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
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).
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.
is uniformly stable.
2. Proofs of Main Results
To prove Theorems 1.4 and 1.6, we need the following lemma.
which shows that (2.17) holds.
There are two cases to consider.
The proof is completed by combining Cases 1 and 2.
Proof of Theorem 1.4.
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.
The author would like to express her thanks to the referees for helpful suggestions. This research is supported by Guangdong College Yumiao Project (2009).
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