ABSTRACT

In this paper, we consider a countable family of set-valued mappings satisfying some quasi-contractive conditions. We also construct a sequence by the quasi-contractive conditions of mappings and the boundary condition of a closed subset of a metrically convex space, and then prove that the unique limit of the sequence is the unique common fixed point of the mappings. Finally, we give more generalized common fixed point theorems for a countable family of single-valued mappings. The main results generalize and improve many common fixed point theorems for a finite or countable family of single valued or set-valued mappings with quasi-contractive conditions.

1. Introduction

There have appeared many fixed point theorems for a single-valued self map of a closed subset of a Banach space. However, in many applications, the mapping under considerations is not a self-mapping on a closed subset. In 1976, Assad [1] gave sufficient condition for such single valued mapping to obtain a fixed point by proving a fixed point theorem for Kannan mappings on a Banach space and putting certain boundary conditions on the mapping. Similar results for multi-valued mappings were respectively given by Assad [2] and Assad and Kirk [3]. On the other hand, many authors discussed common fixed point problems [4-7] for finite single or multi-valued mappings on a complete 2-metric convex space or a complete cone metric space respectively. And some authors also discussed common fixed point problems [8-13] for a countable family of self-single-valued mappings with contractive or quasi-contractive conditions on a metric space or a metrically convex space respectively. These results improved and generalized many previous works.

In this paper, we will discuss the existent problems of common fixed points for a countable family of surjective set-valued mappings, which satisfy certain quasi-contractive condition, defined on a complete metrically convex space and obtain some important theorems. The main results in this paper further generalize and improve many common fixed point theorems for single valued or multi-valued mappings with quasi-contractive type conditions.

Through this paper, (or) is a metric space. Let denote the families of all bounded closed subset of.

Let, the distance between and.

Definition 1.1. ([8-10]) A metric space is said to be metrically convex, if any with, there exists such that, and.

Lemma 1.1. ([3,8]) If is a nonempty closed subset of a complete metrically convex space, then for any and, there exists which satisfies.

Lemma 1.2. ([13]) If is a complete metric space and, then is continuous on. Moreover, we have :

1);

2) if and only if,;

3) for any,.

2. Main Results

Theorem 2.1. Let be a nonempty closed subset of a complete metrically convex space with, a countable family of surjective set-valued mappings with nonempty values such that for any with, any,

(1)

where and is a constant number.

Furthermore, if for all, and for each and and any, there exists such that, then has a unique common fixed point in.

Proof Take. We will construct two sequences and in the following manner. Since is on-to, there exists such that. If, then put; if, then by Lemma 1.1 there exits such that. For, since is on-to, there exists such that. If, then put; if, then by Lemma 1.1 there exists such that. Continuing this way, we obtain and:

1);

2) if, then put;

3) if, then by Lemma 1.1 there exists such that

4) for all

Let and. If there exists such that, then In fact, By 3) and the definition of, we have that, ,. If, then. On the other hand, since and, hence which is a contradiction. If, then and, hence, so, which is another contradiction.

By the definitions and properties of and, we can estimate into three cases:

Case I.. In this case, , , and. And we have

where

If then

hence

If, then

hence

Therefore, in any situation, we have

Case II. and. In this case, , and and. And we have

where

If then

hence

If, then

hence

Therefore, in any situation, we have

But, hence we obtain

Case III. and. In this case, by the property of and, and, , and. And we have

where

Here, we give two basic properties:

1) since so and hence

2) since

hence

If then

hence by 2),

So by Case II, we obtain

If, then

hence by 2),

So by Case II again, we obtain

Hence in any situation, we have

Therefore, from Case I, Case II and Case III, we obtain

Let, then since, hence we have

so

Let, then for,

as. Hence is a Cauchy sequence. Since is complete, has a limit. But is closed and for all, hence.

By the property of and, we can see that there exists an infinite subsequence of such that, hence and

Next, we will prove that is a common fixed point of. Fix any, for each fixed, there exists such that . Take an enough large such that and. By Lemma 1.2 3) and (1), we have

and

where

If then

Let, then since, hence

. So since, therefore by Lemma 1.2 1).

If, then

Let, then since, hence similarly,

So in any situation, for all, so is a common fixed point of.

If and are all common fixed points of, then we will have

where

If, then, hence;

If, thenhence since, so.

Hence in any situation,. So is the unique common fixed points of

If the mappings in Theorem 2.1 are all single-valued, then Theorem 2.1 becomes the next form.

Theorem 2.2. Let be a nonempty closed subset of a complete metrically convex space with, a countable family of surjective single-valued mappings such that for any with, any,

(2)

where and is a constant number.

Furthermore, if for all, and for each and, there exists such that, then has a unique common fixed point in.

From Theorem 2.2, we can obtain the following more generalized common fixed point theorem.

Theorem 2.3. Let be a nonempty closed subset of a complete metrically convex space with, a family of subjective single-valued mappings, a family of positive integral numbers such that for any, ,

(3)

where and is a constant number. Furthermore, if 1) for all, 2) for each andthere exists such that, 3) for each with,. Then has a unique common fixed point in.

Proof Fix, and let, then satisfies all of the conditions of Theorem 2.2, hence has an unique common fixed point in. Now, we will prove that is also unique common fixed point of. In fact, for any fixed,

. This means that is a fixed point of. For any with, there exists such that by 2), and by (3) we have that

where

If, then, hence;

If, then

, hence

Hence in any situation, we have that is a fixed point of for each with. So is a common fixed point of. By uniqueness of common fixed points of, we have for each. Hence is a common fixed point of.

If and are all common fixed points of, then they are also common fixed points of, hence by the uniqueness of common fixed points of, we obtain. This means that for each has a unique common fixed point.

Now, we prove for each. In fact, for any with, since and, so, hence

by 3). Therefore, is a fixed point of for eachi.e., is a common fixed point of. But has a unique common fixe point, hence for each, and therefore is a common fixed point of. But

has a unique common fixed point, hence. Let, then is the common fixed point of. The uniqueness of common fixed points of is obvious.

Funding

This work was supported by the National Natural Science Foundation of China (No. 11361064).

REFERENCES

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NOTES

^*Corresponding author.