Abstract
We prove a set of fixed point theorems for mappings with φcontractive iterate at a point in a class of generalized metric spaces. This is a generalization of some wellknown results for Guseman and Matkowski types of fixed point results in metric and generalized metric spaces.
MSC: 47H10, 54H25.
Keywords:
φcontraction; fixed point; mapping with contractive iterate at a point; generalized metric space1 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,
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 MeirKeeler contraction [2], and it was generalized in different directions [310]. Matkowski generalized his own result proving a theorem of SegalGuseman 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 2metric 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 socalled Dmetric 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:
(G4) (symmetry in all three variables);
Then the function G is called a generalized metric, abbreviated Gmetric, on X, and the pair is called a Gmetric 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 Gmetrics on X by
Example 1.2[12]
and extend G to by using the symmetry in the variables. Then it is clear that is a Gmetric space.
The following useful properties of a Gmetric are readily derived from the axioms.
Proposition 1.1[12]
Letbe aGmetric space, then for anyx, y, zandafromXit follows that:
Definition 1.3[12]
Let be a Gmetric 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 Gconvergent to x.
Proposition 1.2[12]
Letbe aGmetric space, then for a sequenceand a pointthe following are equivalent:
Definition 1.4[12]
Let be a Gmetric space, a sequence is called GCauchy if for every , there is such that , for all , that is, if as .
Proposition 1.3[12]
In aGmetric space, the following are equivalent:
2. for every, there exists ansuch that, for all.
A Gmetric space is Gcomplete (or complete Gmetric), if every GCauchy sequence in is Gconvergent in .
Proposition 1.4[12]
Letbe aGmetric space, then the functionis jointly continuous in all three of its variables.
Definition 1.5 is symmetric Gmetric space if for all .
Fixed point theorems in symmetric Gmetric space are mostly consequences of the related fixed point results in metric spaces. In this paper we discuss the nonsymmetric case.
In [13] it was shown that if is a Gmetric space, putting , is a quasimetric space (δ is not symmetric). It is well known that any quasimetric induces different metrics and mostly used are
The following result is an immediate consequence of above definitions and relations.
Theorem 1.2Letbe aGmetric space and let. Then
1. isGconvergent toif and only ifis convergent toxin;
2. isGCauchy if and only ifis Cauchy in;
3. isGcomplete if and only ifis complete.
Recently, Samet et al.[14] and JleliSamet [13] observed that some fixed point theorems in the context of a Gmetric 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 Gmetric 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 Gmetric 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 JleliSamet’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 Gmetric spaces, we refer the reader to [3,6,9,10,1536].
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:
() if is a sequence such that , then ,
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
(ii) ifφis right continuous, then () ⇔ () ⇔ (),
(iii) () ⇒ () ⇒ () ⇒ (), where,
(vi) () + () ⇏ () 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 completeGmetric space, , where the nondecreasing functionφsatisfies () or () together with () or () and for eachthere exists a positive integersuch that
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 nonsymmetric Gmetric 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 JleliSamet technique [13] is not applicable. We are going to prove our theorem using the Gmetric 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:
We shall prove that is a Cauchy sequence. Let . From (2) we obtain
Since , is a Cauchy sequence in a complete Gmetric 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
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 completeGmetric 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 Gmetric space.
Theorem 2.2LetTbe a selfmapping of a completeGmetric 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 completeGmetric 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 wellknown results in metric spaces.
Proposition 2.1Letbe a completeGmetric space, whereφis a nondecreasing function satisfying (). Ifsatisfies
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 Gmetric 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 aGmetric space and letbe a subadditive mapping satisfying (). If for somethe closure of orbitis complete and for eachthere exists ansuch that
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:
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:
and
By (6)
and by (7)
Next, we claim that , for each . Putting , , , we get
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 Gmetric space.
Theorem 2.4LetbeGmetric space and. Further, letbe the sequence of selfmappings ofXsuch that for all, and for eachthere exists ansuch that
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
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
On the other hand, in that case
that is,
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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