1. Introductions and Preliminaries

Most of fixed point theorems for mappings in metric spaces satisfying different contraction conditions may be extended to the abstract spaces like Hilbert spaces, Banach spaces and locally convex spaces etc., with some modifications Banach fixed point theorem and its applications are well known. Many authors have extended this theorem, introducing more general contractive conditions, which imply the existence of a fixed point. Almost all of conditions imply the asymptotic regularity of the mappings under considerations. So the investigation of the asymptotically regular maps play an important role in fixed point theory.

Sharma and Yuel ^[14] and Guay and Singh ^[7] were among the first who used the concept of asymptotic regularity to prove fixed point theorems for wider class of mappings than a class of mappings introduced and studied by Ćirić ^[6].

The purpose of this paper is to prove some common and coincidences fixed point theorems in Hilbert spaces and we study the well- posedness of their fixed point problem.

Definition 1.1. ^[1]. A self mapping on a closed subset of a Hilbert space is said to be asymptotically regular at a point x in , if

where denotes the nth iterate of at x.

Definition 1.2. ^[1]. Let C be a closed subset of a Hilbert space . A sequence in C is said to be asymptotically T - regular if

Definition 1.3. ^[5]. A pair of mappings (f, T) on a Hilbert space is said to be weakly compatible if f and T commute at their coincidence point (i.e. f T x = T f x whenever f x = T x). A point is called point of coincidence of two self –mappings f and T on if there exists a point such that

The following lemma was given in ^[5] in a metric space setting.

Lemma 1.1. Let X be a non-empty set and the mappings have a unique point of coincidence v in X. If the pair (f, T) is weakly compatible then T and f have a unique common fixed point.

Let H be a Hilbert space,T and f be self –mappings on H with and Choose a point such that This can be done since .Continuing this process, having chosen we choose in such that

The sequence is called a T –sequence with initial point .

Definition1.4. Let T and f be self – mappings on a Hilbert space , with and . A mapping T is said to be asymptotically f – regular at a point if :

Where is a T – sequence with initial point

We know that a Banach space is a Hilbert space iff its norm satisfies the parallelogram law i.e., every ,

(1.1)

which implies,

(1.2)

2. Common Fixed Point Theorems

S. T. Patel and at.al. ^[9] gave the following theorem:

Theorem 2.1. Let C be a closed subset of a Hilbert space H and T be a mapping on H into it self satisfying

(2.1)

for all x, y in H, where are non –negative reals with Then T has a unique fixed point in H.

The purpose of this section is to extend Theorem 2.1 to the case of two mappings in a Hilbert space as the following.

Theorem 2.2. Let C be a closed subset of a Hilbert space H and S,T be mappings on C into itself satisfying:

(2.2)

for all are non – negative reals with . Then S and T have a unique common fixed point in C.

Proof. Let we define a sequence as follows

From (2.2), we have

Thus we have

Putting

Then, we have

Proceeding in this way, we get

For any positive integer p, one gets :

Thus

Hence is a Cauchy sequence in C. Since C is closed subset of H, then there exists an element such that

Now further, we have

As

We have

Then

This implies that v =T v, since .

Similarly we get v = S v. Then v is a common fixed point of S and T.

For the uniqueness, let be another fixed point of S and T, where , then

Since , so u = v i.e., the common fixed point is unique.

Next, we extend Theorem 2.2 to the case of pair , where p and q are some positive integers and to the case of a sequence of mappings satisfying the same contractive condition (2.2).

Theorem 2.3. Let C be a closed subset of a Hilbert space H and S,T be mappings on C into itself satisfying

(2.3)

for all are nonnegative reals with Then S and T have a unique common fixed point.

Proof. Since satisfies all the conditions of theorem 2.2. Hence have a unique common fixed point, we assume that they have a common fixed point v

So S v is a fixed point of Similarly we can show that T v is a fixed point of i.e., . Now we have

Then we have:

So, we have v = Tv.

On the same way we can prove that S v =v. So v is a common fixed point of S and T.

To prove the uniqueness, let be another common fixed point of S and T. Then clearly w is also a common fixed point of . So from Theorem 2.3 have a common fixed point. Therefore w= v. Hence S and T have a unique common fixed point.

Hence we have proved that if is unique common fixed point of for all p, q > 0, then is unique common fixed point of S and T.

Theorem 2.4. Let C be a closed subset of a Hilbert space H and let be a sequence of mapping on C converging point wise to F satisfying

for all x, y in C, where are nonnegative reals in [0,1]^{[, 1]} with If has a fixed point and F has a fixed point v. Then the sequence converges to v.

Proof. Since it is given .

Now

Taking , we have

So .

3. Coincidences Fixed Point Theorems

Our main results in this section are the following theorems

Theorem 3.1. Let H be a Hilbert space and let be such that :

(iv) T is asymptotically f - regular of some point in H.

Then T and f have a point of coincidence

Proof. Let be an asymptotically T – regular sequence in H. Then by Parallelogram law we have

(3.2)

Using (3.1) we have

(3.3)

Taking limit as , and using asymptotically T – regular of gives

Hence we have

It follows that is a Cauchy sequence in H. If f(H) is a complete subspace of H, there exists a point such that:

(this also holds if T(H) is complete with ).

We claim that u is a coincidence point of f and T. If not . From (ii), we obtain

As we get

Hence

a contradiction and so p = f u = T u is a point of coincidence of f and T.

Theorem 3.2. Let H be a Hilbert space and let be such that:

Then f, T have at least a unique point of coincidence.

Proof. Assume there exist in H such that

P = f u = T u and

From (3.4) we obtain

we deduce that

Remark. 3.1. If we put in Theorem 3.1and in Theorem 3.2, we obtain Theorems 3. 1and Theorem 3.2 in ^[1].

Let be functions such that and is continuous at 0 (i=1,2). Ciric ^[5] studied necessary conditions to obtain a fixed point result of asymptotically regular mappings on complete metric spaces. M. Abbas and H.Aydi ^[1] extended the results of Ćirić ^[5] to the case of two mappings satisfying a generalized contractive conditions in a metric space and they proved the following theorem.

Theorem 3.3. ^[5]. Let (X, d) be a metric space. Let be such that Assume that the following condition holds:

(3.5)

for all such that for arbitrary fixed k > 0, and and If is a complete subspace of X and T is asymptotically f- regular at some point in X, then T and f have a unique point of coincidence.

In the following theorem we prove a similar result using a contractive condition (3.5) in a Hilbert space.

Theorem 3.4. Let H be a Hilbert space. Let be such that:

for all such that for arbitrary fixed we have and If is a complete subspace of H and if T is asymptotically f –regular at some point Then f, T have a point of coincidence.

Proof. Let be an arbitrary point in H and let be a T-sequence with initial point . Since T is asymptotically f-regular mappings at therefore Now for m > n, we have

Thus we obtain that

(3.7)

Since T is asymptotically f –regular and are continuous at zero, then the right hand of the inequality (3.7) tends to zero as Thus

It follows that is a Cauchy sequence in H. If f(H) is a complete subsequence of H there exists u, p in H such that (this holds also if T(H) is complete with ).

We claim that u is a coincidence point of f and T. If not From (3.6) we obtain:

which yields that

Taking limit as , we get

a contradiction and so p = fu = T u is a point of a coincidence of f and T.

Lemma 3.5. Let H be a Hilbert space. Let be such that:

for all such that for arbitrary fixed .Then f, T have at most a unique coincidence point

Proof. Assume that there exists, in H such that such that

Hence we obtain

This shows that

From Theorem 3.4 and Lemma 3.5 we obtain the following theorem

Theorem 3.6. Let H be a Hilbert space and let f, T be mappings on H into H such that Assume that T and f satisfy condition 3.6 for all . If f(H) or T(H) is a complete subspace of H such that (T, f) is weakly compatible, then T and f have a unique common fixed point provided that T is asymptotically f- regular at some point in H.

As a consequence of Theorem 3.4, Lemma 3.5 and Theorem 3.6 we obtain the following corollary.

Corollary 3.7. Let H be a Hilbert space. Let be such that the following condition holds:

(3.8)

for all , where such that for an arbitrary fixed we have If T is asymptotically regular at some point in H. Then T has a unique fixed point.

Taking in the inequality 3.6, we have tha following corollary.

Colorary 3.8. Let H be a Hilbert space. Let be such that .Assume that the following condition holds:

for all , where such that for arbitrary fixed , we have If f(H) or T(H) is a complete subspace of H and if T is asymptotically regular at some point in H. Then T and f have a point. of coincidence.

We give an example to support our results

Example 3.9. Let X= and let Le t be defined as

Let =1 and the sequence be given by . Note that is a T-sequence with initial point . Since the mapping T is asymptotically f – regular at the point . Also T(X), T(X) is a complete and the inequality (3.6) holds for all with

Thus satisfy all conditions of Theorem 3.4. Moreover u=0 is the common fixed point of f and T.

Let denote the set of all common fixed points of f and T. Now we have the following result on the continuity on the set of common fixed points

Theorem 3.10. Let H be a Hilbert space. Assume that satisfy condition (3.6) for all . If Then T is continuous at whenever f is continuous at p.

Proof. Fix .Let be any sequence in H converging to p. Then by taking and z:=p in (3.6) we get

which in view of T p = f p, we obtain

Now by letting we get

whenever f is continuous at p. The last inequality is true only if

We get that .

4. Well – Posedness

The notion of well - posedness of a fixed point problem has generated much interest to several mathematicians, for example ^{[2, 3, 8, 11, 12, 13]}. Here, we study well – posedness of a common fixed point problem.

Definition 4.1. Let H be a Hilbert space and be a mapping. The fixed point problem of f is said to be well posed if

(i) f has a unique fixed point in H

(ii) for any sequence , we have

Definition 4.2. A common fixed point problem of self-maps f and T on H, CFP(f,T,H) is called well- posed if CF(f, T) ( the set of all common fixed points of f and T) is singletion and for any sequence in H with implies .

Theorem 4.1. Suppose that T and f be self –maps on H as in Theorem 3.4 and Lemma 3.1. Then the common fixed problem of f and T is well posed.

Proof. From Theorem 3.4 and Lemma 3.1, the mappings f and T have a unique common fixed point, say u . Let be a sequence in H and . With loss of generality, we may suppose that for every nonnegative integer n. Then having in mind fu =T u and from triangle inequality (3.6), we have,

Letting , we get that. We deduce, . This completes the proof of Theorem.

Acknowledgement

The author would like to express his thanks to the referees and editors for helpful comments and suggestions.

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