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# On mappings with φ-contractive iterate at a point on generalized metric spaces

Ljiljana Gajić1 and Mila Stojaković2*

Author Affiliations

1 Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Novi Sad, Serbia

2 Department of Mathematics, Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

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Fixed Point Theory and Applications 2014, 2014:46  doi:10.1186/1687-1812-2014-46

 Received: 18 July 2013 Accepted: 6 February 2014 Published: 21 February 2014

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.

### Abstract

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.

Definition 1.1[1]

Let T be a mapping on a metric space . Then T is called a weak contraction if there exists a function φ from to itself satisfying the following:

(i) φ is nondecreasing,

(ii) for all ,

(iii) for all .

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 [2], and it was generalized in different directions [3-10]. Matkowski generalized his own result proving a theorem of Segal-Guseman type [11].

Theorem 1.1[4]

Letbe a complete metric space, , and. Ifφis nondecreasing, , for, and for eachthere is a positive integersuch that for all,

thenThas a unique fixed point. Moreover, for each, .

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 [12] introduced the correct definition of a generalized metric space as follows.

Definition 1.2[12]

Let X be a nonempty set, and let be a function satisfying the following properties:

(G1) if ;

(G2) , for all , with ;

(G3) , for all , with ;

(G4) (symmetry in all three variables);

(G5) , for all .

Then the function G is called a generalized metric, abbreviated G-metric, on X, and the pair is called a G-metric space.

Clearly these properties are satisfied when is the perimeter of the triangle with vertices at x, y, and . Moreover, taking a in the interior of the triangle shows that (G5) is the best possible.

Example 1.1[12]

Let be an ordinary metric space, then defines G-metrics on X by

Example 1.2[12]

Let . Define G on by

and extend G to by using the symmetry in the variables. Then it is clear that is a G-metric space.

The following useful properties of a G-metric are readily derived from the axioms.

Proposition 1.1[12]

Letbe aG-metric space, then for anyx, y, zandafromXit follows that:

1. if, then,

2. ,

3. ,

4. ,

5. ,

6. .

Definition 1.3[12]

Let be a G-metric space, and let be a sequence of points of X. A point is said to be the limit of the sequence if , and one says that the sequence is G-convergent to x.

Proposition 1.2[12]

Letbe aG-metric space, then for a sequenceand a pointthe following are equivalent:

1. isG-convergent tox,

2. as,

3. as.

Definition 1.4[12]

Let be a G-metric space, a sequence is called G-Cauchy if for every , there is such that , for all , that is, if as .

Proposition 1.3[12]

In aG-metric space, the following are equivalent:

1. the sequenceisG-Cauchy,

2. for every, there exists ansuch that, for all.

A G-metric space is G-complete (or complete G-metric), if every G-Cauchy sequence in is G-convergent in .

Proposition 1.4[12]

Letbe aG-metric space, then the functionis jointly continuous in all three of its variables.

Definition 1.5 is symmetric G-metric space if for all .

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 [13] 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

(μ) ,

(ρ) .

The following result is an immediate consequence of above definitions and relations.

Theorem 1.2Letbe aG-metric space and let. Then

1. isG-convergent toif and only ifis convergent toxin;

2. isG-Cauchy if and only ifis Cauchy in;

3. isG-complete if and only ifis complete.

Recently, Samet et al.[14] and Jleli-Samet [13] 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 [15]). 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 [16], contraction conditions cannot be expressed in two variables. So, in some cases, as is noticed even in Jleli-Samet’s paper [13], when the contraction condition is of nonlinear type, this strategy cannot be always successfully used. This is exactly the case in our paper.

For more fixed point results for mappings defined in G-metric spaces, we refer the reader to [3,6,9,10,15-36].

### 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:

() ,

() , for all ,

() , for all ,

() if is a sequence such that , then ,

() for any there exists a , ,

() ,

() , for all .

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.

Lemma 2.1Letbe a nondecreasing function. Then

(i) () ⇔ () ⇒ (),

(ii) ifφis right continuous, then () ⇔ () ⇔ (),

(iii) () ⇒ () ⇒ () ⇒ (), where,

(iv) () ⇔ (),

(v) () ⇏ (), () ⇏ (),

(vi) () + () ⇏ () and () + () ⇏ (),

(vii) () ⇏ () and () ⇏ ().

Proof (i) () ⇒ (): If for some , , then, knowing that φ is nondecreasing, . It means that , which contradicts ().

() ⇒ (): Let be any sequence such that . Using the implication () ⇒ (), we get and .

() ⇒ (): We assume that for some , . Since () ⇒ (), the sequence satisfies condition , but it converges to . That contradicts ().

(ii) It is enough to prove that () ⇒ (): We assume that for some , . Since is a nonincreasing sequence, by the right continuity of φ, , i.e., which contradicts ().

(iii), (iv) are obvious, so the proof is omitted.

(v) The function

satisfies (), but not (), nor (), since for every , .

(vi) The function

satisfies (), (), and (), but not ().

(vii) The function

satisfies (), but not (), nor (). □

Theorem 2.1Letbe a completeG-metric space, , where the nondecreasing functionφsatisfies () or () together with () or () and for eachthere exists a positive integersuch that

(1)

for all. ThenThas a unique fixed point. Moreover, for each, andis continuous ata.

Proof Let the nondecreasing function φ satisfy () together with () (weak contraction in the sense of Matkowski). Then by Proposition 1.1(3), in a non-symmetric G-metric space we have

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 [13] is not applicable. We are going to prove our theorem using the G-metric G.

We first prove by mathematical induction that, for every , the orbit is bounded.

Fix , fix the integer s, and put

By () there exist c, , such that

The last inequalities imply that . Suppose that there exists a positive integer j such that , but for .

Using (1), we get

i.e., which contradicts the choice of c. Therefore for  , and consequently the orbit is bounded, so .

For any , we define sequence as follows:

(2)

We shall prove that is a Cauchy sequence. Let . From (2) we obtain

With the notation , we have

Since , is a Cauchy sequence in a complete G-metric space, , .

In order to prove that , we assume that . Using the same arguments as in the previous part of the proof, we see that

meaning that there exists such that

Hence,

From the last contradiction we conclude that .

Suppose that there is a point , , such that . Then, by (1),

This contradiction proves that a is a unique fixed point of . According to and from the uniqueness which has been proved already, we deduce that .

Next, we claim that , for each . To prove this, fix , , and put

Then

By (), , which implies that .

To prove continuity of at a, we consider any sequence converging to a. For any

for . Letting , we get . Hence, converges to , meaning that is continuous at a.

In other cases (when we use () instead of () or () instead of ()), by Lemma 2.1, the same conclusion can be drawn. □

Corollary 2.1Letbe a completeG-metric space, and for eachthere exists a positive integersuch that

for alland some. ThenThas a unique fixed point. Moreover, for each, andis continuous ata.

Proof The function , , satisfies () and (), so the corollary is a consequence of Theorem 2.1. □

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 [11]. 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.

Remark 2.1 Taking , , by Theorem 2.2 we obtain the fixed point result from [23] or [24], so Theorem 2.2 is also a generalization of the Guseman fixed point result from [11].

Corollary 2.2Letbe a completeG-metric space, , and for eachthere exists a positive integersuch that

for all. ThenThas a unique fixed point. Moreover, for each, andis continuous ata.

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 [9]. 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.

Proposition 2.1Letbe a completeG-metric space, whereφis a nondecreasing function satisfying (). Ifsatisfies

(3)

for alland some, thenThas a unique fixed point. Moreover, for each, andis continuous ata.

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.

Theorem 2.3Let, whereis aG-metric space and letbe a subadditive mapping satisfying (). If for somethe closure of orbitis complete and for eachthere exists ansuch that

(4)

for all, then the sequence, , converges to some.

If inequality (4) holds for all, thenandfor every. If, thenis a fixed point off.

Proof First, we show that is a Cauchy sequence. For sufficiently large , there exist , such that . Using (4), we get

Putting , for all , the next inequality holds:

(5)

and consequently,

for all . Using the last inequality, for every , , we have

implying that is a Cauchy sequence. Since is complete, and there exists an such that .

In the second part of the theorem, inequality (4) holds for all . Then the elements of the sequence from the previous part of the proof satisfy the next two relations:

(6)

and

(7)

By (6)

and by (7)

Hence, .

Next, we claim that , for each . Putting , , , we get

where . Since () ⇒ (), .

To show that is a unique fixed point of in , we assume that there exists another point with the same property. Then

that is, . Further, if , then , implying . □

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.

Theorem 2.4LetbeG-metric space and. Further, letbe the sequence of selfmappings ofXsuch that for all, and for eachthere exists ansuch that

(8)

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.

Proof First we prove that is a unique point in B with the property that , . If , , , , then

By , , since , we have a contradiction, that is, the assumption is not correct.

Further, since

it follows that for all .

Now, for some , we form the sequence .

If , then and the sequence converges to .

If , in order to prove that the sequence converges to , we consider the sequence , ,

If we choose the option that

it implies that

(9)

On the other hand, in that case

that is,

(10)

It is obvious that (9) contradicts (10). So,

Now, applying that procedure i times and letting , we get

Since , and . The last relation proves that the sequence converges to . □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

Both authors have equal contribution in the paper and they read and approved the final manuscript.

### Acknowledgements

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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