A set of global bounded solutions for a Volterra system of retarded equations on $ {\Bbb R}^3_+ $

Abstract

It is proved the existence of a compact set \( {\cal K} \), invariant under the flow of a Volterra system of retarded equations on \( {\Bbb R}^3_+ \), with lag r > 0; \({\cal K}\) is homeomorphic to a solid tri-dimensional cylinder. The boundary \(\partial {\cal K}\) of \({\cal K}\) is the union of a closed bi-dimensional cylinder \({\cal C ({\cal K})}\) with two open disks (the two basis of the cylinder \({\cal K}\)). \({\cal C ({\cal K})}\) is the union of a continuous one-parameter family of r-periodic orbits of the retarded Volterra system and any r-periodic orbit of the retarded system is contained in \({\cal K}\). The flow, restricted to \({\cal K}\), of the system of retarded equations, is the flow of a C1-vector-field.