#### Abstract

We prove a strong convergence theorem by using a hybrid algorithm in order to find a common
fixed point of Lipschitz pseudocontraction and *κ*-strict pseudocontraction in Hilbert spaces. Our results extend the
recent ones announced by Yao et al. (2009) and many others.

#### 1. Introduction

Let be a real Hilbert space, and let be a nonempty closed convex subset of . Let . Recall that is said to be a pseudocontraction if is equivalent to for all , and is said to be a strict pseudocontraction if there exists a constant such that for all . For the second case, we say that is a -strict pseudocontraction. We use to denote the set of fixed points of .

The class of strict pseudocontractions extend the class of nonexpansive mapping. (A mapping is said to be nonexpansive if , for all ) that is, is nonexpansive if and only if is a 0-strict pseudocontraction. The pseudocontractive mapping includes the strict pseudocontractive mapping.

Iterative methods for finding fixed points of nonexpansive mappings are an important topic in the theory of nonexpansive mappings and have wide applications in a number of applied areas, such as the convex feasibility problem [1–4], the split feasibility problem [5–7] and image recovery and signal processing [3, 8, 9], and so forth. However, the Picard sequence often fails to converge even in the weak topology. Thus, averaged iterations prevail. The Mann iteration [10] is one of the types and is defined by where is chosen arbitrarily and . Reich [11] proved that if is a uniformly convex Banach space with a Fréchet differentiable norm and if is chosen such that , then the sequence defined by (1.4) converges weakly to a fixed point of . However, we note that Mann iterations have only weak convergence even in a Hilbert space (see e.g., [12]). From a practical point of view, strict pseudocontractions have more powerful applications than nonexpansive mappings do in solving inverse problems (see [13]). Therefore, it is important to develop theory of iterative methods for strict pseudocontractions. Indeed, Browder and Petryshyn [14] prove that if the sequence is generated by the following: for any starting point , is a constant such that , converges weakly to a fixed point of strict pseudocontraction. Marino and Xu [15] extended the result of Browder and Petryshyn [14] to Mann iteration (1.4); they proved converges weakly to a fixed point of , provided the control sequence satisfies the conditions that for all and .

The well-known strong convergence theorem for pseudocontractive mapping was proved by Ishikawa [16] in 1974. More precisely, he got the following theorem.

Theorem 1.1 (see [16]). *Let be a convex compact subset of a Hilbert space and let be a Lipschitzian pseudocontractive mapping. For any , suppose the sequence is defined by
**
where , are two real sequences in satisfying *(i)*, , *(ii)*,*(iii)*. **Then converges strongly to a fixed point of .*

*Remark 1.2. *(i) Since , and , the iterative sequence (1.6) could not be reduced to a Mann iterative sequence (1.4). Therefore, the iterative sequence (1.6) has some particular cases.

(ii) The iterative sequence (1.6) is usually called the Ishikawa iterative sequence.

(iii) Chidume and Mutangadura [17] gave an example to show that the Mann iterative sequence failed to be convergent to a fixed point of Lipschitzian pseudocontractive mapping.

In an infinite-dimensional Hilbert spaces, Mann and Ishikawa's iteration algorithms have only weak convergence, in general, even for nonexpansive mapping. In order to obtain a strong convergence theorem for the Mann iteration method (1.4) to nonexpansive mapping, Nakajo and Takahashi [18] modified (1.4) by employing two closed convex sets that are created in order to form the sequence via metric projection so that strong convergence is guaranteed. Later, it is often referred as the hybrid algorithm or the algorithm. After that the hybrid algorithm have been studied extensively by many authors (see e.g., [19–23]). Particularly, Martinez-Yanes and Xu [24] and Plubtieng and Ungchittrakool [20] extended the same results of Nakajo and Takahashi [18] to the Ishikawa iteration process. In 2007, Marino and Xu [15] further generalized the hybrid algorithm from nonexpansive mappings to strict pseudocontractive mappings. In 2008, Zhou [25] established the hybrid algorithm for pseudocontractive mapping in the case of the Ishikawa iteration process.

Recently, Yao et al. [26] introduced the hybrid iterative algorithm which just involved one closed convex set for pseudocontractive mapping in Hilbert spaces as follows.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a pseudocontraction. Let be a sequence in . Let . For and , define a sequence of as follows.

Theorem 1.3 (see [26]). *Let be a nonempty closed convex subset of a real Hilbert space . Let be a L-Lipschitz pseudocontraction such that . Assume the sequence for some . Then the sequence generated by (1.7) converges strongly to .*

Very recently, Tang et al. [27] generalized the hybrid algorithm (1.7) in the case of the Ishikawa iterative precess as follows: Under some appropriate conditions of and , they proved that (1.8) converges strongly to .

Motivated and inspired by the above works, in this paper, we generalize (1.7) to the Ishikawa iterative process in the case of finding the common fixed point of Lipschitz pseudocontraction and -strict pseudocontraction. More precisely, we provide some applications of the main theorem to find the common zero point of the Lipshitz monotone mapping and -inverse strongly monotone mapping in Hilbert spaces.

#### 2. Preliminaries

Let be a real Hilbert space with inner product and norm , and let be a closed convex subset of . For every point , there exists a unique nearest point in , denoted by , such that where is called the metric projection of onto . We know that is a nonexpansive mapping. It is also known that satisfies Opial's condition, that is, for any sequence with , the inequality holds for every with .

For a given sequence , let denote the weak -limit set of .

Now we collect some Lemmas which will be used in the proof of the main result in the next section. We note that Lemmas 2.1 and 2.2 are well known.

Lemma 2.1. *Let be a real Hilbert space. There holds the following identities: *(i)*,
*(ii)* and . *

Lemma 2.2. *Let be a closed convex subset of real Hilbert space . Given and , then if and only if there holds the relation
*

Proposition 2.3 (see [15, Proposition 2.1]). *Assume is a closed convex subset of a Hilbert space ; let be a self-mapping of . If is a -strict pseudocontraction, then satisfies the Lipschitz condition
*

Lemma 2.4 (see [28]). *Let be a real Hilbert space, let be a closed convex subset of , and let be a continuous pseudocontractive mapping, then *(i)* is closed convex subset of , *(ii)* is demiclosed at zero, that is, if is a sequence in such that and , then . *

Lemma 2.5 (see [24]). *Let be a closed convex subset of . Let be a sequence in , and let . Let . If is such that and satisfies the condition
**
then .*

#### 3. Main Result

Theorem 3.1. *Let be a nonempty closed convex subset of a real Hilbert space , let be -Lipschitz pseudocontraction, and let be -strict pseudocontraction with . Let . For and , define a sequence of as follows:
**
Assume the sequence , be such that and for all with . Then converges strongly to .*

*Proof. *By Lemma 2.4(i), we see that and are closed and convex, then is as well. Hence, is well defined. Next, we will prove by induction that for all . Note that . Assume that holds for . Let , thus , and we observe that
Consider the last term of (3.2), we obtain
Substituting (3.3) into (3.2), we obtain
Notice that
Therefore, from (3.4) and (3.5), we get
On the other hand, we found that
Notice that
Combining (3.7) and (3.8) and then it implies that
Since for all , so we get
It follows from (3.6) and (3.10) that we obtain
Therefore, . By mathematical induction, we have for all . It is easy to check that is closed and convex, and then is well defined. From , we have for all . Using , we also have for all . So, for , we have
Hence, . In particular,
This implies that is bounded, and then , , , , and are as well.

From and , we have
Hence
and; therefore,
which implies that exists. From Lemma 2.1 and (3.14), we obtain
Since , we have
therefore, we obtain
We note that
that is,

By Lemma 2.4(ii), and are demiclosed at zero. Together with the fact that is bounded, which guarantees that every weak limit point of is a fixed point of and , that is , therefore, by inequality (3.13) and Lemma 2.5, we know that converges strongly to . This completes the proof.

If , then we obtain the following corollary.

Corollary 3.2 (Yao et al. [26, Theorem 3.1]). *Let be a nonempty closed convex subset of a real Hilbert space . Let be -Lipschitz pseudocontraction such that . Assume the sequence be such that for all . Then the sequence generated by (1.7) converges strongly to .*

If and are nonexpansive, then we also have the following corollary.

Corollary 3.3. *Let be a nonempty closed convex subset of a real Hilbert space , and let be nonexpansive mappings. Suppose that . Assume the sequence be such that and for all with . Let . For and , define a sequence of as follows:
**
Then converges strongly to .*

Recall that a mapping is said to be monotone if for all and inverse strongly monotone if there exists a real number such that for all . For the second case, is said to be -inverse strongly monotone. It follows immediately that if is -inverse strongly monotone, then is monotone and Lipschitz continuous, that is, . It is well known (see e.g., [29]) that if is monotone, then the solutions of the equation correspond to the equilibrium points of some evolution systems. Therefore, it is important to focus on finding the zero point of monotone mappings. The pseudocontractive mapping and strictly pseudocontractive mapping are strongly related to the monotone mapping and inverse strongly monotone mapping, respectively. It is well known that (i) is monotone is pseudocontractive, (ii) is inverse strongly monotone is strictly pseudocontractive.

Indeed, for (ii), we notice that the following equality always holds in a real Hilbert space: Without loss of generality, we can assume that , and then it yields Due to Theorem 3.1, we have the following corollary which generalize the corresponding results of Yao et al. [26].

Corollary 3.4. *Let be -Lipschitz monotone mapping and let be an -inverse strongly monotone which . Assume the sequence be such that , for all with and such that . Let . For and , define a sequence as follows:
**
Then converges strongly to .*

*Proof. *Let and let . Then and are pseudocontractive and -pseudocontractive, respectively. Moreover, is also -Lipschitz, and if we set , is also -Lipschitz, and then . Hence, it follows from Theorem 3.1 that we have the desired result.

If (zero mapping), then and . So, we obtain the following corollary.

Corollary 3.5 (Yao et al. [26, Corollary 3.2]). *Let be a -Lipschitz monotone mapping for which . Assume that the sequence be as in Corollary 3.4. Then the sequence generated by
**
strongly converges to .*

#### Acknowledgments

The author is supported by the Naresuan University and the Thailand Research Fund under Grant no. MRG5380249. Finally, the author would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.