Explicit Conditions for Stability of Nonlinear Scalar Delay Impulsive Difference Equation
© Bo Zheng. 2010
Received: 20 March 2010
Accepted: 2 June 2010
Published: 27 June 2010
Sufficient conditions are obtained for the uniform stability and global attractivity of the zero solution of nonlinear scalar delay impulsive difference equation, which extend and improve the known results in the literature. An example is also worked out to verify that the global attractivity condition is a sharp condition.
1. Introduction and Main Results
Let , , and be the sets of real numbers, natural numbers, and integers, respectively. For any , define and when .
It is well known that the theory of impulsive differential equations is emerging as an important area of investigation, since it is not only richer than the corresponding theory of differential equations without impulse effects but also represents a more natural framework for mathematical modeling of many world phenomena . Moreover, such equations may exhibit several real-world phenomena, such as rhythmical beating, merging of solutions, and noncontinuity of solutions. And hence ordinary differential equations and delay differential equations with impulses have been considered by many authors, and numerous papers have been published on this class of equations and good results were obtained (see, e.g., [1–10] and references therein).
We assume that and , so that is a solution of (1.1), which we call the zero solution.
The zero solution of (1.1) is stable, if, for any and , there exists a such that implies that for . If is independent of , we say that the zero solution of (1.1) is uniformly stable.
The zero solution of (1.1) is globally attractive, if every solution of (1.1) tends to zero as .
where , and is a sequence of real numbers. In , the author studied the stability of the zero solution of (1.5), where for and , and obtained the following result.
then the zero solution of (1.5) is stable.
The main purpose of this paper is to establish the following theorems.
Then the zero solution of (1.1) is uniformly stable.
Theorem 1.4 generalizes and improves Theorem 1.3 greatly.
The next theorem provides a sufficient condition for every solution of (1.1) tends to zero as , that is, the zero solution of (1.1) is globally attractive.
Then every solution of (1.1) tends to zero as .
An example is worked out in Section 3 to verify that the upper bound in (1.10) is best possible, that is, the upper bound in (1.10) cannot be improved.
where , and for , . Then for any , (1.7) holds with . As a consequence of Theorem 1.4, we have the following.
is uniformly stable.
2. Proofs of Main Results
To prove Theorems 1.4 and 1.6, we need the following lemma.
We assume that and . The case where and is similar and is omitted.
It is easy to see that Lemma 2.1 holds for .
which implies that , so by the definition of .
for , where .
which shows that (2.17) holds.
There are two cases to consider.
Case 1 ( ).
Case 2 ( ).
The proof is completed by combining Cases 1 and 2.
Proof of Theorem 1.4.
In fact, we can assume that . The case where is similar and is omitted. If , by the definition of , (2.40) holds. If , then by Lemma 2.1 we have that (2.40) holds.
By repeatedly using (2.40) we get that (2.39) holds.
Combining (2.30) and (2.39), we find that (2.29) holds and the proof of Theorem 1.4 is complete.
Proof of Theorem 1.6.
There are two cases to consider.
is eventually positive, that is, there exists such that for all . Hence, by (1.7) and (1.8), we have for . That is, is eventually nonincreasing and hence . Thus, by (1.11) we see that (2.44) holds. The case when is eventually negative is similar and will be omitted.
In fact, if (2.48) does not hold, we assume that and . The case where and is similar and is omitted.
Equations (2.49) and (2.51) imply that , which contradicts the fact that ; thus (2.48) holds and the proof is complete.
In this section we will give a result which guarantees that the upper bound is best possible in Theorem 1.6.
then every solution of (3.1) tends to zero as .
The following theorem shows that if (3.4) does not hold, then there is a solution of (3.1) which does not tend to zero as . This shows that the upper bound cannot be improved.
Then there exists a solution of (3.1) which does not tend to zero as .
which implies that is a non-decreasing sequence and does not tend to zero as .
By the definition of , we know that as , so as . The proof is complete.
The author would like to express her thanks to the referees for helpful suggestions. This research is supported by Guangdong College Yumiao Project (2009).
- Lakshmikantham V, Baĭnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.View ArticleGoogle Scholar
- Hale J: Theory of Functional Differential Equations, Applied Mathematical Sciences. Volume 3. 2nd edition. Springer, New York, NY, USA; 1977:x+365.View ArticleGoogle Scholar
- Nieto JJ, Tisdell CC: On exact controllability of first-order impulsive differential equations. Advances in Difference Equations 2010, 2010:-9.Google Scholar
- Lakshmikantham V, Liu X, Sathananthan S: Impulsive integro-differential equations and extension of Lyapunov's method. Applicable Analysis 1989,32(3-4):203-214.MathSciNetView ArticleMATHGoogle Scholar
- Xing Y, Wang Q, Chen D: Antiperiodic boundary value problem for second-order impulsive differential equations on time scales. Advances in Difference Equations 2009, 2009:-14.Google Scholar
- Yu JS: Stability for nonlinear delay differential equations of unstable type under impulsive perturbations. Applied Mathematics Letters 2001,14(7):849-857. 10.1016/S0893-9659(01)00055-6MathSciNetView ArticleMATHGoogle Scholar
- Yu JS: Explicit conditions for stability of nonlinear scalar delay differential equations with impulses. Nonlinear Analysis: Theory, Methods & Applications 2001,46(1):53-67. 10.1016/S0362-546X(99)00445-9MathSciNetView ArticleMATHGoogle Scholar
- Xing Y, Han M: A new approach to stability of impulsive functional differential equations. Applied Mathematics and Computation 2004,151(3):835-847. 10.1016/S0096-3003(03)00540-XMathSciNetView ArticleMATHGoogle Scholar
- Yan J: Stability for impulsive delay differential equations. Nonlinear Analysis: Theory, Methods & Applications 2005,63(1):66-80. 10.1016/j.na.2005.05.001MathSciNetView ArticleMATHGoogle Scholar
- Yan J: Global attractivity for impulsive population dynamics with delay arguments. Nonlinear Analysis: Theory, Methods & Applications 2009,71(11):5417-5426. 10.1016/j.na.2009.04.030MathSciNetView ArticleMATHGoogle Scholar
- May RM: Simple mathematical models with very complicated dynamics. Nature 1976,261(5560):459-467. 10.1038/261459a0View ArticleGoogle Scholar
- Mohamad S, Gopalsamy K: Dynamics of a class of discrete-time neural networks and their continuous-time counterparts. Mathematics and Computers in Simulation 2000,53(1-2):1-39. 10.1016/S0378-4754(00)00168-3MathSciNetView ArticleGoogle Scholar
- Mohamad S, Gopalsamy K: Exponential stability of continuous-time and discrete-time cellular neural networks with delays. Applied Mathematics and Computation 2003,135(1):17-38. 10.1016/S0096-3003(01)00299-5MathSciNetView ArticleMATHGoogle Scholar
- Abdullin RZ: The comparison method in the stability of nonlinear difference equations with impulse actions. Automation and Remote Control 2000,61(11, part 1):1796-1807.MathSciNetMATHGoogle Scholar
- Zhang Q: On a linear delay difference equation with impulses. Annals of Differential Equations 2002,18(2):197-204.MathSciNetMATHGoogle Scholar
- Peng M: Oscillation theorems of second-order nonlinear neutral delay difference equations with impulses. Computers & Mathematics with Applications 2002,44(5-6):741-748. 10.1016/S0898-1221(02)00187-6MathSciNetView ArticleMATHGoogle Scholar
- He Z, Zhang X: Monotone iterative technique for first order impulsive difference equations with periodic boundary conditions. Applied Mathematics and Computation 2004,156(3):605-620. 10.1016/j.amc.2003.08.013MathSciNetView ArticleMATHGoogle Scholar
- Geng F, Xu Y, Zhu D: Periodic boundary value problems for first-order impulsive dynamic equations on time scales. Nonlinear Analysis: Theory, Methods & Applications 2008,69(11):4074-4087. 10.1016/j.na.2007.10.038MathSciNetView ArticleMATHGoogle Scholar
- Zhang Y, Sun J, Feng G: Impulsive control of discrete systems with time delay. IEEE Transactions on Automatic Control 2009,54(4):871-875.MathSciNetView ArticleGoogle Scholar
- Zhang QQ, Zhou Z: Stability for difference equations of unstable type under impulsive perturbations. Hunan Daxue Xuebao 2002,29(3):4-9.MathSciNetGoogle Scholar
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