On Asymptotically Non-Expansive Self-Maps in Metric Spaces and Related Stability of Dynamic Systems

Abstract

This paper investigates self-maps which satisfy a distance constraint in a metric space which mixed point-dependent non-expansive properties, or in particular contractive ones, and potentially expansive properties related to some distance threshold. The above mentioned constraint is feasible in certain real -world problems of usefulness, for instance, when discussing ultimate boundedness in dynamic systems guaranteeing Lyapunov stability. This fact makes the proposed analysis to be potentially useful to investigate global stability properties in dynamic systems in the potential presence of some locally unstable equilibrium points. The results can potentially be applied to stability problems of dynamics systems and circuit theory as the given examples suggest