Almost periodic solutions of differential equations with piecewise constant argument of generalized type

We consider the existence and stability of an almost periodic solution of the following hybrid system: dx(t)/dt = A(t)x(t) + f(t, x (theta(beta(t)-p1)), x(theta(beta(t)-p2)), ... , x(theta(beta(t)-pm))), where x is an element of R-n, t is an element of R, beta(t) = i if theta(i) <= t < theta(i+1), i = ... - 2, -1, 0, 1, 2, ... , is an identification function, theta(i) is a strictly ordered sequence of real numbers, unbounded on the left and on the right, p(j), j = 1, 2, ... , m, are fixed integers, and the linear homogeneous system associated with (1) satisfies exponential dichotomy. The deviations of the argument are not restricted by any sign assumption when existence is considered. A new technique of investigation of equations with piecewise argument, based on integral representation, is developed. (c) 2008 Published by Elsevier Ltd