One of the most important ways to investigate the asymptotic structure of two equa- tions is to build asymptotic equivalence between their solutions. Establishing such equiv- alence enables researchers to comment on the asymptotic behavior of solutions of certain nonlinear equations by studying another linear equation whose solutions bear the same character.

## Theory Of Impulsive Differential Equations

The proof of the above statement is straightforward and can be achieved by direct sub- stitution. If Eq.

Let C be a nonempty subset of X. We now have the following lemma. We claim that Eqs.

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Competing interests The authors declare that they have no competing interests. Box , Riyadh, , Saudi Arabia.

Acknowledgements The authors would like to express their sincere thanks to the editor, Professor Dr. Sandra Pinelas, for handling our paper during the reviewing process and to the referees for suggesting some corrections that helped make the contents of the paper more accurate.

Ahmad, S, Stamov, G: On the almost periodic processes in impulsive competitive systems with delay and impulsive perturbations.

- Nonlinear impulsive problems for fractional differential equations and Ulam stability;
- A Survey on Oscillation of Impulsive Ordinary Differential Equations!
- Impulsive differential equations with a small parameter.
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Nonlinear Anal. Allegretto, W, Papini, D, Forti, M: Common asymptotic behavior of solutions and almost periodicity for discontinuous delayed and impulsive networks. IEEE Trans.

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## Impulsive Differential Equations: Periodic Solutions and Applications - CRC Press Book

Zou, L, Xiong, Z, Shu, Z: The dynamics of an eco-epidemic model with distributed time delay and impulsive control strategy. Li, X, Rakkiyappan, R, Balasubramaniam, P: Existence and global stability analysis of equilibrium of fuzzy cellular neural networks with time delay in the leakage term under impulsive perturbations. Plovdiv, Fundam.

A, Pure Appl. Moscow in Russian Pazy, A: Semigroups of Linear Operators. Springer, Berlin B 32, In Chapter Three, singularly perturbed differential equations are considered. A modification of the method of boundary functions is proposed, and asymptotic expansions along the powers of the small parameters of the solutions of the initial value problem, the periodic problem, and some boundary value problems are found. Numerous nonstandard applications to the theory of optimal control are made. The application of some other methods to impulsive singularly perturbed equations is illustrated, such as the numerical-analytical method for finding periodic solutions, the method of differential inequalities and the averaging method.

### Impulsive Hybrid Discrete-Continuous Delay Differential Equations

The book is written clearly, strictly, and understandably. It is intended for mathematicians, physicists, chemists, biologists and economists, as well as for senior students of these specialities. Integral Inequalities for Piecewise Continuous Functions. Regularly perturbed impulsive. Partially Additive Averaging for Impulsive Differential. Periodic and Almost Periodic Solutions of Regularly. Existence of Integral Sets Manifolds and Behaviour.