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.
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 , and it was generalized in different directions [3-10]. Matkowski generalized his own result proving a theorem of Segal-Guseman type .
Let be a complete metric space, , and . Ifφis nondecreasing, , for , and for each there is a positive integer such 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  introduced the correct definition of a generalized metric space as follows.
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.
Let be an ordinary metric space, then defines G-metrics on X by
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.
Let be aG-metric space, then for anyx, y, zandafromXit follows that:
1. if , then ,
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.
Let be aG-metric space, then for a sequence and a point the following are equivalent:
1. isG-convergent tox,
2. as ,
3. as .
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 .
In aG-metric space , the following are equivalent:
1. the sequence isG-Cauchy,
2. 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 .
Let be aG-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.
Theorem 1.2Let be aG-metric space and let . Then
1. isG-convergent to if and only if is convergent toxin ;
2. isG-Cauchy if and only if is Cauchy in ;
3. isG-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.
Lemma 2.1Let be 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 2.1Let be a completeG-metric space, , where the nondecreasing functionφsatisfies ( ) or ( ) together with ( ) or ( ) and for each there exists a positive integer such that
for all . ThenThas a unique fixed point . Moreover, for each , and is 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  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:
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
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
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 2.1Let be a completeG-metric space, and for each there exists a positive integer such that
for all and some . ThenThas a unique fixed point . Moreover, for each , and is 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 . 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 unique such that and for each . Furthermore, ifTsatisfies (1) overX, thenuis unique fixed point inXand for each .
Corollary 2.2Let be a completeG-metric space, , and for each there exists a positive integer such that
for all . ThenThas a unique fixed point . Moreover, for each , and is 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 . 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 2.1Let be a completeG-metric space, whereφis a nondecreasing function satisfying ( ). If satisfies
for all and some , thenThas a unique fixed point . Moreover, for each , and is 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 , where is aG-metric space and let be a subadditive mapping satisfying ( ). If for some the closure of orbit is complete and for each there exists an such that
for all , then the sequence , , converges to some .
If inequality (4) holds for all , then and for every . If , then is 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:
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 (7)
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 2.4Let beG-metric space and . Further, let be the sequence of selfmappings ofXsuch that for all , and for each there exists an such that
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 inBand 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.
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
On the other hand, in that case
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 . □
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.
Meir, A, Keeler, E: A theorem on contraction mappings. J. Math. Anal. Appl.. 28, 326–329 (1969). Publisher Full Text
Suzuki, T, Vetro, C: Three existence theorems for weak contraction of Matkowski type. Int. J. Math. Stat.. 6, 110–120 (2010). Publisher Full Text
Guseman, LF: Fixed point theorems for mappings with a contractive iterate at a point. Proc. Am. Math. Soc.. 26, 615–618 (1970). Publisher Full Text
Article ID 210BioMed Central Full Text
Article ID 154BioMed Central Full Text
Abbas, M, Rhoades, BE: Common fixed point results for noncommuting mapping without continuity in generalized metric space. Appl. Math. Comput.. 215, 262–269 (2009). Publisher Full Text
Aydi, H, Shatanawi, W, Vetro, C: On generalized weakly G-contraction mapping in G-metric spaces. Comput. Math. Appl.. 62, 4222–4229 (2011). Publisher Full Text
Article ID 458086Publisher Full Text
Gajić, L, Lozanov-Crvenković, Z: A fixed point result for mappings with contractive iterate at a point in G-metric spaces. Filomat. 25(2), 53–58 (2011). Publisher Full Text
Gajić, L, Stojaković, M: On Ćirić generalization of mappings with a contractive iterate at a point in G-metric space. Appl. Math. Comput.. 219, 435–441 (2012). Publisher Full Text
Gholizadeh, L, Saadati, R, Shatanawi, W, Vaezpour, SM: Contractive mapping in generalized, ordered metric spaces with application in integral equations. Math. Probl. Eng.. 2011, Article ID 380784 (2011)
Article ID 380784Publisher Full Text
Article ID 283028Publisher Full Text
Mustafa, Z, Aydi, H, Karapinar, E: On common fixed points in G-metric spaces using (E.A) property. Comput. Math. Appl. (2012). Publisher Full Text
Article ID 815870Publisher Full Text
Article ID 126205Publisher Full Text
Tahat, N, Aydi, H, Karapinar, E, Shatanawi, W: Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces. Fixed Point Theory Appl.. 2012, Article ID 48 (2012)
Article ID 48BioMed Central Full Text
An, TV, Dung, NV, Hang, VTL: A new approach to fixed point theorems on G-metric spaces. Topol. Appl.. 160(12), 1486–1493 (2013). Publisher Full Text