Development of the new approach for existence of bounded solutions for point-type functional differential equations

This paper is a continuations of the studies of [1] and [2] on existence of periodic bounded solutions for point-type functional differential equations, where deviations of the argument are defined in terms of a cyclic group of shifts on the real line. We prove an existence theorem for a bounded solution for equations in which the deviations of the argument are given by elements of a finitely generated group of orientation preserving diffeomorphisms of the real line.