A stationary point theorem characterizing metric completeness

Abstract

a b s t r a c tWe 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.

Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces

“…An element x X is said to be an endpoint (invariant or stationary point) of F, if Fx = {x}. The investigation of the existence and uniqueness of endpoints of set-valued contraction maps in metric spaces have received much attention in recent years [21][22][23][24][25][26].…”

Fixed points of multi-valued Subrahmanyan contractions

“…A special fixed point for a multi-valued operator F : X → P(X) is the so-called strict fixed point or end-point for F. In this context, we have the following strict fixed point result for multi-valued Subrahmanyan contractions, which generalize one of the main theorem in [14].…”

Equivalents of Ordered Fixed Point Theorems of Kirk, Caristi, Nadler, Banach, and others

“…In 2011, Jachymski [17] gave a stationary point theorem for some multimap on a metric space. The existence of fixed points of such maps characterizes the metric completeness and yields the order-theoretic Cantor theorem and the Ekeland variational principle.…”

“…Proof. Recall that (i) is true by [17,Theorem 3]. Now Metatheorem with X = A works for x ̸ ≼ y instead of G(x, y).…”

“…This article is organized as follows: In Section 2, we introduce our Metatheorem. Sections 3-10 are concerned to apply Metatheorem to theorems due to Browder-Göhde-Kirk [4,11,18], Brøndsted [3], Caristi-Kirk [6], Caristi [5], Nadler [8,20], Banach [2], Tarski-Kantorovitch [9], and Jachymski [13][14][15][16][17], respectively. Finally, Section 11 is for certain remarks as the conclusion.…”