Suzuki’s fixed point results from (Suzuki, Proc. Am. Math. Soc. 136:1861-1869, 2008) and (Suzuki, Nonlinear Anal. 71:5313-5317, 2009) are extended to the case of metric type spaces and cone metric type spaces. Examples are given to distinguish our results from the known ones.
MSC: 47H10, 54H25.
Keywords:metric type space; cone metric space; normal cone; fixed point
1 Introduction and preliminaries
In 2008 Suzuki proved the following refinement of Banach’s fixed point principle.
Theorem 1 (, Theorem 2])
Suzuki proved also the following version of Edelstein’s fixed point theorem.
Theorem 2 (, Theorem 3])
ThenThas a unique fixed point inX.
This theorem was generalized in .
Let E be a real Banach space with the zero vector θ. A subset P of E is called a cone if: (a) P is closed, non-empty and ; (b) , , imply that ; (c) . Given a cone P, we define the partial ordering ⪯ with respect to P by if and only if . We shall write for , where intP stands for the interior of P and use for and . If , then P is called a solid cone. It is said to be normal if there is a number such that for all , implies . Such a minimal constant K is called the normal constant of P.
Huang and Zhang re-introduced cone metric spaces in  (this notion was known under various names since the mid of the 20th century, see a survey in ), replacing the set of real numbers by an ordered Banach space as the codomain for a metric. Cone metric spaces over normal cones inspired another generalization of metric spaces that were called metric type spaces by Khamsi  (see also [9-12]; note that, in fact, spaces of this kind were used earlier under the name of b-spaces by Czerwik ). Cvetković et al. and Shah et al. extended Khamsi’s definition and defined cone metric type spaces as follows:
Of course, for we get the cone metric space (CMS) of , resp. the usual metric space.
Example 1 ()
It was shown in  that is a solid cone in and that is a CMTS. In particular, for we get an MTS and for a CMS.
Let be any CMS over a normal cone with normal constant . Then is an MTS, where . In this case the spaces and have the same topologies (see , Theorem 2.7]).
Notions such as convergent and Cauchy sequences, as well as completeness, are introduced in (cone) metric type spaces in the standard way. The following obviously holds in an arbitrary (cone) metric type space:
We will sometimes need the continuity of metric-type function D in one variable:
or in two variables:
Recall that a selfmap is said to have the property (P)  if for each , where is the set of fixed points of T.
In this paper, we extend Suzuki’s Theorems 1 and 2, as well as Popescu’s results from  to the case of metric type spaces and cone metric type spaces. Examples are given to distinguish our results from the known ones.
2.1 Results in metric type spaces
Note that for , Theorem 3 reduces to a special case of Theorem 2.1 by Popescu .
and, by induction,
Using , Lemma 3.1] we conclude that is a Cauchy sequence, tending to some z in the complete space X. Obviously, also .
Let us prove now that
It is easy to see that (2.6) follows from the previous relation.
We will prove now that
Using (2.8) and the induction hypothesis, we get that
and (2.9) is proved by induction.
Assumption (2.2) implies that
It follows from (2.11) that
which is impossible. Hence one of the following holds for each n:
Thus, we have proved that z is a fixed point of T. The uniqueness of the fixed point follows easily from (2.6). Indeed, if yz are two fixed points of T, then (2.6) implies that
wherefrom . The property (P) follows from (2.3) (see ). □
Adapting , Example 1] we give now an example of a mapping satisfying the conditions of Theorem 3 (and having a unique fixed point) but not satisfying the respective classical (non-Suzuki-type) condition in metric type spaces (see, e.g., , Theorem 3.4]).
The following is a metric-type version of Theorem 2.
ThenThas a unique fixed point inX.
holds true. Then and assumption (2.13) implies that for . Passing to the limit, we obtain that . If , the last inequality is impossible by the definition of β. If , it is possible only if (recall that we have supposed that ). But in this case and (2.13) implies that , which is again impossible by the definition of β. Hence, in all cases we obtain a contradiction and it follows that and so .
It follows that
Suppose now that
holds. In other words, there exists a sequence such that holds for each , or there exists a sequence such that holds for each . In both cases, passing to the limit, we obtain that , i.e., , a contradiction with the assumption that T has no fixed points.
2.2 Results in cone metric type spaces
In this subsection, we formulate cone-metric-type versions of the results from the previous subsection.
Proof Since the cone P is normal, without loss of generality, we can assume that the normal constant of P is and that the given norm in E is monotone, i.e. (see , Lemma 2.1]). Denote . Then D is a (real-valued) metric-type function and the space is compact (together with , see , Theorem 2.7]). Let us prove that the mapping T satisfies for some the condition
Hence, condition (2.15) is satisfied, and the conclusion follows. □
In a similar way, the following corollaries and the theorem can be proved.
Example 4 can be easily adapted to a CMTS.
ThenThas a unique fixed point inX.
Note that for the above theorem reduces to , Theorem 3.8].
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
The first author gratefully acknowledges the support provided by the Deanship of Scientific Research (DSR), King Abdulaziz University during this research. The second, third and fourth authors are thankful to the Ministry of Science and Technological Development of Serbia.
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