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Fixed Point Theory and Applications

, 2009:657914 | Cite as

Some Generalizations of Fixed Point Theorems in Cone Metric Spaces

  • J. O. Olaleru
Open Access
Review Article

Abstract

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.

Keywords

Fixed Point Theorem Contractive Condition Normal Constant Normal Cone Cauchy Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

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 [3] 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, 5, 6, 7]). This paper generalizes, extends and improves the results of all those authors.

The following definitions are given in [3].

Let Open image in new window be a real Banach space and Open image in new window a subset of Open image in new window . Open image in new window is called a cone if and only if

(i) Open image in new window is closed, nonempty, and Open image in new window ;

(ii) Open image in new window , Open image in new window , Open image in new window ;

(iii) Open image in new window .

For a given cone Open image in new window , we can define a partial ordering Open image in new window with respect to Open image in new window by Open image in new window if and only if Open image in new window . Open image in new window will stand for Open image in new window and Open image in new window , while Open image in new window will stand for Open image in new window , where Open image in new window denotes the interior of Open image in new window .

The cone Open image in new window is called Open image in new window if there is Open image in new window such that for all Open image in new window , Open image in new window implies Open image in new window .

The least positive number Open image in new window satisfying the above is called the normal constant of Open image in new window .

The cone Open image in new window is called Open image in new window if every increasing sequence which is bounded from above is convergent. That is, if Open image in new window is a sequence such that Open image in new window for some Open image in new window , then there is Open image in new window such that Open image in new window . Equivalently, the cone Open image in new window is regular if and only if every decreasing sequence which is bounded from below is convergent. In [5] it was shown that every regular cone is normal.

In the sequel we will suppose that Open image in new window is a metrizable linear topological space whose topology is defined by a real-valued function Open image in new window called Open image in new window (see [8]). We will assume that Open image in new window is a cone in Open image in new window with Open image in new window and Open image in new window is partial ordering with respect to Open image in new window .

Metrizable linear topological spaces contain metrizable locally convex spaces and normed linear spaces [9]. Therefore our Open image in new window generalizes the Open image in new window as a normed linear space used in all the previous results on cone metric spaces.

A cone Open image in new window is therefore called normal if there is Open image in new window such that for all Open image in new window , Open image in new window implies Open image in new window .

Definition 1.1.

Let Open image in new window be a nonempty set. Suppose that Open image in new window satisfies

(i) Open image in new window for all Open image in new window and Open image in new window if and only if Open image in new window ,

(ii) Open image in new window for all Open image in new window ,

(iii) Open image in new window for all Open image in new window .

Then Open image in new window is called a cone metric on Open image in new window , and Open image in new window is called a cone metric space.

Example 1.2 (see [3]).

Let Open image in new window , Open image in new window , Open image in new window and Open image in new window defined by Open image in new window , where Open image in new window is a constant. Then Open image in new window is a cone metric space.

Clearly, this example shows that cone metric spaces generalize metric spaces.

We now give another example where Open image in new window is a metrizable linear topological vector space that is not a normed linear space.

Example 1.3.

Let Open image in new window , ( Open image in new window ), Open image in new window , Open image in new window a metric space and Open image in new window defined by Open image in new window . Then Open image in new window is a cone metric space.

Definition 1.4.

Let Open image in new window be a cone metric space. Let Open image in new window be a sequence in Open image in new window . If for every Open image in new window with Open image in new window there is Open image in new window such that for all Open image in new window , Open image in new window , then Open image in new window is said to be convergent to Open image in new window , that is, Open image in new window .

Definition 1.5.

Let Open image in new window be a cone metric space. Let Open image in new window be a sequence in Open image in new window . If for every Open image in new window with Open image in new window there is Open image in new window such that for all Open image in new window , Open image in new window , then Open image in new window is called a Cauchy sequence in Open image in new window .

It is shown in [3] that a convergent sequence in a cone metric space Open image in new window is a Cauchy sequence.

Definition 1.6.

Let Open image in new window be a cone metric space. If for any sequence Open image in new window in Open image in new window , there is a subsequence Open image in new window of Open image in new window such that Open image in new window is convergent in Open image in new window , then Open image in new window is called a sequentially compact metric space. Furthermore, Open image in new window is compact if and only if Open image in new window is sequentially compact. (see also [10]).

Proposition 1.7 (see [3]).

Let Open image in new window be a cone metric space, Open image in new window a normal cone. Let Open image in new window and Open image in new window be two sequences in Open image in new window and Open image in new window , Open image in new window as Open image in new window . Then

(iii) Open image in new window is a Cauchy sequence if and only if Open image in new window as Open image in new window

(iv) Open image in new window as Open image in new window

Huang and Zhang [3] proved the following theorems for Open image in new window a Banach space.

Theorem 1.8.

Let Open image in new window be a complete metric space, Open image in new window a normal cone with normal constant Open image in new window . Suppose that the mapping Open image in new window satisfies the contractive condition

where Open image in new window is a constant. Then Open image in new window has a unique fixed point in Open image in new window . And for any Open image in new window , iterative sequence Open image in new window converges to the fixed point.

Theorem 1.9.

Let Open image in new window be a complete metric space, Open image in new window a normal cone with normal constant Open image in new window . Suppose that the mapping Open image in new window satisfies the contractive condition

where Open image in new window is a constant. Then Open image in new window has a unique fixed point in Open image in new window . And for any Open image in new window , iterative sequence Open image in new window converges to the fixed point.

Theorem 1.10.

Let Open image in new window be a complete metric space, Open image in new window a normal cone with normal constant Open image in new window . Suppose that the mapping Open image in new window satisfies the contractive condition

where Open image in new window is a constant. Then Open image in new window has a unique fixed point in Open image in new window . And for any Open image in new window , iterative sequence Open image in new window converges to the fixed point.

Rezapour and Hamlbarani [5] improved on Theorems (1.8–1.10) by proving the same results without the assumption that Open image in new window is a normal cone. They gave examples of non-normal cones and showed that there are no normal cones with normal constant Open image in new window . Observe that the normal constant Open image in new window for Example 1.3 is 1.

Vetro [7] 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.

Theorem 1.11.

Let Open image in new window be a cone metric space, Open image in new window a normal cone with normal constant Open image in new window . Let Open image in new window be mappings such that

and Open image in new window and Open image in new window or Open image in new window is a complete subspace of Open image in new window , then the mappings Open image in new window and Open image in new window have a unique common fixed point. Moreover, for any Open image in new window , the sequence Open image in new window of the initial point Open image in new window , where Open image in new window is defined by Open image in new window for all Open image in new window , converges to the fixed point.

Remark 1.12.

The two maps Open image in new window and Open image in new window are said to be Open image in new window if they satisfy condition (1.5). This concept was introduced by Huang and Zhang [3] 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 Open image in new window is actually Open image in new window . 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 Open image in new window in all our results and extend Open image in new window to a metrizable linear topological space. Some other consequences follow.

2. Theorems on Single Maps

Theorem 2.1.

Let Open image in new window be a complete cone metric space and Open image in new window be mappings such that

for all Open image in new window where Open image in new window and Open image in new window . Then the mappings Open image in new window have a unique fixed point. Moreover, for any Open image in new window , the sequence Open image in new window converges to the fixed point.

Proof.

We adapt the technique in [13]. Without loss of generality we may assume that Open image in new window and Open image in new window so that from (2.1), we have

Set Open image in new window in (2.1) and simplify to obtain

By the triangle inequality, Open image in new window and so from (2.3) we get
which on simplifying gives
Substituting (2.5) into (2.3) we obtain
where Open image in new window . Let Open image in new window , then in view of (2.8), we obtain
for Open image in new window . Therefore, Open image in new window is a Cauchy sequence in Open image in new window . Since Open image in new window is complete, there exists Open image in new window such that Open image in new window . Choose a natural number Open image in new window such that for all Open image in new window ,

Thus, Open image in new window , for all Open image in new window . So Open image in new window , for all Open image in new window . Since Open image in new window as Open image in new window , and Open image in new window is closed, Open image in new window . But Open image in new window and so Open image in new window . Hence Open image in new window . The uniqueness follows from the contractive definition of Open image in new window in (2.1).

Remark 2.2.

The theorem is valid if we replace the completeness of Open image in new window with the condition that Open image in new window is complete. If Open image in new window is restricted to a normed linear space and Open image in new window in Theorem 2.1 we have [5, Theorem ?2.3]; if Open image in new window in Theorem 2.1, we obtain [5, Theorem ?2.6]; if Open image in new window , we obtain [5, Theorem ?2.7] and if Open image in new window , 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.

Thus Theorem 2.1 is both an extension generalization and an improvement of the results of [3, 5].

We now consider the situation where Open image in new window in Theorem 2.1.

Theorem 2.3.

Let Open image in new window be a sequentially compact cone metric space and Open image in new window be a continuous mapping such that

for all Open image in new window where Open image in new window and Open image in new window . Then the mappings Open image in new window have a unique fixed point.

Proof.

We follow the same argument as Theorem 2.1. Without loss of generality, we may assume that Open image in new window and Open image in new window are less than 1. Hence (2.8) becomes

Since Open image in new window is sequentially compact, then it is compact [10]. The fact that Open image in new window is continuous and Open image in new window is compact implies that Open image in new window is compact and hence Open image in new window exists and Open image in new window for some Open image in new window . From (2.14), it can be infered that Open image in new window is fixed under Open image in new window and uniqueness follows from (2.13).

Remark 2.4.

If Open image in new window , with the additional assumption that Open image in new window is a regular cone in Theorem 2.3, we obtain [3, Theorem ?2]. Thus Theorem 2.3 is both an extension and improvement of [3, Theorem ?2].

3. Common Fixed Points

Theorem 3.1.

Let Open image in new window be a cone metric space and let Open image in new window be mappings such that

for all Open image in new window where Open image in new window and Open image in new window . Suppose Open image in new window and Open image in new window are weakly compatible and Open image in new window such that Open image in new window or Open image in new window is a complete subspace of Open image in new window , then the mappings Open image in new window and Open image in new window have a unique common fixed point. Moreover, for any Open image in new window , the sequence Open image in new window defined by Open image in new window for all Open image in new window , converges to the fixed point.

Proof.

Observe that if Open image in new window satisfies (3.1), it also satisfies

for all Open image in new window where Open image in new window and Open image in new window , Open image in new window .

If Open image in new window for all Open image in new window , then Open image in new window is a Cauchy sequence. Suppose Open image in new window for all Open image in new window . Using (3.2) and the fact that Open image in new window for all Open image in new window , we have

Consequently

where Open image in new window .

Let Open image in new window be given and choose a natural number Open image in new window such that Open image in new window for all Open image in new window . Thus,

Next we show that Open image in new window . Suppose Open image in new window , from (3.2), we have

This is a contradiction and hence Open image in new window . Thus Open image in new window is a common fixed point of Open image in new window and Open image in new window . The uniqueness follows from (3.1).

Remark 3.2.
  1. (i)

    If Open image in new window and Open image in new window is restricted to normed linear spaces in Theorem 3.1, with the additional normality assumption, we obtain the common fixed point Theorem of Vetro [7].

     
  2. (ii)

    Suppose Open image in new window is restricted to normed linear spaces, with the additional normality assumption, if Open image in new window , then Theorem 3.1 gives [4, Theorem ?2.1]; if Open image in new window , we obtain [4, Theorem ?2.3], and if Open image in new window , we obtain [4, Theorem ?2.4]. Thus our theorem is both an extension, generalization and an improvement of the results of [4, 7].

     
  3. (iii)

    If Open image in new window is restricted to normed linear spaces, Theorem 3.1 reduces to [14, Theorem ?2.8].

     
  4. (iv)

    If in Theorem 3.1 we choose choose Open image in new window the identity mapping on Open image in new window , we have Theorem 2.1.

     

Open Question

Theorem 2.3 was proved for the usual metric space by the author in [15] without the assumptions that Open image in new window is continuous and Open image in new window is compact. Is the above Theorem 2.3 still valid if we remove the assumption that Open image in new window is continuous and Open image in new window is compact?.

Notes

Acknowledgments

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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Copyright information

© J. O. Olaleru. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of LagosYaba, LagosNigeria

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