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In this paper, we establish an iterative algorithm by combining Yamada’s hybrid steepest descent method and Wang’s algorithm for finding the common solutions of variational inequality problems and split feasibility problems. The strong convergence of the sequence generated by our suggested iterative algorithm to such a common solution is proved in the setting of Hilbert spaces under some suitable assumptions imposed on the parameters. Moreover, we propose iterative algorithms for finding the common solutions of variational inequality problems and multiple-sets split feasibility problems. Finally, we also give numerical examples for illustrating our algorithms.

In 2005, Censor et al. [

When

Denote by

Assume that the SFP is consistent (i.e., (

The variational inequality problem (VIP) was introduced by Stampacchia [

In [

Moreover, it is well known that if

Since SFP and VIP include some special cases (see [

(C1)

(C2)

Moreover, Buong [

Motivated by the aforementioned works, we establish an iterative algorithm by combining algorithms (

In order to solve our results, we now recall the following definitions and preliminary results that will be used in the sequel. Throughout this section, let

A mapping

Nonexpansive, if (i) holds with

Firmly nonexpansive, if

In [

We collect some basic properties of averaged mappings in the following results.

We have

The composite of finitely many averaged mappings is averaged. In particular, if

If the mappings

Let

For every

The following properties of the nonexpansive mappings are very convenient and helpful to use.

Assume that

Let

The following results play a crucial role in the next section.

Let

Let

Let

In this section, we consider the following iterative algorithm by combining Yamada’s hybrid steepest descent method [

Throughout our results, unless otherwise stated, we assume that

Let

From Lemma

In [

Let

In the proof of Theorem

Moreover, we obtain the following results which are solving the common solution of variational inequality problem and multiple-sets split feasibility problem, i.e., find a point

Let

Then, as

(A1)

(A2)

(A3)

(A4)

Let

In the case of (A1),

If

Cases (A3) and (A4) are similar. This implies that

Then, iterative algorithm (

Let

In the proof of Theorem

In this section, we present the numerical example comparing algorithm (

We consider test problem (

So, we have that

Computational results for Example

Initial point | |||||
---|---|---|---|---|---|

Buong method | 29461 | 0.364595 | 2946204 | 31.362283 | |

New method | 11784 | 0.241371 | 1178481 | 23.411679 | |

Buong method | 30632 | 0.565431 | 3063343 | 33.468210 | |

New method | 12252 | 0.324808 | 1225336 | 25.570356 |

The convergence behavior of

The convergence behavior of

From the numerical analysis of our results in Table

In this example, we consider algorithm (

Computational results for Example

Initial point | A1 | A2 | A3 | A4 | |
---|---|---|---|---|---|

28577 | 24264 | 28577 | 24264 | ||

1.491225 | 1.355074 | 1.534414 | 1.282528 | ||

33407 | 31438 | 33407 | 31438 | ||

1.746868 | 1.693069 | 1.816897 | 1.690618 |

The convergence behavior of

The convergence behavior of

Computational results for Example

0.1 | 0.2 | 0.3 | ||
---|---|---|---|---|

9675 | 19200 | 28577 | ||

0.669508 | 1.245136 | 1.666702 | ||

11311 | 22447 | 33407 | ||

1.372600 | 1.958486 |

We observe from the numerical analysis of Table

No data were used to support this study.

The authors declare that there are no conflicts of interest.

The first author is thankful to the Science Achievement Scholarship of Thailand. The authors would like to thank the Department of Mathematics, Faculty of Science, Naresuan University (grant no. R2564E049), for the support.