Abstract
In this paper, using the concept of the common property, we prove a common fixed point theorem for a class of twice power type weakly compatible mappings in generalized metric space. Our results do not rely on any commuting or continuity condition of the mappings. We also state some examples to illustrate our new results in symmetric and nonsymmetric generalized metric spaces. It should be pointed out that this is the first time to use common properties to discuss common fixed point problems of contractive mappings for twice power type in generalized metric spaces.
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1 Introduction and preliminaries
In 2006, Mustafa and Sims [1] introduced a new structure of generalized metric space which is called a G-metric. Based on the notion of generalized metric spaces, Mustafa et al. [2–5], Obiedat and Mustafa [6], Aydi et al. [7, 8], Gajić and Stojaković [9], Zhou and Gu [10], Shatanawi [11] obtained some fixed point results for mappings satisfying different contractive conditions. Chugh et al. [12] obtained some fixed point results for maps satisfying property P in G-metric spaces. The study of common fixed point problems in G-metric spaces was initiated by Abbas and Rhoades [13]. Subsequently, many authors obtained many common fixed point theorems for the mappings satisfying different contractive conditions; see [14–31] for more details. Recently, some authors using property in generalized metric space to prove common fixed point, such as Abbas et al. [32], Mustafa et al. [33], Long et al. [34], Gu and Yin [35], Gu and Shatanawi [36].
Recently, Jleli and Samet [37] and Samet et al. [38] 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 [39]).
Very recently, Karapınar and Agarwal suggest new contraction conditions in G-metric space in a way that the techniques in [37, 38] are not applicable. In this approach [40], contraction conditions cannot be expressed in two variables. So, in some cases, as is noticed even in Jleli-Samet’s paper [37], when the contraction condition is of nonlinear type, this strategy cannot be always successfully used. This is exactly the case in our paper.
The purpose of this paper is to use the concept of the common property and weakly compatible mappings to discuss common fixed point problem for a class of twice power type contractive mappings in the framework of a generalized metric space. Our results do not rely on any commuting or continuity condition of the mappings. We also state some examples to illustrate our new results in the framework of symmetric and nonsymmetric generalized metric spaces.
As far as we know, this is the first time to use common properties to discuss common fixed point problems of contractive mappings for twice power type in generalized metric spaces.
Now we give preliminaries and basic definitions which are used throughout the paper.
Definition 1.1 [1]
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 a G-metric on X, and the pair is called a G-metric space.
Definition 1.2 [1]
Let be a G-metric space, and let a sequence of points in X, a point x in X is said to be the limit of the sequence , , and one says that sequence is G-convergent to x.
Thus, if or in a G-metric space , then if for each , there exists a positive integer N such that for all .
Proposition 1.1 [1]
Let be a G-metric space. Then the following are equivalent:
-
(1)
is G-convergent to x.
-
(2)
as .
-
(3)
as .
-
(4)
as .
Definition 1.3 [1]
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 1.2 [1]
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 1.3 [1]
Let be a G-metric space. Then the function is jointly continuous in all three of its variables.
Definition 1.4 [1]
A G-metric space is said to be G-complete if every G-Cauchy sequence in is G-convergent in X.
Definition 1.5 [41]
Let f and g be self-maps of a set X. If for some x in X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g.
Definition 1.6 [41]
Two self-mappings f and g on X are said to be weakly compatible if they commute at coincidence points.
Definition 1.7 [32]
Let X be a G-metric space. Self-maps f and g on X are said to satisfy the G- property if there exists a sequence in X such that and are G-convergent to some .
Definition 1.8 [32]
Let be a G-metric space and A, B, S, and T be four self-maps on X. The pairs and are said to satisfy the common property if there exist two sequences and in X such that
for some .
Definition 1.9 [17]
Self-mappings f and g of a G-metric space are said to be compatible if and , whenever is a sequence in X such that
for some .
2 Main results
Theorem 2.1 Let be a G-metric space. Suppose mappings satisfying the following conditions:
for all , . If one of the following conditions is satisfied, then the pairs , , and have a common point of coincidence in X.
-
(i)
The subspace RX is closed in X, , , and the two pairs of and satisfy the common property.
-
(ii)
The subspace SX is closed in X, , , and the two pairs of and satisfy the common property.
-
(iii)
The subspace TX is closed in X, , , and the two pairs of and satisfy the common property.
Further, if the pairs , , and are weakly compatible, then f, g, h, R, S, and T have a unique common fixed point in X.
Proof First we suppose that RX is closed in X, , , and the two pairs of and satisfy the common property, then by Definition 1.8 we know that there exist two sequences and in X such that
for some .
Since , there exists a sequence in X such that . So we get . By the condition (2.1) we have
Letting , we have
this gives , since . Hence .
Since RX is a closed subspace of X, and , there exists a point such that . By the condition (2.1) we have
Letting , we have , hence . Thus , so u is the coincidence point of the pair .
Since and , there exists a point such that . By the condition (2.1) we have
Letting , we have , hence . Thus , so v is the coincidence point of the pair .
Since and , there exists a point such that . By the condition (2.1) we have
hence , since . Thus , so w is the coincidence point of the pair .
In the above proof we get . Then we get , , and , since the pairs , , and are weakly compatible. By the condition (2.1), we have
hence , since . Thus . Similarly, it can be shown that and , which means that t is a common fixed point of f, g, h, R, S, and T.
Now we prove the uniqueness of the common fixed point t.
Let t and p be two common fixed point of f, g, h, R, S, and T, then using the condition (2.1), we have
hence , since . Thus . So common fixed point is unique. □
Example 2.1 Let be a G-metric space with
We define mappings f, g, h, R, S, and T on X by
Clearly, from the above functions we know that the subspace RX is closed in X, , , and the pairs , , be weakly compatible. The pairs and satisfy the common property, let and for each be the required sequences.
Now we prove that the mappings f, g, h, R, S, and T are satisfying the condition (2.1) of Theorem 2.1 with . Let
Case (1) If , then we have
Thus we have
Case (2) If , , then we have
If , then
If , then
So we know . Thus we have
Case (3) If , , then we have
If , then
If , then
So we know . Thus we have
Case (4) If , , then we have
If , then
If , then
So we know . Thus we have
Case (5) , , then we have
If , then
If , then
So we know . Thus we have
Case (6) , , then we have
If , then
If , then
So we know . Thus we have
Case (7) , , then we have
If , , then
If , , then
If , , then
If , then
So we know . Thus we have
Case (8) If , then
Then in all the above cases, the mappings f, g, h, R, T, and S are satisfying the condition (2.1) of Theorem 2.1 with , so that all the conditions of Theorem 2.1 are satisfied. Moreover, is the unique common fixed point of f, g, h, R, T, and S.
The following example supports the usability of our results for nonsymmetric generalized metric spaces.
Example 2.2 Let be a set with G-metric defined by Table 1. It is easy to see that is a nonsymmetric generalized metric space. Let the maps be defined by Table 2.
Clearly, the subspace RX, SX, and TX are closed in X, , , and with the pairs , , and being weakly compatible. Also two pairs and satisfy the common property, indeed, and for each are the required sequences.
To check the contractive condition (2.1) for all , we consider the following cases.
Note that for Case (1) , (2) , , (3) , , (4) , , , (5) , , (6) , , , (7) , , (8) , , (9) , , (10) , , (11) , , , and (12) , .
We have , and hence (2.1) is obviously satisfied.
Case (13) If , , then , , , , hence we have
Case (14) If , , then , , , hence we have
Case (15) If , , then , , , hence we have
Case (16) If , , , then , , , , hence we have
Case (17) If , , then , , , hence we have
Case (18) If , , then , , , hence we have
Case (19) , then , , , hence we have
Case (20) If , , , then , , , hence we have
Case (21) If , , then , , , hence we have
Case (22) If , , then , , , hence we have
Case (23) If , , , then , , , hence we have
Case (24) If , , then , , , hence we have
Case (25) , , then , , , hence we have
Case (26) , , then , , , hence we have
Case (27) If , then , , , hence we have
Hence, all of the conditions of Theorem 2.1 are satisfied. Moreover, 0 is the unique common fixed point of f, g, h, R, S, and T.
Corollary 2.1 Let be a G-metric space. Suppose mappings satisfying the following conditions:
for all . Here () and . If one of the following conditions is satisfied, then the pairs , , and have a common point of coincidence in X.
-
(i)
The subspace RX is closed in X, , , and the two pairs of and satisfy the common property.
-
(ii)
The subspace SX is closed in X, , , and the two pairs of and satisfy the common property.
-
(iii)
The subspace TX is closed in X, , , and the two pairs of and satisfy the common property.
Further, if the pairs , , and are weakly compatible, then f, g, h, R, S, and T have a unique common fixed point in X.
Proof Suppose that
Then
So, if the condition (2.2) holds, then . Taking in Theorem 2.1, the conclusion of Corollary 2.1 can be obtained from Theorem 2.1, since . □
References
Mustafa Z, Sims B: A new approach to a generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7: 289-297.
Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorems for mappings on complete G -metric space. Fixed Point Theory Appl. 2008. Article ID 189870, 2008: 10.1155/2008/189870
Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory Appl. 2009. Article ID 917175, 2009:
Mustafa Z, Shatanawi W, Bataineh M: Existence of fixed points results in G -metric spaces. Int. J. Math. Math. Sci. 2009. Article ID 283028, 2009:
Mustafa Z, Khandagji M, Shatanawi W: Fixed point results on complete G -metric spaces. Studia Sci. Math. Hung. 2011,48(3):304-319.
Obiedat H, Mustafa Z: Fixed point results on a nonsymmetric G -metric spaces. Jordan J. Math. Stat. 2010,3(2):65-79.
Aydi H, Shatanawi W, Vetro C: On generalized weakly G -contraction mapping in G -metric spaces. Comput. Math. Appl. 2011,62(11):4222-4229. 10.1016/j.camwa.2011.10.007
Aydi H: A fixed point result involving a generalized weakly contractive condition in G -metric spaces. Bull. Math. Anal. Appl. 2011,3(4):180-188.
Gajić L, Stojaković M: On Ćirić generalization of mappings with a contractive iterate at a point in G -metric spaces. Appl. Math. Comput. 2012,219(1):435-441. 10.1016/j.amc.2012.06.041
Zhou SH, Gu F: Some new fixed points in G -metric spaces. J. Hangzhou Norm. Univ., Nat. Sci. Ed. 2012,11(1):47-50.
Shatanawi W: Fixed point theory for contractive mappings satisfying Φ-maps in G -metric spaces. Fixed Point Theory Appl. 2010. Article ID 181650, 2010:
Chugh R, Kadian T, Rani A, Rhoades BE: Property P in G -metric spaces. Fixed Point Theory Appl. 2010. Article ID 401684, 2010:
Abbas M, Rhoades BE: Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. Appl. Math. Comput. 2009,215(1):262-269. 10.1016/j.amc.2009.04.085
Abbas M, Nazir T, Saadati R: Common fixed point results for three maps in generalized metric space. Adv. Differ. Equ. 2011. Article ID 49, 2011:
Abbas M, Nazir T, Radenović S: Some periodic point results in generalized metric spaces. Appl. Math. Comput. 2010,217(8):4094-4099. 10.1016/j.amc.2010.10.026
Abbas M, Khan SH, Nazir T: Common fixed points of R -weakly commuting maps in generalized metric spaces. Fixed Point Theory Appl. 2011. Article ID 784595, 2011:
Vats RK, Kumar S, Sihag V: Some common fixed point theorems for compatible mappings of type in complete G -metric space. Adv. Fuzzy Math. 2011,6(1):27-38.
Abbas M, Nazir T, Vetro P: Common fixed point results for three maps in G -metric spaces. Filomat 2011,25(4):1-17. 10.2298/FIL1104001A
Gu F: Common fixed point theorems for six mappings in generalized metric spaces. Abstr. Appl. Anal. 2012. Article ID 379212, 2012: 10.1155/2012/379212
Gu F: Some new common coupled fixed point results in two generalized metric spaces. Fixed Point Theory Appl. 2013. Article ID 181, 2013: 10.1186/1687-1812-2013-181
Gu F, Yang Z: Some new common fixed point results for three pairs of mappings in generalized metric spaces. Fixed Point Theory Appl. 2013. Article ID 174, 2013: 10.1186/1687-1812-2013-174
Gu F, Ye H: Common fixed point theorems of Altman integral type mappings in G -metric spaces. Abstr. Appl. Anal. 2012. Article ID 630457, 2012: 630457 10.1155/2012/630457
Ye H, Gu F: Common fixed point theorems for a class of twice power type contraction maps in G -metric spaces. Abstr. Appl. Anal. 2012. Article ID 736214, 2012: 736214
Yin Y, Gu F: Common fixed point theorem about four mappings in G -metric spaces. J. Hangzhou Norm. Univ., Nat. Sci. Ed. 2012,11(6):511-515.
Ye HQ, Lu J, Gu F: A new common fixed point theorem for noncompatible mappings of type in G -metric space. J. Hangzhou Norm. Univ., Nat. Sci. Ed. 2013,12(1):50-56.
Shen YJ, Lu J, Zheng HH: Common fixed point theorem for converse commuting mappings in generalized metric spaces. J. Hangzhou Norm. Univ., Nat. Sci. Ed. 2014,13(5):542-547.
Hussain N, Parvaneh V, Hoseini Ghoncheh SJ: Generalized contractive mappings and weakly α -admissible pairs in G -metric spaces. Sci. World J. 2014. Article ID 941086, 2014:
Hussain N, Parvaneh V, Roshan JR: Fixed point results for G - α -contractive maps with application to boundary value problems. Sci. World J. 2014. Article ID 585964, 2014:
Hussain N, Roshan JR, Parvaneh V, Latif A: A unification of G -metric, partial metric and b -metric spaces. Abstr. Appl. Anal. 2014. Article ID 180698, 2014:
Tahat N, Aydi H, Karapınar 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: 10.1186/1687-1812-2012-48
Mustafa Z: Common fixed points of weakly compatible mappings in G -metric spaces. Appl. Math. Sci. 2012,6(92):4589-4600.
Abbas M, Nazir T, Dorić D:Common fixed point of mappings satisfying property in generalized metric spaces. Appl. Math. Comput. 2012,218(14):7665-7670. 10.1016/j.amc.2011.11.113
Mustafa Z, Aydi H, Karapınar E: On common fixed points in G -metric spaces using property. Comput. Math. Appl. 2012,64(6):1944-1956. 10.1016/j.camwa.2012.03.051
Long W, Abbas M, Nazir T, Radenović S:Common fixed point for two pairs of mappings satisfying property in generalized metric spaces. Abstr. Appl. Anal. 2012. Article ID 394830, 2012: 10.1155/2012/394830
Gu F, Yin Y:Common fixed point for three pairs of self-maps satisfying common property in generalized metric spaces. Abstr. Appl. Anal. 2013. Article ID 808092, 2013: 10.1155/2013/808092
Gu F, Shatanawi W: Common fixed point for generalized weakly G -contraction mappings satisfying common property in G -metric spaces. Fixed Point Theory Appl. 2013. Article ID 48, 2013: 10.1186/1687-1812-2013-309
Jleli M, Samet B: Remarks on G -metric spaces and fixed point theorems. Fixed Point Theory Appl. 2012. Article ID 210, 2012: 10.1186/1687-1812-2012-210
Samet B, Vetro C, Vetro F: Remarks on G -metric spaces. Int. J. Anal. 2013. Article ID 917158, 2013:
Mustafa Z, Obiedat H, Awawdeh H: Some fixed point theorem for mappings on complete G -metric spaces. Fixed Point Theory Appl. 2008. Article ID 189870, 2008:
Karapınar E, Agarval R: Further remarks on G -metric spaces. Fixed Point Theory Appl. 2013. Article ID 154, 2013: 10.1186/1687-1812-2013-154
Jungck G, Rhoades BE: Fixed point for set valued functions without continuity. Indian J. Pure Appl. Math. 1998, 29: 227-238.
Acknowledgements
The present studies are supported by the National Natural Science Foundation of China (11071169, 11361070), the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030), the Natural Science Foundation of Shandong Province (ZR2013AL015) and the Innovation Foundation of Graduate Student of Hangzhou Normal University.
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Gu, F., Shen, Y. & Wang, L. Common fixed point results under a new contractive condition without using continuity. J Inequal Appl 2014, 464 (2014). https://doi.org/10.1186/1029-242X-2014-464
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DOI: https://doi.org/10.1186/1029-242X-2014-464