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: (1) d x ( t ) d t = A ( t ) x ( t ) + f ( t , x ( θ β ( t ) − p 1 ) , x ( θ β ( t ) − p 2 ) , … , x ( θ β ( t ) − p m ) ) , where x ∈ R n , t ∈ R , β ( t ) = i if θ i ≤ t < θ i + 1 , i = … − 2 , − 1 , 0 , 1 , 2 , … , is an identification function, θ 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.