On cone metric spaces: A survey
Introduction
There are many generalizations of metric spaces: Menger spaces, fuzzy metric spaces, generalized metric spaces, abstract (cone) metric spaces or -metric and -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 -metric, and -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 of real numbers was replaced by an ordered Banach space . However, Huang and Zhang went further and defined the convergence via interior points of the cone by which the order in 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 be a complete cone metric space over a normal solid cone. Suppose that a mapping satisfies the contractive conditionfor all , where is a constant. Then has a unique fixed point in and, for any , the iterative sequence 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 be a vector space over the real field . A nonempty convex subset of is called a cone if for all . Clearly a cone in determines a transitive and reflexive relation “” by moreover this relation is compatible with the vector structure, i.e.,
- (a)
if and then ,
- (b)
if then , for all .
The relation “” determined by the cone is called the vector (partial) ordering in , and the pair (or ) is referred to as a (partially) ordered vector
Cone metric spaces
Let be a Banach space with a solid cone . We shall write if (this notation was introduced in [97]). Definition 3.1 Let be a nonempty set. Suppose that a mapping satisfies the following conditions: for all , and if and only if ; , for all ; , for all .[1]
Then is called a cone metric on and 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 and topologies and
If is a cone metric space with a normal solid cone (with the normal constant satisfying ), then is a symmetric on the set , that is, a mapping from into with the following properties:
- (s1)
and if and only if ;
- (s2)
.
It satisfies also:
- (s3)
.
The cone metric and the associated symmetric in generate two topologies: and . Their bases of neighborhoods consist of the sets
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 are applicable. In fact, in every contractive condition the letter “” becomes “”, while the relation “” in the cone 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, – hold. Further, the vector cone metric is not continuous in the general case, i.e., from , it need not follow that (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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