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Iterative methods for solving variational inclusions and fixed-point problems have been considered and investigated by many scholars. In this paper, we use the Halpern-type method for finding a common solution of variational inclusions and fixed-point problems of pseudocontractive operators. We show that the proposed algorithm has strong convergence under some mild conditions.

Let

Let

Setting

Problem (

Let

Problem (

Problem (

Recently, there has been increasing interest for studying common solution problems relevant to (

Motivated by the results in this direction, the main purpose of this paper is to research a common solution problem of variational inclusions and fixed point of pseudocontractions. We suggest a Halpern-type algorithm for solving such problem. We show that the proposed algorithm has strong convergence under some mild conditions.

Let

A monotone operator

For a maximal monotone operator

Set

Denote its resolvent by

It is known that

Let

If

Pseudocontractive if

Inverse-strongly monotone if

Recall that the projection

Let

For

Let

Assume that a real number sequence

Then,

Let

Let

Next, we introduce a Halpern-type algorithm for finding a common solution of variational inclusion (

Let

Next, we prove the convergence of Algorithm

Suppose that

Then, the sequence

Let

By the nonexpansivity of

Using Lemma

This together with (

According to (

Then, the sequence

Again, by (

It follows that

Since

Combining (

Next, we analyze two cases. (i)

In case of (i),

It follows from (

On the basic of (

Note that

However,

We have

This together with (

Set

Since

From (

Owing to (

Since

Applying Lemma

It follows that

Thanks to (

Noting that

By Lemma

From (

Applying Lemma

In case of (ii), let

Note that

It follows that

Combining (

From (

Since the pseudocontractive operator is nonexpansive, Theorem

Assumption

No data were used to support this study.

The authors declare that they have no conflicts of interest.

Zhangsong Yao was partially supported by the Grant 19KJD100003. Ching-Feng Wen was partially supported by the Grant of MOST 109-2115-M-037-001.