Four Mappings Satisfying Ψ-Contractive Type Condition and Having Unique Common Fixed Point on 2-Metric Spaces

Abstract

In this paper, we introduce a class Ψ of real functions defined on the set of non-negative real numbers, and obtain a new unique common fixed point theorem for four mappings satisfying Ψ-contractive condition on a non-complete 2-metric space and give the versions of the corresponding result for two and three mappings.

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H. Jin and Y. 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. doi: 10.4236/apm.2013.32039.

1. Introduction and Preliminaries

Using subsidiary conditions [1,2] such as commutability of mappings or uniform boundless of mappings at some point and so on, many authors have discussed and obtained many unique common fixed point theorems of mappings with some contractive or quasi-contractive condition on 2-metric spaces. The author [3-7] obtained similar results for infinite mappings with contractive conditions or quasicontractive conditions under removing the above subsidiary conditions. These results generalized and improved many same type unique common fixed point theorems. Recently, the author [8] discussed the existence of coincidence points and common fixed points for four mappings with -contractive conditions on 2-metric spaces and give some corresponding results.

Here, by introducing a new class of real functions defined on, we will discuss the existence problem of unique common fixed points for four mappings with -contractive type condition on non-complete 2-metric spaces and give some corresponding forms.

The following definitions and lemmas are well known.

Definition1.1. ([3]) 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]) 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. ([3,4]) A sequence is said to be convergent to, if for each,. And we write that and callthe limit of.

Definition 1.4. ([3,4]) A 2-metric space is said to be complete, if every Cauchy sequence inis convergent.

Definition 1.5. ([9,10]) Let andbe self-maps on a set. If for some, then is called a coincidence point of and, and is called a point of coincidence ofand.

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

Lemma 1.7. ([5-7]) 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. ([5-7]) If is a 2-metric space and sequence, then for each.

Lemma 1.9. ([9,10]) Let be weakly compatible. If have a unique point of coincidence, thenis the unique common fixed point of.

2. Main Results

Denoted by the set of functions satisfying the following: is continuous and non-decreasing, for all.

Remark if and only if is continuous and increasing in each coordinate variable and satisfy that and for all, see [8]. Obviously, the set is vary different from the set.

Example Let be defined by

Then, obviously,.

The following is the main result in this paper.

Theorem 2.1. Let be a 2-metric space, four single valued 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 fixed, by (1) and and (iv)

in definition 1.1, we obtain that

Suppose that.

Take, then by (1) and definition 1.1 and, we obtain that

Hence, so we have that

(2)

If for some, then (2) becomes that

This is a contradiction. Hence for all, so we have that

Similarly, we can obtain that

.

Hence we have that

.

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 know that.

For any,

Let，then by and Lemma 1.8, the above becomes

(3)

If for some, then we obtain from (3) 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.

Since, there exists such that. For any,

Let, then by and Lemma 1.8, we obtain that

If for some, then the above becomes that, which is a contradiction since, so for all. Hence, i.e., is a point of coincidence of and, and is coincidence point of and. Suppose that is another point of coincidence of 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, is also the unique point of coincidence of and.

By Lemma 1.9, is the unique common fixed point of 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 will omit this part.

Using Theorem 2.1 and in Example, we will obtain the next particular result.

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

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.

The following two theorems are the contractive and quasi-contractive versions of theorem 2.1 for two mappings.

Theorem 2.3. 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.

Proof Let, then by Theorem 2.1, there existsuch thatis the unique point of coincidence of and. But obviously and are weakly compatible, so is the unique fixed point of by Lemma 1.9. Similarly, is also unique fixed point of, hence is the unique common fixed point of and.

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

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

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

Finally we give two coincidence point theorems for three mappings.

Theorem 2.5. Let be a 2-metric space, three mappings satisfying that . Suppose that for each ,

where and. If one of and is complete, then and and have an unique point of coincidence in. Further, is one to one mapping, then have an unique point of coincidence.

Proof Let, then by Theorem 2.1, there exist a unique element and such that and, hence, which implies that, so we obtain that. This means thatis point of coincidence of. If is also point of coincidence of, then is also point of coincidence of, hence by uniqueness of points of coincidence of and, we have that. Hence is the unique point of coincidence of.

Theorem 2.6. Let be a 2-metric space, three mappings satisfying that . Suppose that for each ,

where and. If one of and is complete, then and and have an unique point of coincidence. Further, is one to one mapping, then have an unique point of coincidence.

Proof The proof is similar to that of Theorem 2.5. So we will omit it.

NOTES

Conflicts of Interest

The authors declare no conflicts of interest.

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