On cone metric spaces: A survey

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Abstract

Using an old M. Krein’s result and a result concerning symmetric spaces from [S. Radenović, Z. Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, Banach J. Math. Anal. 5 (1) (2011), 38–50], we show in a very short way that all fixed point results in cone metric spaces obtained recently, in which the assumption that the underlying cone is normal and solid is present, can be reduced to the corresponding results in metric spaces. On the other hand, when we deal with non-normal solid cones, this is not possible. In the recent paper [M.A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory Appl. 2010, 7 pages, Article ID 315398, doi:10.1115/2010/315398] the author claims that most of the cone fixed point results are merely copies of the classical ones and that any extension of known fixed point results to cone metric spaces is redundant; also that underlying Banach space and the associated cone subset are not necessary. In fact, Khamsi’s approach includes a small class of results and is very limited since it requires only normal cones, so that all results with non-normal cones (which are proper extensions of the corresponding results for metric spaces) cannot be dealt with by his approach.

Introduction

There are many generalizations of metric spaces: Menger spaces, fuzzy metric spaces, generalized metric spaces, abstract (cone) metric spaces or K-metric and K-normed spaces, rectangular metric and rectangular cone metric spaces

In 2007, Huang and Zhang [1] introduced cone metric spaces, being unaware that they already existed under the name K-metric, and K-normed spaces that were introduced and used in the middle of the 20th century in [2], [3], [4], [5], [6], [7], [8]. In both cases the set R of real numbers was replaced by an ordered Banach space E. However, Huang and Zhang went further and defined the convergence via interior points of the cone by which the order in E is defined. This approach allows the investigation of cone spaces in the case when the cone is not necessarily normal. And yet, they continued with results concerned with the normal cones only. One of the main results from [1] is the following Banach Contraction Principle in the setting of normal cone spaces:

Theorem 1.1 [1]

Let (X,d) be a complete cone metric space over a normal solid cone. Suppose that a mapping T:XX satisfies the contractive conditiond(Tx,Ty)λd(x,y)for all x,yX, where λ[0,1) is a constant. Then T has a unique fixed point in X and, for any xX, the iterative sequence {Tnx} converges to the fixed point.

Soon after this work many articles concerned with cone spaces with normal cones appeared (see [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53]). The results were focused on fixed points of one or two self-mappings of the cone space, accompanied mostly with the usual contractive conditions known from the setting of ordinary metric spaces (Kannan, Zamfirescu, Jungck, Rhoades, …).

The purpose of this paper is to show, using some old and well known results (Lemma 2.6 below) on cones in Banach spaces ([54], 1940), that most of the results of these articles can be reduced to the corresponding results from metric space theory. In other words, the fixed point problem in the setting of cone metric spaces is appropriate only in the case when the underlying cone is non-normal but just has interior that is nonempty. In this case only, proper generalizations of results from the ordinary metric spaces can be obtained. We present some examples showing that theorems from ordinary metric spaces cannot be applied in the setting of cone metric spaces, when the cone is non-normal. Such is Theorem 1.1. Namely, applying Lemma 2.6 in the case of a normal cone, this theorem reduces to the classical Banach Contraction Principle, proved in 1922. However, if in Theorem 1.1 only nonemptiness of the interior of cone is assumed, then the obtained result is more general and it is a proper extension of the classical Banach Contraction Principle. Similar approach was applied in [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83].

We note that there exist other ways of reducing some cone metric results to the respective results in metric spaces, see [84], [85], [86], [87], [88], [89], and a remark at the end of the paper.

Section snippets

Cones

Let E be a vector space over the real field R. A nonempty convex subset P of E is called a cone if λPP for all λ0. Clearly a cone P in E determines a transitive and reflexive relation “” by xyif yxP; moreover this relation is compatible with the vector structure, i.e.,

  • (a)

    if θx and θy then θx+y,

  • (b)

    if θx then θλx, for all λ0.

The relation “” determined by the cone P is called the vector (partial) ordering in E, and the pair (E,P) (or (E,)) is referred to as a (partially) ordered vector

Cone metric spaces

Let E be a Banach space with a solid cone P. We shall write xy if yxintP (this notation was introduced in [97]).

Definition 3.1

[1]

Let X be a nonempty set. Suppose that a mapping d:X×XE satisfies the following conditions:

  • (d1)

    θd(x,y) for all x,yX, and d(x,y)=θ if and only if x=y;

  • (d2)

    d(x,y)=d(y,x), for all x,yX;

  • (d3)

    d(x,y)d(x,z)+d(z,y), for all x,y,zX.

Then d is called a cone metric on X and (X,d) is called a cone metric space.

The concept of a cone metric space is more general than that of a metric space, because

Symmetric space (X,D) and topologies td and tD

If (X,d) is a cone metric space with a normal solid cone (with the normal constant satisfying K1), then D(x,y)=d(x,y) is a symmetric on the set X, that is, a mapping from X×X into [0,+) with the following properties:

  • (s1)

    D(x,y)0 and D(x,y)=0 if and only if x=y;

  • (s2)

    D(x,y)=D(y,x).

    It satisfies also:

  • (s3)

    D(x,y)K(D(x,z)+D(z,y)).

The cone metric d and the associated symmetric D=d in X generate two topologies: td and tD. Their bases of neighborhoods consist of the sets Bc(y)={xX:d(x,y)c}andBε(y)={xX:D(x,y

Application to the known results

In what follows we present some results from [1], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], to which Lemma 2.6 and the metric space (X,D) are applicable. In fact, in every contractive condition the letter “d” becomes “D”, while the relation “” in the cone P becomes the

Non-normal solid cones

Results concerning fixed points and other results, in the case of cone spaces with non-normal solid cones, cannot be proved by reducing to metric spaces, because in this case neither of the conditions from Lemma 2.6, 14 hold. Further, the vector cone metric is not continuous in the general case, i.e., from xnx, yny it need not follow that d(xn,yn)d(x,y) (see Example 6.1. below).

To replace standard properties of a metric, the following properties of cone metrics are often useful while

Acknowledgements

The authors are very grateful to the referees for the valuable suggestions and comments that enabled us to revise this paper. The authors are thankful to the Ministry of Science and Technological Development of Serbia.

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