**Applied Mathematics** Vol.4 No.11A(2013), Article ID:38833,6 pages DOI:10.4236/am.2013.411A1001

An Improvement of a Known Unique Common Fixed Point Result for Four Mappings on 2-Metric Spaces^{*}

Department of mathematics, College of Science, Yanbian University, Yanji, China

Email: ^{#}pyj6216@hotmail.com

Copyright © 2013 Ailian Jin, Yongjie Piao. 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.

Received February 28, 2013; revised March 28, 2013; accepted April 5, 2013

**Keywords:** 2-Metric Space; Class; Class; Common Fixed Point

ABSTRACT

In this paper, we introduce a new class, which is weaker than a known class, of real continuous functions defined on, and use another method to prove the known unique common fixed point theorem for four mappings with -contractive condition instead of -contractive condition on 2-metric spaces.

1. Introduction

The second author has obtained an unique common fixed point theorem for four mappings with -contractive condition [1,2] on 2-metric spaces in [1], where is a continuous and non-decreasing real function on satisfying that for all. The result generalizes and improves many corresponding results.

Here, we introduce a new class of real functions defined on, and reprove the well known unique common fixed point theorem for four mappings with -contractive condition replaced by -contractive condition on 2-metric spaces. The method used in this paper is very different from that in [1].

At first, we give well known definitions and results.

Definition 1.1. ([3,4]) A 2-metric space consists of a nonempty set and a function

such that

1) for distant elements, there exists an such that;

2) if and only if at least two elements in are equal;

3), where is any permutation of;

4) for all.

**Definition 1.2. **([3,4]) A sequence in 2-metric space is said to be cauchy sequence, if for each there exists a positive integer such that for all and.

**Definition 1.3.** ([5,6]) A sequence is said to be convergent to, if for each,

.

And write and call the limit of.

**Definition 1.4.** ([5,6]) A 2-metric space is said to be complete, if every cauchy sequence in is convergent.

**Definition 1.5. **([7,8]) Let and be two selfmappings on a set. If for some, then is called a coincidence point of and, and is called a point of coincidence of and.

**Definition 1.6.** ([9]) Two mappings are said to be weakly compatible if, for every, holds whenever

The following three lemmas are known results.

**Lemma 1.7. **([3-6]) Let be a 2-metric space and a sequence. If there exists such that

for all and, then for all, and is a cauchy sequence.

**Lemma 1.8.** ([3-6]) If is a 2-metric space and sequence, then

for each.

**Lemma 1.9.** ([7,8]) Let be weakly compatible. If and have a unique point of coincidence, then is the unique common fixed point of and.

2. Main Results

Denote by the set of functions satisfying the following:

(1) is continuous; (2) for all.

Denote by the set of functions

satisfying the following:

(1) is continuous and non-decreasing; (2) for all.

Obviously, is stronger than.

**Example 2.1.** Define as follow:

Obviously, , but since, so.

The following is the main conclusion in this paper.

**Theorem 2.2.** Let be a 2-metric space,

four mappings satisfying that

and.

Suppose that for each,

(1)

where and. If one of

and is complete, then and, and have an unique point of coincidence in. Further, and are weakly compatible respectively, then have an unique common fixed point in

.

**Proof** Take any element, then in view of the conditions and, we can construct two sequences and as follows:

For any,

(2)

If

for some, then, hence we have that

Hence we can assume now that

for all.

If

for some, then (2) becomes that

which is a contradiction since. Hence we have that

for all.

If for some, then from (2),

(3)

If for some, then from (2),

(4)

If, then

which is a contradiction since. hence.

So (4) becomes that

(5)

Hence we obtain that

(6)

By (3) and (6), we obtain that

(7)

Similarly, we can obtain that for each

. (8)

Combining (7) and (8), we have that

. (9)

Hence is Cauchy sequence by Lemma 1.7.

Suppose that is complete, then there exists and such that

(If is complete, then there exists ,hence the conclusions remains the same).

Since

and is Cauchy sequence and, we know that.

For any,

Let, then by Lemma 1.8, the above becomes

If for some, then we obtain that

which is a contradiction since. Hence for all, so, i.e., is a point of coincidence of and, and is a coincidence point of and.

On the other hand, since, there exists such that By (1), for any,

Let, then we obtain that

If for some, then the above becomes that

which is a contradiction since 0 < q < 1, so for all. Hence, i.e, is a point of coincidence of and, and is a coincidence point of and.

If is another point of coincidence of S and, then there exists such that, and we have that

which is a contradiction. So for all, hence, i.e, is the unique point of coincidence of and. Similarly, we can prove that is also the unique point of coincidence of and.

By Lemma 1.9, is the unique common fixed point and respectively, hence is the unique common fixed point of.

If or is complete, then we can also use similar method to prove the same conclusion. We omit the part.

The following particular form of Theorem 2.2 for -condition is the main result in [1]. The detailed proof can be found in [1].

**Theorem 2.3. **Let be a 2-metric space,

four mappings satisfying that

anwd. Suppose that for each

,

(10)

where 0 < q < 1 and. If one of

and is complete, then and, and have an unique point of coincidence in.

Further, and are weakly compatible respectively, then have an unique common fixed point in.

Using Theorem 2.2, we can give many different type fixed point or common fixed point theorems. But we give only the next two contractive or quasi-contractive versions of Theorem 2.2 for two mappings.

Theorem 2.4. Let be a 2-metric space, two mappings satisfying that for each,

where and. If one of and is complete, then and have an unique common fixed point in.

Theorem 2.5. Let be a complete 2-metric space, two surjective mappings. If for each,

where and. Then and have an unique common fixed point in.

REFERENCES

- H. L. Jin and Y. J. Piao, “Four Mappings Satisfying Ψ-Contractive Type Condition and Having Unique Common Fixed Point on 2-Metric Spaces ,” Advances in Pure Mathematics, Vol. 3, No. 2, 2013, pp. 277-281. http://dx.doi.org/10.4236/apm.2013.32039
- B. Samet, M. Rajovic, R. Lazovi and R. Stoilikovic, “Common Fixed Point Results for Nonlinear Contractions in Ordered Partial Metric Spaces,” Fixed Point and Applications, Vol. 71, No. 1, 2011. http://dx.doi.org/10.1186/1687-1812-2011-71
- S. L. Singh, “Some Contractive Type Principles on 2-Metric Spaces and Applications,” Mathematics Seminar Notes (Kobe University), Vol. 7, No. 1, 1979, pp. 1-11.
- Y. J. Piao, “A Family of Quasi-Contractive Type NonCommutative Self-Maps Having an Unique Common Fixed Point in 2-Metric Spaces,” Journal of Heilongjiang University (Science Edition), Vol. 23, No. 5, 2006, pp. 655-657.
- Y. J. Piao, “Unique Common Fixed Point for a Family of Self-Maps with Same Type Contractive Condition in 2-Metric Spaces,” Analysis in Theory and Applications, Vol. 24, No. 4, 2008, pp. 316-320. http://dx.doi.org/10.1007/s10496-008-0316-9
- Y. J. Piao, “Unique Common Fixed Point for a Family of Self-Maps with Same Quasi-Contractive Type Condition in 2-Metric Space,” Journal of Nanjing University (Mathematical Biquarterly), Vol. 27, No. 1, 2010, pp. 82-87.
- M. Abbas and G. Jungck, “Common Fixed Point Results for Noncommuting Mappings without Continuity in Cone Metric Spaces,” Journal of Mathematical Analysis and Applications, Vol. 341, No. 1, 2008, pp. 416-420. http://dx.doi.org/10.1016/j.jmaa.2007.09.070
- Y. Han and S. Y. Xu, “New Common Fixed Point Results for Four Maps on Cone Metric Spaces,” Applied Mathematics, Vol. 2, No. 9, 2011, pp. 1114-1118. http://dx.doi.org/10.4236/am.2011.29153
- C. D. Bari and P. Vetro, “Φ-Pairs and Common Fixed Points in Cone Metric Spaces,” Rendiconti del Circolo Matematico Palermo, Vol. 57, 2008, pp. 279-285.

NOTES

^{*}This paper is supported by the NNSF of China (No. 11261062) and (N0. 11361064).

^{#}Corresponding author.