On mappings with φ-contractive iterate at a point on generalized metric spaces
© Gaji¿ and Stojakovi¿; licensee Springer. 2014
Received: 18 July 2013
Accepted: 6 February 2014
Published: 21 February 2014
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.
1 Introduction and preliminaries
In 1975 Matkowski introduced the following class of mappings.
Definition 1.1 
φ is nondecreasing,
for all ,
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 , and it was generalized in different directions [3–10]. Matkowski generalized his own result proving a theorem of Segal-Guseman type .
Theorem 1.1 
then T has 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  introduced the correct definition of a generalized metric space as follows.
Definition 1.2 
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 
Example 1.2 
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 
if , then ,
Definition 1.3 
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 
is G-convergent to x,
Definition 1.4 
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 
the sequence is G-Cauchy,
for every , there exists an such 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 
Let be a G-metric space, then the function is 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  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.
is G-convergent to if and only if is convergent to x in ;
is G-Cauchy if and only if is Cauchy in ;
is G-complete if and only if is complete.
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:
() , 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.
() ⇔ () ⇒ (),
if φ is right continuous, then () ⇔ () ⇔ (),
() ⇒ () ⇒ () ⇒ (), where ,
() ⇔ (),
() ⇏ (), () ⇏ (),
() + () ⇏ () and () + () ⇏ (),
() ⇏ () 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.
- (vi)The function
- (vii)The function
satisfies (), but not (), nor (). □
for all . Then T has a unique fixed point . Moreover, for each , and is continuous at a.
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.
We first prove by mathematical induction that, for every , the orbit is bounded.
The last inequalities imply that . Suppose that there exists a positive integer j such that , but for .
i.e. , which contradicts the choice of c. Therefore for , and consequently the orbit is bounded, so .
Since , is a Cauchy sequence in a complete G-metric space, , .
From the last contradiction we conclude that .
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 .
By (), , which implies that .
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. □
for all and some . Then T has a unique fixed point . Moreover, for each , and is continuous at a.
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 . The next theorem is a Guseman type of fixed point theorem in a G-metric space.
Theorem 2.2 Let T be a selfmapping of a complete G-metric space . If there exists a subset B of X such that , T satisfies (1) over B and for some , , then there exists a unique such that and for each . Furthermore, if T satisfies (1) over X, then u is unique fixed point in X and for each .
for all . Then T has a unique fixed point . Moreover, for each , and is continuous at a.
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.
for all and some , then T has a unique fixed point . Moreover, for each , and is continuous at a.
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.
for all , then the sequence , , converges to some .
If inequality (4) holds for all , then and for every . If , then is a fixed point of f.
implying that is a Cauchy sequence. Since is complete, and there exists an such that .
where . Since () ⇒ (), .
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.
for all , , and all , where is a nondecreasing right continuous function satisfying (). If there exists such that for all , then is a unique common fixed point for in B and for every , the sequence , , converges to .
By , , since , we have a contradiction, that is, the assumption is not correct.
it follows that for all .
Now, for some , we form the sequence .
If , then and the sequence converges to .
Since , and . The last relation proves that the sequence converges to . □
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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