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Common fixed point theorems for expansion mappings in various abstract spaces using the concept of weak reciprocal continuity
Fixed Point Theory and Applications volume 2012, Article number: 221 (2012)
Abstract
In this paper, we prove expansion mapping theorems using the concept of compatible maps, weakly reciprocal continuity, R-weakly commuting mappings, R-weakly commuting of type , and in metric spaces and in G-metric spaces.
MSC:54H25, 47H10.
1 Metric spaces
In 1922, Banach proved a common fixed point theorem which ensures, under appropriate conditions, the existence and uniqueness of a fixed point. This result of Banach is known as Banach’s fixed point theorem or Banach contraction principle. Many authors have extended, generalized and improved Banach’s fixed point theorem in different ways.
Jungck [1] proved a common fixed point theorem for commuting maps, which generalized the Banach fixed point theorem. This theorem has had many applications but suffers from one drawback that the continuity of a map throughout the space is needed. Jungck [2] defined the concept of compatible mappings.
Definition 1 ([2], see also [3])
A pair of self-mappings of a metric space is said to be compatible if whenever is a sequence in X such that for some z in X.
In 1994, Pant [4] introduced the notion of pointwise R-weak commutativity in metric spaces.
Definition 2 ([4], see also [5])
A pair of self-mappings of a metric space is said to be R-weakly commuting at a point x in X if for some .
Definition 3 ([4])
Two self-maps f and g of a metric space are called pointwise R-weakly commuting on X if, given x in X, there exists such that .
In 1997, Pathak et al. [6] generalized the notion of R-weakly commuting mappings to R-weakly commuting mappings of type and of type .
Definition 4 ([6])
Two self-maps f and g of a metric space are called R-weakly commuting of type if there exists some such that for all x in X.
Similarly, two self-mappings f and g of a metric space are called R-weakly commuting of type if there exists some such that for all x in X.
It is obvious that pointwise R-weakly commuting maps commute at their coincidence points and pointwise R-weak commutativity is equivalent to commutativity at coincidence points. It may be noted that both compatible and non-compatible mappings can be R-weakly commuting of type or but converse may not be true.
Definition 5 ([6])
Two self-maps f and g of a metric space are called R-weakly commuting of type if there exists some such that for all x in X.
In 1999, Pant [7] introduced a new continuity condition, known as reciprocal continuity, and obtained a common fixed point theorem by using the compatibility in metric spaces. He also showed that in the setting of common fixed point theorems for compatible mappings satisfying contraction conditions, the notion of reciprocal continuity is weaker than the continuity of one of the mappings. The notion of pointwise R-weakly commuting mappings increased the scope of the study of common fixed point theorems from the class of compatible to the wider class of pointwise R-weakly commuting mappings. Subsequently, several common fixed point theorems have been proved by combining the ideas of R-weakly commuting mappings and reciprocal continuity of mappings in different settings.
Definition 6 ([7])
Two self-mappings f and g are called reciprocally continuous if and , whenever is a sequence such that for some z in X.
If f and g are both continuous, then they are obviously reciprocally continuous, but the converse is not true.
Recently, Pant et al. [8] generalized the notion of reciprocal continuity to weak reciprocal continuity as follows.
Definition 7 ([8])
Two self-mappings f and g are called weakly reciprocally continuous if or whenever is a sequence such that for some z in X.
If f and g are reciprocally continuous, then they are obviously weak reciprocally continuous, but the converse is not true. Now, as an application of weak reciprocal continuity, we prove common fixed point theorems under contractive conditions that extend the scope of the study of common fixed point theorems from the class of compatible continuous mappings to a wider class of mappings which also includes non-compatible mappings.
Theorem 1 Let f and g be two weakly reciprocally continuous self-mappings of a complete metric space satisfying the following conditions:
for any and , we have that
If f and g are either compatible or R-weakly commuting of type or R-weakly commuting of type or R-weakly commuting of type , then f and g have a unique common fixed point.
Proof Let be any point in X. Since , there exists a sequence of points such that .
Define a sequence in X by
Now, we will show that is a Cauchy sequence in X. For proving this, from (1.2), we have
Hence,
Therefore, for all (a set of natural numbers), , we have
Thus, is a Cauchy sequence in X. Since X is complete, there exists a point z in X such that . Therefore, by (1.3), we have .
Suppose that f and g are compatible mappings. Now, by the weak reciprocal continuity of f and g, we obtain or . Let , then the compatibility of f and g gives , that is, .
Hence, . From (1.3), we get .
Therefore, from (1.2), we get
Taking the limit as , we get
Hence, . Again, the compatibility of f and g implies the commutativity at a coincidence point. Hence, . Using (1.2), we obtain
which proves that . We also get and then gz is a common fixed point of f and g.
Next, suppose that . The assumption implies that for some and therefore, .
The compatibility of f and g implies that . By virtue of (1.3), we have . Using (1.2), we get
Taking the limit as , we get
Then we get . The compatibility of f and g yields . Finally, using (1.2), we obtain
that is, . We also have and gu is a common fixed point of f and g.
Now, suppose that f and g are R-weakly commuting of type . Now, the weak reciprocal continuity of f and g implies that or . Let us first assume that . Then the R-weak commutativity of type of f and g yields
and therefore
This proves that . Again, using (1.2), we get
Taking the limit as , we get
Hence, we get . Again, by using the R-weak commutativity of type , we have
This yields . Therefore, . Using (1.2), we get
that is, . Then we also get and gz is a common fixed point of f and g.
Similar proof works in the case where .
Suppose that f and g are R-weakly commuting of type . Again, as done above, we can easily prove that fz is a common fixed point of f and g.
Finally, suppose that f and g are R-weakly commuting of type . The weak reciprocal continuity of f and g implies that or . Let us assume that . Then the R-weak commutativity of type of f and g yields
Taking the limit as , we get
that is, .
Using (1.1) and (1.3), we have, as , which gives as . Also, using (1.2), we get
Taking the limit as , we get
Hence, . Again, by using the R-weak commutativity of type ,
This yields .
Therefore, . Using (1.2), we get
This proves that . Hence, and gz is a common fixed point of f and g.
Similar proof works in the case where .
Uniqueness of the common fixed point theorem follows easily in each of the four cases by using (1.2). □
2 G-metric spaces
In 1963, Gahler [9] introduced the concept of 2-metric spaces and claimed that a 2-metric is a generalization of the usual notion of a metric, but some authors proved that there is no relation between these two functions. It is clear that in 2-metric, is to be taken as the area of the triangle with vertices x, y and z in . However, Hsiao [10] showed that for every contractive definition, with , every orbit is linearly dependent, thus rendering fixed point theorems in such spaces trivial.
In 1992, Dhage [11] introduced the concept of a D-metric space. The situation for a D-metric space is quite different from that for 2-metric spaces. Geometrically, a D-metric represents the perimeter of the triangle with vertices x, y and z in . Recently, Mustafa and Sims [10] have shown that most of the results concerning Dhage’s D-metric spaces are invalid. Therefore, they introduced an improved version of the generalized metric space structure, which they called G-metric spaces.
In 2006, Mustafa and Sims [12] introduced the concept of G-metric spaces as follows.
Definition 8 ([12])
Let X be a nonempty set, and let be a function satisfying the following axioms:
(G1) if ,
(G2) for all with ,
(G3) for all with ,
(G4) (symmetry in all three variables),
(G5) for all (rectangle inequality).
Then the function G is called a generalized metric or, more specifically, a G-metric on X and the pair is called a G-metric space.
Definition 9 ([12])
Let be a G-metric space and let be a sequence of points in X. A point x in X is said to be the limit of the sequence , , and one says that the sequence is G-convergent to x.
Thus, , or in a G-metric space if for each , there exists a positive integer N such that for all .
Now, we state some results from the papers [10, 12–15] which are helpful for proving our main results.
Proposition 1 ([12])
Let be a G-metric space. Then the following are equivalent:
-
(1)
is G convergent to x,
-
(2)
as ,
-
(3)
as ,
-
(4)
as .
Definition 10 ([12])
Let be a G-metric space. A sequence is called G-Cauchy if, for each , there exists a positive integer N such that for all ; i.e., if as .
Proposition 2 ([15])
If is a G-metric space, then the following are equivalent:
-
(1)
the sequence is G-Cauchy,
-
(2)
for each , there exists a positive integer N such that for all .
Proposition 3 ([12])
Let be a G-metric space. Then the function is jointly continuous in all three of its variables.
Definition 11 ([12])
A G-metric space is called a symmetric G-metric space if
Proposition 4 ([14])
Every G-metric space will define a metric space by
-
(i)
for all x, y in X.
If is a symmetric G-metric space, then
-
(ii)
for all x, y in X.
However, if is not symmetric, then it follows from the G-metric properties that
-
(iii)
for all x, y in X.
Definition 12 ([14])
A G-metric space is said to be G-complete if every G-Cauchy sequence in is G-convergent in X.
Proposition 5 ([14])
A G-metric space is G-complete if and only if is a complete metric space.
Proposition 6 ([15])
Let be a G-metric space. Then, for any , it follows that
-
(i)
if , then ,
-
(ii)
,
-
(iii)
,
-
(iv)
,
-
(v)
,
-
(vi)
.
Definition 13 ([16])
A pair of self-mappings of a G-metric space is said to be compatible if or whenever is a sequence in X such that for some z in X.
Definition 14 ([17])
A pair of self-mappings of a G-metric space is said to be R-weakly commuting at a point x in X if for some .
Definition 15 ([17])
Two self-maps f and g of a G-metric space are called pointwise R-weakly commuting on X if, given x in X, there exists such that .
Definition 16 ([6])
Two self-maps f and g of a G-metric space are called R-weakly commuting of type if there exists some such that for all x in X. Similarly, two self-mappings f and g of a G-metric space are called R-weakly commuting of type if there exists some such that for all x in X.
Definition 17 ([6])
Two self-mappings f and g of a G-metric space are called R-weakly commuting of type if there exists some such that for all x in X.
Theorem 2 Let f and g be two weakly reciprocally continuous self-mappings of a complete G-metric space satisfying the following conditions:
for any and , we have that
If f and g are either compatible or R-weakly commuting of type or R-weakly commuting of type or R-weakly commuting of type , then f and g have a unique common fixed point.
Proof Let be any point in X. Since , there exists a sequence of points such that .
Define a sequence in X by
Now, we will show that is a G-Cauchy sequence in X. For proving this, by (2.2) take , , , we have
Continuing in the same way, we have
Therefore, for all (a set of natural numbers), , we have by using (G5)
Thus, is a G-Cauchy sequence in X. Since is a complete G-metric space, there exists a point z in X such that . Therefore, by (2.3), we have .
Suppose that f and g are compatible mappings. Now, the weak reciprocal continuity of f and g implies that or . Let , then the compatibility of f and g gives , that is, .
Hence, . From (2.3), we get .
Therefore, by (2.2), we get
Taking the limit as , we get
Hence, . Again, the compatibility of f and g implies the commutativity at a coincidence point. Hence, . Now, we claim that . Suppose not, then by using (2.2), we obtain
and
which gives contradiction because . Hence, . Hence, and gz is a common fixed point of f and g.
Next suppose that . The assumption implies that for some and therefore, .
The compatibility of f and g implies that . By virtue of (2.3), this gives . Using (2.2), we get
Taking the limit as , we get
which gives . The compatibility of f and g yields . Finally, we claim that . Suppose not, then by using (2.2), we obtain
and
that is, . Hence, and gu is a common fixed point of f and g.
Now, suppose that f and g are R-weakly commuting of type . Now, the weak reciprocal continuity of f and g implies that or . Let us first assume that . Then the R-weak commutativity of type of f and g yields
Taking the limit as , we get
This proves that . Again, using (2.2), we get
Taking the limit as , we get
Hence, we get . Again, by using the R-weak commutativity of type ,
This yields . Therefore, . We claim that . Suppose not, using (2.2), we get
and
a contradiction, that is, . Hence, and gz is a common fixed point of f and g.
Similar proof works in the case where .
Suppose that f and g are R-weakly commuting of type . Again, as done above, we can easily prove that gz is a common fixed point of f and g.
Finally, suppose f and g are R-weakly commuting of type . The weak reciprocal continuity of f and g implies that or . Let us assume that . Then the R-weak commutativity of type of f and g yields
Taking the limit as , we get
That is,
Using (2.1) and (2.3), we have as , which gives as . Also, using (2.2), we get
Taking the limit as , we get
Hence, . Again, by using the R-weak commutativity of type ,
This yields .
Therefore, . Finally, we claim that . Suppose not, using (2.2), we get
and
a contradiction, we get . Hence, and gz is a common fixed point of f and g.
Similar proof works in the case where .
Uniqueness of the common fixed point theorem follows easily in each of the four cases by using (2.2). □
We now give an example (see also [18]) to illustrate Theorem 2.
Example 1 Let be a G-metric space, where and
for all . Define by
Let be a sequence in X such that either or for each n.
Then, clearly, f and g satisfy all the conditions of Theorem 2 and have a unique common fixed point at .
References
Jungck G: Commuting mappings and fixed points. Am. Math. Mon. 1976, 83: 261–263. 10.2307/2318216
Jungck G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci. 1986, 9: 771–779. 10.1155/S0161171286000935
Sintunavarat W, Kumam P: Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces. J. Appl. Math. 2011., 2011: Article ID 637958
Pant RP: Common fixed points of non-commuting mappings. J. Math. Anal. Appl. 1994, 188: 436–440. 10.1006/jmaa.1994.1437
Sintunavarat W, Kumam P: Common fixed points for R -weakly commuting in fuzzy metric spaces. Ann. Univ. Ferrara 2012, 58: 389–406. 10.1007/s11565-012-0150-z
Pathak HK, Cho YJ, Kang SM: Remarks of R -weakly commuting mappings and common fixed point theorems. Bull. Korean Math. Soc. 1997, 34: 247–257.
Pant RP: Common fixed points of four mappings. Bull. Calcutta Math. Soc. 1998, 90: 281–286.
Pant RP, Bisht RK, Arora D: Weak reciprocal continuity and fixed point theorems. Ann. Univ. Ferrara 2011, 57: 181–190. 10.1007/s11565-011-0119-3
Gahler S: 2-metrices Raume und ihre topologische Struktur. Math. Nachr. 1963, 26: 115–148. 10.1002/mana.19630260109
Mustafa Z, Sims B: Some remarks concerning D -metric spaces. In Proceedings of International Conference on Fixed Point Theory and Applications. Yokohama Publishers, Yokohama; 2004:189–198. Valencia, Spain, 13-19 July
Dhage BC: Generalized metric spaces and mappings with fixed point. Bull. Calcutta Math. Soc. 1992, 84: 329–336.
Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7: 289–297.
Manro S, Kumar S, Bhatia SS: Weakly compatible maps of typein G -metric spaces. Demonstr. Math. 2012, 45(4):901–908.
Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorems for mappings on complete G -metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 18970
Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 917175
Manro S, Bhatia SS, Kumar S: Expansion mapping theorems in G -metric spaces. Int. J. Contemp. Math. Sci. 2010, 5(51):2529–2535.
Manro S, Kumar S, Bhatia SS: R weakly commuting maps in G metric spaces. Fasc. Math. 2011, 47: 11–17.
Pant RP: A common fixed point theorem under a new condition. Indian J. Pure Appl. Math. 1999, 30: 147–152.
Acknowledgements
The authors thank the editor and the referees for their useful comments and suggestions. The second author would like to thank the Higher Education Research Promotion and National Research University Project of Thailand’s Office of the Higher Education Commission for financial support.
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An erratum to this article is available at http://dx.doi.org/10.1186/1687-1812-2013-8.
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Manro, S., Kumam, P. Common fixed point theorems for expansion mappings in various abstract spaces using the concept of weak reciprocal continuity. Fixed Point Theory Appl 2012, 221 (2012). https://doi.org/10.1186/1687-1812-2012-221
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DOI: https://doi.org/10.1186/1687-1812-2012-221