LIPSCHITZ AND ASYMPTOTIC STABILITY FOR PERTURBED FUNCTIONAL DIFFERENTIAL SYSTEMS
LIPSCHITZ AND ASYMPTOTIC STABILITY FOR PERTURBED FUNCTIONAL DIFFERENTIAL SYSTEMS
Journal of Applied Mathematics & Informatics. 2015. Jan, 33(1_2): 219-228
• Received : June 15, 2014
• Accepted : October 16, 2014
• Published : January 30, 2015
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SANG IL CHOI
YOON HOE GOO

Abstract
In this paper, we investigate Lipschitz and asymptotic stability for perturbed functional differential systems AMS Mathematics Subject Classification : 34D10.
Keywords
1. Introduction
The notion of uniformly Lipschitz stability (ULS) was introduced by Dannan and Elaydi [9] . This notion of ULS lies somewhere between uniformly stability on one side and the notions of asymptotic stability in variation of Brauer [4] and uniformly stability in variation of Brauer and Strauss [3] on the other side. An important feature of ULS is that for linear systems, the notion of uniformly Lipschitz stability and that of uniformly stability are equivalent. However, for nonlinear systems, the two notions are quite distinct. Also, Elaydi and Farran [10] introduced the notion of exponential asymptotic stability(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. Gonzalez and Pinto [11] proved theorems which relate the asymptotic behavior and boundedness of the solutions of nonlinear differential systems.
In this paper, we investigate Lipschitz and asymptotic stability for solutions of the functional differential systems. To do this we need some integral inequalities. 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.
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, consider the perturbed differential system of (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) 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
PPT Slide
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The fundamental matrix Փ( t , t 0 , x 0 ) of (4) is given by
PPT Slide
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and Փ( 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 [9] .
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,
(US) uniformly stable if the δ in (S) is independent of the time t 0 ,
(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
(ULSV) uniformly Lipschitz stable in variation if there exist M > 0 and δ > 0 such that |Փ( t , t 0 , x 0 ) ≤ M for | 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
PPT Slide
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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
PPT Slide
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provided that | x 0 | < ∞.
Remark 2.1 ( [11] ). The last definition implies that for | x 0 | ≤ δ
PPT Slide
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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
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 (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 such that x ( t , t 0 , y 0 ) ∈ ℝ n ,
PPT Slide
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Lemma 2.2 ( [8] ). Let u , λ 1 , λ 1 , w C (ℝ + ), w ( u ) be nondecreasing in u and
PPT Slide
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for some v > 0. If, for some c > 0,
PPT Slide
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then
PPT Slide
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where
PPT Slide
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is the inverse of W ( u ), and
PPT Slide
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Lemma 2.3 ( [15] ). Let u, p, q, w, and r C (ℝ + ) and suppose that, for some c ≥ 0, we have
PPT Slide
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Then
PPT Slide
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Lemma 2.4 ( [13] ). 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
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Then
PPT Slide
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where W , W −1 are the same functions as in Lemma 2.2, and
PPT Slide
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Lemma 2.5 ( [13] ). Let u, p, q, w, r C (ℝ + ), w C ((0, ∞)) and w ( u ) be nondecreasing in u, u w ( u ). Suppose that for some c ≥ 0,
PPT Slide
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Then
PPT Slide
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where W , W −1 are the same functions as in Lemma 2.2, and
PPT Slide
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Lemma 2.6 ( [6] ). 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
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Then
PPT Slide
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where W , W −1 are the same functions as in Lemma 2.2, and
PPT Slide
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Lemma 2.7 ( [5] ). Let u, λ 1 , λ 2 , λ 3 , λ 4 , w C (ℝ + ), w C ((0, ∞)) and w ( u ) be nondecreasing in u, u w ( u ). Suppose that for some c ≥ 0,
PPT Slide
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for t t 0 ≥ 0 and for some c ≥ 0. Then
PPT Slide
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for t 0 t < b 1 , where W, W −1 are the same functions as in Lemma 2.2, and
PPT Slide
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3. Main results
In this section, we investigate Lipschitz and asymptotic stability for solutions of the perturbed functional differential systems.
Theorem 3.1. For the perturbed (2), we asssume that
PPT Slide
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and
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where a, b, c, k C (ℝ + ), a, b, c, k L 1 (ℝ + ), w C ((0, ∞), and w ( u ) is nondecreasing in
PPT Slide
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PPT Slide
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where M ( t 0 ) < ∞ and b 1 = ∞. Then the zero solution of (2) is ULS whenever the zero solution of (1) is ULSV.
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 x = 0 of (1) is ULSV, it is ULS ( [9] ,Theorem 3.3). Applying Lemma 2.1, condition (6), and condition (7), we have
PPT Slide
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Set u ( t ) = | y ( t )|| y 0 | −1 . Now an application of Lemma 2.4 yields
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By condition (8), we have | y ( t )| ≤ M ( t 0 )| y 0 | for some M ( t 0 ) > 0 whenever | y 0 | < δ . This completes the proof. □
Remark 3.1. Letting c ( t ) = 0 in Theorem 3.1, we obtain the same result as that of Theorem 3.6 in [12] .
Theorem 3.2. For the perturbed (2), we asssume that
PPT Slide
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and
PPT Slide
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where a, b, c, k C (ℝ + ), a, b, c, k L 1 (ℝ + ), w C ((0, ∞), and w ( u ) is nondecreasing in
PPT Slide
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PPT Slide
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where M ( t 0 ) < ∞ and b 1 = ∞. Then the zero solution of (2) is ULS whenever the zero solution of (1) is ULSV.
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 x = 0 of (1) is ULSV, it is ULS . Using the nonlinear variation of constants formula , condition (9), and condition (10), we have
PPT Slide
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Set u ( t ) = | y ( t )|| y 0 | −1 . Then, it follows from Lemma 2.7 that
PPT Slide
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By condition (11), we have | y ( t )| ≤ M ( t 0 )| y 0 | for some M ( t 0 ) > 0 whenever | y 0 | < δ , and so the proof is complete. □
Remark 3.2. Letting c ( t ) = 0 in Theorem 3.2, we obtain the same result as that of Theorem 3.7 in [12] .
Theorem 3.3. For the perturbed (2), we asssume that
PPT Slide
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and
PPT Slide
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where a, b, c, k C (ℝ + ), a, b, c, k L 1 (ℝ + ), w C ((0, ∞), and w ( u ) is nondecreasing in
PPT Slide
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PPT Slide
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where M ( t 0 ) < ∞ and b 1 = ∞. Then the zero solution of (2) is ULS whenever the zero solution of (1) is ULSV.
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 x = 0 of (1) is ULSV, it is ULS. Using the nonlinear variation of constants formula, condition (12), and condition (13), we have
PPT Slide
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Set u ( t ) = | y ( t )|| y 0 | −1 . Now an application of Lemma 2.5 yields
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t t 0 . From condition (14) we get | y ( t )| ≤ M ( t 0 )| y 0 | for some M ( t 0 ) > 0 whenever | y 0 | < δ , and so the proof is complete. □
Remark 3.3. Letting c ( s ) = 0 in Theorem 3.3, we obtain the same result as that of Theorem 3.7 in [12] .
Theorem 3.4. Let the solution x = 0 of (1) be EASV. Suppose that the perturbing term g(t, y) satisfies
PPT Slide
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and
PPT Slide
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where α > 0, a, b, c, k C (ℝ + ), a, b, c, k L 1 (ℝ + ), w ( u ) is nondecreasing in u. If
PPT Slide
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where c = | y 0 | Me α t 0 , then all solutions of (2) approch 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, by remark 2.1, it is EAS. Using Lemma 2.1, condition (15), and condition (16), we have
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Set u ( t ) = | y ( t )| e αt . By Lemma 2.3, we obtain
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where c = M | y 0 | e αt 0 . By condition (17), we have | y ( t )| ≤ ce αt M ( t 0 ). This estimation yields the desired result. □
Remark 3.4. Letting c ( s ) = 0 in Theorem 3.4, we obtain the same result as that of Theorem 3.8 in [12] .
Theorem 3.5. Let the solution x = 0 of (1) be EASV. Suppose that the perturbed term g ( t, y ) 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 L 1 (ℝ + ), w ( u ) is nondecreasing in u, and
PPT Slide
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PPT Slide
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where c = M | y 0 | e αt 0 , then all solutions of (2) approch 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, condition (18), and condition (19), we have
PPT Slide
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Set u ( t ) = | y ( t )| e αt . Since w ( u ) is nondecreasing, an application of Lemma 2.6 obtains
PPT Slide
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where c = M | y 0 | e αt 0 . By condition (20), we have | y ( t )| ≤ e αt M ( t 0 ). From this estimation, we obtain the desired result. □
Remark 3.5. Letting c ( t ) = 0 in Theorem 3.5, we obtain the same result as that of Theorem 3.9 in [12] .
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
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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