We prove a set of fixed point theorems for mappings with φ-contractive iterate at a point in a class of generalized metric spaces. This is a generalization of some well-known results for Guseman and Matkowski types of fixed point results in metric and generalized metric spaces.
MSC: 47H10, 54H25.
Keywords:φ-contraction; fixed point; mapping with contractive iterate at a point; generalized metric space
1 Introduction and preliminaries
In 1975 Matkowski introduced the following class of mappings.
(i) φ is nondecreasing,
In the same paper he proved the existence and uniqueness of a fixed point for such type of mappings. This result is significant because the concept of weak contraction of Matkowski type is independent of the Meir-Keeler contraction , and it was generalized in different directions [3-10]. Matkowski generalized his own result proving a theorem of Segal-Guseman type .
The aim of this paper is to show that this result is valid in a more general class of spaces and wide class of functions φ.
In 1963, Gähler introduced 2-metric spaces, but other authors proved that there is no relation between the two distance functions and there is no easy relationship between results obtained in the two settings. Dhage introduced a new concept of the measure of nearness between three or more objects. But the topological structure of so-called D-metric spaces was incorrect. Finally, Mustafa and Sims  introduced the correct definition of a generalized metric space as follows.
The following useful properties of a G-metric are readily derived from the axioms.
Fixed point theorems in symmetric G-metric space are mostly consequences of the related fixed point results in metric spaces. In this paper we discuss the non-symmetric case.
In  it was shown that if is a G-metric space, putting , is a quasi-metric space (δ is not symmetric). It is well known that any quasi-metric induces different metrics and mostly used are
The following result is an immediate consequence of above definitions and relations.
Recently, Samet et al. and Jleli-Samet  observed that some fixed point theorems in the context of a G-metric space can be proved (by simple transformation) using related existing results in the setting of a (quasi-) metric space. Namely, if the contraction condition of the fixed point theorem on G-metric space can be reduced to two variables, then one can construct an equivalent fixed point theorem in setting of usual metric space. This idea is not completely new, but it was not successfully used before (see ). Very recently, Karapinar and Agarwal suggest new contraction conditions in G-metric space in a way that the techniques in [13,14] are not applicable. In this approach , contraction conditions cannot be expressed in two variables. So, in some cases, as is noticed even in Jleli-Samet’s paper , when the contraction condition is of nonlinear type, this strategy cannot be always successfully used. This is exactly the case in our paper.
2 Main result
A generalization of the contraction principle can be obtained using a different type of a nondecreasing function . The most usual additional properties imposed on φ are given using a combination of the next seven conditions:
Some of the noted properties of φ are equivalent, some of them imply others, some of them are incompatible. The next lemma discusses some of the relations between properties ()-(), especially those which are used in this paper to define a generalized contraction.
(iii), (iv) are obvious, so the proof is omitted.
(v) The function
(vi) The function
(vii) The function
The last inequality together with (1) implies
for all . So, one can apply the Matkowski fixed point theorem if the function satisfies the conditions () and (). Since there exist functions φ which satisfy () and (), but 2φ does not (for example , ), the Jleli-Samet technique  is not applicable. We are going to prove our theorem using the G-metric G.
Using (1), we get
From the proof of Theorem 2.1 we can see that it would be enough to impose certain assumptions not for all elements from X but only over some subset B of X, just as was done by Guseman . The next theorem is a Guseman type of fixed point theorem in a G-metric space.
Theorem 2.2LetTbe a selfmapping of a completeG-metric space. If there exists a subsetBofXsuch that, Tsatisfies (1) overBand for some, , then there exists a uniquesuch thatandfor each. Furthermore, ifTsatisfies (1) overX, thenuis unique fixed point inXandfor each.
Proof Since the function , , satisfies (), (), (), and (), we can apply Theorem 2.1. Also for that φ, the appropriate version of Theorem 2.2 can be formulated in a similar way as was done in this corollary. □
If , for each , it is easy to see that condition () or () in Theorem 2.1 can be omitted. This version of Theorem 1.2 is an improvement and another proof of Theorem 3.1 (Corollary 3.2) from . But in that case it would be more appropriate to use the metric ρ, which reduces (1) to , and enables the use of well-known results in metric spaces.
The next theorem is also a Guseman type of fixed point theorem in a G-metric space. The assumptions about the contractor φ is different with respect to Theorem 2.2. Similarly as in previous analysis, the next theorem can be applied in a metric space and in cases where some special form of function φ is used.
and by (7)
In the last theorem in this paper we consider a common fixed point for a family of selfmappings with the property of a contractive iterate at a point. The generalized contractive condition is imposed over a subset of a G-metric space.
for all, , and all, whereis a nondecreasing right continuous function satisfying (). If there existssuch thatfor all, thenis a unique common fixed point forinBand for every, the sequence, , converges to.
If we choose the option that
it implies that
On the other hand, in that case
It is obvious that (9) contradicts (10). So,
The authors declare that they have no competing interests.
Both authors have equal contribution in the paper and they read and approved the final manuscript.
The authors are very grateful to the anonymous referees for their careful reading of the paper and suggestions, which have contributed to the improvement of the paper. This work is supported by Ministry of Science and Technological Development, Republic of Serbia.
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