#### Abstract

The aim of this paper is to provide some existence theorems of a strict pseudocontraction by the way of a hybrid shrinking projection method, involving some necessary and sufficient conditions. The method allows us to obtain a strong convergence iteration for finding some fixed points of a strict pseudocontraction in the framework of real Hilbert spaces. In addition, we also provide certain applications of the main theorems to confirm the existence of the zeros of an inverse strongly monotone operator along with its convergent results.

#### 1. Introduction

There are several attempts to establish an iteration method to find a fixed point of some well-known nonlinear mappings, for instant, nonexpansive mapping. We note that Mann's iterations [1] have only weak convergence even in a Hilbert space (see, e.g., [2]). Nakajo and Takahashi [3] modified the Mann iteration method so that strong convergence is guaranteed, later well known as a hybrid projection method. Since then, the hybrid method has received rapid developments. For more details, the readers are referred to papers [4–23] and the references cited therein. In 2008, Takahashi et al. [18] introduced an alternative projection method, subsequently well known as the shrinking projection method, and they showed several strong convergence theorems for a family of nonexpansive mappings; see also [24]. In 2009, Aoyama et al. [25] applied the hybrid shrinking projection method along with creating some necessary and sufficient conditions to confirm the existence of a fixed point of firmly nonexpansive mapping.

Let be a real Hilbert space; a mapping with domain and range in is called firmly nonexpansive if
nonexpansive if
Throughout this paper, stands for an identity mapping. The mapping is said to be a strict pseudocontraction if there exists a constant such that
In this case, may be called as a -strict pseudocontraction. It is not hard to verify that (3) is equivalent to
If we set that satisfies (4), then is said to be *inverse strongly monotone*. For such a case, may be called as -inverse strongly monotone (let us see Section 4).

We use to denote the set of fixed points of (i.e., ). is said to be a quasi-strict pseudocontraction if the set of fixed points is nonempty and there exists a constant such that

The class of strict pseudocontractions extends the class of nonexpansive mappings and firmly nonexpansive mappings. That is is nonexpansive if and only if is a 0-strict pseudocontraction.

By definition, it is clear that However, the following examples show that the converse is not true.

*Example 1. *Let be a real Hilbert space and . Define by
Then, is a strict pseudocontraction but not a nonexpansive mapping.

Indeed, it is clear that is not nonexpansive. On the other hand, let us consider for all . Thus, is a strict pseudocontraction.

*Example 2. *Take , and let : it is not hard to verify that is nonexpansive but not firmly nonexpansive.

From a practical point of view, strict pseudocontractions have more powerful applications than nonexpansive mappings do in solving inverse problems (see Scherzer [26]). Therefore, it is important to develop theory of iterative methods for strict pseudocontractions. Within the past several decades, many authors have been devoted to the studies on the existence and convergence of fixed points for strict pseudocontractions. In 1967, Browder and Petryshyn [27] introduced a convex combination method to study strict pseudocontractions in Hilbert spaces. On the other hand, Marino and Xu [11] and Zhou [28] developed some iterative scheme for finding a fixed point of a strict pseudocontraction mapping.

In 2009, Yao et al. [29] introduced the hybrid iterative algorithm for pseudo-contractive 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 3 (Yao et al. [29]). *Let be a nonempty closed convex subset of a real Hilbert space . Let be an L-Lipschitz pseudocontraction such that . Assume that the sequence for some . Then, the sequence generated by (9) converges strongly to . *

In 2009, Aoyama et al. [25] provided the useful and interesting lemma to confirm that the sequence generated by the shrinking projection method is well defined even if the firmly nonexpansive mapping has no fixed points.

Lemma 4 (Aoyama et al. [25, Lemma 4.2]). *Let be a Hilbert space, a nonempty closed convex subset of , a firmly nonexpansive mapping, and . Let be a sequence in and a sequence of closed convex subsets of generated by and
**
for all . Then, is nonempty for every , and, consequently, is well defined. *

By using the lemma mentioned above, they proved the following theorem.

Theorem 5 (Aoyama et al. [25, Theorem 4.3]). *Let be a Hilbert space, a nonempty closed convex subset of , a firmly nonexpansive mapping and . Let be a sequence in and a sequence of closed convex subsets of generated by and
**
for all . Then, the following are equivalent: *(i)* is nonempty; *(ii)* is bounded; *(iii)* is nonempty. *

Motivated and inspired by the results mentioned above, in this paper, we provide some existence theorems of a strict pseudocontraction by the way of the shrinking projection method, involving some necessary and sufficient conditions. Then, we prove a strong convergence theorem and present its applications to confirm the existence of the zeros of an inverse strongly monotone operator along with its convergent results.

Throughout the paper, we will using the following notations: (i) for strong convergence and for weak convergence; (ii) denotes the weak -limit set of .

#### 2. Preliminaries

In this section, some definitions are provided, and some relevant lemmas which are useful to prove in the next section are collected. Most of them are known and others are not hard to prove.

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
The mapping is called the *metric projection* of onto . It is well known that is a firmly nonexpansive mapping of onto , that is,
Furthermore, for any and ,
Moreover, is characterized by the following:
It is obvious that the following equality holds for all :

Proposition 6 (see [11, Proposition 2.1]). *Assuming that is a closed convex subset of a Hilbert space , let be a self-mapping of . *(i)*If ** is a **-strict pseudocontraction, then ** satisfies the Lipschitz condition*(ii)*If ** is a **-strict pseudocontraction, then ** is demiclosed at zero; that is, if ** is a sequence in ** such that ** and **, then **.*(iii)*If ** is a **-quasi-strict strict pseudocontraction, then the set of fixed points ** is a closed convex subset of **.*

Lemma 7. *Assuming that is a closed convex subset of a Hilbert space , let be a self-mapping of . Then, the following are equivalent: *(i)* is a -strict pseudocontraction; *(ii)* is -inverse strongly monotone. *

Lemma 8 (see [30, Theorem 7.1.8]). *Let be a bounded closed convex subset of a Hilbert space and a continuous monotone mapping. Then, there exists an element such that for all . *

#### 3. Main Result

In this section, motivated by Aoyama et al. [25] (see also, Matsushita and Takahashi [31]), we discuss the existence of fixed point of a strict pseudocontraction by using the shrinking projection technique acting as the tool to guarantee the existence of fixed point of a strict pseudocontraction.

Every iteration process generated by the shrinking projection method for a -strict pseudocontraction is well defined even if is a fixed point free.

Lemma 9. *Let be a Hilbert space, a nonempty closed convex subset of , a -strict pseudocontraction and . Let be a sequence in and a sequence of closed convex subsets of generated by and
**
for all . Then, is nonempty for every , and, consequently, is well defined. *

*Proof. *Clearly, is nonempty. Suppose that is nonempty for some . Since , we have that are nonempty and hence is well defined. Put and . Obviously, is a nonempty bounded closed convex subset of . Let denote the identity mapping on . Since is continuous and monotone, it follows from Lemma 8 that there exists such that
In particular, we have
for every . On the other hand, by employing the identity (16) and then adding and subtracting the terms and , we obtain
By using the identity (16) again, it follows that
Substituting (22) in (21), we have
By the virtue of Lemma 7 and some simple calculations, we obtain
Joining (23) for the term with (24) and by , the monotonicity of , and (20), we have
Notice that
Combining (25) and (26), we have
for every . This shows that . By induction on , we obtain the desired result.

The following theorem provides some necessary and sufficient conditions to confirm the existence of a fixed point of a strict pseudocontraction in Hilbert spaces.

Theorem 10. *Let all the assumptions be as in Lemma 9 and for all . Then, the following are equivalent: *(i)* is nonempty; *(ii)* is bounded; *(iii)* is nonempty. *

*Proof. *[(i)*⇒*(ii)] By letting , it follows from the nonexpansiveness of that
This shows that is bounded.

[(ii)*⇒*(iii)] Suppose that is bounded; we observe that
This shows that is nondecreasing and then with the boundedness of , we have that exists. By using (29), we obtain
Since and , we have
Since , we obtain
Furthermore, Proposition 6 (i) allows us to have
By simple calculation, we have
Since is bounded, the reflexivity of allows a subsequence of such that as . By using (34) and the demicloseness of , we obtain ; that is, .

[(iii)*⇒*(i)] Suppose that . We will show that for every . Let ; then we have . Let us replace in the proof of Lemma 9 with ; it is not difficult to see that all equalities and inequalities are satisfied until (27). This implies that for all . Therefore, .

Theorem 11. *Let all the assumptions be as in Theorem 10. If ( is bounded ), then the sequence generated by (18) converges strongly to some points of , and its strong limit point is a member of ; that is, . *

*Proof. *If , then Theorem 10 guarantees that is bounded and exists. So, there is such that as . By using (34) and the demicloseness of , we obtain . On the other hand, noticing that , we observe that
for every . Since is weakly lower semicontinuous and is convergent, it follows from (35) that
Taking into account , we obtain . This implies that and . Hence, by using (16), we obtain
This completes the proof.

#### 4. Deduced Theorems and Applications

In this section, some deduced theorems and applications of the main theorem are provided in order to guarantee the existence of fixed points of a nonexpansive mapping and the existence of the zeros of an inverse strongly monotone operator. Moreover, we also have the methods that can be used to find fixed points and zero points as mentioned above.

If is nonexpansive ( is a 0-strict pseudocontraction), then we have the following corollaries.

Corollary 12. *Let be a Hilbert space, a nonempty closed convex subset of , a nonexpansive mapping, and . Let be a sequence in and a sequence of closed convex subsets of generated by and
**
for all . Then, the following are equivalent: *(i)* is nonempty; *(ii)* is bounded; *(iii)* is nonempty. *

Corollary 13. *Let be a Hilbert space, a nonempty closed convex subset of , a nonexpansive mapping, and . Let be a sequence in and a sequence of closed convex subsets of generated by and
**
for all . If ( is bounded), then converges strongly to some points of , and its strong limit point is a member of ; that is, (i.e., ). *

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 the Lipschitz continuous; that is, . It is well known (see, e.g., [32]) 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 pseudo-contractive mapping and strictly pseudo-contractive mapping are strongly related to the monotone mapping and the inverse strongly monotone mapping, respectively. It is well known that (i) is monotone is pseudo-contractive. (ii) is inverse strongly monotone is strictly pseudo-contractive.

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 that Every iteration process generated by the shrinking projection method for a -inverse strongly monotone is well defined even if has no zeros.

Lemma 14. *Let be a Hilbert space, and let be -inverse strongly monotone (without loss of generality, one can assume that ). Let , and a sequence generated by
**
for all . Then, is nonempty for every , and, consequently, is well defined. *

*Proof. *Let . Then, is -pseudocontraction,
and , where . Hence, by applying Lemma 9, we have the desired result.

The following theorem provides some necessary and sufficient conditions to confirm the existence of the zeros of -inverse strongly monotone in Hilbert spaces.

Theorem 15. *Let all the assumptions be as in Lemma 14 and for all . Then, the following are equivalent: *(i)* is nonempty; *(ii)* is bounded; *(iii)* is nonempty. *

*Proof. *Let . Then, is -pseudocontraction; it is not difficult to show that and , where . Hence, by applying Theorem 10, we have the desired result.

Theorem 16. *Let all the assumptions be as in Theorem 15. If ( is bounded ), then the sequence generated by (42) converges strongly to some points of , and its strong limit point is a member of ; that is, . *

*Proof. *Let , and by applying Theorem 11, we have the desired result.

#### Acknowledgments

The author would like to thank Professor Dr. Naseer Shahzad and an anonymous referee for their valuable comments and suggestions, which were helpful in improving the paper. Moreover, the author would like to thank the Centre of Excellence in Mathematics under the Commission of Higher Education, Ministry of Education, Thailand. The project was supported by the Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand.