#### Abstract

Assume that *F* is a nonlinear operator which is Lipschitzian and strongly monotone on a nonempty closed convex subset *C* of a real Hilbert space *H*. Assume also that is the intersection of the fixed point sets of a finite number of Lipschitzian pseudocontractive self-mappings on *C*. By combining hybrid steepest-descent method, Mann’s iteration method and projection method, we devise a hybrid iterative algorithm with perturbation *F*, which generates two sequences from an arbitrary initial point . These two sequences are shown to converge in norm to the same point under very mild assumptions.

#### 1. Introduction and Preliminaries

Let be a real Hilbert space with inner product and norm and a nonempty closed convex subset of . Let be a self-mapping of . Recall that is said to be a pseudocontractive mapping if and is said to be a strictly pseudo-contractive mapping if there exists a constant such that For such cases, we also say that is a -strict pseudo-contractive mapping. We use to denote the set of fixed points of .

It is well known that the class of strictly pseudo-contractive mappings strictly includes the class of nonexpansive mappings which are the mappings on such that

Iterative methods for nonexpansive mappings have been extensively investigated; see [1–16] and the references therein.

However, iterative methods for strictly pseudo-contractive mappings are far less developed than those for nonexpansive mappings though Browder and Petryshyn initiated their work in 1967; the reason is probably that the second term appearing on the right-hand side of (1.2) impedes the convergence analysis for iterative algorithms used to find a fixed point of the strictly pseudo-contractive mapping . However, on the other hand, strictly pseudo-contractive mappings have more powerful applications than nonexpansive mappings do in solving inverse problems; see Scherzer [17]. Therefore, it is interesting to develop iterative methods for strictly pseudo-contractive mappings. As a matter of fact, Browder and Petryshyn [18] showed that if a -strict pseudo-contractive mapping has a fixed point in , then starting with an initial , the sequence generated by the recursive formula: where is a constant such that converges weakly to a fixed point of .

Recently, Marino and Xu [19] have extended Browder and Petryshyn's result by proving that the sequence generated by the following Mann's algorithm: converges weakly to a fixed point of , provided that the control sequence satisfies the condition that for all and . However, this convergence is in general not strong. It is well known that if is a bounded and closed convex subset of , and is a demicontinuous pseudocontraction, then has a fixed point in (Theorem 2.3 in [20]). However, all efforts to approximate such a fixed point by virtue of the normal Mann's iteration algorithm failed.

In 1974, Ishikawa [21] introduced a new iteration algorithm and proved the following convergence theorem.

Theorem I (see [21]). *If is a compact convex subset of a Hilbert space is a Lipschitzian pseudocontraction and is chosen arbitrarily, then the sequence converges strongly to a fixed point of , where is defined iteratively for each positive integer by
**
where and are sequences of real numbers satisfying the conditions (i) ; (ii) as ; (iii) .*

Since its publication in 1974, it remains an open question whether or not Mann's iteration algorithm converges under the setting of Theorem I to a fixed point of if the mapping is Lipschitzian pseudo-contractive. In [22], Chidume and Mutangadura gave an example of a Lipschitzian pseudocontraction with a unique fixed point for which Mann's iteration algorithm fails to converge.

In an infinite-dimensional Hilbert space, Mann and Ishikawa's iteration algorithms have only weak convergence, in general, even for nonexpansive mapping. So, in order to get strong convergence for strictly pseudo-contractive mappings, several attempts have been made based on the CQ method (see, e.g., [19, 23, 24]). The last scheme, in such a direction, seems for us to be the following due to Zhou [25]: He proved, under suitable choice of the parameters and , that the sequence generated by (1.7) strongly converges to .

Among classes of nonlinear mappings, the class of pseudocontractions is one of the most important. This is due to the relation between the class of pseudocontractions and the class of monotone mappings (we recall that a mapping is monotone if ). A mapping is monotone if and only if is pseudo-contractive. It is well known (see, e.g., [26]) that if is monotone, then the solutions of the equation correspond to the equilibrium points of some evolution systems. Consequently, considerable research efforts, especially within the past 30 years or so, have been devoted to iterative methods for approximating fixed points of a pseudo-contractive mapping (see e.g., [27–32] and the references therein).

Very recently, motivated by the work in [19, 25, 33] and the related work in the literature, Yao et al. [34] suggested and analyzed a hybrid algorithm for pseudo-contractive mappings in Hilbert spaces. Further, they proved the strong convergence of the proposed iterative algorithm for Lipschitzian pseudo-contractive mappings.

Theorem YLM (see [34]). *Let be a nonempty closed convex subset of a real Hilbert space . Let be a -Lipschitzian pseudo-contractive mapping such that . Assume that the sequence for some . Let . For and , let be the sequence in generated iteratively by
**
Then converges strongly to .*

Inspired by the above research work of Yao et al. [34], in this paper we will continue this direction of research. Let be a nonempty closed convex subset of a real Hilbert space . We will propose a new hybrid iterative scheme with perturbed mapping for approximating fixed points of a Lipschitzian pseudo-contractive self-mapping on . We will establish a strong convergence theorem for this hybrid iterative scheme. To be more specific, let be a -Lipschitzian pseudo-contractive mapping and a mapping such that for some constants is -Lipschitzian and -strong monotone. Let and take a fixed number . We introduce the following hybrid iterative process with perturbed mapping . Let . For and , two sequences are generated as follows: It is clear that if , then the hybrid iterative scheme (1.9) reduces to the hybrid iterative process (1.8). Under very mild assumptions, we obtain a strong convergence theorem for the sequences and generated by the introduced method. Our proposed hybrid method with perturbation is quite general and flexible and includes the hybrid method considered in [34] and several other iterative methods as special cases. Our results represent the modification, supplement, extension, and improvement of [34, Algorithm 3.1 and Theorem 3.1]. Further, we consider the more general case, where are -Lipschitzian pseudo-contractive self-mappings on with an integer. In this case, we propose another hybrid iterative process with perturbed mapping for approximating a common fixed point of . Let . For and , two sequences are generated as follows: where , for integer , with the mod function taking values in the set (i.e., if for some integers and , then if and if ). It is clear that if , then the hybrid iterative scheme (1.10) reduces to the hybrid iterative process (1.9). Under quite appropriate conditions, we derive a strong convergence theorem for the sequences and generated by the proposed method.

We now give some preliminaries and results which will be used in the rest of this paper. A Banach space is said to satisfy Opial's condition if whenever is a sequence in which converges weakly to , then It is well known that every Hilbert space satisfies Opial's condition (see, e.g., [35]). Throughout this paper, we shall use the notations: “” and “” standing for the weak convergence and strong convergence, respectively. Moreover, we shall use the following notation: for a given sequence denotes the weak -limit set of , that is, In addition, for each point , there exists a unique nearest point in , denoted by , such that where is called the metric projection of onto . It is known that is a nonexpansive mapping.

Now we collect some lemmas which will be used in the proof of the main result in the next section. We note that Lemmas 1.1 and 1.2 are well known.

Lemma 1.1. *Let be a real Hilbert space. There holds the following identity:*

Lemma 1.2. *Let be a nonempty closed convex subset of a real Hilbert space . Given and . Then if and only if there holds the relation: *

Lemma 1.3 (see [23]). *Let be a nonempty closed convex subset of . Let be a sequence in and . Let . If is such that and satisfies the condition:
**
Then .*

Lemma 1.4 (see [27]). *Let be a real reflexive Banach space which satisfies Opial's condition. Let be a nonempty closed convex subset of , and be a continuous pseudo-contractive mapping. Then, is demiclosed at zero.*

Let be a nonexpansive mapping and be a mapping such that for some constants is -Lipschitzian and -strongly monotone, that is, satisfies the following conditions: respectively. For any given numbers and , we define the mapping :

Lemma 1.5 (see [36]). *If and , then there holds for :
**
where .*

In particular, whenever the identity operator of , we have

#### 2. Main Result

In this section, we introduce a hybrid iterative algorithm with perturbed mapping for pseudo-contractive mappings in a real Hilbert space .

*Algorithm 2.1. *Let be a nonempty closed convex subset of a real Hilbert space . Let be a pseudo-contractive mapping and be a mapping such that for some constants is -Lipschitzian and -strong monotone. Let and take a fixed number . Let . For and , define two sequences: of as follows:

Now we prove the strong convergence of the above iterative algorithm for Lipschitzian pseudo-contractive mappings.

Theorem 2.2. *Let be a nonempty closed convex subset of a real Hilbert space . Let be a -Lipschitzian pseudo-contractive mapping such that , and let be a mapping such that for some constants is -Lipschitzian and -strong monotone. Assume that for some and such that . Take a fixed number . Then the sequences generated by (2.1) converge strongly to the same point .*

* Proof. *
Firstly, we observe that and are well defined. From [19, 27], we note that is closed and convex. Indeed, by [27], we can define a mapping by for every . It is clear that is a nonexpansive self-mapping such that . Hence, by [23, Proposition 2.1 (iii)], we conclude that is a closed convex set. This implies that the projection is well defined. It is obvious that is closed and convex. Thus, is also well defined.

Now, we show that for all . Indeed, taking , we note that , and (1.1) is equivalent to
Using Lemma 1.1 and (2.2), we obtain
Since is -Lipschitzian, utilizing Lemma 1.5 we derive
From (2.1), we observe that . Hence, utilizing Lemma 1.5 and (2.4) we obtain
Combining (2.3) and (2.5), we get
At the same time, we observe that
Therefore, from (2.6) and (2.7) we have
which implies that
that is,

From , we have
Utilizing , we also have
So, for all we have
which hence implies that
Thus, is bounded and so are and .

From and , we have
Hence,
and therefore
This implies that exists.

From Lemma 1.1 and (2.15), we obtain
Since , from and it follows that
Noticing that for some , thus, we obtain
Also, we note that . Therefore, we get

On the other hand, utilizing Lemma 1.5 we deduce that
that is,
Meantime, it is clear that
Consequently,
Now (2.25) and Lemma 1.4 guarantee that every weak limit point of is a fixed point of , that is, . In fact, the inequality (2.14) and Lemma 1.3 ensure the strong convergence of to . Since , it is immediately known that converges strongly to . This completes the proof.

Corollary 2.3. *Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping such that , and let be a mapping such that for some constants is -Lipschitzian and -strong monotone. Assume that for some and such that . Take a fixed number . Then the sequences generated by (2.1) converge strongly to the same point .*

Corollary 2.4. *Let be a nonempty closed convex subset of a real Hilbert space . Let be a -Lipschitzian pseudo-contractive mapping such that . Assume that for some and such that . Then the sequences generated by the scheme
**
converge strongly to the same point .*

* Proof. *Put and in Theorem 2.2. Then, in this case we have and hence
This implies that . Meantime, it is easy to see that the scheme (2.1) reduces to (2.26). Therefore, by Theorem 2.2, we obtain the desired result.

Corollary 2.5 ([34, Corollary 3.2]). *Let be a -Lipschitzian monotone mapping for which . Assume that the sequence for some . Then the sequence generated by the scheme
**
strongly converges to .*

* Proof. *Put and in Corollary 2.4. Then, it is easy to see that the scheme (2.26) reduces to (2.28). Therefore, by Corollary 2.4, we derive the desired result.

Next, consider the more general case where is expressed as the intersection of the fixed-point sets of pseudo-contractive mappings with an integer, that is, In this section, we propose another hybrid iterative algorithm with perturbed mapping for a finite family of pseudo-contractive mappings in a real Hilbert space .

*Algorithm 2.6. *Let be a nonempty closed convex subset of a real Hilbert space . Let be pseudo-contractive self-mappings on with an integer, and let be a mapping such that for some constants is -Lipschitzian and -strong monotone. Let , and take a fixed number . Let . For and , define two sequences of as follows:
where
for integer , with the mod function taking values in the set (i.e., if for some integers and , then if and if ).

Theorem 2.7. *Let be a nonempty closed convex subset of a real Hilbert space . Let be -Lipschitzian pseudo-contractive self-mappings on such that , and let be a mapping such that for some constants is -Lipschitzian and -strong monotone. Assume that for some and such that . Take a fixed number . Then the sequences generated by (2.30) converge strongly to the same point .*

* Proof. *Firstly, as stated in the proof of Theorem 2.2, we can readily see that each is closed and convex for . Hence, is closed and convex. This implies that the projection is well defined. It is clear that the sequence is closed and convex. Thus, is also well defined.

Now let us show that for all . Indeed, taking , we note that and
Using Lemma 1.1 and (2.32), we obtain
Since each is -Lipschitzian for , utilizing Lemma 1.5 we derive
From (2.30), we observe that . Hence, utilizing Lemma 1.5 and (2.34) we obtain
Combining (2.33) and (2.35), we get
Meantime, we observe that
Therefore, from (2.36) and (2.37) we have
which implies that
that is,

From , we have
Utilizing , we also have
So, for all we have
which hence implies that
Thus is bounded and so are and .

From and , we have
Hence,
and therefore
This implies that exists.

From Lemma 1.1 and (2.45), we obtain
Thus,
Obviously, it is easy to see that for each . Since , from and