The Caristi\u2013Kirk Fixed Point Theorem from the point of view of ball spaces

Franz-Viktor Kuhlmann,Matthias Paulsen,Katarzyna Kuhlmann

doi:10.1007/s11784-018-0576-8

Franz-Viktor Kuhlmann, Matthias Paulsen + Show 1 more

Open Access

https://doi.org/10.1007/s11784-018-0576-8

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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.

Highlights

IntroductionThe present paper owes its existence to the discovery that other sets which came up in proofs of the Caristi–Kirk Fixed Point Theorem (discussed below) fit much better to the ball spaces framework
We consider a metric space (X, d) with a function f : X → X and ask for the existence of a fixed point, that is, a point x ∈ X such that f (x) = x
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

Summary

Introduction

The present paper owes its existence to the discovery that other sets which came up in proofs of the Caristi–Kirk Fixed Point Theorem (discussed below) fit much better to the ball spaces framework. While the first two are avoided in [12] and in [2,6], the axiom of choice, or at least the axiom of dependent choice, is still present (cf [6, Section 3]) In this connection, we should point out that the generic fixed point theorems in the theory of ball spaces are making essential use of Zorn’s Lemma. Working with these balls directly is more natural than the detour of defining the partial order explicitly

A modification of Penot’s proof of the Caristi–Kirk Theorem

Proof of Proposition 3

Proofs of Theorem 1 and Theorem 4