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.
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