The Caristi–Kirk Fixed Point Theorem from the point of view of ball spaces

Abstract

Abstract. We take a fresh look at the important Caristi-Kirk Fixed Point Theorem and link it to the recently developed theory of ball spaces, which provides generic fixed point theorems for contracting functions in a number of applications including, but not limited to, metric spaces. The connection becomes clear from a proof of the Caristi-Kirk Theorem given by J.-P. Penot in 1976. We define Caristi-Kirk ball spaces and use a generic fixed point theorem to reprove the Caristi-Kirk Theorem. Further, we show that a metric space is complete if and only if all of its Caristi-Kirk ball spaces are spherically complete.Mathematics Subject Classification. Primary 54H25, Secondary 47H09, 47H10.

Caristi–Kirk and Oettli–Théra ball spaces and applications

“…The corresponding balls B ϕ x have been introduced in [8] as the Caristi-Kirk balls and B ϕ is the induced Caristi-Kirk ball space. For brevity, we will write CK in place of Caristi-Kirk. A number of remarkable properties of the balls defined above, given in the following lemma, can be found in [8].…”

Hyperbolic spaces and directional contractions

“…(IV) New versions Caristi's theorem continue to emerge. One of the most recent is the approach of Kuhlmann et al in [69]. Suppose (X, d) is a metric space and ϕ : X → R is a mapping which is lower semicontinuous (from above) and bounded below (as before, a function ϕ is lower semicontinuous from above if given any net {x i } i∈I in X the following condition holds:…”

“…These sets are called Caristi-Kirk balls (CK-balls for short) in [69]. In general, they are not balls in the usual sense, but the definition is motivated by properties they share with ordinary balls in an ultrametric space.…”

“…If (X, B) is a ball space a function f : X → X is said to be contracting on orbits if there is a function that associates to every x ∈ X some ball B x ∈ B such that for all x ∈ X the following conditions hold: The following non-metric extension of Caristi's Theorem is due to Khulmann et al [70] (this, of course, represents a departure from the underlying metric theme of this paper). In [69], it is shown that Caristi's Theorem (Theorem 6.1) can be derived fairly easily from this theorem. B) has a fixed point.…”

Chain intersection closures

“…In [2,3,4,5,6,7], ball spaces are studied in order to provide a general framework for fixed point theorems that in some way or the other work with contractive functions. A ball space (X, B) is a nonempty set X together with any nonempty collection of nonempty subsets of X.…”