Existence of solutions for a mixed type differential equation with state-dependence

In this paper we study a general n-dimensional mixed type differential equation with state dependence. Many known works give the existence of solutions with the so-called Return Condition. In this paper, without requiring the Return Condition, we prove a non-local existence of solutions by using a technique of compactification and the dependence of the maximum of functions on the size of their domains. We apply our result to differential equations with iterates.