ABSTRACT

In this paper, some new unique common fixed points for four mappings satisfying Ф-contractive conditions on noncomplete 2-metric spaces are obtained, in which the mappings do not satisfy continuity and commutation. The main results generalize and improve many well-known and corresponding conclusions in the literatures.

1. Introduction

There have appeared many unique common fixed point theorems of mappings with some contractive condition on 2-metric spaces. But most of them held under subsidiary conditions [1-3], for examples: commutativity of mappings or uniform boundness of mappings at some point, and so on. In [4-8], the author obtained similar results for infinite mappings with generalized contractive or quasi-contractive conditions under removing the above subsidiary conditions. These results generalized and improved many same type unique common fixed point theorems.

In this paper, by introducing a new class Ф, we will discuss the existence problem of unique common fixed points for four mappings with Ф-contractive type on noncomplete 2-metric spaces without any subsidiary conditions. The obtained main results in this paper further generalize and improve the corresponding results.

Here, we give some well known concepts and results.

Definition 1.1. ([4]) A 2-metric space consists of a nonempty set X 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. ([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. ([4,5]) A sequence is said to be convergent to, if for each,

.

And write and call x the limit of.

Definition 1.4. ([4,5]) A 2-metric space is said to be complete, if every cauchy sequence in X is convergent.

Definition 1.5. ([9,10]) Let f and g be self-maps on a set X. If for some, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g.

Definition 1.6. ([11]) Two mappings are weakly compatible if, for every, holds fgx = gfx whenever

Lemma 1.7. ([6-8]) 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. ([6-8]) If is a 2-metric space and sequence, then

for each.

Lemma 1.9. ([9,10]) Let be weakly compatible. If f and g have a unique point of coincidence, then w is the unique common fixed point of f and g.

2. Main Results

Denote Ф the set of functions satisfying the following conditions: is continuous and increasing in each coordinate variable, and and for all.

Examples Let be defined by

where are non-negative real numbers satisfying

.

Then obviously,.

The following theorem is the main result in this present paper.

Theorem 2.1. Let be a 2-metric space, S, T, I, four single valued mappings satisfying that and. Suppose that for each,

(1)

where and. If one of and is complete, then T and I, S and J have an unique point of coincidence in X. Further, and are weakly compatible respectively, then S, T, I, J have an unique common fixed point in X.

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

(2)

For any

(3)

If, then by (1) and Ф, we have that

(4)

which is a contradiction since, hence . And therefore, (3) becomes that

(5)

If there exists an such that

, then (5) becomes

which is a contradiction since, hence we have that for all. So by (5) and Ф, we obtain that

(6)

Similarly, we can prove that

(7)

Hence we have that

(8)

So is a Cauchy sequence by Lemma 1.7.

Suppose that is complete, then there exists and such that

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

Since

and is Cauchy sequence and, we obtain that.

For any,

Let, then the above becomes

If for some, then we obtain that

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

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., u is a point of coincidence of S and J, and w is a coincidence point of S and J.

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

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

By Lemma 1.9, u is the unique common fixed point and respectively, hence u is the unique common fixed point of S, T, I, J.

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

Here, we give only one of particular forms of theorem 2.1, which itself also generalize and improve many known results.

Theorem 2.2. Let be a 2-metric space, S, T, I, four single valued mappings satisfying that and. Suppose that for each,

where are non-negative real numbers satisfying

.

If one of, , and is complete, then T and I, S and J have an unique point of coincidence in X. Further, and are weakly compatible respectively, then S, T, I, J have an unique common fixed point in X.

Proof Take satisfying

and let

.

Then obviously, , hence q and satisfy all conditions of Theorem 2.1, so the conclusion follows from theorem 2.1 Using Theorem 2.1, we give the following contractive or quasi-contractive versions of Theorem 2.1 for two mappings.

Corollary 2.3 Let be a 2-metric space, two single valued mappings satisfying that for each,

where 0 < q < 1 and. If one of and is complete, then T and S have an unique common fixed point in X.

Proof Let, then by Theorem 2.1, there exist such that u is the unique point of coincidence of S and J. But obviously S and J are weakly compatible, so u is the unique fixed point of S by Lemma 1.9. Similarly, u is also unique fixed point of T, hence u is the unique common fixed point of S and T.

Corollary 2.4 Let be a complete 2-metric space, two single valued surjective mappings satisfying for each,

where and, then I and J have an unique common fixed point in X.

Proof Let, then by Theorem 2.1, there exist such that u is the unique point of coincidence of S and J. But obviously S and J are weakly compatible, so u is the unique fixed point of J by Lemma 1.9. Similarly, u is also unique fixed point of I, hence u is the unique common fixed point of I and J.

REFERENCES

NOTES

^*This paper is supported by the Foundation of Education Ministry, Jilin Province, China (No: 2011[434]).