Research Article | Open Access

# Convergence of Path and Approximation of Common Element of Null Spaces of Countably Infinite Family of -Accretive Mappings in Uniformly Convex Banach Spaces

**Academic Editor:**Gelu Popescu

#### Abstract

We prove path convergence theorems and introduce a new iterative sequence for a countably infinite family of -accretive mappings and prove strong convergence of the sequence to a common zero of these operators in uniformly convex real Banach space. Consequently, we obtain strong convergence theorems for a countably infinite family of pseudocontractive mappings. Our theorems extend and improve some important results which are announced recently by various authors.

#### 1. Introduction

Let be a real Banach space with dual . We denote by the normalized duality mapping from to defined by where denotes the generalized duality pairing between members of and . It is well known that if is strictly convex, then is single-valued (see, e.g., [1, 2]). In the sequel, we will denote the single-valued normalized duality mapping by .

A mapping is called *accretive* if for all there exists such that
By the result of Kato [3], (1.2) is equivalent to
If is a Hilbert space, accretive operators are also called *monotone*. An operator is called *m-accretive* if it is accretive and , range of , is for all ; and is said to satisfy the range condition if , for all , where denotes the closure of the domain of . It is easy to see that every -accretive operator satisfies the range condition. An operator is said to be *maximal accretive* if it is accretive and the inclusion implies , where denotes the graph of and is an accretive operator.

A mapping is said to be *nonexpansive* if
It is not difficult to deduce from (1.3) that a mapping is accretive if and only if its resolvent , for all , is nonexpansive and single valued on the range of . Thus, in particular, is nonexpansive and single valued on the range of . Furthermore, . For more details see, for example, [4, 5].

Closely related to the class of accretive operators is the class of pseudocontractive maps. An operator with domain in and range in is called *pseudocontractive* if is accretive. The importance of these operators in application is well known (see, e.g., [6–9] and the references contained therein).

It is well known that the class of pseudocontractive mappings properly contains the class of nonexpansive mappings (see, e.g., [4]). Construction of fixed points of nonexpansive mappings is an important subject in nonlinear operator theory and its applications, in particular, in image recovery and signal processing (see, e.g., [10]).

Iterative approximation of fixed points and zeros of nonlinear operators have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequality problems (see, e.g., [11–15]). The iterative scheme (where is the resolvent of an -accretive operator ) for example, has been extensively studied over the past forty years or so for construction of zeros of accretive operators (see, e.g., [16–20]).

Kim and Xu [21] introduced a modification of Mann iterative scheme in a reflexive Banach space having weakly continuous duality mapping for finding a zero of an -accretive operator as follows: They proved that the sequence generated by (1.6) converges to a zero of -accretive operator under the following conditions:

(i)(ii) (equivalently, ),(iii); for some and for all , (iv)(v) for some and for all and .In 2007, Qin and Su [22] also considered the following iterative scheme in either a uniformly smooth Banach space or a reflexive Banach space having a weakly sequentially continuous duality mapping: where and are sequences in . They proved that the sequence generated by (1.7) converges strongly to a zero of -accretive operator provided that and satisfy conditions (i), (ii), and (iii), and satisfies condition (v).

Chen and Zhu [23] considered the following viscosity iterative scheme for resolvent of -accretive mapping : where is a contraction mapping defined on . Under the assumption that satisfies condition (v), Chen and Zhu [23] proved in a reflexive Banach space having weakly sequentially continuous duality mapping that the sequence generated by (1.8) converges strongly to a zero of , which solves a certain variational inequality.

Recently, Jung [24] introduced the following viscosity iterative method: Under certain appropriate conditions on the parameters , , and the sequence ; Jung [24] established strong convergence of the sequence generated by (1.9) to a zero of , which is a unique solution of a certain variational inequality problem, in either a reflexive Banach space having a weakly sequentially continuous duality mapping or a reflexive Banach space having a uniformly Gâteaux differentiable norm such that every weakly compact convex subset of has the fixed point property for nonexpansive mappings.

In [5], Zegeye and Shahzad proved the following theorem.

Theorem 1 ZS. *Let be a strictly convex reflexive real Banach space which has uniformly Gâteaux differentiable norm and let be a nonempty closed convex subset of . Assume that every nonempty closed convex and bounded subset of has the fixed point property for nonexpansive mappings. Let , be a finite family of -accretive mappings with . For given , let be generated by the algorithm
**
where , with , , , and is a sequence in satisfying the following conditions:*(i)*; *(ii)*; *(iii)* or . **Then, converges strongly to a common solution of the equation for .*

Motivated by the results of the authors mentioned above, it is our purpose in this paper to prove new path convergnce theorems and introduce a new iteration process for a countably infinite family of -accretive mappings and prove strong convergence of the sequence to a common zero of these operators in uniformly convex real Banach spaces. As a result, we obtain strong convergence theorems for a countably infinite family of pseudocontractive mappings. Our theorems extend and improve some important results which are announced recently by various authors.

#### 2. Preliminaries

Let be a real normed linear space. Let . is said to have a *Gâteaux differentiable* norm (and is called *smooth*) if the limit
exists for each ; is said to have a *uniformly Gâteaux differentiable* norm if for each the limit is attained uniformly for . Furthermore, is said to be *uniformly smooth* if the limit exists uniformly for .

Let be a real normed linear space. The modulus of convexity of is the function defined by
The space is said to be uniformly convex if and only if for all . is said to be *strictly convex* if for all such that and for all we have . It is well known that every uniformly convex Banach space is strictly convex.

A mapping with domain and range in is said to be *demiclosed* at if whenever is a sequence in such that and then .

A mapping is said to be *demicompact* at if for any bounded sequence in such that as , there exists a subsequence say of and such that converges strongly to and .

We need the following lemmas in the sequel.

Lemma 2.1. *Let be a real normed space, then
**
for all and for all .*

Lemma 2.2 (Lemma 3 of Bruck [25]). *Let be a nonempty closed convex subset of a strictly convex real Banach space . Let be a sequence of nonexpansive mappings from to such that . Let be a sequence of positive numbers such that , then a mapping on defined by for all is well defined, nonexpansive, and .*

Lemma 2.3 (Xu [26]). *Let be a sequence of nonnegative real numbers satisfying the following relation:
**
where and is a sequence in satisfying (i) ; (ii) . Then, as .*

Lemma 2.4 (Suzuki [27]). *Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose that for all integers and . Then, .*

Lemma 2.5 (Cioranescu [28]). *Let be a continuous accretive operator defined on a real Banach space with . Then, is -accretive.*

Lemma 2.6 (C. E. Chidume and C. O. Chidume [29]). *Let be a nonempty closed convex subset of a real Banach space . For arbitrary , let . Then, there exists a continuous strictly increasing function , such that for every and for , the following inequality holds:
*

#### 3. Path Convergence Theorems

We begin with the following lemma.

Lemma 3.1. *Let be a nonempty closed convex subset of a reflexive strictly convex Banach space . Let be a nonexpansive mapping. Let , a bounded sequence in , be an approximate fixed point sequence of , that is . Let , for all and let , where is any bounded closed convex nonempty subset of such that for all . Then has a fixed point in , provided that .*

*Proof. *Since is a reflexive Banach space, then is a bounded closed convex nonempty subset of . Since , we have that for all ,
Hence, , for all , that is, is invariant under . Let . Then since every closed convex nonempty subset of a reflexive strictly convex Banach space is a Chebyshev set (see, e.g., [30, Corollary 5.1.19]), there exists a unique such that
but and . Thus,
So, . Hence, . This completes the proof.

Proposition 3.2. *Let be a nonempty closed convex subset of a real Banach space . Let , , be a countably infinite family of -accretive mappings and define , Let , , be sequences in such that . Fix , for some . For arbitrary fixed , define a map by
**
Then, is a strict contraction on .*

*Proof. *Let , then

Thus, for each , there is a unique satisfying

Lemma 3.3. *Let be a nonempty closed convex subset of a real Banach space . For each , let be a countably infinite family of -accretive mappings. For , let be a sequence satisfying (3.6) and assume . Then, is bounded.*

*Proof. *Let . Then, using (3.6), we obtain
which implies that . Thus, is bounded.

Lemma 3.4. *Let be a nonempty closed convex subset of a uniformly convex real Banach space . For each , let be a countably infinite family of -accretive mappings such that . Let be a sequence in such that , for all , . Let be a sequence satisfying (3.6). Then, , for all . Furthermore, if is a sequence in such that ; and define , where , then .*

*Proof. *We start by showing that , for all . For this, let , where is the identity operator on . Since is bounded, then for each and , we have the following using (2.5):
Hence,
and so,
Using (3.6), we have
which implies
Using this and (3.10), we get
Since is bounded, we have that
for some constant . This yields
Thus, since is continuous, strictly increasing, and , for all , we have
So, , for all , but
Thus,
Next, we show that . Observe that
So,
for some . Hence,
This completes the proof.

Theorem 3.5. *Let be a nonempty closed convex subset of a uniformly convex real Banach space with uniformly Gâteaux differentiable norm. Let , , be a countably infinite family of -accretive mappings such that . Let be a sequence satisfying (3.6). Let be a sequence in such that and . Let , where . Then, converges strongly to an element in .*

*Proof. *Observe that by Lemma 2.2, is well defined, nonexpansive, and . Furthermore, it is easy to see that is nonexpansive and that . Now, since is bounded and , we have by Lemma 3.1 that there exists a unique in the set such that , where is a bounded closed convex nonempty subset of such that for all . Thus, . Let , then by convexity of , we have that . Thus, . Moreover, we have, by Lemma 2.1 that
This implies that . Furthermore, since has uniformly Gâteaux differentiable norm, we obtain that
Thus, given , there exists such that for all and for all ,
Taking Banach limit on both sides of this inequality, we obtain
and since is arbitrary, we have that
Now, using (3.6), we have that
So,
Again, taking Banach limit, we obtain
so that . Hence, there exists a subsequence of such that . We now show that actually converges to . Suppose there is another subsequence of such that . Then, since and is continuous for all , we have that . *Claim 1 (). *Suppose for contradiction that , then , but using (3.6), we have that
Thus,
Using the fact that is bounded and that has a uniformly Gâteaux differentiable norm, we obtain from (3.31) that
Similarly, we also obtain that or rather
Adding (3.32) and (3.33), we have that , a contradiction. Thus, . Hence, converges strongly to . This completes the proof.

Theorem 3.6. *Let be a nonempty closed convex subset of a uniformly convex real Banach space with uniformly Gâteaux differentiable norm. Let , , be a countably infinite family of -accretive mappings such that . Let be a sequence satisfying (3.6). If at least one of the mappings is demicompact, then converges strongly to an element of .*

*Proof. *For fixed , let be demicompact. Since , there exists a subsequence say of that converges strongly to some point . By continuity of and the fact that , , we have that . Assuming that there is another subsequence of that converges strongly to a point say, , then following the argument of the last part of proof of Theorem 3.5, we obtain that converges strongly to . This completes the proof.

Theorem 3.7. *Let be a nonempty closed convex subset of a uniformly convex real Banach space with uniformly Gâteaux differentiable norm and also admits weakly sequential continuous duality mapping. Let , , be a countably infinite family of -accretive mappings such that . Let be a sequence satisfying (3.6). Then, converges strongly to an element of .*

*Proof. *Since is bounded, there exists a subsequence say of that converges weakly to some point . Using the demiclosedness property of at 0 for each (since is nonexpansive for each , see, e.g., [31]) and the fact that , we get that . We also observe from (3.6) that
This implies that . Using the fact that is weakly sequential continuous, then we have from the last inequality that converges strongly to . Then following the argument of the last part of proof of Theorem 3.5, we obtain that converges strongly to . This completes the proof.

#### 4. Convergence Theorems for Countably Infinite Family of -Accretive Mappings

For the rest of this paper, and are in satisfying the following additional conditions:

(i), (ii), (iii).Theorem 4.1. *Let be a nonempty closed convex subset of a uniformly convex real Banach space with uniformly Gâteaux differentiable norm. Let , , be a countably infinite family of -accretive mappings such that . For fixed , for some , , let be generated by
**
then . Furthermore, if is such that
**
then converges strongly to a common zero of .*

*Proof. *Using mathematical induction, it is easy to see that for fixed
Hence, is bounded and so is also bounded.

Now, define the sequences and by and . Then,
Observe that is bounded and that
for some . Thus,
Hence, by Lemma 2.4, we have . Consequently, we have
From (4.1), we have that
which implies that and thus
Let be a sequence satisfying (3.6). Then, by Theorem 3.5, . Using Lemma 2.1, we have that
This implies that
and hence,
Moreover, we have that
Using the boundedness of , we have
Also, since is norm-to- uniformly continuous on bounded subsets of , we have that
From (4.12) and (4.13), we obtain that
Finally, using Lemma 2.1 on (4.1), we get
Using (4.16) and Lemma 2.3 in (4.17), we get that converges strongly to common zero of the family of -accretive operators.

*Remark 4.2. *If is replaced with in Theorems 3.5, 3.6, 3.7, and 4.1, then by Lemma 2.5, the assumption that is -accretive for each could be replaced with is continuous for each .

Hence, we have the following theorem.

Theorem 4.3. *Let be a uniformly convex real Banach space with uniformly Gâteaux differentiable norm. Let , , be a countably infinite family of continuous accretive operators such that . For arbitrary but fixed , , let be generated by ,
**
then, converges strongly to a common zero of .*

*Proof. *By Lemma 2.5, we have that is -accretive for each . Then, the rest of the proof follows from Theorem 4.1.

We also have the following theorems.

Theorem 4.4. *Let , , and be as an Theorem 4.1. Suppose that at least one of is demicompact, then converges strongly to a common zero of , *

*Proof. *The proof follows as in the proof of Theorem 4.1 but using Theorem 3.6.

Theorem 4.5. *Let , , and be as an Theorem 4.1. Suppose that, in addition, admits weakly sequential continuous duality mapping, then converges strongly to a common zero of , *

*Proof. *The proof follows as in the proof of Theorem 4.1 but using Theorem 3.7.

#### 5. Convergence Theorems for Countably Infinite Family of Pseudocontractive Mappings

Theorem 5.1. *Let be a nonempty closed convex subset of a uniformly convex real Banach space with uniformly Gâteaux differentiable norm. Let , , be a countably infinite family of pseudocontractive mappings such that for each , is -accretive and . Let , . For fixed , for some and , let be generated by :
**
then converges strongly to a common fixed point of .*

*Proof. *Put , . It is then obvious that , for all and hence . Furthermore, is -accretive for each . Thus, we obtain the conclusion from Theorem 4.1 with in the definition of replaced with .

Theorem 5.2. *Let be a uniformly convex real Banach space with uniformly Gâteaux differentiable norm. For each , let be a countably infinite family of continuous pseudocontractive mappings such that for each and . Let , . For arbitrary but fixed , for some and , let be generated by :
**
then converges strongly to a common fixed point of .*

*Remark 5.3. *Theorems similar to Theorems 4.4 and 4.5 could also be obtained for countably infinite family of pseudocontractive mappings.

*Remark 5.4. *Prototypes for our iteration parameters are

*Remark 5.5. *The addition of bounded error terms in any of our recursion formulas leads to no further generalization.

*Remark 5.6. *If is a contraction map and we replace by in the recursion formulas of our theorems, we obtain what some authors now call *viscosity* iteration process. We observe that all our theorems in this paper carry over trivially to the so-called viscosity process. One simply replaces by , repeats the argument of this paper, using the fact that is a contraction map.

#### Acknowledgments

The first author's research was supported by the Japanese Mori Fellowship of UNESCO at the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. The second author undertook this work when he was visiting the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

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