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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.

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 [

Let

We use

The class of strict pseudocontractions extends the class of nonexpansive mappings and firmly nonexpansive mappings. That is

By definition, it is clear that

Let

Indeed, it is clear that

Take

From a practical point of view, strict pseudocontractions have more powerful applications than nonexpansive mappings do in solving inverse problems (see Scherzer [

In 2009, Yao et al. [

Let

Let

In 2009, Aoyama et al. [

Let

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

Let

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:

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

Assuming that

Assuming that

Let

In this section, motivated by Aoyama et al. [

Every iteration process generated by the shrinking projection method for a

Let

Clearly,

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

Let all the assumptions be as in Lemma

[(i)

[(ii)

[(iii)

Let all the assumptions be as in Theorem

If

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

Let

Let

Recall that a mapping

Indeed, for (ii), we notice that the following equality always holds in a real Hilbert space:

Let

Let

The following theorem provides some necessary and sufficient conditions to confirm the existence of the zeros of

Let all the assumptions be as in Lemma

Let

Let all the assumptions be as in Theorem

Let

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.