Monotone skew-product semiflows for Carath\'{e}odory differential equations and applications

TL;DR

This paper investigates the continuous dependence and long-term behavior of solutions to Carathéodory differential equations with delay, establishing conditions for equilibria and attractors, and applying these results to specific examples.

Contribution

It introduces new results on the continuity of skew-product semiflows and the existence of equilibria for Carathéodory delay differential equations under cooperative and sublinear conditions.

Findings

Continuity of solutions with respect to initial data and vector fields.

Existence of a unique continuous equilibrium under sublinearity.

Construction of semicontinuous semiequilibria and attractors.

Abstract

The first part of the paper is devoted to studying the continuous dependence of the solutions of Carath\'eodory constant delay differential equations where the vector fields satisfy classical cooperative conditions. As a consequence, when the set of considered vector fields is invariant with respect to the time-translation map, the continuity of the respective induced skew-product semiflows is obtained. These results are important for the study of the long-term behavior of the trajectories. In particular, the construction of semicontinuous semiequilibria and equilibria is extended to the context of ordinary and delay Carath\'eodory differential equations. Under appropriate assumptions of sublinearity, the existence of a unique continuous equilibrium, whose graph coincides with the pullback attractor for the evolution processes, is shown. The conditions under which such a solution is the forward attractor of the considered problem are outlined. Two examples of application of the developed tools are also provided.