- Research Article
- Open Access
Common Fixed Points of Generalized Contractive Hybrid Pairs in Symmetric Spaces
© M. Abbas and A. R. Khan. 2009
- Received: 16 April 2009
- Accepted: 10 November 2009
- Published: 3 December 2009
Several fixed point theorems for hybrid pairs of single-valued and multivalued occasionally weakly compatible maps satisfying generalized contractive conditions are established in a symmetric space.
- Point Theorem
- Symmetric Space
- Triangle Inequality
- Coincidence Point
- Contractive Type
In 1968, Kannan  proved a fixed point theorem for a map satisfying a contractive condition that did not require continuity at each point. This paper was a genesis for a multitude of fixed point papers over the next two decades. Sessa  coined the term weakly commuting maps. Jungck  generalized the notion of weak commutativity by introducing compatible maps and then weakly compatible maps . Al-Thagafi and Shahzad  gave a definition which is proper generalization of nontrivial weakly compatible maps which have coincidence points. Jungck and Rhoades  studied fixed point results for occasionally weakly compatible (owc) maps. Recently, Zhang  obtained common fixed point theorems for some new generalized contractive type mappings. Abbas and Rhoades  obtained common fixed point theorems for hybrid pairs of single-valued and multivalued owc maps defined on a symmetric space (see also ). For other related fixed point results in symmetric spaces and their applications, we refer to [10–15]. The aim of this paper is to obtain fixed point theorems involving hybrid pairs of single-valued and multivalued owc maps satisfying a generalized contractive condition in the frame work of a symmetric space.
Maps and are said to becompatible if for each and whenever is a sequence in such that ( ) and for some .
It can be easily verified that is coincidence point of and but and are not weakly compatible there, as . Hence and are not compatible. However, the pair is occasionally weakly compatible, since the pair is weakly compatible at
For some examples of mappings which satisfy we refer to .
Since (2.7) is a special case of (2.1), the result follows from Theorem 2.1.
So, (2.9) is a special case of (2.1) and hence the result follows from Theorem 2.1.
which, from and implies that this further implies that Hence the claim follows. Similarly, it can be shown that which proves that is a common fixed point of , and . Uniqueness follows from (2.21) and ( ).
The following theorem generalizes [16, Theorem ].
Clearly, but and but they show that is not weakly compatible. On the other hand, gives that Hence is occasionally weakly compatible. Note that , , , and they imply that is not weakly compatible Now gives that . Hence is occasionally weakly compatible. As and so is the unique common fixed point of , and
Remark 2.10 s.
Weakly compatible maps are occasionally weakly compatible but converse is not true in general. The class of symmetric spaces is more general than that of metric spaces. Therefore the following results can be viewed as special cases of our results:
(d)[28, Theorem ] becomes special case of Corollary 2.4.
The authors are thankful to the referees for their critical remarks to improve this paper. The second author gratefully acknowledges the support provided by King Fahad University of Petroleum and Minerals during this research.
- Kannan R: Some results on fixed points. Bulletin of the Calcutta Mathematical Society 1968, 60: 71–76.MathSciNetMATHGoogle Scholar
- Sessa S: On a weak commutativity condition of mappings in fixed point considerations. Publications de l'Institut Mathématique. Nouvelle Série 1982, 32(46): 149–153.MathSciNetMATHGoogle Scholar
- Jungck G: Compatible mappings and common fixed points. International Journal of Mathematics and Mathematical Sciences 1986,9(4):771–779. 10.1155/S0161171286000935MathSciNetView ArticleMATHGoogle Scholar
- Jungck G: Common fixed points for noncontinuous nonself maps on nonmetric spaces. Far East Journal of Mathematical Sciences 1996,4(2):199–215.MathSciNetMATHGoogle Scholar
- Al-Thagafi MA, Shahzad N: Generalized -nonexpansive selfmaps and invariant approximations. Acta Mathematica Sinica 2008,24(5):867–876. 10.1007/s10114-007-5598-xMathSciNetView ArticleMATHGoogle Scholar
- Jungck G, Rhoades BE: Fixed point theorems for occasionally weakly compatible mappings. Fixed Point Theory 2006,7(2):287–296.MathSciNetMATHGoogle Scholar
- Zhang X: Common fixed point theorems for some new generalized contractive type mappings. Journal of Mathematical Analysis and Applications 2007,333(2):780–786. 10.1016/j.jmaa.2006.11.028MathSciNetView ArticleMATHGoogle Scholar
- Abbas M, Rhoades BE: Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings defined on symmetric spaces. Panamerican Mathematical Journal 2008,18(1):55–62.MathSciNetMATHGoogle Scholar
- Abbas M, Rhoades BE: Common fixed point theorems for occasionally weakly compatible mappings satisfying a generalized contractive condition. Mathematical Communications 2008,13(2):295–301.MathSciNetMATHGoogle Scholar
- Aliouche A: A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type. Journal of Mathematical Analysis and Applications 2006,322(2):796–802. 10.1016/j.jmaa.2005.09.068MathSciNetView ArticleMATHGoogle Scholar
- Chandra H, Bhatt A: Some fixed point theorems for set valued maps in symmetric spaces. International Journal of Mathematical Analysis 2009,3(17):839–846.MathSciNetMATHGoogle Scholar
- Cho S-H, Lee G-Y, Bae J-S: On coincidence and fixed-point theorems in symmetric spaces. Fixed Point Theory and Applications 2008, 2008:-9.Google Scholar
- Hicks TL, Rhoades BE: Fixed point theory in symmetric spaces with applications to probabilistic spaces. Nonlinear Analysis: Theory, Methods & Applications 1999,36(3):331–344. 10.1016/S0362-546X(98)00002-9MathSciNetView ArticleMATHGoogle Scholar
- Imdad M, Ali J: Common fixed point theorems in symmetric spaces employing a new implicit function and common property (E.A). Bulletin of the Belgian Mathematical Society. Simon Stevin 2009, 16: 421–433.MathSciNetMATHGoogle Scholar
- Pathak HK, Tiwari R, Khan MS: A common fixed point theorem satisfying integral type implicit relations. Applied Mathematics E-Notes 2007, 7: 222–228.MathSciNetMATHGoogle Scholar
- Beg I, Abbas M: Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition. Fixed Point Theory and Applications 2006, 2006:-7.Google Scholar
- Chang TH: Common fixed point theorems for multivalued mappings. Mathematica Japonica 1995,41(2):311–320.MathSciNetMATHGoogle Scholar
- Shrivastava PK, Bawa NPS, Nigam SK: Fixed point theorems for hybrid contractions. Varāhmihir Journal of Mathematical Sciences 2002,2(2):275–281.MathSciNetMATHGoogle Scholar
- Azam A, Beg I: Coincidence points of compatible multivalued mappings. Demonstratio Mathematica 1996,29(1):17–22.MathSciNetMATHGoogle Scholar
- Kamran T: Common coincidence points of R -weakly commuting maps. International Journal of Mathematics and Mathematical Sciences 2001,26(3):179–182. 10.1155/S0161171201005245MathSciNetView ArticleMATHGoogle Scholar
- Jungck G, Rhoades BE: Fixed points for set valued functions without continuity. Indian Journal of Pure and Applied Mathematics 1998,29(3):227–238.MathSciNetMATHGoogle Scholar
- Hadžić O: Common fixed point theorems for single-valued and multivalued mappings. Review of Research. Faculty of Science. Mathematics Series 1988,18(2):145–151.MathSciNetMATHGoogle Scholar
- Kaneko H, Sessa S: Fixed point theorems for compatible multi-valued and single-valued mappings. International Journal of Mathematics and Mathematical Sciences 1989,12(2):257–262. 10.1155/S0161171289000293MathSciNetView ArticleMATHGoogle Scholar
- Kaneko H: A common fixed point of weakly commuting multi-valued mappings. Mathematica Japonica 1988,33(5):741–744.MathSciNetMATHGoogle Scholar
- Fisher B: Common fixed points for set-valued mappings. Indian Journal of Mathematics 1983,25(3):265–270.MathSciNetMATHGoogle Scholar
- Sessa S, Fisher B: On common fixed points of weakly commuting mappings and set-valued mappings. International Journal of Mathematics and Mathematical Sciences 1986,9(2):323–329. 10.1155/S0161171286000406MathSciNetView ArticleMATHGoogle Scholar
- Fisher B: Common fixed point theorem for commutative mappings and set valued mappings. Journal of University of Kuwait 1984, 11: 15–21.MATHGoogle Scholar
- Dhage BC: Common fixed point theorems for coincidentally commuting pairs of nonself mappings in metrically convex metric spaces. Analele Ştiinţifice ale Universităţii Al. I. Cuza din Iaşi. Serie Nouă. Matematică 2003,49(1):45–60.MathSciNetMATHGoogle Scholar
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