Strong convergence theorems by hybrid and shrinking projection methods for sums of two monotone operators
- Tadchai Yuying^{1} and
- Somyot Plubtieng^{1, 2}Email author
https://doi.org/10.1186/s13660-017-1338-7
© The Author(s) 2017
Received: 12 January 2017
Accepted: 22 March 2017
Published: 11 April 2017
Abstract
In this paper, we introduce two iterative algorithms for finding the solution of the sum of two monotone operators by using hybrid projection methods and shrinking projection methods. Under some suitable conditions, we prove strong convergence theorems of such sequences to the solution of the sum of an inverse-strongly monotone and a maximal monotone operator. Finally, we present a numerical result of our algorithm which is defined by the hybrid method.
Keywords
1 Introduction
In this paper motivated by the iterative schemes considered in the present paper, we will introduce two iterative algorithms for finding zero points of the sum of an inverse-strongly monotone and a maximal monotone operator by using hybrid projection methods and shrinking projection methods. Under some suitable conditions, we obtained strong convergence theorems of the iterative sequences generated by the our algorithms. The organization of this paper is as follows: Section 2, we recall some definitions and lemmas. Section 3, we prove a strong convergence theorem by using hybrid projection methods. Section 4, we prove a strong convergence theorem by using shrinking projection methods. Section 5, we report a numerical example which indicate that the hybrid projection method is effective.
2 Preliminaries
Lemma 2.1
[18]
Lemma 2.2
[13]
Lemma 2.3
[18]
Lemma 2.4
[19]
3 Hybrid projection methods
In this section, we introduce a new iterative hybrid projection method and prove a strong convergence theorem for finding a solution of the sum of an α-inverse-strongly monotone (single-value) operator and a maximal monotone (multi-valued) operator.
Theorem 3.1
Proof
If we take \(A=0\) and \(\alpha_{n}=0\) for all \(n\in\mathbb{N}\cup\{0\}\) in Theorem 3.1, then we obtain the following result.
Corollary 3.2
4 Shrinking projection methods
In this section, we introduce a new iterative shrinking projection method and prove a strong convergence theorem for finding a solution of the sum of an α-inverse-strongly monotone (single-value) operator and a maximal monotone (multi-valued) operator.
Theorem 4.1
Proof
If we take \(A=0\) and \(\alpha_{n}=0\) for all \(n\in\mathbb{N}\cup\{0\}\) in Theorem 4.1, then we obtain the following result.
Corollary 4.2
5 Numerical results
Let \(R^{2}\) be the two dimensional Euclidean space with usual inner product \(\langle x,y\rangle=x_{1}y_{1}+x_{2}y_{2}\) for all \(x=(x_{1},x_{2})^{T}\), \(y=(y_{1},y_{2})^{T}\in R^{2}\) and denote \(\Vert x \Vert =\sqrt{{x_{1}}^{2}+{x_{2}}^{2}}\).
\(\boldsymbol{x^{(0)}}\) | Iter. | \(\boldsymbol{x=(x_{1},x_{2})^{T}}\) | E ( x ) |
---|---|---|---|
(4,3) | 4520 | (1.573198640818142,1.573198530023523) | 3.521317011074167e − 08 |
(−2,8) | 5420 | (0.944819548758385,0.944819526356611) | 1.185505467234501e − 08 |
(3,−4) | 3307 | (99.631392375764780,99.631402116509490) | 4.888391102078766e − 08 |
(−1,−3) | 4110 | (−0.781555402714756,−0.781555394005797) | 5.571556279844247e − 09 |
6 Conclusions
We have proposed two new iterative algorithms for finding the common solution of the sum of two monotone operators by using hybrid methods and shrinking projection methods. The convergence of the proposed algorithms is obtained and the numerical result of the hybrid iterative algorithm is also effective.
Declarations
Acknowledgements
The first author would like to thanks the Thailand Research Fund through the Royal Golden Jubilee PH.D. Program for supporting by grant fund under Grant No. PHD/0032/2555 and Naresuan University.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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