Research Article | Open Access

# Strong Convergence of a Unified General Iteration for *k*-Strictly Pseudononspreading Mapping in Hilbert Spaces

**Academic Editor:**Poom Kumam

#### Abstract

We introduce a unified general iterative method to approximate a fixed point of *k*-strictly pseudononspreading mapping. Under some suitable conditions, we prove that the iterative sequence generated by the proposed method converges strongly to a fixed point of a *k*-strictly pseudononspreading mapping with an idea of mean convergence, which also solves a class of variational inequalities as an optimality condition for a minimization problem. The results presented in this paper may be viewed as a refinement and as important generalizations of the previously known results announced by many other authors.

#### 1. Introduction

Let be a nonempty closed convex subset of a real Hilbert space with inner product and norm , respectively. Recall that a mapping is said to be -strict pseudocontractive if there exists a constant such that If , is said to be nonexpansive mapping; that is, The set of fixed points of is denoted by ; that is, . Recall also that a mapping is said to be nonspreading if It is shown in the study by Iemoto and Takahashi [1] that (3) is equivalent to Observe that every nonspreading mapping is quasinonexpansive; that is, for all and all . Following the terminology of Browder and Petryshyn [2], a mapping is called -strictly pseudononspreading if there exists a constant such that Clearly, every nonspreading mapping is -strictly pseudononspreading, but the converse is not true. This shows that the class of -strictly pseudononspreading mappings is more general than the class of nonspreading mappings. Moreover, we remark also that the class of -strictly pseudononspreading mappings is independent of the class of -strict pseudo-contractions.

Fixed point problem of nonlinear mappings recently becomes an attractive subject because of its application in solving variational inequalities and equilibrium problems arising in various fields of pure and applied sciences. Moreover, various iterative schemes and methods have been developed for finding fixed points of nonlinear mappings. It is worth mentioning that iterative methods for nonexpansive and nonspreading mappings have been extensively investigated. However, iterative methods for strict pseudo-contractions are far less developed than those for nonexpansive mappings though Browder and Petryshyn [2] initiated their work in 1967; the reason is probably that the second term appearing in the right-hand side of (1) impedes the convergence analysis for iterative algorithms used to find a fixed point of the strict pseudo-contraction. This case is aggravated by adding another inner product to the right-hand side of (5) for -strictly pseudononspreading mapping; see, for example, [3–13] and the references therein. On the other hand, -strictly pseudononspreading mappings have more powerful applications than nonexpansive mappings do in solving mean ergodic problems; see, for example, [14, 15]. Therefore, it is interesting to develop the effective numerical methods for approximating fixed point of -strictly pseudononspreading mapping.

In 2006, Marino and Xu [10] introduced a general iterative method and proved that, for a given , the sequence generated by where is a self-nonexpansive mapping on , is a contraction of into itself, satisfies certain conditions, and is a strongly positive bounded linear operator on , converges strongly to , which is the unique solution of the following variational inequality: and is also the optimality condition of problem , where is a potential function for (i.e., ,). Thereafter, the general iterative method is used to find a common element of the set of fixed point problems and the set of solutions of variational inequalities and equilibrium problems (see, e.g., [11–13]).

Recently, Kurokawa and Takahashi [14] obtained a weak mean ergodic theorem for nonspreading mappings in Hilbert spaces. Furthermore, they proved a strong convergence theorem using an idea of mean convergence. In 2011, Osilike and Isiogugu [15] introduced a more general -strictly pseudononspreading mapping and considered the following iterative schemes: where auxiliary mapping . They proved that the sequences and converge strongly to , which is the metric projection of onto . Moreover, they considered the following Halpern type iterative scheme: They also proved that generated by (9) converges strongly to under some suitable conditions and hence resolved in the affirmative the open problem raised by Kurokawa and Takahashi [14] in their final remark for the case where the mapping is averaged.

In 2013, Kangtunyakarn [16] further studied variational inequalities and fixed point problem of -strictly pseudononspreading mapping by modifying the auxiliary mapping with projection technique. To be more precise, he introduced the following iterative scheme: where such that and and is a nonexpansive mapping generated by a finite family of defining operators, whose fixed point problems are equivalent to variational inequalities. Moreover, under some suitable conditions, he proved that the sequence converges strongly to , where is the intersection of the set of fixed point problems and the set of solutions for variational inequalities.

Inspired and motivated by research going on in this area, we introduce a modified general iterative method for -strictly pseudononspreading mapping, which is defined in the following way: where with and sequences and in . Note that, if , scheme (11) reduces to general iterative method (6), which is mainly due to Marino and Xu [10]. If , , and , scheme (11) reduces to viscosity approximate method introduced by Moudafi [17] and developed by Inchan [18], which also extends the Halpern type results of [19, 20] with an idea of mean convergence for -strictly pseudononspreading mapping.

Our purpose is not only to modify the general iterative method (6) and projection method (10) to the case of a -strictly pseudononspreading mapping, but also to establish a new strong convergence theorem with an idea of mean convergence for a -strictly pseudononspreading mapping, which also solves a class of variational inequalities as an optimality condition for a minimization problem. Our main results presented in this paper improve and extend the corresponding results of [10, 14–17] and many others.

#### 2. Preliminaries

Let be a nonempty closed convex subset of real Hilbert space with inner product and norm , respectively. For every point , there exists a unique nearest point in , denoted by , such that Then is called the metric projection of onto . It is well known that is a nonexpansive mapping and the following inequality holds: if and only if for given and .

Let be a mapping from into . The normal variational inequality problem is to find a point such that The set of all solutions of the variational inequality is denoted by . Note that if and only if for some .

Recall that an operator is strongly positive if there exists a constant with the property Recall also that an operator is a contraction, if there exists a constant such that

In order to prove our main results, we need the following lemmas and propositions.

Lemma 1. *Let be a real Hilbert space. There hold the following well-known results: *(i)*, ;*(ii)*, , .*

Lemma 2 (see [6]). *Let and be bounded sequences in Banach space and let be a sequence in such that . Suppose and
**
Then .*

Lemma 3 (see [10]). *Let be a strongly positive linear bounded operator on a Hilbert space with a coefficient and . Then .*

Lemma 4 (see [10]). *Let be a nonempty closed convex subset of a Hilbert space . Assume that is a contraction with a coefficient and is a strongly positive linear bounded operator with a coefficient . Then, for ,
**
That is, is strongly monotone with coefficient .*

Lemma 5 (see [15]). *Let be a nonempty closed convex subset of a real Hilbert space , and let be a -strictly pseudononspreading mapping. Then is demiclosed at zero.*

Lemma 6 (see [15]). *Let be a nonempty closed convex subset of a real Hilbert space , and let be a -strictly pseudononspreading mapping. If , then it is closed and convex.*

Lemma 7 (see [16]). *Let be a nonempty closed convex subset of a real Hilbert space , and let be a -strictly pseudononspreading mapping with . Then .*

Lemma 8 (see [21]). *Assume is a sequence of nonnegative real numbers such that
**
where is a sequence in (0,1) and is a sequence such that *(i)*;*(ii)* or .**Then .*

#### 3. Main Results

Theorem 9. *Let be a nonempty closed convex subset of a Hilbert space and let be a -strictly pseudononspreading mapping such that . Let be a contraction with a coefficient and let be a strongly positive bounded linear operator with . For a given point and , assume that satisfying the following conditions: *(i)* and ;*(ii)*;*(iii)* and .**Then the sequence generated by (11) converges strongly to , which is the unique solution of the following variational inequality:
*

*Proof. *First, we show that sequences and are bounded. Indeed, from the property of -strictly pseudononspreading mapping defined on and , we have
which implies that
From and (22), we obtain
By (i) and Lemma 3, we have that is positive and for all (see, i.e., [8]). It follows from (11) and (23) that
By induction, we have that
Therefore, is bounded and so is . On the other hand, we estimate
which implies that
From (27), we can obtain
It follows that
Combining (25) and (29), we conclude that is bounded.

Next, we will show that . To do this, define a sequence by
Observe that
where , and
From (31) and (32), we obtain
It follows from conditions (i)–(iii) and Lemma 2 that
From (30) and (34) and condition (ii), we have
Moreover, note that and
which implies that
Combining conditions (i)-(ii) and (35), we obtain
That is,

Next, we will prove that , where . To show this inequality, take a subsequence of such that
Without loss of generality, we may assume that converges weakly to ; that is, as , where . We will show that . From Lemmas 5 and 7, we have . Assume that . By condition (iii), (38), and Opial's property, we obtain
This is a contradiction. Then . Since as , we have
On the other hand, we will show the uniqueness of a solution of the variational inequality
Suppose and both are solutions to (43); then
Adding up (44), we get
From Lemma 4, the strong monotonicity of , we obtain and the uniqueness is proved.

Finally, we show that converges strongly to as . From (11), (23), and Lemma 1, we have (note that )
It follows that
Together with , condition (i), and (42), we can arrive at the desired conclusion by Lemma 8. This completes the proof.

Theorem 10. *Let be a nonempty closed convex subset of a Hilbert space and let be a -strictly pseudononspreading mapping such that . Let be a contraction with a coefficient . Let be a sequence generated by in the following manner:
**
where , , and are sequences in (0,1) satisfying the following conditions: *(i)* and ;*(ii)*;*(iii)* and .**Then the sequence converges strongly to , which is the unique solution of the following variational inequality:
*

*Proof. *Putting and , general iterative scheme (11) reduces to viscosity iteration (48). The desired conclusion follows immediately from Theorem 9. This completes the proof.

Theorem 11. *Let be a nonempty closed convex subset of a Hilbert space and let be a nonspreading mapping (or quasinonexpansive) such that . Let be a contraction with a coefficient and let be a strongly positive bounded linear operator with and . Let be a sequence generated by in the following manner:
**
where and are two sequences in (0,1) satisfying the following conditions: *(i)* and ;*(ii)*;*(iii)* and .**Then the sequence converges strongly to , which is the unique solution of the following variational inequality:
*

*Proof. *Clearly, every nonspreading mapping is -strictly pseudononspreading, which is also quasinonexpansive. Therefore, the desired conclusion follows immediately from Theorem 9. This completes the proof.

*Remark 12. *Theorems 9 and 10 extend the Halpern type methods of [14, 15] and viscosity methods of Moudafi [17] to more general unified general iterative methods for -strictly pseudononspreading mapping, which also solves a class of variational inequalities related to an optimality problem.

*Remark 13. *Theorems 9 and 10 improve and extend the main results of Kangtunyakarn [16] for -strictly pseudononspreading mapping in different directions.

*Remark 14. *The auxiliary mapping of Osilike and Isiogugu [15] is generalized to the averaged mapping presented in scheme (11) with variable coefficient and projection operator based on the equivalence between variational inequality and fixed point problem.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Science Foundation of China (11001287), Natural Science Foundation Project of Chongqing (CSTC 2012jjA00039), and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ130712).

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#### Copyright

Copyright © 2014 Dao-Jun Wen 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.