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 ).
Let be a real Banach space. A subset of is called a Cone if and only if
(1) is nonempty, closed and ;
(2) , and ;
For a given cone , a partial ordering is defined as on with respect to by , if and only if . It is denoted as to indicate that but , while will stand for , where denotes the interior of .
The cone is called normal, if there is a number such that for all , implies
The least positive number satisfying (2.1) is called the normal constant of . It is clear that . There are nonnormal cones also.
The definition of a cone metric space given by Huang and Zhang  is as follows.
Definition 2.2 (see ).
Let be a nonempty set. Suppose that is a real Banach space, is a cone with and is a partial ordering with respect to .
If the mapping satisfies the following:
(1) for all and if and only if ;
(2) for all ;
(3) for all ;
then is called a cone metric on and is called a cone metric space.
Example 2.3 (see ).
Let , and such that , where is a constant. Then is a cone metric space.
Definition 2.4 (see ).
Let be a cone metric space and a sequence in . Then, one has the following.
It is denoted by or , .
then is called a Cauchy sequence in .
(3) is a complete cone metric space, if every Cauchy sequence in is convergent.
(4)A self mapping is said to be continuous at a point , if implies that for every in .
The following two lemmas of Huang and Zhang  will be required in the sequel.
Lemma 2.5 (see ).
Let be a cone metric space and a normal cone with normal constant . A sequence in converges to if and only if as .
Lemma 2.6 (see ).
Let be a cone metric space and a normal cone with normal constant . A sequence in is a Cauchy sequence if and only if as .
The following Corollary of Rezapour  will be needed in the sequel.
Corollary 2.7 (see ).
Let , the real Banach space.
(i)If and , then .
(ii)If and , then .
(iii)If for each , then .
The following remarks of Radenović and Rhoades  will be needed in the sequel.
Remark 2.8 (see ).
If , and , then there exists such that for all , it follows that .
Remark 2.9 (see ).
If and , then , where is a sequence and is a given point in .
Remark 2.10 (see ).
If and , , then for each cone .
Remark 2.11 (see ).
If is a real Banach space with a cone and if , where and , then .
3. Main Results
In the following, we suppose that is a Banach space, is a cone in with and is partial ordering with respect to .
Also assume that
(iii) is closed in ;
Then there exists a coincidence point of and in . Moreover, if and are coincidentally commuting, then and have a unique common fixed point in .
Two sequences and are constructed in the following way. Let . As , by (i) there exists a point such that . Since , from (ii) it follows that . Let be such that . Since , there exists such that .
If , then which implies that there exists a point such that . Otherwise, if , then there exists a point such that
Assume that .
Thus repeating the arguments, two sequences and are obtained such that
(iii) whenever , then there exists such that
Obviously, two consecutive terms cannot lie in . Note that, if , then and belong to . Now, three cases are distinguished.
Now again four cases arise.
Let be given with . From as and Definition 2.4(1),
Therefore, for each . Then by (iii) of Corollary 2.7, we have , that is, which implies that is the coincidence point of and .
Since and are coincidentally commuting, for which implies . Consider
Thus and . Two cases arise.
Since by (3.3), it follows from Remark 2.11 that which implies that . Thus .
Uniqueness: if is another common fixed point of and in , then . Now by (3.2), it follows that
Thus and . Two cases arise.
If and , then (3.60) becomes . Since , by Remark 2.11 we have which implies that is the unique common fixed point of and .
Since , by Remark 2.11 we have which implies is the unique common fixed point of and . Hence is the unique common fixed point of and in .
The following example illustrates Theorem 3.1.
Let , , , and . Define two nonself mappings as and for all .
Now let us see that conditions (i)–(iii) in Theorem 3.1 are satisfied.
It may be seen that and . Then and . Also, as . Moreover is closed in .
Next, we shall see that inequality (3.2) is satisfied by taking and . It is easy to see that .
Now, LHS of inequality (3.2) is . Taking and , it follows that .
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 .
and are nonnegative real numbers such that . Also assume that . Then there exists a unique fixed point of in .
The proof of this corollary follows by taking , the identity mapping of in Theorem 3.1.
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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