There have appeared many unique common fixed point theorems for self-maps with some contractive condition on 2-metric spaces. But most of them held under subsidiary conditions [1-4], for examples: commutativity of or uniform boudedness of at some point, and so on. In [5], the author obtained similar results under removing the above subsidiary conditions. The result generalized and improved many same type unique common fixed point theorems. Recently, the author discussed unique common fixed point theorems for a family of contractive or quasi-contractive type mappings on 2-metric spaces, see [6-8], these results improve the above known common fixed point theorems.

In this paper, in order to generalize and unify further these results, we will prove that a family of self-maps satisfying -contractive condition on 2-metric spaces have an unique common fixed point if satisfy the condition 2.

The following definitions are well known results.

Definition 1.1. [4] 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. [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 the limit of.

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

Let denotes a family of mappings such that each, is continuous and increasing in each coordinate variable, and for all.

There are many functions which belongs to:

Example 1.5. Let be defined by

Then obviously,

Example 1.6. Let be defined by

Then obviously, is continuous and increasing in each coordinate variable, and

Hence

The following two lemmas are known.

Lemma 1.7. [1-4] 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. [1-4] If is a 2-metric space and sequence, then for each.

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

Theorem 2.1. Let be a complete 2-metric space, a family of maps from into itself, a family of positive integers, and and for each. If the following - contractive conditions hold

   (1)

and for all with. Then have an unique common fixed point in X.

Proof Fix and let for each, then (1) becomes the following

   (2)

Take an and define a sequence as follows

Then

 (3)

If, then

 (4)

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

  (5)

If there exists an such that, then (5) becomes

which is a contradiction since and , hence he have that for all. In this case, (5) becomes

(6) implies that is a cauchy sequence by Lemma 1, hence by the completeness of, converges to some element.                                                                       (7)

Now, we prove that is the unique common fixed point of. In fact, for any fixed and any with and any,

Let, then by Lemma 2, the continuity of and (7), the above becomes

.

But, hence for all

, and therefore, for all.

This completes that is a common fixed point of

.

Let be a common fixed point of, If there exists an such that, then

which is a contradiction since, hence for all, and therefore. This completes that has an unique common fixed point for all.

Next, we will prove that is the unique common fixed point of for each fixed. Indeed, for fixed, Since for each, hence

for each, which means that is a fixed point of for each. Now, fix and let with, if there exists an such that , then

which is a contradiction since, hence

for all, and therefore. This means that

is a common fixed point of. But

is the unique common fixed point of, hence

for all, which means that is a common fixed point of for all.

If is a common fixed point of, then for all, which means that is a common fixed point of

. But is the unique common fixed point of, hence. This completes that

has the unique common fixed point for each.

Finally, we will prove that for all. In fact, for any fixed with, since and, hence

by condition 2). Which means that is a common fixed point of for all. But the unique common fixed point of is, hence

for all, this means that is a common fixed point of, and therefore

since is the unique common fixed point of. Let, then is the unique common fixed point of.

The following is a particular form of Theorem 2.1:

Theorem 2.2. Let be a complete 2-metric space, a family of maps from into itself and and. If the following -contractive condition holds

then has an unique common fixed point in.

Next theorem is the main result in [5].

Theorem 2.3. Let be a complete 2-metric space, a family of maps from into itself. If there exist a family non-negative integers and nonnegative real numbers with such that for all and all natural numbers with, the following holds

Then have an unique common fixed point in.

Remark. Obviously, Theorem 2.3 is a very particular form of Theorem 2.1. In fact, Let

, and take

satisfying, then and satisfy all conditions of Theorem 2.1. Hence we sure that our main result generalized and improve many corresponding common fixed point theorems in 2-metric spaces.