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OUNDEDNESS IN NONLINEAR PERTURBED DIFFERENTIAL SYSTEMS
OUNDEDNESS IN NONLINEAR PERTURBED DIFFERENTIAL SYSTEMS
Journal of Applied Mathematics & Informatics. 2014. Jan, 32(1_2): 247-254
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : August 24, 2013
  • Accepted : November 17, 2013
  • Published : January 28, 2014
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HOE GOO YOON

Abstract
In this paper, we investigate bounds for solutions of nonlinear perturbed differential systems. AMS Mathematics Subject Classification : 34D10.
Keywords
1. Introduction
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 : Lyapunov’s second method, the use of integral inequalities, and the method of variation of constants formula. The method incorporating integral inequalities takes an important place among the methods developed for the qualitative analysis of solutions to linear and nonlinear system of differential equations. In the presence the method of integral inequalities is as efficient as the direct Lyapunov’s 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. Using this notion, Choi and Ryu [3 , 5] investigated bounds of solutions for nonlinear perturbed systems and nonlinear functional differential systems. Also, Goo et al. [8] studied the boundedness of solutions for nonlinear perturbed systems.
In this paper, we obtain some results on boundedness of solutions of nonlinear perturbed differential systems under suitable conditions on perturbed term. To do this we need some integral inequalities.
2. Preliminaries
We are interested in the relations of the unperturbed system
PPT Slide
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and the solutions of the perturbed system
PPT Slide
Lager Image
where x , y , f and g are elements of ℝ n , an n -dimensional real Euclidean space.
We assume that f , g C (ℝ + × ℝ n ,ℝ n ), ℝ + = [0,∞), and that f is continuously differentiable with respect to the components of x on ℝ + × ℝ n , f ( t , 0) = 0 for all t ∈ ℝ + . The symbol |·| will be used to denote arbitrary vector norm in ℝ n .
Let x ( t , t 0 , x 0 ) denote the unique solutions of (1) and (2), satisfying the initial conditions x ( t 0 , t 0 , x 0 ) = x 0 , and y ( t 0 , t 0 , y 0 ) = y 0 , existing on [ t 0 ,∞), respectively. Then we can consider the associated variational systems around the zero solution of (1) and around x ( t ), respectively,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Here, fx ( t , x ) is the matrix whose element in the i th row, j th column is the partial derivative of the i th component of f with respect to the j th component of x . The fundamental matrix Փ( t , t 0 , x 0 ) of (4) is given by
PPT Slide
Lager Image
and Փ( t , t 0 , 0) is the fundamental matrix of (3).
We recall some notions of h -stability [15] .
Definition 2.1. The system (1)(the zero solution x = 0 of (1)) is called an h - system if there exist a constant c ≥ 1, and a positive continuous function h on ℝ + such that
PPT Slide
Lager Image
for t t 0 ≥ 0 and | x 0 | small enough(here
PPT Slide
Lager Image
Definition 2.2. The system (1) (the zero solution x = 0 of (1)) is called h - stable ( hS ) if there exists δ > 0 such that (1) is an h -system for | x 0 | ≤ δ and h is bounded.
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 [6] .
Definition 2.3. 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
Lager Image
such that
PPT Slide
Lager Image
for some S ( t ) ∈ N .
We give some related properties that we need in the sequel.
Lemma 2.1 ( [16] ). The linear system
PPT Slide
Lager Image
where A ( t ) is an n × n continuous matrix, is an h-system( h-stable, respectively) if and only if there exist c ≥ 1 and a positive continuous( bounded, repectively) function h defined on + such that
PPT Slide
Lager Image
for t × t 0 × 0, where ϕ ( t, t 0 ) is a fundamental matrix of (6).
The following is a generalization to nonlinear system of the variation of constants formula due to Alekseev [1] .
Lemma 2.2. Let x ( t ) = x ( t, t 0 , y 0 ) and y ( t ) = y ( t, t 0 , y 0 ) be solutions of (1) and (2), respectively. If y 0 ∈ ℝ n , then for all t such that x ( t, t 0 , y 0 ) ∈ ℝ n ,
PPT Slide
Lager Image
Theorem 2.3 ( [3] ). If the zero solution of (1) is hS, then the zero solution of (3) is hS.
Theorem 2.4 ( [4] ). 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 (3) is hS, then the solution z = 0 of (4) is hS.
Lemma 2.5 ( [13] ). Let u, f, g C (ℝ + ), for which the inequality
PPT Slide
Lager Image
holds, where u 0 is a nonnegative constant. Then,
PPT Slide
Lager Image
Lemma 2.6 ( [5] ). Let u, λ 1 , λ 2 , w C (ℝ + ), w ( u ) be nondecreasing in u and
PPT Slide
Lager Image
for some v > 0. If ,for some c > 0,
PPT Slide
Lager Image
then
PPT Slide
Lager Image
where
PPT Slide
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is the inverse of W ( u ) and
PPT Slide
Lager Image
Lemma 2.7 ( [11] ). Let u, λ 1 , λ 2 , λ 3 C (ℝ + ), w C ((0,∞)) and w ( u ) be nondecreasing in u,u w ( u ). Suppose that for some c > 0,
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
where
PPT Slide
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is the inverse of W ( u ) and
PPT Slide
Lager Image
3. Main results
In this section, we investigate bounds for the nonlinear differential systems. Also, we examine the bounded property for the perturbed system of (1)
PPT Slide
Lager Image
where g C (ℝ + × ℝ n ,ℝ n ) and g ( t , 0) = 0.
The generalization of a function h ’s condition and the strong condition of a function g in Theorem 3.1 [10] are the following result.
Theorem 3.1. 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) is hS with a positive continuous function h, and g in (9) satisfies
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Then, the solution y = 0 of (9) is hS.
Proof . Using the nonlinear variation of Alekseev [1] , any solution y ( t ) = y ( t, t 0 , y 0 ) of (9) passing through ( t 0 , y 0 ) is given by
PPT Slide
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By Theorem 2.3, since the solution x = 0 of (1) is hS, the solution v = 0 of (3) is hS. Therefore, by Theorem 2.4, the solution z = 0 of (4) is hS. By Lemma 2.1 and (10) , we have
PPT Slide
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Set u ( t ) = | y ( t )| h ( t ) −1 . Then, by Lemma 2.5, we obtain
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It follows that y = 0 of (9) is hS. Hence, the proof is complete.
Remark 3.1. In the linear case, we can obtain that if the zero solution x = 0 of (6) is hS, then the perturbed system
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is also hS under the same hypotheses in Theorem 3.1 except the condition of t -similarity.
Remark 3.2. Letting k ( t ) = 0 in Theorem 3.1, we obtain the same result as that of Theorem 3.3 in [9] .
The weak condition of a function h and the strong condition of a function g in Theorem 3.3 [8] are the following result.
Theorem 3.2. Let a, b, k, u, w C (ℝ + ), w ( u ) be nondecreasing in u, 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) is hS with a positive continuous function h, and g in (9) satisfies
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Then, any solution y ( t ) = y ( t, t 0 , y 0 ) of (9) is bounded on [ t 0 ,∞) and it satisfies
PPT Slide
Lager Image
where c = c 1 | y 0 | h ( t 0 ) −1 and W, W −1 are the same functions as in Lemma 2.6 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) and (9), respectively. By Theorem 2.3, since the solution x = 0 of (1) is hS, the solution v = 0 of (3) is hS. Therefore, by Theorem 2.4, the solution z = 0 of (4) is hS. Using Lemma 2.1 and (10), we have
PPT Slide
Lager Image
Set u ( t ) = | y ( t )| h ( t ) −1 . Now an application of Lemma 2.6 yields
PPT Slide
Lager Image
where c = c 1 | y 0 | h ( t 0 ) −1 . The above estimation yields the desired result since the function h is bounded, and the theorem is proved.
The generalization of a function h ’s condition and a slight modification of a function g ’s condition in Theorem 3.4 [11] are the following result.
Theorem 3.3. Let a, b, k, u, w C (ℝ + ), w ( u ) be nondecreasing in u, 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) is hS with the positive continuous function h, and g in (9) satisfies
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Then, any solution y ( t ) = y ( t, t 0 , y 0 ) of (9) is bounded on [ t 0 ,∞) and it satisfies
PPT Slide
Lager Image
where W, W -1 are the same functions as in Lemma 2.6 and
PPT Slide
Lager Image
Proof . It is known that the solution of (9) is represented by the integral equation(10). By Theorem 2.3, since the solution x = 0 of (1) is hS, the solution v = 0 of (3) is hS. Therefore, by Theorem 2.4, the solution z = 0 of (4) is hS. Using Lemma 2.1 and (10), we have
PPT Slide
Lager Image
Set u ( t ) = | y ( t )| h ( t ) −1 . Now an application of Lemma 2.7 yields
PPT Slide
Lager Image
where c = c 1 | y 0 | h ( t 0 ) −1 . The above estimation implies the boundedness of y ( t ), and the proof is complete.
Remark 3.3. Letting k ( t ) = 0 in Theorem 3.5 and adding the increasing condition of the function h , we obtain the same result as that of Theorem 3.2 in [8] .
Acknowledgements
The author is very grateful for the referee’s valuable comments.
BIO
Yoon Hoe Goo received the BS from Cheongju University and Ph.D at Chungnam National University under the direction of Chin-Ku Chu. Since 1993 he has been at Hanseo University as a professor. His research interests focus on topologival dynamical systems and differential equations.
Department of Mathematics, Hanseo University, Seasan 356-706, Korea.
e-mail: yhgoo@hanseo.ac.kr
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