A stationary point theorem characterizing metric completeness
We give a stationary point theorem for some set-valued mappings on a metric space. The existence of fixed points of such mappings characterizes the metric completeness. Our result easily yields the order-theoretic Cantor theorem of Granas and Horvath, and the famous Ekeland variational principle.
Generalized contractions of Meir–Keeler type, endpoints, set-valued dynamical systems and generalized pseudometrics in uniform spaces
Abstract 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.
Some asymptotic fixed point theorems
By an asymptotic fixed point theorem we mean a theorem in functional analysis in which the existence of fixed points of a map f is established with the aid of assumptions on the iterates f" of f. We prove below some new theorems of this type, and we obtain as corollaries results of F. E. Browder, G. Darbo, R. L. Frum-Ketkov, W. A. Horn and others. We also state a number of conjectures about fixed point theorems at the end of the paper. Our interest in the results here is two-fold. First, asymptotic fixed point theorems have proved useful in the theory of ordinary and functional differential equations (see [17], [18], [19] and [34]), and in fact we hope to indicate in a future paper some applications of our results to functional differential equations of neutral type (see [11] or [15]). Second, and perhaps more relevant to our immediate line of development, asymptotic fixed point theorems provide a framework for unifying and generalizing many of the known fixed point theorems of functional anal-