# Iterative Algorithms for Systems of Generalized Equilibrium Problems with the Constraints of Variational Inclusion and Fixed Point Problems

Authors: Lu-Chuan Ceng; Abdul Latif; Abdullah E. Al-Mazrooei
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We introduce and analyze a hybrid extragradient-like viscosity iterative algorithm for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems.


---

## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, C be a nonempty closed convex subset of H and PC be the metric projection of H onto C. Let S:C→H be a nonlinear mapping on C. We denote by Fix⁡(S) the set of fixed points of S and by R the set of all real numbers. A mapping V is called strongly positive on H if there exists a constant γ¯∈(0,1] such that
(1)〈Vx,x〉≥γ¯∥x∥2,∀x∈H.
A mapping S:C→H is called L-Lipschitz continuous if there exists a constant L≥0 such that
(2)∥Sx-Sy∥≤L∥x-y∥,∀x,y∈C.
In particular, if L=1 then S is called a nonexpansive mapping; if L∈[0,1) then A is called a contraction.LetA:C→H be a nonlinear mapping on C. We consider the following variational inequality problem (VIP) [1] which is to find a point x∈C such that
(3)〈Ax,y-x〉≥0,∀y∈C.
The solution set of VIP (3) is denoted by VI⁡(C,A).Letφ:C→R be a real-valued function, A:H→H be a nonlinear mapping and Θ:C×C→R be a bifunction. In 2008, Peng and Yao [2] introduced the following generalized mixed equilibrium problem (GMEP) of finding x∈C such that
(4)Θ(x,y)+φ(y)-φ(x)+〈Ax,y-x〉≥0,∀y∈C.
We denote the set of solutions of GMEP (4) by GMEP⁡(Θ,φ,A). The system of equilibrium problems or generalized equilibrium problems is a tool to study Nash eequilibrium problems, see for example [3–8]. In fact, the GMEP (4) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. The GMEP is further considered and studied; see for example, [9–15]. Here we also consider a system of two generalized equilibrium problem that could be usefull to study the Two players game problem, see [16].Throughout this paper, it is assumed as in [2] that Θ:C×C→R is a bifunction satisfying conditions (H1)–(H4) and φ:C→R is a lower semicontinuous and convex function with restriction (H5), where(H1)
Θ(x,x)=0 for all x∈C;(H2)
Θ is monotone, that is, Θ(x,y)+Θ(y,x)≤0 for any x,y∈C;(H3)
Θ is upper-hemicontinuous, that is, for each x,y,z∈C,
(5)limsup⁡t→0+Θ(tz+(1-t)x,y)≤Θ(x,y);(H4)
Θ(x,·) is convex and lower semicontinuous for each x∈C;(H5)
for eachx∈H and r>0, there exists a bounded subset Dx⊂C and yx∈C such that for any z∈C∖Dx,
(6)Θ(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0.Given a positive numberr>0. Let Sr(Θ,φ):H→C is the solution set of the auxiliary mixed equilibrium problem, that is, for each x∈H,
(7)Sr(Θ,φ)(x)∶={+1r〈K′(y)-K′(x),z-y〉≥0,∀z∈Cy∈C:Θ(y,z)+φ(z)-φ(y)+1r〈K′(y)-K′(x),z-y〉≥0,∀z∈C}.
In particular, whenever K(x)=(1/2)∥x∥2,∀x∈H, Sr(Θ,φ) is rewritten as Tr(Θ,φ).LetΘ1,Θ2:C×C→R be two bifunctions, and A1,A2:C→H be two nonlinear mappings. Consider the following system of generalized equilibrium problems (SGEP): find (x*,y*)∈C×C such that
(8)Θ1(x*,x)+〈A1y*,x-x*〉+1ν1〈x*-y*,x-x*〉≥0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀x∈C,Θ2(y*,y)+〈A2x*,y-y*〉+1ν2〈y*-x*,y-y*〉≥0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀y∈C,
where ν1>0 and ν2>0 are two constants. It is introduced and studied in [17]. Whenever Θ1≡Θ2≡0, the SGEP reduces to a system of variational inequalities, which is considered and studied in [18]. It is worth to mention that the system of variational inequalities is a tool to solve the Nash equilibrium problem for noncooperative games.In 2010, Ceng and Yao [17] transformed the SGEP into a fixed point problem in the following way.Proposition CY (see [17]).
LetΘ1,Θ2:C×C→R be two bifunctions satisfying conditions (H1)–(H4) and let Ak:C→H be ζk-inverse-strongly monotone for k=1,2. Let νk∈(0,2ζk) for k=1,2. Then, (x*,y*)∈C×C is a solution of SGEP (8) if and only if x* is a fixed point of the mapping G:C→C defined by G=Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2) where y*=Tν2Θ2(I-ν2A2)x*. Here, we denote the fixed point set of G by SGEP(G).Let{Tn}n=1∞ be an infinite family of nonexpansive mappings on H and {λn}n=1∞ be a sequence of nonnegative numbers in [0,1]. For any n≥1, define a mapping Wn on H as follows:
(9)Un,n+1=I,Un,n=λnTnUn,n+1+(1-λn)I,Un,n-1=λn-1Tn-1Un,n+(1-λn-1)I,⋯Un,k=λkTkUn,k+1+(1-λk)I,Un,k-1=λk-1Tk-1Un,k+(1-λk-1)I,⋯Un,2=λ2T2Un,3+(1-λ2)I,Wn=Un,1=λ1T1Un,2+(1-λ1)I.
Such a mapping Wn is called the W-mapping generated by Tn,Tn-1,…,T1 and λn,λn-1,…,λ1.In 2011, for the case whereC=H, Yao et al. [14] proposed the following hybrid iterative algorithm
(10)Θ(yn,z)+φ(z)-φ(yn)+1r〈K′(yn)-K′(xn),z-yn〉≥0,z∈H,xn+1=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μV))Wnyn,∀n≥1,
where f:H→H be a contraction, K:H→R is differentiable and strongly convex, {αn},{βn}⊂(0,1) and x0,u∈H are given, for finding a common element of the set MEP⁡(Θ,φ) and the fixed point set ∩n=1∞Fix⁡(Tn) of an infinite family of nonexpansive mappings {Tn}n=1∞ on H. They proved the strong convergence of the sequence generated by the hybrid iterative algorithm (10) to a point x*∈∩n=1∞Fix⁡(Tn)∩MEP⁡(Θ,φ) under some appropriate conditions. This point x* also solves the following optimization problem:
(OP0)min⁡x∈∩n=1∞Fix⁡(Tn)∩MEP⁡(Θ,φ)μ2〈Vx,x〉+12∥x-u∥2-h(x),
where h:H→R is the potential function of γf.Letf:H→H be a contraction and V be a strongly positive bounded linear operator on H. Assume that φ:H→R is a lower semicontinuous and convex functional, that Θ,Θ1,Θ2:H×H→R satisfy conditions (H1)–(H4), and that A,A1,A2:H→H are inverse-strongly monotone. Let the mapping G be defined as in Proposition CY. Very recently, Ceng et al. [11] introduced the following hybrid extragradient-like iterative algorithm
(11)zn=Srn(Θ,φ)(xn-rnAxn),xn+1=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μV))WnGzn,∀n≥0,
for finding a common solution of GMEP (4), SGEP (8) and the fixed point problem of an infinite family of nonexpansive mappings {Tn}n=1∞ on H, where {rn}⊂(0,∞),{αn},{βn}⊂(0,1),νk∈(0,2ζk),k=1,2, and x0,u∈H are given. The authors proved the strong convergence of the sequence generated by the hybrid iterative algorithm (11) to a point x*∈∩n=1∞Fix⁡(Tn)∩GMEP⁡(Θ,φ,A)∩SGEP(G) under some suitable conditions. This point x* also solves the following optimization problem:
(OP1)min⁡x∈∩n=1∞Fix⁡(Tn)∩GMEP⁡(Θ,φ,A)∩SGEP(G)μ2〈Vx,x〉+12∥x-u∥2-h(x),
where h:H→R is the potential function of γf.On the other hand, letB be a single-valued mapping of C into H and R be a set-valued mapping with D(R)=C. Consider the following variational inclusion: find a point x∈C such that
(12)0∈Bx+Rx.
We denote by I(B,R) the solution set of the variational inclusion (12). In particular, if B=R=0, then I(B,R)=C. If B=0, then problem (12) becomes the inclusion problem introduced by Rockafellar [19]. It is known that problem (12) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, and so forth. Let a set-valued mapping R:D(R)⊂H→2H be maximal monotone. We define the resolvent operator JR,λ:H→D(R)¯ associated with R and λ as follows:
(13)JR,λ=(I+λR)-1,∀x∈H,
where λ is a positive number.In this paper, we will introduce and analyze an iterative algorithm by hybrid extragradient-like viscosity method for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems. Such solution also solves an optimization problem. Several special cases are also discussed. The results presented in this paper are the supplement, extension, improvement and generalization of the previously known results in this area.


## 2. Preliminaries

Throughout this paper, we assume thatH is a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let C be a nonempty closed convex subset of H. We write xn⇀x to indicate that the sequence {xn} converges weakly to x and xn→x to indicate that the sequence {xn} converges strongly to x. Moreover, we use ωw(xn) to denote the weak ω-limit set of the sequence {xn}, that is,
(14)ωw(xn)∶={forsomesubsequence{xni}of{xn}{xni}x∈H:xni⇀xforsomesubsequence{xni}of{xn}}.Definition 1.
A mappingA:C→H is called
(i) monotone if(15)〈Ax-Ay,x-y〉≥0,∀x,y∈C,
(ii)η-strongly monotone if there exists a constant η>0 such that
(16)〈Ax-Ay,x-y〉≥η∥x-y∥2,∀x,y∈C,
(iii)ζ-inverse-strongly monotone if there exists a constant ζ>0 such that
(17)〈Ax-Ay,x-y〉≥ζ∥Ax-Ay∥2,∀x,y∈C.It is easy to see that the projectionPC is 1-ism. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.Definition 2.
A differentiable functionK:H→R is called:
(i) convex, if(18)K(y)-K(x)≥〈K′(x),y-x〉,∀x,y∈H,
where K′(x) is the Frechet derivative of K at x;
(ii) strongly convex, if there exists a constantσ>0 such that
(19)K(y)-K(x)-〈K′(x),y-x〉≥σ2∥x-y∥2,∀x,y∈H.It is easy to see that ifK:H→R is a differentiable strongly convex function with constant σ>0 then K′:H→H is strongly monotone with constant σ>0.The metric (or nearest point) projection fromH onto C is the mapping PC:H→C which assigns to each point x∈H the unique point PCx∈C satisfying the property
(20)∥x-PCx∥=inf⁡y∈C∥x-y∥=:d(x,C).Some important properties of projections are gathered in the following proposition.Proposition 3.
For given
x
∈
H
and
z
∈
C:(i)
z=PCx⇔〈x-z,y-z〉≤0,∀y∈C;(ii)
z=PCx⇔∥x-z∥2≤∥x-y∥2-∥y-z∥2, ∀y∈C;(iii)
〈PCx-PCy,x-y〉≥∥PCx-PCy∥2, ∀y∈H. (This implies thatPCis nonexpansive and monotone.)By using the technique of [20], we can readily obtain the following elementary result.Proposition 4 (see [11, Lemma 1 and Proposition 1]).
Let
C
be a nonempty closed convex subset of a real Hilbert space
H
and let
φ
:
C
→
R
be a lower semicontinuous and convex function. Let
Θ
:
C
×
C
→
R
be a bifunction satisfying the conditions (H1)–(H4). Assume that
(i)
K
:
H
→
R
is strongly convex with constant
σ
>
0
and the function
x
↦
〈
y
-
x
,
K
′
(
x
)
〉
is weakly upper semicontinuous for each
y
∈
H;
(ii)
for each
x
∈
H
and
r
>
0,there exists a bounded subsetDx⊂Candyx∈Csuch that for anyz∈C∖Dx,
(21)Θ(z,yx)+φ(yx)-φ(z)+1r〈K′(z)-K′(x),yx-z〉<0.
Then the following hold:(a) for each
x
∈
H
,
S
r
(
Θ
,
φ
)
(
x
)
≠
∅;(b)Sr(Θ,φ)is single-valued;(c)Sr(Θ,φ)is nonexpansive ifK′is Lipschitz continuous with constantν>0and(22)〈K′(x1)-K′(x2),u1-u2〉≥〈K′(u1)-K′(u2),u1-u2〉,∀(x1,x2)∈H×H,whereui=Sr(Θ,φ)(xi)fori=1,2;(d) for alls,t>0andx∈H(23)〈K′(Ss(Θ,φ)x)-K′(St(Θ,φ)x),Ss(Θ,φ)x-St(Θ,φ)x〉≤s-ts〈K′(Ss(Θ,φ)x)-K′(x),Ss(Θ,φ)x-St(Θ,φ)x〉;(e)Fix(Sr(Θ,φ))=MEP(Θ,φ);(f)MEP(Θ,φ)is closed and convex.
In particular, whenever
Θ
:
C
×
C
→
R
is a bifunction satisfying the conditions (H1)–(H4) and
K
(
x
)
=
(
1
/
2
)
∥
x
∥
2
,
∀
x
∈
H,then that is, for anyx,y∈H,
(24)∥Sr(Θ,φ)x-Sr(Θ,φ)y∥2≤〈Sr(Θ,φ)x-Sr(Θ,φ)y,x-y〉(Sr(Θ,φ)is firmly nonexpansive) and(25)∥Ss(Θ,φ)x-St(Θ,φ)x∥≤|s-t|s∥Ss(Θ,φ)x-x∥,hhhh∀s,t>0,x∈H.In this case,Sr(Θ,φ)is rewritten asTr(Θ,φ).If, in addition,φ≡0,thenTr(Θ,φ)is rewritten asTrΘ.Remark 5.
SupposeK:H→R is strongly convex with constant σ>0 and K′:H→H is Lipschitz continuous with constant ν>0. Then K′:H→H is σ-strongly monotone and ν-Lipschitz continuous with positive constants σ,ν>0. Utilizing Proposition 4 (d) we can show that for all s,t>0 and x∈H,
(26)∥Ss(Θ,φ)x-St(Θ,φ)x∥≤|s-t|s·νσ∥Ss(Θ,φ)x-x∥.We need some facts and tools in a real Hilbert spaceH which are listed as lemmas below.Lemma 6.
LetX be a real inner product space. Then there holds the following inequality
(27)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈X.Lemma 7.
LetH be a real Hilbert space. Then the following hold:(a)
∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,y〉 for all x,y∈H;(b)
∥λx+μy∥2=λ∥x∥2+μ∥y∥2-λμ∥x-y∥2 for all x,y∈H and λ,μ∈[0,1] with λ+μ=1;(c)
If{xn} is a sequence in H such that xn⇀x, it follows that
(28)limsup⁡n→∞∥xn-y∥2=limsup⁡n→∞∥xn-x∥2+∥x-y∥2,∀y∈H.We have the following crucial lemmas concerning the W-mappings defined by (9).Lemma 8 (see [21, Lemma 3.2]).
Let{Tn}n=1∞ be a sequence of nonexpansive self-mappings on H such that ∩n=1∞Fix⁡(Tn)≠∅ and let {λn} be a sequence in (0,b] for some b∈(0,1). Then, for every x∈H and k≥1 the limit lim⁡n→∞Un,kx exists, where Un,k is defined by (9).Remark 9 (see [22, Remark 3.1]).
It can be known from Lemma8 that if D is a nonempty bounded subset of H, then for ϵ>0 there exists n0≥k such that for all n>n0(29)sup⁡x∈D∥Un,kx-Ukx∥≤ϵ.Remark 10 (see [22, Remark 3.2]).
Utilizing Lemma8, we define a mapping W:H→H as follows:
(30)Wx=lim⁡n→∞Wnx=lim⁡n→∞Un,1x,∀x∈H.
Such a W is called the W-mapping generated by T1,T2,… and λ1,λ2,…. Since Wn is nonexpansive, W:H→H is also nonexpansive. Indeed, observe that for each x,y∈H(31)∥Wx-Wy∥=lim⁡n→∞∥Wnx-Wny∥≤∥x-y∥.
If {xn} is a bounded sequence in H, then we put D={xn:n≥1}. Hence, it is clear from Remark 5 that for an arbitrary ϵ>0 there exists N0≥1 such that for all n>N0(32)∥Wnxn-Wxn∥=∥Un,1xn-U1xn∥≤sup⁡x∈D∥Un,1x-U1x∥≤ϵ.
This implies that
(33)lim⁡n→∞∥Wnxn-Wxn∥=0.Lemma 11 (see [21, Lemma 3.3]).
Let{Tn}n=1∞ be a sequence of nonexpansive self-mappings on H such that ∩n=1∞Fix⁡(Tn)≠∅, and let {λn} be a sequence in (0,b] for some b∈(0,1). Then, Fix⁡(W)=∩n=1∞Fix⁡(Tn).Lemma 12 (see [23, Demiclosedness principle]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C. Then I-T is demiclosed. That is, whenever {xn} is a sequence in C weakly converging to some x∈C and the sequence {(I-T)xn} strongly converges to some y, it follows that (I-T)x=y. Here I is the identity operator of H.Lemma 13.
LetA:C→H be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 3 (i)) implies
(34)u∈VI⁡(C,A)⟺u=PC(u-λAu),λ>0.Lemma 14 (see [24]).
Let{xn} and {wn} be bounded sequences in a real Banach space X and {βn} be a sequence in [0,1] with 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1. Suppose
(35)xn+1=βnxn+(1-βn)wn,∀n≥0,limsup⁡n→∞(∥wn+1-wn∥-∥xn+1-xn∥)≤0.
Then, lim⁡n→∞∥wn-xn∥=0.Lemma 15 (see [25]).
Assume that{an} is a sequence of nonnegative real numbers such that
(36)an+1≤(1-γn)an+σnγn,∀n≥1,
where {γn} is a sequence in [0,1] and {σn} is a real sequence such that(i)
∑n=1∞‍γn=∞;(ii)
limsup⁡n→∞σn≤0 or ∑n=1∞‍|σnγn|<∞. Then lim⁡n→∞an=0.Recall that a set-valued mappingT:D(T)⊂H→2H is called monotone if for all x,y∈D(T),f∈Tx and g∈Ty imply
(37)〈f-g,x-y〉≥0.
A set-valued mapping T is called maximal monotone if T is monotone and (I+λT)D(T)=H for each λ>0, where I is the identity mapping of H. We denote by G(T) the graph of T. It is known that a monotone mapping T is maximal if and only if, for (x,f)∈H×H,〈f-g,x-y〉≥0 for every (y,g)∈G(T) implies f∈Tx. Next we provide an example to illustrate the concept of maximal monotone mapping.LetA:C→H be a monotone, k-Lipschitz-continuous mapping and let NCv be the normal cone to C at v∈C, that is,
(38)NCv={w∈H:〈v-u,w〉≥0,∀u∈C}.
Define
(39)Tv={Av+NCv,ifv∈C,∅,ifv∉C.
Then, T is maximal monotone and 0∈Tv if and only if v∈VI⁡(C,A); see [19].Assume thatR:D(R)⊂H→2H is a maximal monotone mapping. Let λ>0. In terms of Huang [26] (see also [27]), there holds the following property for the resolvent operator JR,λ:H→D(R)¯.Lemma 16.
J
R
,
λ is single-valued and firmly nonexpansive, that is,
(40)〈JR,λx-JR,λy,x-y〉≥∥JR,λx-JR,λy∥2,∀x,y∈H.
Consequently, JR,λ is nonexpansive and monotone.Lemma 17 (see [28]).
LetR be a maximal monotone mapping with D(R)=C. Then for any given λ>0,u∈C is a solution of problem (12) if and only if u∈C satisfies
(41)u=JR,λ(u-λBu).Lemma 18 (see [27]).
LetR be a maximal monotone mapping with D(R)=C and let B:C→H be a strongly monotone, continuous and single-valued mapping. Then for each z∈H, the equation z∈(B+λR)x has a unique solution xλ for λ>0.Lemma 19 (see [28]).
LetR be a maximal monotone mapping with D(R)=C and B:C→H be a monotone, continuous and single-valued mapping. Then (I+λ(R+B))C=H for each λ>0. In this case, R+B is maximal monotone.Lemma 20 (see [29]).
LetC be a nonempty closed convex subset of a real Hilbert space H, and g:C→R∪+∞ be a proper lower semicontinuous differentiable convex function. If x* is a solution the minimization problem
(42)g(x*)=inf⁡x∈Cg(x),
then,
(43)〈g′(x),x-x*〉≥0,∀x∈C.
In particular, if x* solves (OP), then
(44)〈u+(γf-(I+μV))x*,x-x*〉≤0.


## 3. Main Results

In this section, we introduce and analyze an iterative algorithm by hybrid extragradient-like viscosity method for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under appropriate conditions imposed on the parameter sequences we will prove strong convergence of the proposed algorithm.Theorem 21.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R be a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let A,Ak:H→H and Bi:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone and ηi-inverse strongly monotone, respectively, where k∈{1,2} and i∈{1,2,…,N}. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and {λn} be a sequence in (0,b] for some b∈(0,1). Let V be a γ¯-strongly positive bounded linear operator and f:H→H be an l-Lipschitzian mapping with γl<(1+μ)γ¯. Let Wn be the W-mapping defined by (9). Assume that Ω:=∩n=1∞Fix⁡(Tn)∩GMEP⁡(Θ,φ,A)∩SGEP(G)∩∩i=1NI(Bi,Ri)≠∅ where G is defined as in Proposition CY. Let {αn},{βn} and {δn} be three sequences in [0,1]. Assume that:(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(45)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
lim⁡n→∞αn=0,∑n=1∞‍αn=∞, 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞δn≤limsup⁡n→∞δn<1;(iv)
νk∈(0,2ζk),k∈{1,2},μi∈(0,2ηi),i∈{1,2,…,N}, and {rn}⊂[0,2ζ] satisfies
(46)0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ;(v)
lim⁡n→∞(|δn+1-δn|+|rn+1-rn|)=0.
Givenx1∈H arbitrarily, then the sequence {xn} generated iteratively by
(47)un=Srn(Θ,φ)(I-rnA)xn,zn=JRN,μN(I-μNBN)JRN-1,μN-1×(I-μN-1BN-1)⋯JR1,μ1(I-μ1B1)un,yn=δnGzn+(1-δn)Wnzn,xn+1=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μV))Wnyn,∀n≥1,
converges strongly to x*∈Ω which solves the following optimization problem provided Sr(Θ,φ) is firmly nonexpansive:
(OP2)min⁡x∈Ωμ2〈Ax,x〉+12∥x-u∥2-h(x),
where h is the potential function of γf.Proof.
Sincelim⁡n→∞αn=0 and 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1, we may assume, without loss of generality, that αn≤(1-βn)(1+μ∥V∥)-1. Since V is a γ¯-strongly positive bounded linear operator on H, we know that
(48)∥V∥=sup⁡{〈Vu,u〉:u∈H,∥u∥=1}.
Observe that
(49)〈((1-βn)I-αn(I+μV))u,u〉=1-βn-αn-αnμ〈Vu,u〉≥1-βn-αn-αnμ∥V∥≥0
that is, (1-βn)I-αn(I+μV) is positive. It follows that
(50)∥(1-βn)I-αn(I+μV)∥=sup⁡{〈((1-βn)I-αn(I+μV))u,u〉:u∈H,∥u∥=1}=sup⁡{1-βn-αn-αnμ〈Vu,u〉:u∈H,∥u∥=1}≤1-βn-αn-αnμγ¯.
Put
(51)Λi=JRi,μi(I-μiBi)JRi-1,μi-1×(I-μi-1Bi-1)⋯JR1,μ1(I-μ1B1)
for all i∈{1,2,…,N}, and Λ0=I, where I is the identity mapping on H. Then we have that zn=ΛNun.
We divide the rest of the proof into several steps.
Step 1. We show that {xn} is bounded. Indeed, take p∈Ω arbitrarily. Since p=Srn(Θ,φ)(p-rnAp), A is ζ-inverse strongly monotone and 0≤rn≤2ζ, we have, for any n≥1,
(52)∥un-p∥2=∥Srn(Θ,φ)(I-rnA)xn-Srn(Θ,φ)(I-rnA)p∥2≤∥(I-rnA)xn-(I-rnA)p∥2=∥(xn-p)-rn(Axn-Ap)∥2=∥xn-p∥2-2rn〈xn-p,Axn-Ap〉+rn2∥Axn-Ap∥2≤∥xn-p∥2-2rnζ∥Axn-Ap∥2+rn2∥Axn-Ap∥2=∥xn-p∥2+rn(rn-2ζ)∥Axn-Ap∥2≤∥xn-p∥2.
Since p=JRi,μi(I-μiBi)p,Λip=p and Bi is ηi-inverse strongly monotone, where μi∈(0,2ηi),i∈{1,2,…,N}, by Lemma 16 we deduce that for each n≥1(53)∥zn-p∥2=∥JRN,μN(I-μNBN)ΛN-1un-JRN,μN(I-μNBN)ΛN-1p∥2≤∥(I-μNBN)ΛN-1un-(I-μNBN)ΛN-1p∥2=∥(ΛN-1un-ΛN-1p)-μN(BNΛN-1un-BNΛN-1p)∥2≤∥ΛN-1un-ΛN-1p∥2+μN(μN-2ηN)∥BNΛN-1un-BNΛN-1p∥2≤∥ΛN-1un-ΛN-1p∥2⋯≤∥Λ0un-Λ0p∥2=∥un-p∥2.
Combining (52) and (53), we have
(54)∥zn-p∥≤∥xn-p∥.
Since p=Gp=Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)p, Ak is ζk-inverse-strongly monotone for k=1,2, and 0≤νk≤2ζk for k=1,2, we deduce that, for any n≥1,
(55)∥Gzn-p∥2=∥Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)zn-Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)p∥2≤∥(I-ν1A1)Tν2Θ2(I-ν2A2)zn-(I-ν1A1)Tν2Θ2(I-ν2A2)p∥2=∥[Tν2Θ2(I-ν2A2)zn-Tν2Θ2(I-ν2A2)p]-ν1[A1Tν2Θ2(I-ν2A2)zn-A1Tν2Θ2(I-ν2A2)p]∥2≤∥Tν2Θ2(I-ν2A2)zn-Tν2Θ2(I-ν2A2)p∥2+ν1(ν1-2ζ1)×∥A1Tν2Θ2(I-ν2A2)zn-A1Tν2Θ2(I-ν2A2)p∥2≤∥Tν2Θ2(I-ν2A2)zn-Tν2Θ2(I-ν2A2)p∥2≤∥(I-ν2A2)zn-(I-ν2A2)p∥2=∥(zn-p)-ν2(A2zn-A2p)∥2≤∥zn-p∥2+ν2(ν2-2ζ2)∥A2zn-A2p∥2≤∥zn-p∥2.
(This shows that G is nonexpansive.) Thus, from (54), we get
(56)∥yn-p∥=∥δn(Gzn-p)+(1-δn)(Wnzn-p)∥≤δn∥Gzn-p∥+(1-δn)∥Wnzn-p∥≤δn∥zn-p∥+(1-δn)∥zn-p∥=∥zn-p∥≤∥xn-p∥.
Set V¯=I+μV. Then from (47) we have
(57)∥xn+1-p∥=∥αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μV))Wnyn-p∥=∥αnu+αn(γf(xn)-V¯p)+βn(xn-p)+((1-βn)I-αnV¯)(Wnyn-p)∥≤∥(1-βn)I-αnV¯∥∥Wnyn-p∥+βn∥xn-p∥+αn∥u∥+αn∥γf(xn)-V¯p∥≤(1-βn-αn-αnμγ¯)∥yn-p∥+βn∥xn-p∥+αn∥u∥+αn∥γf(xn)-V¯p∥≤(1-αn-αnμγ¯)∥xn-p∥+αn∥u∥+αn(γ∥f(xn)-f(p)∥+∥γf(p)-V¯p∥)≤(1-αn-αnμγ¯)∥xn-p∥+αn∥u∥+αn(γl∥xn-p∥+∥γf(p)-V¯p∥)≤[1-((1+μ)γ¯-γl)αn]∥xn-p∥+αn(∥γf(p)-V¯p∥+∥u∥)=[1-((1+μ)γ¯-γl)αn]∥xn-p∥+((1+μ)γ¯-γl)αn×∥γf(p)-V¯p∥+∥u∥(1+μ)γ¯-γl≤max⁡{∥xn-p∥,∥γf(p)-V¯p∥+∥u∥(1+μ)γ¯-γl}.
By induction, we get
(58)∥xn-p∥≤max⁡{∥x0-p∥,∥γf(p)-V¯p∥+∥u∥(1+μ)γ¯-γl}.
Therefore, {xn} is bounded and so are the sequences {un},{zn},{yn},{f(xn)} and {Wnyn}.
Step 2. We show that ∥xn+1-xn∥→0 as n→∞.
Indeed, define(59)xn+1=βnxn+(1-βn)wn,∀n≥1.
Then from the definition of wn, we obtain
(60)wn+1-wn=xn+2-βn+1xn+11-βn+1-xn+1-βnxn1-βn=αn+1(u+γf(xn+1))+((1-βn+1)I-αn+1V¯)Wn+1yn+11-βn+1-αn(u+γf(xn))+((1-βn)I-αnV¯)Wnyn1-βn=αn+11-βn+1(u+γf(xn+1))-αn1-βn(u+γf(xn))+Wn+1yn+1-Wnyn+αn1-βnV¯Wnyn-αn+11-βn+1V¯Wn+1yn+1=αn+11-βn+1[u+γf(xn+1)-V¯Wn+1yn+1]+αn1-βn[V¯Wnyn-u-γf(xn)]+Wn+1yn+1-Wn+1yn+Wn+1yn-Wnyn.
It follows that
(61)∥wn+1-wn∥-∥xn+1-xn∥≤αn+11-βn+1(∥u∥+∥γf(xn+1)∥+∥V¯Wn+1yn+1∥)+αn1-βn(∥V¯Wnyn∥+∥u∥+∥γf(xn)∥)+∥Wn+1yn+1-Wn+1yn∥+∥Wn+1yn-Wnyn∥-∥xn+1-xn∥≤αn+11-βn+1(∥u∥+∥γf(xn+1)∥+∥V¯Wn+1yn+1∥)+αn1-βn(∥V¯Wnyn∥+∥u∥+∥γf(xn)∥)+∥Wn+1yn-Wnyn∥+∥yn+1-yn∥-∥xn+1-xn∥.
From (9), since Wn,Tn and Un,i are all nonexpansive, we have
(62)∥Wn+1zn-Wnzn∥=∥λ1T1Un+1,2zn-λ1T1Un,2zn∥≤λ1∥Un+1,2zn-Un,2zn∥=λ1∥λ2T2Un+1,3zn-λ2T2Un,3zn∥≤λ1λ2∥Un+1,3zn-Un,3zn∥⋯≤λ1λ2⋯λn∥Un+1,n+1zn-Un,n+1zn∥≤M∏i=1nλi,(62)′∥Wn+1yn-Wnyn∥=∥λ1T1Un+1,2yn-λ1T1Un,2yn∥≤λ1∥Un+1,2yn-Un,2yn∥=λ1∥λ2T2Un+1,3yn-λ2T2Un,3yn∥≤λ1λ2∥Un+1,3yn-Un,3yn∥⋯≤λ1λ2⋯λn∥Un+1,n+1yn-Un,n+1yn∥≤M∏i=1nλi,
where M is a constant such that
(63)sup⁡n≥1{∥Un+1,n+1zn∥+∥Un,n+1zn∥}≤M,sup⁡n≥1{∥Un+1,n+1yn∥+∥Un,n+1yn∥}≤M.
On the other hand, we estimate ∥yn+1-yn∥. Taking into account that 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1,0<liminf⁡n→∞δn≤limsup⁡n→∞δn<1 and 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ, we may assume, without loss of generality, that {rn}⊂[c,d]⊂(0,2ζ) and {βn},{δn}⊂[c^,d^]⊂(0,1). Utilizing Remark 5 and Lemma 16, we have
(64)∥zn+1-zn∥2=∥JRN,μN(I-μNBN)ΛN-1un+1-JRN,μN(I-μNBN)ΛN-1un∥2≤∥(I-μNBN)ΛN-1un+1-(I-μNBN)ΛN-1un∥2=∥(ΛN-1un+1-ΛN-1un)-μN(BNΛN-1un+1-BNΛN-1un)∥2≤∥ΛN-1un+1-ΛN-1un∥2+μN(μN-2ηN)∥BNΛN-1un+1-BNΛN-1un∥2≤∥ΛN-1un+1-ΛN-1un∥2⋯≤∥Λ0un+1-Λ0un∥2=∥un+1-un∥2,(65)∥(I-rn+1A)xn+1-(I-rnA)xn∥=∥xn+1-xn-rn+1(Axn+1-Axn)+(rn-rn+1)Axn∥≤∥xn+1-xn-rn+1(Axn+1-Axn)∥+|rn+1-rn|∥Axn∥≤∥xn+1-xn∥+|rn+1-rn|∥Axn∥,(66)∥un+1-un∥=∥Srn+1(Θ,φ)(I-rn+1A)xn+1-Srn(Θ,φ)(I-rnA)xn∥=∥Srn+1(Θ,φ)(I-rn+1A)xn+1-Srn+1(Θ,φ)(I-rnA)xn+Srn+1(Θ,φ)(I-rnA)xn-Srn(Θ,φ)(I-rnA)xn∥≤∥Srn+1(Θ,φ)(I-rn+1A)xn+1-Srn+1(Θ,φ)(I-rnA)xn∥+∥Srn+1(Θ,φ)(I-rnA)xn-Srn(Θ,φ)(I-rnA)xn∥≤∥(I-rn+1A)xn+1-(I-rnA)xn∥+∥Srn+1(Θ,φ)(I-rnA)xn-Srn(Θ,φ)(I-rnA)xn∥≤∥xn+1-xn∥+|rn+1-rn|∥Axn∥+∥Srn+1(Θ,φ)(I-rnA)xn-Srn(Θ,φ)(I-rnA)xn∥≤∥xn+1-xn∥+|rn+1-rn|∥Axn∥+|rn+1-rn|rn+1·νσ∥Srn+1(Θ,φ)(I-rnA)xn-(I-rnA)xn∥≤∥xn+1-xn∥+|rn+1-rn|×(∥Axn∥+νcσ∥Srn+1(Θ,φ)(I-rnA)xn-(I-rnA)xn∥)≤∥xn+1-xn∥+|rn+1-rn|M1,
where sup⁡n≥1{∥Axn∥+(ν/cσ)∥Srn+1(Θ,φ)(I-rnA)xn-(I-rnA)xn∥}≤M1 for some M1>0.
Note that(67)yn+1-yn=δn(Gzn+1-Gzn)+(δn+1-δn)(Gzn+1-Wn+1zn+1)+(1-δn)(Wn+1zn+1-Wnzn).
Since G is nonexpansive, from (62), (64) and (66) it follows that
(68)∥yn+1-yn∥≤δn∥Gzn+1-Gzn∥+|δn+1-δn|∥Gzn+1-Wn+1zn+1∥+(1-δn)∥Wn+1zn+1-Wnzn∥≤δn∥zn+1-zn∥+|δn+1-δn|∥Gzn+1-Wn+1zn+1∥+(1-δn)(∥Wn+1zn+1-Wn+1zn∥+∥Wn+1zn-Wnzn∥)≤δn∥zn+1-zn∥+|δn+1-δn|∥Gzn+1-Wn+1zn+1∥+(1-δn)(∥zn+1-zn∥+M∏i=1nλi)≤∥zn+1-zn∥+|δn+1-δn|∥Gzn+1-Wn+1zn+1∥+M∏i=1nλi≤∥un+1-un∥+|δn+1-δn|∥Gzn+1-Wn+1zn+1∥+M∏i=1nλi≤∥xn+1-xn∥+|rn+1-rn|M1+|δn+1-δn|∥Gzn+1-Wn+1zn+1∥+M∏i=1nλi.
Utilizing (61), (62)′ and (68), we have
(69)∥wn+1-wn∥-∥xn+1-xn∥≤αn+11-βn+1(∥u∥+∥γf(xn+1)∥+∥V¯Wn+1yn+1∥)+αn1-βn(∥V¯Wnyn∥+∥u∥+∥γf(xn)∥)+∥Wn+1yn-Wnyn∥+∥yn+1-yn∥-∥xn+1-xn∥≤αn+11-βn+1(∥u∥+∥γf(xn+1)∥+∥V¯Wn+1yn+1∥)+αn1-βn(∥V¯Wnyn∥+∥u∥+∥γf(xn)∥)+M∏i=1nλi+∥xn+1-xn∥+|rn+1-rn|M1+|δn+1-δn|∥Gzn+1-Wn+1zn+1∥+M∏i=1nλi-∥xn+1-xn∥≤αn+11-d^(∥u∥+∥γf(xn+1)∥+∥V¯Wn+1yn+1∥)+αn1-d^(∥V¯Wnyn∥+∥u∥+∥γf(xn)∥)+2M∏i=1nλi+|rn+1-rn|M1+|δn+1-δn|∥Gzn+1-Wn+1zn+1∥≤M2(αn+1+αn+∏i=1nλi+|rn+1-rn|+|δn+1-δn|αn+1+αn+∏i=1nλi)≤M2(αn+1+αn+bn+|rn+1-rn|+|δn+1-δn|),
where sup⁡n≥1{(1/(1-d^))(∥u∥+∥γf(xn)∥+∥V¯Wnyn∥)+∥Gzn-Wnzn∥+M1+2M}≤M2 for some M2>0. Since b∈(0,1),lim⁡n→∞αn=0 and lim⁡n→∞(|δn+1-δn|+|rn+1-rn|)=0, we deduce from (69) that
(70)limsup⁡n→∞(∥wn+1-wn∥-∥xn+1-xn∥)=0.
Since xn+1=βnxn+(1-βn)wn for all n≥1, by Lemma 14 we obtain from 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 that
(71)lim⁡n→∞∥wn-xn∥=0,
which immediately yields
(72)lim⁡n→∞∥xn+1-xn∥=lim⁡n→∞(1-βn)∥wn-xn∥=0.
Step 3. ∥yn-Gzn∥→0 as n→∞.
Indeed, utilizing Lemmas6 and 7 (b) we obtain from (47) and (54) that
(73)∥xn+1-p∥2=∥αn((u+γf(xn))-V¯Wnyn)+βn(xn-p)+(1-βn)(Wnyn-p)((u+γf(xn))-V¯Wnyn)∥2≤∥βn(xn-p)+(1-βn)(Wnyn-p)∥2+2αn〈(u+γf(xn))-V¯Wnyn,xn+1-p〉=βn∥xn-p∥2+(1-βn)∥Wnyn-p∥2-βn(1-βn)∥xn-Wnyn∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤βn∥xn-p∥2+(1-βn)∥yn-p∥2-βn(1-βn)∥xn-Wnyn∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤βn∥xn-p∥2+(1-βn)∥xn-p∥2-βn(1-βn)∥xn-Wnyn∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥=∥xn-p∥2-βn(1-βn)∥xn-Wnyn∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥,
which leads to
(74)c^(1-d^)∥xn-Wnyn∥2≤βn(1-βn)∥xn-Wnyn∥2≤∥xn-p∥2-∥xn+1-p∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥)+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
Since lim⁡n→∞∥xn-xn+1∥=0 and lim⁡n→∞αn=0, we deduce from the boundedness of {xn},{yn},{f(xn)} and {Wnyn} that
(75)lim⁡n→∞∥xn-Wnyn∥=0.
Also, by Lemma 7 (b) we deduce from (47) and (54) that
(76)∥yn-p∥2=δn∥Gzn-p∥2+(1-δn)∥Wnzn-p∥2-δn(1-δn)∥Gzn-Wnzn∥2≤δn∥zn-p∥2+(1-δn)∥zn-p∥2-δn(1-δn)∥Gzn-Wnzn∥2=∥zn-p∥2-δn(1-δn)∥Gzn-Wnzn∥2≤∥xn-p∥2-δn(1-δn)∥Gzn-Wnzn∥2.
From (73) and (76) we get
(77)∥xn+1-p∥2≤βn∥xn-p∥2+(1-βn)∥yn-p∥2-βn(1-βn)∥xn-Wnyn∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤βn∥xn-p∥2+(1-βn)∥yn-p∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤βn∥xn-p∥2+(1-βn)×[∥xn-p∥2-δn(1-δn)∥Gzn-Wnzn∥2]+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥=∥xn-p∥2-(1-βn)δn(1-δn)∥Gzn-Wnzn∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥,
which immediately implies that
(78)c^(1-d^)2∥Gzn-Wnzn∥2≤(1-βn)δn(1-δn)∥Gzn-Wnzn∥2≤∥xn-p∥2-∥xn+1-p∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥)+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
Since lim⁡n→∞αn=0 and lim⁡n→∞∥xn+1-xn∥=0, we deduce from the boundedness of {xn},{yn},{f(xn)} and {Wnyn} that
(79)lim⁡n→∞∥Gzn-Wnzn∥=0.
So, it follows that
(80)lim⁡n→∞∥yn-Gzn∥=lim⁡n→∞(1-δn)∥Wnzn-Gzn∥=0.
Step
4. ∥xn-un∥→0,∥un-zn∥→0,∥zn-Gzn∥→0 and ∥zn-Wzn∥→0 as n→∞.
Indeed, forp∈Ω, we find that
(81)∥un-p∥2=∥Srn(Θ,φ)(I-rnA)xn-Srn(Θ,φ)(I-rnA)p∥2≤∥(I-rnA)xn-(I-rnA)p∥2=∥xn-p-rn(Axn-Ap)∥2≤∥xn-p∥2+rn(rn-2ζ)∥Axn-Ap∥2.
From (47), (53) and (81), we obtain
(82)∥yn-p∥2≤δn∥Gzn-p∥2+(1-δn)∥Wnzn-p∥2≤∥zn-p∥2≤∥un-p∥2≤∥xn-p∥2+rn(rn-2ζ)∥Axn-Ap∥2,
which together with (73), implies that
(83)∥xn+1-p∥2≤βn∥xn-p∥2+(1-βn)∥yn-p∥2-βn(1-βn)∥xn-Wnyn∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤βn∥xn-p∥2+(1-βn)∥yn-p∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤βn∥xn-p∥2+(1-βn)×[∥xn-p∥2+rn(rn-2ζ)∥Axn-Ap∥2]+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥=∥xn-p∥2+(1-βn)rn(rn-2ζ)∥Axn-Ap∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
So, it follows that
(84)(1-d^)c(2ζ-d)∥Axn-Ap∥2≤(1-βn)rn(2ζ-rn)∥Axn-Ap∥2≤∥xn-p∥2-∥xn+1-p∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥)+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
Since lim⁡n→∞αn=0 and lim⁡n→∞∥xn-xn+1∥=0, from the boundedness of {xn},{yn},{Wnyn} and {f(xn)} we get
(85)lim⁡n→∞∥Axn-Ap∥=0.
Furthermore, from the firm nonexpansivity of Srn(Θ,φ), we have
(86)∥un-p∥2=∥Srn(Θ,φ)(I-rnA)xn-Srn(Θ,φ)(I-rnA)p∥2≤〈(I-rnA)xn-(I-rnA)p,un-p〉=12[∥(I-rnA)xn-(I-rnA)p∥2+∥un-p∥2-∥(I-rnA)xn-(I-rnA)p-(un-p)∥2]≤12[∥xn-p∥2+∥un-p∥2-∥xn-un-rn(Axn-Ap)∥2]=12[∥xn-p∥2+∥un-p∥2-∥xn-un∥2+2rn〈Axn-Ap,xn-un〉-rn2∥Axn-Ap∥2],
which implies that
(87)∥un-p∥2≤∥xn-p∥2-∥xn-un∥2+2rn∥Axn-Ap∥∥xn-un∥.
From (47) and (87), we have
(88)∥yn-p∥2≤δn∥Gzn-p∥2+(1-δn)∥Wnzn-p∥2≤∥zn-p∥2≤∥un-p∥2≤∥xn-p∥2-∥xn-un∥2+2rn∥Axn-Ap∥∥xn-un∥,
which together with (73), implies that
(89)∥xn+1-p∥2≤βn∥xn-p∥2+(1-βn)∥yn-p∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤βn∥xn-p∥2+(1-βn)×[∥xn-p∥2-∥xn-un∥2+2rn∥Axn-Ap∥∥xn-un∥]+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤∥xn-p∥2-(1-βn)∥xn-un∥2+2rn∥Axn-Ap∥∥xn-un∥+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
So, it follows that
(90)(1-d^)∥xn-un∥2≤(1-βn)∥xn-un∥2≤∥xn-p∥2-∥xn+1-p∥2+2rn∥Axn-Ap∥∥xn-un∥+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥)+2rn∥Axn-Ap∥∥xn-un∥+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
Since lim⁡n→∞αn=0 and lim⁡n→∞∥xn-xn+1∥=0, from (85) and the boundedness of {xn},{yn},{Wnyn} and {f(xn)} we get
(91)lim⁡n→∞∥xn-un∥=0.
Next we show thatlim⁡n→∞∥AiΛiun-Aip∥=0,   i=1,2,…,N. Observe that
(92)∥Λiun-p∥2=∥JRi,μi(I-μiBi)Λi-1un-JRi,μi(I-μiBi)p∥2≤∥(I-μiBi)Λi-1un-(I-μiBi)p∥2≤∥Λi-1un-p∥2+μi(μi-2ηi)∥BiΛi-1un-Bip∥2≤∥un-p∥2+μi(μi-2ηi)∥BiΛi-1un-Bip∥2≤∥xn-p∥2+μi(μi-2ηi)∥BiΛi-1un-Bip∥2.
From (47) and (92), we have
(93)∥yn-p∥2≤δn∥Gzn-p∥2+(1-δn)∥Wnzn-p∥2≤∥zn-p∥2≤∥Λiun-p∥2≤∥xn-p∥2+μi(μi-2ηi)∥BiΛi-1un-Bip∥2,
which together with (73), implies that
(94)∥xn+1-p∥2≤βn∥xn-p∥2+(1-βn)∥yn-p∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤βn∥xn-p∥2+(1-βn)×[∥xn-p∥2+μi(μi-2ηi)∥BiΛi-1un-Bip∥2]+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥=∥xn-p∥2+(1-βn)μi(μi-2ηi)∥BiΛi-1un-Bip∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
So, it follows that
(95)(1-d^)μi(2ηi-μi)∥BiΛi-1un-Bip∥2≤(1-βn)μi(2ηi-μi)∥BiΛi-1un-Bip∥2≤∥xn-p∥2-∥xn+1-p∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥)+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
Since μi∈(0,2ηi),   i=1,2,…,N,  lim⁡n→∞αn=0 and lim⁡n→∞∥xn-xn+1∥=0, from the boundedness of {xn},{yn},{Wnyn} and {f(xn)} we get
(96)lim⁡n→∞∥BiΛi-1un-Bip∥=0,∀i∈{1,2,…,N}.
By Lemmas 7 (a) and 16, we obtain
(97)∥Λiun-p∥2=∥JRi,μi(I-μiBi)Λi-1un-JRi,μi(I-μiBi)p∥2≤〈(I-μiBi)Λi-1un-(I-μiBi)p,Λiun-p〉=12(∥(I-μiBi)Λi-1un-(I-μiBi)p∥2+∥Λiun-p∥2-∥(I-μiBi)Λi-1un-(I-μiBi)p-(Λiun-p)∥2)≤12(∥Λi-1un-p∥2+∥Λiun-p∥2-∥Λi-1un-Λiun-μi(BiΛi-1un-Bip)∥2)≤12(∥un-p∥2+∥Λiun-p∥2-∥Λi-1un-Λiun-μi(BiΛi-1un-Bip)∥2)≤12(∥xn-p∥2+∥Λiun-p∥2-∥Λi-1un-Λiun-μi(BiΛi-1un-Bip)∥2),
which implies that
(98)∥Λiun-p∥2≤∥xn-p∥2-∥Λi-1un-Λiun-μi(BiΛi-1un-Bip)∥2=∥xn-p∥2-∥Λi-1un-Λiun∥2-μi2∥BiΛi-1un-Bip∥2+2μi〈Λi-1un-Λiun,BiΛi-1un-Bip〉≤∥xn-p∥2-∥Λi-1un-Λiun∥2+2μi∥Λi-1un-Λiun∥∥BiΛi-1un-Bip∥.
From (47) and (98), we have
(99)∥yn-p∥2≤δn∥Gzn-p∥2+(1-δn)∥Wnzn-p∥2≤∥zn-p∥2≤∥Λiun-p∥2≤∥xn-p∥2-∥Λi-1un-Λiun∥2+2μi∥Λi-1un-Λiun∥∥BiΛi-1un-Bip∥,
which together with (73), implies that
(100)∥xn+1-p∥2≤βn∥xn-p∥2+(1-βn)∥yn-p∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤βn∥xn-p∥2+(1-βn)×[∥xn-p∥2-∥Λi-1un-Λiun∥2+2μi∥Λi-1un-Λiun∥∥BiΛi-1un-Bip∥∥xn-p∥2-∥Λi-1un-Λiun∥2]+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤∥xn-p∥2-(1-βn)∥Λi-1un-Λiun∥2+2μi∥Λi-1un-Λiun∥∥BiΛi-1un-Bip∥+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
So, it follows that
(101)(1-d^)∥Λi-1un-Λiun∥2≤(1-βn)∥Λi-1un-Λiun∥2≤∥xn-p∥2-∥xn+1-p∥2+2μi∥Λi-1un-Λiun∥∥BiΛi-1un-Bip∥+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥)+2μi∥Λi-1un-Λiun∥∥BiΛi-1un-Bip∥+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
Since lim⁡n→∞αn=0 and lim⁡n→∞∥xn-xn+1∥=0, from (96) and the boundedness of {xn},{yn},{Wnyn} and {f(xn)} we get
(102)lim⁡n→∞∥Λi-1un-Λiun∥=0,∀i∈{1,2,…,N}.
From (102) we get
(103)∥un-zn∥=∥Λ0un-ΛNun∥≤∥Λ0un-Λ1un∥+∥Λ1un-Λ2un∥+⋯+∥ΛN-1un-ΛNun∥⟶0asn⟶∞.
By (91) and (103), we have
(104)∥xn-zn∥≤∥xn-un∥+∥un-zn∥⟶0asn⟶∞.
On the other hand, for simplicity, we writep~=Tν2Θ2(I-ν2A2)p,vn=Tν2Θ2(I-ν2A2)zn and v~n=Gzn=Tν1Θ1(I-ν1A1)vn for all n≥1. Then
(105)p=Gp=Tν1Θ1(I-ν1A1)p~=Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)p.
We now show that lim⁡n→∞∥Gzn-zn∥=0, that is, lim⁡n→∞∥v~n-zn∥=0. As a matter of fact, for p∈Ω, it follows from (47), (54) and (55) that
(106)∥yn-p∥2≤δn∥Gzn-p∥2+(1-δn)∥Wnzn-p∥2≤δn∥v~n-p∥2+(1-δn)∥zn-p∥2≤δn[∥vn-p~∥2+ν1(ν1-2ζ1)∥A1vn-A1p~∥2]+(1-δn)∥zn-p∥2≤δn[∥zn-p∥2+ν2(ν2-2ζ2)∥A2zn-A2p∥2+ν1(ν1-2ζ1)∥A1vn-A1p~∥2]+(1-δn)∥zn-p∥2=∥zn-p∥2+δn(ν2(ν2-2ζ2)∥A2zn-A2p∥2+ν1(ν1-2ζ1)∥A1vn-A1p~∥2)≤∥xn-p∥2+δn(ν2(ν2-2ζ2)∥A2zn-A2p∥2+ν1(ν1-2ζ1)∥A1vn-A1p~∥2),
which together with (73), implies that
(107)∥xn+1-p∥2≤βn∥xn-p∥2+(1-βn)∥yn-p∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤βn∥xn-p∥2+(1-βn)×[∥xn-p∥2+δn(ν2(ν2-2ζ2)∥A2zn-A2p∥2+ν1(ν1-2ζ1)∥A1vn-A1p~∥2)]+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥=∥xn-p∥2+(1-βn)δn(ν2(ν2-2ζ2)∥A2zn-A2p∥2+ν1(ν1-2ζ1)∥A1vn-A1p~∥2)+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
So, it follows that
(108)(1-d^)c^(ν2(2ζ2-ν2)∥A2zn-A2p∥2+ν1(2ζ1-ν1)∥A1vn-A1p~∥2)≤(1-βn)δn(ν2(2ζ2-ν2)∥A2zn-A2p∥2+ν1(2ζ1-ν1)∥A1vn-A1p~∥2)≤∥xn-p∥2-∥xn+1-p∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥)+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
Since νk∈(0,2ζk),   k=1,2,   lim⁡n→∞αn=0 and lim⁡n→∞∥xn-xn+1∥=0, from the boundedness of {xn},{yn},{Wnyn} and {f(xn)} we get
(109)lim⁡n→∞∥A2zn-A2p∥=0,lim⁡n→∞∥A1vn-A1p~∥=0.
Also, in terms of the firm nonexpansivity of TνkΘk and the ζk-inverse strong monotonicity of Ak for k=1,2, we obtain from νk∈(0,2ζk),k∈{1,2} and (54)-(55) that
(110)∥vn-p~∥2=∥Tν2Θ2(I-ν2A2)zn-Tν2Θ2(I-ν2A2)p∥2≤〈(I-ν2A2)zn-(I-ν2A2)p,vn-p~〉=12[∥(I-ν2A2)zn-(I-ν2A2)p∥2+∥vn-p~∥2-∥(I-ν2A2)zn-(I-ν2A2)p-(vn-p~)∥2]≤12[∥zn-p∥2+∥vn-p~∥2-∥(zn-vn)-ν2(A2zn-A2p)-(p-p~)∥2]≤12[∥xn-p∥2+∥vn-p~∥2-∥(zn-vn)-(p-p~)∥2+2ν2〈(zn-vn)-(p-p~),A2zn-A2p〉-ν22∥A2zn-A2p∥2],∥v~n-p∥2=∥Tν1Θ1(I-ν1A1)vn-Tν1Θ1(I-ν1A1)p~∥2≤〈(I-ν1A1)vn-(I-ν1A1)p~,v~n-p〉=12[∥(I-ν1A1)vn-(I-ν1A1)p~∥2+∥v~n-p∥2-∥(I-ν1A1)vn-(I-ν1A1)p~-(v~n-p)∥2]≤12[∥vn-p~∥2+∥v~n-p∥2-∥(vn-v~n)+(p-p~)∥2+2ν1〈A1vn-A1p~,(vn-v~n)+(p-p~)〉-ν12∥A1vn-A1p~∥2]≤12[∥xn-p∥2+∥v~n-p∥2-∥(vn-v~n)+(p-p~)∥2+2ν1〈A1vn-A1p~,(vn-v~n)+(p-p~)〉∥xn-p∥2+∥v~n-p∥2].
Thus, we have
(111)∥vn-p~∥2≤∥xn-p∥2-∥(zn-vn)-(p-p~)∥2+2ν2〈(zn-vn)-(p-p~),A2zn-A2p〉-ν22∥A2zn-A2p∥2,(112)∥v~n-p∥2≤∥xn-p∥2-∥(vn-v~n)+(p-p~)∥2+2ν1∥A1vn-A1p~∥∥(vn-v~n)+(p-p~)∥.
Consequently, from (47), (54), (55) and (111) it follows that
(113)∥yn-p∥2≤δn∥Gzn-p∥2+(1-δn)∥Wnzn-p∥2≤δn∥v~n-p∥2+(1-δn)∥zn-p∥2≤δn∥vn-p~∥2+(1-δn)∥zn-p∥2≤δn[∥xn-p∥2-∥(zn-vn)-(p-p~)∥2+2ν2〈(zn-vn)-(p-p~),A2zn-A2p〉∥xn-p∥2]+(1-δn)∥xn-p∥2≤∥xn-p∥2-δn∥(zn-vn)-(p-p~)∥2+2ν2∥(zn-vn)-(p-p~)∥∥A2zn-A2p∥,
which together with (73), implies that
(114)∥xn+1-p∥2≤βn∥xn-p∥2+(1-βn)∥yn-p∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤βn∥xn-p∥2+(1-βn)×[∥xn-p∥2-δn∥(zn-vn)-(p-p~)∥2+2ν2∥(zn-vn)-(p-p~)∥∥A2zn-A2p∥∥xn-p∥2]+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤∥xn-p∥2-(1-βn)δn∥(zn-vn)-(p-p~)∥2+2ν2∥(zn-vn)-(p-p~)∥∥A2zn-A2p∥+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
So, it follows that
(115)(1-d^)c^∥(zn-vn)-(p-p~)∥2≤(1-βn)δn∥(zn-vn)-(p-p~)∥2≤∥xn-p∥2-∥xn+1-p∥2+2ν2∥(zn-vn)-(p-p~)∥∥A2zn-A2p∥+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥)+2ν2∥(zn-vn)-(p-p~)∥∥A2zn-A2p∥+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
Since lim⁡n→∞αn=0 and lim⁡n→∞∥xn-xn+1∥=0, from (109) and the boundedness of {xn},{yn},{Wnyn} and {f(xn)} we get
(116)lim⁡n→∞∥(zn-vn)-(p-p~)∥=0.
Furthermore, from (47), (54) and (112) it follows that
(117)∥yn-p∥2≤δn∥Gzn-p∥2+(1-δn)∥Wnzn-p∥2≤δn∥v~n-p∥2+(1-δn)∥zn-p∥2≤δn[∥xn-p∥2-∥(vn-v~n)+(p-p~)∥2+2ν1∥A1vn-A1p~∥∥(vn-v~n)+(p-p~)∥∥xn-p∥2-∥(vn-v~n)+(p-p~)∥2]+(1-δn)∥xn-p∥2≤∥xn-p∥2-δn∥(vn-v~n)+(p-p~)∥2+2ν1∥A1vn-A1p~∥∥(vn-v~n)+(p-p~)∥,
which together with (73), implies that
(118)∥xn+1-p∥2≤βn∥xn-p∥2+(1-βn)∥yn-p∥2+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤βn∥xn-p∥2+(1-βn)×[∥xn-p∥2-δn∥(vn-v~n)+(p-p~)∥2+2ν1∥A1vn-A1p~∥∥(vn-v~n)+(p-p~)∥∥xn-p∥2]+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤∥xn-p∥2-(1-βn)δn∥(vn-v~n)+(p-p~)∥2+2ν1∥A1vn-A1p~∥∥(vn-v~n)+(p-p~)∥+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
So, it follows that
(119)(1-d^)c^∥(vn-v~n)+(p-p~)∥2≤(1-βn)δn∥(vn-v~n)+(p-p~)∥2≤∥xn-p∥2-∥xn+1-p∥2+2ν1∥A1vn-A1p~∥∥(vn-v~n)+(p-p~)∥+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥≤∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥)+2ν1∥A1vn-A1p~∥∥(vn-v~n)+(p-p~)∥+2αn∥(u+γf(xn))-V¯Wnyn∥∥xn+1-p∥.
Since lim⁡n→∞αn=0 and lim⁡n→∞∥xn-xn+1∥=0, from (109) and the boundedness of {xn},{yn},{Wnyn} and {f(xn)} we get
(120)lim⁡n→∞∥(vn-v~n)+(p-p~)∥=0.
Note that
(121)∥zn-v~n∥≤∥(zn-vn)-(p-p~)∥+∥(vn-v~n)+(p-p~)∥.
Hence from (116) and (120) we get
(122)lim⁡n→∞∥zn-v~n∥=lim⁡n→∞∥zn-Gzn∥=0,
which together with (79), implies that
(123)∥zn-Wnzn∥≤∥zn-Gzn∥+∥Gzn-Wnzn∥⟶0asn⟶∞.
Also, observe that
(124)∥zn-Wzn∥≤∥zn-Wnzn∥+∥Wnzn-Wzn∥.
From (122), Remark 10 and the boundedness of {zn} we immediately obtain
(125)lim⁡n→∞∥zn-Wzn∥=0.
Step 5. We show that
(126)limsup⁡n→∞〈u+(γf-V¯)x*,xn-x*〉≤0,
where x* is a solution of (OP2).
Indeed, we note thatV is a γ¯-strongly positive bounded linear operator and f:H→H is an l-Lipschitzian mapping with γl<(1+μ)γ¯. It is clear that
(127)〈(V¯x-(u+γf(x)))-(V¯y-(u+γf(y))),x-y〉≥((1+μ)γ¯-γl)∥x-y∥2,∀x,y∈H.
Hence we deduce that V¯x-(u+γf(x)) is ((1+μ)γ¯-γl)-strongly monotone. In the meantime, it is easy to see that V¯x-(u+γf(x)) is (∥V¯∥+γl)-Lipschitzian with constant ∥V¯∥+γl>0. Thus, there exists a unique solution x* in Ω to the VIP
(128)〈u+(γf-V¯)x*,u-x*〉≤0,∀u∈Ω.
Equivalently, x*∈Ω solves (OP2) (due to Lemma 20).
First, we observe that there exists a subsequence{xni} of {xn} such that
(129)limsup⁡n→∞〈u+(γf-V¯)x*,xn-x*〉=lim⁡i→∞〈u+(γf-V¯)x*,xni-x*〉.
Since {xni} is bounded, there exists a subsequence {xnij} of {xni} which converges weakly to some w. Without loss of generality, we may assume that xni⇀w. From (91) and (102)–(104), we have that uni⇀w,Λmuni⇀w and zni⇀w, where m∈{1,2,…,N}. By (122) and (125) we have that ∥Gzn-zn∥→0 and ∥Wzn-zn∥→0 as n→∞. Utilizing the similar arguments to those of (55), we know that G is nonexpansive. Hence, by Lemma 12 we obtain w∈Fix⁡(G)=SGEP(G) and w∈Fix⁡(W)=∩n=1∞Fix⁡(Tn) (due to Lemma 11). Next, we prove that w∈∩m=1NI(Bm,Rm). As a matter of fact, since Bm is ηm-inverse strongly monotone, Bm is a monotone and Lipschitz continuous mapping. It follows from Lemma 19 that Rm+Bm is maximal monotone. Let (v,g)∈G(Rm+Bm), that is, g-Bmv∈Rmv. Again, since Λmun=JRm,μm(I-μmBm)Λm-1un,n≥1,m∈{1,2,…,N}, we have
(130)Λm-1un-μmBmΛm-1un∈(I+μmRm)Λmun.
that is,
(131)1μm(Λm-1un-Λmun-μmBmΛm-1un)∈RmΛmun.
In terms of the monotonicity of Rm, we get
(132)〈-1μmv-Λmun,g-Bmv-1μm(Λm-1un-Λmun-μmBmΛm-1un)〉≥0,
and hence
(133)〈v-Λmun,g〉≥〈+1μm(Λm-1un-Λmun-μmBmΛm-1un)v-Λmun,Bmv+1μm(Λm-1un-Λmun-μmBmΛm-1un)〉=〈+1μm(Λm-1un-Λmun)v-Λmun,Bmv-BmΛmun+BmΛmun-BmΛm-1un+1μm(Λm-1un-Λmun)〉≥〈v-Λmun,BmΛmun-BmΛm-1un〉+〈v-Λmun,1μm(Λm-1un-Λmun)〉.
In particular,
(134)〈v-Λmuni,g〉≥〈v-Λmuni,BmΛmuni-BmΛm-1uni〉+〈v-Λmuni,1μm(Λm-1uni-Λmuni)〉.
Since ∥Λmun-Λm-1un∥→0 (due to (102)) and ∥BmΛmun-BmΛm-1un∥→0 (due to the Lipschitz continuity of Bm), we conclude from Λmuni⇀w and μm∈(0,2ηm),m∈{1,2,…,N} that
(135)lim⁡i→∞〈v-Λmuni,g〉=〈v-w,g〉≥0.
It follows from the maximal monotonicity of Bm+Rm that 0∈(Rm+Bm)w, that is, w∈I(Bm,Rm). Therefore, w∈∩m=1NI(Bm,Rm).
Next, we show thatw∈GMEP⁡(Θ,φ,A). In fact, from un=Srn(Θ,φ)(I-rnA)xn, we know that
(136)Θ(un,y)+φ(y)-φ(un)+〈Axn,y-un〉+1rn〈K′(un)-K′(xn),y-un〉≥0,∀y∈C.
From (H2) it follows that
(137)φ(y)-φ(un)+〈Axn,y-un〉+1rn〈K′(un)-K′(xn),y-un〉≥Θ(y,un),∀y∈C.
Replacing n by ni, we have
(138)φ(y)-φ(uni)+〈Axni,y-uni〉+〈K′(uni)-K′(xni)rni,y-uni〉≥Θ(y,uni),∀y∈C.
Put ut=ty+(1-t)w for all t∈(0,1] and y∈C. Then, from (138) we have
(139)〈ut-uni,Aut〉≥〈ut-uni,Aut〉-φ(ut)+φ(uni)-〈ut-uni,Axni〉-〈K′(uni)-K′(xni)rni,ut-uni〉+Θ(ut,uni)≥〈ut-uni,Aut-Auni〉+〈ut-uni,Auni-Axni〉-φ(ut)+φ(uni)-〈K′(uni)-K′(xni)rni,ut-uni〉+Θ(ut,uni).
Since ∥uni-xni∥→0 as i→∞, we deduce from the Lipschitz continuity of A and K′ that ∥Auni-Axni∥→0 and ∥K′(uni)-K′(xni)∥→0 as i→∞. Further, from the monotonicity of A, we have 〈ut-uni,Aut-Auni〉≥0. So, from (H4), the weakly lower semicontinuity of φ,(K′(uni)-K′(xni))/rni→0 and uni⇀w, we have
(140)〈ut-w,Aut〉≥-φ(ut)+φ(w)+Θ(ut,w),asi⟶∞.
From (H1), (H4) and (140) we also have
(141)0=Θ(ut,ut)+φ(ut)-φ(ut)≤tΘ(ut,y)+(1-t)Θ(ut,w)+tφ(y)+(1-t)φ(w)-φ(ut)=t[Θ(ut,y)+φ(y)-φ(ut)]+(1-t)[Θ(ut,w)+φ(w)-φ(w)-φ(ut)]≤t[Θ(ut,y)+φ(y)-φ(ut)]+(1-t)〈ut-w,Aut〉=t[Θ(ut,y)+φ(y)-φ(ut)]+(1-t)t〈y-w,Aut〉,
and hence
(142)0≤Θ(ut,y)+φ(y)-φ(ut)+(1-t)〈y-w,Aut〉.
Letting t→0, we have, for each y∈C,
(143)0≤Θ(w,y)+φ(y)-φ(w)+〈Aw,y-w〉.
This implies that w∈GMEP⁡(Θ,φ,A). Therefore, w∈∩n=1∞Fix⁡(Tn)∩GMEP⁡(Θ,φ,A)∩SGEP(G)∩∩i=1NI(Bi,Ri):=Ω. This shows that ωw(xn)⊂Ω. Consequently, from (128) and (129) we have
(144)limsup⁡n→∞〈u+(γf-V¯)x*,xn-x*〉=〈u+(γf-V¯)x*,w-x*〉≤0.
Step 6. Finally, we show that xn→x*∈Ω as n→∞.
Indeed, from (47) and (54), we have
(145)∥yn-x*∥2≤δn∥Gzn-x*∥2+(1-δn)∥Wnzn-x*∥2≤∥zn-x*∥2≤∥xn-x*∥2.
In terms of Lemma 6 we have
(146)∥xn+1-x*∥2=∥+((1-βn)I-αnV¯)(Wnyn-x*)αn(u+γf(xn)-V¯x*)+βn(xn-x*)+((1-βn)I-αnV¯)(Wnyn-x*)∥2≤∥βn(xn-x*)+((1-βn)I-αnV¯)(Wnyn-x*)∥2+2αn〈u+γf(xn)-V¯x*,xn+1-x*〉≤[∥((1-βn)I-αnV¯)(Wnyn-x*)∥+βn∥xn-x*∥]22+2αnγ〈f(xn)-f(x*),xn+1-x*〉+2αn〈u+γf(x*)-V¯x*,xn+1-x*〉≤[(1-βn-αn(1+μ)γ¯)∥yn-x*∥+βn∥xn-x*∥]2+2αnγl∥xn-x*∥∥xn+1-x*∥+2αn〈u+γf(x*)-V¯x*,xn+1-x*〉≤(1-αn(1+μ)γ¯)2∥xn-x*∥2+αnγl(∥xn-x*∥2+∥xn+1-x*∥2)+2αn〈u+γf(x*)-V¯x*,xn+1-x*〉,
which leads to
(147)∥xn+1-x*∥2≤1-2αn(1+μ)γ¯+αn2(1+μ)2γ¯2+αnγl1-αnγl×∥xn-x*∥2+2αn1-αnγl〈u+γf(x*)-V¯x*,xn+1-x*〉=[1-2((1+μ)γ¯-γl)αn1-αnγl]∥xn-x*∥2+((1+μ)αnγ¯)21-αnγl∥xn-x*∥2+2αn1-αnγl〈u+γf(x*)-V¯x*,xn+1-x*〉≤[1-2((1+μ)γ¯-γl)αn1-αnγl]∥xn-x*∥2+2((1+μ)γ¯-γl)αn1-αnγl×{(αn(1+μ)2γ¯2)M02((1+μ)γ¯-γl)+1(1+μ)γ¯-γl×〈u+γf(x*)-V¯x*,xn+1-x*〉(αn(1+μ)2γ¯2)M02((1+μ)γ¯-γl)}=(1-γn)∥xn-x*∥2+σnγn,
where M0=sup⁡{∥xn-x*∥2:n≥1},γn=2((1+μ)γ¯-γl)αn/(1-αnγl) and σn=((αn(1+μ)2γ¯2)M0/2((1+μ)γ¯-γl))+(1/((1+μ)γ¯-γl))〈u+γf(x*)-V¯x*,xn+1-x*〉. It is easy to see that ∑n=1∞‍γn=∞ and limsup⁡n→∞σn≤0. Hence by Lemma 15, we infer that the sequence {xn} converges strongly to x*. This completes the proof.Corollary 22.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ be a bifunction from C×C to R satisfying (H1)–(H4) and φ:C→R be a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let A:H→H and Bi:C→H be ζ-inverse strongly monotone and ηi-inverse strongly monotone, respectively, where i∈{1,2,…,N}. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and {λn} be a sequence in (0,b] for some b∈(0,1). Let V be a γ¯-strongly positive bounded linear operator and f:H→H be an l-Lipschitzian mapping with γl<(1+μ)γ¯. Let Wn be the W-mapping defined by (9). Assume that Ω:=∩n=1∞Fix⁡(Tn)∩GMEP⁡(Θ,φ,A)∩∩i=1NI(Bi,Ri)≠∅. Let {αn},{βn} and {δn} be three sequences in [0,1]. Assume that:
(i)K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;
(ii) for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(148)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;
(iii)lim⁡n→∞αn=0,∑n=1∞‍αn=∞, 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞δn≤limsup⁡n→∞δn<1;
(iv)μi∈(0,2ηi),i∈{1,2,…,N}, and {rn}⊂[0,2ζ] satisfies 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ;
(v)lim⁡n→∞(|δn+1-δn|+|rn+1-rn|)=0.
Givenx1∈H arbitrarily, then the sequence {xn} generated iteratively by
(149)un=Srn(Θ,φ)(I-rnA)xn,zn=JRN,μN(I-μNBN)JRN-1,μN-1×(I-μN-1BN-1)⋯JR1,μ1(I-μ1B1)un,yn=δnzn+(1-δn)Wnzn,xn+1=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μV))Wnyn,∀n≥1,
converges strongly to x*∈Ω which solves the following optimization problem provided Sr(Θ,φ) is firmly nonexpansive:
(OP3)min⁡x∈Ωμ2〈Ax,x〉+12∥x-u∥2-h(x),
where h is the potential function of γf.Proof.
In Theorem21, putting Θ1≡Θ2≡0 and A1≡A2≡0, we get Gzn=zn and SGEP(G)=C. Utilizing Theorem 21 we derive the desired result.Corollary 23.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R be a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let A,Ak:H→H and Bi:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone and ηi-inverse strongly monotone, respectively, for i=1,2 and k=1,2. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and {λn} be a sequence in (0,b] for some b∈(0,1). Let V be a γ¯-strongly positive bounded linear operator and f:H→H be an l-Lipschitzian mapping with γl<(1+μ)γ¯. Let Wn be the W-mapping defined by (9). Assume that Ω:=∩n=1∞Fix⁡(Tn)∩GMEP⁡(Θ,φ,A)∩SGEP(G)∩I(B2,R2)∩I(B1,R1)≠∅ where G is defined as in Proposition CY. Let {αn},{βn} and {δn} be three sequences in [0,1]. Assume that:
(i)K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;
(ii) for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(150)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;
(iii)lim⁡n→∞αn=0,∑n=1∞‍αn=∞, 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞δn≤limsup⁡n→∞δn<1;
(iv)νk∈(0,2ζk),   k=1,2,   μi∈(0,2ηi),  i=1,2, and {rn}⊂[0,2ζ] satisfies 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ;
(v)lim⁡n→∞(|δn+1-δn|+|rn+1-rn|)=0.
Givenx1∈H arbitrarily, then the sequence {xn} generated iteratively by
(151)un=Srn(Θ,φ)(I-rnA)xn,zn=JR2,μ2(I-μ2B2)JR1,μ1(I-μ1B1)un,yn=δnGzn+(1-δn)Wnzn,xn+1=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μV))Wnyn,∀n≥1,
converges strongly to x*∈Ω which solves the following optimization problem provided Sr(Θ,φ) is firmly nonexpansive:
(OP4)min⁡x∈Ωμ2〈Ax,x〉+12∥x-u∥2-h(x),
where h is the potential function of γf.Corollary 24.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R be a lower semicontinuous and convex functional. Let R:C→2H be a maximal monotone mapping and let A,Ak:H→H and B:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone and η-inverse strongly monotone, respectively, for k=1,2. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and {λn} be a sequence in (0,b] for some b∈(0,1). Let V be a γ¯-strongly positive bounded linear operator and f:H→H be an l-Lipschitzian mapping with γl<(1+μ)γ¯. Let Wn be the W-mapping defined by (9). Assume that Ω:=∩n=1∞Fix⁡(Tn)∩GMEP⁡(Θ,φ,A)∩SGEP(G)∩I(B,R)≠∅ where G is defined as in Proposition CY. Let {αn},{βn} and {δn} be three sequences in [0,1]. Assume that:
(i)K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;
(ii) for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(152)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;
(iii)lim⁡n→∞αn=0,∑n=1∞‍αn=∞, 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞δn≤limsup⁡n→∞δn<1;
(iv)νk∈(0,2ζk), k=1,2,λ∈(0,2η), and {rn}⊂[0,2ζ] satisfies 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ;
(v)lim⁡n→∞(|δn+1-δn|+|rn+1-rn|)=0.
Givenx1∈H arbitrarily, then the sequence {xn} generated iteratively by
(153)un=Srn(Θ,φ)(I-rnA)xn,yn=δnGJR,λ(I-λB)un+(1-δn)WnJR,λ(I-λB)un,xn+1=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μV))Wnyn,∀n≥1,
converges strongly to x*∈Ω which solves the following optimization problem provided Sr(Θ,φ) is firmly nonexpansive:
(OP5)min⁡x∈Ωμ2〈Ax,x〉+12∥x-u∥2-h(x),
where h is the potential function of γf.Corollary 25.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R be a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let Ak:H→H and Bi:C→H be ζk-inverse strongly monotone and ηi-inverse strongly monotone, respectively, where k∈{1,2} and i∈{1,2,…,N}. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and {λn} be a sequence in (0,b] for some b∈(0,1). Let V be a γ¯-strongly positive bounded linear operator and f:H→H be an l-Lipschitzian mapping with γl<(1+μ)γ¯. Let Wn be the W-mapping defined by (9). Assume that Ω:=∩n=1∞Fix⁡(Tn)∩MEP⁡(Θ,φ)∩SGEP(G)∩∩i=1NI(Bi,Ri)≠∅ where G is defined as in Proposition CY. Let {αn},{βn} and {δn} be three sequences in [0,1]. Assume that:
(i)K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;
(ii) for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(154)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;
(iii)lim⁡n→∞αn=0,∑n=1∞‍αn=∞, 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞δn≤limsup⁡n→∞δn<1;
(iv)νk∈(0,2ζk),  k∈{1,2},  μi∈(0,2ηi),  i∈{1,2,…,N}, and {rn} is a bounded sequence in (0,∞) satisfying liminf⁡n→∞rn>0;
(v)lim⁡n→∞(|δn+1-δn|+|rn+1-rn|)=0.
Givenx1∈H arbitrarily, then the sequence {xn} generated iteratively by
(155)Θ(un,y)+φ(y)-φ(un)+1rn〈k′(un)-k′(xn),y-un〉≥0,∀y∈C,zn=JRN,μN(I-μNBN)JRN-1,μN-1×(I-μN-1BN-1)⋯JR1,μ1(I-μ1B1)un,yn=δnGzn+(1-δn)Wnzn,xn+1=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μV))Wnyn,∀n≥1,
converges strongly to x*∈Ω which solves the following optimization problem provided Sr(Θ,φ) is firmly nonexpansive:
(OP6)min⁡x∈Ωμ2〈Ax,x〉+12∥x-u∥2-h(x),
where h is the potential function of γf.Proof.
In Theorem21, for all n≥1,  un=Srn(Θ,φ)(I-rnA)xn is equivalent to
(156)Θ(un,y)+φ(y)-φ(un)+〈Axn,y-un〉+1rn〈k′(un)-k′(xn),y-un〉≥0,∀y∈C.
Put A≡0. Then it follows that
(157)Θ(un,y)+φ(y)-φ(un)+1rn〈k′(un)-k′(xn),y-un〉≥0,∀y∈C.
Observe that for all ζ∈(0,∞)(158)〈Ax-Ay,x-y〉≥ζ∥Ax-Ay∥2,∀x,y∈H.
So, whenever 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ for some ζ∈(0,∞), we obtain the desired result by using Theorem 21.LetT:H→H be a κ-strictly pseudocontractive mapping. For recent convergence result for strictly pseudocontractive mappings, we refer to [16]. Putting A=I-T, we know that for all x,y∈H(159)∥(I-A)x-(I-A)y∥2≤∥x-y∥2+κ∥Ax-Ay∥2.
Note that
(160)∥(I-A)x-(I-A)y∥2=∥x-y∥2+∥Ax-Ay∥2-2〈Ax-Ay,x-y〉.
Hence we have for all x,y∈H(161)〈Ax-Ay,x-y〉≥1-κ2∥Ax-Ay∥2.
Consequently, if T:H→H is a κ-strictly pseudocontractive mapping, then the mapping A=I-T is (1-κ)/2-inverse strongly monotone.Corollary 26.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R be a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let T,Ak:H→H and Bi:C→H be κ-strictly pseudocontractive, ζk-inverse strongly monotone and ηi-inverse strongly monotone, respectively, where k∈{1,2} and i∈{1,2,…,N}. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and {λn} be a sequence in (0,b] for some b∈(0,1). Let V be a γ¯-strongly positive bounded linear operator and f:H→H be an l-Lipschitzian mapping with γl<(1+μ)γ¯. Let Wn be the W-mapping defined by (9). Assume that Ω:=∩n=1∞Fix⁡(Tn)∩GMEP⁡(Θ,φ,A)∩SGEP(G)∩∩i=1NI(Bi,Ri)≠∅ where A=I-T and G is defined as in Proposition CY. Let {αn},{βn} and {δn} be three sequences in [0,1]. Assume that:
(i)K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;
(ii) for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(162)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;
(iii)lim⁡n→∞αn=0,∑n=1∞‍αn=∞, 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞δn≤limsup⁡n→∞δn<1;
(iv)νk∈(0,2ζk),  k∈{1,2},  μi∈(0,2ηi),  i∈{1,2,…,N}, and {rn}⊂[0,1-κ] satisfies
(163)0<liminf⁡n→∞rn≤limsup⁡n→∞rn<1-κ;
(v)lim⁡n→∞(|δn+1-δn|+|rn+1-rn|)=0.
Givenx1∈H arbitrarily, then the sequence {xn} generated iteratively by
(164)un=Srn(Θ,φ)((1-rn)xn+rnTxn),zn=JRN,μN(I-μNBN)JRN-1,μN-1×(I-μN-1BN-1)⋯JR1,μ1(I-μ1B1)un,yn=δnGzn+(1-δn)Wnzn,xn+1=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μV))Wnyn,∀n≥1,
converges strongly to x*∈Ω which solves the following optimization problem provided Sr(Θ,φ) is firmly nonexpansive:
(OP7)min⁡x∈Ωμ2〈Ax,x〉+12∥x-u∥2-h(x),
where h is the potential function of γf.Proof.
SinceT is a κ-strictly pseudocontractive mapping, the mapping A=I-T is (1-κ)/2-inverse strongly monotone. In this case, put ζ=(1-κ)/2. Moreover, we obtain that
(165)un=Srn(Θ,φ)(I-rnA)xn=Srn(Θ,φ)(xn-rn(I-T)xn)=Srn(Θ,φ)((1-rn)xn+rnTxn).
So, from Theorem 21, we obtain the desired result.Corollary 27.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R be a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let A,Ak:H→H and Bi:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone and ηi-inverse strongly monotone, respectively, where k∈{1,2} and i∈{1,2,…,N}. Let V be a γ¯-strongly positive bounded linear operator and f:H→H be an l-Lipschitzian mapping with γl<(1+μ)γ¯. Assume that Ω:=GMEP⁡(Θ,φ,A)∩SGEP(G)∩∩i=1NI(Bi,Ri)≠∅ where G is defined as in Proposition CY. Let {αn},{βn} and {δn} be three sequences in [0,1]. Assume that:
(i)K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;
(ii) for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(166)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;
(iii)lim⁡n→∞αn=0,∑n=1∞‍αn=∞, 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞δn≤limsup⁡n→∞δn<1;
(iv)νk∈(0,2ζk),  k∈{1,2},  μi∈(0,2ηi),  i∈{1,2,…,N}, and {rn}⊂[0,2ζ] satisfies
(167)0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ;
(v)lim⁡n→∞(|δn+1-δn|+|rn+1-rn|)=0.
Givenx1∈H arbitrarily, then the sequence {xn} generated iteratively by
(168)un=Srn(Θ,φ)(I-rnA)xn,zn=JRN,μN(I-μNBN)JRN-1,μN-1×(I-μN-1BN-1)⋯JR1,μ1(I-μ1B1)un,yn=δnGzn+(1-δn)zn,xn+1=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μV))yn,∀n≥1,
converges strongly to x*∈Ω which solves the following optimization problem provided Sr(Θ,φ) is firmly nonexpansive:
(OP8)min⁡x∈Ωμ2〈Ax,x〉+12∥x-u∥2-h(x),
where h is the potential function of γf.Proof.
PutTnx=x for all integers n≥1 and all x∈H. Then, the desired result follows from Theorem 21.Corollary 28.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ1 and Θ2 be two bifunctions from C×C to R satisfying (H1)–(H4). Let Ri:C→2H be a maximal monotone mapping and let Ak:H→H and Bi:C→H be ζk-inverse strongly monotone and ηi-inverse strongly monotone, respectively, where k∈{1,2} and i∈{1,2,…,N}. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and {λn} be a sequence in (0,b] for some b∈(0,1). Let V be a γ¯-strongly positive bounded linear operator and f:H→H be an l-Lipschitzian mapping with γl<(1+μ)γ¯. Let Wn be the W-mapping defined by (9). Assume that Ω:=∩n=1∞Fix⁡(Tn)∩SGEP(G)∩∩i=1NI(Bi,Ri)≠∅ where G is defined as in Proposition CY. Let {αn},{βn} and {δn} be three sequences in [0,1]. Assume that:
(i)lim⁡n→∞αn=0,∑n=1∞‍αn=∞, 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞δn≤limsup⁡n→∞δn<1;
(ii)νk∈(0,2ζk),k∈{1,2},  μi∈(0,2ηi),  i∈{1,2,…,N}, and {rn}⊂[0,2ζ] satisfies
(169)0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ;
(iii)lim⁡n→∞(|δn+1-δn|+|rn+1-rn|)=0.
Givenx1∈C arbitrarily, then the sequence {xn} generated iteratively by
(170)zn=JRN,μN(I-μNBN)JRN-1,μN-1×(I-μN-1BN-1)⋯JR1,μ1(I-μ1B1)xn,yn=δnGzn+(1-δn)Wnzn,xn+1=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μV))Wnyn,∀n≥1,
converges strongly to x*∈Ω which solves the following optimization problem provided Sr(Θ,φ) is firmly nonexpansive:
(OP9)min⁡x∈Ωμ2〈Ax,x〉+12∥x-u∥2-h(x),
where h is the potential function of γf.Proof.
PutΘ(x,y)=0,  φ(x)=0 for all x,y∈C, Ax=0 for all x∈H and rn=1. Take K(x)=(1/2)∥x∥2 for all x∈H. Then we get un=xn in Theorem 21 and the conclusion follows.



---

*Source: 540381-2014-03-12.xml*