
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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
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 it satisfies
PPT Slide
Lager Image
where 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), (2.7), and (2.8), we have
PPT Slide
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Defining u ( t ) = | y ( t )|| h ( t )| −1 , then, by Lemma 1.10, we have
PPT Slide
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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
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
Lager Image
and
PPT Slide
Lager Image
Where
PPT Slide
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and
PPT Slide
Lager Image
. Then, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is bounded on [ t 0 , ∞) and it satisfies
PPT Slide
Lager Image
where W , W −1 are the same functions as in Lemma 1.8, and
PPT Slide
Lager Image
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
PPT Slide
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Set u ( t ) = | y ( t )| |h ( t )| −1 . Then, by Lemma 1.9, we have
PPT Slide
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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
Lager Image
Then
PPT Slide
Lager Image
where W, W −1 are the same functions as in Lemma 1.8, and
PPT Slide
Lager Image
Proof. Define a function v ( t ) by the right member of (2.11). 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
PPT Slide
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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
Lager Image
and
PPT Slide
Lager Image
Where
PPT Slide
Lager Image
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
Lager Image
where W , W −1 are the same functions as in Lemma 1.8, and
PPT Slide
Lager Image
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
PPT Slide
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Set u ( t ) = | y ( t )|| h ( t )| −1 . Then, it follows from Lemma 2.7 that we have
PPT Slide
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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
Lager Image
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
Lager Image
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
PPT Slide
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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
Lager Image
and
PPT Slide
Lager Image
Where
PPT Slide
Lager Image
and
PPT Slide
Lager Image
. Then, any solution y = 0 of (1.2) is bounded on [ t 0 , ∞) and it satisfies
PPT Slide
Lager Image
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
PPT Slide
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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
PPT Slide
Lager Image
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