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LIPSCHITZ AND ASYMPTOTIC STABILITY OF PERTURBED FUNCTIONAL DIFFERENTIAL SYSTEMS
LIPSCHITZ AND ASYMPTOTIC STABILITY OF PERTURBED FUNCTIONAL DIFFERENTIAL SYSTEMS
The Pure and Applied Mathematics. 2015. Feb, 22(1): 1-11
Copyright © 2015, Korean Society of Mathematical Education
  • Received : July 10, 2014
  • Accepted : November 13, 2014
  • Published : February 28, 2015
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About the Authors
SANG IL CHOI
DEPARTMENT OF MATHEMATICS, HANSEO UNIVERSITY, SEOSAN, CHUNGNAM, 356-706, REPUBLIC OF KOREAEmail address:schoi@hanseo.ac.kr
YOON HOE GOO
DEPARTMENT OF MATHEMATICS, HANSEO UNIVERSITY, SEOSAN, CHUNGNAM, 356-706, REPUBLIC OF KOREAEmail address:yhgoo@hanseo.ac.kr

Abstract
The present paper is concerned with the notions of Lipschitz and asymptotic for perturbed functional differential system knowing the corresponding stability of functional differential system. We investigate Lipschitz and asymptotic stability for perturbed functional differential systems. The main tool used is integral inequalities of the Bihari-type, and all that sort of things.
Keywords
1. INTRODUCTION
Dannan and Elaydi introduced a new notion of uniformly Lipschitz stability (ULS) [8] . 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 [9] 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 [10] 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 et al. [11 , 13] investigated Lipschitz and asymptotic stability for perturbed differential systems.
In this paper, we investigate Lipschitz and asymptotic stability for solutions of the functional differential systems using 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
PPT Slide
Lager Image
where
PPT Slide
Lager Image
and
PPT Slide
Lager Image
is the Euclidean n -space. We assume that the Jacobian matrix fx = ∂f / ∂x exists and is continuous on
PPT Slide
Lager Image
and f ( t , 0) = 0. Also, consider the perturbed functional differential system of (2.1)
PPT Slide
Lager Image
where
PPT Slide
Lager Image
, g ( t , 0, 0) = h ( t , 0, 0) = 0 and
PPT Slide
Lager Image
is a continuous operator .
For
PPT Slide
Lager Image
. 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 (2.1) 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 (2.1) and around x ( t ), respectively,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
The fundamental matrix Φ( t , t 0 , x 0 ) of (2.4) is given by
PPT Slide
Lager Image
and Φ( t , t 0 , 0) is the fundamental matrix of (2.3).
Before giving further details, we give some of the main definitions that we need in the sequel [8] .
Definition 2.1. The system (2.1) (the zero solution x = 0 of (2.1)) 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 |Φ( 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
  • |x(t)| ≤K|x0|e−c(t-t0), 0 ≤t0≤t
provided that | x 0 | < δ ,
(EASV) exponentially asymptotically stable in variation if there exist constants K > 0 and c > 0 such that
  • |Φ(t,t0,x0)| ≤Ke−c(t−t0), 0 ≤t0≤t
provided that | x 0 | < ∞.
Remark 2.2 ( [10] ). The last deffnition implies that for | x 0 | ≤ δ
  • |x(t)| ≤K|x0|e−c(t−t0), 0 ≤t0≤t
We give some related properties that we need in the sequel.
We need Alekseev formula to compare between the solutions of (2.1) and the solutions of perturbed nonlinear system
PPT Slide
Lager Image
where
PPT Slide
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and g ( t , 0) = 0. Let y ( t ) = y ( t , t 0 , y 0 ) denote the solution of (2.5) passing through the point ( t 0 , y 0 ) in
PPT Slide
Lager Image
.
The following is a generalization to nonlinear system of the variation of constants formula due to Alekseev [1] .
Lemma 2.3. Let x and y be a solution of (2.1) and (2.5), respectively . If
PPT Slide
Lager Image
then for all t such that
PPT Slide
Lager Image
,
PPT Slide
Lager Image
where Φ( t , s , y ( s )) is a fundamental matrix of (2.4) .
Lemma 2.4 ( [14] ). Let
PPT Slide
Lager Image
and suppose that, for some c ≥ 0, we have
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
Lemma 2.5 ( [7] ). (Bihari – type Inequality) Let
PPT Slide
Lager Image
, 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
PPT Slide
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is the inverse of W ( u ), and
PPT Slide
Lager Image
Lemma 2.6 ( [12] ). Let
PPT Slide
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, 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 W , W −1 are the same functions as in Lemma 2.5, and
PPT Slide
Lager Image
Lemma 2.7 ( [12] ). Let
PPT Slide
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, 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 W , W −1 are the same functions as in Lemma 2.5, and
PPT Slide
Lager Image
Lemma 2.8 ( [5] ). Let
PPT Slide
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, 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 Lipschitz and asymptotic stability for solutions of the perturbed functional differential systems.
We need the lemma to prove the following theorem.
Lemma 3.1. Let
PPT Slide
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, 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 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 2.5, and
PPT Slide
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Proof . Define a function v ( t ) by the right member of (3.1) . 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, (3.3) yields the estimate (3.2).
PPT Slide
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Theorem 3.2. For the perturbed (2.2), we assume that
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where
PPT Slide
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, w C ((0, ∞)), and w ( u ) is nondecreasing in u , u w ( u ), and
PPT Slide
Lager Image
for some v > 0,
PPT Slide
Lager Image
where M ( t 0 ) < ∞ and b 1 = ∞. Then the zero solution of (2.2) is ULS whenever the zero solution of (2.1) is ULSV .
Proof . Using the nonlinear variation of constants formula of Alekseev [1] , the solu- tions of (2.1) and (2.2) with the same initial value are related by
PPT Slide
Lager Image
Since x = 0 of (2.1) is ULSV, it is ULS( [8] ,Theorem 3.3) . Using the ULSV condition of x = 0 of (2.1), (3.4), and (3.5), we have
PPT Slide
Lager Image
Set u ( t ) = | y ( t )|| y 0 | −1 . Now an application of Lemma 3.1 yields
PPT Slide
Lager Image
Thus, by (3.6), we have | y ( t )| ≤ M ( t 0 )| y 0 | for some M ( t 0 ) > 0 whenever | y 0 | < δ . So, the proof is complete.
PPT Slide
Lager Image
Remark 3.3. Letting c ( t ) = 0 in Theorem 3.2, we obtain the same result as that of Theorem 3.6 in [13] .
Theorem 3.4. For the perturbed (2.2), we assume that
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where
PPT Slide
Lager Image
, w C ((0, ∞)), and w ( u ) is nondecreasing in u , u w ( u ), and
PPT Slide
Lager Image
for some v > 0,
PPT Slide
Lager Image
where M ( t 0 ) < ∞ and b 1 = ∞. Then the zero solution of (2.2) is ULS whenever the zero solution of (2.1) is ULSV .
Proof . Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) be solutions of (2.1) and (2.2), respectively. Since x = 0 of (2.1) is ULSV, it is ULS . Applying Lemma 2.3, (3.7), and (3.8), we have
PPT Slide
Lager Image
Set u ( t ) = | y ( t )|| y 0 | −1 . Now an application of Lemma 2.6 yields
PPT Slide
Lager Image
Hence, by (3.9), we have | y ( t )| ≤ M ( t 0 )| y 0 | for some M ( t 0 ) > 0 whenever | y 0 | < δ . This completes the proof.
PPT Slide
Lager Image
Remark 3.5. Letting c ( t ) = 0 in Theorem 3.4, we obtain the same result as that of Theorem 3.5 in [13] .
Theorem 3.6. For the perturbed (2.2), we assume that
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where
PPT Slide
Lager Image
, w C ((0, ∞)), and w ( u ) is nondecreasing in u , u w ( u ), and
PPT Slide
Lager Image
for some v > 0,
PPT Slide
Lager Image
where M ( t 0 ) < ∞ and b 1 = ∞. Then the zero solution of (2.2) is ULS whenever the zero solution of (2.1) is ULSV .
Proof . Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y( t , t 0 , y 0 ) be solutions of (2.1) and (2.2), respectively. Since x = 0 of (2.1) is ULSV, it is ULS. Using the nonlinear variation of constants formula and the ULSV condition of x = 0 of (2.1), (3.10), and (3.11), we have
PPT Slide
Lager Image
Set u ( t ) = | y ( t )|| y 0 | −1 . Now an application of Lemma 2.7 and (3.12) yield
PPT Slide
Lager Image
Thus 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.
PPT Slide
Lager Image
Remark 3.7. Letting c ( t ) = 0 in Theorem 3.6, we obtain the same result as that of Theorem 3.6 in [13] .
Theorem 3.8. Let the solution x = 0 of (2.1) be EASV. Suppose that the perturbing term g(t, y, Ty) satisfies
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where α > 0,
PPT Slide
Lager Image
. If
PPT Slide
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where c = | y 0 | Me αt0 , then all solutions of (2.2) 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 (2.1) and (2.2), respectively. Since the solution x = 0 of (2.1) is EASV, it is EAS by remark 2.2. Using Lemma 2.3, (3.13), and (3.14), we have
PPT Slide
Lager Image
Set u ( t ) = | y ( t )| e αt . An application of Lemma 2.4 and (3.15) obtain
PPT Slide
Lager Image
t t 0 , where c = M | y 0 | e αt0 . Hence, all solutions of (2.2) approch zero as t → ∞.
PPT Slide
Lager Image
Theorem 3.9. Let the solution x = 0 of (2.1) be EASV. Suppose that the perturbed term g ( t , y , Ty ) satisfies
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where α > 0,
PPT Slide
Lager Image
and w ( u ) is nondecreasing in u, and
PPT Slide
Lager Image
for some v > 0. If
PPT Slide
Lager Image
where c = M | y 0 | e αt0 , then all solutions of (2.2) 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 (2.1) and (2.2), respectively. Since the solution x = 0 of (2.1) is EASV, it is EAS. Using Lemma 2.3, (3.16), and (3.17), we have
PPT Slide
Lager Image
Set u ( t ) = | y ( t )| eαt . Since w ( u ) is nondecreasing, an application of Lemma 2.8 and (3.18) obtain
PPT Slide
Lager Image
where c = M | y 0 | e αt0 . Therefore, all solutions of (2.2) approch zero as t → ∞.
PPT Slide
Lager Image
Acknowledgements
The authors are very grateful for the referee’s valuable comments.
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