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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Highly Accessed Research

Generalized metrics and Caristi’s theorem

William A Kirk1 and Naseer Shahzad2*

Author Affiliations

1 Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

2 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

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Fixed Point Theory and Applications 2013, 2013:129  doi:10.1186/1687-1812-2013-129


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/129


Received:4 February 2013
Accepted:29 April 2013
Published:15 May 2013

© 2013 Kirk and Shahzad; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A ‘generalized metric space’ is a semimetric space which does not satisfy the triangle inequality, but which satisfies a weaker assumption called the quadrilateral inequality. After reviewing various related axioms, it is shown that Caristi’s theorem holds in complete generalized metric spaces without further assumptions. This is noteworthy because Banach’s fixed point theorem seems to require more than the quadrilateral inequality, and because standard proofs of Caristi’s theorem require the triangle inequality.

MSC: 54H25, 47H10.

Keywords:
fixed points; contraction mappings metric spaces; semimetric spaces; generalized metric spaces; Caristi’s theorem

1 Introduction

In an effort to generalize Banach’s contraction mapping principle, which holds in all complete metric spaces, to a broader class of spaces, Branciari [1] conceived of the notion to replace the triangle inequality with a weaker assumption he called the quadrilateral inequality. He called these spaces ‘generalized metric spaces’. These spaces retain the fundamental notion of distance. However, as we shall see, the quadrilateral inequality, while useful in some sense, ignores the importance of such things as the continuity of the distance function, uniqueness of limits, etc. In fact it has been asserted (see, e.g., [2]) that for an accurate generalization of Banach’s fixed point theorem along the lines envisioned by Branciari, one needs the quadrilateral inequality in conjunction with the assumption that the space is Hausdorff.

We begin by discussing the relationship of Branciari’s concept to the classical axioms of semimetric spaces. Then we show that Caristi’s fixed point theorem holds within Branciari’s framework without any additional assumptions. This is possibly surprising. All proofs of Caristi’s theorem that the writers are aware of rely in some way on use of the triangle inequality. (In contrast, it has been noted that the proof of the first author’s fundamental fixed point theorem for nonexpansive mappings does not require the triangle inequality; see [3].)

2 Semimetric spaces

In the absence of relevant examples, it is not clear whether Branciari’s concept of weakening the triangle inequality will prove useful in analysis. However, the notion of assigning a ‘distance’ between each two points of an abstract set is fundamental in geometry. According to Blumenthal [[4], p.31], this notion has its origins in the late nineteenth century in axiomatic studies of de Tilly [5]. In his 1928 treatise [6], Karl Menger used the term halb-metrischer Raume, or semimetric space, to describe the same concept. We begin by summarizing the results of Wilson’s seminal paper [7] on semimetric spaces.

Definition 1 Let X be a set and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M1">View MathML</a> be a mapping satisfying for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M2">View MathML</a>:

I. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M3">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M4">View MathML</a>;

II. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M5">View MathML</a>. Then the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6">View MathML</a> is called a semimetric space.

In such a space, convergence of sequences is defined in the usual way: A sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M7">View MathML</a> is said to converge to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M8">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M9">View MathML</a>. Also, a sequence is said to be Cauchy (or d-Cauchy) if for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M10">View MathML</a> there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M11">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M12">View MathML</a>. The space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M13">View MathML</a> is said to be complete if every Cauchy sequence has a limit.

With such a broad definition of distance, three problems are immediately obvious: (i) There is nothing to assure that limits are unique (thus the space need not be Hausdorff); (ii) a convergent sequence need not be a Cauchy sequence; (iii) the mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M14">View MathML</a>need not even be continuous. Therefore it is unlikely there could be an effective topological theory in such a setting.

With the introduction of the triangle inequality, problems (i), (ii), and (iii) are simultaneously eliminated.

VI. (Triangle inequality) WithXanddas in Definition 1, assume also that for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M15">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M16">View MathML</a>

Definition 2 A pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6">View MathML</a> satisfying Axioms I, II, and VI is called a metric space.a

In his study [7], Wilson introduces three axioms in addition to I and II which are weaker than VI. These are the following.

III. For each pair of (distinct) points<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M2">View MathML</a>, there is a number<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M19">View MathML</a>such that for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M20">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M21">View MathML</a>

IV. For each point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M22">View MathML</a>and each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M23">View MathML</a>, there is a number<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M24">View MathML</a>such that if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M25">View MathML</a> satisfies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M26">View MathML</a>, then for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M20">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M28">View MathML</a>

V. For each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M23">View MathML</a>, there is a number <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M30">View MathML</a> such that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M2">View MathML</a> satisfy <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M26">View MathML</a>, then for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M20">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M34">View MathML</a>

Obviously, if Axiom V is strengthened to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M35">View MathML</a>, then the space becomes metric. Chittenden [8] has shown (using an equivalent definition) that a semimetric space satisfying Axiom V is always homeomorphic to a metric space.

Axiom III is equivalent to the assertion that there do not exist distinct points <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M2">View MathML</a> and a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M37">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M38">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M39">View MathML</a>. Thus, as Wilson observes, the following is self-evident.

Proposition 1In a semimetric space, Axiom III is equivalent to the assertion that limits are unique.

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M40">View MathML</a>, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M41">View MathML</a>. Then Axiom III is also equivalent to the assertion that X is Hausdorff in the sense that given any two distinct points <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M2">View MathML</a>, there exist positive numbers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M43">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M44">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M45">View MathML</a>. This suggests the presence of a topology.

Definition 3 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6">View MathML</a> be a semimetric space. Then the distance function d is said to be continuous if for any sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M47">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M48">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M49">View MathML</a>.

Remark Some writers call a space satisfying Axioms I and II a ‘symmetric space’ and reserve the term semimetric space for a symmetric space with a continuous distance function (see, e.g., [9]; cf. also [10,11]). Here we use Menger’s original terminology.

A point p in a semimetric space X is said to be an accumulation point of a subset E of X if, given any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M50">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M51">View MathML</a>. A subset of a semimetric space is said to be closed if it contains each of its accumulation points. A subset of a semimetric space is said to be open if its complement is closed. With these definitions, if X is a semimetric space with a continuous distance function, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M52">View MathML</a> is an open set for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M53">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M40">View MathML</a> and, moreover, X is a Hausdorff topological space [4].

We now turn to the concept introduced by Branciari.

Definition 4 ([1])

Let X be a nonempty set, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M55">View MathML</a> be a mapping such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M56">View MathML</a> and all distinct points <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M57">View MathML</a>, each distinct from x and y:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M58">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M59">View MathML</a>;

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M60">View MathML</a> (quadrilateral inequality).

Then X is called a generalized metric space (g.m.s.).

Proposition 2If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6">View MathML</a>is a generalized metric space which satisfies Axiom III, then the distance function is continuous.

Proof Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M47">View MathML</a> satisfy <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M48">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M64">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M65">View MathML</a>. Also assume that for n arbitrarily large, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M66">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M67">View MathML</a>. In view of Axiom III, we may also assume that for n sufficiently large, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M68">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M69">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M70">View MathML</a>

Together these inequalities imply

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M71">View MathML</a>

Thus <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M72">View MathML</a>. □

Therefore if a generalized metric space satisfies Axiom III, it is a Hausdorff topological space. However, the following observation shows that the quadrilateral inequality implies a weaker but useful form of distance continuity. (This is a special case of Proposition 1 of [12].)

Proposition 3Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M73">View MathML</a>is a Cauchy sequence in a generalized metric spaceXand suppose<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M74">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M75">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M53">View MathML</a>. In particular, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M73">View MathML</a>does not converge topif<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M65">View MathML</a>.

Proof We may assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M65">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M80">View MathML</a> for arbitrarily large n, it must be the case that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M81">View MathML</a>. So, we may also assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M82">View MathML</a> for all n. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M67">View MathML</a> for infinitely many n; otherwise, the result is trivial. So, we may assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M84">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M85">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M86">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M87">View MathML</a>. Then, by the quadrilateral inequality,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M88">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M89">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M73">View MathML</a> is a Cauchy sequence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M91">View MathML</a>. Therefore, letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M39">View MathML</a> in the above inequalities,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M93">View MathML</a>

 □

We now come to Branciari’s extension of Banach’s contraction mapping theorem. Although in his proof Branciari makes the erroneous assertion that a g.m.s. is a Hausdorff topological space with a neighborhood basis given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M94">View MathML</a>

with the aid of Proposition 3, Branciari’s proof carries over with only a minor change. The assertion in [2] that the space needs to be Hausdorff is superfluous, a fact first noted in [12]. See also the example in [13].

Theorem 1 ([1])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6">View MathML</a>be a complete generalized metric space, and suppose that the mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M96">View MathML</a>satisfies<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M97">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M56">View MathML</a>and fixed<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M99">View MathML</a>. Thenfhas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M100">View MathML</a>, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M101">View MathML</a>for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M8">View MathML</a>.

It is possible to prove this theorem by following the proof given by Branciari up to the point of showing that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M103">View MathML</a> is a Cauchy sequence for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M8">View MathML</a>. Then, by completeness of X, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M105">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M106">View MathML</a>. But <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M107">View MathML</a>, so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M108">View MathML</a>. In view of Proposition 3, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M109">View MathML</a>.

3 Caristi’s theorem

We now turn to a proof of Caristi’s theorem in a complete g.m.s.

Theorem 2 (cf. Caristi [14])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6">View MathML</a>be a complete g.m.s. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M96">View MathML</a>be a mapping, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M112">View MathML</a>be a lower semicontinuous function. Suppose that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M113">View MathML</a>

Thenfhas a fixed point.

Typically, proofs of Caristi’s theorem (and there have been many) involve assigning a partial order ⪯ to X by setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M114">View MathML</a>, and then either using Zorn’s lemma or the Brézis-Browder order principle (see Section 4). However, the triangle inequality is needed for these approaches in order to show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M115">View MathML</a> is transitive. The proof we give below is based on Wong’s modification [15] of Caristi’s original transfinite induction argument [14]. (Recall that if M is a metric space, a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M116">View MathML</a> is said to be lower semicontinuous (l.s.c.) if given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M8">View MathML</a> and a net <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M118">View MathML</a> in M, the conditions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M119">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M120">View MathML</a> imply <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M121">View MathML</a>.)

Proof of Theorem 2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M122">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M123">View MathML</a>

Hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M124">View MathML</a>

so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M125">View MathML</a>

This proves that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M126">View MathML</a> is a Cauchy sequence. If f were continuous, one could immediately conclude that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M105">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M128">View MathML</a>. (The quadrilateral inequality is not needed in this case, but it is necessary for Cauchy sequences to have unique limits.)

Let Γ denote the set of countable ordinals. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M129">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M130">View MathML</a>, we use <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M131">View MathML</a> to denote the cardinality of the set

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M132">View MathML</a>

Now let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M105">View MathML</a>, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M134">View MathML</a>, and suppose that the net <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M135">View MathML</a> has been defined so that

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M136">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M137">View MathML</a>;

(ii) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M138">View MathML</a> is a limit ordinal, then the net <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M139">View MathML</a> converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M140">View MathML</a>;

(iii) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M141">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M142">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M143">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M144">View MathML</a>, define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M145">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M130">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M147">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M148">View MathML</a> and by the quadrilateral inequality,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M149">View MathML</a>

Thus if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M150">View MathML</a>, by the inductive assumption,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M151">View MathML</a>

Otherwise, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M152">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M153">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M154">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M155">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M156">View MathML</a> and we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M157">View MathML</a>

Finally, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M158">View MathML</a>, we can write (here order 3 is needed!)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M159">View MathML</a>

Now suppose β is a limit ordinal. We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M160">View MathML</a> is a Cauchy net. If not, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M10">View MathML</a> and a strictly increasing sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M162">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M163">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M164">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M165">View MathML</a>. This leads to the contradiction

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M166">View MathML</a>

Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M167">View MathML</a> is a Cauchy net and, since X is complete, it is possible to take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M168">View MathML</a>.

Since β is a limit ordinal, the cardinality of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M169">View MathML</a> is infinite for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M130">View MathML</a>. Consequently, since φ is lower semicontinuous,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M171">View MathML</a>

Therefore a net <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M118">View MathML</a> has been defined satisfying (i), (ii), and (iii) for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M173">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M174">View MathML</a> denote the set of limit ordinals in Γ. If f has no fixed point, the net <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M175">View MathML</a> is strictly decreasing. This is a contradiction because <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M176">View MathML</a> is uncountable and any strictly decreasing net of real numbers must be countable. □

4 Another approach

We now examine an easy proof of Caristi’s original theorem based on Zorn’s lemma. (A more constructive proof which uses the Brézis-Browder order principle is given in [16].)

Theorem 3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M6">View MathML</a>be a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M96">View MathML</a>be a mapping, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M179">View MathML</a>be a lower semicontinuous function. Suppose that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M180">View MathML</a>

(C)

Thenfhas a fixed point.

Proof Introduce the Brøndsted partial order on X by setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M114">View MathML</a>. Let I be a totally ordered set, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M182">View MathML</a> be a chain in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M183">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M184">View MathML</a>. Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M185">View MathML</a> is decreasing. Since φ is bounded below, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M186">View MathML</a>. This implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M187">View MathML</a>; hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M188">View MathML</a> is a Cauchy net. Since X is complete, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M8">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M190">View MathML</a>. Thus for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M191">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M192">View MathML</a>

Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M193">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M191">View MathML</a>, so x is an upper bound for the chain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M195">View MathML</a>. By Zorn’s lemma, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M196">View MathML</a> has a maximal element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M197">View MathML</a>. But condition (C) implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M198">View MathML</a>, so it must be the case that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M199">View MathML</a>. □

The above argument fails in the setting of Theorem 2 because it is not possible to show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M196">View MathML</a> is transitive in a g.m.s. In a metric space, transitivity follows directly from the triangle inequality. A way to circumvent this difficulty is to only consider points of X that are limits of nontrivial Cauchy sequences. The proof of Theorem 2 implies that nontrivial Cauchy sequences exist. So, let

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M201">View MathML</a>

and define

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M202">View MathML</a>

Now let x, y, and z be three distinct points in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M203">View MathML</a>, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M204">View MathML</a> be a Cauchy sequence converging to z. Then, by the quadrilateral inequality,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M205">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M39">View MathML</a> and applying Proposition 3, we see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M207">View MathML</a>. Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M208">View MathML</a> is a metric space. In the proof of Theorem 3 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M209">View MathML</a>. To show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M198">View MathML</a>, it is necessary to show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M211">View MathML</a>. Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M212">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M213">View MathML</a> is a Cauchy sequence. So, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M214">View MathML</a>.

By induction,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M215">View MathML</a>

Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M216">View MathML</a>

This leads to the contradiction <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M217">View MathML</a>. The other alternative is that there exists a periodic point. This is impossible because

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M218">View MathML</a>

Remark In view of Proposition 3, it seems reasonable to introduce the following definition.

Definition 5 A point p in a generalized metric space X is said to be an accumulation point of a subset E of X if some infinite Cauchy sequence in E converges to p. A set E in X is said to be closed if it contains all of its accumulation points.

Observe that with convergence defined as above, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M219">View MathML</a> is a Cauchy sequence and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/129/mathml/M220">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

We thank a referee for pointing out some oversights in the original draft of this manuscript. The research of N. Shahzad was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

End notes

  1. The term ‘metric space’ for spaces satisfying Axioms I, II, and VI is apparently due to Hausdorff [17].

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