# Viscosity Approximation Method for System of Variational Inclusions Problems and Fixed-Point Problems of a Countable Family of Nonexpansive Mappings

Authors: Chaichana Jaiboon; Poom Kumam
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We propose new iterative schemes for finding the common element of the set of common fixed points of countable family of nonexpansive mappings, the set of solutions of the variational inequality problem for relaxed cocoercive and Lipschitz continuous, the set of solutions of system of variational inclusions problem, and the set of solutions of equilibrium problems in a real Hilbert space by using the viscosity approximation method. We prove strong
convergence theorem under some parameters. The results in this paper unify and generalize some well-known results in the literature.


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## Body

## 1. Introduction

LetC be a nonempty closed convex subset of a real Hilbert space H. A mapping S of C into itself is called nonexpansive if ∥Sx-Sy∥≤∥x-y∥ for all x,y∈C. We denote by F(S) the set of fixed points of S; that is, F(S)={x∈C:Sx=x}. If C⊂H is nonempty, closed and convex and let S:C→C be a nonexpansive mapping, then F(S) is closed and convex and F(S)≠∅, when C is bounded; see, for example, [1, 2]. The metric projection, PC, onto a given nonempty, closed and convex subset C, satisfies the nonexpansive with F(PC)=C. A mapping B:C→C is called monotone if 〈Bx-By,x-y〉≥0 for all x,y∈C. A mapping B:C→C is called β-inverse-strongly monotone if there exists a constant β>0 such that 〈Bx-By,x-y〉≥β∥x-y∥2, for all x,y∈C. A mapping B:C→C is called relaxed (ϕ,ω)-cocoercive if there exists ϕ,ω>0 such that(1.1)〈Bx-By,x-y〉≥(-ϕ)‖Bx-By‖2+ω‖x-y‖2,∀x,y∈C.A mappingB:C→C is said to be ξ-Lipschitz continuous if there exists ξ≥0 such that(1.2)‖Bx-By‖≤ξ‖x-y‖,∀x,y∈C.LetB:H→H be a single-valued nonlinear mapping and M:H→2H a multivalued mapping. The variational inclusion problem is to find x̃∈H such that(1.3)θ∈B(x̃)+M(x̃),
where θ is the zero vector in H. The set of solutions of problem (1.3) is denoted by I(B,M). If M=∂ψC, where C is a nonempty closed convex subset of H and ∂ψC:H→[0,+∞] is the indicator function of C; that is,(1.4)ψC(x)={0,x∈C,+∞,x∉C,then, the variational inclusion problem (1.3) is equivalent to the variational inequality problems denoted by VI(C,B) which is to find x̃∈C such that(1.5)〈Bx̃,y-x̃〉≥0,∀y∈C.In 2003, Takahashi and Toyoda [3] to find x*∈F(S)∩VI(C,B) introduced the following iterative scheme:(1.6)x0∈C chosen arbitrary,xn+1=αnxn+(1-αn)SPC(xn-ξnBxn),∀n≥0,
where B is a β-inverse-strongly monotone mapping, {αn} is a sequence in (0, 1), and {ξn} is a sequence in (0,2β). They showed that if F(S)∩VI(C,B) is nonempty, then the sequence {xn} generated by (1.6) converges weakly to some x*∈F(S)∩VI(C,B).In 2008, Zhang et al. [4] to find x*∈F(S)∩I(M,B). They introduced the following new iterative scheme:(1.7)x0∈C chosen arbitrary,yn=JM,λ(xn-λBxn),xn+1=αnx+(1-αn)Syn,∀n≥0,
where JM,λ=(I+λM)-1 is the resolvent operator associated with M and a positive number λ,{αn} is a sequence in the interval [0,1].LetF be a bifunction of C×C into ℝ, where ℝ is the set of real numbers. The equilibrium problem for F:C×C→ℝ is to find x̃∈C such that(1.8)F(x,̃y)≥0,∀y∈C.
The set of solutions of (1.8) is denoted by EP(F). Many problems in applied sciences, such as monotone inclusion problems, variational inequality problems, saddle point problems, Nash equilibria in noncooperative games, as well as certain fixed-point problems reduce to finding some element to EP(F) in Hilbert and Banach spaces (see [5–14]).Given anyr>0. The operator Tr:H→C defined by(1.9)Tr(x)={z∈C:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C},
is called the resolvent of F (see [5, 6]).It is shown in [6] that, under suitable hypotheses on F (to be stated precisely in Section 2), Tr:H→C is single valued and firmly nonexpansive and satisfies(1.10)F(Tr)=EP⁡(F),∀r>0.Using this result, for finding an element ofF(S)∩VI(C,B)∩EP(F), Su et al. [15] introduced the following iterative scheme by the viscosity approximation method in Hilbert spaces:(1.11)x0∈Cchosenarbitrary,xn+1=αnf(xn)+(1-αn)SPC(I-ξnB)Trnxn,∀n≥0,
where f:C→C is a contraction (i.e., ∥f(x)-f(y)∥≤ψ∥x-y∥,forallx,y∈Cand0≤ψ<1) and {αn}⊂(0,1), ξn⊂(0,2β), and rn⊂(0,∞) satisfy some appropriate conditions. Furthermore, they prove {xn} converges strongly to the same point x*∈F(S)∩VI(C,B)∩EP(F), where x*=PF(S)∩VI(C,B)∩EP(F)f(x*).In this paper, motivated and inspired by the above facts, we introduce a new iterative scheme for finding a common element of the set of solutions of the variational inequalities forμ-Lipschitz continuous and relaxed (ϕ,ω)-cocoercive mapping, the set of solutions to the variational inclusion for family of α-inverse strongly monotone mappings, the set of fixed points of a countable family of nonexpansive mappings, and the set of solutions of an equilibrium problem in a real Hilbert space by using the viscosity approximation method. Strong convergence results are derived under suitable conditions in a real Hilbert space.


## 2. Preliminaries

In this section, we will recall some basic notations and collect some conclusions that will be used in the next section.LetH be a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. We denote strong convergence of {xn} to x∈H by xn→x and weak convergence by xn⇀x. Let C be nonempty closed convex subset of H. Recall that for all x∈H there exists a unique nearest point in C to x denoted PCx; that is, ∥x-PCx∥≤∥x-y∥,forally∈C. The mapping PC is nonexpansive; that is, ∥PCx-PCy∥≤∥x-y∥,forallx,y∈H. The mapping PC is firmly nonexpansive; that is, ∥PCx-PCy∥2≤〈PCx-PCy,x-y〉,forallx,y∈H. It is well known that(2.1)x̃∈VI⁡(C,B)⇔x̃=PC(x̃-λBx̃),∀λ>0.
A set-valued mapping M:H→2H is called monotone if, for all x,y∈H, f∈Mx and g∈My imply 〈x-y,f-g〉≥0. A monotone mapping M:H→2H is called maximal, if its graph of any Graph (M):={(x,f)∈H×H|f∈M(x)} of M is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping M is maximal if and only if for all (x,f)∈H×H,〈x-y,f-g〉≥0,forall(y,g)∈ Graph (M) (the graph of mapping M) implies that f∈Mx.Definition 2.1.
LetM:H→2H be a multivalued maximal monotone mapping; then the set-valued mapping JM,λ:H→H defined by
(2.2)JM,λ(x̃)=(I+λM)-1(x̃),∀x̃∈H,
is called the resolvent operator associated with M, where λ is any positive number and I is the identity mapping.Lemma 2.2 (see [16]).
LetM:H→2H be a maximal monotone mapping and let B:H→H be a Lipschitz continuous mapping. Then the mapping M+B:H→2H is a maximal monotone mapping.Lemma 2.3 (see [16, 17]).
(1)
The resolvent operatorJM,λ is single valued and nonexpansive for all λ>0; that is,(2.3)‖JM,λ(x)-JM,λ(y)‖≤‖x-y‖,∀x,y∈H,∀λ>0.(2)
The resolvent operatorJM,λ is 1-inverse-strongly monotone; that is,(2.4)‖JM,λ(x)-JM,λ(y)‖2≤〈x-y,JM,λ(x)-JM,λ(y)〉,∀x,y∈H.Lemma 2.4 (see [17]).
(1)
Letx̃∈H is a solution of problem (1.3) if and only if x̃=JM,λ(I-λB) for all λ>0; that is,(2.5)I(B,M)=F(JM,λ(I-λB)),∀λ>0.(2)
Ifλ∈[0,2β], then I(B,M) is a closed convex subset in H.Lemma 2.5 (see [18]).
Each Hilbert spaceH satisfies Opial’s condition; that is, for any sequence {xn}⊂H with xn⇀x, the inequality
(2.6)liminf⁡n→∞‖xn-x‖<liminf⁡n→∞‖xn-y‖
holds for each y∈H with y≠x.Lemma 2.6 (see [19]).
Let{xn} and {zn} be bounded sequences in a Banach space E, and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose xn+1=(1-βn)zn+βnxn for all integers n≥1 and limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0. Then, limn→∞∥zn-xn∥=0.Lemma 2.7 (see [20]).
Assume{an} is a sequence of nonnegative real numbers such that
(2.7)an+1≤(1-bn)an+δn,n≥0,
where {bn} is a sequence in (0,1) and {δn} is a sequence in ℝ such that(1)
∑n=1∞bn=∞,(2)
limsupn→∞δn/bn≤0 or ∑n=1∞|δn|<∞. Then limn→∞an=0.Lemma 2.8.
LetH be a real Hilbert space. Then hold the following identities:(i)
∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2,∀t∈[0,1],∀x,y∈H,(ii)
∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.Lemma 2.9 (see [21]).
LetC be a nonempty closed subset of a Banach space, and let {Sn} be a sequence of mappings of C into itself. Suppose that ∑n=1∞sup{∥Sn+1z-Snz∥:z∈C}<∞. Then, for each y∈C, {Sny} converges strongly to some point of C. Moreover, let S be a mapping of C into itself defined by
(2.8)Sy=lim⁡n→∞Sny,∀y∈C.
Then limn→∞sup{∥Sz-Snz∥:z∈C}=0.For solving the equilibrium problem for a bifunctionF:C×C→ℝ, let us assume that F satisfies the following conditions:(A1)
F(x,x)=0 for all x∈C;(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0,∀x,y∈C;(A3)
for eachx,y,z∈C,limt↓0F(tz+(1-t)x,y)≤F(x,y);(A4)
for eachx∈C,y↦F(x,y) is convex and lower semicontinuous.Lemma 2.10 (see [5]).
LetC be a nonempty closed convex subset of H, and let F be a bifunction of C×C into ℝ satisfying (A1)–(A4). Let r>0 and x∈H. Then, there exists z∈C such that
(2.9)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.Lemma 2.11 (see [6]).
Assume thatF:C×C→ℝ satisfies (A1)–(A4). For r>0 and x∈H, define a mapping Tr:H→C as follows:
(2.10)Tr(x)={z∈C:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C},
for all x∈H. Then, the following hold:(i)
Tr is single valued;(ii)
Tr is firmly nonexpansive; that is, for any x,y∈H∥Trx-Try∥2≤〈Trx-Try,x-y〉;(iii)
F(Tr)=EP(F);(iv)
EP(F) is closed and convex.Lemma 2.12 (see [22]).
LetH be a Hilbert space and M a maximal monotone on H. Then, the following holds:
(2.11)‖JM,rx-JM,sx‖2≤r-sr〈JM,rx-JM,sx,JM,rx-x〉,∀s,r>0,x∈H,
where JM,r=(I+rM)-1 and JM,s=(I+sM)-1.


## 3. Main Results

In this section, we will use the viscosity approximation method to prove a strong convergence theorem for finding a common element of the set of fixed points of a countable family of nonexpansive mappings, the set of solutions of the variational inequality problem for relaxed cocoercive and Lipschitz continuous mappings, the set of solutions of system of variational inclusions, and the set of solutions of equilibrium problem in a real Hilbert space.Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H, and let B:C→H be relaxed (ϕ,ω)-cocoercive and μ-Lipschitz continuous with ω>ϕμ2, for some ϕ,ω,μ>0. Let 𝒢={Gk:k=1,2,3,…,N} be a finite family of β-inverse strongly monotone mappings from C into H, and let F be a bifunction from C×C→ℝ satisfying (A1)–(A4). Let f:C→C be a contraction with coefficient ψ(0≤ψ<1), and let {Sn} be a sequence of nonexpansive mappings of C into itself such that
(3.1)Ω:⋂n=1∞F(Sn)∩(⋂k=1NI(Gk,Mk))∩VI⁡(C,B)∩EP⁡(F)≠∅.
Let the sequences {xn} and {yn} be generated by
(3.2)x1=x∈Cchosen arbitrarily,yn=JMN,λN,n(I-λN,nGn)…JM2,λ2,n(I-λ2,nG2)JM1,λ1,n(I-λ1,nG1)Trnxn,xn+1=αnf(xn)+βnxn+γnSnPC(yn-ξnByn),∀n≥1,
where {αn},{βn}, {γn}⊂(0,1) and {ξn},{rn}⊂(0,∞) satisfy the following conditions:(C1)
αn+βn+γn=1,(C2)
limn→∞αn=0,∑n=1∞αn=∞,(C3)
0<liminfn→∞βn≤limsupn→∞βn<1,(C4)
{ξn}⊂[a,b] for some a, b with 0≤a≤b≤2(ω-ϕμ2)/μ2 and limn→∞|ξn+1-ξn|=0,(C5)
{λk,n}k=1N⊂[c,d]⊂(0,2β) and limn→∞|λk,n+1-λk,n|=0, for each k∈{1,2,…,N},(C6)
liminfn→∞rn>0 and limn→∞|rn+1-rn|=0.Suppose that ∑n=1∞sup{∥Sn+1z-Snz∥:z∈K}<∞ for any bounded subset K of C. Let S be a mapping of C into itself defined by Sy=limn→∞Snyforally∈C and suppose that F(S)=⋂n=1∞F(Sn). Then, the sequences {xn} and {yn} converge strongly to the same point x*∈Ω, where x*=PΩf(x*).Proof.
First, we prove that the mappingPΩf:H→C has a unique fixed point.
In fact, sincef:C→C is a contraction with ψ∈[0,1) and PΩf:H→Ω is also a contraction, we obtain
(3.3)‖PΩf(x)-PΩf(y)‖≤‖f(x)-f(y)‖≤ψ‖x-y‖,∀x,y∈C.
Therefore, there exists a unique element x*∈C such that x*=PΩf(x*), where
(3.4)Ω:⋂n=1∞F(Sn)∩(⋂k=1NI(Gk,Mk))∩VI⁡(C,B)∩EP⁡(F).
Now, we prove that (I-ξnB) is nonexpansive.
Indeed, for anyx,y∈C, since B:C→H is a μ-Lipschitz continuous and relaxed (ϕ,ω)-cocoercive mappings with ω>ϕμ2 and ξn≤2(ω-ϕμ2)/μ2, we obtain
(3.5)‖(I-ξnB)x-(I-ξnB)y‖2=‖(x-y)-ξn(Bx-By)‖2=‖x-y‖2-2ξn〈x-y,Bx-By〉+ξn2‖Bx-By‖2≤‖x-y‖2-2ξn{-ϕ‖Bx-By‖2+ω‖x-y‖2}+ξn2‖Bx-By‖2≤‖x-y‖2+2ξnϕμ2‖x-y‖2-2ξnω‖x-y‖2+ξn2μ2‖x-y‖2=(1+2ξnϕμ2-2ξnω+ξn2μ2)‖x-y‖2=(1-ξnμ2[2(ω-ϕμ2)μ2-ξn])‖x-y‖2≤(1-ξnμ2[2(ω-ϕμ2)μ2-b])‖x-y‖2.
Setting
(3.6)ζ=μ22[2(ω-ϕμ2)μ2-b]>0,
thus,
(3.7)‖(I-ξnB)x-(I-ξnB)y‖2≤(1-2ξnζ)‖x-y‖2≤(1-ξnζ)2‖x-y‖2,
which implies that
(3.8)‖(I-ξnB)x-(I-ξnB)y‖≤(1-ξnζ)‖x-y‖≤‖x-y‖.
Hence (I-ξnB) is nonexpansive.
We divide the proof of Theorem3.1 into five steps.
Step 1.
We show that the sequence{xn} is bounded.
Now, letx̃∈Ω and if {Trn} is a sequence of mappings defined as in Lemma 2.11, then x̃=PC(x̃-λnBx̃)=Trnx̃, and let un=Trnxn. So, we have
(3.9)‖un-x̃‖=‖Trnxn-Trnx̃‖≤‖xn-x̃‖.
For k∈{1,2,…,N} and for any positive integer number n, we define the operator Υnk:C→H as follows:
(3.10)Υn0x=x,Υnkx=JMk,λk,n(I-λk,nGk)…JM2,λ2,n(I-λ2,nG2)JM1,λ1,n(I-λ1,nG1)x,
for all n, we get yn=ΥnNun. On the other hand, since Gk:C→H is β-inverse strongly monotone and λk,n⊂[c,d]⊂(0,2β), then JMk,λk,n(I-λk,nGk) is nonexpansive. Thus Υnk is nonexpansive. From Lemma 2.4(1), we have x̃=ΥnNx̃. It follows that
(3.11)‖yn-x̃‖=‖ΥnNun-ΥnNx̃‖≤‖un-x̃‖≤‖xn-x̃‖.
Setting vn=PC(yn-ξnByn) and I-ξnB is a nonexpansive mapping, we obtain
(3.12)‖vn-x̃‖=‖PC(yn-ξnByn)-PC(x̃-ξnBx̃)‖≤‖(yn-ξnByn)-(x̃-ξnBx̃)‖=‖(I-ξnB)yn-(I-ξnB)x̃‖≤‖yn-x̃‖≤‖xn-x̃‖.
From (3.2) and (3.12), we deduce that
(3.13)‖xn+1-x̃‖=‖αnf(xn)+βnxn+γnSnvn-x̃‖≤αn‖f(xn)-x̃‖+βn‖xn-x̃‖+γn‖vn-x̃‖≤αn‖f(xn)-f(x̃)‖+αn‖f(x̃)-x̃‖+βn‖xn-x̃‖+γn‖xn-x̃‖≤αnψ‖xn-x̃‖+αn‖f(x̃)-x̃‖+(1-αn)‖xn-x̃‖≤(1-αn(1-ψ))‖xn-x̃‖+αn‖f(x̃)-x̃‖=(1-αn(1-ψ))‖xn-x̃‖+αn(1-ψ)‖f(x̃)-x̃‖(1-ψ)≤max⁡{‖xn-x̃‖,‖f(x̃)-x̃‖1-ψ}.
It follows from induction that
(3.14)‖xn-x̃‖≤max⁡{‖x1-x̃‖,‖f(x̃)-x̃‖1-ψ},∀n≥1.
Therefore, {xn} is bounded and hence so are {vn}, {yn}, {un}, {Byn}, and {Snvn}.Step 2.
We claim thatlimn→∞∥xn+1-xn∥=0.
By the definition ofTr, un=Trnxn and un+1=Trn+1xn+1, we get
(3.15)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈H,(3.16)F(un+1,y)+1rn+1〈y-un+1,un+1-xn+1〉≥0,∀y∈H.
Taking y=un+1 in (3.15) and y=un in (3.16), we have
(3.17)F(un,un+1)+1rn〈un+1-un,un-xn〉≥0,
and hence
(3.18)F(un+1,un)+1rn+1〈un-un+1,un+1-xn+1〉≥0.
So, from (A2) we have
(3.19)〈un+1-un,un-xnrn-un+1-xn+1rn+1〉≥0,
and hence
(3.20)〈un+1-un,un-un+1+un+1-xn-rnrn+1(un+1-xn+1)〉≥0.
Without loss of generality, let us assume that there exists a real number c such that rn>c>0 for all n∈ℕ. Then, we have
(3.21)‖un+1-un‖2≤〈un+1-un,xn+1-xn+(1-rnrn+1)(un+1-xn+1)〉≤‖un+1-un‖{‖xn+1-xn‖+|1-rnrn+1|‖un+1-xn+1‖},
and hence
(3.22)‖un+1-un‖≤‖xn+1-xn‖+1rn+1|rn+1-rn|‖un+1-xn+1‖≤‖xn+1-xn‖+M1c|rn+1-rn|,
where M1=sup{∥un-xn∥:n∈ℕ}.
Notice from Lemma2.12 that
(3.23)‖yn+1-yn‖=‖Υn+1Nun+1-ΥnNun‖≤‖(un+1-λk,n+1GkΥn+1kun+1)-(un-λk,nGkΥnkun)‖+‖JMk,λk,n+1(un-λk,nGkΥnkun)-JMk,λk,n(un-λk,nGkΥnkun)‖≤‖un+1-un‖+|λk,n+1-λk,n|‖GkΥnkun‖+|λk,n+1-λk,n|λk,n+1‖JMk,λk,n+1(un-λk,nGkΥnkun)-(un-λk,nGkΥnkun)‖≤‖un+1-un‖+2M2|λk,n+1-λk,n|≤‖xn+1-xn‖+M1c|rn+1-rn|+2M2|λk,n+1-λk,n|,
where M2 is an appropriate constant such that
(3.24)M2=max⁡{sup⁡n≥1{‖GkΥnkun‖},sup⁡n≥1{‖JMk,λk,n+1(un-λk,nGkΥnkun)-(un-λk,nGkΥnkun)‖JMk,λk,n+1}}.
Since I-ξnB is nonexpansive mappings, we have the following estimates:
(3.25)‖vn+1-vn‖≤‖PC(yn+1-ξn+1Byn+1)-PC(yn-ξnByn)‖≤‖(yn+1-ξn+1Byn+1)-(yn-ξnByn)‖=‖(yn+1-ξn+1Byn+1)-(yn-ξn+1Byn)+(ξn-ξn+1)Byn‖≤‖(yn+1-ξn+1Byn+1)-(yn-ξn+1Byn)‖+|ξn-ξn+1|‖Byn‖=‖(I-ξn+1B)yn+1-(I-ξn+1B)yn‖+|ξn-ξn+1|‖Byn‖≤‖yn+1-yn‖+|ξn-ξn+1|‖Byn‖.
Substituting (3.23) into (3.25), we obtain
(3.26)‖vn+1-vn‖≤‖xn+1-xn‖+M1c|rn+1-rn|+2M2|λk,n+1-λk,n|+|ξn-ξn+1|‖Byn‖.
Indeed, define xn+1=(1-βn)zn+βnxn for all n∈ℕ. It follows that
(3.27)zn=xn+1-βnxn1-βn=αnf(xn)+γnSnvn1-βn.
Thus, we have
(3.28)‖zn+1-zn‖=‖αn+1f(xn+1)+γn+1Sn+1vn+11-βn+1-αnf(xn)+γnSnvn1-βn‖=‖αn+11-βn+1(f(xn+1)-f(xn))+γn+11-βn+1(Sn+1vn+1-Snvn)+(αn+11-βn+1-αn1-βn)f(xn)+(γn+11-βn+1-γn1-βn)Snvn‖≤αn+11-βn+1‖f(xn+1)-f(xn)‖+γn+11-βn+1‖Sn+1vn+1-Snvn‖+|αn+11-βn+1-αn1-βn|‖f(xn)-Snvn‖≤ψαn+11-βn+1‖xn+1-xn‖+γn+11-βn+1‖Sn+1vn+1-Snvn‖+|αn+11-βn+1-αn1-βn|‖f(xn)-Snvn‖.
Now, compute
(3.29)‖Sn+1vn+1-Snvn‖≤‖Sn+1vn+1-Sn+1vn‖+‖Sn+1vn-Snvn‖≤‖vn+1-vn‖+‖Sn+1vn-Snvn‖≤‖xn+1-xn‖+M1c|rn+1-rn|+|ξn-ξn+1|‖Byn‖+2M2|λk,n+1-λk,n|+‖Sn+1vn-Snvn‖.
Combining (3.28) and (3.29), we have
(3.30)‖zn+1-zn‖≤ψαn+11-βn+1‖xn+1-xn‖+γn+11-βn+1{‖xn+1-xn‖+M1c|rn+1-rn|+|ξn-ξn+1|‖Byn‖+2M2|λk,n+1-λk,n|+‖Sn+1vn-Snvn‖M1c}+|αn+11-βn+1-αn1-βn|‖f(xn)-Snvn‖≤‖xn+1-xn‖+γn+11-βn+1{M1c|rn+1-rn|+|ξn-ξn+1|‖Byn‖+2M2|λk,n+1-λk,n|M1c}+γn+11-βn+1‖Sn+1vn-Snvn‖+|αn+11-βn+1-αn1-βn|‖f(xn)-Snvn‖.
It follows that
(3.31)‖zn+1-zn‖-‖xn+1-xn‖≤γn+11-βn+1{M1c|rn+1-rn|+|ξn-ξn+1|‖Byn‖+2M2|λk,n+1-λk,n|}+γn+11-βn+1‖Sn+1vn-Snvn‖+|αn+11-βn+1-αn1-βn|‖f(xn)-Snvn‖≤γn+11-βn+1{M1c|rn+1-rn|+|ξn-ξn+1|‖Byn‖+2M2|λk,n+1-λk,n|}+γn+11-βn+1sup⁡{‖Sn+1z-Snz‖:z∈{vn}}+|αn+11-βn+1-αn1-βn|‖f(xn)-Snvn‖.
This together with conditions (C1)–(C6) and limn→∞sup{∥Sn+1z-Snz∥:z∈{vn}}=0 implies that
(3.32)limsup⁡n→∞(‖zn+1-zn‖-‖xn+1-xn‖)≤0.
Hence, by Lemma 2.6, we obtain ∥zn-xn∥→0 as n→∞. It then follows that
(3.33)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞(1-βn)‖zn-xn‖=0.
By (3.26), we also have
(3.34)lim⁡n→∞‖vn+1-vn‖=0.Step 3.
We claim thatlimn→∞∥Svn-vn∥=0.
Since{Gk:k=1,2,3,…,N} is β-inverse strongly monotone mappings, by the choice of {λk,n} for given x̃∈Ω and k∈{0,1,2,…,N-1}, we also have
(3.35)‖Υnk+1un-x̃‖2=‖JMk+1,λk+1,n(I-λk+1,nGk+1)Υnkun-JMk+1,λk+1,n(I-λk+1,nGk+1)x̃‖2≤‖(I-λk+1,nGk+1)Υnkun-(I-λk+1,nGk+1)x̃‖2=‖(Υnkun-λk+1,nGk+1Υnkun)-(x̃-λk+1,nGk+1x̃)‖2=‖(Υnkun-x̃)-λk+1,n(Gk+1Υnkun-Gk+1x̃)‖2=‖Υnkun-x̃‖2-2λk+1,n〈Υnkun-x̃,Gk+1Υnkun-Gk+1x̃〉+λk+1,n2‖Gk+1Υnkun-Gk+1x̃‖2≤‖Υnkun-x̃‖2-2λk+1,nβ‖Gk+1Υnkun-Gk+1x̃‖+λk+1,n2‖Gk+1Υnkun-Gk+1x̃‖2≤‖un-x̃‖2-2λk+1,nβ‖Gk+1Υnkun-Gk+1x̃‖+λk+1,n2‖Gk+1Υnkun-Gk+1x̃‖2≤‖xn-x̃‖2+λk+1,n(λk+1,n-2β)‖Gk+1Υnkun-Gk+1x̃‖2.
Form (3.13), we have
(3.36)‖xn+1-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖vn-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖yn-x̃‖2=αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖ΥnNun-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖Υnk+1un-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn{‖xn-x̃‖2+λk+1,n(λk+1,n-2β)‖Gk+1Υnkun-Gk+1x̃‖2}≤αn‖f(xn)-x̃‖2+‖xn-x̃‖2+γnλk+1,n(λk+1,n-2β)‖Gk+1Υnkun-Gk+1x̃‖2.
It follows that
(3.37)γnλk+1,n(2β-λk+1,n)‖Gk+1Υnkun-Gk+1x̃‖2≤γnc(2β-d)‖Gk+1Υnkun-Gk+1x̃‖2≤‖xn-xn+1‖(‖xn-x̃‖+‖xn+1-x̃‖)+αn‖f(xn)-x̃‖2.
By condition (C2), (3.33), and liminfn→∞γn>0, we obtain
(3.38)lim⁡n→∞‖Gk+1Υnkun-Gk+1x̃‖=0.
From Lemma 2.3(2) and as I-λk+1,nGk+1 is nonexpansive, we have
(3.39)‖Υnk+1un-x̃‖2=‖JMk+1,λk+1,n(I-λk+1,nGk+1)Υnkun-JMk+1,λk+1,n(I-λk+1,nGk+1)x̃‖2≤〈(I-λk+1,nGk+1)Υnkun-(I-λk+1,nGk+1)x̃,Υnk+1un-x̃〉=12{‖(I-λk+1,nGk+1)Υnkun-(I-λk+1,nGk+1)x̃‖2+‖Υnk+1un-x̃‖2-‖(I-λk+1,nGk+1)Υnkun-(I-λk+1,nGk+1)x̃-(Υnk+1un-x̃)‖2}≤12{‖Υnkun-x̃‖2+‖Υnk+1un-x̃‖2-‖(Υnkun-Υnk+1un)-λk+1,n(Gk+1Υnkun-Gk+1x̃)‖2}≤12{‖Υnkun-x̃‖2+‖Υnk+1un-x̃‖2-‖Υnkun-Υnk+1un‖2-λk+1,n2‖Gk+1Υnkun-Gk+1x̃‖2+2λk+1,n〈Υnkun-Υnk+1un,Gk+1Υnkun-Gk+1x̃〉},
which yields that
(3.40)‖Υnk+1un-x̃‖2≤‖Υnkun-x̃‖2-‖Υnkun-Υnk+1un‖2+2λk+1,n‖Υnkun-Υnk+1un‖‖Gk+1Υnkun-Gk+1x̃‖≤‖un-x̃‖2-‖Υnkun-Υnk+1un‖2+2λk+1,n‖Υnkun-Υnk+1un‖‖Gk+1Υnkun-Gk+1x̃‖≤‖xn-x̃‖2-‖Υnkun-Υnk+1un‖2+2λk+1,n‖Υnkun-Υnk+1un‖‖Gk+1Υnkun-Gk+1x̃‖.
Substituting (3.40) into (3.36), we obtain
(3.41)‖xn+1-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖Υnk+1un-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn{‖xn-x̃‖2-‖Υnkun-Υnk+1un‖2+2λk+1,n‖Υnkun-Υnk+1un‖‖Gk+1Υnkun-Gk+1x̃‖}≤αn‖f(xn)-x̃‖2+‖xn-x̃‖2-γn‖Υnkun-Υnk+1un‖2+2λk+1,nγn‖Υnkun-Υnk+1un‖‖Gk+1Υnkun-Gk+1x̃‖.
It follows that
(3.42)γn‖Υnkun-Υnk+1un‖2≤‖xn-xn+1‖(‖xn-x̃‖+‖xn+1-x̃‖)+αn‖f(xn)-x̃‖2+2λk+1,nγn‖Υnkun-Υnk+1un‖‖Gk+1Υnkun-Gk+1x̃‖.
By condition (C2), (3.33), (3.38), and liminfn→∞γn>0, we obtain
(3.43)lim⁡n→∞‖Υnkun-Υnk+1un‖=0.
For x̃∈Ω, we obtain
(3.44)‖vn-x̃‖2=‖PC(yn-ξnByn)-PC(x̃-ξnBx̃)‖2≤‖(yn-ξnByn)-(x̃-ξnBx̃)‖2=‖(yn-x̃)-ξn(Byn-Bx̃)‖2≤‖yn-x̃‖2-2ξn〈yn-x̃,Byn-Bx̃〉+ξn2‖Byn-Bx̃‖2≤‖yn-x̃‖2-2ξn{-ϕ‖Byn-Bx̃‖2+ω‖yn-x̃‖2}+ξn2‖Byn-Bx̃‖2≤‖yn-x̃‖2+2ξnϕ‖Byn-Bx̃‖2-2ξnω‖yn-x̃‖2+ξn2‖Byn-Bx̃‖2≤‖yn-x̃‖2+2ξnϕ‖Byn-Bx̃‖2-2ξnωμ2‖Byn-Bx̃‖2+ξn2‖Byn-Bx̃‖2≤‖xn-x̃‖2+(2ξnϕ+ξn2-2ξnωμ2)‖Byn-Bx̃‖2.
On the other hand, we have
(3.45)‖xn+1-x̃‖2=‖αnf(xn)+βnxn+γnSnvn-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖Snvn-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖vn-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn{‖xn-x̃‖2+(2ξnϕ+ξn2-2ξnωμ2)‖Byn-Bx̃‖2}=αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖xn-x̃‖2+γn(2ξnϕ+ξn2-2ξnωμ2)‖Byn-Bx̃‖2≤αn‖f(xn)-x̃‖2+‖xn-x̃‖2+γn(2ξnϕ+ξn2-2ξnωμ2)‖Byn-Bx̃‖2.
It follows that
(3.46)(2aωμ2-b2-2bϕ)γn‖Byn-Bx̃‖2≤(2ξnωμ2-ξn2-2ξnϕ)γn‖Byn-Bx̃‖2≤‖xn-x̃‖2-‖xn+1-x̃‖2+αn‖f(xn)-x̃‖2≤‖xn-xn+1‖(‖xn-x̃‖+‖xn+1-x̃‖)+αn‖f(xn)-x̃‖2.
It now follows from the last inequality, conditions (C2), (3.33), and liminfn→∞γn>0 that
(3.47)lim⁡n→∞‖Byn-Bx̃‖=0.
Since PC is firmly nonexpansive, we have
(3.48)‖vn-x̃‖2=‖PC(yn-ξnByn)-PC(x̃-ξnBx̃)‖2=‖PC(I-ξnB)yn-PC(I-ξnB)x̃‖2≤〈(I-ξnB)yn-(I-ξnB)x̃,vn-x̃〉=12{‖(I-αnB)yn-(I-ξnB)x̃‖2+‖vn-x̃‖2-‖(I-ξnB)yn-(I-ξnB)x̃-(vn-x̃)‖2}≤12{‖yn-x̃‖2+‖vn-x̃‖2-‖(yn-vn)-ξn(Byn-Bx̃)‖2}≤12{‖yn-x̃‖2+‖vn-x̃‖2-‖yn-vn‖2-ξn2‖Byn-Bx̃‖2+2ξn〈yn-vn,Byn-Bx̃〉},
which yields that
(3.49)‖vn-x̃‖2≤‖yn-x̃‖2-‖yn-vn‖2+2ξn‖yn-vn‖‖Byn-Bx̃‖≤‖xn-x̃‖2-‖yn-vn‖2+2ξn‖yn-vn‖‖Byn-Bx̃‖.
Substituting (3.49) into (3.45), we obtain
(3.50)‖xn+1-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖vn-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn{‖xn-x̃‖2-‖yn-vn‖2+2ξn‖yn-vn‖‖Byn-Bx̃‖}=αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖xn-x̃‖2-γn‖yn-vn‖2+2γnξn‖yn-vn‖‖Byn-Bx̃‖≤αn‖f(xn)-x̃‖2+‖xn-x̃‖2-γn‖yn-vn‖2+2γnξn‖yn-vn‖‖Byn-Bx̃‖.
It follows that
(3.51)γn‖yn-vn‖2≤‖xn-xn+1‖(‖xn-x̃‖+‖xn+1-x̃‖)+αn‖f(xn)-x̃‖2+2γnξn‖yn-vn‖‖Byn-Bx̃‖.
By condition (C2), (3.33), (3.47), and liminfn→∞γn>0, we obtain
(3.52)lim⁡n→∞‖yn-vn‖=0.
On the other hand, in the light of Lemma 2.11(ii) Trn is firmly nonexpansive; so we have
(3.53)‖un-x̃‖2=‖Trnxn-Trnx̃‖2≤〈Trnxn-Trnx̃,xn-x̃〉=〈un-x̃,xn-x̃〉=12(‖un-x̃‖2+‖xn-x̃‖2-‖xn-un‖2),
which implies that
(3.54)‖un-x̃‖2≤‖xn-x̃‖2-‖xn-un‖2.
Form (3.45), we have
(3.55)‖xn+1-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖vn-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖yn-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖un-x̃‖2≤αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn{‖xn-x̃‖2-‖xn-un‖2}=αn‖f(xn)-x̃‖2+βn‖xn-x̃‖2+γn‖xn-x̃‖2-γn‖xn-un‖2≤αn‖f(xn)-x̃‖2+‖xn-x̃‖2-γn‖xn-un‖2.
It follows that
(3.56)γn‖xn-un‖2≤‖xn-xn+1‖(‖xn-x̃‖+‖xn+1-x̃‖)+αn‖f(xn)-x̃‖2.
By condition (C2), (3.33), and liminfn→∞γn>0, we obtain
(3.57)lim⁡n→∞‖xn-un‖=0.
Observe that
(3.58)xn+1-xn=αn(f(xn)-xn)+γn(Snvn-xn).
By condition (C2) and (3.33), we have
(3.59)lim⁡n→∞⁡γn‖Snvn-xn‖=lim⁡n→∞⁡(‖xn+1-xn‖-αn‖f(xn)-xn‖)=0.
Since
(3.60)‖Svn-un‖≤‖Svn-xn‖+‖xn-un‖,
from (3.57) and (3.59), we have
(3.61)lim⁡n→∞⁡‖Snvn-un‖=0.
Form (3.55), we have
(3.62)‖xn+1-x̃‖2≤αn‖f(xn)-x̃‖2+‖xn-x̃‖2-γn‖xn-un‖2≤αn‖f(xn)-x̃‖2+‖xn-x̃‖2-γn‖(xn-yn)+(yn-un)‖2≤αn‖f(xn)-x̃‖2+‖xn-x̃‖2-γn{‖xn-yn‖2+2‖xn-yn‖‖yn-un‖+‖yn-un‖2}=αn‖f(xn)-x̃‖2+‖xn-x̃‖2-γn‖xn-yn‖2-2γn‖xn-yn‖‖yn-un‖-γn‖yn-un‖2≤αn‖f(xn)-x̃‖2+‖xn-x̃‖2-γn‖xn-yn‖2.
It follows that
(3.63)γn‖xn-yn‖2≤‖xn-xn+1‖(‖xn-x̃‖+‖xn+1-x̃‖)+αn‖f(xn)-x̃‖2.
By condition (C2), (3.33), and liminfn→∞γn>0, we obtain
(3.64)lim⁡n→∞‖xn-yn‖=0.
Since
(3.65)‖un-yn‖≤‖un-xn‖+‖xn-yn‖,
from (3.57) and (3.64), we have
(3.66)lim⁡n→∞⁡‖un-yn‖=0.
Furthermore, by the triangular inequality we also have
(3.67)‖Snvn-vn‖≤‖Snvn-un‖+‖un-yn‖+‖yn-vn‖.
From (3.52), (3.61), and (3.66), we have
(3.68)lim⁡n→∞⁡‖Snvn-vn‖=0.
Applying Lemma 2.9 and (3.68), we have
(3.69)‖Svn-vn‖≤‖Svn-Snvn‖+‖Snvn-vn‖≤sup⁡{‖Sz-Snz‖:z∈{vn}}+‖Snvn-vn‖⟶0.Step 4.
We claim thatlimsupn→∞〈f(x*)-x*,xn-x*〉≤0.
Indeed, we choose a subsequence{vni} of {vn} such that
(3.70)limsup⁡n→∞〈f(x*)-x*,Svn-x*〉=lim⁡i→∞〈f(x*)-x*,Svni-x*〉.
Without loss of generality, let {vni}⇀z∈C. From ∥Svn-vn∥→0, we obtain Svni⇀z. Then, (3.70) reduces to
(3.71)limsup⁡n→∞〈f(x*)-x*,Svn-x*〉=〈f(x*)-x*,z-x*〉.
In order to show 〈f(x*)-x*,z-x*〉≤0, it suffices to show that
(3.72)z∈Ω:⋂n=1∞F(Sn)∩(⋂k=1NI(Gk,Mk))∩VI⁡(C,B)∩EP⁡(F).
Firstly, we will show z∈F(S)=⋂n=1∞F(Sn).
Assumez∉F(S). By Opial’s theorem (Lemma 2.5) and ∥Svn-vn∥→0, we have
(3.73)liminf⁡i→∞‖vni-z‖<liminf⁡i→∞‖vni-Sz‖=liminf⁡i→∞‖vni-Svni+Svni-Sz‖=liminf⁡i→∞‖Svni-Sz‖≤liminf⁡i→∞‖vni-z‖.
This is a contradiction. Thus, we obtain z∈F(S).
Next, we will show thatz∈VI(C,B).
Let(3.74)Tw1={Bw1+NCw1,w1∈C;∅,w1∉C.
Since B is relaxed (ϕ,ω)-cocoercive, μ-Lipschitz continuous with ω>ϕμ2, we obtain
(3.75)〈Bx-By,x-y〉≥(-ϕ)‖Bx-By‖2+ω‖x-y‖2≥(ω-ϕμ2)‖x-y‖2≥0,
which yields that B is monotone. Then T is maximal monotone (see [23]). Let (w1,w2)∈G(T). Since w2-Bw1∈NC(w1) and vn∈C, we have 〈w1-vn,w2-Bw1〉≥0. On the other hand, from vn=PC(yn-ξnByn), we have
(3.76)〈w1-vn,vn-(yn-ξnByn)〉≥0;
that is,
(3.77)〈w1-vn,vn-ynξn+Byn〉≥0.
Therefore, we obtain
(3.78)〈w1-vni,w2〉≥〈w1-vni,Bw1〉≥〈w1-vni,Bw1〉-〈w1-vni,vni-yniξni+Byni〉=〈w1-vni,Bw1-Byni-vni-yniξni〉=〈w1-vni,Bw1-Bvni〉+〈w1-vni,Bvni-Byni〉-〈w1-vni,vni-yniξni〉≥〈w1-vni,Bvni〉-〈w1-vni,vni-yniξni+Byni〉=〈w1-vni,Bvni-Byni〉-〈w1-vni,vni-yniξni〉.
Noting that ∥vni-yni∥→0 and B is relaxed (ϕ,ω)-cocoercive and (3.78), we obtain
(3.79)〈w1-z,w2〉≥0.
Since T is maximal monotone, we have z∈T-10, and hence z∈VI(C,B).
Now, we will show thatz∈⋂k=1NI(Gk,Mk).
For this purpose, letk∈{1,2,3,…,N} and Gk is β-inverse strongly monotone, Gk is an 1/β-Lipschitz continuous monotone mapping. From Lemma 2.2, we know that Mk+Gk is maximal monotone. Let (v,g)∈G(Mk+Gk); that is, g-Gkv∈Mk(v). On the other hand, since Υnikuni=JMk,λk,ni(Υnik-1uni-λk,niGkΥnik-1uni), we have
(3.80)Υnikuni-λk,niGkΥnikuni∈(I+λk,niMk)(Υnikuni);
that is,
(3.81)1λk,ni(Υnik-1uni-Υnikuni-λk,niGkΥnik-1uni)∈Mk(Υnikuni).
By virtue of the maximal monotonicity of Mk+Gk, we have
(3.82)〈v-Υnikuni,g-Gkv-1λk,ni(Υnik-1uni-Υnikuni-λk,niGkΥnik-1uni)〉≥0,
and so
(3.83)〈v-Υnikuni,g〉≥〈v-Υnikuni,Gkv+1λk,ni(Υnik-1uni-Υnikuni-λk,niGkΥnik-1uni)〉=〈1λk,niv-Υnikuni,Gkv-GkΥnikuni+GkΥnikuni-GkΥnik-1uni+1λk,ni(Υnik-1uni-Υnikuni)〉≥0+〈v-Υnikuni,GkΥnikuni-GkΥnik-1uni〉+〈v-Υnikuni,1λk,ni(Υnik-1uni-Υnikuni)〉.
From ∥Υnkun-Υnk+1un∥→0, we also obtain that Υnikuni⇀z and {Gk:k=1,2,3,…,N} are Lipschitz continuous; we have
(3.84)lim⁡n→∞〈v-Υnikuni,g〉=〈v-z,g〉≥0.
Since Mk+Gk is maximal monotone, we have θ∈(Mk+Gk)(z); that is, z∈⋂k=1NI(Gk,Mk).
Finally, we will show thatz∈EP(F).
Sinceun=Trnxn, we have
(3.85)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C.
If follows from (A2) that
(3.86)1rn〈y-un,un-xn〉≥-F(un,y)≥F(y,un),
and hence
(3.87)〈y-uni,uni-xnirni〉≥F(y,uni).
Since (uni-xni)/rni→0 and uni⇀z, it follows by (A4) that F(y,z)≤0 for all y∈H. For t with 0<t≤1 and y∈H, let yt=ty+(1-t)z. Since y∈H and z∈H, we have yt∈H, and hence F(yt,z)≤0. So, from (A1) and (A4) we have
(3.88)0=F(yt,yt)≤tF(yt,y)+(1-t)F(yt,z)≤tF(yt,y),
and hence F(yt,y)≥0. From (A3), we have F(z,y)≥0 for all y∈H and hence z∈EP(F). Therefore, it follows that z∈Ω.
Sincex*=PΩf(x*), we have
(3.89)limsup⁡n→∞〈f(x*)-x*,xn-x*〉=limsup⁡n→∞〈f(x*)-x*,Svn-x*〉=lim⁡i→∞〈f(x*)-x*,Svni-x*〉=〈f(x*)-x*,z-x*〉≤0.
On the other hand, we have
(3.90)limsup⁡n→∞〈f(x*)-x*,xn+1-x*〉≤limsup⁡n→∞〈f(x*)-x*,xn+1-xn〉+limsup⁡n→∞〈f(x*)-x*,xn-x*〉.
Since ∥xn+1-xn∥→0 as n→∞ and (3.91), we have
(3.91)limsup⁡n→∞〈f(x*)-x*,xn+1-x*〉≤0.Step 5.
We claim thatlimn→∞∥xn-x*∥=0.
Indeed, from (3.2) and (3.12), we obtain
(3.92)‖xn+1-x*‖2=〈αnf(xn)+βnxn+γnSnvn-x*,xn+1-x*〉=αn〈f(xn)-x*,xn+1-x*〉+βn〈xn-x*,xn+1-x*〉+γn〈Snvn-x*,xn+1-x*〉≤12βn(‖xn-x*‖2+‖xn+1-x*‖2)+12γn(‖vn-x*‖2+‖xn+1-x*‖2)+αn〈f(xn)-f(x*),xn+1-x*〉+αn〈f(x*)-x*,xn+1-x*〉≤12(1-αn)(‖xn-x*‖2+‖xn+1-x*‖2)+12αn(‖f(xn)-f(x*)‖2+‖xn+1-x*‖2)+αn〈f(x*)-x*,xn+1-x*〉≤12[1-αn(1-ψ2)]‖xn-x*‖2+12(1-αn)‖xn+1-x*‖2+12αn‖xn+1-x*‖2+αn〈f(x*)-x*,xn+1-x*〉,
which implies that
(3.93)‖xn+1-x*‖2≤[1-αn(1-ψ2)]‖xn-x*‖2+2αn〈f(x*)-x*,xn+1-x*〉=(1-bn)‖xn-x*‖2+δn,
where bn=αn(1-ψ2) and δn=2αn〈f(x*)-x*,xn+1-x*〉. It is easy to see that bn→0, ∑n=1∞bn=∞, and limsupn→∞δn/bn≤0. Applying Lemma 2.7 to (3.93), we conclude that
(3.94)xn→x*=PΩf(x*).
Consequently, also {yn} converges strongly to x*. The proof is now complete.As in [21, Theorem  4.1], we can generate a sequence {Sn} of nonexpansive mappings satisfying condition ∑n=1∞sup{∥Sn+1z-Snz∥:z∈K}<∞ for any bounded subset K of C by using convex combination of general sequence {Tk} of nonexpansive mappings with a common fixed point.Corollary 3.2.
LetC be a nonempty closed convex subset of a real Hilbert space H, and let B:C→H be relaxed (ϕ,ω)-cocoercive and μ-Lipschitz continuous with ω>ϕμ2, for some ϕ,ω,μ>0. Let 𝒢={Gk:k=1,2,3,…,N} be a finite family of β-inverse strongly monotone mappings from C into H, and let F be a bifunction from C×C→ℝ satisfying (A1)–(A4). Let f:C→C be a contraction with coefficient ψ(0≤ψ<1), and let {δnk} be a family of nonnegative numbers with indices n,k∈ℕ with k≤n such that
(3.95)Ω:F(⋂k=1∞F(Tk))∩(⋂k=1NI(Gk,Mk))∩VI⁡(C,B)∩EP⁡(F)≠∅.
Let the sequences {xn} and {yn} be generated by
(3.96)x1=x∈C chosen arbitrarily,yn=JMN,λN,n(I-λN,nGn)⋅⋅⋅JM2,λ2,n(I-λ2,nG2)JM1,λ1,n(I-λ1,nG1)Trnxn,xn+1=αnf(xn)+βnxn+γn∑k=1nδnkTkPC(yn-ξnByn),∀n≥1,
where {αn},{βn}, {γn}⊂(0,1) and {ξn},{rn}⊂(0,∞) satisfy the following conditions:(C1)
αn+βn+γn=1,(C2)
limn→∞αn=0,∑n=1∞αn=∞,(C3)
0<liminfn→∞βn≤limsupn→∞βn<1,(C4)
{ξn}⊂[a,b] for some a,b with 0≤a≤b≤2(ω-ϕμ2)/μ2 and limn→∞|ξn+1-ξn|=0,(C5)
{λk,n}k=1N⊂[c,d]⊂(0,2β) and limn→∞|λk,n+1-λk,n|=0, for each k∈{1,2,…,N},(C6)
liminfn→∞rn>0 and limn→∞|rn+1-rn|=0,(C7)
∑k=1nδnk,foralln∈ℕ, limn→∞δnk>0,forallk∈ℕ and ∑n=1k∑k=1n|δn+1k-δnk|<∞.Then, the sequences {xn} and {yn} converge strongly to the same point x*∈Ω, where x*=PΩf(x*).In Theorem3.1, taking N=1 and Sn=S, then we have the following corollary.Corollary 3.3.
LetC be a nonempty closed convex subset of a real Hilbert space H, and let B:C→H be relaxed (ϕ,ω)-cocoercive and μ-Lipschitz continuous with ω>ϕμ2, for some ϕ,ω,μ>0. Let G be an β-inverse strongly monotone mappings from C into H, and let F be a bifunction from C×C→ℝ satisfying (A1)–(A4). Let f:C→C be a contraction with coefficient ψ(0≤ψ<1), and let S be a nonexpansive mappings of C into itself such that
(3.97)Ω:F(S)∩I(G,M)∩VI⁡(C,B)∩EP⁡(F)≠∅.
Let the sequences {xn} and {yn} be generated by
(3.98)x1=x∈C chosen arbitrarily,F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=JM,λn(I-λnG)un,xn+1=αnf(xn)+βnxn+γnSPC(yn-ξnByn),∀n≥1,
where {αn},{βn},{γn}⊂(0,1) and {ξn},{rn}⊂(0,∞) satisfy the following conditions:(C1)
αn+βn+γn=1,(C2)
limn→∞αn=0,∑n=1∞αn=∞,(C3)
0<liminfn→∞βn≤limsupn→∞βn<1,(C4)
{ξn}⊂[a,b] for some a,b with 0≤a≤b≤2(ω-ϕμ2)/μ2 and limn→∞|ξn+1-ξn|=0,(C5)
{λn}⊂[c,d]⊂(0,2β) and limn→∞|λn+1-λn|=0,(C6)
liminfn→∞rn>0 and limn→∞|rn+1-rn|=0. Then, the sequences {xn} and {yn} converge strongly to the same point x*∈Ω, where x*=PΩf(x*).



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*Source: 816529-2012-05-13.xml*