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ASYMPTOTIC PROPERTY FOR PERTURBED NONLINEAR FUNCTIONAL DIFFERENTIAL SYSTEMS
ASYMPTOTIC PROPERTY FOR PERTURBED NONLINEAR FUNCTIONAL DIFFERENTIAL SYSTEMS
Journal of Applied Mathematics & Informatics. 2015. Sep, 33(5_6): 687-697
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : March 29, 2015
  • Accepted : June 29, 2015
  • Published : September 30, 2015
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DONG MAN IM
YOON HOE GOO

Abstract
This paper shows that the solutions to the perturbed nonlinear functional differential system y = f ( t , y ) + , Ty ( s )) ds , f ( t , 0) = 0, g ( t , 0, 0) = 0 go to zero as t goes to infinity. To show asymptotic property, we impose conditions on the perturbed part , Ty ( s )) ds and the fundamental matrix of the unperturbed system y = f ( t , y ). AMS Mathematics Subject Classification : 34D05.
Keywords
1. Introduction
Elaydi and Farran [8] introduced the notion of exponential asymptotic stabilit y (EAS) which is a stronger notion than that of ULS. They investigated some analytic criteria for an autonomous differential system and its perturbed systems to be EAS. Pachpatte [13] investigated the stability and asymptotic behavior of solutions of the functional differential equation. Gonzalez and Pinto [9] proved theorems which relate the asymptotic behavior and boundedness of the solutions of nonlinear differential systems. Choi et al. [6 , 7] examined Lipschitz and exponential asymptotic stability for nonlinear functional systems. Also, Goo [11] and Choi and Goo [2 , 4] investigated Lipschitz and asymptotic stability for perturbed differential systems.
In this paper we will obtain some results on asymptotic property for nonlinear perturbed differential systems. We will employ the theory of integral inequalities to study asymptotic property for solutions of the nonlinear differential systems. 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.
2. 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 fx = ∂ f /∂ x exists and is continuous on ℝ + × ℝ n and f ( t , 0) = 0. Also, we consider the perturbed differential system of (1)
PPT Slide
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where g C (ℝ + × ℝ n × ℝ n , ℝ n ), g ( 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) 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) and around x ( t ), respectively,
PPT Slide
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and
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The fundamental matrix Φ( t , t 0 , x 0 ) of (4) is given by
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and Φ( t , t 0 , 0) is the fundamental matrix of (3).
Before giving further details, we give some of the main definitions that we need in the sequel [8] .
Definition 2.1. The system (1) (the zero solution x = 0 of (1)) is called (S) stable if for any ϵ > 0 and t 0 ≥ 0, there exists δ = δ ( t 0 , ϵ ) > 0 such that if | x 0 | < δ , then | x ( t )| < ϵ for all t t 0 ≥ 0,
(AS) asymptotically stable if it is stable and if there exists δ = δ ( t 0 ) > 0 such that if | x 0 | < δ , then | x ( t )| → 0 as t → ∞,
(ULS) uniformly Lipschitz stable if there exist M > 0 and δ > 0 such that | x ( t )| ≤ M | x 0 | whenever | x 0 | ≤ δ and t t 0 ≥ 0,
(EAS) exponentially asymptotically stable if there exist constants K > 0 , c > 0, and δ > 0 such that
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provided that | x 0 | < δ ,
(EASV) exponentially asymptotically stable in variation if there exist constants K > 0 and c > 0 such that
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provided that | x 0 | < ∞.
Remark 2.1 ( [9] ). The last definition implies that for | x 0 | ≤ δ
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We give some related properties that we need in the sequel. We need Alekseev formula to compare between the solutions of (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 (5) 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 2.1. Let x and y be a solution of (1) and (5), respectively. If y 0 ∈ ℝ n, then for all t t 0 such that x ( t , t 0 , y 0 ) ∈ ℝ n , y ( t , t 0 , y 0 ) ∈ ℝ n ,
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Lemma 2.2 (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
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where t 0 t < b 1 ,
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, W −1 ( u ) is the inverse of W ( u ) , and
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Lemma 2.3 ( [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
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where t 0 t < b 1 , W , W −1 are the same functions as in Lemma 2.2, and
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Lemma 2.4 ( [3] ). Let u , λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 , λ 7 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 t 0 t < b 1 , W , W −1 are the same functions as in Lemma 2.2, and
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Proof . Define a function v ( t ) by the right member of (6). Then, we have v ( t 0 ) = c and
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t t 0 , since v ( t ) is 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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It follows from Lemma 2.2 that (8) yields the estimate (7). □
For the proof we need the following corollary from Lemma 2.4.
Corollary 2.5. Let u , λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 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 ,
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Then
PPT Slide
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where t 0 t < b 1 , W , W −1 are the same functions as in Lemma 2.2, and
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Lemma 2.6 ( [5] ). Let u , λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 C (ℝ + ), w C ((0,∞)) and w ( u ) be nondecreasing in u, u w ( u ). Suppose that for some c > 0,
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Then
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where t 0 t < b 1 , W , W −1 are the same functions as in Lemma 2.2, and
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Proof . Define a function z ( t ) by the right member of (9). Then, we have z ( t 0 ) = c and
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since z ( t ) and w ( u ) are nondecreasing, u w ( u ), and u ( t ) ≤ z ( t ). Therefore, by integrating on [ t 0 , t ], the function z satisfies
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It follows from Lemma 2.2 that (11) yields the estimate (10). □
We prepare two corollaries from Lemma 2.6 that are used in proving the theorems.
Corollary 2.7. Let u , λ 1 , λ 2 , λ 3 C (ℝ + ), w C ((0,∞)) and w ( u ) be nonde-creasing in u, u w ( u ). Suppose that for some c > 0,
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Then
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where t 0 t < b 1 , W , W −1 are the same functions as in Lemma 2.2, and
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Corollary 2.8. Let u , λ 1 , λ 2 , λ 3 , λ 4 C (ℝ + ), w C ((0,∞)) and w ( u ) be non-decreasing in u, u w ( u ). Suppose that for some c > 0,
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Then
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where t 0 t < b 1 , W , W −1 are the same functions as in Lemma 2.2, and
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3. Main results
In this section, we investigate asymptotic property for solutions of perturbed nonlinear functional differential systems.
Theorem 3.1. Let the solution x = 0 of ( 1) be EASV. Suppose that the per-turbing term g ( t , y , Ty ) satisfies
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and
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where α > 0, a , b , k , w C (ℝ + ), a , b , k , w L 1 (ℝ + ), w ( u ) is nondecreasing in u, and u w ( u ). If
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where t t 0 and c = | y 0 | Me αt0 , then all solutions of (2) approach zero as t → ∞.
Proof . 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. Since the solution x = 0 of (1) is EASV, it is EAS by remark 2.1.
Using Lemma 2.1, (12), and (13), we have
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Then, we obtain
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since w is nondecreasing. Set u ( t ) = | y ( t )| eαt . By Lemma 2.3 and (14) we have
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where c = M | y 0 | e αt0 . The above estimation yields the desired result. □
Remark 3.1. Letting b ( t ) = 0 in Theorem 3.1, we obtain the similar result as that of Theorem 3.5 in [4] .
Theorem 3.2. Let the solution x = 0 of (1) be EASV. Suppose that the per-turbing term g ( t , y , Ty ) satisfies
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and
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where α > 0, a , b , c , k , w C (ℝ + ), a , b , c , k , w L 1 (ℝ + ), w ( u ) is nondecreasing in u , u w ( u ). If
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where t t 0 and c = | y 0 | Me αt0 , then all solutions of (2) approach zero as t → ∞.
Proof . 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. Since the solution x = 0 of (1) is EASV, it is EAS by Remark 2.1. Applying Lemma 2.1, (15), and (16), we have
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Since w is nondecreasing, we obtain
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Set u ( t ) = | y ( t )| eαt . By Corollary 2.5 and (17), we have
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where c = M | y 0 | e αt0 . From the above estimation, we obtain the desired result. □
Remark 3.2. Letting w ( u ) = u , b ( t ) = c ( t ) = 0 in Theorem 3.2, we obtain the similar result as that of Corollary 3.6 in [4] .
Theorem 3.3. Let the solution x = 0 of (1) be EASV. Suppose that the perturbed term g ( t , y , Ty ) satisfies
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and
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where α > 0, a , b , k , w C (ℝ + ), a , b , k , w L 1 (ℝ + ) and w ( u ) is nondecreasing in u, u w ( u ). If
PPT Slide
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where b 1 = ∞ and c = M | y 0 | e αt0 , then all solutions of (2) approach zero as t → ∞.
Proof . 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. Since the solution x = 0 of (1) is EASV, it is EAS. By conditions, Lemma 2.1, (18), and (19), we have
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Since w is nondecreasing, we have
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Then, it follows from Corollary 2.7 with u ( t ) = | y ( t )| eαt and (20) that
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where c = M | y 0 | e αt0 . Hence, all solutions of (2) approach zero as t → ∞ , and so the proof is complete. □
Remark 3.3. Letting b ( t ) = 0 in Theorem 3.3, we obtain the similar result as that of Theorem 3.7 in [4] .
Theorem 3.4. Let the solution x = 0 of (1) be EASV. Suppose that the perturbed term g ( t , y , Ty ) satisfies
PPT Slide
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and
PPT Slide
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where α > 0, a , b , c , k , w C (ℝ + ), a , b , c , k , w L 1 (ℝ + ) and w ( u ) is nonde-creasing in u, u w ( u ). If
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where b 1 = ∞ and c = M | y 0 | e αt0 , then all solutions of (2) approach zero as t → ∞.
Proof . 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. Since the solution x = 0 of (1) is EASV, it is EAS. By means of Lemma 2.1, (21), and (22), we have
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Since w is nondecreasing, we have
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Set u ( t ) = | y ( t )| eαt . Then, it follows from Corollary 2.8 and (23) that
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where c = M | y 0 | e αt0 . From the above inequality, we obtain the desired result. □
Remark 3.4. Letting w ( u ) = u , b ( t ) = c ( t ) = 0 in Theorem 3.4, we obtain the similar result as that of Corolary 3.8 in [4] .
Acknowledgements
The authors are very grateful for the referee’s valuable comments.
BIO
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, 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 topologival dynamical systems and differential equations.
Department of Mathematics, Hanseo University, Seosan 356-706, Korea.
e-mail: yhgoo@hanseo.ac.kr
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