Dear Ashraf:

I suggest some interesting books for time delay such as:

Hale, J. K. & Lunel, S. M. V. Introduction to Functional Differential Equations

Springer-Verlag, 1993, 447

Smith, H. L. An introduction to delay differential equations with applications to the life sciences Springer, 2010, 57

Lakshmanan, M., & Senthilkumar, D. V. (2011). Dynamics of nonlinear time-delay systems. Springer Science & Business Media.‏ 

Dear Ayman thank you very much, also I found another useful book suitable for teaching:

Introduction to Time-Delay Systems: Analysis and Control

Authors: Fridman, Emilia

http://www.springer.com/us/book/9783319093925

As a co-author (with H. Gorecki, A. Korytowski and S. Fuksa) I may recommend: Analysis and Synthesis of Time-Dalay Systems. Wiley. Chichester. 1989

Thank you Dear Khan, yes these two books are one of the best introductory books for the problem, especially Emilia's book.

Best Regards

Ashraf

 Dear Piotr,

Thank you very much from the title of the book, it is very interesting. I will read it and give you my feedback.

Best Regards

Ashraf

Dear Gabriel,

Thank you very much for the valuable files, I really appreciate your assistance.

Best Regards

Ashraf

While old, I suggest these 4 books useful to your query: 

1. R. Bellman and K. Cooke, Differential Difference Equations, Academic Press 1963

2. A. Halanay, Differential Equations. Stability. Oscillations. Time Lags, Academic Press, 1966.

3. L. E. El'sgol'ts, Introduction to the theory of differential equations with deviated argument, Academic Press, 1973.

4. V. B. Kolmanovskii and A.D. Myshkis, Introduction to the theory and Applications of Functional Differential Equations, Kluwer Academic (Springer, 1999).

Dear Prof. Vladimir, thank you very much for the valuable references. I think they are the best sources for the literature and cover the differential difference equations in general .

Best Regards

Dear Dr. Corentin, thank you very much for sharing your book, it looks very interesting.

Best Regards

Ashraf

Dear Ashraf

There are lots of papers abd books of Prof. U. Shaked and Prof. E. Fridman in the field of time delay systems. Also, please see:

Introduction to Time-Delay Systems, Analysis, and Control

Authors: Fridman, Emilia

Time-Delay Systems, Lyapunov Functionals, and Matrices

Authors: Vladimir L. Kharitonov

Regards,

Hamid