# A Viscosity Hybrid Steepest Descent Method for Generalized Mixed Equilibrium Problems and Variational Inequalities for Relaxed Cocoercive Mapping in Hilbert Spaces

Authors: Wanpen Chantarangsi; Chaichana Jaiboon; Poom Kumam
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We present an iterative method for fixed point problems, generalized mixed equilibrium problems,
and variational inequality problems. Our method is based on the so-called viscosity hybrid steepest descent
method. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping, the
set of solutions of generalized mixed equilibrium problems, and the set of solutions of variational inequality problems for
a relaxed cocoercive mapping in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative
scheme to the unique solution of variational inequality. The results presented in this paper generalize and extend some
well-known strong convergence theorems in the literature.


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## Body

## 1. Introduction

Throughout this paper, unless otherwise specified, we considerH to be a real Hilbert space with inner product 〈·,·〉 and its induced norm ∥·∥. Let C be a nonempty closed convex subset of H and let PC be the metric projection of H onto the closed convex subset C. Let S:C→C be a nonexpansive mapping, that is, ∥Sx-Sy∥≤∥x-y∥ for all x,y∈C. The fixed point set of S is defined by
(1.1)F(S)={x∈C:Sx=x}.
If C⊂H is nonempty, bounded, closed, and convex and S is a nonexpansive mapping of C into itself, then F(S) is nonempty; see, for example, [1, 2]. A mapping f:C→C is a contraction on C if there exists a constant η∈(0,1) such that ∥f(x)-f(y)∥≤η∥x-y∥ for all x,y∈C. In addition, let Ψ:C→H be a nonlinear mapping. Let φ:C→ℝ∪{+∞} be a real-valued function and let Θ:C×C→ℝ be a bifunction such that C∩domφ≠∅, where ℝ is the set of real numbers and domφ={x∈C:φ(x)<+∞}.The generalized mixed equilibrium problem for findingx∈C(1.2)Θ(x,y)+〈Ψx,y-x〉+φ(y)-φ(x)≥0,∀y∈C.
The set of solutions of (1.2) is denoted by GMEP(Θ,φ,Ψ), that is,
(1.3)GMEP(Θ,φ,Ψ)={x∈C:Θ(x,y)+〈Ψx,y-x〉+φ(y)-φ(x)≥0,∀y∈C}.
We see that if x is a solution of a problem (1.2), then x∈domφ.Special Examples
(1)
IfΨ=0, then the problem (1.2) is reduced into the mixed equilibrium problem for finding x∈C such that
(1.4)Θ(x,y)+φ(y)-φ(x)≥0,∀y∈C.
The set of solutions of (1.4) is denoted by MEP(Θ,φ).(2)
Ifφ=0, then the problem (1.2) is reduced into the generalized equilibrium problem for finding x∈C such that
(1.5)Θ(x,y)+〈Ψx,y-x〉≥0,∀y∈C.
The set of solutions of (1.5) is denoted by GEP(Θ,Ψ).(3)
IfΨ=0 and φ=0, then the problem (1.2) is reduced into the equilibrium problem for finding x∈C such that
(1.6)Θ(x,y)≥0,∀y∈C.
The set of solutions of (1.6) is denoted by EP(Θ).(4)
IfΘ=0,φ=0, and Ψ=B, then the problem (1.2) is reduced into the variational inequality problem for finding x∈C such that
(1.7)〈Bx,y-x〉≥0,∀y∈C.
The set of solutions of (1.7) is denoted by VI(C,B).The generalized mixed equilibrium problem is very general in the sense that it includes, as special cases, fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems in noncooperative games, the equilibrium problem, and Numerous problems in physics, economics, and others. Some methods have been proposed to solve problem (1.2); see, for instance, [3, 4] and the references therein.LetB:C→H be a nonlinear mapping. Now, we recall the following definitions.(d1)
B is said to be monotone if for each x,y∈C(1.8)〈Bx-By,x-y〉≥0.(d2)
B is said to be ρ-strongly monotone if there exists a positive real number ρ such that
(1.9)〈Bx-By,x-y〉≥ρ∥x-y∥2,∀x,y∈C.(d3)
B is said to be ω-Lipschitz continuous if there exists a positive real number ω such that
(1.10)∥Bx-By∥≤ω∥x-y∥,∀x,y∈C.(d4)
B is said to be ξ-inverse-strongly monotone if there exists a constant ξ>0 such that
(1.11)〈Bx-By,x-y〉≥ξ∥Bx-By∥2,∀x,y∈C.(d5)
B is said to be relaxed (u,v)-cocoercive if there exist positive real numbers u,v such that
(1.12)〈Bx-By,x-y〉≥(-u)∥Bx-By∥2+v∥x-y∥2,∀x,y∈C.(d6)
A set-valued mappingQ:H→2H is called monotone if for all x,y∈H, f∈Qx and g∈Qy imply 〈x-y,f-g〉≥0.(d7)
A monotone mappingQ:H→2H is called maximal if the graph G(Q) of Q is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping Q is maximal if and only if for (x,f)∈H×H, 〈x-y,f-g〉≥0 for every (y,g)∈G(Q) implies f∈Qx.For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequalities for aξ-inverse-strongly monotone mapping, Takahashi and Toyoda [5] introduced the following iterative scheme:
(1.13)x0∈Cchosenarbitrary,xn+1=γnxn+(1-γn)SPC(xn-αnBxn),∀n≥0,
where B is a ξ-inverse-strongly monotone mapping, {γn} is a sequence in (0,1), and {αn} is a sequence in (0,2ξ). They showed that if F(S)∩VI(C,B) is nonempty, then the sequence {xn} generated by (1.13) converges weakly to some z∈F(S)∩VI(C,B).For finding an element ofVI(C,B), Iiduka et al. [6] introduced the following iterative scheme:(1.14)x0∈Cchosenarbitrary,xn+1=PC(γnxn+(1-γn)PC(xn-αnBxn)),∀n≥0,
where B is a ξ-inverse-strongly monotone mapping, {γn} is a sequence in (-1,1), and {αn} is a sequence in (0,2ξ). They showed that if VI(C,B) is nonempty, then the sequence {xn} generated by (1.14) converges weakly to some z∈VI(C,B).For finding a common element ofF(S)∩VI(C,B), let S:H→H be a nonexpansive mapping. Yamada [7] introduced the following iterative scheme called the hybrid steepest descent method:
(1.15)xn+1=Sxn-αnμBSxn,∀n≥1,
where x1=x∈H,{αn}⊂(0,1),B:H→H is a strongly monotone and Lipschitz continuous mapping, and μ is a positive real number. He proved that the sequence {xn} generated by (1.15) converges strongly to the unique solution of the F(S)∩VI(C,B).The hybrid steepest descent method is constructed by blending important ideas in the steepest descent method and in the fixed point theory. The remarkable applicability of this method to the convexly constrained generalized pseudoinverse problem as well as to the convex feasibility problem is demonstrated by constructing nonexpansive mappings whose fixed point sets are the feasible sets of the problems.On the other hand, Shang et al. [8] introduced a new iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequalities for relaxed (u,v)-cocoercive mappings in a real Hilbert space by using viscosity approximation method. Let S:C→C be a nonexpansive mapping and let f:C→C be a contraction mapping. Starting with arbitrary initial x1∈C and define sequences {xn} recursively by
(1.16)xn+1=ϵnf(xn)+βnxn+γnSPC(xn-αnBxn),∀n≥1.
They proved that under certain appropriate conditions imposed on {ϵn},{βn},{γn}, and {αn}, the sequence {xn} converges strongly to z∈F(S)∩VI(C,B), where z=PF(S)∩VI(C,B)f(z).For finding a common element ofF(S)∩GEF(Θ,Ψ), let C be a nonempty closed convex subset of a real Hilbert space H. Let Ψ be a ξ-inverse-strongly monotone mapping of C into H and let S be a nonexpansive mapping of C into itself. S. Takahashi and W. Takahashi [9] introduced the following iterative scheme:
(1.17)Θ(un,y)+〈Ψxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=αnx+(1-αn)un,xn+1=γnxn+(1-γn)Syn,
where {αn}⊂[0,1],{γn}⊂[0,1], and {rn}⊂[0,2ξ] satisfy some parameters controlling conditions. They proved that the sequence {xn} defined by (1.17) converges strongly to a common element of F(S)∩GEF(Θ,Ψ).Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [7, 10–12] and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences.A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping defined on a real Hilbert spaceH:
(1.18)minx∈F12〈Ax,x〉-〈x,b〉,
where F is the fixed point set of a nonexpansive mapping S defined on H and b is a given point in H.A linear bounded operatorA is strongly positive if there exists a constant γ¯>0 with the property
(1.19)〈Ax,x〉≥γ¯∥x∥2,∀x∈H.Recently, Marino and Xu [13] introduced a new iterative scheme by the viscosity approximation method:
(1.20)xn+1=ϵnγf(xn)+(1-ϵnA)Sxn.
They proved that the sequence {xn} generated by (1.20) converges strongly to the unique solution of the variational inequality:
(1.21)〈γfz-Az,x-z〉≤0,∀x∈F(S),
which is the optimality condition for the minimization problem:
(1.22)minx∈F(S)12〈Ax,x〉-h(x),
where h is a potential function for γf.In 2008, Qin et al. [14] proposed the following iterative algorithm:
(1.23)Θ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈H,xn+1=ϵnγf(xn)+(I-ϵnA)SPC(un-αnBun),
where A is a strongly positive linear bounded operator and B is a relaxed cocoercive mapping of C into H. They proved that if the sequences {ϵn},{αn}, and {rn} of parameters satisfy appropriate condition, then the sequence {xn} defined by (1.23) converges strongly to the unique solution z of the variational inequality:
(1.24)〈γfz-Az,x-z〉≤0,∀x∈F(S)∩VI(C,B)∩EP(Θ),
which is the optimality condition for the minimization problem:
(1.25)minx∈F(S)∩VI(C,B)∩EP(Θ)12〈Ax,x〉-h(x),
where h is a potential function for γf.In this paper, we introduce an iterative scheme by using a viscosity hybrid steepest descent method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of a nonexpansive mapping, and the set of solutions of variational inequality problem for a relaxed cocoercive mapping in a real Hilbert space. The results shown in this paper improve and extend the recent ones announced by many others.


## 2. Preliminaries

Throughout this paper, we always assume thatH is a real Hilbert space and C is a nonempty closed convex subset of H. For a sequence {xn}, the notation of xn⇀x and xn→x means that the sequence {xn} converges weakly and strongly to x, respectively.The following lemmata give some characterizations and useful properties of the metric projectionPC in a real Hilbert space. The metric (or nearest point) projection from H onto C is the mapping PC:H→C which assigns to each point x∈H the unique point PCx∈C satisfying the following property:
(2.1)∥x-PCx∥=infy∈C∥x-y∥.Lemma 2.1.
It is well known that the metric projectionPC has the following properties:  (m1)
for eachx∈H and z∈C,
(2.2)z=PCx⇔〈x-z,y-z〉≤0,∀y∈C;(m2)
PC:H→C is nonexpansive, that is,
(2.3)∥PCx-PCy∥≤∥x-y∥,∀x,y∈H;(m3)
PC is firmly nonexpansive, that is,
(2.4)∥PCx-PCy∥2≤〈PCx-PCy,x-y〉∀x,y∈H.In order to prove our main results, we also need the following lemmata.Lemma 2.2 (see [2]).
LetH be a Hilbert space, let C be a nonempty closed convex subset of H, and let B be a mapping of C into H. Let x*∈C. Then, for λ>0,
(2.5)x*∈VI(C,B)⇔x*=PC(x*-λBx*),
that is,
(2.6)x*∈VI(C,B)⇔x*∈F(PC(I-λB)),
where PC is the metric projection of H onto C.Lemma 2.3 (see [15]).
LetB be a monotone mapping of C into H and let NCw1 be the normal cone to C at w1∈C, that is,
(2.7)NCw1={w∈H:〈w1-w2,w〉≥0,∀w2∈C},
and define a mapping Q on C by
(2.8)Qw1={Bw1+NCw1,w1∈C,∅,w1∉C.
Then Q is maximal monotone and 0∈Qw1 if and only if 〈Bw1,w2-w1〉≥0 for all w2∈C.Lemma 2.4 (see [16]).
Each Hilbert spaceH satisfies Opials condition; that is, for any sequence {xn}⊂H with xn⇀x, the inequality
(2.9)lim infn→∞∥xn-x∥<lim infn→∞∥xn-y∥
holds for each y∈H with y≠x.Lemma 2.5 (see [13]).
LetC be a nonempty closed convex subset of H, let f be a contraction of H into itself with coefficient η∈(0,1), and let A be a strongly positive linear bounded operator on H with coefficient γ¯>0. Then, for 0<γ<γ¯/η,
(2.10)〈x-y,(A-γf)x-(A-γf)y〉≥(γ¯-ηγ)∥x-y∥2,x,y∈H.
That is, A-γf is strongly monotone with coefficient γ¯-ηγ.Lemma 2.6 (see [13]).
Assume thatA is a strongly positive linear bounded operator on H with coefficient γ¯>0 and 0<ρ≤∥A∥-1. Then ∥I-ρA∥≤1-ργ¯.Lemma 2.7 (see [17]).
Let{xn} and {yn} be bounded sequences in a Banach space X and let {γn} be a sequence in [0,1] with
(2.11)0<lim infn→∞γn≤lim supn→∞γn<1.
Suppose
(2.12)xn+1=(1-γn)yn+γnxn,∀n≥0,lim supn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0.
Then, limn→∞∥yn-xn∥=0.Lemma 2.8 (see [18]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.13)an+1≤(1-ϱn)an+σn,n≥0,
where {ϱn} is a sequence in (0,1) and {σn} is a sequence in ℝ such that(1)
∑n=1∞ϱn=∞,(2)
lim supn→∞(σn/ϱn)≤0 or ∑n=1∞|σn|<∞.Then limn→∞an=0.For solving the generalized mixed equilibrium problem and the mixed equilibrium problem, let us give the following assumptions for the bifunctionΘ, the function φ, and the set C:  (H1)
Θ(x,x)=0,∀x∈C;(H2)
Θ is monotone, that is, Θ(x,y)+Θ(y,x)≤0,∀x,y∈C;(H3)
for eachy∈C,x↦Θ(x,y) is weakly upper semicontinuous;(H4)
for eachx∈C,y↦Θ(x,y) is convex;(H5)
for eachx∈C,y↦Θ(x,y) is lower semicontinuous;(B1)
for eachx∈H and r>0, there exist abounded subsets Dx⊆C and yx∈C such that for any z∈C∖Dx,
(2.14)Θ(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0;(B2)
C is a bounded set.Lemma 2.9 (see [19]).
LetC be a nonempty closed convex subset of H. Let Θ:C×C→ℝ be a bifunction that satisfies (H1)–(H5) and let φ:C→ℝ∪{+∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r>0 and x∈H, define a mapping Tr(Θ,φ):H→C as follows:
(2.15)Tr(Θ,φ)(x)={z∈C:Θ(z,y)+φ(y)-φ(z)+1r〈y-z,z-x〉≥0,∀y∈C}
for all z∈H. Then, the following properties hold:(i)
for eachx∈H,Tr(Θ,φ)(x)≠∅;(ii)
Tr(Θ,φ) is single-valued;(iii)
Tr(Θ,φ) is firmly nonexpansive; that is, for any x,y∈H,(2.16)∥Tr(Θ,φ)x-Tr(Θ,φ)y∥2≤〈Tr(Θ,φ)x-Tr(Θ,φ)y,x-y〉;(iv)
F(Tr(Θ,φ))=MEP(Θ,φ);(v)
MEP(Θ,φ) is closed and convex.Remark 2.10.
Ifφ=0, then Tr(Θ,φ) is rewritten as TrΘ.Lemma 2.11 (see [9]).
LetC,H,Θ, and TrΘ be as in Remark 2.10. Then the following holds:
(2.17)∥TsΘx-TtΘx∥2≤s-ts〈TsΘx-TtΘx,TsΘx-x〉
for all s,t>0 and x∈H.The following lemma is an immediate consequence of an inner product.Lemma 2.12.
LetH be a real Hilbert space, let x and y be elements in H, and let λ∈[0,1]. Then  (1)
∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)∥x-y∥2,(2)
∥x+y∥2≤∥x∥2+2〈y,x+y〉.


## 3. Main Results

In this section, we will introduce an iterative scheme by using a viscosity hybrid steepest descent method for finding a common element of the set of fixed points for nonexpansive mappings, the set of solutions of a generalized mixed equilibrium problem, and the set of solutions of variational inequality problem for a relaxed cocoercive mapping in a real Hilbert space. We show that the iterative sequence converges strongly to a common element of the three sets.In order to prove our main results, we first prove the following lemmata.Lemma 3.1.
LetC,H,Θ,φ, and Tr(Θ,φ) be as in Lemma 2.9. Then the following holds:
(3.1)∥Ts(Θ,φ)x-Tt(Θ,φ)x∥2≤s-ts〈Ts(Θ,φ)x-Tt(Θ,φ)x,Ts(Θ,φ)x-x〉
for all s,t>0 and x∈H.Proof.
By similar argument as in the proof of Lemma  2.11 in [9], for s,t>0 and x∈H. Observing that ℑ=Ts(Θ,φ)x and ϑ=Tt(Θ,φ)x, we have
(3.2)Θ(ℑ,y)+φ(y)-φ(ℑ)+1s〈y-ℑ,ℑ-x〉≥0,∀y∈C,(3.3)Θ(ϑ,y)+φ(y)-φ(ϑ)+1t〈y-ϑ,ϑ-x〉≥0,∀y∈C.
Putting y=ϑ in (3.2) and y=ℑ in (3.3), we obtain
(3.4)Θ(ℑ,ϑ)+φ(ϑ)-φ(ℑ)+1s〈ϑ-ℑ,ℑ-x〉≥0,Θ(ϑ,ℑ)+φ(ℑ)-φ(ϑ)+1t〈ℑ-ϑ,ϑ-x〉≥0.
So, summing up these two equalities and using the monotonicity of Θ (H2), we get
(3.5)1s〈ϑ-ℑ,ℑ-x〉+1t〈ℑ-ϑ,ϑ-x〉≥0,
and hence
(3.6)〈ℑ-ϑ,ϑ-xt-ℑ-xs〉≥0.
We derive from (3.6) that
(3.7)〈ℑ-ϑ,ϑ-x-ts(ℑ-x)〉≥0,
and so
(3.8)-∥ℑ-ϑ∥2+〈ℑ-ϑ,(1-ts)(ℑ-x)〉≥0.
This indicates that
(3.9)(1-ts)〈ℑ-ϑ,ℑ-x〉≥∥ℑ-ϑ∥2.
In other words,
(3.10)∥ℑ-ϑ∥2≤s-ts〈ℑ-ϑ,ℑ-x〉,
and thus the claim holds.Lemma 3.2.
LetH be a real Hilbert space, let C be a nonempty closed convex subset of H, let S:C→C be a nonexpansive mapping, and let B:C→H be an ω-Lipschitz continuous and relaxed (u,v)-cocoercive mappings with v>uω2. If 0≤αn≤2(v-uω2)/ω2, then S-αnBS is a nonexpansive mapping in H.Proof.
Letαn≤2(v-uω2)/ω2,v>uω2. Then, for every x,y∈C, we have
(3.11)∥(S-αnBS)x-(S-αnBS)y∥2=∥(Sx-Sy)-αn(BSx-BSy)∥2=∥Sx-Sy∥2-2αn〈Sx-Sy,BSx-BSy〉+αn2∥BSx-BSy∥2≤∥Sx-Sy∥2-2αn{-u∥BSx-BSy∥2+v∥Sx-Sy∥2}+αn2∥BSx-BSy∥2≤∥Sx-Sy∥2+2αnuω2∥Sx-Sy∥2-2αnv∥Sx-Sy∥2+αn2ω2∥Sx-Sy∥2=(1+2αnuω2-2αnv+αn2ω2)∥Sx-Sy∥2≤(1-αnω2[2(v-uω2)ω2-αn])∥x-y∥2.
Now, since (1-αnω2[2(v-uω2)/ω2-αn])<1, thus ∥(S-αnBS)x-(S-αnBS)y∥≤∥x-y∥.
Thus,S-αnBS is a nonexpansive mapping of C into H.Now we can prove that a strong convergence theorem is a real Hilbert space.Theorem 3.3.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ1 and Θ2 be two bifunctions from C×C to ℝ satisfying (H1)–(H5) and let φ:C→ℝ∪{+∞} be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let(i)
Ψ1:C→H be a ξ-inverse-strongly monotone mapping,(ii)
Ψ2:C→H be a β-inverse-strongly monotone mapping,(iii)
B:C→H be an ω-Lipschitz continuous and relaxed (u,v)-cocoercive mappings,(iv)
f:C→C be a contraction mapping with coefficient η∈(0,1) and let A be a strongly positive linear bounded self-adjoint operator with the coefficient γ¯>0 and 0<γ<γ¯/η.Let S:C→C be a nonexpansive mapping with F(S)≠∅.
Assume that(3.12)ℱ:=F(S)∩GMEP(Θ1,φ,Ψ1)∩GMEP(Θ2,φ,Ψ2)∩VI(C,B)≠∅.
Let {xn}, {yn}, {zn}, {vn}, and {un} be the sequences generated by
(3.13)un=Trn(Θ1,φ)(xn-rnΨ1xn),vn=Tsn(Θ2,φ)(un-snΨ2un),zn=PC(Svn-αnBSvn),yn=ϵnγf(xn)+βnxn+((1-βn)I-ϵnA)zn,xn+1=γnxn+(1-γn)yn,∀n≥1,
where {rn}⊂[a,b]⊂[0,2ξ],{sn}⊂[c,d]⊂[0,2β],{γn}⊂[h,j]⊂(0,1), and {γn},{ϵn}, and {βn} are three sequences in (0,1) satisfying the following conditions:  (C1)
limn→∞ϵn=0 and ∑n=1∞ϵn=∞,(C2)
0<lim infn→∞βn≤lim supn→∞βn<1, and limn→∞βn=0,(C3)
0<lim infn→∞rn≤lim supn→∞rn<2ξ, and limn→∞|rn+1-rn|=0,(C4)
0<lim infn→∞sn≤lim supn→∞sn<2β, and limn→∞|sn+1-sn|=0,(C5)
{αn}⊂[e,g]⊂(0,2(v-uω2)/ω2),v>uω2, and limn→∞|αn+1-αn|=0.Then, {xn} converges strongly to z=Pℱ(γf+(I-A))(z), which is the unique solution of the variational inequality:
(3.14)〈γf(z)-Az,x-z〉≤0,∀x∈ℱ.Proof.
From the restrictions on control sequences, we may assume, without loss of generality, thatϵn≤(1-βn)∥A∥-1 for all n≥1. Since A is a strongly positive linear bounded self-adjoint operator on H, we have
(3.15)∥A∥=sup{|〈Ax,x〉|:x∈H,∥x∥=1}.
Observe that
(3.16)〈((1-βn)I-ϵnA)x,x〉=1-βn-ϵn〈Ax,x〉≥1-βn-ϵn∥A∥≥0.
That is, (1-βn)I-ϵnA is positive. It follows that
(3.17)∥(1-βn)I-ϵnA∥=sup{|〈((1-βn)I-ϵnA)x,x〉|:x∈H,∥x∥=1}=sup{1-βn-ϵn〈Ax,x〉:x∈H,∥x∥=1}≤1-βn-ϵnγ¯.
We will split the proof of Theorem 3.3 into six steps.Step 1.
We claim that the sequence{xn} is bounded.
Indeed, letx*∈ℱ, by Lemmas 2.2 and 2.9, we obtain
(3.18)x*=Sx*=PC(x*-αnBx*)=Trn(Θ1,φ)(x*-rnΨ1x*)=Tsn(Θ2,φ)(x*-snΨ2x*).
Since un=Trn(Θ1,φ)(xn-rnΨ1xn)∈domφ,Ψ1 is ξ-inverse-strongly monotone, and 0≤rn≤2ξ, we know that, for any n∈ℕ, we have
(3.19)∥un-x*∥2=∥Trn(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(x*-rnΨ1x*)∥2≤∥(xn-rnΨ1xn)-(x*-rnΨ1x*)∥2=∥(xn-x*)-rn(Ψ1xn-Ψ1x*)∥2=∥xn-x*∥2-2rn〈xn-x*,Ψ1xn-Ψ1x*〉+rn2∥Ψ1xn-Ψ1x*∥2≤∥xn-x*∥2-2rnξ∥Ψ1xn-Ψ1x*∥2+rn2∥Ψ1xn-Ψ1x*∥2=∥xn-x*∥2+rn(rn-2ξ)∥Ψ1xn-Ψ1x*∥2≤∥xn-x*∥2.
Similarly, from (3.19), vn=Tsn(Θ2,φ)(un-snΨ2un)∈domφ, and 0≤sn≤2β, we can prove that
(3.20)∥vn-x*∥2=∥Tsn(Θ2,φ)(un-snΨ2un)-Tsn(Θ2,φ)(x*-snΨ2x*)∥2≤∥un-x*∥2≤∥xn-x*∥2,
and hence
(3.21)∥vn-x*∥≤∥un-x*∥≤∥xn-x*∥.
Let x*∈ℱ, and form Lemma 3.2  S-αnBS is a nonexpansive mapping and from Lemma  2.2 x*=PC(x*-αnBx*), we have
(3.22)∥zn-x*∥=∥PC(Svn-αnBSvn)-PC(x*-αnBx*)∥≤∥(Svn-αnBSvn)-(x*-αnBx*)∥=∥(Svn-αnBSvn)-(Sx*-αnBSx*)∥=∥(S-αnBS)vn-(S-αnBS)x*∥≤∥vn-x*∥≤∥xn-x*∥,∥yn-x*∥=∥ϵn(γf(xn)-Ax*)+βn(xn-x*)+((1-βn)I-ϵnA)(zn-x*)∥≤(1-βn-ϵnγ¯)∥zn-x*∥+βn∥xn-x*∥+ϵn∥γf(xn)-Ax*∥≤(1-βn-ϵnγ¯)∥xn-x*∥+βn∥xn-x*∥+ϵn∥γf(xn)-Ax*∥≤(1-ϵnγ¯)∥xn-x*∥+ϵnγ∥f(xn)-f(x*)∥+ϵn∥γf(x*)-Ax*∥≤(1-ϵnγ¯)∥xn-x*∥+ϵnγη∥xn-x*∥+ϵn∥γf(x*)-Ax*∥=(1-(γ¯-ηγ)ϵn)∥xn-x*∥+ϵn∥γf(x*)-Ax*∥,
which yields that
(3.23)∥xn+1-x*∥≤γn∥xn-x*∥+(1-γn)∥yn-x*∥≤γn∥xn-x*∥+(1-γn){(1-(γ¯-ηγ)ϵn)∥xn-x*∥+ϵn∥γf(x*)-Ax*∥}=γn∥xn-x*∥+(1-γn)∥xn-x*∥-(1-γn)(γ¯-ηγ)ϵn∥xn-x*∥+(1-γn)ϵn∥γf(x*)-Ax*∥=(1-(1-γn)(γ¯-ηγ)ϵn)∥xn-x*∥+(1-γn)ϵn∥γf(x*)-Ax*∥.
By mathematical induction, putting D=max{∥x1-x*∥,∥γf(x*)-Ax*∥/γ¯-ηγ}, we have that ∥xn-x*∥≤D for all n≥1. Indeed, we can easily see that ∥x1-x*∥≤D. Suppose that ∥xk-x*∥≤D for some positive integral k. Then we have that
(3.24)∥xk+1-x*∥≤(1-(1-γk)(γ¯-ηγ)ϵk)∥xk-x*∥+(1-γk)ϵk∥γf(x*)-Ax*∥≤(1-(1-γk)(γ¯-ηγ)ϵk)D+(1-γk)ϵk∥γf(x*)-Ax*∥=(1-(1-γk)(γ¯-ηγ)ϵk)D+(1-γk)(γ¯-ηγ)ϵk∥γf(x*)-Ax*∥(γ¯-ηγ)≤(1-(1-γk)(γ¯-ηγ)ϵk)D+(1-γk)(γ¯-ηγ)ϵkD=D.
This shows that {xn} is bounded in H. From (3.21), we know that {un} and {vn} are bounded in C and so {yn}, {Ψ1un}, {Ψ2xn}, {Svn}, {BSvn}, and {f(xn)} are bounded sequence in H.Step 2.
We claim thatlimn→∞∥xn+1-xn∥=0.
SinceS-αnBS is nonexpansive, we have
(3.25)∥zn+1-zn∥=∥PC(Svn+1-αn+1BSvn+1)-PC(Svn-αnBSvn)∥≤∥(Svn+1-αn+1BSvn+1)-(Svn-αnBSvn)∥=∥(Svn+1-αn+1BSvn+1)-(Svn-αn+1BSvn)+(αn-αn+1)BSvn∥≤∥(Svn+1-αn+1BSvn+1)-(Svn-αn+1BSvn)∥+|αn-αn+1|∥BSvn∥≤∥vn+1-vn∥+|αn-αn+1|∥BSvn∥.
Next, we estimate ∥un+1-un∥. Observing that un=Trn(Θ1,φ)(xn-rnΨ1xn) and un+1=Trn+1(Θ1,φ)(xn+1-rn+1Ψ1xn+1), we have
(3.26)∥un+1-un∥=∥Trn+1(Θ1,φ)(xn+1-rn+1Ψ1xn+1)-Trn(Θ1,φ)(xn-rnΨ1xn)∥=∥Trn+1(Θ1,φ)(xn+1-rn+1Ψ1xn+1)-Trn+1(Θ1,φ)(xn-rnΨ1xn)+Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥≤∥Trn+1(Θ1,φ)(xn+1-rn+1Ψ1xn+1)-Trn+1(Θ1,φ)(xn-rnΨ1xn)∥+∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥≤∥(xn+1-rn+1Ψ1xn+1)-(xn-rnΨ1xn)∥+∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥=∥xn+1-xn-rn+1(Ψ1xn+1-Ψ1xn)+(rn+1-rn)Ψ1xn∥+∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥≤∥xn+1-xn-rn+1(Ψ1xn+1-Ψ1xn)∥+|rn+1-rn|∥Ψ1xn∥+∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥≤∥xn+1-xn∥+|rn+1-rn|∥Ψ1xn∥+∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥.
Similarly, we can prove that
(3.27)∥vn+1-vn∥≤∥un+1-un∥+|sn+1-sn|∥Ψ2un∥+∥Tsn+1(Θ2,φ)(un-snΨ2un)-Tsn(Θ2,φ)(un-snΨ2un)∥.
Substitution (3.26) into (3.27), we derive
(3.28)∥vn+1-vn∥≤∥xn+1-xn∥+|rn+1-rn|∥Ψ1xn∥+|sn+1-sn|∥Ψ2un∥+∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥+∥Tsn+1(Θ2,φ)(un-snΨ2un)-Tsn(Θ2,φ)(un-snΨ2un)∥.
Since {Ψ1xn},{Ψ2un}, and {BSvn} are bounded, K is an appropriate constant such that
(3.29)K≥max{supn≥1{∥Ψ1xn∥},supn≥1{∥Ψ2un∥},supn≥1{∥BSvn∥}}.
Substitution (3.28) into (3.25), we obtain
(3.30)∥zn+1-zn∥≤∥vn+1-vn∥+|αn-αn+1|∥BSvn∥≤∥xn+1-xn∥+|rn+1-rn|∥Ψ1xn∥+|sn+1-sn|∥Ψ2un∥+|αn-αn+1|∥BSvn∥+∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥+∥Tsn+1(Θ2,φ)(un-snΨ2un)-Tsn(Θ2,φ)(un-snΨ2un)∥≤∥xn+1-xn∥+K(|rn+1-rn|+|sn+1-sn|+|αn-αn+1|)+∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥+∥Tsn+1(Θ2,φ)(un-snΨ2un)-Tsn(Θ2,φ)(un-snΨ2un)∥.
From (3.13), we have
(3.31)yn=ϵnγf(xn)+βnxn+((1-βn)I-ϵnA)zn,yn+1=ϵn+1γf(xn+1)+βn+1xn+1+((1-βn+1)I-ϵn+1A)zn+1.
Simple calculations show that
(3.32)yn+1-yn=ϵn+1γ(f(xn+1)-f(xn))+(ϵn+1-ϵn)γf(xn)+βn+1(xn+1-xn)+(βn+1-βn)xn+[(1-βn+1)I-ϵn+1A](zn+1-zn)-(βn+1-βn)zn-(ϵn+1-ϵn)Azn=ϵn+1γ(f(xn+1)-f(xn))+(ϵn+1-ϵn)(γf(xn)-Azn)+βn+1(xn+1-xn)+(βn+1-βn)(xn-zn)+[(1-βn+1)I-ϵn+1A](zn+1-zn),
which yields that
(3.33)∥yn+1-yn∥≤ϵn+1γ∥f(xn+1)-f(xn)∥+|ϵn+1-ϵn|∥γf(xn)-Azn∥+βn+1∥xn+1-xn∥+|βn+1-βn|∥xn-zn∥+(1-βn+1-ϵn+1γ¯)∥zn+1-zn∥≤ϵn+1γη∥xn+1-xn∥+|ϵn+1-ϵn|∥γf(xn)-Azn∥+βn+1∥xn+1-xn∥+|βn+1-βn|∥xn-zn∥+(1-βn+1-ϵn+1γ¯)∥zn+1-zn∥.
Substitution (3.30) into (3.33) yields that
(3.34)∥yn+1-yn∥≤ϵn+1γη∥xn+1-xn∥+|ϵn+1-ϵn|∥γf(xn)-Azn∥+βn+1∥xn+1-xn∥+|βn+1-βn|∥xn-zn∥+(1-βn+1-ϵn+1γ¯)×{∥Trn+1(Θ1,φ)(xn-rnΨ1xn)∥∥xn+1-xn∥+K(|rn+1-rn|+|sn+1-sn|+|αn-αn+1|)+∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥+∥Tsn+1(Θ2,φ)(un-snΨ2un)-Tsn(Θ2,φ)(un-snΨ2un)∥}=(1-ϵn+1(γ¯-γη))∥xn+1-xn∥+|ϵn+1-ϵn|∥γf(xn)-Azn∥+|βn+1-βn|∥xn-zn∥+(1-βn+1-ϵn+1γ¯)K×(|rn+1-rn|+|sn+1-sn|+|αn-αn+1|)+(1-βn+1-ϵn+1γ¯)×∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥+(1-βn+1-ϵn+1γ¯)∥Tsn+1(Θ2,φ)(un-snΨ2un)-Tsn(Θ2,φ)(un-snΨ2un)∥≤(1-ϵn+1(γ¯-γη))∥xn+1-xn∥+M(|ϵn+1-ϵn|+|βn+1-βn|+|rn+1-rn|+|sn+1-sn|+|αn-αn+1|)+∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥+∥Tsn+1(Θ2,φ)(un-snΨ2un)-Tsn(Θ2,φ)(un-snΨ2un)∥,
where M is an appropriate constant such that
(3.35)M≥max{supn≥1{∥γf(xn)-Azn∥},supn≥1{∥xn-zn∥},K}.
Since xn+1=γnxn+(1-γn)yn and
(3.36)∥yn+1-yn∥-∥xn+1-xn∥≤M(|ϵn+1-ϵn|+|βn+1-βn|+|rn+1-rn|+|sn+1-sn|+|αn-αn+1|)+∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥+∥Tsn+1(Θ2,φ)(un-snΨ2un)-Tsn(Θ2,φ)(un-snΨ2un)∥,
next, we estimate
(3.37)∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥.
Note that lim infn→∞rn>0; there exists a constant r̃>0 such that rn≥r̃>0 for all n≥1. From Lemma 3.1, we get
(3.38)∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥2≤rn+1-rnrn+1〈Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn),Trn+1(Θ1,φ)(xn-rnΨ1xn)-(xn-rnΨ1xn)〉=〈rn+1-rnrn+1(Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)),Trn+1(Θ1,φ)(xn-rnΨ1xn)-(xn-rnΨ1xn)(Trn+1(Θ1,φ)(xn-rnΨ1xn))〉≤|rn+1-rn|rn+1∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-(xn-rnΨ1xn)∥.
It follows that
(3.39)∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥≤|rn+1-rn|r̃∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-(xn-rnΨ1xn)∥.
Since limn→∞|rn+1-rn|=0, we obtain
(3.40)limn→∞∥Trn+1(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(xn-rnΨ1xn)∥=0.
Similarly, we can prove that
(3.41)limn→∞∥Tsn+1(Θ2,φ)(un-snΨ2un)-Tsn(Θ2,φ)(un-snΨ2un)∥=0.
Consequently, from (3.40), (3.41), and conditions in Theorem 3.3, we obtain
(3.42)limn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0.
It follows from Lemma 2.7 that
(3.43)limn→∞∥yn-xn∥=0.
In view of (3.13), we see that
(3.44)∥xn+1-xn∥=(1-γn)∥xn-yn∥,∀n≥1,
which, combining with (3.43) and 0<h≤γn≤j<1, yields that
(3.45)limn→∞∥xn+1-xn∥=0.Step 3.
We claim thatlimn→∞∥Szn-zn∥=0.
Observing thatyn=ϵnγf(xn)+βnxn+((1-βn)I-ϵnA)zn, we have
(3.46)∥yn-zn∥≤ϵn∥γf(xn)-Azn∥+βn∥xn-zn∥,
which, combining with the conditions (C1) and (C2), gives
(3.47)limn→∞∥yn-zn∥=0.
From (3.43) and (3.47), we have
(3.48)limn→∞∥xn-zn∥=0.
For any x*∈ℱ, we see that
(3.49)∥zn-x*∥2=∥PC(Svn-αnBSvn)-PC(x*-αnBx*)∥2≤∥(Svn-αnBSvn)-(x*-αnBx*)∥2=∥(Svn-x*)-αn(BSvn-Bx*)∥2=∥Svn-x*∥2-2αn〈Svn-x*,BSvn-Bx*〉+αn2∥BSvn-Bx*∥2≤∥vn-x*∥2-2αn〈Svn-x*,BSvn-Bx*〉+αn2∥BSvn-Bx*∥2≤∥xn-x*∥2-2αn{-u∥BSvn-Bx*∥2+v∥Svn-x*∥2}+αn2∥BSvn-Bx*∥2≤∥xn-x*∥2+2αnu∥BSvn-Bx*∥2-2αnv∥Svn-x*∥2+αn2∥BSvn-Bx*∥2≤∥xn-x*∥2+2αnu∥BSvn-Bx*∥2-2αnvω2∥BSvn-Bx*∥2+αn2∥BSvn-Bx*∥2=∥xn-x*∥2+(2αnu+αn2-2αnvω2)∥BSvn-Bx*∥2,(3.50)∥yn-x*∥2=∥((1-βn)I-ϵnA)(zn-x*)+βn(xn-x*)+ϵn(γf(xn)-Ax*)∥2=∥((1-βn)I-ϵnA)(zn-x*)+βn(xn-x*)∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤((1-βn-ϵnγ¯)∥zn-x*∥+βn∥xn-x*∥)2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉=(1-βn-ϵnγ¯)2∥zn-x*∥2+βn2∥xn-x*∥2+2(1-βn-ϵnγ¯)βn∥zn-x*∥∥xn-x*∥+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤(1-βn-ϵnγ¯)2∥zn-x*∥2+βn2∥xn-x*∥2+(1-βn-ϵnγ¯)βn{∥zn-x*∥2+∥xn-x*∥2}+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉=[(1-ϵnγ¯)2-2(1-ϵnγ¯)βn+βn2]∥zn-x*∥2+βn2∥xn-x*∥2+((1-ϵnγ¯)βn-βn2)(∥zn-x*∥2+∥xn-x*∥2)+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉=[(1-ϵnγ¯)2-(1-ϵnγ¯)βn]∥zn-x*∥2+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉=(1-ϵnγ¯)(1-βn-ϵnγ¯)∥zn-x*∥2+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤(1-ϵnγ¯)(1-βn-ϵnγ¯){∥xn-x*∥2+(2αnu+αn2-2αnvω2)∥BSvn-Bx*∥}+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉=(1-ϵnγ¯)(1-βn-ϵnγ¯)∥xn-x*∥2+(1-ϵnγ¯)βn∥xn-x*∥2+(1-ϵnγ¯)(1-βn-ϵnγ¯)(2αnu+αn2-2αnvω2)∥BSvn-Bx*∥+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉=(1-ϵnγ¯)2∥xn-x*∥2+(1-ϵnγ¯)(1-βn-ϵnγ¯)(2αnu+αn2-2αnvω2)∥BSvn-Bx*∥+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2εn〈((1-βn)I-εnA)(zn-x*),γf(xn)-Ax*〉≤∥xn-x*∥2+(1-εnγ̅)(1-βn-εnγ̅)(2αnu+αn2-2αnvω2)∥BSvn-Bx*∥+εn2∥γf(xn)-Ax*∥2+2βnεn〈xn-x*,γf(xn)-Ax*〉+2εn〈((1-βn)I-εnA)(zn-x*),γf(xn)-Ax*〉.
Furthermore, from (3.13) and Lemma 2.12(1), we have
(3.51)∥xn+1-x*∥2=∥γnxn+(1-γn)yn-x*∥2≤γn∥xn-x*∥2+(1-γn)∥yn-x*∥2≤γn∥xn-x*∥2+(1-γn)×{∥xn-x*∥2+(1-ϵnγ¯)(1-βn-ϵnγ¯)(2αnu+αn2-2αnvω2)∥BSvn-Bx*∥+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉(2αnu+αn2-2αnvω2)}≤∥xn-x*∥2+(1-γn)(1-ϵnγ¯)(1-βn-ϵnγ¯)(2αnu+αn2-2αnvω2)∥BSvn-Bx*∥+(1-γn)ϵn2∥γf(xn)-Ax*∥2+2(1-γn)βnϵn〈xn-x*,γf(xn)-Ax*〉+2(1-γn)ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉.
It follows that
(3.52)(1-j)(1-ϵnγ¯)(1-βn-ϵnγ¯)(2evω2-2gu-g2)∥BSvn-Bx*∥≤(1-γn)(1-ϵnγ¯)(1-βn-ϵnγ¯)(2αnvω2-2αnu-αn2)∥BSvn-Bx*∥≤∥xn-x*∥2-∥xn+1-x*∥2+(1-γn)ϵn2∥γf(xn)-Ax*∥2+2(1-γn)βnϵn〈xn-x*,γf(xn)-Ax*〉+2(1-γn)ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤∥xn-xn+1∥(∥xn-x*∥+∥xn+1-x*∥)+(1-γn)ϵn2∥γf(xn)-Ax*∥2+2(1-γn)βnϵn〈xn-x*,γf(xn)-Ax*〉+2(1-γn)ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉.
From the condition (C1) and (3.45), we arrive at
(3.53)limn→∞∥BSvn-Bx*∥=0.
On the other hand, we have
(3.54)∥zn-x*∥2=∥PC(Svn-αnBSvn)-PC(x*-αnBx*)∥2≤〈(Svn-αnBSvn)-(x*-αnBx*),zn-x*〉=12{∥(Svn-αnBSvn)-(x*-αnBx*)∥2+∥zn-x*∥2-∥(Svn-αnBSvn)-(x*-αnBx*)-(zn-x*)∥2}≤12{∥vn-x*∥2+∥zn-x*∥2-∥(Svn-zn)-αn(BSvn-Bx*)∥2}≤12{∥xn-x*∥2+∥zn-x*∥2-∥Svn-zn∥2-αn2∥BSvn-Bx*∥2+2αn〈Svn-zn,BSvn-Bx*〉},
which yields that
(3.55)∥zn-x*∥2≤∥xn-x*∥2-∥Svn-zn∥2+2αn∥Svn-zn∥∥BSvn-Bx*∥.
Substituting (3.55) into (3.50), we have
(3.56)∥yn-x*∥2≤(1-ϵnγ¯)(1-βn-ϵnγ¯)∥zn-x*∥2+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤(1-ϵnγ¯)(1-βn-ϵnγ¯){∥xn-x*∥2-∥Svn-zn∥2+2αn∥Svn-zn∥∥BSvn-Bx*∥}+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉=(1-ϵnγ¯)2∥xn-x*∥2-(1-ϵnγ¯)(1-βn-ϵnγ¯)∥Svn-zn∥2+(1-ϵnγ¯)(1-βn-ϵnγ¯)2αn∥Svn-zn∥∥BSxn-Bx*∥+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤∥xn-x*∥2-(1-ϵnγ¯)(1-βn-ϵnγ¯)∥Svn-zn∥2+(1-ϵnγ¯)(1-βn-ϵnγ¯)2αn∥Svn-zn∥∥BSvn-Bx*∥+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉.
Using (3.51) and (3.56), we have
(3.57)∥xn+1-x*∥2≤γn∥xn-x*∥2+(1-γn)∥yn-x*∥2≤γn∥xn-x*∥2+(1-γn)×{∥xn-x*∥2-(1-ϵnγ¯)(1-βn-ϵnγ¯)∥Svn-zn∥2+(1-ϵnγ¯)(1-βn-ϵnγ¯)2αn∥Svn-zn∥∥BSvn-Bx*∥+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(Szn-x*),γf(xn)-Ax*〉∥xn-x*∥2-(1-ϵnγ¯)(1-βn-ϵnγ¯)∥Svn-zn∥2}=∥xn-x*∥2-(1-γn)(1-ϵnγ¯)(1-βn-ϵnγ¯)∥Svn-zn∥2+2(1-γn)(1-ϵnγ¯)(1-βn-ϵnγ¯)αn∥Svn-zn∥∥BSvn-Bx*∥+(1-γn)ϵn2∥γf(xn)-Ax*∥2+2(1-γn)βnϵn〈xn-x*,γf(xn)-Ax*〉+2(1-γn)ϵn〈((1-βn)I-ϵnA)(Svn-x*),γf(xn)-Ax*〉.
It follows that
(3.58)(1-γn)(1-ϵnγ¯)(1-βn-ϵnγ¯)∥Svn-zn∥2≤∥xn-x*∥2-∥xn+1-x*∥2+2(1-γn)(1-ϵnγ¯)(1-βn-ϵnγ¯)αn∥Svn-zn∥∥BSvn-Bx*∥+(1-γn)ϵn2∥γf(xn)-Ax*∥2+2(1-γn)βnϵn〈xn-x*,γf(xn)-Ax*〉+2(1-γn)ϵn〈((1-βn)I-ϵnA)(Szn-x*),γf(xn)-Ax*〉≤∥xn-xn+1∥(∥xn-x*∥+∥xn+1-x*∥)+2(1-γn)(1-ϵnγ¯)(1-βn-ϵnγ¯)αn∥Svn-zn∥∥BSvn-Bx*∥+(1-γn)ϵn2∥γf(xn)-Ax*∥2+2(1-γn)βnϵn〈xn-x*,γf(xn)-Ax*〉+2(1-γn)ϵn〈((1-βn)I-ϵnA)(Szn-x*),γf(xn)-Ax*〉.
From condition (C1), (3.45), and (3.53), we obtain
(3.59)limn→∞∥Svn-zn∥=0.
Consequently, from (3.50) we derive that
(3.60)∥yn-x*∥2≤(1-ϵnγ¯)(1-βn-ϵnγ¯)∥zn-x*∥2+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤(1-ϵnγ¯)(1-βn-ϵnγ¯)∥vn-x*∥2+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉=(1-ϵnγ¯)(1-βn-ϵnγ¯)∥Tsn(Θ2,φ)(un-snΨ2un)-Tsn(Θ2,φ)(x*-snΨ2x*)∥2+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤(1-ϵnγ¯)(1-βn-ϵnγ¯)∥(un-snΨ2un)-(x*-snΨ2x*)∥2+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤(1-ϵnγ¯)(1-βn-ϵnγ¯){∥un-x*∥2+sn(sn-2β)∥Ψ2un-Ψ2x*∥2}+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤(1-ϵnγ¯)(1-βn-ϵnγ¯){∥xn-x*∥2+sn(sn-2β)∥Ψ2un-Ψ2x*∥2}+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤∥xn-x*∥2+(1-ϵnγ¯)(1-βn-ϵnγ¯)sn(sn-2β)∥Ψ2un-Ψ2x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉.
From (3.51) and (3.60), we obtain
(3.61)∥xn+1-x*∥2≤γn∥xn-x*∥2+(1-γn)∥yn-x*∥2≤γn∥xn-x*∥2+(1-γn)×{∥xn-x*∥2+(1-ϵnγ¯)(1-βn-ϵnγ¯)sn(sn-2β)∥Ψ2un-Ψ2x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉∥xn-x*∥2+(1-ϵnγ¯)(1-βn-ϵnγ¯)sn(sn-2β)∥Ψ2un-Ψ2x*∥2}=∥xn-x*∥2+(1-γn)(1-ϵnγ¯)(1-βn-ϵnγ¯)sn(sn-2β)∥Ψ2un-Ψ2x*∥2+(1-γn)ϵn2∥γf(xn)-Ax*∥2+2(1-γn)βnϵn〈xn-x*,γf(xn)-Ax*〉+2(1-γn)ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉.
So, we obtain
(3.62)(1-j)(1-ϵnγ¯)(1-βn-ϵnγ¯)c(2β-d)∥Ψ2un-Ψ2x*∥2≤(1-γn)(1-ϵnγ¯)(1-βn-ϵnγ¯)sn(2β-sn)∥Ψ2un-Ψ2x*∥2≤∥xn-x*∥2-∥xn+1-x*∥2+(1-γn)ϵn2∥γf(xn)-Ax*∥2+2(1-γn)βnϵn〈xn-x*,γf(xn)-Ax*〉+2(1-γn)ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤∥xn-xn+1∥(∥xn-x*∥+∥xn+1-x*∥)+(1-γn)ϵn2∥γf(xn)-Ax*∥2+2(1-γn)βnϵn〈xn-x*,γf(xn)-Ax*〉+2(1-γn)ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉.
Since limn→∞∥xn+1-xn∥=0 and limn→∞ϵn=0, we obtain
(3.63)limn→∞∥Ψ2un-Ψ2x*∥=0.
Similarly, we can prove that
(3.64)limn→∞∥Ψ1xn-Ψ1x*∥=0.
In addition, from the firmly nonexpansivity of Trn(Θ1,φ), we have
(3.65)∥un-x*∥2=∥Trn(Θ1,φ)(xn-rnΨ1xn)-Trn(Θ1,φ)(x*-rnΨ1x*)∥2≤〈(xn-rnΨ1xn)-(x*-rnΨ1x*),un-x*〉=12{∥(xn-rnΨ1xn)-(x*-rnΨ1x*)∥2+∥un-x*∥2-∥(xn-rnΨ1xn)-(x*-rnΨ1x*)-(un-x*)∥2}≤12{∥xn-x*∥2+∥un-x*∥2-∥xn-un-rn(Ψ1xn-Ψ1x*)∥2}=12{∥xn-x*∥2+∥un-x*∥2-∥xn-un∥2+2rn〈xn-un,Ψ1xn-Ψ1x*〉-rn2∥Ψ1xn-Ψ1x*∥2}.
So, we obtain
(3.66)∥un-x*∥2≤∥xn-x*∥2-∥xn-un∥2+2rn∥xn-un∥∥Ψ1xn-Ψ1x*∥.
Similarly, we can prove that
(3.67)∥vn-x*∥2≤∥xn-x*∥2-∥un-vn∥2+2sn∥un-vn∥∥Ψ2un-Ψ2x*∥.
Substituting (3.66) into (3.50), we have
(3.68)∥yn-x*∥2≤(1-ϵnγ¯)(1-βn-ϵnγ¯)∥zn-x*∥2+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤(1-ϵnγ¯)(1-βn-ϵnγ¯)∥un-x*∥2+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤(1-ϵnγ¯)(1-βn-ϵnγ¯){∥xn-x*∥2-∥xn-un∥2+2rn∥xn-un∥∥Ψ1xn-Ψ1x*∥}+(1-ϵnγ¯)βn∥xn-x*∥2+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉=(1-ϵnγ¯)2∥xn-x*∥2-(1-ϵnγ¯)(1-βn-ϵnγ¯)∥xn-un∥2+(1-ϵnγ¯)(1-βn-ϵnγ¯)2rn∥xn-un∥∥Ψ1xn-Ψ1x*∥+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉≤∥xn-x*∥2-(1-ϵnγ¯)(1-βn-ϵnγ¯)∥xn-un∥2+(1-ϵnγ¯)(1-βn-ϵnγ¯)2rn∥xn-un∥∥Ψ1xn-Ψ1x*∥+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉.
Using (3.51) and (3.68), we have
(3.69)∥xn+1-x*∥2≤γn∥xn-x*∥2+(1-γn)∥yn-x*∥2≤γn∥xn-x*∥2+(1-γn)×{∥xn-x*∥2-(1-ϵnγ¯)(1-βn-ϵnγ¯)∥xn-un∥2+(1-ϵnγ¯)(1-βn-ϵnγ¯)2rn∥xn-un∥∥Ψ1xn-Ψ1x*∥+ϵn2∥γf(xn)-Ax*∥2+2βnϵn〈xn-x*,γf(xn)-Ax*〉+2ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉∥xn-x*∥2-(1-ϵnγ¯)(1-βn-ϵnγ¯)∥xn-un∥2}=∥xn-x*∥2-(1-γn)(1-ϵnγ¯)(1-βn-ϵnγ¯)∥xn-un∥2+2(1-γn)(1-ϵnγ¯)(1-βn-ϵnγ¯)rn∥xn-un∥∥Ψ1xn-Ψ1x*∥+(1-γn)ϵn2∥γf(xn)-Ax*∥2+2(1-γn)βnϵn〈xn-x*,γf(xn)-Ax*〉+2(1-γn)ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉.
It follows that
(3.70)(1-γn)(1-ϵnγ¯)(1-βn-ϵnγ¯)∥xn-un∥2≤∥xn-xn+1∥(∥xn-x*∥+∥xn+1-x*∥)+2(1-γn)(1-ϵnγ¯)(1-βn-ϵnγ¯)rn∥xn-un∥∥Ψ1xn-Ψ1x*∥+(1-γn)ϵn2∥γf(xn)-Ax*∥2+2(1-γn)βnϵn〈xn-x*,γf(xn)-Ax*〉+2(1-γn)ϵn〈((1-βn)I-ϵnA)(zn-x*),γf(xn)-Ax*〉.
Since limn→∞ϵn=0,limn→∞∥Ψ1xn-Ψ1x*∥=0, and limn→∞∥xn+1-xn∥=0, we obtain
(3.71)limn→∞∥xn-un∥=0.
Similarly, we can prove that
(3.72)limn→∞∥un-vn∥=0.
Furthermore, by the triangular inequality, we also have
(3.73)∥xn-vn∥≤∥xn-un∥+∥un-vn∥.
Applying (3.71) and (3.72), we have
(3.74)limn→∞∥xn-vn∥=0.
Since
(3.75)∥zn-vn∥≤∥zn-xn∥+∥xn-vn∥,
so we get
(3.76)limn→∞∥zn-vn∥=0.
Also, observe that
(3.77)∥Szn-zn∥≤∥Szn-Svn∥+∥Svn-zn∥≤∥zn-vn∥+∥Svn-zn∥.
Consequently, we obtain
(3.78)limn→∞∥Szn-zn∥=0.Step 4.
We prove that the mappingPℱ(γf+(I-A)) has a unique fixed point.
Sincef is a contraction of C into itself with coefficient η∈(0,1), then, we have
(3.79)∥Pℱ(γf+(I-A))(x)-Pℱ(γf+(I-A))(y)∥≤∥(γf+(I-A))(x)-(γf+(I-A))(y)∥≤γ∥f(x)-f(y)∥+∥I-A∥∥x-y∥≤γη∥x-y∥+(1-γ¯)∥x-y∥=(1-(γ¯-ηγ))∥x-y∥,∀x,y∈C.
Since 0<1-(γ¯-ηγ)<1, it follows that Pℱ(γf+(I-A)) is a contraction of C into itself. Therefore by the Banach Contraction Mapping Principle, it has a unique fixed point, say z∈C, that is,
(3.80)z=Pℱ(γf+(I-A))(z).Step 5.
We claim thatlim supn→∞〈γfz-Az,xn-z〉≤0, where z is the unique solution of the variational inequality 〈γf(z)-Az,x-z〉≤0, for all x∈ℱ.
Sincez=Pℱ(γf+(I-A))(z) is a unique solution of the variational inequality (3.14), to show this inequality, we choose a subsequence {xni} of {xn} such that
(3.81)lim supn→∞〈γfz-Az,xn-z〉=limi→∞〈γfz-Az,xni-z〉.
Correspondingly, there exists a subsequence {zni} of {zn} such that
(3.82)lim supn→∞〈γfz-Az,zn-z〉=limi→∞〈γfz-Az,zni-z〉.
Since {zni} is bounded, there exists a subsequence {znij} of {zni} which converges weakly to w. Without loss of generality, we can assume that zni⇀w. From ∥Szn-zn∥→0, we obtain Szni⇀w.
Next, we show thatw∈F(S)∩GMEP(Θ1,φ,Ψ1)∩GMEP(Θ2,φ,Ψ2)∩VI(C,B).
First, we show thatw∈F(S).
Assumew∉F(S). Since zni⇀w and w≠Sw, it follows by Opial's condition (Lemma 2.4) that
(3.83)lim infi→∞∥zni-w∥<lim infi→∞∥zni-Sw∥≤lim infi→∞{∥zni-Szni∥+∥Szni-Sw∥}=lim infi→∞∥Szni-Sw∥≤lim infi→∞∥zni-w∥.
This is a contradiction. Thus, we have w∈F(S).
Next, we prove thatw∈GMEP(Θ1,φ,Ψ1).
For anyy∈C, we have
(3.84)0≤Θ1(un,y)+φ(y)-φ(un)+〈Ψ1xn,y-un〉+1rn〈y-un,un-xn〉=Θ1(un,y)+φ(y)-φ(un)+1rn〈rnΨ1xn,y-un〉+1rn〈y-un,un-xn〉=Θ1(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn+rnΨ1xn〉,
which yield that
(3.85)Θ1(un,y)+φ(y)-φ(un)+1rn〈y-un,un-(xn-rnΨ1xn)〉≥0,∀y∈C.
It follows from Lemma 2.9 that un=Trn(Θ1,φ)(xn-rnΨ1xn) for all n≥1.
Thus, we conclude thatun=Trn(Θ1,φ)(xn-rnΨ1xn) is equivalent to
(3.86)Θ1(un,y)+φ(y)-φ(un)+〈Ψ1xn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C.
From (H2), we also have
(3.87)φ(y)-φ(un)+〈Ψ1xn,y-un〉+1rn〈y-un,un-xn〉≥-Θ1(un,y)≥Θ1(y,un).
Replacing n by ni, we obtain
(3.88)φ(y)-φ(uni)+〈Ψ1xni,y-uni〉+〈y-uni,uni-xnirni〉≥Θ1(y,uni).
Let ut=ty+(1-t)w for all t∈(0,1] and y∈C. Since y∈C and w∈C, we obtain ut∈C. So, from (3.88) we have
(3.89)〈ut-uni,Ψ1ut〉≥〈ut-uni,Ψ1ut〉-φ(ut)+φ(uni)-〈Ψ1xni,ut-uni〉-〈ut-uni,uni-xnirni〉+Θ1(ut,uni)≥〈ut-uni,Ψ1ut-Ψ1uni〉+〈ut-uni,Ψ1uni-Ψ1xni〉-φ(ut)+φ(uni)-〈ut-uni,uni-xnirni〉+Θ1(ut,uni).
Since ∥uni-xni∥→0, we have ∥Ψ1uni-Ψ1xni∥→0. Further, from the inverse strong monotonicity of Ψ1, we have
(3.90)〈ut-uni,Ψ1ut-Ψ1uni〉≥0.
So, from (H4), (H5), and the weak lower semicontinuity of φ,(uni-xni)/rni→0, and uni⇀w, we have
(3.91)〈ut-w,Ψ1ut〉≥-φ(ut)+φ(w)+Θ1(ut,w),asi→∞.
From (H1), (H4), and (3.91), we also get
(3.92)0=Θ1(ut,ut)+φ(ut)-φ(ut)≤tΘ1(ut,y)+(1-t)Θ1(ut,w)+tφ(y)+(1-t)φ(w)-φ(ut)=t[Θ1(ut,y)+φ(y)-φ(ut)]+(1-t)[Θ1(ut,w)+φ(w)-φ(ut)]≤t[Θ1(ut,y)+φ(y)-φ(ut)]+(1-t)〈ut-w,Ψ1ut〉=t[Θ1(ut,y)+φ(y)-φ(ut)]+(1-t)t〈y-w,Ψ1ut〉.
Dividing by t, we get
(3.93)Θ1(ut,y)+φ(y)-φ(ut)+(1-t)〈y-w,Ψ1ut〉≥0.
Letting t→0 in the above inequality, we arrive that
(3.94)Θ1(w,y)+φ(y)-φ(w)+〈y-w,Ψ1w〉≥0.
Thus, w∈GMEP(Θ1,φ,Ψ1). Similarly, we can prove that w∈GMEP(Θ2,φ,Ψ2).
Finally, now we prove thatw∈VI(C,B).
We define the maximal monotone operator:(3.95)Qw1={Bw1+NCw1,w1∈C,∅,w1∉C.
Since B is relaxed (u,v)-cocoercive and condition (C5), we have
(3.96)〈Bx-By,x-y〉≥(-u)∥Bx-By∥2+v∥x-y∥2≥(v-uω2)∥x-y∥2≥0,
which yields that B is monotone. Thus, Q is maximal monotone. Let (w1,w2)∈G(Q). Since w2-Bw1∈NCw1 and zn∈C, we have
(3.97)〈w1-zn,w2-Bw1〉≥0.
On the other hand, from zn=PC(Svn-αnBSvn), we have
(3.98)〈w1-zn,zn-(Svn-αnBSvn)〉≥0,
and hence
(3.99)〈w1-zn,zn-Svnαn+BSvn〉≥0.
It follows that
(3.100)〈w1-zni,w2〉≥〈w1-zni,Bw1〉≥〈w1-zni,Bw1〉-〈w1-zni,zni-Svniαni+BSvni〉=〈w1-zni,Bw1-BSvni-zni-Svniαni〉=〈w1-zni,Bw1-Bzni〉+〈w1-zni,Bzni-BSvni〉-〈w1-zni,zni-Svniαni〉≥〈w1-zni,Bzni-BSvni〉-〈w1-zni,zni-Svniαni〉,
which implies that
(3.101)〈w1-w,w2〉≥0.
Since Q is maximal monotone, we obtain that w∈Q-10. From Lemma 2.3, we get that w∈VI(C,B). That is, w∈ℱ. Since z=Pℱ(γf+(I-A))(z), we have
(3.102)lim supn→∞〈γfz-Az,xn-z〉=lim supn→∞〈γfz-Az,xn-z〉=limi→∞〈γfz-Az,xni-z〉=〈γfz-Az,w-z〉≤0.
On the other hand, we have
(3.103)〈γfz-Az,yn-z〉=〈γfz-Az,yn-xn〉+〈γfz-Az,xn-z〉≤∥γfz-Az∥∥yn-xn∥+〈γfz-Az,xn-z〉.
From (3.43) and (3.102), we obtain that
(3.104)lim supn→∞〈γfz-Az,yn-z〉≤0.Step 6.
Finally, we show that{xn} converges strongly to z=Pℱ(γf+(I-A))(z). Indeed, by (3.13) and using Lemmas 2.6 and 2.12(2), we observe that
(3.105)∥yn-z∥2=∥ϵnγf(xn)+βnxn+((1-βn)I-ϵnA)zn-z∥2=∥((1-βn)I-ϵnA)(zn-z)+βn(xn-z)+ϵn(γf(xn)-Az)∥2≤∥((1-βn)I-ϵnA)(zn-z)+βn(xn-z)∥2+2ϵn〈γf(xn)-Az,((1-βn)I-ϵnA)(zn-z)+βn(xn-z)+ϵn(γf(xn)-Az)〉=∥((1-βn)I-ϵnA)(zn-z)+βn(xn-z)∥2+2ϵn〈γf(xn)-Az,ϵnγf(xn)+βnxn+((1-βn)I-ϵnA)zn-z〉=∥(1-βn)((1-βn)I-ϵnA)(1-βn)(zn-z)+βn(xn-z)∥2+2ϵn〈γf(xn)-Az,yn-z〉≤(1-βn)∥((1-βn)I-ϵnA)1-βn(zn-z)∥2+βn∥xn-z∥2+2ϵnγ〈f(xn)-f(z),yn-z〉+2ϵn〈γf(z)-Az,yn-z〉≤(1-βn)∥((1-βn)I-ϵnA)1-βn(zn-z)∥2+βn∥xn-z∥2+2ϵnγη∥xn-z∥∥yn-z∥+2ϵn〈γf(z)-Az,yn-z〉≤∥(1-βn)I-ϵnA∥21-βn∥zn-z∥2+βn∥xn-z∥2+ϵnγη(∥xn-z∥2+∥yn-z∥2)+2ϵn〈γf(z)-Az,yn-z〉≤∥(1-βn)I-ϵnA∥21-βn∥xn-z∥2+βn∥xn-z∥2+ϵnγη(∥xn-z∥2+∥yn-z∥2)+2ϵn〈γf(z)-Az,yn-z〉=(((1-βn)-γ¯ϵn)21-βn+βn+ϵnγη)∥xn-z∥2+ϵnγη∥yn-z∥2+2ϵn〈γf(z)-Az,yn-z〉=(1-(2γ¯-ηγ)ϵn+γ¯2ϵn21-βn)∥xn-z∥2+ϵnγη∥yn-z∥2+2ϵn〈γf(z)-Az,yn-z〉,
which implies that
(3.106)∥yn-z∥2≤(1-2(γ¯-ηγ)ϵn1-ηγϵn)∥xn-z∥2+ϵn1-ηγϵn[γ¯2ϵn1-βn∥xn-z∥2+2〈γf(z)-Az,yn-z〉].
On the other hand, we have
(3.107)∥xn+1-z∥2=∥γn(xn-z)+(1-γn)(yn-z)∥2≤γn∥xn-z∥2+(1-γn)∥yn-z∥2.
Substituting (3.106) into (3.107) yields that
(3.108)∥xn+1-z∥2≤γn∥xn-z∥2+(1-γn)∥yn-z∥2≤γn∥xn-z∥2+(1-γn)×{(1-2(γ¯-ηγ)ϵn1-ηγϵn)∥xn-z∥2+ϵn1-ηγϵn[γ¯2ϵn1-βn∥xn-z∥2+2〈γf(z)-Az,yn-z〉]}=(1-2(1-γn)(γ¯-ηγ)ϵn1-ηγϵn)∥xn-z∥2+ϵn(1-γn)1-ηγϵn[γ¯2ϵn1-βn∥xn-z∥2+2〈γf(z)-Az,yn-z〉].
Taking
(3.109)ϱn=2(1-γn)(γ¯-ηγ)ϵn1-ηγϵn,σn=ϵn(1-γn)1-ηγϵn{γ¯2ϵn1-βn∥xn-z∥2+2〈γf(z)-Az,yn-z〉},
then, we can rewrite (3.108) as
(3.110)∥xn+1-z∥2≤(1-ϱn)∥xn-z∥2+σn,
It follows from condition (C1) and (3.104) that
(3.111)limn→∞ϱn=0,∑n=1∞ϱn=∞.
Since
(3.112)lim supn→∞〈γfz-Az,yn-z〉≤0.
and {xn-z} is bounded, we have
(3.113)lim supn→∞σn=lim supn→∞ϵn(1-γn)1-ηγϵn{γ¯2ϵn1-βn∥xn-z∥2+2〈γf(z)-Az,yn-z〉}≤0.
Applying Lemma 2.8 to (3.110), we conclude that {xn} converges strongly to z in norm. This completes the proof.Corollary 3.4.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ1 and Θ2 be two bifunctions from C×C to ℝ satisfying (H1)–(H5) and let φ:C→ℝ∪{+∞} be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let(i)
Ψ1:C→H be a ξ-inverse-strongly monotone mapping,(ii)
Ψ2:C→H be a β-inverse-strongly monotone mapping,(iii)
f:C→C be a contraction mapping with coefficient η∈(0,1).Let S:C→C be a nonexpansive mapping with F(S)≠∅.
Assume that(3.114)ℱ:=F(S)∩GMEP(Θ1,φ,Ψ1)∩GMEP(Θ2,φ,Ψ2)≠∅.
Let {xn},{yn},{vn}, and {un} be the sequences generated by
(3.115)un=Trn(Θ1,φ)(xn-rnΨ1xn),vn=Tsn(Θ2,φ)(un-snΨ2un),yn=ϵnf(xn)+βnxn+δnSvn,xn+1=γnxn+(1-γn)yn,∀n≥1,
where {rn}⊂[a,b]⊂[0,2ξ],{sn}⊂[c,d]⊂[0,2β],{γn}⊂[h,j]⊂(0,1), and {γn},{ϵn},{βn}, and {δn} are three sequences in (0,1) satisfying the following conditions:(C1)
ϵn+βn+δn=1,(C2)
limn→∞ϵn=0 and ∑n=1∞ϵn=∞,(C3)
0<lim infn→∞βn≤lim supn→∞βn<1 and limn→∞βn=0,(C4)
0<lim infn→∞rn≤lim supn→∞rn<2ξ and limn→∞|rn+1-rn|=0,(C5)
0<lim infn→∞sn≤lim supn→∞sn<2β and limn→∞|sn+1-sn|=0.Then, {xn} converges strongly to z=Pℱf(z), which is the unique solution of the variational inequality:
(3.116)〈f(z)-z,x-z〉≤0,∀x∈ℱ.Proof.
In Theorem3.3, put A=I,γ=1, and δn=1-ϵn-βn. Let B=0 in Theorem 3.3; then we have VI(C,B)=C and
(3.117)zn=PC(Svn-αnBSvn)=Svn,
and we can obtain the desired conclusion from Theorem 3.3 immediately.Corollary 3.5.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ1 and Θ2 be two bifunctions from C×C to ℝ satisfying (H1)–(H5) and let φ:C→ℝ∪{+∞} be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let f:C→C be a contraction mapping with coefficient η∈(0,1) and let S:C→C be a nonexpansive mapping with F(S)≠∅.
Assume that(3.118)ℱ:=F(S)∩MEP(Θ1,φ)∩MEP(Θ2,φ)≠∅.
Let {xn},{yn},{vn}, and {un} be the sequences generated by
(3.119)Θ1(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,Θ2(vn,y)+φ(y)-φ(vn)+1sn〈y-vn,vn-un〉≥0,∀y∈C,yn=ϵnf(xn)+βnxn+δnSvn,xn+1=γnxn+(1-γn)yn,∀n≥1,
where {rn}⊂[a,b]⊂(0,∞),{sn}⊂[c,d]⊂(0,∞),{γn}⊂[h,j]⊂(0,1), and {γn},{ϵn},{βn}, and {δn} are three sequences in (0,1) satisfying the following conditions:  (C1)
ϵn+βn+δn=1,(C2)
limn→∞ϵn=0 and ∑n=1∞ϵn=∞,(C3)
0<lim infn→∞βn≤lim supn→∞βn<1 and limn→∞βn=0,(C4)
0<lim infn→∞rn≤lim supn→∞rn<∞ and limn→∞|rn+1-rn|=0,(C5)
0<lim infn→∞sn≤lim supn→∞sn<∞ and limn→∞|sn+1-sn|=0.Then, {xn} converges strongly to z=Pℱf(z), which is the unique solution of the variational inequality:
(3.120)〈f(z)-z,x-z〉≤0,∀x∈ℱ.Proof.
In Theorem3.3, put un=Trn(Θ1,φ)(xn-rnΨ1xn) to be equivalent to
(3.121)Θ1(un,y)+φ(y)-φ(un)+〈Ψ1xn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,
and put vn=Tsn(Θ2,φ)(un-snΨ2un) to be equivalent to
(3.122)Θ2(vn,y)+φ(y)-φ(vn)+〈Ψ2xn,y-vn〉+1sn〈y-vn,vn-un〉≥0,∀y∈C.
Now, put Ψ1=Ψ2=0. Then, it follows that
(3.123)Θ1(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,Θ2(vn,y)+φ(y)-φ(vn)+1sn〈y-vn,vn-un〉≥0,∀y∈C.
Observe that for all ξ>0, we see that
(3.124)〈Ψ1x-Ψ2y,x-y〉≥ξ∥Ψ1x-Ψ2y∥2,∀x,y∈C.
Thus, let {rn} be a sequence satisfying the restriction: a≤rn≤b, where a,b∈(0,∞) with 0<lim infn→∞rn≤lim supn→∞rn<∞. Similarly, we obtain c≤sn≤d, where c,d∈(0,∞) with 0<lim infn→∞sn≤lim supn→∞sn<∞, and we obtain the desired result by Corollary 3.4.



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*Source: 390972-2010-07-20.xml*