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: equation(1)dx(t)dt=A(t)x(t)+f(t,x(θβ(t)−p1),x(θβ(t)−p2),…,x(θβ(t)−pm)), where x∈Rn,t∈R,β(t)=ix∈Rn,t∈R,β(t)=i if θi≤t<θi+1,i=…−2,−1,0,1,2,…θi≤t<θi+1,i=…−2,−1,0,1,2,…, is an identification function, θiθi is a strictly ordered sequence of real numbers, unbounded on the left and on the right, pj,j=1,2,…,mpj,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.