Article
Kyungpook Mathematical Journal 2022; 62(1): 69-88
Published online March 31, 2022
Copyright © Kyungpook Mathematical Journal.
The Three-step Intermixed Iteration for Two Finite Families of Nonlinear Mappings in a Hilbert Space
Sarawut Suwannaut, Atid Kangtunyakarn*
Department of Mathematics, Faculty of Science, Lampang Rajabhat University, Lampang 52100, Thailand
e-mail : sarawut-suwan@hotmail.co.th
Department of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand
e-mail : beawrock@hotmail.com
Received: June 16, 2019; Revised: August 21, 2020; Accepted: November 16, 2020
Abstract
In this work, the three-step intermixed iteration for two finite families of nonlinear mappings is introduced. We prove a strong convergence theorem for approximating a common fixed point of a strict pseudo-contraction and strictly pseudononspreading mapping in a Hilbert space. Some additional results are obtained. Finally, a numerical example in a space of real numbers is also given and illustrated.
Keywords: fixed point, the intermixed algorithm, strictly pseudo-contraction, strictly pseudononspreading, strong convergence theorem
1. Introduction
Let
We denote the fixed point set of a mapping
Definition 1.1. Let
(i) a mapping
(ii)
Note that the class of κ-strict pseudo-contractions strictly includes the class of nonexpansive mappings, that is, a nonexpansive mapping is a 0-strictly pseudo-contractive mapping.
In 2008, Kohsaka and Takahashi [6] introduced
In 2009, it is shown by Iemoto and Takahashi [2] that (1.1) is equivalent to the following equation.
Later, in 2011, Osilike and Isiogugu [13] proposed a
Obviously, every nonspreading mapping is a κ-strictly pseudononspreading mapping, that is, a nonspreading mapping is a 0-stricly pseudononspreading mapping.
Many mathematicians tried to proposed iterative algorithms and proved the strong convergence theorems for a nonspreading mapping and a strictly pseudononspreading mapping in Hilbert space to find their fixed points, see, for instance, [7, 13, 8, 1].
Over the past decades, many others have constructed various types of iterative methods to approximate fixed points. The first one is the Mann iteration introduced by Mann [9] in 1953 and is defined as follows:
where
By modification of Mann's iteration (1.1), the next iteration process is referred to as Ishikawa's iteration process [3] which is defined recursively as follows:
where
In 2000, Moudafi [11] introduced the viscosity approximation method for nonexpansive mapping
Let
where
In 2006, using the concept of the viscosity approximation method (1.4), Marino and Xu [10] introduced the general iterative method and obtained the strong convergence theorem.
Let
where
In 2015, Yao
Algorithm 1.2. For arbitrarily given
where
Furthermore, under some control conditions, they proved that the iterative sequences
Motivated by Yao
Algorithm 1.3. Starting with
where
Under appropriate conditions, they prove a strong convergence theorem for finding a common solution of two finite families of equilibrium problems.
Inspired by the previous work, we introduce the new iterative method called
Algorithm 1.4. Starting with
where
Remark 1.5. From Algorithm 1.2 and 1.4, we observe that Algorithm 1.4 can be seen as a modification and extension of Algorithm 1.2 in senses that we choose to consider the three-step intermixed algorithm for approximating fixed points of two finite families of nonlinear mappings and we study the general iterative method without a constant
Remark 1.6. If we take
The iteration (1.9) is a modification and improvement of iteration (1.7) in sense that it extends to three-step iteration for three nonlinear mappings.
Inspired by the previous research, we introduce the three-steps intermixed iteration for two finite families of nonlinear mappings. Under appropriate conditions, we prove a strong convergence theorem for finding a common fixed point of a strictly pseudo-contractive mapping and a strictly pseudononspreading mapping. Finally, we give a numerical example for the main theorem in a space of real numbers.
2. Preliminaries
We denote weak convergence and strong convergence by notations
Such an operator
We now recall the following definition and well-known lemmas.
Lemma 2.1. ([16]) For a given
Furthermore,
Lemma 2.2. ([12]) Each Hilbert space
holds for every
Lemma 2.3. ([13]) Let
(i) For all
(ii)
Lemma 2.4. ([17]) Let
where
(1)
(2)
Then,
Lemma 2.5. ([10]) Assume
Lemma 2.6. ([4, 14]) Let
(i)
(ii) For every
By applying Remark 2.10 in [5], we easily obtain the following result:
Lemma 2.7. Let
(i)
(ii)
(ii) Let
This implies that
3. Strong Convergence Theorem
Let
for
(i)
(ii)
(iii)
(iv)
Then the sequences
Step 1. We show that
Since
Let
Similarly, we get
and
Combining (3.1), (3.2) and (3.3), we have
By induction, we can derive that
for every
Step 2. Claim that
First, we let
and
Then, observe that
By the definition of
Using the same method as derived in (3.5), we have
and
From (3.5), (3.6) and (3.7), then we get
Applying Lemma 2.4 and the condition(iii), (iv), we can conclude that
Step 3. Prove that
To show this, take
which implies that
By (3.8), the condition (i) and (ii), thus we get
Observe that
This follows that
From (3.8) and (3.9), we obtain
Observe that
Hence, by (3.9), (3.10) and the condition (iii), we obtain
Applying the same argument as (3.11), we also obtain
Consider
Since
from (3.8), (3.13) and the condition (i), we get
Consider
Therefore, by (3.10), (3.14) and the condition (iii), we have
Applying the same method as (3.15), we also have
Step 4. Claim that
First, take a subsequence
Since
Next, assume
By nonexpansiveness of
This is a contradiction. Therefore
Assume that
From (3.15) and the Opial's condition, we deduce that
This is a contradiction. Thus we obtain
By (3.18) and (3.19), this yields that
Since
Following the same method as (3.21), we easily obtain that
Step 5. Finally, Prove that the sequence
By firmly-nonexpansiveness of
which yields that
From the definition of
Similarly, as derived above, we also have
and
From (3.24), (3.25) and (3.26), we deduces that
By (3.21), (3.22), the condition (i) and Lemma 2.4, this implies by (3.27) that the sequences
The following Corollary is a direct consequence of Theorem 3.1.
Corollary 3.2. Let
for
-
(i)
and ; -
(ii)
, for some ; -
(iii)
; -
(iv)
, , ,
Then the sequences
4. A Numerical Example
In this section, we give a numerical example to support our main theorem.
Example 4.1. For
Let
Solution. For every
Moreover,
Thus, we get
Clearly, all sequences and parameters are satisfied all conditions of Theorem 3.1. Hence, by Theorem 3.1, we can conclude that the sequences
Table 1 and Figure 1 show the numerical results of sequences
-
The values of {
xn }, {yn } and {zn } with initial valuesx1 = 0 ,y1 = 5 ,z1 = 0 andn=100 ..n xn yn zn 1 0.000000 5.000000 0.000000 2 -0.796797 4.561817 1.374887 3 -1.553784 4.159760 2.449299 4 -2.262385 3.796830 3.231195 5 -2.923918 3.470288 3.783815 ⋮ ⋮ ⋮ ⋮ 50 -5.000000 0.132362 5.000000 ⋮ ⋮ ⋮ ⋮ 96 -5.000000 0.008590 5.000000 97 -5.000000 0.008146 5.000000 98 -5.000000 0.007729 5.000000 99 -5.000000 0.007335 5.000000 100 -5.000000 0.006965 5.000000
-
Figure 1. An independent convergence of {
xn }, {yn } and {zn } with initial valuesx1 = 0 ,y1 = 5 ,z1 = 0 andn=100 .
Remark 4.2. From the above numerical results, we can conclude that Table 1 and Figure 1 show that the sequences
References
- B. C. Deng, T. Chen and F. L. Li,
Viscosity iteration algorithm for a ρ-strictly pseudononspreading mapping in a Hilbert space , J. Inequal. Appl.,80 (2013). - S. Iemoto and W. Takahashi,
Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space , Nonlinear Anal. Theory Methods Appl.,71 (2009), 2082-2089. - S. Ishikawa,
Fixed point by a new iterative method , Proc. Am. Math. Soc.,44 (1974), 147-150. - A. Kangtunyakarn,
Convergence theorem of κ-strictly pseudo-contractive mapping and a modification of genealized equilibrium problems , Fixed Point Theory Appl.,89 (2012), 1-17. - W. Khuangsatung and A. Kangtunyakarn,
Algorithm of a new variational inclusion problem and strictly pseudononspreding mapping with application , Fixed Point Theory Appl.,30+ (2014). - F. Kohsaka and W. Takahashi,
Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces , Arch. Math.,91 (2008), 166-177. - Y. Kurokawa and W. Takahashi,
Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces , Nonlinear Anal. Theory Methods Appl.,73 (2010), 1562-1568. - H. Liu, J. Wang and Q. Feng,
Strong convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space , Abstr. Appl. Anal., (2012). - W. R. Mann,
Mean value methods in iteration , Proc. Am. Math. Soc.,4 (1953), 506-510. - G. Marino and H. K. Xu,
A general iterative method for nonexpansive mappings in Hilbert spaces , J. Math. Anal. Appl.,318 (2006), 43-52. - A. Moudafi,
Viscosity approximation methods for fixed-points problems , J. Math. Anal. Appl.,241 (2000), 46-55. - Z. Opial,
Weak convergence of the sequence of successive approximation of nonexpansive mappings , Bull. Amer. Math. Soc.,73 (1967), 591-597. - M. O. Osilike and F. O. Isiogugu,
Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces , Nonlinear Anal.,74 (2011), 1814-1822. - S. Suwannaut and A. Kangtunyakarn,
Convergence theorem for solving the combination of equilibrium problems and fixed point problems in Hilbert spaces , Thai J. Math.,14 (2016), 77-79. - S. Suwannaut,
The S-intermixed iterative method for equilibrium problems , Thai J. Math.,17 (2019), 60-74. - W. Takahashi. Nonlinear Functional Analysis. Yokohama: Yokohama Publishers; 2000.
- H. K. Xu,
An iterative approach to quadric optimization , J. Optim Theory Appl.,116 (2003), 659-678. - Z. Yao, S. M. Kang and H. J. Li,
An intermixed algorithm for strict pseudo-contractions in Hilbert spaces , Fixed Point Theory Appl.,206 (2015).