Integral manifolds of differential equations with piecewise constant argument of generalized type (Q858659)

Equations with piecewise constant delayed argument are studied, like \[ \dot x(t) = f(t, x(t), x([t])). \] One main assumption is that there is a linear part \(\dot x(t) = A(t) x(t)\) of the equation which has an exponential dichotomy. Manifolds of solutions converging to zero in forward/backward time are constructed using a Perron-type approach. Existence and uniqueness of bounded/periodic solutions is obtained (as a consequence of the exponential dichotomy). The author uses successive approximations instead of the contraction theorem.