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
(i) φ is nondecreasing,
(ii)
(iii)
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]
Let
thenThas a unique fixed point
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
(G1)
(G2)
(G3)
(G4)
(G5)
Then the function G is called a generalized metric, abbreviated Gmetric, on X, and the pair
Clearly these properties are satisfied when
Example 1.1[12]
Let
Example 1.2[12]
Let
and extend G to
The following useful properties of a Gmetric are readily derived from the axioms.
Proposition 1.1[12]
Let
1. if
2.
3.
4.
5.
6.
Definition 1.3[12]
Let
Proposition 1.2[12]
Let
1.
2.
3.
Definition 1.4[12]
Let
Proposition 1.3[12]
In aGmetric space
1. the sequence
2. for every
A Gmetric space
Proposition 1.4[12]
Let
Definition 1.5
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
(μ)
(ρ)
The following result is an immediate consequence of above definitions and relations.
Theorem 1.2Let
1.
2.
3.
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
(
(
(
(
(
(
(
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 (
Lemma 2.1Let
(i) (
(ii) ifφis right continuous, then (
(iii) (
(iv) (
(v) (
(vi) (
(vii) (
Proof (i) (
(
(
(ii) It is enough to prove that (
(iii), (iv) are obvious, so the proof is omitted.
(v) The function
satisfies (
(vi) The function
satisfies (
(vii) The function
satisfies (
Theorem 2.1Let
for all
Proof Let the nondecreasing function φ satisfy (
The last inequality together with (1) implies
for all
We first prove by mathematical induction that, for every
Fix
By (
The last inequalities imply that
Using (1), we get
i.e.
For any
We shall prove that
With the notation
Since
In order to prove that
meaning that there exists
Hence,
From the last contradiction we conclude that
Suppose that there is a point
This contradiction proves that a is a unique fixed point of
Next, we claim that
Then
By (
To prove continuity of
for
In other cases (when we use (
Corollary 2.1Let
for all
Proof The function
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
Remark 2.1 Taking
Corollary 2.2Let
for all
Proof Since the function
If
Proposition 2.1Let
for all
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
for all
If inequality (4) holds for all
Proof First, we show that
Putting
and consequently,
for all
implying that
In the second part of the theorem, inequality (4) holds for all
and
By (6)
and by (7)
Hence,
Next, we claim that
where
To show that
that is,
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 2.4Let
for all
Proof First we prove that
By
Further, since
it follows that
Now, for some
If
If
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
Since
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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