# Iterative Scheme for Split Variational Inclusion and a Fixed-Point Problem of a Finite Collection of Nonexpansive Mappings

Authors: M. Dilshad; A. F. Aljohani; M. Akram
Journal: Journal of Function Spaces
 (2020)

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## Abstract

This article is aimed at introducing an iterative scheme to approximate the common solution of split variational inclusion and a fixed-point problem of a finite collection of nonexpansive mappings. It is proven that under some suitable assumptions, the sequences achieved by the proposed iterative scheme converge strongly to a common element of the solution sets of these problems. Some consequences of the main theorem are also given. Finally, the convergence analysis of the sequences achieved from the iterative scheme is illustrated with the help of a numerical example.


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## Body

## 1. Introduction

LetH1 and H2 be two real Hilbert spaces endowed with inner product ⋅,⋅ and induced norm ⋅. A mapping T:H1⟶H1 is called contraction, if ∃κ∈0,1 such that Tφ−Tψ≤κφ−ψ, ∀φ, ψ∈H1. If κ=1, then T becomes nonexpansive. A mapping T is said to have a fixed point, if ∃φ∈H1 such that Tφ=φ. Further, if Tn:H1⟶H1, n=1,⋯,M is a finite collection of nonexpansive mappings. Then, the fixed-point problem (FPP) is defined as find φ∈H1 such that
(1)⋂n=1MTnφ=φ.It is easy to show that if⋂n=1MFixTn≠0, then ⋂n=1MFixTn is closed and convex. Many iterative methods have been adopted to examine the solution of a fixed-point problem for nonexpansive mappings and its variant forms, see [1–5] and references therein.We know that most of the techniques for solving the fixed-point problems can be acquired from Mann’s iterative technique [3], namely, for arbitrary x0∈C, compute
(2)xk+1=αkxk+1−αkTxk,k≥0,where T is a nonexpansive mapping from a nonempty closed convex subset C of Hilbert space H1 to itself and αn is a control sequence, which force xk to converge (weak) to a fixed point of T. To obtain the strong convergence result, Moudafi [4] proposed the viscosity approximation method by combining the nonexpansive mapping T with a contraction of given mapping f over C. For an arbitrary x0∈C, compute the sequence xk generated by
(3)xk+1=αkfxk+1−αkTxk,k≥0,where αn∈0,1 goes slowly to zero. The sequence xk+1 achieved from this iterative method converges strongly to a fixed point of T.On the other hand, let us recall some work about split variational inequality/inclusion problems. A multivalued mappingG:H1⟶2H1 is called maximal monotone, if its graph gphG=φ,ψ∈H1×H1:ψ∈Gφ is not properly contained by the graph of any other monotone mapping. A monotone mapping G is maximal monotone if and only if for φ,ζ∈H1×H1, φ−ψ,ζ−ϑ≥0 for every ψ,ϑ∈gphG implies that ζ∈Gφ. If G is maximal monotone, then operator JλG=I+λG−1 is well defined, nonexpansive, and known as the resolvent of G with parameter λ>0, which is defined at every point of the domain.The idea of split variational inequality problem (SVIP) given by Censor et al. [6], which amounts to saying find a solution of variational inequality whose image, under a given bounded linear operator, solves another variational inequality. Find φ∗∈C such that
(4)hφ∗,φ−φ∗≥0,∀φ∈C,and such that
(5)ψ∗=Aφ∗∈Dsolvesgψ∗,ψ−ψ∗≥0,∀ψ∈D,where C and D are closed, convex subsets of Hilbert spaces H1 and H2, respectively; A:H1⟶H2 is a bounded linear operator, and h:H1⟶H1 and g:H2⟶H2 are two operators. They studied the weak convergent result to solve SVIP.Moudafi [7] generalized SVIP and introduced split monotone variational inclusion problem (SpMVIP): find φ∗∈H1 such that
(6)0∈hφ∗+G1φ∗,and such that
(7)ψ∗=Aφ∗∈H2solves0∈gψ∗+G2ψ∗,where G1:H1⟶2H1 and G2:H2⟶2H2 are multivalued monotone mappings, A:H1⟶H2 is a bounded linear operator, h:H1⟶H1 and g:H2⟶H2 are two single-valued operators. The author also composed an iterative algorithm to solve (SpMVIP) and showed that the sequence achieved by the proposed algorithm converges weakly to the solution of (SpMVIP). Numerous iterative methods have been investigated for split variational inequality/inclusion problems, split common fixed-point problems, split feasibility problems, and split zero problems and their generalizations, see [6, 8–16] and references therein.Ifh=g=0 in SpMVIP, then we obtain the split variational inclusion problem (SpVIP) considered in [8], stated as find φ∗∈H1 such that
(8)0∈G1φ∗such that
(9)ψ∗=Aφ∗∈H2solves0∈G2ψ∗.Byrne et al. [8] proposed the following iterative scheme for SpVIP and studied the strong and weak convergence. For arbitrary xo∈H1, compute the iterative sequence achieved by the following scheme:
(10)xk+1=JλG1xk+μA∗JλG2−IAxk,for λ>0.Recently, Kazmi and Rizvi [17] suggested and examined an iterative algorithm to estimate the common solution for SpVIP and a fixed-point problem of a nonexpansive mapping in Hilbert spaces. Puangpee and Sauntai [18] studied the split variational inclusion problem and fixed-point problem in Banach spaces. Haitao and Li [19] investigated the split variational inclusion problem and fixed-point problem of nonexpansive semigroup without prior calculation of operator norm. Later, many authors studied the common solution of split variational inequality/inclusion problem and fixed-point problem of nonexpansive mappings in the framework of Hilbert/Banach spaces, see for example [18–24] and references therein.Following the works in [4, 7, 8, 17] and by the current research in this flow, we propose an iterative scheme to approximate a common solution of FPP and SpVIP. We prove that the sequences achieved by the proposed iterative scheme strongly converge to the common solution of FPP and SpVIP. The iterative scheme and results discussed in this article are new and can be viewed as generalization and refinement of the previously published work in this area.


## 2. Prelude and Auxiliary Results

In this section, we assembled some underlying definitions and supporting results.Definition 1.
LetCC⊂H1, the metric projection PC onto the set C is defined as PCϑ∈C and ϑ−PCϑ=infϑ∈Cϑ−ω,∀ω∈H1.
PC is also characterised by the facts that PCϑ∈C,
(11)ϑ−PCϑ,ω−PCϑ≤0and
(12)ϑ−ω2≥ϑ−PCϑ2+ω−PCϑ2,∀ϑ∈H1,ω∈C.Remark 2 (see [25, 26]).
For a nonexpansive mappingT and projection PCϑ onto C, the following results hold in Hilbert spaces:
(i)
(13)ϑ−ω,PCϑ−PCω≥PCϑ−PCω2,∀ϑ,ω∈H1(ii)
For allϑ,ω∈H1×H1,
(14)ϑ−Tϑ−ω−Tω,Tω−Tϑ≤12Tϑ−ϑ−Tω−ω2.Thus, for allϑ,ω∈H1×FixT, we get
(15)ϑ−Tϑ,ω−Tω≤12Tϑ−ϑ2(iii)
For allϑ,ω∈H1, t∈0,1(16)tϑ−1−tω2=tϑ2+1−tω2−t1−tϑ−ω2Definition 3.
A mappingT:H1⟶H1 is said to be
(i)
monotone, if(17)Tϑ−Tω,ϑ−ω≥0,forallϑ,ω∈H1(ii)
τ-strongly monotone, if there exists a constant τ>0 such that
(18)Tϑ−Tω,ϑ−ω≥τϑ−ω2,∀ϑ,ω∈H1(iii)
γ-inverse strongly monotone, if there exists a constant γ>0 such that
(19)Tϑ−Tω,ϑ−ω≥γTϑ−Tω2,∀ϑ,ω∈H1(iv)
firmly nonexpansive, if(20)Tϑ−Tω,ϑ−ω≥Tϑ−Tω2,∀ϑ,ω∈H1
Some important characteristics of an averaged operator are mentioned below; for more details, we refer to [7, 27, 28].Definition 4.
A mappingT:H1⟶H1 is called an averaged if and only if T is the average of identity mapping and a nonexpansive mapping, that is, T=1−tI+tS, where t∈0,1 and S:H1⟶H1 is nonexpansive.Thus, firmly nonexpansive mappings are averaged. It can also be seen that averaged mappings are nonexpansive.Proposition 5.
(i)
LetS:H1⟶H1 be an averaged and V:H1⟶H1 be a nonexpansive mapping, then T=1−tS+tV is averaged for t∈0,1(ii)
If the compositeTnn=1M is averaged and have a nonempty common fixed point, then
(21)∩n=1MFixTn=FixT1T2⋯⋯⋯TM(iii)
IfT is γ-inverse strongly monotone, then for r>0, rT is r/γ-inverse strongly monotone(iv)
T is averaged if its compliment I−T is γ-inverse strongly monotone for some γ>1/2Lemma 6 (see [29]).
Assume thatT is nonexpansive self-mapping of a closed convex subset D of a Hilbert space H1. If T has a fixed point, then I−T is demiclosed, i.e., whenever ωn is a sequence in D converging weakly to some ω∈D and the sequence I−Tωn converges strongly to some ϖ, then I−Tω=ϖ, where I is the identity mapping on H1.Lemma 7 (see [5]).
Ifvk is a sequence of nonnegative real numbers such that
(22)vk+1≤1−ξnvk+ωk,k=0,1,2⋯,where ξk is a sequence in (0,1) and ωk is a sequence in ℝ such that
(i)
∑k=1∞ξk=∞(ii)
limsupk→∞ωk/ξk≤0orlimsupk→∞ωk<∞
Then,limk→∞vk=0.
We denote the solution set of SpVIP by Ξ=φ∗∈H1:0∈G1φ∗and0∈G2Aφ∗ and of FPP by ∩n=1MFixTn.


## 3. Iterative Scheme and Its Convergence

In this section, we present the iterative scheme and show that the sequences obtained from the proposed iterative scheme converge strongly to the common solution of FPP and SpVIP.For integerK≥1, we define the mapping TK=TKmodM with the mod function, which is taking values from the set 1,2,⋯,M, that is, if K=aM+b for some integer a≥0 and 0≤b≤M, then TK=TM if b=0 and TK=Tb if 0<b<M.Iterative Scheme 8.
Step 0. Takeαk⊂0,1. Choose u0∈H1 arbitrary and let k=0.Step 1.
Givenuk∈H1, compute uk+1∈H1 as
(23)vk=JλG1uk+μA∗JλG2−IAuk,uk+1=αkfuk+1−αkTk+1vk,update k=k+1 and go to Step 1.Condition C.
We assume thatTn:H1⟶H1, n=1,2,⋯,M, is a finite number of nonexpansive mappings such that ∩n=1MFixTn≠0 and
(24)∩n=iMFixTn=FixT1∘T2∘⋯∘TM=FixTM∘T1∘⋯∘TM−1=⋯=FixT2∘T3∘⋯∘TM∘T1.Lemma 9.
φ∗∈H1 and ψ∗=Aφ∗ are solutions of SpVIP, if and only if φ∗=JλG1φ∗ and ψ∗=Aφ∗=JλG2ψ∗, for some λ>0.Proof.
The proof of the lemma follows immediately from the definitions of resolvent operators.Remark 10.
IfJλG is the resolvent of maximal monotone mapping G, A∗ is the adjoint operator of A and R is the spectral radius of AA∗. Then, using the properties of averaged mapping, one can easily show that the operator I+μA∗JλG−IA is averaged with λ>0, μ∈0,1/R.Now, we prove the following lemma which guarantees the contractivity ofL, which is needed to prove our main result.Lemma 11.
LetH1 and H2 be two real Hilbert spaces and A:H1⟶H2 be a bounded linear operator. Suppose that G1:H1⟶H1 and G2:H2⟶H2 are maximal monotone operators and T:H1⟶H1 be a nonexpansive mapping. Let f:H1⟶H1 be a κ-contraction mapping with constant κ>0. For any θ∈0,1, we define a mapping on H1 by
(25)Lϑ=θfϑ+1−θTJλG1ϑ+μA∗JλG2−IAϑ,∀ϑ∈H1,where μ∈0,1/R, R is the spectral radius of the operator AA∗, and A∗ is the adjoint operator of A. Then, the mapping L is a contraction with constant 0<1−θ1−κ<1; hence, L has a unique fixed point.Proof.
The operatorsJλG1 and JλG2 are averaged being firmly nonexpansive. For μ∈0,1/R, the operators I+μA∗JλG2−IA and JλG1I+μA∗JλG2−IA are averaged and hence nonexpansive. Thus, for all u, v∈H1, we have
(26)Lϑ−Lω=θfϑ+1−θTJλG1ϑ+μA∗JλG2−IAϑ−θfω−1−θTJλG1ω+μA∗JλG2−IAω≤θfϑ−fω+1−θTϑ−Tω≤θκϑ−ω+1−θϑ−ω=1−θ1−κϑ−ω.
Since0<1−θ1−κ<1 implies that L is a contraction, hence L has a unique fixed point.Theorem 12.
LetH1 and H2 be two real Hilbert spaces and A:H1⟶H2 be a bounded linear operator. Assume that G1:H1⟶2H1 and G2:H2⟶2H2 are two maximal monotone operators and f:H1⟶H1 is a contraction with constant κ∈0,1. Let Tn:H1⟶H1, n=1,2,⋯,M, be a finite collection of nonexpansive mappings satisfying the Condition C such that ∩n=1MFixTn∩Ξ≠0. Let R be a spectral radius of A⋆A, where A⋆ is the adjoint of A such that μ∈0,1/R and αk be a sequence in (0,1) such that limk→∞αk=0, ∑k=1∞αk=∞, and ∑k=1∞αk−αk−1<∞. Then, the iterative sequences vk and uk generated by Iterative Scheme 8 converge strongly to v¯∈∩n=1MFixTn∩Ξ, where v¯=P∩n=1MFixTn∩Ξfv¯.Proof.
Letu⋆∈∩n=1MFixTn∩Ξ, then we have JλG1u⋆=u⋆, JλG2Au⋆=Au⋆, and Tnu⋆=u⋆, n=1,2,⋯M, then using Iterative Scheme 8, we evaluate
(27)vk−u⋆2=JλG1uk+μA∗JλG2−IAuk−u⋆2=JλG1uk+μA∗JλG2−IAuk−JλG1u⋆2≤uk−u⋆2+μ2A⋆JλG2−IAuk2+2μuk−u⋆,A∗JλG2−IAuk=uk−u⋆2+μ2JλG2−IAuk,AA∗JλG2−IAuk+2μuk−u⋆,A∗JλG2−IAuk≤uk−u⋆2+μ2RJλG2−IAuk2+2μuk−u⋆,A∗JλG2−IAuk.
Denoting∇=2μuk−u⋆,A∗JλG2−IAuk and using (16), we have
(28)∇=2μAuk−u⋆,JλG2−IAuk=2μAuk−u⋆+JλG2−IAuk−JλG2−IAuk,JλG2−IAuk=2μJλG2Auk−Au⋆,JλG2−IAuk−JλG2−IAuk2≤2μ12JλG2−IAuk2−JλG2−IAuk2≤−μJλG2−IAuk2,using (28), (27) becomes
(29)vk−u⋆2≤uk−u⋆2+μRμ−1JλG2−IAuk2.
Sinceμ∈0,1/R, we obtain
(30)vk−u⋆2≤uk−u⋆2.
Now, we show thatuk is bounded.
(31)uk+1−u⋆=αkfuk+1−αkTk+1vk−u⋆≤αkfuk−u⋆+1−αkTk+1vk−u⋆≤αkfuk−fu⋆+αkfu⋆−u⋆+1−αkTk+1vk−Tk+1u⋆≤αkκuk−u⋆+αkfu⋆−u⋆+1−αkvk−u⋆=1−αk1−κuk−u⋆+αkfu⋆−u⋆≤maxuk−u⋆,fu⋆−u⋆1−κ⋯≤maxu0−u⋆,fu⋆−u⋆1−κ.
Hence,uk is bounded, which implies that the sequences vk, fuk, and Tk+1vk are also bounded. It follows from nonexpansiveness of Tn, n=1,⋯M, and Lipschitz continuity of f with constant κ that
(32)uk+M−uk=αk+M−1fuk+M−1+1−αk+M−1Tk+Mvk+M−1−αk−1fuk−1+1−αk−1Tkvk−1=αk+M−1fuk+M−1−αk+M−1fuk−1+αk+M−1fuk−1+1−αk+M−1Tkvk−1−1−αk+M−1Tkvk−1+1−αk+M−1Tk+Mvk+M−1−αk−1fuk−1+1−αk−1Tkvk−1≤αk+M−1fuk+M−1−fuk−1+fuk−1αk+M−1−αk−1+1−αk+M−1Tk+Mvk+M−1−Tkvk−1+Tkvk−1αk−1−αk+M−1,that is,
(33)uk+M−uk=καk+M−1uk+M−1−uk−1+1−αk+M−1vk+M−1−vk−1+2αk+M−1−αk−1M1,where M1=supfuk−1+Tkvk−1:k∈ℕ. Since, μ∈0,1/R, the operator JλG1I+μA∗JλG2−IA is average and hence nonexpansive, then we have
(34)vk+M−1−vk−1=JλG1uk+M−1+μA∗JλG2−IAuk+M−1−JλG1uk−1+μA∗JλG2−IAuk−1≤JλG1I+μA∗JλG2−IAuk+M−1−JλG1I+μA∗JλG2−IAuk−1≤uk+M−1−uk−1.
From (34), (33) becomes
(35)uk+M−uk≤1−αk+M−11−κuk+M−1−uk−1+2αk+M−1−αk−1M1,let ξk=αk+M−11−κ, ωk=2αk+M−1−αk−1M1, by using Lemma 7, we conclude that
(36)limk→∞uk+M−uk=0.
Now, we show thatuk−vk⟶0 as k⟶∞. From (29), it follows that
(37)uk+1−u⋆2=αkfuk+1−αkTk+1vk−u⋆2≤αkfuk−u⋆2+1−αkTk+1vk−Tk+1u⋆2≤αkfuk−u⋆2+1−αkvk−u⋆2≤αkfuk−u⋆2+1−αkuk−u⋆2+μRμ−1JλG2−IAuk2.
Therefore,(38)μ−Rμ+1JλG2−IAuk2≤αkfuk−u⋆2+uk−u⋆2−uk+1−u⋆2=αkfuk−u⋆2+uk+1−ukuk−u⋆+uk+1−u⋆.
Since1−Rμ>0, αk⟶0, and uk+1−uk⟶0, we get
(39)JλG2−IAuk⟶0,ask⟶∞.
Sinceμ∈01/R and using (27) and (29), we obtain
(40)vk−u⋆2=JλG1uk+μA∗JλG2−IAuk−u⋆2=JλG1uk+μA∗JλG2−IAuk−JλG1u⋆2≤vk−u⋆,uk+μA∗JλG2−IAuk−u⋆=12vk−u⋆2+uk+μA∗JλG2−IAuk−u⋆2−vk−u⋆−uk+μA∗JλG2−IAuk−u⋆2=12vk−u⋆2+uk−u⋆2+μRμ−1JλG2−IAuk2−vk−uk−μA∗JλG2−IAuk2≤12vk−u⋆2+uk−u⋆2−vk−uk2+μ2A∗JλG2−IAuk2−2μvk−uk,A∗JλG2−IAuk≤12vk−u⋆2−vk−uk2+uk−u⋆2+2μAvk−ukJλG2−IAuk.
Thus, we get(41)vk−u⋆2≤uk−u⋆2+2μAvk−ukJλG2−IAuk−vk−uk2.
By (37) and (41), we have
(42)uk+1−u⋆2≤αkfuk−u⋆2+1−αk2μAvk−ukJλG2−IAuk+uk−u⋆2−vk−uk2≤αkfuk−u⋆2+2μAvk−ukJλG2−IAuk+uk−u⋆2−vk−uk2,that is,
(43)vk−uk2≤αkfuk−u⋆2+2μAvk−ukJλG2−IAuk+uk−u⋆2−uk+1−u⋆2≤αkfuk−u⋆2+2μAvk−ukJλG2−IAuk+uk−u⋆+uk+1−u⋆uk+1−uk.
Sinceαk⟶0, uk+1−uk⟶0, JλG2−IAuk⟶0 as k⟶∞, we get
(44)limk→∞vk−uk=0.
We recognized that the following relation holds:(45)uk+M−uk=uk+M−Tk+Mvk+M−1+Tk+Mvk+M−1−Tk+Muk+M−1+Tk+Muk+M−1−Tk+MTk+M−1vk+M−2+⋯+Tk+M∘⋯∘Tk+2uk+1−Tk+M∘⋯∘Tk+2vk+1+Tk+M∘⋯∘Tk+2vk+1−Tk+M∘⋯∘Tk+1uk+Tk+M∘⋯∘Tk+1uk−Tk+M∘⋯∘Tk+1vk+Tk+M∘⋯∘Tk+1vk−uk.
By Iterative Scheme8, we can easily see that uk+1−Tk+1vk⟶0 as k⟶∞. From (44) and nonexpansiveness of Tnn=1,2,⋯,M, it follows that
(46)uk+M−Tk+Mvk+M−1⟶0Tk+Mvk+M−1−Tk+Muk+M−1⟶0Tk+Muk+M−1−Tk+M∘Tk+M−1vk+M−2⟶0⋮Tk+M∘⋯∘Tk+2vk+1−Tk+M∘⋯∘Tk+1uk⟶0Tk+M∘⋯∘Tk+1uk−Tk+M∘⋯∘Tk+1vk⟶0.
By using (36) and (45), we conclude that
(47)limk→∞Tk+M∘Tk+M−1∘⋯∘Tk+2∘Tk+1vk−uk⟶0.
Now, using (47) and (44), we write
(48)Tk+M∘Tk+M−1∘⋯∘Tk+1vk−vk≤Tk+M∘Tk+M−1∘⋯∘Tk+1vk−uk+vk−uk⟶0,as k⟶∞, that is,
(49)Tk+M∘Tk+M−1∘⋯∘Tk+1vk−uk⟶0ask⟶∞.
Boundedness ofvk implies that there exists a subsequence vki of vk, converging weakly to w. Because the collection of mappings Tn:1≤n≤M is finite, we can say for some integer K∈1,2,⋯,N(50)Tkn≡TK,∀n≥1.
Thus, from (49), we have
(51)Tn+M∘Tn+M−1∘⋯∘Tn+1vkn−vkn⟶0.
Therefore, using Lemma6, we conclude that
(52)w∈FixTn+M∘⋯Tn+1.
Thus, by the assumptions of ConditionC, we have w∈∩n=1MFixTn. On the other hand,
(53)vki=JλG1uki+μA∗JλG2−IAukiuki+μA∗JλG2−IAuki∈I+λG1vkiuki−vki+μA∗JλG2−IAukiλ∈G1vki.
We know that the graph of a maximal monotone operator is weakly strongly closed; hence, by takingi⟶∞ and using (37) and (44), we get
(54)0∈G1w.
Sinceuk, vk have the same asymptotical behaviour, Auki converges weakly to Aw. Therefore, by (39), the nonexpansive property of JλG2 and Lemma 6, we get 0∈G2Aw. Thus, w∈∩n=1MFixTn∩Ξ.
Now, we need to show thatlimsupk→∞fv¯−v¯,uk−v¯≤0, where v¯=P∩n=1MFixTn∩Ξfv¯.
We have(55)limsupk→∞fv¯−v¯,uk−v¯=limsupk→∞fv¯−v¯,Tk+1vk−v¯≤fv¯−v¯,w−v¯≤0,since v¯∈∩n=1MFixTn∩Ξ.
Finally, we show thatxk⟶v¯(56)uk+1−v¯2=αkfuk+1−αkTk+1vk−v¯2=αkfuk+1−αkTk+1vk−v¯,uk+1−v¯=αkfuk−v¯,uk+1−v¯+1−αkTk+1vk−v¯,uk+1−v¯≤αkfuk−v¯,uk+1−v¯+1−αkvk−v¯,uk+1−v¯≤αkfuk−fv¯,uk+1−v¯+αkfuk−v¯,uk+1−v¯+1−αkvk−v¯,uk+1−v¯≤αk2fuk−fv¯2+uk+1−v¯2+αkfv¯−v¯,uk+1−v¯+1−αk2uk−v¯2+uk+1−v¯2≤121−αk1−κ2uk+1−v¯2+1−αk2uk−v¯2+αk2uk+1−v¯2+αkfv¯−v¯,uk+1−v¯,which implies that
(57)uk+1−v¯2≤1−αk1−κ2uk+1−v¯2+2αkfv¯−v¯,uk+1−v¯.
From Lemma7 and (55), we conclude that uk⟶v¯ and from vk−uk⟶0, vk⇀w∈∩n=1MFixTn∩Ξ, and uk⟶v¯ as k⟶∞, we achieve that v¯=w. This completes the proof.


## 4. Consequences

SupposeC and D are closed convex subsets of Hilbert spaces H1 and H2, respectively. Then, find u∈H1 such that
(58)u∈CandAu∈D,is called the split feasibility problem (SFP), where A:H1⟶H2 is a bounded linear operator. Byrne [9] introduced the CD algorithm to approximate the solution of (58):
(59)uk+1=PCuk+μA∗PD−IAuk,where PC and PD are orthogonal projections onto C and D, respectively.The split common fixed-point problem (SCFPP) is an extension of Problem (58), which has been widely investigated in the present scenario. The SCFPP is the inverse problem design to search a vector in a fixed-point set so that its image under a bounded linear operator corresponds to other fixed-point set, that is, find u∈H1 such that
(60)u=WuandAu=VAu,where W:H1⟶H1 and W:H2⟶H2 are nonexpansive mappings. By putting W=PC and W=PD, in (59), we can have an iterative scheme, which converges to the solution of SCFPP.We denote the solution set of SFP (58) and SCFPP (60) by Ψ, and Ω, respectively. The following corollaries are given as consequences of Theorem 12.Corollary 13.
LetC and D be two closed convex subsets of Hilbert spaces H1 and H2, respectively. Let A:H1⟶H2 be a bounded linear operator and f:H1⟶H1 be a contraction mapping with constant κ∈0,1. Let Tn:H1⟶H1, n=1,2,⋯,M, be a finite collection of nonexpansive mappings satisfying the condition C such that ∩n=1MFixTn∩Ψ≠0. Let R be the spectral radius of A⋆A, where A⋆ is the adjoint of A such that μ∈0,1/R and αk be a sequence in (0,1) with limk→∞αk=0, ∑k=1∞αk=∞, and ∑k=1∞αk−αk−1<∞. Then, the iterative sequences vk and uk generated by Iterative Scheme 8 with JλG1=PC and JλG2=PD converge to v¯∈∩n=1MFixTn∩Ψ, where v¯=P∩n=1MFixTn∩Ψfv¯.Corollary 14.
LetH1 and H2 be two real Hilbert spaces and A:H1⟶H2 be a bounded linear operator. Assume that G1:H1⟶2H1 and G2:H2⟶2H2 are maximal monotone operators and f:H1⟶H1 is a κ-contraction mapping with constant κ∈0,1. Let Tn:H1⟶H1, n=1,2,⋯,M, be a finite collection of nonexpansive mappings satisfying the condition C such that ∩n=1MFixTn∩Ω≠0. Let R be spectral radius of A⋆A, where A⋆ is the adjoint of A such that μ∈0,1/R and αk be a sequence in (0,1) with limn→∞αk=0, ∑k=1∞αk=∞, and ∑k=1∞αk−αk−1<∞. Then, the iterative sequences vk and uk obtained from Iterative Scheme 8 with JλG1=W and JλG2=V converge to v¯∈∩n=1MFixTn∩Ω, where v¯=P∩n=1MFixTn∩Ωfv¯.Remark 15.
If we takeT1=T2=⋯TM=T, a nonexpansive mapping, then we can obtain the iterative scheme and its convergence theorem for the common solution of SpVIP and a nonexpansive mapping T, studied in [17].
At last, we illustrate the convergence analysis of the proposed iterative scheme with the help of the following numerical example.


## 5. Numerical Example

LetH1=H2=ℝ and G1:H1⟶H1 defined as G1u=2u+1 and G2:H2⟶H2 defined as G2u=−4/5u+12/5. For λ=1/4, we compute the resolvents of G1 and G2 as
(61)JλG1u=I+λG1−1u=23u−13,JλG2u=I+λG2−1u=54u−34.It can be easily seen that, here,Ξ=−1. Further, let T1,T2, and T3:H1⟶H1 are three nonexpansive mappings, defined by
(62)T1u=sinu+1−1,T2u=−u−32,T3u=cosπu+u2such that
(63)FixT1∩FixT2∩FixT3=−1.Letf:H1⟶H1 be a contraction mapping defined as fu=u/2 and A be a bounded linear operator defined as Au=−3u with adjoint operator A∗ such that A=A∗=3.Sinceμ∈0,1/9 and αk∈0,1, so we choose μ=1/18 and αk=1/3k; then, the sequences vk and uk generated by the iterative scheme are evaluated as
(64)vk=JλG1uk+μA∗JλG2−IAuk=912uk−312,uk+1=αkfuk+1−αkTk+1vk=16k+1uk+1−13k+1Tk+1vkor for some positive integer a≥0, and M=3, we can write
(65)uk+1=uk6k+1+1−13k+1sinvk+1−1,ifk=3a+1,uk6k+1+1−13k+1−vk−32,ifk=3a+2,uk6k+1+1−13k+1cosπvk+vk2,ifk=3a.From Table1, we conclude that for two arbitrary different initial points u0=5 and u0=−3, the sequences vk and uk converge approximately to a point u⋆=−1∈∩n=1MFixTn∩Ξ.Table 1
Computation of iterative sequences of Iterative Scheme8 for the choices of parameters λ=1/4 and αn=1/3k and different initial points u0=5 and u0=−3.
λ=1/4u0=5u0=−3No. iter.vkuk+1vkuk+103.5-0.48502-0.25-1.8316631-1.613765-1.034683-1.623747-0.7260772-1.26012-0.956449-0.794558-0.7485193-0.967337-0.926583-0.8113989-0.7759854-0.944937-0.989916-0.8319989-1.0376055-0.992437-0.968237-1.028204-0.9847336-0.976178-0.948843-0.998550-0.9649227-0.961632-0.996486-0.973792-0.9910428-0.997364-0.980131-0.993282-0.9779749-0.985098-0.968597-0.983481-0.96699910-0.976448-0.995792-0.975249-0.99634911-0.986844-0.984495-0.997262-0.98471112-0.984495-0.975650-0.988533-0.97581113-0.981737-0.996719-0.981858-0.99666214-0.997539-0.997635-0.997496-0.987612



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*Source: 3567648-2020-10-09.xml*