Generalized contractions of Meir-Keeler type, endpoints, set-valued dynamical systems and generalized pseudometrics in uniform spaces

This paper gives the concepts of W-families of generalized pseudodistances, Φ-families of generalized gauge maps and set-valued dynamical systems of (W;Φ)-contractions of Meir-Keeler type in Hausdorff complete uniform spaces and a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions. The definitions, the results and the method presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. The paper includes examples which show a fundamental difference between our results and the well-known ones.