Asymptotically non-expansive self-maps and global stability with ultimate boundedness of dynamic systems

This paper investigates self-maps T : X → X which satisfy a distance constraint in a metric space with 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 which guarantees 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 be applied to stability problems of dynamics systems and circuit theory as the given examples suggest.