Abstract and Applied Analysis

Volume 2012 (2012), Article ID 736214, 19 pages

doi:10.1155/2012/736214

Research Article

Common Fixed Point Theorems for a Class of Twice Power Type Contraction Maps in G-Metric Spaces

Hongqing Ye and Feng Gu

Institute of Applied Mathematics and Department of Mathematics, Hangzhou Normal University, Hangzhou Zhejiang 310036, China

Received 5 February 2012; Accepted 27 July 2012

Academic Editor: Svatoslav Staněk

Copyright © 2012 Hongqing Ye and Feng Gu. 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.

Abstract.

We introduce a new twice power type contractive condition for three mappings in G-metric spaces, and several new common fixed point theorems are established in complete G-metric space. An example is provided to support our result. The results obtained in this paper differ from other comparable results already known.

1. Introduction

The study of fixed points of mappings satisfying certain contractive conditions has been in the center of rigorous research activity. In 2006, a new structure of generalized metric space was introduced by Mustafa and Sims [1] as an appropriate notion of generalized metric space called -metric space. Abbas and Rhoades [2] initiated the study of common fixed point in generalized metric space. Recently, many fixed point theorems for certain contractive conditions have been established in -metric spaces, and for more details one can refer to [3–27]. Fixed point problems have also been considered in partially ordered -metric spaces [28–31], cone metric spaces [32], and generalized cone metric spaces [33].

In 2006, Gu and He [34] introduced a class of twice power type contractive condition in metric space, proving some common fixed point theorems for four self-maps with twice power type -contractive condition.

In this paper, motivated and inspired by the above results, we introduce a new twice power type contractive condition in -metric space, and we prove some new common fixed point theorems in complete -metric spaces. Our results obtained in this paper differ from other comparable results already known.

Throughout the paper, we mean by the set of all natural numbers. Consistent with Mustafa and Sims [1], the following definitions and results will be needed in the sequel.

Definition 1.1 (see [1]). Let be a nonempty set, and let be a function satisfying the following axioms:

  if ; , for all with ;   , for all with y;    (symmetry in all three variables);  for all , (rectangle inequality);

then the function is called a generalized metric, or, more specifically, a -metric on and the pair are called a -metric space.

Definition 1.2 (see [1]). Let be a -metric space, and let be a sequence of points in , a point in is said to be the limit of the sequence if , and one says that sequence is -convergent to .

Thus, if in a -metric space , then for any , there exists such that , for all .

Proposition 1.3 (see [1]). Let be a -metric space, then the followings are equivalent.

(1) is -convergent to .(2) as .(3) as .(4) as .

Definition 1.4 (see [1]). Let be a -metric space. A sequence is called -Cauchy sequence if for each there exists a positive integer such that for all ; that is, if as .

Definition 1.5 (see [1]). A -metric space is said to be -complete if every -Cauchy sequence in is -convergent in .

Proposition 1.6 (see [1]). Let be a -metric space. Then the following are equivalent.

(1)The sequence is -Cauchy.(2)For every , there exists such that , for all .

Proposition 1.7 (see [1]). Let be a -metric space. Then the function is jointly continuous in all three of its variables.

Definition 1.8 (see [1]). Let and be -metric space, and be a function. Then is said to be -continuous at a point if and only if for every , there is such that and implies . A function is -continuous at if and only if it is -continuous at all .

Proposition 1.9 (see [1]). Let and be -metric space. Then is -continuous at if and only if it is -sequentially continuous at ; that is, whenever is -convergent to , is -convergent to .

Proposition 1.10 (see, [1]). Let be a -metric space. Then, for any in it follows that:

(i)if , then ;(ii); (iii); (iv); (v);(vi).

2. Main Results

Theorem 2.1. Let be a complete -metric space. Suppose the three self-mappings satisfy the following condition:

for all , where are nonnegative real numbers and . Then , and have a unique common fixed point (say ) and are all -continuous at .

Proof . We will proceed in two steps.

Step 1. We prove any fixed point of is a fixed point of and and conversely. Assume that is such that . However, by (2.1), we have

Now we discuss the above inequality in three cases.

Case (i). If and , then, by , we have

So, the above inequality becomes

Since , hence we have ; however, it contradicts with , so we get .

Case (ii). If , then we have

Hence we have and so .

Case (iii). If , we can also get . Hence we have . Therefore is a common fixed point of and .

The same conclusion holds if or .

Step 2. We prove that , , and have a unique common fixed point.

Let be an arbitrary point, and define the sequence by , . If , for some , with , then is a fixed point of and, by the first step, is a common fixed point of , , and . The same holds if or . Without loss of generality, we can assume that , for all .

Next, we prove sequence is a -Cauchy sequence. In fact, by (2.1) and , we have

Which gives that

It follows that

From we know that . Then, we have

On the other hand, by using (2.1) and , we have

Which implies that

It follows that

Form the condition , we know that . Therefore, we have

Again, using (2.1) and , we can get

Which implies that

It follows that

By the condition , we know that . Hence, we have

Let , then from we know that . Combining (2.9), (2.13), and (2.17), we have

Thus, by and , for every , noting that , we have

Which implies that , as . Thus is a -Cauchy sequence. Due to the completeness of , there exists , such that is -convergent to .

Next we prove is a common fixed point of , and . By using (2.1), we have

Letting , and using the fact that is continuous on its variables, we can get

Which gives that , that is is a fixed point of . By using (2.1) again, we have

Letting at both sides, for is continuous on its variables, it follows that

Therefore, ; that is, is a fixed point of . Similarly, by (2.1), we can also get

On taking at both sides, since is continuous on its variables, we get that

Which gives that , therefore, is fixed point of . Consequently, we have , and is a common fixed point of and . Suppose is another common fixed point of , and , and we have , then by (2.1), we have

Which implies that , hence, . Then we know the common fixed point of , and is unique.

To show that is -continuous at , let be any sequence in such that is -convergent to . For , we have

Which implies that . Hence is -convergent to . So is -continuous at . Similarly, we can also prove that are -continuous at . Therefore, we complete the proof.

Corollary 2.2. Let be a complete -metric space. Suppose the three self-mappings satisfy the condition:

for all , where are nonnegative real numbers and . Then , and have a unique common fixed point (say ) and are all -continuous at .

Proof. From Theorem 2.1 we know that have a unique common fixed point (say ); that is, , and are -continuous at . Since , so is another fixed point of , , so is another fixed point of , and , so is another fixed point of . By and the condition (2.28) in Corollary 2.2, we have

Since , we can get . That means , hence is another common fixed point of -and . Since the common fixed point of -and is unique, we deduce that . By the same argument, we can prove . Thus, we have . Suppose is another common fixed point of , and , then , and by using the condition (2.28) in Corollary 2.2 again, we have

Which implies that , hence . So the common fixed of , and is unique. It is obvious that every fixed point of is a fixed point of and and conversely.

Corollary 2.3. Let be a complete -metric space. Suppose the self-mapping satisfies the following condition:

for all , where are nonnegative real numbers and . Then has a unique fixed point (say ) and is -continuous at .

Proof. Let in Theorem 2.1, we can get this conclusion holds.

Corollary 2.4. Let be a complete -metric space. Suppose the self-mapping satisfies the following condition:

for all , where are nonnegative real numbers and . Then has a unique fixed point (say ) and is -continuous at .

Proof. Let in Corollary 2.2, we can get this conclusion holds.

Theorem 2.5. Let be a complete -metric space, and let be three self-mappings in , which satisfy the following condition.

for all , are nonnegative real numbers and . Then and have a unique common fixed point (say ) and are all -continuous at .

Proof. We will proceed in two steps.

Step 1. We prove any fixed point of is a fixed point of and and conversely. Assume that is such that . Now we prove that and . If it is not the case, then for and , by (2.33) and we have

It follows that

Since , hence we have , however it contradicts with the condition , so we can have , hence is a common fixed point of , and .

Analogously, following the similar arguments to those given above, we can obtain a contradiction for and or and . Hence in all the cases, we conclude that . The same conclusion holds if or .

Step 2. We prove that , and have a unique common fixed point. Let be an arbitrary point, and define the sequence by , . If , for some , with , then is a fixed point of and, by the first step, is a common fixed point of , , and . The same holds if or . Without loss of generality, we can assume that , for all . We first prove the sequence is a -Cauchy sequence. In fact, by using (2.33) and , we have

Which gives that

It follows that

From , we know that . Then, we have

On the other hand, by using (2.33) and , we have

Which implies that

It follows that

Since , we know that . So, we have

Again, using (2.33) and , we can get

Which implies that

It follows that

Since , we know that . So we have

Let , then , and by combining (2.39), (2.43), and (2.47), we have

Thus, by and , for every , if , noting that , we have

Which implies that , as . Thus is a -Cauchy sequence. Due to the completeness of , there exists , such that is -convergent to .

Now we prove is a common fixed point of , and . By using (2.33), we have

Letting , and using the fact that is continuous on its variables and , we can get

Which gives that , hence is a fixed point of . By using (2.33) again, we have

Letting at both sides, for is continuous in its variables, it follows that

For , Therefore, we can get , hence , hence is a fixed point of . Similarly, by (2.33), we can also get

On taking at both sides, since is continuous in its variables, we get that

Since , so we get , hence , therefore, is a fixed point of . Consequently, we have , and is a common fixed point of , and . Suppose is another common fixed point of , and , and we have , then by (2.33), we have

Which gives that

Hence, we can get . By using (2.33) again, we get

Which implies that

Hence, we can get

By combining , we can have

Since , so we have that . Since , we know , so it’s a contradiction. Hence, we get . Then we know the common fixed point of , and is unique.

To show that is -continuous at , let be any sequence in such that is -convergent to . For , we have

Which implies that

On taking at both sides, considering , we get . Hence is -convergent to . So is -continuous at . Similarly, we can also prove that are -continuous at . Therefore, we complete the proof.

Now we introduce an example to support Theorem 2.5.

Example 2.6. Let , and let be a -metric space defined by , for all in . Let , , and be three self-mappings defined by

Next we proof the mappings , , and are satisfying Condition (2.33) of Theorem 2.5 with , and .

Case 1. If , , then

Thus, we have

Case 2. If , then we can get

Thus, we have

Case 3. If , then we have

Thus, we have

Case 4. If , then we have

Thus, we have

Then in all of the above cases, the mappings , and satisfy the contractive condition (2.33) of Theorem 2.5 with . So that all the conditions of Theorem 2.5 are satisfied. Moreover, is the unique common fixed point for all of the three mappings , and .

At last, we prove , and are all -continuous at the common fixed point . Since , and let the sequence and , then there exists such that , for all . Without loss of generality, we can assume that , and so and . Therefore,

Which implies that , and are all -continuous at the common fixed point .

Corollary 2.7. Let be a complete -metric space. Suppose the three self-mappings satisfy the condition:

for all , where , are nonnegative real numbers and . Then , and have a unique common fixed point (say ) and are all -continuous at .

Proof. Since the proof of Corollary 2.7 is very similar to that of Corollary 2.2, so we delete it.

Corollary 2.8. Let be a complete -metric space, and let be a self-mapping in , which satisfies the following condition:

for all , where are nonnegative real numbers and . Then has a unique fixed point (say ) and is -continuous at .

Corollary 2.9. Let be a complete -metric space, and let be a self-mapping in , which satisfies the following condition:

for all , where are nonnegative real numbers and . Then has a unique fixed point (say ) and is -continuous at .

Acknowledgments

The present studies are supported by the National Natural Science Foundation of China (11071169), the Natural Science Foundation of Zhejiang Province (Y6110287, Y12A010095).

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