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 theorem1 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 halbmetrischer 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
I.
II.
In such a space, convergence of sequences is defined in the usual way: A sequence
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
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
Definition 2 A pair
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
IV. For each point
V. For each
Obviously, if Axiom V is strengthened to
Axiom III is equivalent to the assertion that there do not exist distinct points
Proposition 1In a semimetric space, Axiom III is equivalent to the assertion that limits are unique.
For
Definition 3 Let
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
We now turn to the concept introduced by Branciari.
Definition 4 ([1])
Let X be a nonempty set, and let
(i)
(ii)
(iii)
Then X is called a generalized metric space (g.m.s.).
Proposition 2If
Proof Suppose that
and
Together these inequalities imply
Thus
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
Proof We may assume that
and
Since
□
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
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
It is possible to prove this theorem by following the proof given by Branciari up
to the point of showing that
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
Thenfhas a fixed point.
Typically, proofs of Caristi’s theorem (and there have been many) involve assigning
a partial order ⪯ to X by setting
Proof of Theorem 2 Let
Hence
so
This proves that
Let Γ denote the set of countable ordinals. For
Now let
(i)
(ii) if
(iii) if
If
Thus if
Otherwise,
Finally, if
Now suppose β is a limit ordinal. We claim that
Therefore
Since β is a limit ordinal, the cardinality of
Therefore a net
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 BrézisBrowder order principle is given in [16].)
Theorem 3Let
Thenfhas a fixed point.
Proof Introduce the Brøndsted partial order on X by setting
Therefore
The above argument fails in the setting of Theorem 2 because it is not possible to
show that
and define
Now let x, y, and z be three distinct points in
Letting
By induction,
Then
This leads to the contradiction
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,
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

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