- Research Article
- Open Access
Common Fixed Point Theorem for Non-Self Mappings Satisfying Generalized Ćirić Type Contraction Condition in Cone Metric Space
© R. Sumitra et al. 2010
Received: 27 November 2009
Accepted: 9 March 2010
Published: 14 March 2010
We prove common fixed point theorem for coincidentally commuting nonself mappings satisfying generalized contraction condition of Ćirić type in cone metric space. Our results generalize and extend all the recent results related to non-self mappings in the setting of cone metric space.
Recently, Huang and Zhang  introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. The category of cone metric spaces is larger than metric spaces and there are different types of cones. Subsequently, many authors like Abbas and Jungck , Abbas and Rhoades , Ilić and Rakočević , Raja andVaezpour have generalized the results of Huang and Zhang  and studied the existence of common fixed points of a pair of self mappings satisfying a contractive type condition in the framework of normal cone metric spaces. However, authors like Janković et al. , Jungck et al. , Kadelburg et al. [8, 9], Radenović and Rhoades , Rezapour and Hamlbarani  studied the existence of common fixed points of a pair of self and nonself mappings satisfying a contractive type condition in the situation in which the cone does not need be normal.
The study of fixed point theorems for nonself mappings in metrically convex metric spaces was initiated by Assad and Kirk . Utilizing the induction method of Assad and Kirk , many authors like Assad , Ćirić , Hadžić , Hadžić and Gajić , Imdad and Kumar , Rhoades [18, 19] have obtained common fixed point in metrically convex spaces. Recently, Ćirić and Ume  defined a wide class of multivalued nonself mappings which satisfy a generalized contraction condition and proved a fixed point theorem which generalize the results of Itoh  and Khan .
Very recently, Radenović and Rhoades  extended the fixed point theorem of Imdad and Kumar  for a pair of nonself mappings to nonnormal cone metric spaces. Janković et al.  proved new common fixed point results for a pair of nonself mappings defined on a closed subset of metrically convex cone metric space which is not necessarily normal by adapting Assad-Kirk's method.
The aim of this paper is to prove common fixed point theorems for coincidentally commuting nonself mappings satisfying a generalized contraction condition of Ćirić type in the setting of cone metric spaces. Our results generalize mainly results of Ćirić and Ume  and all the recent results related to nonself mappings in the setting of cone metric space.
2. Definitions and Preliminaries
We recall some basic definitions and preliminaries that will be needed in the sequel.
Definition 2.1 (see ).
The definition of a cone metric space given by Huang and Zhang  is as follows.
Definition 2.2 (see ).
Example 2.3 (see ).
Definition 2.4 (see ).
The following two lemmas of Huang and Zhang  will be required in the sequel.
Lemma 2.5 (see ).
Lemma 2.6 (see ).
The following Corollary of Rezapour  will be needed in the sequel.
Corollary 2.7 (see ).
The following remarks of Radenović and Rhoades  will be needed in the sequel.
Remark 2.8 (see ).
Remark 2.9 (see ).
Remark 2.10 (see ).
Remark 2.11 (see ).
3. Main Results
Also assume that
Now again four cases arise.
The following example illustrates Theorem 3.1.
Now let us see that conditions (i)–(iii) in Theorem 3.1 are satisfied.
Next, RHS of inequality (3.2) is , where , and . Then RHS of inequality (3.2) is if and . Thus LHS of inequality (3.2) RHS of inequality (3.2). Similarly, LHS of inequality (3.2) RHS of inequality (3.2) for all possible cases of and . Thus all the conditions of Theorem 3.1 are satisfied. Hence "0" is the unique common fixed point of and in .
The authors would like to thank the referees for their valuable suggestions which lead to the improvement of the presentation of the paper.
- Huang L-G, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications 2007,332(2):1468–1476. 10.1016/j.jmaa.2005.03.087MathSciNetView ArticleMATHGoogle Scholar
- Abbas M, Jungck G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. Journal of Mathematical Analysis and Applications 2008,341(1):416–420. 10.1016/j.jmaa.2007.09.070MathSciNetView ArticleMATHGoogle Scholar
- Abbas M, Rhoades BE: Fixed and periodic point results in cone metric spaces. Applied Mathematics Letters 2009,22(4):511–515. 10.1016/j.aml.2008.07.001MathSciNetView ArticleMATHGoogle Scholar
- Ilić D, Rakočević V: Common fixed points for maps on cone metric space. Journal of Mathematical Analysis and Applications 2008,341(2):876–882. 10.1016/j.jmaa.2007.10.065MathSciNetView ArticleMATHGoogle Scholar
- Raja P, Vaezpour SM: Some extensions of Banach's contraction principle in complete cone metric spaces. Fixed Point Theory and Applications 2008, Article ID 768294 2008:-11 Pages.Google Scholar
- Janković S, Kadelburg Z, Radenović S, Rhoades BE: Assad-Kirk-type fixed point theorems for a pair of nonself mappings on cone metric spaces. Fixed Point Theory and Applications 2009, Article ID 761086 2009:-16 Pages.Google Scholar
- Jungck G, Radenović S, Radojević S, Rakočević V: Common fixed point theorems for weakly compatible pairs on cone metric spaces. Fixed Point Theory and Applications 2009, Article ID 643840 2009:-13 Pages.Google Scholar
- Kadelburg Z, Radenović S, Rakočević V: Remarks on "Quasi-contraction on a cone metric space". Applied Mathematics Letters 2009,22(11):1674–1679. 10.1016/j.aml.2009.06.003MathSciNetView ArticleMATHGoogle Scholar
- Kadelburg Z, Radenović S, Rosić B: Strict contractive conditions and common fixed point theorems in cone metric spaces. Fixed Point Theory and Applications 2009, Article ID 173838 2009:-14 Pages.Google Scholar
- Radenović S, Rhoades BE: Fixed point theorem for two non-self mappings in cone metric spaces. Computers & Mathematics with Applications 2009,57(10):1701–1707. 10.1016/j.camwa.2009.03.058MathSciNetView ArticleMATHGoogle Scholar
- Rezapour Sh, Hamlbarani R: Some notes on the paper: "Cone metric spaces and fixed point theorems of contractive mappings". Journal of Mathematical Analysis and Applications 2008,345(2):719–724. 10.1016/j.jmaa.2008.04.049MathSciNetView ArticleMATHGoogle Scholar
- Assad NA, Kirk WA: Fixed point theorems for set-valued mappings of contractive type. Pacific Journal of Mathematics 1972,43(3):553–562.MathSciNetView ArticleMATHGoogle Scholar
- Assad NA: On a fixed point theorem of Kannan in Banach spaces. Tamkang Journal of Mathematics 1976,7(1):91–94.MathSciNetMATHGoogle Scholar
- Ćirić L: Non-self mappings satisfying non-linear contractive condition with applications. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7–8):2927–2935. 10.1016/j.na.2009.01.174View ArticleMathSciNetMATHGoogle Scholar
- Hadžić O: On coincidence points in convex metric spaces. Univerzitet u Novom Sadu. Zbornik Radova Prirodno-Matematičkog Fakulteta. Serija za Matematiku 1989,19(2):233–240.MathSciNetMATHGoogle Scholar
- Hadžić O, Gajić L: Coincidence points for set-valued mappings in convex metric spaces. Univerzitet u Novom Sadu. Zbornik Radova Prirodno-Matematičkog Fakulteta. Serija za Matematiku 1986,16(1):13–25.MathSciNetMATHGoogle Scholar
- Imdad M, Kumar S: Rhoades-type fixed-point theorems for a pair of nonself mappings. Computers & Mathematics with Applications 2003,46(5–6):919–927. 10.1016/S0898-1221(03)90153-2MathSciNetView ArticleMATHGoogle Scholar
- Rhoades BE: A fixed point theorem for some non-self-mappings. Mathematica Japonica 1978,23(4):457–459.MathSciNetMATHGoogle Scholar
- Rhoades BE: A fixed point theorem for non-self set-valued mappings. International Journal of Mathematics and Mathematical Sciences 1997,20(1):9–12. 10.1155/S0161171297000021MathSciNetView ArticleMATHGoogle Scholar
- Ćirić LB, Ume JS: Multi-valued non-self-mappings on convex metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2005,60(6):1053–1063. 10.1016/j.na.2004.09.057MathSciNetView ArticleMATHGoogle Scholar
- Itoh S: Multivalued generalized contractions and fixed point theorems. Commentationes Mathematicae Universitatis Carolinae 1977,18(2):247–258.MathSciNetMATHGoogle Scholar
- Khan MS: Common fixed point theorems for multivalued mappings. Pacific Journal of Mathematics 1981,95(2):337–347.MathSciNetView ArticleMATHGoogle Scholar
- Rezapour Sh: A review on topological properties of cone metric spaces. Proceedings of the Conference on Analysis, Topology and Applications (ATA '08), May-June 2008, Vrnjacka Banja, SerbiaGoogle Scholar
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