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Common Fixed Point Theorems for a Finite Family of Discontinuous and Noncommutative Maps

Fixed Point Theory and Applications volume 2011, Article number: 847170 (2011) Cite this article

Abstract

We study common fixed point theorems for a finite family of discontinuous and noncommutative single-valued functions defined in complete metric spaces. We also study a common fixed point theorem for two multivalued self-mappings and a stationary point theorem in complete metric spaces. Throughout this paper, we establish common fixed point theorems without commuting and continuity assumptions. In contrast, commuting or continuity assumptions are often assumed in common fixed point theorems. We also give examples to show our results. Results in this paper except those that generalized Banach contraction principle and those improve and generalize recent results in fixed point theorem are original and different from any existence result in the literature. The results in this paper will have some applications in nonlinear analysis and fixed point theory.

1. Introduction and Preliminaries

Let be a metric space and be a multivalued map. We say that is a stationary point of if . The existence theorem of stationary point was first considered by Dancs et al. [1]. If is a self-mapping (multivalued or single valued) defined on , we denote the collection of all the fixed points of . In this paper,we need the following definitions.

Definition 1.1.

A function is called

(i)contraction if there exists such that

(1.1)

(ii)kannan if there exists such that

(1.2)

(iii)quasicontractive if there is a constant such that

(1.3)

where .

(iv)weakly contractive if there exists a lower semicontinuous and nondecreasing function with if and only if such that

(1.4)

It is known that every contraction and every Kannan mapping has a unique fixed point in complete metric spaces Banach [2], Kannan [3] and every quasicontractive mapping has a unique fixed point in Banach spaces Ćirić [4], Rhoades [5]. In 2001, Rhoades [6] proved that every weakly contractive mapping has a unique fixed point in a complete metric space. Let and be self-maps defined on ; the following inequality was considered in the study of common fixed points theorems Rhoades [5], Chang [7]:

(1.5)

for some constant and function .

If and satisfy the inequality (1.5) with

(1.6)

then and are said to be a couple of quasicontractive mappings which is studied by Rhoades [5]. Chang [7] prove that every couple of quasicontractive mappings has a unique common fixed point in Banach spaces. Recently, Zhang and Song [8] proved a common fixed point theorem in complete metric spaces under the following assumption:

(1.7)

where .

The result of Zhang and Song [8] generalized the results in [2, 3, 5, 6]. Motivated by Chang [7], Zhang and Song [8], it is natural to ask whether there is a common fixed point of and in satisfy inequality (1.5) with . In this paper, we give a positive answer to this question in complete metric spaces.

Let be a finite family of self-mappings on . If there is a nondecreasing, lower semicontinuous function with if and only if such that for every ,

(1.8)

where , for all , and

(1.9)

Here, is the identity map defined on and . We show that have a unique common fixed point if is complete. As a special case of this result, we give a common fixed point theorem in complete metric spaces under the assumption that inequality (1.5) holds with . One of our results generalized Banach contraction principle, an example is given (Example 2.12) to show that the maps above need not to be continuous. The assumption of continuity is often used in the existence theorems of fixed points [6, 9–14]. We also give an example to show that the family above is not necessary to be commuting, and in contrast that the commutativity assumption is often used in the existence theorems of common fixed points [9, 10, 13, 15, 16]. Finally, we generalize some of our results to the case of multivalued maps.

Let be multivalued maps satisfy

(1.10)

for some (where denotes the Hausdorff metric). In fact, under the hypothesis that inequality (1.10) holds, we can show that and for all if and have nonempty closed bounded values. Further we give a new stationary point theorem in complete metric spaces and illustrate with examples (Examples 3.4 and 3.8).

2. Fixed Point Theorems

Throughout this paper, let be a complete metric space and let be the set of all positive integers. In this section, all the self-maps on are single valued. The following theorem is the main result in this section.

Theorem 2.1.

Let be a finite family of self-mappings on . If there is a nondecreasing, lower semicontinuous function with if and only if such that for every ,

(2.1)

where

(2.2)

for all , and

(2.3)

is the identity map defined on and .

Then, have a unique common fixed point.

Proof.

For any fixed , take

(2.4)

Continuing in this way, we obtain by induction a sequence in such that , whenever with and . Then, if , we have

(2.5)

If for some , then

(2.6)

Therefore is a decreasing and bounded below sequence,and there exists such that . Since is lower semicontinuous, . Taking upper limits as on two sides of the following inequality

(2.7)

we have

(2.8)

Then, and, hence, .

is a Cauchy sequence in . Indeed, let , . Then is a decreasing sequence. If , we are done. Suppose that , choose small enough and select such that

(2.9)

By the definition of , there exist such that

(2.10)

Since , for all . Replace and if necessary, we may assume that , and

(2.11)

Hence,

(2.12)

Then,

(2.13)

We consider the following two cases:

(i)

(ii).

If , we have

(2.14)

Then, . Since is arbitrary small positive number, if we take . Then,

(2.15)

This yields a contradiction.

If , we have

(2.16)

Then . This also yields a contradiction.

Therefore, and is a Cauchy sequence in . Since is complete, converges to a point in , say .

In order to show that is the unique common fixed point of . We first claim that , for all .

Indeed, for each ,

(2.17)

We consider the following three cases:

(i),

(ii),

(iii).

If , then

(2.18)

If , then

(2.19)

If , then

(2.20)

Continuing in this process, we show that . By the same argument as in the case above, we see that .

Then, we see that , for all . Next, we claim that is the unique fixed point of . Indeed, for any , we have

(2.21)

Then, and . Therefore, is the unique fixed point of and we complete the proof.

Remark 2.2.

  1. (a)

    The sequence approaching to the unique common fixed point in Theorem 2.1 is different from those in [8, 11, 12, 16–19].

  2. (b)

    The finite family of self-mappings in Theorem 2.1 is neither commuting nor continuous, which are often assumed in common fixed point theorems, see [6, 9–16]. In fact, the commuting and continuity assumptions are not needed throughout this paper and we will give examples (Examples 2.12–2.15) to show this fact.

As special cases of Theorem 2.1, we have the following theorems and corollaries.

Theorem 2.3.

Let , be self-mappings on . If there is a nondecreasing, lower semicontinuous function with if and only if such that

(2.22)

for all . Then and have a unique common fixed point.

Proof.

Take , and in Theorem 2.1, then Theorem 2.3 follows from Theorem 2.1.

Corollary 2.4.

Let be self-mappings on . If there is a nondecreasing, lower semicontinuous function with if and only if such that

(2.23)

Then and have a unique common fixed point.

Corollary 2.5.

Let be a self-mapping on . If there is a nondecreasing, lower semicontinuous function with if and only if such that

(2.24)

Then has a unique fixed point.

Proof.

Take in Theorem 2.3, then Corollary 2.4 follows from Corollary 2.4.

Remark 2.6.

  1. (a)

    Since for all implies

    (2.25)

Corollary 2.5 generalizes Theorem  1 in Rhoades [6].

  1. (b)

    Corollary 2.4 is equivalent to Corollary 2.5.

Proof.

It suffices to show that in Corollary 2.4. Indeed, for each , there exists such that . By the hypothesis in Corollary 2.4, and we complete the proof.

  1. (c)

    In Theorem 2.3, the map is not necessary equal , see Example 2.14. In fact, the maps and in Theorem 2.3 are not necessary to be commuting, see Example 2.15.

Theorem 2.7.

Let be a finite family of self-mappings on . If there exists such that for every

(2.26)

where

(2.27)

and is the identity map defined on and .

Then have a unique common fixed point.

Proof.

Take for all , then Theorem 2.7 follows from Theorem 2.1.

Theorem 2.8.

Let be self-mappings on . If there exists such that

(2.28)

for all . Then and have unique common fixed point.

Proof.

Take, , and in Theorem 2.7, then Theorem 2.8 follows from Theorem 2.7.

Corollary 2.9.

Let be self-mappings on and if there exists such that

(2.29)

Then, and have a unique common fixed point.

Corollary 2.10.

Let be a self-map on and if there exists such that

(2.30)

Then has a unique fixed point.

Proof.

Take in Corollary 2.9, then Corollary 2.10 follows from Corollary 2.9.

Remark 2.11.

  1. (i)

    If is contractive, then there exists such that for all , but the converse is not true. It is obvious that Corollary 2.9 is a special case of Corollaries 2.4 and 2.10 is a generalization of Banach contraction principle. Further we see that Corollary 2.9 is equivalent to Corollary 2.10 by the same argument as in Remark 2.6.

  2. (ii)

    A map satisfies for all and for some is neither continuous nor nonexpansive. We give an example (Example 2.12) to show this fact.

Example 2.12.

Let be defined by

(2.31)

Then, for all and is not continuous.

Proof.

We consider the following three cases:

(i),

(ii),

(iii).

If , , then

(2.32)

If , , then

(2.33)

If , , then

(2.34)

and, hence . It is obvious that is not continuous at but for all .

Example 2.13.

Let be the same as in Example 2.12. and take for all . Since for all . By Example 2.12 and Corollary 2.10, we see that has a unique common fixed point. But for each , is not continuous, the results in [13, 16] do not work in this example. Further it is obvious that the family have a unique common fixed point 0.

Example 2.14.

Let and define maps by and . Then and we see that and have a unique common fixed point.

Proof.

It suffices to show that there exists such that

(2.35)

for all .

We have to consider the following two cases:

(i),

(ii).

If , then

(2.36)

If , then

(2.37)

If we take , then by Theorem 2.8, we see that and have a unique common fixed point. In fact, 0 is the unique common fixed point of and .

By the same argument as in Example 2.14, we give the following example to show that the maps and in Theorem 2.8 are not necessary to be commuting.

Example 2.15.

Let and define maps by and . Then and are not commuting and we see that and have a unique common fixed point.

Proof.

It suffices to show that there exists such that

(2.38)

for all .

We have to consider the following two cases:

(i),

(ii).

If , then

(2.39)

If , then

(2.40)

If we take , then by Theorem 2.8, we see and have a unique common fixed point. In fact, 0 is the unique common fixed point of and .

3. A Common Fixed Point Theorem of Set-Valued Maps and a Stationary Point Theorem

In this section, we study a fixed point theorem and a stationary point theorem which generalize a fixed point theorem in Section 2.

In this section, let be the class of all nonempty bounded closed subsets of and for , let be the Hausdorff metric of and and let for all .

Lemma 3.1 (see [20]).

For all , and , there exists such that .

Theorem 3.2.

Let be multivalued maps. If there exists such that

(3.1)

Then and for all .

Proof.

For any fixed and . Take , and let . By Lemma 3.1, we may choose such that , such that , such that . Continuing in this process, we obtain by induction a sequence such that

(3.2)

Therefore,

(3.3)

Therefore, for all and

(3.4)

This shows that

(3.5)

and is a Cauchy sequence. Since is complete, there exists such that . Since

(3.6)

and , . Therefore and .

To complete the proof, it suffices to show the following four cases:

(i) and for all ,

(ii),

(iii) for all ,

(iv).

For any ,

(3.7)

This shows that and . Further

(3.8)

and .

For any ,

(3.9)

This shows that . Till now, we see that and for all .

For any ,

(3.10)

Hence .

It remains to show that .

Indeed, for any ,

(3.11)

and . Then and .

Remark 3.3.

  1. (a)

    If one of and in Theorem 3.2 is single valued, then the set is singleton and the maps and have a unique common fixed point in . Therefore, Theorem 3.2 is a generalization of Corollary 2.9, but Theorem 3.2 is not a generalization of Theorem 5 Nadler [20].

  2. (b)

    The sequence approaches to the common fixed point of and in Theorem 3.2 is different from those in [20–25].

  3. (c)

    By Example 2.12, we see that both and in Theorem 3.2 are neither to be upper semicontinuous nor to be lower semicontinuous (multivalued maps). Further the maps and are not necessary to be commuting. We give an example below.

Example 3.4.

Let and let maps be defined by and . Then we see that and have a unique common fixed point.

Proof.

It suffices to show that there exists such that

(3.12)

We have to consider the following two cases:

(i)

(ii)

If , we have

(3.13)

If , we have

(3.14)

If we take , then by Theorem 3.2, we see and have a unique common fixed point. In fact, 0 is the unique common fixed point of and .

Corollary 3.5.

Let be a multivalued map with nonempty compact values and such that

(3.15)

Then and for all .

Similarly, we have the following existence theorem of stationary points.

Theorem 3.6.

Let be a multivalued map, be a single valued function. If is a nondecreasing, lower semicontinuous function with for all and if and only if . Suppose that

(3.16)

Then has a unique stationary point, say . In fact, and .

Proof.

For any fixed , let , , , . Continuing in this process, we obtain by induction a sequence such that and . Since

(3.17)

Then, is a decreasing and bounded below sequence, and hence there exist such that . Since is lower semicontinuous, . Taking upper limits as on two sides of the following inequality

(3.18)

we have

(3.19)

Then, and hence .

is a Cauchy sequence in . Indeed, let . Then is a decreasing sequence. If , we are done. Suppose that , choose small enough and select such that

(3.20)

By the definition of , there exists such that

(3.21)

Since , for all . Replace and if necessary, we may assume that , and

(3.22)

Hence,

(3.23)

Then,

(3.24)

and . Since is arbitrary small positive number, if we take . Then

(3.25)

This yields a contradiction. Therefore and is a Cauchy sequence in . By the completeness of , converges to a point in , say .

Since,

(3.26)

It follows that .

Further for all ,

(3.27)

Then, and .

For all ,

(3.28)

Then, and hence .

Remark 3.7.

  1. (a)

    The single valued map in Theorem 3.6 is not necessary to be continuous (see Example 2.12), but the continuity assumption is used in Theorem 3.2 [21, 22, 25] and Theorem 2.1 Ćirić and Ume [23]. We give an example to show that and in Theorem 3.6 are not necessary to be commuting.

  2. (b)

    Theorems 3.2 and 3.6 are different and Theorem 3.6 is also a generalization of Corollary 2.9.

Example 3.8.

Let and let maps and be defined by and . Then we see that has a unique stationary point.

Proof.

It suffices to show that there exists such that

(3.29)

We have to consider the following two cases:

(i),

(ii).

If , we have

(3.30)

If , we have

(3.31)

If we take . Then by Theorem 3.6, we see has a unique stationary point. In fact, 0 is the unique stationary point of .

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    Lai-Jiu Lin & Sung-Yu Wang

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Lin, LJ., Wang, SY. Common Fixed Point Theorems for a Finite Family of Discontinuous and Noncommutative Maps. Fixed Point Theory Appl 2011, 847170 (2011). https://doi.org/10.1155/2011/847170

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