Generalized metrics and Caristi’s theorem

  • William A Kirk1 and

    Affiliated with

    • Naseer Shahzad2Email author

      Affiliated with

      Fixed Point Theory and Applications20132013:129

      DOI: 10.1186/1687-1812-2013-129

      Received: 4 February 2013

      Accepted: 29 April 2013

      Published: 15 May 2013

      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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq1_HTML.gif be a mapping satisfying for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq2_HTML.gif :

      I. http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq3_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq4_HTML.gif ;

      II. http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq5_HTML.gif . Then the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq6_HTML.gif is called a semimetric space.

      In such a space, convergence of sequences is defined in the usual way: A sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq7_HTML.gif is said to converge to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq8_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq9_HTML.gif . Also, a sequence is said to be Cauchy (or d-Cauchy) if for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq10_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq11_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq12_HTML.gif . The space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq13_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq14_HTML.gif 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) With X and d as in Definition 1, assume also that for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq15_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equa_HTML.gif

      Definition 2 A pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq6_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq2_HTML.gif , there is a number http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq16_HTML.gif such that for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq17_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equb_HTML.gif
      IV. For each point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq18_HTML.gif and each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq19_HTML.gif , there is a number http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq20_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq21_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq22_HTML.gif , then for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq17_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equc_HTML.gif
      V. For each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq19_HTML.gif , there is a number http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq23_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq2_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq22_HTML.gif , then for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq17_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equd_HTML.gif

      Obviously, if Axiom V is strengthened to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq24_HTML.gif , 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq2_HTML.gif and a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq25_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq26_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq27_HTML.gif . Thus, as Wilson observes, the following is self-evident.

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

      For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq28_HTML.gif , let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq29_HTML.gif . Then Axiom III is also equivalent to the assertion that X is Hausdorff in the sense that given any two distinct points http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq2_HTML.gif , there exist positive numbers http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq30_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq31_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq32_HTML.gif . This suggests the presence of a topology.

      Definition 3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq6_HTML.gif be a semimetric space. Then the distance function d is said to be continuous if for any sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq35_HTML.gif .

      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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq37_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq38_HTML.gif is an open set for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq39_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq28_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq40_HTML.gif be a mapping such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq41_HTML.gif and all distinct points http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq42_HTML.gif , each distinct from x and y:
      1. (i)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq43_HTML.gif ;

         
      2. (ii)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq44_HTML.gif ;

         
      3. (iii)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq45_HTML.gif (quadrilateral inequality).

         

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

      Proposition 2 If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq6_HTML.gif is a generalized metric space which satisfies Axiom III, then the distance function is continuous.

      Proof Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq33_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq46_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq47_HTML.gif . Also assume that for n arbitrarily large, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq48_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq49_HTML.gif . In view of Axiom III, we may also assume that for n sufficiently large, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq50_HTML.gif . Then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Eque_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equf_HTML.gif
      Together these inequalities imply
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equg_HTML.gif

      Thus http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq51_HTML.gif . □

      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 3 Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq52_HTML.gif is a Cauchy sequence in a generalized metric space X and suppose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq53_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq54_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq39_HTML.gif . In particular, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq52_HTML.gif does not converge to p if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq47_HTML.gif .

      Proof We may assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq47_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq55_HTML.gif for arbitrarily large n, it must be the case that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq56_HTML.gif . So, we may also assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq57_HTML.gif for all n. Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq49_HTML.gif for infinitely many n; otherwise, the result is trivial. So, we may assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq58_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq59_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq60_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq61_HTML.gif . Then, by the quadrilateral inequality,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equh_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equi_HTML.gif
      Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq52_HTML.gif is a Cauchy sequence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq62_HTML.gif . Therefore, letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq27_HTML.gif in the above inequalities,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equj_HTML.gif

       □

      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
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equk_HTML.gif

      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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq6_HTML.gif be a complete generalized metric space, and suppose that the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq63_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq64_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq41_HTML.gif and fixed http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq65_HTML.gif . Then f has a unique fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq66_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq67_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq8_HTML.gif .

      It is possible to prove this theorem by following the proof given by Branciari up to the point of showing that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq68_HTML.gif is a Cauchy sequence for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq8_HTML.gif . Then, by completeness of X, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq69_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq70_HTML.gif . But http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq71_HTML.gif , so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq72_HTML.gif . In view of Proposition 3, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq73_HTML.gif .

      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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq6_HTML.gif be a complete g.m.s. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq63_HTML.gif be a mapping, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq74_HTML.gif be a lower semicontinuous function. Suppose that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equl_HTML.gif

      Then f has a fixed point.

      Typically, proofs of Caristi’s theorem (and there have been many) involve assigning a partial order ⪯ to X by setting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq75_HTML.gif , 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq76_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq77_HTML.gif is said to be lower semicontinuous (l.s.c.) if given http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq8_HTML.gif and a net http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq78_HTML.gif in M, the conditions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq79_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq80_HTML.gif imply http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq81_HTML.gif .)

      Proof of Theorem 2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq82_HTML.gif . Then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equm_HTML.gif
      Hence
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equn_HTML.gif
      so
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equo_HTML.gif

      This proves that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq83_HTML.gif is a Cauchy sequence. If f were continuous, one could immediately conclude that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq69_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq84_HTML.gif . (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq85_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq86_HTML.gif , we use http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq87_HTML.gif to denote the cardinality of the set
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equp_HTML.gif
      Now let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq69_HTML.gif , let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq88_HTML.gif , and suppose that the net http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq89_HTML.gif has been defined so that
      1. (i)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq90_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq91_HTML.gif ;

         
      2. (ii)

        if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq92_HTML.gif is a limit ordinal, then the net http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq93_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq94_HTML.gif ;

         
      3. (iii)

        if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq95_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq96_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq97_HTML.gif .

         
      If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq98_HTML.gif , define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq99_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq100_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq101_HTML.gif and by the quadrilateral inequality,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equq_HTML.gif
      Thus if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq102_HTML.gif , by the inductive assumption,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equr_HTML.gif
      Otherwise, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq103_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq104_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq105_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq106_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq107_HTML.gif and we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equs_HTML.gif
      Finally, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq108_HTML.gif , we can write (here order 3 is needed!)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equt_HTML.gif
      Now suppose β is a limit ordinal. We claim that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq109_HTML.gif is a Cauchy net. If not, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq10_HTML.gif and a strictly increasing sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq110_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq111_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq113_HTML.gif . This leads to the contradiction
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equu_HTML.gif

      Therefore http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq114_HTML.gif is a Cauchy net and, since X is complete, it is possible to take http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq115_HTML.gif .

      Since β is a limit ordinal, the cardinality of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq116_HTML.gif is infinite for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq86_HTML.gif . Consequently, since φ is lower semicontinuous,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equv_HTML.gif

      Therefore a net http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq78_HTML.gif has been defined satisfying (i), (ii), and (iii) for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq117_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq118_HTML.gif denote the set of limit ordinals in Γ. If f has no fixed point, the net http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq119_HTML.gif is strictly decreasing. This is a contradiction because http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq120_HTML.gif 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 3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq6_HTML.gif be a complete metric space. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq63_HTML.gif be a mapping, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq121_HTML.gif be a lower semicontinuous function. Suppose that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equ1_HTML.gif
      (C)

      Then f has a fixed point.

      Proof Introduce the Brøndsted partial order on X by setting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq75_HTML.gif . Let I be a totally ordered set, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq122_HTML.gif be a chain in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq123_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq124_HTML.gif . Therefore http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq125_HTML.gif is decreasing. Since φ is bounded below, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq126_HTML.gif . This implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq127_HTML.gif ; hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq128_HTML.gif is a Cauchy net. Since X is complete, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq8_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq129_HTML.gif . Thus for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq130_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equw_HTML.gif

      Therefore http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq131_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq130_HTML.gif , so x is an upper bound for the chain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq132_HTML.gif . By Zorn’s lemma, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq133_HTML.gif has a maximal element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq134_HTML.gif . But condition (C) implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq135_HTML.gif , so it must be the case that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq136_HTML.gif . □

      The above argument fails in the setting of Theorem 2 because it is not possible to show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq133_HTML.gif 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
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equx_HTML.gif
      and define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equy_HTML.gif
      Now let x, y, and z be three distinct points in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq137_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq138_HTML.gif be a Cauchy sequence converging to z. Then, by the quadrilateral inequality,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equz_HTML.gif

      Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq27_HTML.gif and applying Proposition 3, we see that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq139_HTML.gif . Therefore http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq140_HTML.gif is a metric space. In the proof of Theorem 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq141_HTML.gif . To show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq135_HTML.gif , it is necessary to show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq142_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq143_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq144_HTML.gif is a Cauchy sequence. So, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq145_HTML.gif .

      By induction,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equaa_HTML.gif
      Then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equab_HTML.gif
      This leads to the contradiction http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq146_HTML.gif . The other alternative is that there exists a periodic point. This is impossible because
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_Equac_HTML.gif

      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, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq147_HTML.gif is a Cauchy sequence and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-129/MediaObjects/13663_2013_464_IEq148_HTML.gif .

      1

      Footnotes
      1

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

       

      Declarations

      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.

      Authors’ Affiliations

      (1)
      Department of Mathematics, University of Iowa
      (2)
      Department of Mathematics, King Abdulaziz University

      References

      1. Branciari A: A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. Publ. Math. (Debr.) 2000, 57:31–37.MathSciNetMATH
      2. Sarma IR, Rao JM, Rao SS: Contractions over generalized metric spaces. J. Nonlinear Sci. Appl. 2009, 2:180–182.MathSciNetMATH
      3. Kirk WA, Kang BG: A fixed point theorem revisited. J. Korean Math. Soc. 1997,34(2): 285–291.MathSciNetMATH
      4. Blumenthal LM: Theory and Applications of Distance Geometry. 2nd edition. Chelsea, New York; 1970.MATH
      5. de Tilly, J: ‘Essai de géométrie analytique gén érale’, Mémoires couronnés et autres mémoires publiés par l’Académie Royale de Belgique, 47, mémoire 5 (1892–93)de Tilly, J: ‘Essai de géométrie analytique gén érale’, Mémoires couronnés et autres mémoires publiés par l’Académie Royale de Belgique, 47, mémoire 5 (1892–93) de Tilly, J: ‘Essai de géométrie analytique gén érale’, Mémoires couronnés et autres mémoires publiés par l’Académie Royale de Belgique, 47, mémoire 5 (1892-93)
      6. Menger K: Untersuchungen über allgemeine Metrik. Math. Ann. 1928, 100:75–163.MathSciNetMATHView Article
      7. Wilson WA: On semimetric spaces. Am. J. Math. 1931,53(2): 361–373.View Article
      8. Chittenden EW: On the equivalence of ecart and voisinage. Trans. Am. Math. Soc. 1917,18(2): 161–166.MathSciNetMATH
      9. Jachymski J, Matkowski J, Świątkowski T: Nonlinear contractions on semimetric spaces. J. Appl. Anal. 1995,1(2): 125–134.MathSciNetView ArticleMATH
      10. Hicks TL, Rhoades BE: Fixed point theory in symmetric spaces with applications to probabilistic spaces. Nonlinear Anal., Theory Methods Appl. 1999,36(3): 331–344.MathSciNetMATHView Article
      11. Miheţ DL: A note on a paper of T. L. Hicks and B. E. Rhoades: ‘Fixed point theory in symmetric spaces with applications to probabilistic spaces’ [Nonlinear Anal. 36 (1999), no. 3, Ser. A: Theory Methods, 331–344; MR1688234]. Nonlinear Anal. 2006,65(7): 1411–1413.MathSciNetMATHView Article
      12. Turinici, M: Functional contractions in local Branciari metric spaces. arXiv:1208.4610v1 [math.GN] 22 Aug 2012Turinici, M: Functional contractions in local Branciari metric spaces. arXiv:1208.4610v1 [math.GN] 22 Aug 2012
      13. Samet B: Discussion on: a fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces by A. Branciari. Publ. Math. (Debr.) 2010,76(4): 493–494.MathSciNetMATH
      14. Caristi J: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 1976, 215:241–251.MathSciNetMATHView Article
      15. Wong CS: On a fixed point theorem of contractive type. Proc. Am. Math. Soc. 1976,57(2): 283–284.MATHView Article
      16. Brézis H, Browder FE: A general principle on ordered sets in nonlinear functional analysis. Adv. Math. 1976,21(3): 355–364.MATHView Article
      17. Hausdorff, F: Grundzüge der Mengenlehre. Leipzig (1914)Hausdorff, F: Grundzüge der Mengenlehre. Leipzig (1914)

      Copyright

      © Kirk and Shahzad; licensee Springer 2013

      This article is published under license to BioMed Central Ltd. 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.