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We establish some results on coincidence and common fixed point for a two pair of multi-valued and single-valued maps in ultra metric spaces.

Roovij in [

J. Kubiaczyk and A. N. Mostafa [

In this article, we are going to establish some results on coincidence and common fixed point for two pair of multi-valued and single-valued maps in ultra metric spaces.

First we introducing a notation.

Let denote the class of all non empty compact subsets of. For, the Hausdorff metric is defined as

where.

The following definitions will be used later.

Definition 2.1 ([

Then is called an ultra metric on and is called an ultra metric space.

Example. Let, , then

is a ultra metric space.

Definition 2.2 ([

Definition 2.3 An elementis said to be a coincidence point of and if. We denote

the set of coincidence points of and.

Definition 2.4 ([

Definition 2.5 ([

The following results are the main result of this paper.

Theorem 3.1 Let be an ultra metric space. Let be a pair of multi-valued maps and a pair of single-valued maps satisfying

(a) is spherically complete;

for all, with;

(c) ;

(d).

Then there exist point and in, such that

.

Proof. Let

denote the closed sphere with centeredand radius

.

Let be the collection of all the spheres for all.

Then the relation

if

is a partial order on.

Consider a totally ordered sub family of.

Since is spherically complete, we have

Let where and.

Then. Hence

If then. Assume that.

Let, then

Sinceis nonempty compact set, then such that

;

is a nonempty compact set, then such that.

from (a) (b) and Equation (1)

Now

So, we have just proved that for every. Thus is an upper bound in for the family and hence by Zorn’s Lemma, there is a maximal element in, say. There exists such that.

Suppose

.

Since are nonempty compact sets, then such that

From (b), (c) and Equation (2), we have

From (b), (c) and Equations (2)-(5)

From Equation (4) and Equation (6) we have

From Equation (5) and Equation (7) we have

If

Then from Equation (8),. Hence. It is a contradiction to the maximality of in, since

If

Then from Equation (9),. Hence.It is a contradiction to the maximality of in, since.

So

In addition,.

Using (b), (c) and Equation (10), we obtain

Hence.

Then the proof is completed.

Theorem 3.2 Let be an ultra metric space. Let be a pair of multi-valued maps and be a single-valued maps satisfying

(a) is spherically complete;

for all,with;

(c);

(d).

Then and have a coincidence point in.

Moreover, if and, and are coincidentally commuting at and, then and have a common fixed point in.

Proof. If in Theorem 2.1, we obtain that there exist points and in such that

.

As, and ipipare coincidentally commuting at and.

Write, then.

Then we have

and

.

Now, since also, and are coincidentally commuting at and, so we obtain

.

Thus, we have proved that, that is, is a common fixed point of and.

Corollary 3.3 Let be a spherically complete ultra metric space. Let be a pair of multi-valued maps satisfying

(a) for all,with;

(b).

Then, there exists a pointinsuch that and.

Remark 1 If in Corollary 3.3, then we obtain the Theorem of Ljiljana Gajic [

Remark 2 If in Theorem 3.1, , we obtain Theorem 9 of K. P. R. Rao at [

Remark 3 Ifandin Theorem 3.1 are single-valued maps, then: 1) we obtain the results of K. P. R. Rao [

In this paper, we get coincidence point theorems and common fixed point theorems for two pair of multi-valued and single-valued maps satisfying different contractive conditions on spherically complete ultra metric space, which is generalized results of [3-7].

Foundation item: Science and Technology Foundation of Educational Committee of Tianjin (11026177).