Some Generalizations of Fixed Point Theorems in Cone Metric Spaces
© J. O. Olaleru. 2009
Received: 17 March 2009
Accepted: 29 August 2009
Published: 27 September 2009
We generalize, extend, and improve some recent fixed point results in cone metric spaces including the results of H. Guang and Z. Xian (2007); P. Vetro (2007); M. Abbas and G. Jungck (2008); Sh. Rezapour and R. Hamlbarani (2008). In all our results, the normality assumption, which is a characteristic of most of the previous results, is dispensed. Consequently, the results generalize several fixed results in metric spaces including the results of G. E. Hardy and T. D. Rogers (1973), R. Kannan (1969), G. Jungck, S. Radenovic, S. Radojevic, and V. Rakocevic (2009), and all the references therein.
The recently discovered applications of ordered topological vector spaces, normal cones and topical functions in optimization theory have generated a lot of interest and research in ordered topological vector spaces (e.g., see [1, 2]). Recently, Huang and Zhang  introduced cone metric spaces, which is a generalization of metric spaces, by replacing the real numbers with ordered Banach spaces. They later proved some fixed point theorems for different contractive mappings. Their results have been generalized by different authors (e.g. see [4–7]). This paper generalizes, extends and improves the results of all those authors.
The following definitions are given in .
The cone is called if every increasing sequence which is bounded from above is convergent. That is, if is a sequence such that for some , then there is such that . Equivalently, the cone is regular if and only if every decreasing sequence which is bounded from below is convergent. In  it was shown that every regular cone is normal.
In the sequel we will suppose that is a metrizable linear topological space whose topology is defined by a real-valued function called (see ). We will assume that is a cone in with and is partial ordering with respect to .
Metrizable linear topological spaces contain metrizable locally convex spaces and normed linear spaces . Therefore our generalizes the as a normed linear space used in all the previous results on cone metric spaces.
Example 1.2 (see ).
Clearly, this example shows that cone metric spaces generalize metric spaces.
It is shown in  that a convergent sequence in a cone metric space is a Cauchy sequence.
Let be a cone metric space. If for any sequence in , there is a subsequence of such that is convergent in , then is called a sequentially compact metric space. Furthermore, is compact if and only if is sequentially compact. (see also ).
Proposition 1.7 (see ).
Huang and Zhang  proved the following theorems for a Banach space.
Rezapour and Hamlbarani  improved on Theorems (1.8–1.10) by proving the same results without the assumption that is a normal cone. They gave examples of non-normal cones and showed that there are no normal cones with normal constant . Observe that the normal constant for Example 1.3 is 1.
Vetro  recently combined the results of Theorems 1.8 and 1.9 and generalized them to two maps satisfying certain conditions, to obtain the following theorem.
and and or is a complete subspace of , then the mappings and have a unique common fixed point. Moreover, for any , the sequence of the initial point , where is defined by for all , converges to the fixed point.
The two maps and are said to be if they satisfy condition (1.5). This concept was introduced by Huang and Zhang  and it is known to be the most general among all commutativity concepts in fixed point theory. For example every pair of weakly commuting self-maps and each pair of compatible self-maps are weakly compatible, but the converse is not always true. In fact, the notion of weakly compatible maps is more general than compatibility of type (A), compatibility of type (B), compatibility of type (C), and compatibility of type (P). For a review of those notions of commutativity, see ([11, 12]).
In Theorem 2.1, we unify Theorems 1.8–1.10 into a single theorem and generalize. In Theorem 2.3, we examine the situation where the sum of the coefficients, rather than less than is actually . Theorem 3.1 generalizes Theorem 2.1 to two weakly compatible maps thus extending Theorem 1.11. Furthermore, we remove the assumption of normality of cone in all our results and extend to a metrizable linear topological space. Some other consequences follow.
2. Theorems on Single Maps
The theorem is valid if we replace the completeness of with the condition that is complete. If is restricted to a normed linear space and in Theorem 2.1 we have [5, Theorem ?2.3]; if in Theorem 2.1, we obtain [5, Theorem ?2.6]; if , we obtain [5, Theorem ?2.7] and if , we obtain [5, Theorem ?2.8]. Furthermore, if we add the normality assumption to Theorem 2.1, then [3, Theorems ?1, 2, and 4] there are special cases of Theorem 2.1.
Since is sequentially compact, then it is compact . The fact that is continuous and is compact implies that is compact and hence exists and for some . From (2.14), it can be infered that is fixed under and uniqueness follows from (2.13).
3. Common Fixed Points
for all where and . Suppose and are weakly compatible and such that or is a complete subspace of , then the mappings and have a unique common fixed point. Moreover, for any , the sequence defined by for all , converges to the fixed point.
If and is restricted to normed linear spaces in Theorem 3.1, with the additional normality assumption, we obtain the common fixed point Theorem of Vetro .
Suppose is restricted to normed linear spaces, with the additional normality assumption, if , then Theorem 3.1 gives [4, Theorem ?2.1]; if , we obtain [4, Theorem ?2.3], and if , we obtain [4, Theorem ?2.4]. Thus our theorem is both an extension, generalization and an improvement of the results of [4, 7].
If is restricted to normed linear spaces, Theorem 3.1 reduces to [14, Theorem ?2.8].
Theorem 2.3 was proved for the usual metric space by the author in  without the assumptions that is continuous and is compact. Is the above Theorem 2.3 still valid if we remove the assumption that is continuous and is compact?.
The author is grateful to the referees for careful readings and corrections. He is also grateful to Professor Stojan Radenvonic for giving him all the papers on cone metric spaces used in this paper and the African Mathematics Millennium Science Initiative (AMMSI) for financial support.
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