Solvability and asymptotic stability of a class of nonlinear functional­integral equation with feedback control

Document Type: Original Article

Author

Department of Mathematics and Computer Science, Amirkabir University of Technology (Polytechnic), Hafez Ave., P. O. Box 15914, Tehran, Iran.

Abstract

Using the technique of measure of noncompactness we prove the existence, asymptotic stability and global
attractivity of a class of nonlinear functional-integral equation with feedback control. We will also include
a class of examples in order to indicate the validity of the assumptions.

Keywords


[1] I. K. Aregon, Quadratic equations and applications to Chandrasekhar's and related equations, Bull. Austral. Math. Soc.,
32 (1985), 275-292.

[2] J. Banas, Measure of noncompactness in the space of continuous temperate functions, Demonstratio Math., 14 (1981), 127-133.

[3] J. Banas, B. C. Dhage, Global asymptotic stability of solutions of a functional integral equation, Nonlinear Anal., 69 (2008), 1945-1952.

[4] J. Banas, K. Goebel, Measure of noncompactness in the Banach space. Lecture Notes in Pure and Applied Mathematics, vol. 60. New York:Dekker, (1980).

[5] J. Banas, B. Rzepka, On existance and asymptotic stability of solutions of a nonlinear integral equation, J. Math. Anal. Appl., 284 (2003), 165-173.

[6] J. Banas, B. Rzepka, An application of a measure of noncompactness in the study of asymptotic stability, App. Math. Lett., 16 (2003), 1-6.

[7] J. Banas, B. Rzepka, On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation,
Appl. Math. Comput., 213 (2009), 102-111.

[8] J. Banas, D. O'Regan, On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order, J. Math. Anal. Appl., 34 (2008), 573-582.

[9] F. Chen, The permanence and global attractivity of Lotka-Volterra competition system with feedback control, Nonlinear
Anal., 7 (2006), 133-143.

[10] K. Deimling, Nonlinear Functional analysis, Springer-Verlag, Berlin, (1985). 1

[11] Z. Liu, S. M. Kang, J. S. Ume, Solvability and asymptotic stability of a nonlinear functional-integral equation, App. Math. Lett., 24 (2011), 911-917.