Cone uniform, cone locally convex and cone metric spaces, endpoints, set-valued dynamic systems and quasi-asymptotic contractions

The concepts of cone pseudometrics, cone uniform spaces generated by these pseudometrics, cone pseudodistances, set-valued quasi-asymptotic contractions with respect to these pseudodistances and cone closed maps are introduced and studied. Conditions guaranteeing the existence and uniqueness of endpoints (stationary points) of these contractions and conditions ensuring that for each starting point each generalized sequence of iterations of these contractions (in particular, each dynamic process) converges and then the limit is an endpoint are all established. Also the concept of the cone locally convex space as a special case of the cone uniform space is introduced and examples of quasi-asymptotic contractions in cone metric spaces are constructed. The definitions, results, ideas and methods are new for set-valued dynamic systems in cone uniform, cone locally convex and cone metric spaces and even for single-valued maps in these spaces.