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### Article

Kyungpook Mathematical Journal 2022; 62(1): 69-88

Published online March 31, 2022

### 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

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 C be a nonempty closed convex subset of a real Hilbert space H. The fixed point problem for the mapping $T:C→C$ is to find x ∈ C such that

$x=Tx.$

We denote the fixed point set of a mapping T by Fix(T).

Definition 1.1. Let $T:C→C$ be a mapping. Then

(i) a mapping T is called nonexpansive if

$Tx−Ty≤x−y,∀x,y∈C;$

(ii) T is said to be κ-strictly pseudo-contractive if there exists a constant $κ∈[0,1)$such that

$Tx−Ty2≤x−y2+κ(I−T)x−(I−T)y2,∀x,y∈C.$

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 a nonspreading mapping T in Hilbert space H as follows:

$2‖Tx−Ty‖2≤‖Tx−y‖2+‖x−Ty‖2,∀x,y∈C.$

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 κ-strictly pseudononspreading mapping, that is, a mapping $T:C→C$ is said to be a κ-strictly pseudononspreading mapping if there exists $κ∈[0,1)$ such that

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 C is a nonempty closed convex subset of a normed space, $T:C→C$ is a mapping and the sequence $αn$ is in the interval (0,1). But this algorithm has only weak convergence. Thus, many mathematicians have been trying to modify Mann's iteration (1.1) and construct new iterative method to obtain the strong convergence theorem.

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 $αn$ and $βn$ are real sequences in [0,1]. He also obtain the strong convergence theorem for the iterative method (1.3) converging to a fixed point of mapping T. Observe that if βn=1, then the Ishikawa's iteration (1.3) reduces to the Mann's iteration (1.2).

In 2000, Moudafi [11] introduced the viscosity approximation method for nonexpansive mapping S as follows:

Let C be a closed convex subset of a real Hilbert space H and let $S:C→C$ be a nonexpansive mapping such that Fix(S) is nonempty. Let $f:C→C$ be a contraction, that is, there exists $α∈(0,1)$ such that $fx−fy≤αx−y,∀x,y∈C$, and let $xn$ be a sequence defined by

where $εn⊂(0,1)$ satisfies certain conditions. Then the sequence $xn$ converges strongly to $z∈Fix(S)$, where $z=PFix(S)f(z)$ and $PFix(S)$ is the metric projection of H onto Fix(S).

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 $T:H→H$ be a nonexpansive mapping with $Fix(T)≠∅$. Let $f:H→H$ be a contractive mapping on H and let $xn$ be generated by

where $αn$ is a sequence in (0,1) satisfying the appropriate conditions. Then $xn$ converges strongly to a fixed point $x˜$ of T which solves the variational inequality:

$(A−γf)x˜,x˜−z≤0,z∈Fix(T).$

In 2015, Yao et al. [18] proposed the intermixed algorithm for two strict pseudocontractions S and T as follows:

Algorithm 1.2. For arbitrarily given $x0∈C,y0∈C$, let the sequences $xn$ and $yn$ be generated iteratively by

$xn+1=1−βnxn+βnPCαnfyn+1−k−αnxn+kTxn,n≥0,yn+1=1−βnyn+βnPCαngxn+1−k−αnyn+kSyn,n≥0,$

where $T:C→C$ is a λ-strict pseudo-contraction, $f:C→H$ is a ρ1-contraction and $g:C→H$ is a $ρ2$-contraction, $k∈(0,1−λ)$ is a constant and $αn$, $βn$ are two real number sequences in (0,1).

Furthermore, under some control conditions, they proved that the iterative sequences $xn$ and $yn$ defined by (1.6) converges independently to $PFix(T)fy*$ and $PFix(S)gx*$, respectively, where $x*∈Fix(T)$ and $y*∈Fix(S)$.

Motivated by Yao et al. [18], in 2018, Suwannaut [15] introduce the S-intermixed iteration for two finite families of nonlinear mappings without considering the constant k as in the following algorithm:

Algorithm 1.3. Starting with $x1,y1,z1∈C$, let the sequences $xn$, $yn$ and $zn$ be defined by

$xn+1=1−βnxn+βnαnf1yn+1−αnSxn,yn+1=1−βnyn+βnαnf2xn+1−αnTyn,n≥1,$

where $S,T:C→C$, is a nonlinear mapping with $Fix(S)∩FixT≠∅$, $fi:C→C$ is a contractive mapping with coefficients $αi;i=1,2$ and $βn$,$αn$ are real sequences in (0,1), $∀n≥1$.

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 the three-step intermixed iteration for two finite families of nonlinear mappings as the following algorithm:

Algorithm 1.4. Starting with $x1,y1,z1∈C$, let the sequences $xn$, $yn$ and $zn$ be defined by

$xn+1=δnxn+ηnS1xn+μnPCαnγ1f1(yn)+(I−αnA1)T1xn,yn+1=δnyn+ηnS2yn+μnPCαnγ2f2(zn)+(I−αnA2)T2yn,zn+1=δnzn+ηnS3zn+μnPCαnγ3f3(xn)+(I−αnA3)T3zn,n≥1,$

where $Si,Ti:C→C$, where i=1,2,3, is nonlinear mappings with $Fix(Si)∩FixTi≠∅,∀i=1,2,3$, fi is a contractive mapping with coefficients $ξi,Ai:C→C$ is a strongly positive linear bounded operator with coefficient $βi>0$ and $0<γ<βξ$, where $γ=maxi∈{1,2,3}γi$, $ξ=maxi∈{1,2,3}ξi$ and $β=mini∈{1,2,3}βi$, $δn$, $ηn$, $μn$ and $αn$ are real sequences in (0,1) and $δn+ηn+μn=1,∀n≥1$.

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 k.

Remark 1.6. If we take $Si≡I$, $γi=1$ and $Ai≡I$ for i=1,2,3, then the iterative method (1.8) reduces to

$xn+1=1−μnxn+μnαnf1(yn)+(1−αn)T1xn,yn+1=1−μnyn+μnαnf2(zn)+(1−αn)T2yn,zn+1=1−μnzn+μnαnf3(xn)+(1−αn)T3zn.$

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 $‵‵⇀′′$ and $‵‵→′′$, respectively. For every x ∈ H, there is a unique nearest point PCx in C such that

$‖x−PCx‖≤‖x−y‖,∀y∈C.$

Such an operator PC is called the metric projection of H onto C.

We now recall the following definition and well-known lemmas.

Lemma 2.1. ([16]) For a given z ∈ H and u ∈ C,

$u=PCz⇔⟨u−z,v−u⟩≥0,∀v∈C.$

Furthermore, PC is a firmly nonexpansive mapping of H onto C and satisfies

$PCx−PCy2≤PCx−PCy,x−y,∀x,y∈H.$

Lemma 2.2. ([12]) Each Hilbert space H satisfies Opial's condition, i.e., for any sequence $xn⊂H$ with $xn⇀x$, the inequality

$liminfn→∞xn−x

holds for every y ∈ H with y ≠ x.

Lemma 2.3. ([13]) Let H be a real Hilbert space. Then the following results hold:

(i) For all x,y ∈ H and $α∈[0,1]$,

$αx+(1−α)y2=αx2+(1−α)y2−α(1−α)x−y2,$

(ii) $‖x+y‖2≤‖x‖2+2⟨y,x+y⟩,$ for each $x,y∈H.$

Lemma 2.4. ([17]) Let ${sn}$ be a sequence of nonnegative real numbers satisfying

$sn+1≤(1−αn)sn+δn,∀n≥0,$

where αn is a sequence in (0,1) and ${δn}$ is a sequence such that

(1) $∑ n=1∞αn=∞$,

(2) $limsupn→∞δnαn≤0$ or $∑ n=1∞|δn|<∞$.

Then, $limn→∞sn=0$.

Lemma 2.5. ([10]) Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient $β>0$ and $0<δ<‖A‖−1$. Then $‖I−δA‖≤1−δβ$.

Lemma 2.6. ([4, 14]) Let C be a nonempty closed convex subset of a real Hilbert space H and let $T:C→C$ be a κ-strictly pseudo-contractive mapping with $Fix(T)≠∅$. Then, we there hold the following statement:

(i) $Fix(T)=VI(C,I−T)$;

(ii) For every u ∈ C and $v∈Fix(T)$,

By applying Remark 2.10 in [5], we easily obtain the following result:

Lemma 2.7. Let $S:C→C$ be a κ-strictly pseudo nonspreading mapping with $Fix(S)≠∅$. Define $T:C→C$ by $Tx:=(1−λ)x+λSx$, where $λ∈(0,1−κ)$. Then there hold the following statement:

(i) Fix(S) = Fix(T)

(ii) T is a quasi-nonexpansive mapping, that is,

Proof. It is clear to prove that (i) holds.

(ii) Let x ∈ H and $y∈Fix(S)$. Then we derive

$‖Tx−y‖2=‖(1−λ)(x−y)+λ(Sx−y)‖2 =(1−λ)‖x−y‖2+λ‖Sx−y‖2−λ(1−λ)‖Sx−x‖2 ≤(1−λ)‖x−y‖2+λ‖x−y‖2+κ‖x−Sx‖2−λ(1−λ)‖Sx−x‖2 =‖x−y‖2+κλ‖x−Sx‖2−λ(1−λ)‖Sx−x‖2 =‖x−y‖2−λ((1−κ)−λ)‖x−Sx‖2 ≤‖x−y‖2.$

This implies that T is a quasi-nonexpansive mapping.

### 3. Strong Convergence Theorem

Let C be a nonempty closed convex subset of a real Hilbert space H. For i=1,2,3, let $fi:C→C$ be a contractive mappings with a coefficient ξi and $ξ=maxi∈{1,2,3}ξi$, let $Si:C→C$ be a κi-strictly pseudo-contractive mapping and $Wi:C→C$ be ρi-strictly pseudo-nonspreading mapping with $Ωi=Fix(Si)∩Fix(Wi)≠∅$. For each i=1,2,3, define a mapping $Tni:C→C$ by $Tnix=1−ωnx+ωnWix,$ for all x ∈ C, and let $Ai:C→C$ be a strongly positive linear bounded operator with a coefficient $βi>0$ and $0<γ<βξ$, where $γ=maxi∈{1,2,3,}γi$ and $β=mini∈{1,2,3}βi$. Let $xn$, $yn$ and $zn$ be sequences generated by $x1,y1,z1∈C$ and

$xn+1=δnxn+ηnPCI−λnI−S1xn +μnPCαnγ1f1(yn)+(I−αnA1)Tn1xn,yn+1=δnyn+ηnPCI−λnI−S2yn +μnPCαnγ2f2(zn)+(I−αnA2)Tn2yn,zn+1=δnzn+ηnPCI−λnI−S3zn +μnPCαnγ3f3(xn)+(I−αnA3)Tn3zn,$

for n ≥ 1, where $αn$, $δn$, $ηn$, $μn⊂(0,1)$, $λn⊂(0,1−κ)$, $κ=mini∈{1,2,3}$, $ωn⊂(0,1−ρ)$, where $ρ=mini∈{1,2,3}$ and $δn+μn+ηn=1$ satisfying the following conditions:

(i) $limn→∞αn=0$ and $∑ n=1∞αn=∞$;

(ii) $0<τ≤δn,ηn,μn,≤υ<1$, for some $τ,υ>0$;

(iii) $∑ n=1∞λn<∞$, $∑ n=1∞ωn<∞$;

(iv) $∑ n=1∞α n+1−αn<∞$, $∑ n=1∞δ n+1−δn<∞$, $∑ n=1∞μ n+1−μn<∞$,

$∑ n=1∞η n+1−ηn<∞$, $∑ n=1∞λ n+1−λn<∞$, $∑ n=1∞ω n+1−ωn<∞$.

Then the sequences $xn$, $yn$ and $zn$ converge strongly to $x˜=PΩ1(I−A1)x˜+γf1 y ˜$, $y˜=PΩ2(I−A2)y˜+γf2 z ˜$ and $z˜=PΩ3(I−A3)z˜+γf3 x ˜$, respectively.

Proof. The proof of this theorem will be divided into five steps.

Step 1. We show that $xn$ is bounded.

Since $αn→0$ as $n→∞$, without loss of generality, we may assume that $αn<1Ai,$ for all i=1,2,3 and $n∈ℕ$.

Let $x*∈Ω1$, $y*∈Ω2$, $z*∈Ω3$, $β=mini∈{1,2,3}βi$, $ξ=maxi∈{1,2,3}ξi$ and $γ=maxi∈{1,2,3}γi$. Then we have

$xn+1−x*≤‖δnxn−x*+ηnPCI−λnI−S1xn−x* +μnPCαnγ1f1(yn)+(I−αnA1)Tn1xn−x*‖≤δnxn−x*+ηnPCI−λnI−S1xn−x* +μnPCαnγ1f1(yn)+(I−αnA1)Tn1xn−x*≤1−μnxn−x*+μnαnγ1f1yn−A1x*+I−αnA1Tn1xn−x*≤1−μnxn−x* +μnαnγ1ξ1yn−y*+αnγ1f1y*−A1x*+1−αnβxn−x*≤1−μnαnβxn−x*+μnαnγξyn−y*+μnαnγ1f1y*−A1x*.$

Similarly, we get

$yn+1−y*≤1−μnαnβyn−y*+μnαnγξzn−z*+μnαnγ2f2z*−A2y*$

and

$zn+1−z*≤1−μnαnβzn−z*+μnαnγξxn−x*+μnαnγ3f3x*−A3z*.$

Combining (3.1), (3.2) and (3.3), we have

$xn+1−x*+yn+1−y*+zn+1−z*≤1−μnαnβ−γξxn−x*+yn−y*+zn−z* +μnαnγ1f1y*−A1x*+γ2f2z*−A2y*+γ3f3x*−A3z*.$

By induction, we can derive that

$xn−x*+yn−y*+zn−z*≤max{x1−x*+y1−y*+z1−z*, γ1f1y*−A1x*+γ2f2z*−A2y*+γ3f3x*−A3z*β−γξ},$

for every $n∈ℕ$. This implies that $xn,yn$ and $zn$ are bounded.

Step 2. Claim that $limn→∞xn+1−xn=0$.

First, we let

$un=PCαnγ1f1(yn)+(I−αnA1)Tn1xn,$ $vn=PCαnγ2f2(zn)+(I−αnA2)Tn2yn$

and

$wn=PCαnγ3f3(xn)+(I−αnA3)Tn3zn.$

Then, observe that

$un−un−1=∥PCαnγ1f1yn+I−αnA1Tn1xn −PCαn−1γ1f1yn−1+I−αn−1A1Tn−11xn−1∥≤αnγ1f1yn−f1yn−1+γ1αn−αn−1f1yn−1 +I−αnA1Tn1xn−Tn−11xn−1 +I−αnA1Tn−11xn−1−I−αn−1A1Tn−11xn−1≤αnγ1ξ1yn−yn−1+γ1αn−αn−1f1yn−1 +1−αnβ1(1−ωnxn−xn−1 +ωn−ωn−1xn−1+ωnW1xn−W1xn−1+ωn−ωn−1W1xn−1) +αn−αn−1A1Tn1xn−1≤αnγξyn−yn−1+αn−αn−1γf1yn−1+A1Tn1xn−1 +1−αnβ(1−ωnxn−xn−1+ωn−ωn−1xn−1+W1xn−1 +ωnW1xn−W1xn−1).$

By the definition of xn, we obtain

$xn+1−xn≤δnxn−xn−1+δn−δn−1xn−1 +ηnPCI−λn(I−S1)xn−PCI−λn−1(I−S1)xn−1 +ηn−ηn−1PCI−λn−1(I−S1)xn−1+μnun−un−1 +μn−μn−1un−1≤δnxn−xn−1+δn−δn−1xn−1+ηnxn−xn−1 +ηnλn(I−S1)xn−λn−1(I−S1)xn−1 +ηn−ηn−1PCI−λn−1(I−S1)xn−1 +μn[αnγξyn−yn−1+αn−αn−1γf1yn−1+A1Tn1xn−1 +1−αnβ(1−ωnxn−xn−1+ωn−ωn−1xn−1+W1xn−1 +ωnW1xn−W1xn−1)]+μn−μn−1un−1$ $≤1−μnxn−xn−1+δn−δn−1xn−1 +ηnλn(I−S1)xn−(I−S1)xn−1 +ηnλn−λn−1(I−S1)xn−1 +ηn−ηn−1PCI−λn−1(I−S1)xn−1 +μnαnγξyn−yn−1+αn−αn−1γf1yn−1+A1Tn1xn−1 +μn1−αnβxn−xn−1+ωn−ωn−1xn−1+W1xn−1 +ωnW1xn−W1xn−1+μn−μn−1un−1$ $≤1−μnαnβxn−xn−1+μnαnγξyn−yn−1+δn−δn−1xn−1 +λn(I−S1)xn−(I−S1)xn−1+λn−λn−1(I−S1)xn−1 +ηn−ηn−1PCI−λn−1(I−S1)xn−1 +αn−αn−1γf1yn−1+A1Tn1xn−1 +ωn−ωn−1xn−1+W1xn−1+ωnW1xn−W1xn−1 +μn−μn−1un−1.$

Using the same method as derived in (3.5), we have

$yn+1−yn≤1−μnαnβyn−yn−1+μnαnγξzn−zn−1+δn−δn−1yn−1 +λn(I−S2)yn−(I−S2)yn−1+λn−λn−1(I−S2)yn−1 +ηn−ηn−1PCI−λn−1(I−S2)yn−1 +αn−αn−1γf2zn−1+A2Tn2yn−1 +ωn−ωn−1yn−1+W2yn−1+ωnW2yn−W2yn−1 +μn−μn−1vn−1$

and

$zn+1−zn≤1−μnαnβzn−zn−1+μnαnγξxn−xn−1+δn−δn−1zn−1 +λn(I−S3)zn−(I−S3)zn−1+λn−λn−1(I−S3)zn−1 +ηn−ηn−1PCI−λn−1(I−S3)zn−1 +αn−αn−1γf3xn−1+A3Tn3zn−1 +ωn−ωn−1zn−1+W3zn−1+ωnW3zn−W3zn−1 +μn−μn−1wn−1.$

From (3.5), (3.6) and (3.7), then we get

$xn+1−xn+yn+1−yn+zn+1−zn≤1−μnαnβ−γξxn−xn−1+yn−yn−1+zn−zn−1 +δn−δn−1xn−1+yn−1+zn−1+λn((I−S1)xn−(I−S1)xn−1 +(I−S2)yn−(I−S2)yn−1+(I−S3)zn−(I−S3)zn−1) +λn−λn−1(I−S1)xn−1+(I−S2)yn−1+(I−S3)zn−1 +ηn−ηn−1(PCI−λn−1(I−S1)xn−1+PCI−λn−1(I−S2)yn−1$ $+PCI−λn−1(I−S3)zn−1)+αn−αn−1(γ(f1xn−1+f2yn−1+f3zn−1)+A1Tn1xn−1+A2Tn2yn−1+A3Tn3zn−1)+ωn−ωn−1(xn−1+yn−1+zn−1+W1xn−1+W2yn−1+W3zn−1)+ωnW1xn−W1xn−1+W2yn−W2yn−1+W3zn−W3zn−1+μn−μn−1un−1+vn−1+wn−1.$

Applying Lemma 2.4 and the condition(iii), (iv), we can conclude that

Step 3. Prove that $limn→∞un−PCI−λn(I−S1)un=limn→∞un−Tn1un=0$.

To show this, take $u˜n=αnγ1f1(yn)+(I−αnA1)Tn1xn$. Then we derive that

$xn+1−x*2=δnxn−x*+ηnPCI−λn(I−S1)xn−x*+μnun−x*2≤δnxn−x*2+ηnPCI−λn(I−S1)xn−x*2+μnun−x*2 −δnηnxn−PCI−λn(I−S1)xn2≤1−μnxn−x*2+μnαnγ1f1yn−A1Tn1xn+Tn1xn−x*2 −δnηnxn−PCI−λn(I−S1)xn2≤1−μnxn−x*2 +μn[Tn1xn−x*2+2αnγ1f1 y n −A1Tn1xn,u˜ n−x*] −δnηnxn−PCI−λn(I−S1)xn2≤xn−x*2+2μnαnγ1f1 y n −A1Tn1xnu˜ n−x* −δnηnxn−PCI−λn(I−S1)xn2,$

which implies that

$δnηn x n −PC I−λ n(I−S1)x n 2≤ x n −x* 2− xn+1 −x* 2+2μnαnγ1f1 yn −A1Tn1xnu˜ n−x*≤xn−x n+1 xn − x * + x n+1 − x * +2μnαnγ1f1 yn −A1Tn1xnu˜ n−x*.$

By (3.8), the condition (i) and (ii), thus we get

Observe that

$xn+1−xn=ηnPCI−λn (I−S1 )xn−xn+μnun−xn.$

This follows that

$μnun−xn≤ηnPCI−λn (I−S1 )xn−xn+xn+1−xn.$

From (3.8) and (3.9), we obtain

Observe that

$un−PCI−λn(I−S1)un≤un−xn+xn−PCI−λn(I−S1)xn+PCI−λn(I−S1)xn−PCI−λn(I−S1)un≤2un−xn+xn−PCI−λn(I−S1)xn+λnI−S1xn−I−S1un.$

Hence, by (3.9), (3.10) and the condition (iii), we obtain

Applying the same argument as (3.11), we also obtain

Consider

$xn+1−un≤xn+1−xn+xn−un,$

by (3.8) and (3.10), we have

Since

$xn−Tn1xn≤xn−xn+1+xn+1−un+un−Tn1xn ≤xn−xn+1+xn+1−un+ u˜n−Tn1xn =xn−xn+1+xn+1−un+αnγ1f1yn−A1Tn1xn,$

from (3.8), (3.13) and the condition (i), we get

Consider

$un−Tn1un≤un−xn+xn−Tn1xn+Tn1xn−Tn1un ≤2un−xn+xn−Tn1xn+ωnW1xn−W1un.$

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 $unk$ of $un$ such that

$limsupn→∞γ1f1y˜−A1x˜,un−x˜=limk→∞γ1f1y˜−A1x˜,unk−x˜.$

Since $xn$ is bounded, then we can assume that $xnk⇀x^$ as $k→∞$. From (3.10), we obtain $unk⇀x^$ as $k→∞$.

Next, assume $x^∉FixS1$. Since $FixS1=FixPCI−λnk(I−S1)$, then we get $x^≠PCI−λnk(I−S1)x^$.

By nonexpansiveness of PC, (3.11), the condition (iii) and the Opial's condition, we obtain

$liminfk→∞unk−x^

$x^∈FixS1.$

Assume that $x^∉FixW1$. Because $FixW1=FixTnk1$, then we have $x^≠Tnk1x^$.

From (3.15) and the Opial's condition, we deduce that

$liminfk→∞unk−x^

This is a contradiction. Thus we obtain

$x^∈FixW1.$

By (3.18) and (3.19), this yields that

$x^∈Ω1=FixS1∩FixW1.$

Since $xnk⇀x^$ as $k→∞$, (3.20) and Lemma 2.1, we can derive that

$limsupn→∞γ1f1y˜−A1x˜,un−x˜=limk→∞γ1f1y˜−A1x˜,unk−x˜ =γ1f1y˜−A1x˜,x^−x˜ =γ1f1y˜−A1x˜+x˜−x˜,x^−x˜ ≤0.$

Following the same method as (3.21), we easily obtain that

Step 5. Finally, Prove that the sequence $xn$, $yn$ and $zn$ converge strongly to $x˜=PΩ1(I−A1)x˜+γ1f1 y ˜$, $y˜=PΩ2(I−A2)y˜+γ2f2 z ˜$ and $z˜=PΩ3(I−A3)z˜+γ3f3 x ˜$, respectively.

By firmly-nonexpansiveness of PC, we derive that

$un−x˜2=PCun˜−x˜2 ≤ u n ˜−x˜,un−x˜ =αn γ1 f1 y n −A1 x ˜ + I−α n A1 T n1 x n − x ˜ ,un−x˜ =αnγ1f1 y n −A1x˜,un−x˜+ I−α n A1 T n1 x n − x ˜ ,un−x˜ ≤αnγ1f1 y n −f1y˜ ,un−x˜+αnγ1f1y˜ −A1x˜,un−x˜ + I−α n A1 T n1 x n − x ˜ un−x˜ ≤αnγ1ξ1yn−y˜un−x˜+αnγ1f1y˜ −A1x˜,un−x˜ +1−αnβ1xn−x˜un−x˜ ≤αnγξ2 y n −y ˜ 2+ u n − x ˜ 2+αnγ1f1y˜ −A1x˜,un−x˜ +1−αnβ2 x n − x ˜ 2+ u n − x ˜ 2 =αnγξ2yn−y˜2+1−αnβ2xn−x˜2+1−αn(β−γξ)2un−x˜2 +αnγ1f1y˜ −A1x˜,un−x˜,$

which yields that

$un−x˜2≤αnγξ1+αn(β−γξ)yn−y˜2+1−αnβ1+αn(β−γξ)xn−x˜2 +2αn1+αn(β−γξ)γ1f1y˜ −A1x˜,un−x˜.$

From the definition of xn and (3.23), we get

$xn+1−x˜2≤δnx n−x˜2+ηnPCI−λ n(I−S1)x n−x˜2+μnu n−x˜2≤1−μnx n−x˜2+μn[αnγξ1+αn(β−γξ)y n−y˜2+1−αnβ1+αn(β−γξ)x n−x˜2 +2αn1+αn(β−γξ)γ1f1y˜ −A1x˜,un−x˜]=1−μn+ μn 1−αn β 1+αn (β−γξ)x n−x˜2+μnαnγξ1+αn(β−γξ)y n−y˜2 +2μnαn1+αn(β−γξ)γ1f1y˜ −A1x˜,un−x˜=1− μn 1+αn (β−γξ)−μn 1−αn β 1+αn (β−γξ)x n−x˜2+μnαnγξ1+αn(β−γξ)y n−y˜2=1− μn αn (2β−γξ) 1+αn (β−γξ)x n−x˜2+μnαnγξ1+αn(β−γξ)y n−y˜2 +2μnαn1+αn(β−γξ)γ1f1y˜ −A1x˜,un−x˜.$

Similarly, as derived above, we also have

$yn+1−y˜2≤1− μn αn (2β−γξ) 1+αn (β−γξ)yn−y˜2+μnαnγξ1+αn(β−γξ)zn−z˜2 +2μnαn1+αn(β−γξ)γ2f2z˜ −A2y˜,vn−y˜$

and

$zn+1−z˜2≤1− μn αn (2β−γξ) 1+αn (β−γξ)zn−z˜2+μnαnγξ1+αn(β−γξ)xn−x˜2 +2μnαn1+αn(β−γξ)γ3f3x˜ −A3z˜,wn−z˜.$

From (3.24), (3.25) and (3.26), we deduces that

$xn+1−x˜2+yn+1−y˜2+zn+1−z˜2≤1− 2μn αn (β−γξ) 1+αn (β−γξ) xn − x ˜ 2+ yn − y ˜ 2+ zn − z ˜ 2+2μnαn1+αn(β−γξ)(γ1f1y˜ −A1x˜,un−x˜+γ2f2z˜ −A2y˜,vn−y˜+γ3f3x˜ −A3z˜,wn−z˜).$

By (3.21), (3.22), the condition (i) and Lemma 2.4, this implies by (3.27) that the sequences $xn$, $yn$ and $zn$ converge strongly to $x˜=PΩ1(I−A1)x˜+γ1f1 y ˜$, $y˜=PΩ2(I−A2)y˜+γ2f2 z ˜$ and $z˜=PΩ3(I−A3)z˜+γ3f3 x ˜$, respectively. This completes the proof.

The following Corollary is a direct consequence of Theorem 3.1.

Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. For i=1,2,3, let $fi:C→C$ be a contractive mappings with a coefficient ξi and $ξ=maxi∈1,2,3ξi$ and let $Wi:C→C$ be ρi-strictly pseudo-nonspreading mapping with $Fix(Wi)≠∅$. Define a mapping $Tni:C→C$ by $Tnix=1−ωnx+ωnWix,$ for all x ∈ C and i=1,2,3. Let $xn$, $yn$ and $zn$ be sequences generated by $x1,y1,z1∈C$ and

$xn+1=1−μnxn+μnαnf1(yn)+(1−αn)Tn1xn,yn+1=1−μnyn+μnαnf2(zn)+(1−αn)Tn2yn,zn+1=1−μnzn+μnαnf3(xn)+(1−αn)Tn3zn,$

for $n≥1$, where $αn$, $μn⊂(0,1)$ and $ωn⊂(0,1−ρ)$, where $ρ=mini∈{1,2,3}ρi$, satisfying the following conditions:

• (i) $limn→∞αn=0$ and $∑ n=1∞αn=∞$;

• (ii) $0<τ≤μn,≤υ<1$, for some $τ,υ>0$;

• (iii) $∑ n=1∞ωn<∞$;

• (iv) $∑ n=1∞α n+1−αn<∞$, $∑ n=1∞μ n+1−μn<∞$, $∑ n=1∞λ n+1−λn<∞$,$∑ n=1∞ω n+1−ωn<∞$

Then the sequences $xn$, $yn$ and $zn$ converge strongly to $x˜=PFix(W1)f1 y ˜$, $y˜=PFix(W2)f2 z ˜$ and $z˜=PFix(W3)f3 x ˜$, respectively.

Proof. For each i=1,2,3, put $Si≡I$, $γi=1$ and $Ai≡I$. Then, by Theorem 3.1, we obtain the desired result.

### 4. A Numerical Example

In this section, we give a numerical example to support our main theorem.

Example 4.1. For i=1,2,3, let $γ1=3$, $γ2=0.0001$, γ3=7 and the mappings $Ai:[−5,5]→[−5,5]$, $fi:[−5,5]→[−5,5]$, $Si:[−5,5]→[−5,5]$ and $Wi:[−5,5]→[−5,5]$ be defined by

Let $αn=1n0.2+1$, $δn=n+26n+5$, $ηn=3n+26n+5$, $μn=2n+16n+5$, $λn=1n2+100$ and $ωn=1n2+100$ for every $n∈ℕ$. Then, the sequences $xn$, $yn$, $zn$ converge strongly to -5, 0, 5, respectively.

Solution. For every i=1,2,3, it is obvious to check that Si is a 0-strictly pseudo-contractive mapping, where $FixS1={−5}$, $FixS2={0}$, $FixS3={5}$.

Moreover, Wi is a κi-strictly pseudononspreading mapping with

Thus, we get

$Ω1=FixS1∩FixW1={−5}Ω2=FixS2∩FixW2={0}Ω3=FixS3∩FixW3={5}.$

Clearly, all sequences and parameters are satisfied all conditions of Theorem 3.1. Hence, by Theorem 3.1, we can conclude that the sequences $xn$, $yn$, $zn$ converge strongly to -5, 0, 5, respectively.

Table 1 and Figure 1 show the numerical results of sequences $xn$, $yn$ and $xn$ with x1 = 0, y1 = 5, z1 = 0 and n=100.

The values of {xn}, {yn} and {zn} with initial values x1 = 0, y1 = 5, z1 = 0 and n=100..

nxnynzn
10.0000005.0000000.000000
2-0.7967974.5618171.374887
3-1.5537844.1597602.449299
4-2.2623853.7968303.231195
5-2.9239183.4702883.783815
50-5.0000000.1323625.000000
96-5.0000000.0085905.000000
97-5.0000000.0081465.000000
98-5.0000000.0077295.000000
99-5.0000000.0073355.000000
100-5.0000000.0069655.000000

Figure 1. An independent convergence of {xn}, {yn} and {zn} with initial values x1 = 0, y1 = 5, z1 = 0 and n=100.

Remark 4.2. From the above numerical results, we can conclude that Table 1 and Figure 1 show that the sequences $xn$, $yn$ and $zn$ converge independently to $−5∈Ω1$, $0∈Ω2$ and $5∈Ω3$, respectively, and the convergence of $xn$, $yn$, and $zn$ can be guaranteed by Theorem 3.1

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