Common Fixed Points for Generalized -Pair Mappings on Cone Metric Spaces

Abstract

We define the concept of generalized -pair mappings and prove some common fixed point theorems for this type of mappings. Our results generalize some recent results.

1. Introduction

Huang and Zhang [1] recently introduced the concept of cone metric spaces and established some fixed point theorems for contractive mappings in these spaces. Afterwards, Rezapour and Hamlbarani [2] studied fixed point theorems of contractive type mappings by omitting the assumption of normality in cone metric spaces. Also, other authors proved the existence of points of coincidence, common fixed point, and coupled fixed point for mappings satisfying different contraction conditions in cone metric spaces (see [1–12]). In [6] Di Bari and Vetro introduced the concept of -map and proved a main theorem generalizing some known results. We define the concept of generalized -mappings and prove some results about common fixed points for such mappings. Our results generalize some results of Huang and Zhang [1], Di Bari and Vetro [6], and Abbas and Jungck [3]. First, we recall some standard notations and definitions in cone metric spaces.

Let be a real Banach space and let denote the zero element in . A cone is a subset of such that

(i) is closed, nonempty, and ,

(ii)if are nonnegative real numbers and , then ,

(iii).

For a given cone , the partial ordering with respect to is defined by if and only if . The notation will stand for but . Also, we will use to indicate that where denotes the interior of . Using these notations, we have the following definition of a cone metric space.

Definition 1.1 (see [1]).

Let be a nonempty set and let be a real Banach space equipped with the partial ordering with respect to the cone . Suppose that the mapping satisfies the following conditions:

for all and if and only if ,

for all ,

for all .

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

The cone is called normal if there exists a constant such that for every if then . The least positive number satisfying this inequality is called the normal constant of . The cone is called regular if every increasing (decreasing) and bounded above (below) sequence is convergent in . It is known that every regular cone is normal [1] (see also [2, Lemma ]).

Definition 1.2 (see [1]).

Let be a cone metric space, let be a sequence in and let .

(i) is said to be Cauchy sequence if for every with there exists such that for all , .

(ii) is said to be convergent to , denoted by or as if for every with there exists such that for all , .

(iii) is said to be complete if every Cauchy sequence in is convergent in .

(iv) is said to be sequentially compact if for every sequence in there exists a subsequence of such that is convergent in .

Clearly, every sequentially compact cone metric space is complete (see [1–12]) for more related results about complete cone metric spaces). We also note that the relations and always hold true.

Definition 1.3 (see [13]).

Let and be self-mappings of a cone metric space . One says that and are compatible if , whenever is a sequence in such that for some .

The concept of weakly compatible mappings is introduced as follows.

Definition 1.4 (see [13]).

The self-mappings and of a cone metric space are said to be weakly compatible if they commute at their coincidence points, that is, if for some , then .

2. Main Results

In this section, we introduce the notation of generalized -mapping and a contractive condition called generalized -pair. We prove some results on common fixed points of these mappings on cone metric spaces.

Let be a cone. A nondecreasing mapping is called a -mapping [6] if

and for ,

for every ,

for every .

Definition 2.1.

Let be a cone and let be a sequence in . One says that if for every with there exists such that for all .

For a nondecreasing mapping we define the following conditions which will be used in the sequal:

if and only if ,

for every , if and only if

for every , .

Definition 2.2.

The self-mappings are called generalized -pair if there exist a -mapping and a mapping satisfying the conditions , and such that

(2.1)

for every .

Now, we are in the position to state the following theorem.

Theorem 2.3.

Let be a cone metric space and let be a generalized -pair. Suppose that and are weakly compatible with such that or is complete. Then the self-mappings and have a unique common fixed point in .

Proof.

Let and choose such that . This can be done, since . Continuing this process, after choosing , we choose such that . Since and are generalized -pair, by Definition 1.2, there exist a -mapping and a mapping satisfying the conditions and the inequality of (2.1). By (2.1), we deduce

(2.2)

Let then, by , . By ,

(2.3)

Therefore, one can find that such that, for all , . We show that

(2.4)

for a fixed and . This holds when . Now let (2.4) hold for some , then we have

(2.5)

Therefore, by induction and we deduce that is a Cauchy sequence. Suppose that is a complete subspace of , then there exists such that and also (This holds also if is complete with ). Let be such that . We show that . By for one can choose a natural number such that and for all . Then,

(2.6)

Thus, for every . This implies that and hence, . So applying we get which implies that , that is, is a point of coincidence of and . Now, we use the hypothesis that and are weakly compatible to deduce that is a common fixed point of and . From , by compatibility of and , it follows that . If , then we have

(2.7)

which implies that . So is a common fixed point of and . The uniqueness of the common fixed point is clear.

Example 2.4.

Let and let be a normal cone. Let with usual metric . Define by and , for all . Also, define by and , for all . Then

(1) and are weakly compatible,

(2),

(3)we have ,

(4).

Example 2.5.

Let and let be a normal cone. Let with metric . Define by and , for all . Also, define by and , for all . Then

(1) and are weakly compatible,

(2),

(3)we have ,

(4).

Example 2.6.

Let and let be a normal cone. Let with metric . Define by and , for all . Also, define by and , for all . Then

(1) and are weakly compatible,

(2),

(3)we have ,

(4).

If we let the mapping be the identity mapping in Theorem 2.3, then we obtain the following corollary.

Corollary 2.7.

Let be a cone metric space. Suppose that the mappings satisfy

(2.8)

for all . If , and are weakly compatible, and or is complete, then and have a unique common fixed point in .

Remark 2.8.

Corollary 2.7 generalizes Theorem in [6]. Also, if we choose the -mapping defined by , where is a constant, then Theorem 2.3 generalizes Theorem in [3]. Furthermore, if we let be the identity map of , then we obtain Theorem in [1], that is, the extension of the Banach fixed point theorem for cone metric spaces.

If we replace the condition with the following condition:

there exists such that for and , then we have the following theorems.

Theorem 2.9.

Let be a cone metric space and let be self-mappings such that

(2.9)

for all where is a nondecreasing mapping from into satisfying the conditions , and and is a nondecreasing mapping satisfying the conditions . Suppose that and are weakly compatible, and or is complete. Then the mappings and have a unique common fixed point in .

Proof.

Let be an arbitrary point in . Choose a point such that . This can be done since . Continuing this process, after choosing with , by (2.9) and we have

(2.10)

Consequently,

(2.11)

where . For we have

(2.12)

Then as and hence, by , is a Cauchy sequence. Suppose that is a complete subspace of , then there exists such that and also (this holds if is complete). Let be such that . By , for a fixed and every there exists a natural number such that and for all . Hence,

(2.13)

which implies that

(2.14)

Thus, for all . This implies that and therefore, . Since and are weakly compatible, . If , then

(2.15)

which gives and hence, . So is a common fixed point for and . The uniqueness of common fixed point is clear.

If in Theorem 2.9 we let be and let the -mapping be , where is a constant, then we obtain the following corollary.

Corollary 2.10.

Let be a cone metric space and let be self-mappings such that

(2.16)

for all , where is a constant. Suppose that and are weakly compatible, the range of contains the range of and or is complete. Then the mappings and have a unique common fixed point in .

Remark 2.11.

Corollary 2.10 generalizes Theorem of [3]. If in Corollary 2.10 we let be the identity map on , then we obtain Theorem of [1].

Theorem 2.12.

Let be a cone metric space and let be self-mappings such that

(2.17)

for all . Suppose that and are weakly compatible, the range of contains the range of and or is complete. Then the mappings and have a unique common fixed point in .

Proof.

Let be an arbitrary point in . Choose a point in such that . This can be done since . Continuing this process having chosen in such that , we have

(2.18)

So,

(2.19)

where . Now let with . Then,

(2.20)

Following an argument similar to that one given in Theorem 2.9, we obtain a unique common fixed point of and .

If in Theorem 2.12 we let be the identity map on and let the -map be , where is a constant, then we obtain the following corollary.

Corollary 2.13.

Let be a cone metric space and let be self-mappings such that

(2.21)

for all , where is a constant. Suppose that and are weakly compatible, the range of contains the range of and or is complete. Then the mappings and have a unique common fixed point in .

Remark 2.14.

Corollary 2.13 generalizes Theorem of [3] and if in Corollary 2.13 we let be the identity map on , then we obtain Theorem of [1].