Some Generalizations of Fixed Point Theorems in Cone Metric Spaces

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.

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 [4–7]). This paper generalizes, extends and improves the results of all those authors.

The following definitions are given in [3].

Let be a real Banach space and a subset of . is called a cone if and only if

(i) is closed, nonempty, and ;

(ii), , ;

(iii).

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

The cone is called if there is such that for all , implies .

The least positive number satisfying the above is called the normal constant of .

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 [5] 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 [8]). 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 [9]. Therefore our generalizes the as a normed linear space used in all the previous results on cone metric spaces.

A cone is therefore called normal if there is such that for all , implies .

Definition 1.1.

Let be a nonempty set. Suppose that satisfies

(i) for all and if and only if ,

(ii) for all ,

(iii) for all .

Then is called a cone metric on , and is called a cone metric space.

Example 1.2 (see [3]).

Let , , and defined by , where is a constant. Then is a cone metric space.

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

We now give another example where is a metrizable linear topological vector space that is not a normed linear space.

Example 1.3.

Let , (), , a metric space and defined by . Then is a cone metric space.

Definition 1.4.

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

Definition 1.5.

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

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

Definition 1.6.

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 [10]).

Proposition 1.7 (see [3]).

Let be a cone metric space, a normal cone. Let and be two sequences in and , as . Then

(i) converges to if and only if as

• (ii) The limit of is unique

(iii) is a Cauchy sequence if and only if as

(iv) as

Huang and Zhang [3] proved the following theorems for a Banach space.

Theorem 1.8.

Let be a complete metric space, a normal cone with normal constant . Suppose that the mapping satisfies the contractive condition

(1.1)

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

Theorem 1.9.

Let be a complete metric space, a normal cone with normal constant . Suppose that the mapping satisfies the contractive condition

(1.2)

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

Theorem 1.10.

Let be a complete metric space, a normal cone with normal constant . Suppose that the mapping satisfies the contractive condition

(1.3)

where is a constant. Then has a unique fixed point in . And for any , iterative sequence 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 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 [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 be a cone metric space, a normal cone with normal constant . Let be mappings such that

(1.4)

for all where and . Suppose

(1.5)

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.

Remark 1.12.

The two maps and are said to be 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 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

Theorem 2.1.

Let be a complete cone metric space and be mappings such that

(2.1)

for all where and . Then the mappings have a unique fixed point. Moreover, for any , the sequence converges to the fixed point.

Proof.

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

(2.2)

Set in (2.1) and simplify to obtain

(2.3)

By the triangle inequality, and so from (2.3) we get

(2.4)

which on simplifying gives

(2.5)

Substituting (2.5) into (2.3) we obtain

(2.6)

and by symmetry, we may exchange with and with in (2.6) to obtain

(2.7)

If = min, then

(2.8)

where . Let , then in view of (2.8), we obtain

(2.9)

Let be given and choose a natural number such thatfor all . Thus,

(2.10)

for . Therefore, is a Cauchy sequence in . Since is complete, there exists such that . Choose a natural number such that for all ,

(2.11)

Then

(2.12)

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

Remark 2.2.

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.

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

We now consider the situation where in Theorem 2.1.

Theorem 2.3.

Let be a sequentially compact cone metric space and be a continuous mapping such that

(2.13)

for all where and . Then the mappings have a unique fixed point.

Proof.

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

(2.14)

Since is sequentially compact, then it is compact [10]. 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).

Remark 2.4.

If , with the additional assumption that 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 be a cone metric space and let be mappings such that

(3.1)

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.

Proof.

Observe that if satisfies (3.1), it also satisfies

(3.2)

for all where and , .

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

(3.3)

Consequently

(3.4)

Now, for all , with , we have

(3.5)

where .

Let be given and choose a natural number such that for all . Thus,

(3.6)

for . Therefore, is a Cauchy sequence. Since or is complete, then there exists such that and . Let such that . We claim that . From (3.2), we have

(3.7)

As we obtain

(3.8)

Since and and are weakly compatible, then

(3.9)

Next we show that . Suppose , from (3.2), we have

(3.10)

This is a contradiction and hence . Thus is a common fixed point of and . The uniqueness follows from (3.1).

Remark 3.2.

• (i) 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 [7].

• (ii) 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].

• (iii) If is restricted to normed linear spaces, Theorem 3.1 reduces to [14, Theorem ?2.8].

• (iv) If in Theorem 3.1 we choose choose the identity mapping on , 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 is continuous and is compact. Is the above Theorem 2.3 still valid if we remove the assumption that is continuous and is compact?.