Abstract and Applied Analysis

Volume 2012 (2012), Article ID 956950, 35 pages

doi:10.1155/2012/956950

Research Article

Strong Convergence Theorems for a Common Fixed Point of Two Countable Families of Relatively Quasi Nonexpansive Mappings and Applications

Jingling Zhang, Yongfu Su, and Qingqing Cheng

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Received 19 June 2012; Revised 3 August 2012; Accepted 7 August 2012

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Jingling Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The purpose of this paper is to prove strong convergence theorems for common fixed points of two countable families of relatively quasi nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalized -projection operator. In order to get the strong convergence theorems, a new iterative scheme by monotone hybrid method is presented and is used to approximate the common fixed points. Then, two examples of countable families of uniformly closed nonlinear mappings are given. The results of this paper modify and improve the results of Li et al. (2010), the results of Takahashi and Zembayashi (2008), and many others.

1. Introduction

Let be a real Banach space with the dual . We denote by the normalized duality mapping from to defined by

where denotes the generalized duality pairing. The duality mapping has the following properties:

(i)if is smooth, then is single-valued;(ii)if is strictly convex, then is one-to-one;(iii)if is reflexive, then is surjective;(iv)if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of ;(v)if is uniformly convex, then is uniformly continuous on bounded subsets of and is single valued and also one-to-one (see [1–4]).

Let be a smooth Banach space with the dual . The functional is defined by

for all .

Let be a closed convex subset of , and let be a mapping from into itself. We denote by the set of fixed points of . A point in is said to be an asymptotic fixed point of [5] if contains a sequence which converges weakly to such that the strong . The set of asymptotic fixed points of will be denoted by . A mapping from into itself is called nonexpansive if for all and relatively nonexpansive [1, 6–8] if and for all and . The asymptotic behavior of relatively nonexpansive mapping was studied in [1, 6–8].

Three classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping. The first one is introduced in 1953 by Mann [9] which is well known as Mann’s iteration process and is defined as follows:

where the sequence is chosen in [0, 1]. Fourteen years later, Halpern [10] proposed the new innovation iteration process which resembles Manns iteration (1.3), it is defined by

where the element is fixed. Seven years later, Ishikawa [2] enlarged and improved Mann's iteration (1.3) to the new iteration method, it is often cited as Ishikawa's iteration process which is defined recursively by

where and are sequences in the interval [0, 1].

In both Hilbert space [10–12] and uniformly smooth Banach space [13–15] the iteration process (1.4) has been proved to be strongly convergent if the sequence satisfies the following conditions:

(i);(ii);(iii) or .

By the restriction of condition (ii), it is widely believed that Halpern’s iteration process (1.4) has slow convergence though the rate of convergence has not been determined. Halpern [10] proved that conditions (i) and (ii) are necessary in the strong convergence of (1.4) for a nonexpansive mapping on a closed convex subset of a Hilbert space . Moreover, Wittmann [12] showed that (1.4) converges strongly to when satisfies (i), (ii), and (iii), where is the metric projection onto .

Both iterations processes (1.3) and (1.5) have only weak convergence, in general Banach space (see [16] for more details). As a matter of fact, process (1.3) may fail to converge while process (1.5) can still converge for a Lipschitz pseudo contractive mapping in a Hilbert space [17]. For example, Reich [18] proved that if is a uniformly convex Banach space with Frechet differentiable norm and if is chosen such that , then the sequence defined by (1.3) converges weakly to a fixed point of . However, we note that Manns iteration process (1.3) has only weak convergence even in a Hilbert space [16].

Some attempts to modify the Mann’s iteration method so that strong convergence guaranteed has recently been made. Nakajo and Takahashi [19] proposed the following modification of the Mann iteration method for a single nonexpansive mapping in a Hilbert space :

where is a closed convex subset of , denotes the metric projection from onto a closed convex subset of . They proved that if the sequence is bounded above from one then the sequence generated by (1.6) converges strongly to , where denotes the fixed points set of .

The ideas to generalize the process (1.6) from Hilbert space to Banach space have recently been made. By using available properties on uniformly convex and uniformly smooth Banach space, Matsushita and Takahashi [8] presented their ideas as the following method for a single relatively nonexpansive mapping in a Banach space :

They proved the following convergence theorem.

Theorem MT. Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of , and let be a relatively nonexpansive mapping from into itself, and let be a sequence of real numbers such that and . Suppose that is given by (1.7), where is the duality mapping on . If is nonempty, then converges strongly to , where is the generalized projection from onto .

In 2007, Plubtieng and Ungchittrakool [20] proposed the following hybrid algorithms for two relatively nonexpansive mappings in a Banach space and proved the following convergence theorems.

Theorem SK 1. Let be a uniformly convex and uniformly smooth real Banach space, let be a nonempty closed convex subset of , and let be two relatively nonexpansive mappings from into itself with is nonempty. Let a sequence be defined by

with the following restrictions:

(i);(ii).

Then the converges strongly to , where is the generalized projection from onto .

Theorem SK 2. Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of , and let be two relatively nonexpansive mappings from into itself with is nonempty. Let a sequence be defined by

with the following restrictions:

(i);(ii).

Then the converges strongly to , where is the generalized projection from onto .

In 2010, Su et al. [21] proposed the following hybrid algorithms for two countable families of weak relatively nonexpansive mappings in a Banach space and proved the following convergence theorems.

Theorem SXZ 1. Let be a uniformly convex and uniformly smooth real Banach space, let be a nonempty closed convex subset of , and let be two countable families of weak relatively nonexpansive mappings from into itself such that . Define a sequence in by the following algorithm:

with the conditions:

(i);(ii); (iii);(iv) for some .

Then converges strongly to , where is the generalized projection from onto .

Theorem SXZ 2. Let be a uniformly convex and uniformly smooth real Banach space, let be a nonempty closed convex subset of , and let be two countable families of weak relatively nonexpansive mappings from into itself such that . Define a sequence in by the following algorithm:

with the conditions:

(i); (ii);(iii);(iv).

Then converges strongly to .

Recently, Li et al. [22] introduced the following hybrid iterative scheme for approximation of fixed points of a relatively nonexpansive mapping using the properties of generalized -projection operator in a uniformly smooth real Banach space which is also uniformly convex: ,

They proved a strong convergence theorem for finding an element in the fixed points set of . We remark here that the results of Li et al. [22] extended and improved on the results of Matsushita and Takahashi [8].

Motivated by the above-mentioned results and the ongoing research, it is our purpose in this paper to prove a strong convergence theorem for two countable families of relatively quasi nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalized -projection operator. Our results extend the results of Li et al. [22], Takahashi and Zembayashi [23], and many other recent known results in the literature.

2. Preliminaries

Let be a smooth Banach space with the dual . The functional is defined by

for all . Observe that, in a Hilbert space , (2.1) reduces to .

Recall that if is a nonempty, closed, and convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive. This is true only when is a real Hilbert space. In this connection, Alber [24] has recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces. The generalized projection is a map that assigns to an arbitrary point , the minimum point of the functional , that is, , where is the solution to the minimization problem

existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping . In Hilbert space, . It is obvious from the definition of the functional that

for all . See [25] for more details.

This section collects some definitions and lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive.

Remark 2.1. If is a reflexive strictly convex and smooth Banach space, then for , if and only if . It is sufficient to show that if , then . From (2.3), we have . This implies . From the definition of , we have . Since is one-to-one, then we have ; see [12, 15, 26] for more details.

Let be a closed convex subset of , and let be a countable family of mappings from into itself. We denote by the set of common fixed points of , that is, , where denote the set of fixed points of , for all . A point in is said to be an asymptotic fixed point of if contains a sequence which converges weakly to such that . The set of asymptotic fixed point of will be denoted by . A point in is said to be a strong asymptotic fixed point of if contains a sequence which converges strongly to such that . The set of strong asymptotic fixed point of will be denoted by [21].

Definition 2.2. Countable family of mappings is said to be countable family of relatively nonexpansive mappings if the following conditions are satisfied:

(i) is nonempty;(ii), ;(iii).

Definition 2.3. Countable family of mappings is said to be countable family of weak relatively nonexpansive mappings if the following conditions are satisfied:

(i) is nonempty;(ii), ;(iii).

Definition 2.4. Countable family of mappings is said to be countable family of relatively quasi nonexpansive mappings if the following conditions are satisfied:

(i) is nonempty;(ii), .

Definition 2.5. A mapping is said to be relatively nonexpansive mappings if the following conditions are satisfied:

(i); (ii);(iii).

Definition 2.6. A mapping is said to be weak relatively nonexpansive mappings if the following conditions are satisfied:

(i); (ii);(iii).

Definition 2.7. A mapping is said to be relatively quasi nonexpansive mappings if the following conditions are satisfied:

(i) is nonempty;(ii),  .

The Definition 2.5 (Definitions 2.6 and 2.7) is a special form of the Definition 2.2 (Definitions 2.3 and 2.4) as for all . The following conclusions are obvious: (1) relatively nonexpansive mapping must be weak relatively nonexpansive mapping; (2) weak relatively nonexpansive mapping must be relatively quasi nonexpansive mapping.

The hybrid algorithms for fixed point of relatively nonexpansive mappings and applications have been studied by many authors, for example, [1, 6, 7, 17, 27, 28]. In recent years, the definition of relatively quasi nonexpansive mapping has been presented and studied by many authors [7, 17, 26, 28]. Now we give an example which is a countable family of relatively quasi nonexpansive mappings but not a countable family of relatively nonexpansive mappings.

Example 2.8. Let , where

It is well known that is a Hilbert space, so that . Let be a sequence defined by

where

for all .

Define a countable family of mappings as follows:

for all .

Conclusion 2.9. converges weakly to .

Proof. For any , we have

as . That is, converges weakly to

Conclusion 2.10. is not a Cauchy sequence, so that, it does not converge strongly to any element of .

Proof. In fact, we have for any . Then is not a Cauchy sequence.

Conclusion 2.11. has a unique fixed point , that is, , for all .

Proof. The conclusion is obvious.

Conclusion 2.12. is an asymptotic fixed point of .

Proof. Since converges weakly to and

as , so that, is an asymptotic fixed point of .

Conclusion 2.13. is a countable family of relatively quasi nonexpansive mappings.

Proof. Since is a Hilbert space, for any we have

then is a countable family of relatively quasi nonexpansive mappings

Conclusion 2.14. is not a countable family of relatively nonexpansive mappings.

Proof. From Conclusions 2.11 and 2.12, we have , so that, is not a countable family of relatively nonexpansive mapping.

Next, we recall the concept of generalized -projector operator, together with its properties. Let be a functional defined as follows:

where is a positive number, and is proper, convex, and lower semicontinuous. From the definitions of and , it is easy to see the following properties:

(i) is convex and continuous with respect to when is fixed;(ii) is convex and lower semi-continuous with respect to when is fixed.

Definition 2.15 (see [29]). Let be a real Banach space with its dual . Let be a nonempty, closed and convex subset of . We say that is a generalized -projection operator if

For the generalized -projection operator, Wu and Huang [42] proved the following theorem basic properties

Lemma 2.16 (see [29]). Let be a real reflexive Banach space with its dual . Let be a nonempty, closed, and convex subset of . Then the following statements hold:

(i) is a nonempty closed convex subset of for all ;(ii)if is smooth, then for all , if and only if

(iii)if is strictly convex and is positive homogeneous such that , then is a single valued mapping.

Fan et al. [30] showed that the condition is a positive homogeneous which appeared in Lemma 2.13 can be removed.

Lemma 2.17 (see [30]). Let be a real reflexive Banach space with its dual and a nonempty, closed, and convex subset of . Then if is strictly convex, then is a single-valued mapping.

Recall that is a single-valued mapping when is a smooth Banach space. There exists a unique element . such that for each . This substitution in (2.15) gives

Now, we consider the second generalized -projection operator in a Banach space.

Definition 2.18. Let be a real Banach space and a nonempty, closed, and convex subset of . We say that is a generalized -projection operator if

Obviously, the definition of relatively quasi nonexpansive mapping is equivalent to

(R1);(R2).

Lemma 2.19 (see [31]). Let be a Banach space and let be a lower semi-continuous convex functional. Then there exists and such that

We know that the following lemmas hold for operator .

Lemma 2.20 (see [22]). Let be a nonempty, closed, and convex subset of a smooth and reflexive Banach space . Then the following statements hold:

(i) is a nonempty closed and convex subset of for all ;(ii)for all , if and only if

(iii)if is strictly convex, then is a single valued mapping.

Lemma 2.21 (see [22]). Let be a nonempty, closed, and convex subset of a smooth and reflexive Banach space . let and , then

The fixed points set of a relatively quasi nonexpansive mapping is closed convex as given in the following lemma.

Lemma 2.22 (see [32, 33]). Let be a nonempty, closed, and convex subset of a smooth and reflexive Banach space . let be a closed relatively quasi nonexpansive mapping of into itself. Then is closed and convex.

Also, this following lemma will be used in the sequel.

Lemma 2.23 (see [25]). Let be a uniformly convex and smooth real Banach space and let , be two sequences of . If and either or is bounded, then .

Lemma 2.24 (see [24, 25, 27]). Let be a nonempty closed convex subset of a smooth real Banach space and . Then, if and only if

Lemma 2.25 (see [28]). Let be a uniformly convex Banach space and let be a closed ball of . Then there exists a continuous strictly increasing convex function with such that

for all and with .

It is easy to prove the following result.

Lemma 2.26. Let be a strictly convex and smooth real Banach space, let be a closed convex subset of , and let be a relatively quasi nonexpansive mapping from into itself. Then is closed and convex.

Lemma 2.27. Let be a uniformly convex Banach space with . Then, for all , and ,

where is the generalized duality mapping from into and is the uniformly convexity constant of .

Observe that an infinite family of operators in a Banach space is said to be uniformly closed, if , then (i.e., ). Obviously, a countable family of uniformly closed of relatively quasi nonexpansive mappings is a countable family of weak relatively nonexpansive mappings.

3. Main Results

Theorem 3.1. Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of , and let be two countable families of uniformly closed of relatively quasi nonexpansive mappings of into itself such that . Let be a convex and lower semicontinuous mapping with . For any given gauss , define a sequence in by the following algorithm:

with the conditions

(i);(ii);(iii);(iv).

Then converges strongly to , where .

Proof. Step 1. We show that and are closed and convex for each .

From the definitions of and , it is obvious that is closed and convex and is closed for each . Moreover, since is equivalent to

and is equivalent to

it follows that is convex for each . So, is a closed convex subset of for all .

Step 2. We show that for all . Observe that

Hence from the definition of and the convexity of we have, for all , that

By the similar reason we have, for all , that

That is, for all .

Next, we show that for all , we prove this by induction. For , we have . Assume that . Since , by Definition 2.15 we have

It is equivalent to

and is equivalent to

As , by the induction assumptions, the last inequality holds, in particular, for all . This together with the definition of implies that .

Step 3. We show that as , and .

We now show that exists. Since is a convex and lower semi-continuous, applying Lemma 2.19, we see that there exists and such that

It follows that

Since , for each , it follows from (3.12) that

This implies that is bounded and so is .

Since , by Lemma 2.21 we have

It is obvious that

and so is nondecreasing. It follows that the limit of exists.

By the fact that and , by Lemma 2.21 we obtain

Taking the limit as in (3.16), we obtain

which holds uniformly for all . By using Lemma 2.21, we get that

which holds uniformly for all . Then is a Cauchy sequence, therefore there exists a point such that . In particular, we have

Since , from the definition of , we know that

is equivalent to

So, .

By using Lemma 2.23, we get that

as .

Hence as . Since is uniformly norm-to-norm continuous on bounded sets, we have

Since , then

Since is also uniformly norm-to-norm continuous on bounded sets, we have

so that as .

Since is convergent, then is bounded, so are , , and . From the definition of and

we have, for all that

Therefore by using Lemma 2.25 (inequality (2.21)), for all , we have

and hence

as . By using the same way, we can prove that

as . From the properties of the mapping , we have

as , and

as . Since is also uniformly norm-to-norm continuous on any bounded sets, we have

as , and

as . Since and , are uniformly closed, , and .

Step 4. we show that .

Since is a closed and convex set, from Lemma 2.19, we know that is single valued and denote . Since and , we have

We know that is convex and lower semicontinuous with respect to when is fixed. This implies that

From the definition of and , we see that . This completes the proof.

Based on Theorem 3.1, we have the following.

Corollary 3.2. Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of , and let be two countable families of weak relatively nonexpansive mappings of into itself such that . For any given gauss , define a sequence in by the following algorithm:

with the conditions:

(i);(ii);(iii);(iv).

Then converges strongly to , where .

Proof. Putting , we can conclude from Theorem 3.1 the desired conclusion immediately.

Theorem 3.3. Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of , let be two countable families of uniformly closed of relatively quasi nonexpansive mappings of into itself such that . Let be a convex and lower semicontinuous mapping with . For any given gauss , define a sequence in by the following algorithm:

with the conditions:

(i);(ii); (iii); (iv).

Then converges strongly to , where .

Proof. Step 1. We show that and are closed and convex for each .

From the definitions of and , it is obvious that is closed and convex and is closed for each . Moreover, since is equivalent to

it follows that is convex for each . So, is a closed convex subset of for all .

Step 2. We show that for all . Observe that

Hence from the definition of and the convexity of we have, for all , that

By the similar reason we have, for all , that

That is, for all .

Next, we show that for all , we prove this by induction. For , we have . Assume that . Since ,  by Definition 2.15 we have

It is equivalent to

and is equivalent to

As , by the induction assumptions, the last inequality holds, in particular, for all . This together with the definition of implies that .

Step 3. We show that as , and .

We now show that exists. Since is a convex and lower semi-continuous, applying Lemma 2.19, we see that there exists and such that

It follows that

Since , for each , it follows from (3.47) that

This implies that is bounded and so is .

Since , by Lemma 2.21, we have

It is obvious that

and so is nondecreasing. It follows that the limit of exists.

By the fact that and , by Lemma 2.21 we obtain

Taking the limit as in (3.51), we obtain

which holds uniformly for all . By using Lemma 2.23, we get that

which holds uniformly for all . Then is a Cauchy sequence, therefore there exists a point such that . In particular, we have

Since , from the definition of , we know that

and is equivalent to

and is equivalent to

So, .

By using Lemma 2.23, we get that

as ,

and hence as . Since is uniformly norm-to-norm continuous on bounded sets, we have

Since , then

Since is also uniformly norm-to-norm continuous on bounded sets, we have

so that as .

Since is convergent, then is bounded, so are , , and . From the definition of and

we have, for all that

Therefore by using Lemma 2.25 (inequality (2.21)), for all , we have

and hence

From and , ,we have

as . From the properties of the mapping , we have

as .

Since

then we have

Therefore

which leads to

Since , and . Then from above inequality we obtain

On the other hand, by using the property of norm , we have

which leads to the following inequality

Therefore, by using (3.67) and (3.72) we have

This together with condition (ii) of Theorem 3.3 implies that

Since is uniformly norm-to-norm continuous on bounded sets, then we have

as .  By using the same way, we can prove that

as . Since and , are uniformly closed, , and , so that .

Step 4. we show that .

Since is a closed and convex set, from Lemma 2.19, we know that is single valued and denote . Since and we have

We know that is convex and lower semicontinuous with respect to when is fixed. This implies that

From the definition of and , we see that . This completes the proof.

Based on Theorem 3.3, we have the following.

Corollary 3.4. Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of , and let be two countable families of weak relatively nonexpansive mappings of into itself such that . For any given gauss , define a sequence in by the following algorithm:

with the conditions:

(i); (ii); (iii); (iv).

Then converges strongly to , where .

Proof. Putting , we can conclude from Theorem 3.3 the desired conclusion immediately.

4. Applications

Let be a real Banach space and let be the dual space of . Let be a closed convex subset of . Let be a bifunction from to . The equilibrium problem is to find such that

The set of solutions of (4.1) is denoted by . Given a mapping let for all . Then, if and only if for all , that is, is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (4.1). Some methods have been proposed to solve the equilibrium problem in Hilbert spaces.

For solving the equilibrium problem, let us assume that a bifunction satisfies the following conditions:

(A1);(A2) is monotone, that is, ;(A3) for all , ;(A4) for all , is convex and lower semi-continuous.

Lemma 4.1 (see [34]). Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to satisfying –, and let and . Then, there exists such that

Lemma 4.2 (see [34]). Let be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space , and let be a bifunction from to satisfying –. For , define a mapping as follows:

for all . Then, the following hold

(1) is single valued;(2) is a firmly nonexpansive-type mapping, that is, for all ,

(3);(4) is closed and convex;(5) is also a relatively nonexpansive mapping.

Lemma 4.3 (see [34]). Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to satisfying –, and let and , , then the following holds

Lemma 4.4. Let be a uniformly convex with and uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying –. Let be a positive real sequence such that . Then the sequence of mappings is uniformly closed.

Proof. (1) Let be a convergent sequence in . Let for all , then

Putting in (4.6) and in (4.7), we have

So, from (A2) we have

and hence

Thus, we have

which implies that

By using Lemma 2.25, we obtain

Therefore, we get

On the other hand, for any , from , we have

so that is bounded. Since , this together with (4.14) implies that is a Cauchy sequence. Hence is convergent.

(2) By using the Lemma 4.2, we know that

(3) From (1) we know that, exists for all . So, we can define a mapping from into itself by

It is obvious that, is nonexpansive. It is easy to see that

On the other hand, let , we have

By (A2) we know

Since and from (A4), we have , for all . Then, for and ,

Therefore, we have

Letting and using (A3), we get

and hence . From the above two respects, we know that .

Next we show is uniformly closed. Assume and , from the above results we know that . On the other hand, from , we also get , so that . That is, the sequence of mappings is uniformly closed. This completes the proof.

Let be a multi-valued operator from to with domain and range . An operator is said to be monotone if

for each and . A monotone operator A is said to be maximal if it is graph is not properly contained in the graph of any other monotone operator. We know that if is a maximal monotone operator, then is closed and convex. The following result is also well known.

Theorem 4.5. Let be a reflexive, strictly convex, and smooth Banach space and let be a monotone operator from to . Then is maximal if and only if , for all .

Let be a reflexive, strictly convex, and smooth Banach space, and let be a maximal monotone operator from to . Using Theorem 4.5 and strict convexity of , one obtains that for every and , there exists a unique such that

Then one can defines a single-valued mapping by and such a is called the resolvent of , one knows that for all .

Theorem 4.6. Let be a uniformly convex and a uniformly smooth Banach space, and let be a maximal monotone operator from to , let be a resolvent of for . Then for any sequence such that , is a uniformly closed sequence of relatively quasi nonexpansive mappings.

Proof. Firstly, we show that is uniformly closed. Let be a sequence such that and . Since is uniformly norm-to-norm continuous on bounded sets, we obtain

It follows from

and the monotonicity of that

for all and . Letting , we have for all and . Therefore from the maximality of , we obtain for all , that is, .

Next we show is a relatively quasi nonexpansive mapping for all . For any and , from the monotonicity of , we have

This implies that is a relatively quasi nonexpansive mapping for all . This completes the proof.

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