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BOUNDEDNESS IN PERTURBED FUNCTIONAL DIFFERENTIAL SYSTEMS
BOUNDEDNESS IN PERTURBED FUNCTIONAL DIFFERENTIAL SYSTEMS
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(5_6): 697-705
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : April 18, 2014
  • Accepted : May 24, 2014
  • Published : September 30, 2014
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SANG IL CHOI
DONG MAN IM
YOON HOE GOO

Abstract
In this paper, we investigate bounds for solutions of the nonlinear functional differential systems AMS Mathematics Subject Classification : 34D10.
Keywords
1. Introduction
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. 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 investigating the qualitative behavior of the solutions of perturbed nonlinear system of differential systems: the method of variation of constants formula, Lyapunov’s second method, and the use of integral inequalities. 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 [13 , 14] 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. He obtained a general variational h -stability and some properties about asymptotic behavior of solutions of differential systems called h -systems. Also, he studied some general results about asymptotic integration and gave some important examples in [13] . Choi and Ryu [3] , Choi, Koo [5] , and Choi et al. [4] investigated bounds of solutions for nonlinear perturbed systems and nonlinear functional differential systems. Also, Goo [9 , 10] studied the boundedness of solutions for nonlinear functional perturbed systems.
In this paper, we investigate bounds of solutions of the nonlinear functional perturbed differential systems.
2. Preliminaries
We consider the nonlinear functional differential equation
PPT Slide
Lager Image
where t ∈ ℝ + = [0, ), x ∈ ℝ n , f C (ℝ + × ℝ n , ℝ n ), f ( t , 0) = 0, the derivative fx C (ℝ + × ℝ n , ℝ n ), g C ((ℝ + × ℝ n , ℝ n ), g ( t , 0, 0) = 0 and T is a continuous operator mapping from C (ℝ + ,ℝ n ) into C (ℝ + ,ℝ n ). The symbol | · | will be used to denote arbitrary vector norm in ℝ n . We assume that for any two continuous functions u, v × C ( I ) where I is the closed interval, the operator T satisfies the following property:
PPT Slide
Lager Image
implies Tu ( t ) ≤ Tv ( t ), 0 ≤ t t 1 , and | Tu | ≤ T | u |.
Equation (1) can be considered as the perturbed equation of
PPT Slide
Lager Image
Let x ( t , t 0 , x 0 ) be denoted by the unique solution of (2) passing through the point ( t 0 , x 0 ) ∈ ℝ + × ℝ n such that x ( t 0 , t 0 , x 0 ) = x 0 . Also, we can consider the associated variational systems around the zero solution of (2) and around x ( t ), respectively,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
The fundamental matrix Φ( t , t 0 , x 0 ) of (4) is given by
PPT Slide
Lager Image
and Φ( t , t 0 , 0) is the fundamental matrix of (3).
We recall some notions of h -stability [13] .
Definition 2.1. The system (2) (the zero solution x = 0 of (2)) 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
PPT Slide
Lager Image
Definition 2.2. The system (2) (the zero solution x = 0 of (2)) is called (hS) h-stable if there exists δ > 0 such that (2) 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 sequal.
Lemma 2.1 ( [14] ). The linear system
PPT Slide
Lager Image
where A ( t ) is an n × n continuous matrix, is an h-system ( h-stable, respec-tively,) if and only if there exist c ≥ 1 and a positive and continuous ( bounded, respectively,) 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).
We need Alekseev formula to compare between the solutions of (2) and the solutions of perturbed nonlinear system
PPT Slide
Lager Image
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 (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 2.2. 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 (2) 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 ( [5] ). Let u, λ 1 , λ 2 , w C (ℝ + ) and w ( u ) be nondecreasing in u such that
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 2.6 ( [2] ). Let u , λ 1 , λ 2 , λ 3 C (ℝ + ), w C ((0, ), and w ( u ) be nondecreasing in u. Suppose that for some c > 0,
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
where W, W −1 are the same functions as in Lemma 2.5, and
PPT Slide
Lager Image
Lemma 2.7 ( [10] ). Let u, p, q, w, r C (ℝ + ), w C ((0, )) , and w ( u ) be nondecreasing in u. Suppose that for some c > 0,
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
where W, W −1 are the same functions as in Lemma 2.5, and
PPT Slide
Lager Image
3. Main results
In this section, we investigate the bounded property for the nonlinear functional differential systems.
Theorem 3.1. Let a, c, u, w C (ℝ + ), w ( u ) be nondecreasing in 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 (2) is hS with the increasing function h, and g in (1) satisfies
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Then, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1) is bounded on [ t 0 , ) and it satisfies
PPT Slide
Lager Image
where W , W −1 are the same functions as in Lemma 2.5 , β ( t ) = c 2 a ( t ) , k is a positive constant, 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 (2) and (1), respectively. By Theorem 2.3, since the solution x = 0 of (2) 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, Lemma 2.2 and the increasing property of the function h , we have
PPT Slide
Lager Image
Set u ( t ) = | y ( t ) |h ( t ) −1 . Then, by Lemma 2.5, we obtain
PPT Slide
Lager Image
where k = c 1 | y 0 | h ( t 0 ) −1 and β ( t ) = c 2 a ( t ). This completes the proof.
Remark 3.1. Letting c ( τ ) = 0 in Theorem 3.1, we have the similar result as that of Theorem 3.3 in [7] .
Theorem 3.2. Let a, b, c, u, w C (ℝ + ), w ( u ) be nondecreasing in u and
PPT Slide
Lager Image
for some v > 0. Suppose that the solution x = 0 of (2) is hS with a non-decreasing function h and the perturbed term g in (1) satisfies
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Then any solution y ( t ) = y ( t , t 0 , y 0 ) of (1) is bounded on [ t 0 , ) and it satisfies
PPT Slide
Lager Image
where W , W −1 are the same functions as in Lemma 2.5, k is a positive constant, 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 (2) and (1), respectively. By Theorem 2.2, we obtain
PPT Slide
Lager Image
since h is nondecreasing. Set u ( t ) = | y ( t )| h ( t ) −1 . Then, by Lemma 2.6, we have
PPT Slide
Lager Image
where k = c 1 | y 0 | h ( t 0 ) −1 . Therefore, we obtain the result.
Remark 3.2. Letting c ( τ ) = 0 in Theorem 3.2, we have the similar result as that of Theorem 3.1 in [8] .
Theorem 3.3. Let a, b, c, u, w C (ℝ + ), w ( u ) be nondecreasing in 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 (2) is hS with the increasing function h, and g in (1) satisfies
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Then any solution y ( t ) = y ( t , t 0 , y 0 ) of (1) is bounded on [ t 0 , ) and it satisfies
PPT Slide
Lager Image
where W, W −1 are the same functions as in Lemma 2.5, k is a positive constant, 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 (2) and (1), respectively. By Theorem 2.3, since the solution x = 0 of (2) 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, Lemma 2.2 and the increasing property of the function h , we have
PPT Slide
Lager Image
Set u ( t ) = | y ( t ) |h ( t ) −1 . Then, by Lemma 2.6, we obtain
PPT Slide
Lager Image
where k = c 1 | y 0 | h ( t 0 ) −1 . Hence, the proof is complete.
Remark 3.3. Letting c ( τ ) = 0 in Theorem 3.3, we have the similar result as that of Theorem 3.2 in [8] .
Theorem 3.4. Let b, c, u, w C (ℝ + ), w ( u ) be nondecreasing in 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, If the solution x = 0 of (2) is an h-system with a positive continuous function h and g in (1) satisfies
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where a : ℝ + → ℝ + is continuous with
PPT Slide
Lager Image
for all t 0 ≥ 0, then any solution y ( t ) = y ( t , t 0 , y 0 ) of (1) satisfies
PPT Slide
Lager Image
, t 0 t < b 1 , where W, W −1 are the same functions as in Lemma 2.5, k is a positive constant, 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 (2) and (1), respectively. By Theorem 2.3, since the solution x = 0 of (2) is an h -system, the solution v = 0 of (3) is an h -system. Therefore, by Theorem 2.4, the solution z = 0 of (4) is an h -system. By Lemma 2.2, we have
PPT Slide
Lager Image
Setting u ( t ) = | y ( t )| h ( t ) −1 and using Lemma 2.7, we obtain
PPT Slide
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, t 0 t < b 1 , where k = c 1 | y 0 | h ( t 0 ) −1 . Hence, the proof is complete.
Remark 3.4. Letting c ( τ ) = 0 in Theorem 3.4, we have the similar result as that of Theorem 3.5 in [8] .
Acknowledgements
The author is very grateful for the referee’s valuable comments.
BIO
Sang Il Choi received the BS from Korea University and Ph.D at North Carolina State University under the direction of J. Silverstein . Since 1995 he has been at Hanseo University as a professor. His research interests focus on Analysis and Probability theory.
Department of Mathematics, Hanseo University, Seasan 356-706, Republic of Korea
e-mail:schoi@hanseo.ac.kr
Dong Man Im received the BS and Ph.D at Inha University. Since 1982 he has been at Cheongju University as a professor. His research interests focus on Algebra and differential equations.
Department of Mathematics Education Cheongju University Cheongju Chungbuk 360-764, Republic of Korea
e-mail:dmim@cheongju.ac.kr
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 topological dynamical systems and differential equations.
Department of Mathematics, Hanseo University, Seasan 356-706, Republic of Korea
e-mail:yhgoo@hanseo.ac.kr
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