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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:CC 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:CC be a mapping. Then

(i) a mapping T is called nonexpansive if

TxTyxy,x,yC;

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

TxTy2xy2+κ(IT)x(IT)y2,x,yC.

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:

2TxTy2Txy2+xTy2,x,yC.

In 2009, it is shown by Iemoto and Takahashi [2] that (1.1) is equivalent to the following equation.

TxTy2xy2+2xTx,yTy, for all x,yC.

Later, in 2011, Osilike and Isiogugu [13] proposed a κ-strictly pseudononspreading mapping, that is, a mapping T:CC is said to be a κ-strictly pseudononspreading mapping if there exists κ[0,1) such that

TxTy2xy2+κ(IT)x(IT)y2+2xTx,yTy, for all x,yC.

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:

x0H arbitrary chosen,xn+1=1αnxn+αnTxn,n0,

where C is a nonempty closed convex subset of a normed space, T:CC 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:

x0H arbitrary chosen,yn=βnxn+1βnTxn,xn+1=αnxn+1αnTyn,n0,

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:CC be a nonexpansive mapping such that Fix(S) is nonempty. Let f:CC be a contraction, that is, there exists α(0,1) such that fxfyαxy,x,yC, and let xn be a sequence defined by

x1C arbitrary chosen,xn+1=11+ϵnSxn+ϵn1+ϵnfxn,n,

where εn(0,1) satisfies certain conditions. Then the sequence xn converges strongly to zFix(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:HH be a nonexpansive mapping with Fix(T). Let f:HH be a contractive mapping on H and let xn be generated by

x0H arbitrary chosen,xn+1=IαnATxn+αnγfxn,n0,

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:

(Aγf)x˜,x˜z0,zFix(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 x0C,y0C, let the sequences xn and yn be generated iteratively by

xn+1=1βnxn+βnPCαnfyn+1kαnxn+kTxn,n0,yn+1=1βnyn+βnPCαngxn+1kαnyn+kSyn,n0,

where T:CC is a λ-strict pseudo-contraction, f:CH is a ρ1-contraction and g:CH is a ρ2-contraction, k(0,1λ) 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,z1C, let the sequences xn, yn and zn be defined by

xn+1=1βnxn+βnαnf1yn+1αnSxn,yn+1=1βnyn+βnαnf2xn+1αnTyn,n1,

where S,T:CC, is a nonlinear mapping with Fix(S)FixT, fi:CC is a contractive mapping with coefficients αi;i=1,2 and βn,αn are real sequences in (0,1), n1.

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,z1C, let the sequences xn, yn and zn be defined by

xn+1=δnxn+ηnS1xn+μnPCαnγ1f1(yn)+(IαnA1)T1xn,yn+1=δnyn+ηnS2yn+μnPCαnγ2f2(zn)+(IαnA2)T2yn,zn+1=δnzn+ηnS3zn+μnPCαnγ3f3(xn)+(IαnA3)T3zn,n1,

where Si,Ti:CC, 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:CC 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,n1.

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 SiI, γi=1 and AiI for i=1,2,3, then the iterative method (1.8) reduces to

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

xPCxxy,yC.

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=PCzuz,vu0,vC.

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

PCxPCy2PCxPCy,xy,x,yH.

Lemma 2.2. ([12]) Each Hilbert space H satisfies Opial's condition, i.e., for any sequence xnH with xnx, the inequality

liminfnxnx<liminfnxny

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+(1α)y2=αx2+(1α)y2α(1α)xy2,

(ii) x+y2x2+2y,x+y, for each x,yH.

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

sn+1(1αn)sn+δn,n0,

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

(1) n=1αn=,

(2) limsupnδnαn0 or n=1|δn|<.

Then, limnsn=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<A1. Then IδA1δβ.

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

(i) Fix(T)=VI(C,IT);

(ii) For every u ∈ C and vFix(T),

PC(Iλ(IT))uvuv, for uC,vFix(T) and λ(0,1κ).

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

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

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

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

Txyxy, for every xC and yFix(S).

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

(ii) Let x ∈ H and yFix(S). Then we derive

Txy2=(1λ)(xy)+λ(Sxy)2    =(1λ)xy2+λSxy2λ(1λ)Sxx2    (1λ)xy2+λxy2+κxSx2λ(1λ)Sxx2    =xy2+κλxSx2λ(1λ)Sxx2    =xy2λ((1κ)λ)xSx2    xy2.

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:CC be a contractive mappings with a coefficient ξi and ξ=maxi{1,2,3}ξi, let Si:CC be a κi-strictly pseudo-contractive mapping and Wi:CC be ρi-strictly pseudo-nonspreading mapping with Ωi=Fix(Si)Fix(Wi). For each i=1,2,3, define a mapping Tni:CC by Tnix=1ωnx+ωnWix, for all x ∈ C, and let Ai:CC 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,z1C and

xn+1=δnxn+ηnPCIλnIS1xn  +μnPCαnγ1f1(yn)+(IαnA1)Tn1xn,yn+1=δnyn+ηnPCIλnIS2yn  +μnPCαnγ2f2(zn)+(IαnA2)Tn2yn,zn+1=δnzn+ηnPCIλnIS3zn  +μnPCαnγ3f3(xn)+(IαnA3)Tn3zn,

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

(i) limnαn=0 and n=1αn=;

(ii) 0<τδn,ηn,μn,υ<1, for some τ,υ>0;

(iii) n=1λn<, n=1ωn<;

(iv) n=1α n+1αn<, n=1δ n+1δn<, n=1μ n+1μn<,

n=1η n+1ηn<, n=1λ n+1λn<, n=1ω n+1ωn<.

Then the sequences xn, yn and zn converge strongly to x˜=PΩ1(IA1)x˜+γf1 y ˜ , y˜=PΩ2(IA2)y˜+γf2 z ˜ and z˜=PΩ3(IA3)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 αn0 as n, without loss of generality, we may assume that αn<1Ai, for all i=1,2,3 and 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+1x*δnxnx*+ηnPCIλnIS1xnx*+μnPCαnγ1f1(yn)+(IαnA1)Tn1xnx*δnxnx*+ηnPCIλnIS1xnx*+μnPCαnγ1f1(yn)+(IαnA1)Tn1xnx*1μnxnx*+μnαnγ1f1ynA1x*+IαnA1Tn1xnx*1μnxnx*+μnαnγ1ξ1yny*+αnγ1f1y*A1x*+1αnβxnx*1μnαnβxnx*+μnαnγξyny*+μnαnγ1f1y*A1x*.

Similarly, we get

yn+1y*1μnαnβyny*+μnαnγξznz*+μnαnγ2f2z*A2y*

and

zn+1z*1μnαnβznz*+μnαnγξxnx*+μnαnγ3f3x*A3z*.

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

xn+1x*+yn+1y*+zn+1z*1μnαnβγξxnx*+yny*+znz*+μnαnγ1f1y*A1x*+γ2f2z*A2y*+γ3f3x*A3z*.

By induction, we can derive that

xnx*+yny*+znz*max{x1x*+y1y*+z1z*,γ1f1y*A1x*+γ2f2z*A2y*+γ3f3x*A3z*βγξ},

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

Step 2. Claim that limnxn+1xn=0.

First, we let

un=PCαnγ1f1(yn)+(IαnA1)Tn1xn, vn=PCαnγ2f2(zn)+(IαnA2)Tn2yn

and

wn=PCαnγ3f3(xn)+(IαnA3)Tn3zn.

Then, observe that

unun1=PCαnγ1f1yn+IαnA1Tn1xnPCαn1γ1f1yn1+Iαn1A1Tn11xn1αnγ1f1ynf1yn1+γ1αnαn1f1yn1+IαnA1Tn1xnTn11xn1+IαnA1Tn11xn1Iαn1A1Tn11xn1αnγ1ξ1ynyn1+γ1αnαn1f1yn1+1αnβ1(1ωnxnxn1+ωnωn1xn1+ωnW1xnW1xn1+ωnωn1W1xn1)+αnαn1A1Tn1xn1αnγξynyn1+αnαn1γf1yn1+A1Tn1xn1+1αnβ(1ωnxnxn1+ωnωn1xn1+W1xn1+ωnW1xnW1xn1).

By the definition of xn, we obtain

xn+1xnδnxnxn1+δnδn1xn1+ηnPCIλn(IS1)xnPCIλn1(IS1)xn1+ηnηn1PCIλn1(IS1)xn1+μnunun1+μnμn1un1δnxnxn1+δnδn1xn1+ηnxnxn1+ηnλn(IS1)xnλn1(IS1)xn1+ηnηn1PCIλn1(IS1)xn1+μn[αnγξynyn1+αnαn1γf1yn1+A1Tn1xn1+1αnβ(1ωnxnxn1+ωnωn1xn1+W1xn1+ωnW1xnW1xn1)]+μnμn1un1 1μnxnxn1+δnδn1xn1+ηnλn(IS1)xn(IS1)xn1+ηnλnλn1(IS1)xn1+ηnηn1PCIλn1(IS1)xn1+μnαnγξynyn1+αnαn1γf1yn1+A1Tn1xn1+μn1αnβxnxn1+ωnωn1xn1+W1xn1+ωnW1xnW1xn1+μnμn1un1 1μnαnβxnxn1+μnαnγξynyn1+δnδn1xn1+λn(IS1)xn(IS1)xn1+λnλn1(IS1)xn1+ηnηn1PCIλn1(IS1)xn1+αnαn1γf1yn1+A1Tn1xn1+ωnωn1xn1+W1xn1+ωnW1xnW1xn1+μnμn1un1.

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

yn+1yn1μnαnβynyn1+μnαnγξznzn1+δnδn1yn1+λn(IS2)yn(IS2)yn1+λnλn1(IS2)yn1+ηnηn1PCIλn1(IS2)yn1+αnαn1γf2zn1+A2Tn2yn1+ωnωn1yn1+W2yn1+ωnW2ynW2yn1+μnμn1vn1

and

zn+1zn1μnαnβznzn1+μnαnγξxnxn1+δnδn1zn1+λn(IS3)zn(IS3)zn1+λnλn1(IS3)zn1+ηnηn1PCIλn1(IS3)zn1+αnαn1γf3xn1+A3Tn3zn1+ωnωn1zn1+W3zn1+ωnW3znW3zn1+μnμn1wn1.

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

xn+1xn+yn+1yn+zn+1zn1μnαnβγξxnxn1+ynyn1+znzn1+δnδn1xn1+yn1+zn1+λn((IS1)xn(IS1)xn1+(IS2)yn(IS2)yn1+(IS3)zn(IS3)zn1)+λnλn1(IS1)xn1+(IS2)yn1+(IS3)zn1+ηnηn1(PCIλn1(IS1)xn1+PCIλn1(IS2)yn1 +PCIλn1(IS3)zn1)+αnαn1(γ(f1xn1+f2yn1+f3zn1)+A1Tn1xn1+A2Tn2yn1+A3Tn3zn1)+ωnωn1(xn1+yn1+zn1+W1xn1+W2yn1+W3zn1)+ωnW1xnW1xn1+W2ynW2yn1+W3znW3zn1+μnμn1un1+vn1+wn1.

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

xn+1xn0,yn+1yn0 and zn+1zn0 as n.

Step 3. Prove that limnunPCIλn(IS1)un=limnunTn1un=0.

To show this, take u˜n=αnγ1f1(yn)+(IαnA1)Tn1xn. Then we derive that

xn+1x*2=δnxnx*+ηnPCIλn(IS1)xnx*+μnunx*2δnxnx*2+ηnPCIλn(IS1)xnx*2+μnunx*2δnηnxnPCIλn(IS1)xn21μnxnx*2+μnαnγ1f1ynA1Tn1xn+Tn1xnx*2δnηnxnPCIλn(IS1)xn21μnxnx*2+μn[Tn1xnx*2+2αnγ1f1 y n A1Tn1xn,u˜ nx*]δnηnxnPCIλn(IS1)xn2xnx*2+2μnαnγ1f1 y n A1Tn1xnu˜ nx*δnηnxnPCIλn(IS1)xn2,

which implies that

δnηn x n PC Iλ n(IS1)x n 2 x n x* 2 xn+1 x* 2+2μnαnγ1f1 yn A1Tn1xnu˜ nx*xnx n+1 xn x * + x n+1 x * +2μnαnγ1f1 yn A1Tn1xnu˜ nx*.

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

xnPCIλn(IS1)xn0 as n.

Observe that

xn+1xn=ηnPCIλn (IS1 )xnxn+μnunxn.

This follows that

μnunxnηnPCIλn (IS1 )xnxn+xn+1xn.

From (3.8) and (3.9), we obtain

unxn0 as n.

Observe that

unPCIλn(IS1)ununxn+xnPCIλn(IS1)xn+PCIλn(IS1)xnPCIλn(IS1)un2unxn+xnPCIλn(IS1)xn+λnIS1xnIS1un.

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

unPCIλn(IS1)un0 as n.

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

vnPCIλn(IS2)vn0 and wnPCIλn(IS3)wn0 as n.

Consider

xn+1unxn+1xn+xnun,

by (3.8) and (3.10), we have

xn+1un0 as n.

Since

xnTn1xnxnxn+1+xn+1un+unTn1xnxnxn+1+xn+1un+ u˜nTn1xn=xnxn+1+xn+1un+αnγ1f1ynA1Tn1xn,

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

xnTn1xn0 as n.

Consider

unTn1ununxn+xnTn1xn+Tn1xnTn1un2unxn+xnTn1xn+ωnW1xnW1un.

Therefore, by (3.10), (3.14) and the condition (iii), we have

unTn1un0 as n.

Applying the same method as (3.15), we also have

vnTn2vn0 and wnTn3wn0 as n.

Step 4. Claim that

limsupnγ1f1y˜A1x˜,unx˜0, where     x˜=PΩ1(IA1)x˜+γ1f1y˜.

First, take a subsequence unk of un such that

limsupnγ1f1y˜A1x˜,unx˜=limkγ1f1y˜A1x˜,unkx˜.

Since xn is bounded, then we can assume that xnkx^ as k. From (3.10), we obtain unkx^ as k.

Next, assume x^FixS1. Since FixS1=FixPCIλnk(IS1), then we get x^PCIλnk(IS1)x^.

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

liminfkunkx^<liminfkunkPCIλnk(IS1)x^  liminfk[unkPCIλnk(IS1)unk  +PCIλnk(IS1)unkPCIλnk(IS1)x^]  liminfk[unkPCIλnk(IS1)unk+unkx^  +λnk(IS1)unk(IS1)x^]  =liminfkunkx^.

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

liminfkunkx^<liminfkunkTnk1x^      liminfkunkTnk1unk+Tnk1unkTnk1x^      liminfk[unkTnk1unk+unkx^      +ωnkIW1unkIW1x^]      =liminfkunkx^.

This is a contradiction. Thus we obtain

x^FixW1.

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

x^Ω1=FixS1FixW1.

Since xnkx^ as k, (3.20) and Lemma 2.1, we can derive that

limsupnγ1f1y˜A1x˜,unx˜=limkγ1f1y˜A1x˜,unkx˜            =γ1f1y˜A1x˜,x^x˜            =γ1f1y˜A1x˜+x˜x˜,x^x˜            0.

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

limsupnγ2f2z˜A2y˜,vny˜0 and limsupnγ3f3x˜A3z˜,wnz˜0.

Step 5. Finally, Prove that the sequence xn, yn and zn converge strongly to x˜=PΩ1(IA1)x˜+γ1f1 y ˜ , y˜=PΩ2(IA2)y˜+γ2f2 z ˜ and z˜=PΩ3(IA3)z˜+γ3f3 x ˜ , respectively.

By firmly-nonexpansiveness of PC, we derive that

unx˜2=PCun˜x˜2   u n ˜x˜,unx˜  =αn γ1 f1 y n A1 x ˜ + Iα n A1 T n1 x n x ˜ ,unx˜  =αnγ1f1 y n A1x˜,unx˜+ Iα n A1 T n1 x n x ˜ ,unx˜  αnγ1f1 y n f1y˜ ,unx˜+αnγ1f1y˜ A1x˜,unx˜  + Iα n A1 T n1 x n x ˜ unx˜  αnγ1ξ1yny˜unx˜+αnγ1f1y˜ A1x˜,unx˜  +1αnβ1xnx˜unx˜  αnγξ2 y n y ˜ 2+ u n x ˜ 2+αnγ1f1y˜ A1x˜,unx˜  +1αnβ2 x n x ˜ 2+ u n x ˜ 2  =αnγξ2yny˜2+1αnβ2xnx˜2+1αn(βγξ)2unx˜2  +αnγ1f1y˜ A1x˜,unx˜,

which yields that

unx˜2αnγξ1+αn(βγξ)yny˜2+1αnβ1+αn(βγξ)xnx˜2    +2αn1+αn(βγξ)γ1f1y˜ A1x˜,unx˜.

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

xn+1x˜2δnx nx˜2+ηnPCIλ n(IS1)x nx˜2+μnu nx˜21μnx nx˜2+μn[αnγξ1+αn(βγξ)y ny˜2+1αnβ1+αn(βγξ)x nx˜2+2αn1+αn(βγξ)γ1f1y˜ A1x˜,unx˜]=1μn+ μn 1αn β 1+αn (βγξ)x nx˜2+μnαnγξ1+αn(βγξ)y ny˜2+2μnαn1+αn(βγξ)γ1f1y˜ A1x˜,unx˜=1 μn 1+αn (βγξ)μn 1αn β 1+αn (βγξ)x nx˜2+μnαnγξ1+αn(βγξ)y ny˜2=1 μn αn (2βγξ) 1+αn (βγξ)x nx˜2+μnαnγξ1+αn(βγξ)y ny˜2+2μnαn1+αn(βγξ)γ1f1y˜ A1x˜,unx˜.

Similarly, as derived above, we also have

yn+1y˜21 μn αn (2βγξ) 1+αn (βγξ)yny˜2+μnαnγξ1+αn(βγξ)znz˜2+2μnαn1+αn(βγξ)γ2f2z˜ A2y˜,vny˜

and

zn+1z˜21 μn αn (2βγξ) 1+αn (βγξ)znz˜2+μnαnγξ1+αn(βγξ)xnx˜2+2μnαn1+αn(βγξ)γ3f3x˜ A3z˜,wnz˜.

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

xn+1x˜2+yn+1y˜2+zn+1z˜21 2μn αn (βγξ) 1+αn (βγξ) xn x ˜ 2+ yn y ˜ 2+ zn z ˜ 2+2μnαn1+αn(βγξ)(γ1f1y˜ A1x˜,unx˜+γ2f2z˜ A2y˜,vny˜+γ3f3x˜ A3z˜,wnz˜).

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(IA1)x˜+γ1f1 y ˜ , y˜=PΩ2(IA2)y˜+γ2f2 z ˜ and z˜=PΩ3(IA3)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:CC be a contractive mappings with a coefficient ξi and ξ=maxi1,2,3ξi and let Wi:CC be ρi-strictly pseudo-nonspreading mapping with Fix(Wi). Define a mapping Tni:CC by Tnix=1ωnx+ωnWix, for all x ∈ C and i=1,2,3. Let xn, yn and zn be sequences generated by x1,y1,z1C and

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

for n1, where αn, μn(0,1) and ωn(0,1ρ), where ρ=mini{1,2,3}ρi, satisfying the following conditions:

• (i) limnαn=0 and n=1αn=;

• (ii) 0<τμn,υ<1, for some τ,υ>0;

• (iii) n=1ωn<;

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

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 SiI, γi=1 and AiI. 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

A1x=2x5, A2x=3x5, A3x=4x5, f1x=x10050, f2x=x+9520, f3x=x+5035,S1x=x103, S2x=x10, S3x=x+52, W1x=x256, W2x=7x8, if 0x5, x, if 5x<0, W3x=x+154, for all x[5,5]

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. 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

FixW1={5}, FixW2={0}, if 0x5,{x}, if 5x<0, FixW3={5}.

Thus, we get

Ω1=FixS1FixW1={5}Ω2=FixS2FixW2={0}Ω3=FixS3FixW3={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 5Ω1, 0Ω2 and 5Ω3, respectively, and the convergence of xn, yn, and zn can be guaranteed by Theorem 3.1

1. 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).
2. 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.
3. S. Ishikawa, Fixed point by a new iterative method, Proc. Am. Math. Soc., 44(1974), 147-150.
4. 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.
5. W. Khuangsatung and A. Kangtunyakarn, Algorithm of a new variational inclusion problem and strictly pseudononspreding mapping with application, Fixed Point Theory Appl., 30+(2014).
6. 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.
7. 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.
8. 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).
9. W. R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc., 4(1953), 506-510.
10. G. Marino and H. K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318(2006), 43-52.
11. A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241(2000), 46-55.
12. Z. Opial, Weak convergence of the sequence of successive approximation of nonexpansive mappings, Bull. Amer. Math. Soc., 73(1967), 591-597.
13. 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.
14. 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.
15. S. Suwannaut, The S-intermixed iterative method for equilibrium problems, Thai J. Math., 17(2019), 60-74.
16. W. Takahashi. Nonlinear Functional Analysis. Yokohama: Yokohama Publishers; 2000.
17. H. K. Xu, An iterative approach to quadric optimization, J. Optim Theory Appl., 116(2003), 659-678.
18. 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).