Common Fixed Point Theorems for Four Mappings on Cone Metric Type Space
© Aleksandar S. Cvetković et al. 2011
Received: 9 December 2010
Accepted: 3 February 2011
Published: 27 February 2011
In this paper we consider the so called a cone metric type space, which is a generalization of a cone metric space. We prove some common fixed point theorems for four mappings in those spaces. Obtained results extend and generalize well-known comparable results in the literature. All results are proved in the settings of a solid cone, without the assumption of continuity of mappings.
Replacing the real numbers, as the codomain of a metric, by an ordered Banach space we obtain a generalization of metric space. Such a generalized space, called a cone metric space, was introduced by Huang and Zhang in . They described the convergence in cone metric space, introduced their completeness, and proved some fixed point theorems for contractive mappings on cone metric space. Cones and ordered normed spaces have some applications in optimization theory (see ). The initial work of Huang and Zhang  inspired many authors to prove fixed point theorems, as well as common fixed point theorems for two or more mappings on cone metric space, for example, [3–14].
In this paper we consider the so-called a cone metric type space, which is a generalization of a cone metric space and prove some common fixed point theorems for four mappings in those spaces. Obtained results are generalization of theorems proved in . For some special choices of mappings we obtain theorems which generalize results from [1, 8, 15]. All results are proved in the settings of a solid cone, without the assumption of continuity of mappings.
The paper is organized as follows. In Section 2 we repeat some definitions and well-known results which will be needed in the sequel. In Section 3 we prove common fixed point theorems. Also, we presented some corollaries which show that our results are generalization of some existing results in the literature.
2. Definitions and Notation
Analogously with definition of metric type space, given in , we consider cone metric type space.
For in Definition 2.1 we obtain a cone metric space introduced in .
We show that is a solid cone. Let , , with property . Since scalar product is continuous, we get , . Clearly, it must be , , and, hence, , that is, is closed. It is obvious that , and for , and all , we have , . Finally, if we have and , , and it follows that , , and, since is complete, we get . Let us choose . It is obvious that , since if not, for every there exists such that . If we choose , we conclude that it must be , hence , which is contradiction.
The following properties are well known in the case of a cone metric space, and it is easy to see that they hold also in the case of a CMTS.
Definition 2.6 (see ).
Definition 2.7 (see ).
Lemma 2.8 (see ).
3. Main Results
Thus, inequality (3.3) holds in this case.
Thus we have the following three cases:
So, inequality (3.3) is satisfied in this case, too.
Therefore we have the following four cases:
Therefore we have the following four cases:
We give one simple, but illustrative, example.
If is weakly compatible pair, we have , which implies , that is, . Similarly, if is weakly compatible pair, we have , which implies , that is, . Then , and is the unique common fixed point of these four mappings.
The following two theorems can be proved in the same way as Theorem 3.1, so we omit the proofs.
that is, inequality (3.34) holds in this case.
and inequality (3.34) holds in this case, too.
and (because of ) it follows that . Therefore, is the unique point of coincidence of pairs and , and we have . If and are weakly compatible pairs, then is the unique common fixed point of , , , and , by Lemma 2.8.
Theorem 3.8 is a generalization of [13, Theorem 2.8]. Choosing from Theorem 3.8 we get the following corollary.
The authors are indebted to the referees for their valuable suggestions, which have contributed to improve the presentation of the paper. The first two authors were supported in part by the Serbian Ministry of Science and Technological Developments (Grant no. 174015).
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