# An Iterative Method for Finding Common Solution of the Fixed Point Problem of a Finite Family of Nonexpansive Mappings and a Finite Family of Variational Inequality Problems in Hilbert Space

Authors: Shamshad Husain; Nisha Singh
Journal: Journal of Applied Mathematics
 (2019)

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## Abstract

In this paper, a hybrid iterative algorithm is proposed for finding a common element of the set of common fixed points of finite family of nonexpansive mappings and the set of common solutions of the variational inequality for an inverse strongly monotone mapping on the real Hilbert space. We establish the strong convergence of the proposed method for approximating a common element of the above defined sets under some suitable conditions. The results presented in this paper extend and improve some well-known corresponding results in the earlier and recent literature.


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## Body

## 1. Introduction

Throughout, letV be a real Hilbert space whose inner product and norm are denoted as 〈·,·〉 and ·, respectively. Let W be a nonempty closed convex subset of V and PW be the metric projection of V onto W. Let S:W→W be a nonexpansive mapping if(1)Ss-St≤s-t∀s,t∈W. The set of fixed point of S is denoted by FixS.LetG:W→V be a nonlinear mapping. The classical variational inequality problem denoted by VIP associated with the set W is to find t∈W such that(2)Gt,s-t≥0,∀s∈W.The solution set of the VIP (2) is denoted by VI(W,G). The VIP is a fundamental problem in variational analysis which has been extensively studied by many researchers in the past decades, see Yao and Chadli [1], Zeng, Schaible, and Yao [2], and the references therein.TheVIP (2) was first discussed by Lions [3], further many different approaches are given to solve (2) in finite dimensional and infinite dimensional spaces, and the research in this direction is still continued.Korpelevich [4] proposed a modification of an iterative algorithm for solving the VIP (2) in Euclidean space Rn:(3)tp=PWsp-ςGsp,sp+1=PWsp-ςGtp,∀p≥0,with ς>0, called the extragradient algorithm under certain assumptions, and it has received great attention. He proved that the sequences {sp} and {tp} converge strongly.Takahashi and Toyoda [5] introduced an iterative algorithm for finding an element of F(S)∩VI(F,W) under the assumption that S is nonexpansive and F is inverse strongly monotone as(4)s1=s∈Wchoosenarbitrarlilysp+1=apsp+1-apSPWsp-ηpFsp,p≥1 where ap is a sequence in (0,1) and ηp is a sequence in (0,2a).Iiduka and Takahashi [6] proposed another iterative algorithm as(5)s1=s∈Wchoosenarbitrarlilysp+1=aps+1-apSPWsp-ηpFsp,p≥1Marino and Xu [7] introduced the following iterative algorithm based on the viscosity approximation method:(6)sp+1=I-apGSsp+apζgsp,p≥0, where G is a strongly positive bounded linear operator on V. They proved that if the sequence {ap} satisfies appropriate conditions, then the sequence {sp} generated by (4) converges strongly to the unique solution of the variational inequality(7)G-ζgs∗,s-s∗≥0,s∈W,Qin and Cho [8] introduced an extended composite iterative algorithm {sp} defined as follows:(8)s1=s∈W,choosenarbitrarilyzp=ζpsp+1-ζpSsp,tp=bpsp+1-bpSzp,sp+1=apζgsp+cpsp+1-cpI-apGtp,p≥1,where g is a contraction, S is a nonexpansive mapping, and G is a strongly positive linear bounded self-adjoint operator. Under certain appropriate assumptions on the parameters, {sp} defined by (8) converges strongly to a fixed point of S, which solves the variational inequality (7).Yamada [9] considered the hybrid steepest descent method for solving the VIP over the set of fixed points of a nonexpansive mapping. Moudafi [10] proposed the viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping which is also a solution of a variational inequality problem.In this paper, we introduce and analyze an iterative algorithm by combining Korpelevich’s extragradient method, viscosity approximation method, and hybrid steepest method. We prove that under certain conditions the proposed algorithm converges strongly to a common element ofs∈F≔(⋂l=1MFix(Sl))∩(⋂l=1MVI(Fl,W)), which is a unique solution of variational inequality problem.


## 2. Preliminaries

LetV be a real Hilbert space and W be a nonempty closed convex subset of V. For every point s∈V, there exists a unique nearest point in W, denoted by PWs, such that (9)s-PWs≤s-t,∀t∈W.PW is called the metric projection of V onto W. It is well known that PW is a nonexpansive mapping of V onto W and satisfies(10)s-t,PWs-PWt≥PWs-PWt2 for every s,t∈V. It is easy to see that (11)u∈VIF,W⇔u=PWu-λFu,λ>0.Definition 1.
A mappingA:W→V is said to be
(1) monotone, if(12)As-At,s-t≥0,∀s,t∈W;
(2)L-Lipschitz, if there exists a constant L>0 such that(13)As-At≤Ls-t,∀s,t∈W;
(3)α-inverse strongly monotone, if there exists a positive real number α such that(14)As-At,s-t≥αAs-At2,∀s,t∈W.Remark 2.
Anyα-inverse strongly monotone mapping A is monotone and (1/α)-Lipschitz continuous.Definition 3.
A mappingg:W→W is a contraction if there exists a constant τ∈(0,1) such that(15)gs-gt≤τs-t,∀s,t∈W.Lemma 4 (see [7]).
Assume thatA is a strongly positive linear bounded self-adjoint operator on a Hilbert space V with coefficient γ¯>0 and 0<ρ≤A-1. Then I-ρA≤1-ργ¯.Definition 5.
A mappingS:W→V is said to be k-strictly pseudocontractive mapping if there exists k∈[0,1) such that (16)Ss-St2≤s-t2+kI-Ss-I-St2,∀s,t∈W.Lemma 6 (see [11]).
LetV be a Hilbert space and let W be a closed convex subset of V. For any integer N≥1, assume that for each 1≤i≤N,Si:W→V is a ki-strictly pseudocontractive mapping for some 0≤ki<1. Assume that {φi}i=1N is a positive sequence such that ∑i=1N‍φi=1. Then ∑i=1N‍φiSi is a k-strictly pseudocontractive mapping with k=max⁡{ki:1≤i≤N}.Lemma 7 (see [11]).
Let{Si}i=1N and {φi}i=1N be as in Lemma 6. Suppose that {Si}i=1N has a common fixed point in W. Then F(∑i=1N‍φiSi)=⋂i=1NF(Si).Lemma 8 (see [12]).
Assume{an} is a sequence of nonnegative real numbers such that (17)an+1≤1-vnan+δn, where {vn} is a sequence in (0,1) and δn is a sequence such that (i)
∑n=1∞‍vn=∞;(ii)
limsupn→∞δn/vn≤0or∑n=1∞‍δn<∞.
Then,limn→∞an=0.Lemma 9 (see [13] demiclosed principle).
LetW be a nonempty closed convex subset of a real Hilbert space V. Let S be a nonexpansive self-mapping on V with Fix(S)≠∅. Then I-S is demiclosed. That is, whenever {sp} is a sequence in V weakly converging to some s∈V and the sequence {(I-S)sp} strongly converges to some t, it follows that (I-S)s=t. Here I is the identity operator of V.Rockafellar [14] defined set-valued mapping T:V→2V to be called monotone if for all s,t∈V,f∈Ts and g∈Tt imply (18)s-t,f-g≥0. A set-valued mapping T:V→2V is maximal monotone if the graph of H(T) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (s,f)∈V×V, 〈s-t,f-g〉≥0 for every (t,g)∈H(T) implies f∈Ts.LetF:W→V be a monotone mapping and let NWv be the normal cone to W at v∈W, that is (19)NWv=w∈V:u-v,w≥0,∀u∈W. Define(20)Tv=Fv+NWv,v∈W∅,v∉W. Then T is the maximal monotone and (21)0∈Tvifandonlyifv∈VIF,W.


## 3. Main Results

Algorithm 10.
LetW be a closed convex subset of a real Hilbert space V, let g:W→W be a ξ-contraction with coefficient ξ∈[0,1], G be strongly positive linear bounded self-adjoint operator, Fl:W→V be a ςl-inverse strongly monotone mappings for each 1≤l≤M, where M is some positive integer, and Sl:W→W be a finite family of nonexpansive mappings for all 1≤l≤M. Let {bp,l} for all 1≤l≤M,{ap} and {cp} be the sequences in [0,1]. For an arbitrarily given s1=s∈W, we propose the following hybrid iterative algorithm:(22)tp,1=bp,1S1sp+1-bp,1sp,tp,l=bp,lSlsp+1-bp,ltp,l-1,l=1,2,…,Msp+1=apζgsp+cpsp+1-cpI-apGPWtp,l-θlFltp,l,∀p≥1.The sequence {sp} defined by (22) converges strongly to a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of the variational inequalities for an inverse strongly monotone mapping which solves the variational inequality problem.Theorem 11.
LetW be a closed convex subset of a real Hilbert space V, let g:W→W be a ζ-contraction, let Fl:W→V be a ςl-inverse strongly monotone mapping, let Sl:W→W be a finite family of nonexpansive mappings with F≔(⋂l=1MFix(Sl))∩(⋂l=1MVI(Fl,W))≠∅, and let G be a strongly positive linear bounded self-adjoint operator with the coefficient ζ¯>0. Assume that 0<ζ≤ζ¯/ς and let {θl} be real numbers in (0,2ςl). Let {ap},{bp,l} and {cp} be sequences in (0,1) satisfying the following assumptions: (i)
limp→∞ap=0 and ∑p=1∞‍ap=∞.(ii)
∑p=1∞‍ap-ap-1<∞ and ∑p=1∞‍cp-cp-1<∞.(iii)
∑p=1∞‍|bp,l-bp-1,l|<∞ for each l=1,2,…,M.(iv)
0<liminfp→∞bp,l≤limsupp→∞bp,l<1.
Then the sequence{sp} defined by (22) converges strongly to t∈F, where t=PF(ζg+(I-G))t is also the unique solution of the following variational inequality:(23)ζgt-Gq,q-t≤0,∀q∈F.Proof.
For anys,t∈W and θl∈(0,2ςl), we have(24)I-θlFls-I-θlFlt2=s-t-θlFls-Flt2=s-t2-2θlFls-Flt,s-t+θl2Fls-Flt2≤s-t2-θl2ςl-θlFls-Flt2≤s-t2.From (24), (I-θlFl) is nonexpansive. Since G is a linear bounded self-adjoint operator, so (25)G=sup⁡Gs,s:s∈V,s=1. We assume that (26)Gap<1-cp∀p≥0, and since limp→∞ap=0, so without loss of generality, we have (27)1-cpI-apGs,s=1-cp-apGs,s≤1-cp-apG≤0, so (1-cp)I-apG is positive. Now, it follows that (28)1-cpI-apG=sup1-cpI-apGs,s:s∈V,s=1=sup1-cp-apGs,s:s∈V,s=1≤1-cp-apζ¯.Next, we prove that the sequence {sp} is bounded. Fixing s∗∈F, we have (29)tp,1-s∗=bp,1S1sp+1-bp,1sp-s∗≤bp,1sp-s∗+1-bp,1sp-s∗≤sp-s∗. By induction from l=2 to l=M, we have (30)tp,l-s∗=bp,lSlsp+1-bp,ltp,l-1-s∗≤bp,lsp-s∗+1-bp,ltp,l-1-s∗≤sp-s∗. Thus, for every l=1,2,…,M, we get(31)tp,l-s∗≤sp-s∗. From (22) and (31), we have (32)sp+1-s∗=apζgsp+cpsp+1-cpI-apGPWtp,l-θlFltp,l-s∗=apζgsp-Gs∗+cpsp-s∗+1-cpI-apGPWtp,l-θlFltp,l-s∗≤apζgsp-Gs∗+cpsp-s∗+1-cp-apζ¯tp,l-s∗≤apζgsp-gs∗+apζgs∗-Gs∗+cpsp-s∗+1-cp-apζ¯tp,l-s∗=apζτsp-s∗+apζgs∗-Gs∗+cpsp-s∗+1-cp-apζ¯sp-s∗≤1-apζ¯-ζτsp-s∗+apζgs∗-Gs∗=1-apζ¯-ζτsp-s∗+apζ¯-ζτζ¯-ζτζgs∗-Gs∗≤max⁡sp-s∗,1ζ¯-ζτζgs∗-Gs∗. By induction on p, we have (33)sp-s∗≤max⁡so-s∗,1ζ¯-ζτζgs∗-Gs∗,∀p≥0.Hence, the given sequence {sp} is bounded and so is {tp,l} for all l=1,2,…,M.
Next, we show that(34)limp→∞sp+1-sp=0.From the definition of the sequence {tp,l} for each l=1,2,…,M in Algorithm 10, we have(35)tp,l-tp-1,l=bp,lSlsp+1-bp,ltp,l-1-bp-1,lSlsp-1-1-bp-1,ltp-1,l-1=bp,lSlsp-Slsp-1+bp,l-bp-1,lSlsp-1+1-bp,ltp,l-1-tp-1,l-1+bp-1,l-bp,ltp-1,l-1≤bp,lsp-sp-1+1-bp,ltp,l-1-tp-1,l-1+bp,l-bp-1,lSlsp-1+tp-1,l-1.When l=1, we have(36)tp,1-tp-1,1=bp,1S1sp+1-bp,1sp-bp-1,1S1sp-1-1-bp-1,1sp-1=bp,1S1sp-S1sp-1+bp,1-bp-1,1S1sp-1+1-bp,1sp-sp-1+bp-1,1-bp,1sp-1≤bp,1sp-sp-1+1-bp,1sp-sp-1+bp,1-bp-1,1S1sp-1+sp-1≤sp-sp-1+bp,1-bp-1,1S1sp-1+sp-1.From (35) and (36), for l=2,...,M, we get (37)tp,l-tp-1,l≤sp-sp-1+∑i=2lSisp-1-tp-1,i-1bp,i-bp-1,i+S1sp-1+sp-1bp,1-bp-1,1. Next, (38)sp+1-sp=apζgsp+cpsp+1-cpI-apGPWtp,l-θlFltp,l-ap-1ζgsp-1-cp-1sp-1-1-cp-1I-ap-1GPWtp-1,l-θlFltp-1,l≤apζgsp-gsp-1+ap-ap-1ζgsp-1+cpsp-sp-1+cp-cp-1sp-1+1-cpI-apGPWtp,l-θlFltp,l-PWtp-1,l-θlFltp-1,l+1-cpI-apGPWtp-1,l-θlFltp-1,l-1-cp-1I-ap-1GPWtp-1,l-θlFltp-1,l≤apζτsp-sp-1+ap-ap-1ζgsp-1+cpsp-sp-1+cp-cp-1sp-1+1-cp-apζ¯PWtp,l-θlFltp,l-PWtp-1,l-θlFltp-1,l+cp-cp-1PWtp-1,l-θlFltp-1,l+ap-ap-1GPWtp-1,l-θlFltp-1,l≤apζτsp-sp-1+cpsp-sp-1+1-cp-apζ¯tp,l-tp-1,l+ap-ap-1ζgsp-1+GPWtp-1,l-θlFltp-1,l+cp-cp-1sp-1+PWtp-1,l-θlFltp-1,l≤apζτsp-sp-1+cpsp-sp-1+1-cp-apζ¯sp-sp-1+∑i=2lSisp-1-tp-1,i-1bp,i-bp-1,i+S1sp-1+sp-1bp,1-bp-1,1+ap-ap-1(ζgsp-1+GPWtp-1,l-θlFltp-1,l+cp-cp-1sp-1+PWtp-1,l-θlFltp-1,l≤1-apζ¯-ζτsp-sp-1+M1∑i=1lbp,i-bp,l,i+ap-ap-1+cp-cp-1 where (39)M1=max⁡supp≥1∑i=2lSisp-tp,i-1+Slsp-1+tp-1,l-1,supp≥1ζgsp-1+GPWtp-1,l-θlFltp-1,l,supp≥1sp-1+PWtp-1,l-θlFltp-1,l. From conditions (i), (ii), and (iii), we have (40)limp→∞sp+1-sp=0. Setting wp,l=PWtp,l-θlFltp,l, we have (41)sp+1=apζgsp+cpsp+1-cpI-apGwp,l. Next for s∗∈F, we have(42)
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,where (43)dp=ap2ζgsp-Gs∗2+2cpapsp-s∗,ζgsp-Gs∗+2ap1-cpI-apGwp,l-s∗,ζgsp-Gs∗. From (42), we get(44)-θlθl-2ςlFltp,l-Fls∗2≤sp-s∗2-sp+1-s∗2+dp≤sp-sp+1sp-s∗+sp+1-s∗+dp.From condition (i), we get(45)limp→∞dp=0.From (34), (44), and (45), we get(46)limp→∞Fltp,l-Fls∗=0∀l=1,2,…,M.Now, (47)sp-wp,l=sp-sp+1+sp+1-wp,l=sp-sp+1+apζgsp+cpsp+1-cpI-apGwp,l-wp,l=sp-sp+1+apζgsp-Gwp,l+cpsp-wp,l.Further, (48)1-cpsp-wp,l=sp-sp+1+apζgsp-Gwp,l.So, we get (49)1-cpsp-wp,l≤sp-sp+1+apζgsp-Gwp,l.From condition (i) and (34), we have(50)limp→∞sp-wp,l=0.Now from (10), we have (51)wp,l-s∗2=PWtp,l-θlFltp,l-PWs∗-θlFls∗2≤tp,l-θlFltp,l-s∗-θlFls∗,wp,l-s∗=12tp,l-θlFltp,l-s∗-θlFls∗2+wp,l-s∗2-tp,l-θlFltp,l-s∗-θlFls∗-wp,l-s∗2≤12tp,l-s∗2+wp,l-s∗2-tp,l-wp,l-θlFltp,l-Fls∗2=12tp,l-s∗2+wp,l-s∗2-tp,l-wp,l2+2θltp,l-wp,l,Fltp,l-Fls∗-θl2Fltp,l-Fls∗2≤tp,l-s∗2-tp,l-wp,l2+2θltp,l-wp,l,Fltp,l-Fls∗-θl2Fltp,l-Fls∗2. Further, (52)sp+1-s∗2≤1-apζ¯1-cp-apζ¯wp,l-s∗2+1-apζ¯cpsp-s∗2+dp≤1-apζ¯1-cp-apγ¯tp,l-s∗2-tp,l-wp,l2+2θltp,l-wp,l,Fltp,l-Fls∗-θl2Fltp,l-Fls∗2+1-apζ¯cpsp-s∗2+dp≤1-apζ¯sp-s∗2-1-apζ¯1-cp-apζ¯tp,l-wp,l2+2θl1-apζ¯1-cp-apζ¯tp,l-wp,lFltp,l-Fls∗-θl21-apζ¯1-cp-apζ¯Fltp,l-Fls∗2+dp.Further, we have (53)1-apζ¯1-cp-apζ¯tp,l-wp,l2≤sp-s∗2-sp+1-s∗2+2θl1-apζ¯1-cp-apζ¯tp,l-wp,lFltp,l-Fls∗-θl21-apζ¯1-cp-apζ¯Fltp,l-Fls∗2+dp≤sp-sp+1sp-s∗+sp+1-s∗+2θl1-apζ¯1-cp-apζ¯tp,l-wp,lFltp,l-Fls∗-θl21-apζ¯1-cp-apζ¯Fltp,l-Fls∗2+dp. Using (34), (45), and (46), we get(54)limp→∞tp,l-wp,l=0. From (50) and (54), we get (55)sp-tp,l≤sp-wp,l+wp,l-tp,l→0asp→∞.Since PF(ζg+(I-G)) is a contraction. So, by Banach Contraction Principle, PF(ζg+(I-G)) has a unique fixed point, say t∈V such that t=PF(ζg+(I-G))(t). Next, we have to show that (56)limp→∞sup⁡ζgt-Gt,wp,l-t≤0.So, we choose a subsequence {wpk,l} of {wp,l} such that (57)limp→∞sup⁡ζgt-Gt,wp,l-t=limk→∞ζgt-Gt,wpk,l-t.Since {wpk,l} is bounded, so there exists a subsequence {wpkm,l} of {wpk,l} which converges weakly to q∈W. Now without loss of generality, we assume that wpk,l⇀q. Therefore, we have(58)limp→∞sup⁡ζgt-Gt,wp,l-t=limk→∞ζgt-Gt,wpk,l-t=ζgt-Gt,q-t.Now, we have to show that q∈F=(⋂l=1MFix(Sl))∩(⋂l=1MVI(Fl,W))). First, we show that (59)q∈⋂l=1MFixSl=FixS.By using Lemma 9 (Demiclosed Principle) we have q∈Fix(Sl) for each l=1,2,…,M, i.e., q∈⋂l=1MFix(Sl)=Fix(S).
Next, we show thatq∈VI(Fl,W). Let T be the maximal monotone mapping defined as (60)Tv=Fv+NWv,v∈W;∅,v∉W.For any given (v,w)∈H(T), we have w-Flv∈NW(v).
Sincewp,l∈W, we have (61)v-wp,l,w-Flv≥0.Since wp,l=PW(tp,l-θlFltp,l), we have (62)v-wp,l,wp,l-tp,l-θlFltp,l≥0v-wp,l,wp,l-tp,lθl+Fltp,l≥0.So, we have (63)v-wpk,l,w≥v-wpk,l,Flv≥v-wpk,l,Flv-v-wpk,l,wpk,l-tpk,lθlk+Fltpk,l=v-wpk,l,Flv-Fltpk,l-wpk,l-tpk,lθlk=v-wpk,l,Flv-Flwpk,l+v-wpk,l,Flwpk,l-Fltpk,l-v-wpk,l-wpk,l-tpk,lθlk≥v-wpk,l,Flwpk,l-v-wpk,l,Fltpk,l+wpk,l-tpk,lθlk=v-wpk,l,Flwpk,l-Fltpk,l-v-wpk,l,wpk,l-tpk,lθlk.Since, wpk,l-tpk,l→0 as p→∞ and Fl is Lipschitz continuous, therefore we get (64)v-q,w≥0.Since T is maximal monotone, we have q∈T-10 and hence q∈VI(Fl,W). Therefore, (65)q∈F=⋂l=1MFixSl∩⋂l=1MVIFl,W.From (58), we get(66)limp→∞sup⁡ζgt-Gt,wp,l-t=ζgt-Gt,q-t≤0.Since t=PF(ζg+(I-G))(t), from (50) and (66), we get(67)limp→∞sup⁡ζgt-Gt,sp-t=limp→∞sup⁡ζgt-Gt,sp-wp,l+wp,l-t≤limp→∞sup⁡ζgt-Gt,wp,l-t≤0.Next, we show that sp→t. We have, (68)
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Since{sp}, {g(sp)}, and wp,l are bounded, we have (69)ζ¯2sp-t2+ζgsp-Gt2+2Gwp,l-tζgt-Gt<M2, where M2>0. Further, we have(70)sp+1-t2≤1-2ζ¯-apζapsp-t2+apσp,where (71)σp=2cpsp-t,ζgt-Gt+21-cpwp,l-t,ζgt-Gt+apM3.Using, condition (i), (66), and (67), we get limsupp→∞cp≤0. Now applying Lemma 8 in (70), we get (72)sp→t.


## 4. Numerical Example

In this section, we discuss the following example which shows the effectiveness and convergence of iterative sequence{sp} considered under Theorem 11.LetV=R and W=[0,1]. Let the mappings Fl, G, and g be defined by (73)Fls=2s,Gs=s5,gs=s7for all s∈W. Let Sl be the mapping defined by Sl(s)=s/l for all s∈C.Define{ap}, {bp,l}, and {cp} in (0,1) by ap=1/2p, bp,l=l/p2, and cp=1/2p. It is clear that Sl is a nonexpansive mapping and Fl is 2-inverse strongly monotone mapping. g is a 1/7-contraction mapping.Now we show that the sequences{ap}, {bp,l}, and {cp} satisfy the condition given. (74)limp→∞ap=12limp→∞1p=0, and (75)∑p=1∞ap=12∑p=1∞1p=∞.The sequence {ap} satisfies condition (i) of Algorithm 10.Next we compute(76)ap-ap-1=12p-12p-1=121p-1p-1=-12pp-1.So,(77)∑p=1∞ap-ap-1<∞.Similarly, we show that(78)∑p=1∞cp-cp-1<∞.and (79)∑p=1∞bp,l-bp-1,l<∞.The sequences {ap},{bp,l}, and {cp} satisfy conditions (i), (ii), and (iii). Clearly, (80)F≔⋂l=12FixSl∩⋂l=12VIFl,W=0.The iterative algorithm (22) is written as follows.When l=1(81)tp,l=sp;sp+1=11sp35p+sp3. When l=2 (82)tp,l=sp-spp2;sp+1=11sp35p-3sp10p2+sp6p3+sp3.Remark 12.
Table1 shows that the sequences {tp,l} and {sp} converge to 0. Also, {0}∈F.Table 1
The values oftp,l and sp with initial values s1=1 when l=1,2.
l = 1 l = 2 s 1 = 1 s 1 = 1 t p , l s p t p , l s p n=1 1 1 0.5000 1  n=2 0.6476 0.6476 0.3855 0.5141  n=3 0.3176 0.3176 0.1993 0.2243  n=4 0.1391 0.1391 0.0863 0.0921  n=5 0.0572 0.0572 0.0349 0.0364  n=6 0.0226 0.0226 0.0136 0.0140  n=7 0.0087 0.0087 0.0050 0.0052  n=8 0.0032 0.0032 0.0018 0.0019  n=9 0.0011 0.0011 0.0005 0.0006  n=10 0.0004 0.0004 0.0001 0.0002  n=11 0.0001 0.0001 0.0000 0.0000  n=12 0.0000 0.0000 0.0000 0.0000


## 5. Application

In this section, by applying Theorem11 we propose the following results.Theorem 13.
LetW be a closed convex subset of a real Hilbert space V, let g:W→W be a ζ-contraction, let Fl:W→V be a ςl-inverse strongly monotone mapping, let Sl:W→W be a finite family of nonexpansive mappings with F≔(⋂l=1MFix(Sl))∩(⋂l=1MFl-1(0))≠∅, and let G be a strongly positive linear bounded self-adjoint operator with the coefficient ζ¯>0. Assume that 0<ζ≤ζ¯/α and let {θl} be real numbers in (0,2ςl). Let {sp} and {tp,l}∀l=1,2,…,M, be the sequences given by(83)tp,1=bp,1S1sp+1-bp,1sptp,l=bp,lSlsp+1-bp,ltp,l-1,l=1,2,…,Msp+1=apζgsp+cpsp+1-cpI-apGtp,l-θlFltp,l,∀p≥1.where {ap}, {bp,l}, and {cp} are sequences in (0,1) satisfying the following: (i)
limp→∞ap=0 and ∑p=1∞‍ap=∞.(ii)
∑p=1∞‍ap-ap-1<∞ and ∑p=1∞‍cp-cp-1<∞.(iii)
∑p=1∞‍bp,l-bp-1,l<∞ for each l=1,2,…,M.(iv)
0<liminfp→∞bp,l≤limsupp→∞bp,l<1.
Then the sequence{sp} defined by (83) converges strongly to t∈F, where t=PF(ζg+(I-G))t which solves the following variational inequality: (84)ζg-Gt,q-t≤0,∀q∈F.Again, we apply Theorem11., the result to the problem for finding a common element of the set of fixed points of a finite family of nonexpansive mappings and the set of fixed points of a finite family of k-strictly pseudocontractive mappings.Theorem 14.
LetW be a closed convex subset of a real Hilbert space V. For any integer M>1, assume that for each 1≤l≤M, Tl:W→V is a kl-strictly pseudocontractive mapping for some 0≤kl<1. Let g:W→W be a ζ-contraction, let Fl:W→V be a ςl-inverse strongly monotone mapping, let Sl:W→W be a finite family of nonexpansive mappings with F≔(⋂l=1MFix(Sl))∩(⋂l=1MF(Tl))≠∅, and let G be a strongly positive linear bounded self-adjoint operator with the coefficient ζ¯>0. Assume that 0<ζ≤ζ¯/α. Let {sp} and {tp,l}∀l=1,2,…,M, be the sequences given by(85)tp,1=bp,1S1sp+1-bp,1sptp,l=bp,lSlsp+1-bp,ltp,l-1,l=1,2,…,Msp+1=apζgsp+cpsp+1-cpI-apGPW1-λptp,l-λp∑l=1MφlTltp,l,where {ap},{bp,l}, and {cp} are sequences in (0,1) satisfying the following: (i)
limp→∞ap=0 and ∑p=1∞‍ap=∞.(ii)
∑p=1∞‍ap-ap-1<∞ and ∑p=1∞‍cp-cp-1<∞.(iii)
∑p=1∞‍|bp,l-bp-1,l|<∞ for each l=1,2,…,M.(iv)
0<liminfp→∞bp,l≤limsupp→∞bp,l<1.
Then the sequence{sp} defined by (85) converges strongly to t∈F, where t=PF(ζg+(I-G))t which solves the following variational inequality: (86)ζg-Gt,q-t≤0,∀q∈F.Proof.
TakingFl=I-∑l=1M‍φlTl:W→V, we know that Fl:W→V is ς-inverse strongly monotone with ς=1-k/2. Hence, Fl is a monotone L-Lipschitz continuous mapping with L=2/1-k. From Lemma 7, we know that ∑l=1M‍φlTl is a k-strictly pseudocontractive mapping with k=max⁡{kl:1≤l≤M} and then F(∑l=1M‍φlTl)=VI(Fl,W). So, we have(87)PWtp,l-θlFltp,l=PW1-λptp,l-λp∑l=1MφlTltp,l.Using Theorem 11 we get the desired result.Remark 15.
Theorem14 is a generalization of the theorems given by Iiduka and Takahashi [6] and Takahashi and Toyoda [5].



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*Source: 6875789-2019-01-22.xml*