The Existence and Application of Unbounded Connected Components

Abstract

LetXbe a Banach space andCna family of connected subsets ofR×X. We prove the existence of unbounded components in superior limit of{Cn}, denoted bylim¯ Cn, which have prescribed shapes. As applications, we investigate the global behavior of the set of positive periodic solutions to nonlinear first-order differential equations with delay, which can be used for modeling physiological processes.