Existence and stability of almost-periodic solutions of quasi-linear differential equations with deviating argument

3(dx,t____~ =A( t ) x ( t )+ f ( t , x ( t ) ) , x ( t r l ( t ) ) , . . . , x ( t f k (t)), (1) dt such that the associated homogeneous linear system satisfies exponential dichotomy and deviations of the argument axe not restricted by any sign assumption. The exponential stability of the solution when the system is with delay argument is considered. (~) 2004 Elsevier Ltd. All rights reserved. Keywords--Quasi-Lineax system, Almost-periodic solutions, Deviating argument, Stability. 1. I N T R O D U C T I O N A N D P R E L I M I N A R I E S Different aspects of the theory of Mmost-periodic solutions of quasi-linear differential equations with deviating argument, including applications, have been investigated by many authors [1-6]. The assumption tha t the associated linear equation has exponential dichotomy is one of the most impor tan t for the discussion. In [2], the problem was considered for functional differential equation when a matr ix of coefficients is constant or periodic with respect to t ime variable t, and the argument is delay. Paper [3] deals with the existence of almost-periodic solutions of a sys tem with unique and constant deviation. The aim of the present paper is to investigate the problem for system (1), where deviations and the mat r ix of coefficients are almost-periodic functions. Moreover, we assume tha t the equation is of mixed type [}7], tha t is, the derivative of x depends upon past as well as future values. One should emphasize tha t the general theory has not been considered for this type of equation as well as for equations with retarded argument or for systems of the neutral type [2,8]. Let N, ]~ be sets of all natural and real numbers, respectively, and I1" ]1 be the Euclidean norm in ] ~ , n E N. Let s C ]~ be a positive number. We denote Gs = {x E R~llxl] <_ s} and G~ +1 = M. U. Akhmet is previously known as M. U. Akhmetov. The author thanks anonymous referees for helpful criticism. 0893-9659/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. Typeset by ~4AAS-TEX doi:10.1016/j.aml.2003.08.012