# Solvability of the functional integro-differential equation with self-reference and state-dependence

Volume 13, Issue 1, pp 1--8
Publication Date: August 11, 2019 Submission Date: March 08, 2019 Revision Date: July 17, 2019 Accteptance Date: July 23, 2019
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### Authors

A. M. A. El-Sayed - Faculty of Science, Alexandria University, Egypt. Reda Gamal Aahmed - Faculty of Science, Al-Azhar University, Cairo, Egypt.

### Abstract

The existence of solutions of a functional integro-differential equation with self-reference and state-dependence will be studied. The continuous dependence of the solution on the delay $\phi(t)$, the functional $g$ and initial data $x_0$ will be proved.

### Keywords

• Functional equations
• existence of solutions
• continuous dependence
• state-dependence
• self-reference

•  34A12
•  34A30
•  34D20

### References

• [1] C. Bacoţiu, Volterra--fredholm nonlinear systems with modified argument via weakly picard operators theory, Carpathian J. Math., 24 (2008), 1--9

• [2] M. Benchohra, M. A. Darwish, On unique solvability of quadratic integral equations with linear modification of the argument, Miskolc Math. Notes, 10 (2009), 3--10

• [3] V. Berinde, Existence and approximation of solutions of some first order iterative differential equations, Miskolc Math. Notes, 11 (2010), 13--26

• [4] A. Buica, Existence and continuous dependence of solutions of some functional-differential equations, Seminar on Fixed Point Theory (Babes-Bolyai Univ., Cluj-Napoca), 3 (1995), 1--14

• [5] E. Eder, The functional differential equation $x'(t)=x(x(t))$, J. Differential Equations, 54 (1984), 390--400

• [6] M. Fečkan, On a certain type of functional differential equations, Math. Slovaca, 43 (1993), 39--43

• [7] K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambirdge Universty Press, Cambirdge (1990)

• [8] A. N. Kolomogorov, S. V. Fomin, Inroductory real analysis, Dover Publications, New York (1975)

• [9] S. Staněk, Global properties of decreasing solutions of the equation $x'(t)=x(x(t))+x(t)$, Funct. Differ. Equ., 4 (1997), 191--213

• [10] S. Staněk, Globel properties of solutions of the functional differenatial equation $x(t)x'(t)=kx(x(t)),~ 0 <|k|< 1$, Funct. Differ. Equ., 9 (2004), 527--550

• [11] K. Wang, On the equation $x'(t)=f(x(x(t)))$, Funkcial. Ekvac., 33 (1990), 405--425

• [12] P. P. Zhang, X. B. Gong, Existence of solutions for iterative differential equations, Electron. J. Differential Equations, 2014 (2014), 10 pages