# A System of Generalized Mixed Equilibrium Problems, Maximal Monotone  Operators,  and Fixed Point Problems with Application to Optimization Problems

Authors: Pongsakorn Sunthrayuth; Poom Kumam
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce a new iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Furthermore, we prove under some mild conditions that the proposed iterative algorithm converges strongly to a common element of the above four sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results presented in the paper improve and extend the recent ones announced by many others.


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## Body

## 1. Introduction

Throughout this paper, we denoted byℕ and ℝ+ the set of all positive integers and all positive real numbers, respectively. We always assume that H is a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively, and C is a nonempty closed convex subset of H. Let φ:C→ℝ be a real-valued function, Θ:C×C→ℝ be an equilibrium bifunction, and Ψ,Φ:C→H be two nonlinear mappings. The generalized mixed equilibrium problem with perturbed mapping is to find x*∈C such that(1.1)Θ(x*,y)+φ(y)-φ(x*)+〈(Ψ+Φ)x*,y-x*〉≥0,∀y∈C.
The set of solutions of the problem (1.1) is denoted by GMEP(Θ,φ,Ψ+Φ). As special cases of the problem (1.1), we have the following.  (1)
IfΦ=0, then the problem (1.1) reduces to the generalized mixed equilibrium problem of  finding x*∈C such that(1.2)Θ(x*,y)+φ(y)-φ(x*)+〈Ψx*,y-x*〉≥0,∀y∈C,which was introduced and studied by Peng and Yao [1]. The set of solutions of the problem (1.2) is denoted by GMEP(Θ,φ,Ψ).(2)
IfΨ=Φ=0, then the problem (1.1) reduces to the mixed equilibrium problem of finding x*∈C such that(1.3)Θ(x*,y)+φ(y)-φ(x*)≥0,∀y∈C,which was consider by Ceng and Yao [2]. The set of solutions of the problem (1.3) is denoted by MEP(Θ).(3)
IfΦ=φ=0, then the problem (1.1) reduces to the generalized equilibrium problem of finding x*∈C such that(1.4)Θ(x*,y)+〈Ψx*,y-x*〉≥0,∀y∈C,which was consider by S. Takahashi and W. Takahashi [3]. The set of solutions of the problem (1.4) is denoted by GEP(Θ,Ψ).(4)
IfΨ=Φ=φ=0, then the problem (1.1) reduces to the equilibrium problem of finding x*∈C such that(1.5)Θ(x*,y)≥0,∀y∈C.The set of solutions of the problem (1.5) is denoted by EP(Θ).(5)
IfΘ=Φ=φ=0, then the problem (1.1) reduces to the classical variational inequality problem of  finding x*∈C such that(1.6)〈Ψx*,y-x*〉≥0,∀y∈C.The set of solutions of the problem (1.6) is denoted by VI(C,Ψ). It is known that x*∈C is a solution of the problem (1.6) if and only if x* is a fixed point of the mapping PC(I-λΨ), where λ>0 is a constant and I is the identity mapping.The generalized mixed equilibrium problems with perturbation is very general in the sense that it includes fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases (see, e.g., [4, 5]). Numerous problems in physics, optimization, and economics reduce to find a solution of problem (1.2). Several methods have been proposed to solve the fixed point problems, variational inequality problems, and equilibrium problems in the literature (see, e.g., [6–34]).LetA be a strongly positive bounded linear operator on H; that is, there exists a constant γ¯>0 such that(1.7)〈Ax,x〉≥γ¯‖x‖2,∀x∈H.Recall that a mappingf:C→C is said to be contractive if there exists a constant α∈(0,1) such that (1.8)‖f(x)-f(y)‖≤α‖x-y‖,∀x,y∈C.A mappingT:C→C is said to be  (1)
nonexpansive if(1.9)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C,(2)
firmly nonexpansive if(1.10)‖Tx-Ty‖2≤〈Tx-Ty,x-y〉,∀x,y∈C,(3)
k-strictly pseudocontractive if there exists a constant k∈[0,1) such that(1.11)‖Tx-Ty‖2≤‖x-y‖2+k‖(I-T)x-(I-T)y‖2,∀x,y∈C.
We denote by F(T) the set of fixed points of T, that is, F(T)={x∈C:x=Tx}.Recall the following definitions of a nonlinear mappingB:C→H; the following is mentioned.Definition 1.1.
The nonlinear mappingB:C→H is said to be  (i)
monotone if(1.12)〈Bx-By,x-y〉≥0,∀x,y∈C,(ii)
β-strongly monotone if there exists a constant β>0 such that(1.13)〈Bx-By,x-y〉≥β‖x-y‖2,∀x,y∈C,(iii)
ν-inverse-strongly monotone if there exists a constant ν>0 such that(1.14)〈Bx-By,x-y〉≥ν‖Bx-By‖2,∀x,y∈C.LetW:H→2H be a set-valued mapping. The set D(W) defined by D(W)={x∈H:Wx≠∅} is said to be the domain of W. The set R(W) defined by R(W)=⋃x∈HWx is said to be the range of W. The set G(W) defined by G(W)={(x,y)∈H×H:x∈D(W),y∈R(W)} is said to be the graph of W.Recall thatW is said to be monotone if (1.15)〈x-y,f-g〉>0,∀(x,f),(y,g)∈G(W).W is said to be maximal monotone if it is not properly contained in any other monotone operator. Equivalently, W is maximal monotone if R(I+rW)=H for all r>0. For a maximal monotone operator M on H and r>0, we may define the single-valued resolvent Jr=(I+rW)-1:H→D(W). It is known that Jr is firmly nonexpansive W-1(0)=F(Jr), where F(Jr) denotes the fixed point set of Jr.We discuss the following algorithms for solving the solutions of variational inequality problems and fixed point problems for a nonexpansive mapping (see, e.g.,  [29, 35–43]).In 2010, Chantarangsi et al. [44] introduced a new viscosity hybrid steepest descent method for solving the generalized mixed equilibrium problems (1.2), variational inequality problems, and fixed point problems of nonexpansive mappings in a real Hilbert space. More precisely, they proved the following theorem.Theorem CCK [see [44]]
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let Θ1, Θ2 be two bifunctions satisfying condition (H1)–(H5), let  Ψ1, Ψ2 be ξ-inverse-strongly monotone mapping and  β-inverse-strongly monotone mapping, respectively, and let  T:C→C be a nonexpansive mapping. Let B be an ω-Lipschitz continuous and relaxed (v,ν) cocoercive mapping, f:C→C a contraction mapping with coefficient α∈(0,1), and A a strongly positive linear bounded self-adjoint operator with coefficient γ¯>0 and 0<γ<γ¯/α. Suppose that Ω∶=F(T)∩GMEP(Θ1,φ,Ψ1)∩GMEP(Θ2,φ,Ψ2)∩VI(C,B). Let {zn}, {un}, {yn}, and {xn} be generated by
(1.16)un=Vrn(Θ2,φ2)(xn-rnΨ2xn),vn=Vμn(Θ1,φ1)(un-μnΨ1un),zn=PC(vn-αnBTvn),yn=ϵnγf(xn)+βnxn+((1-βn)I-ϵnA)zn,xn+1=γnxn+(1-γn)yn,∀n≥1,
where {γn}⊂[a,b]⊂[0,2ξ], {sn}⊂[c,d]⊂[0,2β], {γn}⊂[h,j]⊂(0,1), {γn}, {ϵn}, and {βn} are three sequences in (0,1) satisfying the following conditions:  (C1)
limn→∞ϵn=0 and ∑n=1∞ϵn=∞,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1,(C3)
0<liminfn→∞μn≤limsupn→∞μn<2β and limn→∞|μn+1-μn|=0,(C4)
0<liminfn→∞rn≤limsupn→∞rn<2ξ and limn→∞|rn+1-rn|=0,(C5)
{αn}⊂[e,g]⊂(0,2(ν-vω2)/ω2), ν>vω2 and limn→∞|αn+1-αn|=0.Then, {xn} converges strongly to x*=PΩ(γf+(I-A))(x*).Very recently, Yu and Liang [45] proved the following convergence theorem of finding a common element in the fixed point set of a strict pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and inverse strongly monotone mapping in a real Hilbert space.Theorem YL [see [45]]
LetH be a real Hilbert space and C a nonempty close and convex subset of H. Let W1:H→2H and W2:H→2H be two maximal monotone operators such that D(W1)⊂C and D(W2)⊂C, respectively. Let S:C→C be a k-strict pseudocontraction mapping, A:C→H an α-inverse-strongly monotone mapping, and B:C→H an β-inverse-strongly monotone mapping. Assume that Ω:=F(S)∩(A+W1)-1(0)∩(B+W2)-1(0)≠∅. Let {xn} be a sequence generated by
(1.17)x1∈C,yn=Jtn(xn-tnBxn),xn+1=αnu+βnxn+γn[δnJsn(yn-snAyn)+(1-δn)Jsn(yn-snAyn)],∀n≥1,
where u∈C is a fixed element, Jsn=(I+snW1)-1, Jtn=(I+tnW2)-1,{sn} is a sequence in (0,2α),{tn} is a sequence in (0,2β), and {αn}, {βn}, {γn}, and {δn} are sequences in [0,1] satisfying the following conditions:  (C1)
limn→∞αn=0 and ∑n=1∞αn=∞,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1,(C3)
0<a≤sn≤b<2α and limn→∞(sn+1-sn)=0,(C4)
0<c≤tn≤d<2β and limn→∞(tn+1-tn)=0,(C5)
0<c≤k≤δn<e<1 and limn→∞(δn+1-δn)=0.Then, the sequence {xn} converges strongly to x*=PΩx*.On the other hand, the following optimization problem has been studied extensively by many authors:(1.18)min⁡x∈Ω¯μ2〈Ax,x〉+12‖x-u‖2-h(x),
where Ω¯=⋂n=1∞Cn,C1,C2,…  are infinitely many closed convex subsets of H such that ⋂n=1∞Cn≠∅, u∈H, μ≥0 is a real number, A is a strongly positive linear bounded operator on H, and h is a potential function for γf (i.e., h′(x)=γf(x) for all x∈H). This kind of optimization problem has been studied extensively by many authors (see, e.g., [5, 46–52]) for when Ω¯=⋂n=1∞Cn and h(x)=〈x,b〉, where b is a given point in H.The following questions naturally arise in connection with above the results.Question 1.
Could we weaken the control conditions of TheoremsCCK and YL in (C3) and (C4)?Question 2.
Can TheoremYL be extended to finding a common element of the set of solutions of a system generalized mixed equilibrium problems and the set of common fixed points of infinite family of nonexpansive mappings?The purpose of this paper is to give the affirmative answers to these questions mentioned above. Motivated by the iterative process (1.16) and (1.17), we introduce a new iterative algorithm (3.2) below, for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Then, we prove the strong convergence theorem of these iterative process in a real Hilbert space. The results presented in the paper improve and extend the recent ones announced by many others.


## 2. Preliminaries

Definition 2.1 (see [53]).
LetC be a nonempty convex subset of a real Hilbert space H. Let Ti, i=1,2,…, be mappings of C into itself. For each  j=1,2,…,  let αj=(α1j,α2j,α3j)∈I×I×I, where I=[0,1] and α1j+α2j+α3j=1. For every n∈ℕ, we define the mapping Sn:C→C as follows:
(2.1)Un,n+1=I,Un,n=α1nTnUn,n+1+α2nUn,n+1+α3nI,Un,n-1=α1n-1Tn-1Un,n+α2n-1Un,n+α3n-1I,⋮Un,k+1=α1k+1Tk+1Un,k+2+α2k+1Un,k+2+α3k+1I,Un,k=α1kTkUn,k+1+α2kUn,k+1+α3kI,⋮Un,2=α12T2Un,3+α22Un,3+α32I,Sn=Un,1=α11T1Un,2+α21Un,2+α31I.
Such a mapping Sn is nonexpansive from C into itself, and it is called S-mapping generated by Tn,Tn-1,…,T1 and αn,αn-1,…,α1.Lemma 2.2 (see [53]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Ti}i=1∞ be nonexpansive mappings of C into itself with F(Ti)≠∅ and let αj=(α1j,α2j,α3j)∈I×I×I, where I=[0,1], α1j+α2j+α3j=1, α1j+α2j≤b<1, and α1j,α2j,α3j∈(0,1) for all j=1,2,…. For all n∈ℕ, let Sn and S be S-mappings generated by Tn,Tn-1,…,T1 and αn,αn-1,…,α1 and Tn,Tn-1,… and αn,αn-1,…, respectively. Then,  (i)
Sn is nonexpansive and F(Sn)=⋂i=1nF(Ti),  for all n≥1,(ii)
for allx∈C and for all positive integer k, the limn→∞Un,k exists,(iii)
the mappingS:C→C defined by(2.2)Sx:=lim⁡n→∞Snx=lim⁡n→∞Un,1x,∀x∈Cis a nonexpansive mapping such thatF(S)=⋂i=1∞F(Ti), and it is called the S-mapping generated by Tn,Tn-1,… and αn,αn-1,…,(iv)
ifK is any bounded subset of C, then(2.3)lim⁡n→∞sup⁡x∈K‖Snx-Sx‖=0.Lemma 2.3 (see [54]).
Let{xn} and {ln} be bounded sequences in a Banach space X and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose that xn+1=(1-βn)ln+βnxn for all integers n≥0 and limsupn→∞(∥ln+1-ln∥-∥xn+1-xn∥)≤0. Then, limn→∞∥ln-xn∥=0.Lemma 2.4 (see [55]).
LetH be a real Hilbert space. Then, the following inequalities hold:  (i)
∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)∥x-y∥2,∀x,y∈Handλ∈[0,1],(ii)
∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.Lemma 2.5 (see [56]).
LetC be a nonempty closed convex subset of a real Hilbert space H, A:C→H a mapping, and W:H→2H a maximal monotone mapping. Then,
(2.4)F(Jr(I-rA))=(A+W)-1(0),∀r>0.Lemma 2.6 (see [57]).
LetH be a real Hilbert space and let M be a maximal monotone operator on H. For r>0 and x∈H, define the resolvent Jrx. Then, the following holds:
(2.5)s-ts〈Jsx-Jtx,Jsx-x〉≥‖Jsx-Jtx‖2
for all s,t>0 and x∈H.For solving the equilibrium problem for bifunctionΘ:C×C→ℝ, let us assume that Θ satisfies the following conditions:  (H1)
Θ(x,x)=0 for all x∈C,(H2)
Θ is monotone; that is, Θ(x,y)+Θ(y,x)≤0 for all x,y∈C,(H3)
for eachy∈C, x↦Θ(x,y) is concave and upper semicontinuous,(H4)
for eachy∈C, x↦Θ(x,y) is convex,(H5)
for eachy∈C, x↦Θ(x,y) is lower semicontinuous.Definition 2.7.
A differentiable functionK:C→ℝ on a convex set C is called (i)
convex[2] if(2.6)K(y)-K(x)≥〈K′(x),y-x〉,∀x,y∈C,whereK′(x) is the Fréchet differentiable of K at x;(ii)
strongly convex [2] if there exists a constant σ>0 such that(2.7)K(y)-K(x)-〈K′(x),x-y〉≥(σ2)‖x-y‖2,∀x,y∈C.It is easy to see that ifK:C→ℝ is a differentiable strongly convex function with constant σ>0, then K′:C→H is strongly monotone with constant σ>0.LetΘ:C×C→ℝ be an equilibrium bifunction satisfying the conditions (H1)–(H5). Let r be any given positive number. For a given point x∈C, consider the auxiliary mixed equilibrium problem to finding y∈C such that(2.8)Θ(y,z)+φ(z)-φ(y)+1r〈K′(y)-K′(x),z-y〉≥0,∀z∈C,
where K′(x) is the Fréchet differentiable of K at x. Let Vr(Θ,φ):C→C be the mapping such that for each x∈C, Vr(Θ,φ)(x) is the set of solutions of MEP(x,y), that is,(2.9)Vr(Θ,φ)(x)={y∈C:Θ(y,z)+φ(z)-φ(y)+1r〈K′(y)-K′(x),z-y〉≥0,∀z∈C}.
Then, the following conclusion holds.Lemma 2.8 (see [58]).
LetC be a nonempty closed convex subset of a real Hilbert space H, and let φ:C→ℝ be a lower semicontinuous and convex functional. Let Θ:C×C→ℝ be a bifunction satisfying the conditions (H1)–(H5). Assume that  (i)
K:C→ℝ is strongly convex with constant σ>0 and the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈C;(ii)
for eachx∈C, there exist a bounded subset Dx⊂C and zx such that for all y∉Dx,(2.10)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0.Then, the following holds:  (a)
Vr(Θ,φ) is single-valued mapping;(b)
Vr(Θ,φ) is nonexpansive if K′ is Lipschitz continuous with constant ν>0 and(2.11)〈K′(x1)-K′(x2),u1-u2〉≥〈K′(u1)-K′(u2),u1-u2〉,∀x1,x2∈C,whereui=Vr(Θ,φ)(xi) for i=1,2;(c)
F(Vr(Θ,φ))=MEP(Θ,φ);(d)
MEP(Θ,φ) is closed and convex.In particular, wheneverΘ:C×C→ℝ is a bifunction satisfying the conditions (H1)–(H5) and K(x)=∥x∥2/2, for all x∈C, then Vr(Θ,φ) is firmly nonexpansive; that is, for any x,y∈C, (2.12)‖Vr(Θ,φ)x-Vr(Θ,φ)y‖2≤〈Vr(Θ,φ)x-Vr(Θ,φ)y,x-y〉.
In this case, Vr(Θ,φ) is rewritten as Tr(Θ,φ). If, in addition, φ≡0, then Tr(Θ,φ) is rewritten as TrΘ (see [59, Lemma 2.1] for more details).Remark 2.9.
We remark that Lemma2.8 is not a consequence of [2, Lemma 3.1] because the condition of the sequential continuity from the weak topology to the strong topology for the derivative K′ of the function K:C→ℝ does not cover the case K(x)=∥x∥2/2.Lemma 2.10.
LetC,H,Θ,φ, and Vr(Θ,φ) be as in Lemma 2.8. Then, the following holds:
(2.13)〈K′(Vs(Θ,φ)x)-K′(Vt(Θ,φ)x),Vs(Θ,φ)x-Vt(Θ,φ)x〉≤s-ts〈K′(Vs(Θ,φ)x)-K′(x),Vs(Θ,φ)x-Vt(Θ,φ)x〉,
for all s,t>0 and x∈C.Proof.
By similar argument as in the proof of Proposition 1 in [58], for all s,t>0 and x∈C, let u=Vs(Θ,φ)x and v=Vt(Θ,φ)x; we have
(2.14)Θ(u,y)+φ(y)-φ(u)+1s〈K′(u)-K′(x),y-u〉≥0,∀x∈C,(2.15)Θ(v,y)+φ(y)-φ(v)+1t〈K′(v)-K′(x),y-v〉≥0,∀x∈C.
Let y=v in (2.14) and y=u in (2.15); we have
(2.16)Θ(u,v)+φ(v)-φ(u)+1s〈K′(u)-K′(x),v-u〉≥0,Θ(v,u)+φ(u)-φ(v)+1t〈K′(v)-K′(x),u-v〉≥0.
Adding up the last two inequalities and from the monotonicity of Θ, we obtain that
(2.17)1s〈K′(u)-K′(x),v-u〉+1t〈K′(v)-K′(x),u-v〉≥0.
It follows that
(2.18)〈K′(v)-K′(x)t-K′(u)-K′(x)s,u-v〉≥0.
We derive from (2.18) that
(2.19)0≤〈K′(v)-K′(x)-ts(K′(u)-K′(x)),u-v〉=〈K′(v)-K′(u)+K′(u)-K′(x)-ts(K′(u)-K′(x)),u-v〉=〈K′(v)-K′(u)+(1-ts)(K′(u)-K′(x)),u-v〉.
Hence, we obtain that
(2.20)〈K′(u)-K′(v),u-v〉≤s-ts〈K′(u)-K′(x),u-v〉.The following lemma can be found in [60, 61] (see also [62, Lemma 2.2]).Lemma 2.11.
LetC be a nonempty closed convex subset of a real Hilbert space H and g:C→ℝ∪{+∞} a proper lower semicontinuous differentiable convex function differentiable convex function. If x* is a solution to the minimization problem
(2.21)g(x*)=inf⁡x∈Cg(x),
then
(2.22)〈g′(x),x-x*〉≥0,x∈C.
In particular, if x* solves the optimization problem
(2.23)min⁡x∈Ωμ2〈Ax,x〉+12‖x-u‖2-h(x),
then
(2.24)〈u+(γf-(I+μA))x*,x-x*〉≤0,x∈C,
where h is a potential function for γf.Lemma 2.12 (see [49]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.25)an+1≤(1-σn)an+δn,
where {σn} is a sequence in (0,1) and {δn} is a sequence in ℝ such that (i)
∑n=0∞σn=∞,(ii)
limsupn→∞δn/σn≤0 or ∑n=0∞|δn|<∞. Then, limn→∞an=0.


## 3.  Main Results

Now, we give our main results in this paper.Theorem 3.1.
LetC be a nonempty closed convex subset of a Hilbert space H. Let φ1,φ2:C→ℝ be two lower semicontinuous and convex functionals, and let Θ1,Θ2:C×C→ℝ be two bifunctions satisfying conditions (H1)–(H5). Let Ψ1,Ψ2,Φ1,Φ2:C→H be β̃1-inverse-strongly monotone mapping, β̃2-inverse-strongly monotone mapping, γ̃1-inverse-strongly monotone mapping, and γ̃2-inverse-strongly monotone mapping, respectively, and let B1,B2:C→H be ξ̃1-inverse-strongly monotone mapping and ξ̃2-inverse-strongly monotone mapping, respectively. Let {Ti}i=1∞:C→C be an infinite family of nonexpansive mappings, and let αi=(α1j,α2j,α3j)∈I×I×I, where I=[0,1], α1j+α2j+α3j=1, α1j+α2j≤b<1, and α1j,α2j,α3j∈(0,1) for all j=1,2,…. For all n∈ℕ, let Sn and S be S-mappings generated by Tn,Tn-1,…,T1 and αn,αn-1,…,α1 and Tn,Tn-1,… and αn,αn-1,…, respectively. Let W1,W2:H→2H be two maximal monotone operators such that D(W1)⊂C and D(W2)⊂C, respectively. Assume that
(3.1)Ω∶=⋂i=1∞F(Ti)∩GMEP⁡(Θ1,φ1,Ψ1+Φ1)∩GMEP⁡(Θ2,φ2,Ψ2+Φ2)∩(B1+W1)-1(0)∩(B2+W2)-1(0)≠∅.
Let f:C→H be a contraction mapping with a coefficient α∈(0,1), and let A:C→H be a strongly positive linear bounded operator with a coefficient γ¯∈(0,1). Let μ>0 and γ>0 be two constants such that 0<γ<(1+μ)γ¯/α. Let {xn} be a sequence defined by u,x1∈H and
(3.2)un=Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn),vn=Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un),yn=Jtn(vn-tnB2vn),xn+1=ϵn(u+γf(Snvn))+βnxn+((1-βn)I-ϵn(I+μA))SnJsn(yn-snB1yn),∀n≥1,
where Jsn=(I+snW1)-1, Jtn=(I+tnW2)-1, {sn}⊂(0,2ξ̃1), {tn}⊂(0,2ξ̃2), {μn}⊂(0,min{β̃1,γ̃1}), {rn}⊂(0,min{β̃2,γ̃2}), {ϵn} and {βn}⊂(0,1). Assume that the following conditions are satisfied:  (C1)
for alli=1,2,   Ki:C→ℝ is strongly convex with constant σi>0 and its derivative Ki′ is Lipschitz continuous with constant νi>0 such that the function x↦〈y-x,Ki′(x)〉 is weakly upper semicontinuous for each y∈C,(C2)
for alli=1,2 and for each x∈C, there exist a bounded subset Dx⊂C and zx∈C such that, for all y∉Dx,(3.3)Θi(y,zx)+φi(zx)-φi(y)+1ri〈Ki′(y)-Ki′(x),zx-y〉<0,(C3)
limn→∞αn1=0,(C4)
limn→∞ϵn=0 and ∑n=1∞ϵn=∞,(C5)
0<liminfn→∞βn≤limsupn→∞βn<1,(C6)
0<liminfn→∞μn≤limsupn→∞μn<min{β̃1,γ̃1} and limn→∞μn/μn+1=1,(C7)
0<liminfn→∞rn≤limsupn→∞rn<min{β̃2,γ̃2} and limn→∞rn/rn+1=1,(C8)
0<liminfn→∞sn≤limsupn→∞sn<2ξ̃1 and limn→∞sn/sn+1=1,(C9)
0<liminfn→∞tn≤limsupn→∞tn<2ξ̃2 and limn→∞tn/tn+1=1. Then, the sequence {xn} defined by (3.2) converges strongly to x*∈Ω, provided Vr,n(Θ,φ) is firmly nonexpansive, where x* solves the following optimization problem:
(3.4)min⁡x∈Ωμ2〈Ax,x〉+12‖x-u‖2-h(x).Proof.
By  the conditions (C4) and (C5), we may assume, without loss of generality, thatϵn≤(1-βn)(1+μ∥A∥)-1 for all n∈ℕ. Since A is a linear bounded self-adjoint operator on C, we have
(3.5)‖A‖=sup⁡{|〈Aũ,ũ〉|:ũ∈C,‖ũ‖=1}.
Observe that
(3.6)〈((1-βn)I-ϵn(I+μA))ũ,ũ〉=1-βn-ϵn-ϵnμ〈Aũ,ũ〉≥1-βn-ϵn-ϵnμ‖A‖≥0.
This shows that (1-βn)I-ϵn(I+μA) is positive. It follows that
(3.7)‖(1-βn)I-ϵn(I+μA)‖=sup⁡{|〈((1-βn)I-ϵn(I+μA))ũ,ũ〉|:ũ∈C,‖ũ‖=1}=sup⁡{1-βn-ϵn-ϵnμ〈Aũ,ũ〉:ũ∈C,‖ũ‖=1}≤1-βn-ϵn(1+μγ¯)<1-βn-ϵn(1+μ)γ¯.
First, we show that {xn} is bounded. Take x¯∈Ω. Since 0<μn<min{β̃1,γ̃1} and by Lemma 2.4  (i), we have
(3.8)‖vn-x¯‖2=‖Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un)-Vμn(Θ1,φ1)(x¯-μn(Ψ1+Φ1)x¯)‖2≤‖(un-μn(Ψ1+Φ1)un)-(x¯-μn(Ψ1+Φ1)x¯)‖2=‖(un-x¯)-μn[(Ψ1+Φ1)un-(Ψ1+Φ1)x¯]‖2=‖12((un-x¯)-2μn(Ψ1un-Ψ1x¯))+12((un-x¯)-2μn(Φ1un-Φ1x¯))‖2≤12‖(un-x¯)-2μn(Ψ1un-Ψ1x¯)‖2+12‖(un-x¯)-2μn(Φ1un-Φ1x¯)‖2=12(‖un-x¯‖2-4μn〈un-x¯,Ψ1un-Ψ1x¯〉+4μn2‖Ψ1un-Ψ1x¯‖2)+12(‖un-x¯‖2-4μn〈un-x¯,Φ1un-Φ1x¯〉+4μn2‖Φ1un-Φ1x¯‖2)≤12(‖un-x¯‖2+4μn(μn-β̃1)‖Ψ1un-Ψ1x¯‖2)+12(‖un-x¯‖2+4μn(μn-γ̃1)‖Φ1un-Φ1x¯‖2)=‖un-x¯‖2+2μn(μn-β̃1)‖Ψ1un-Ψ1x¯‖2+2μn(μn-γ̃1)‖Φ1un-Φ1x¯‖2≤‖un-x¯‖2.
In a similar way, we can get
(3.9)‖un-x¯‖2=‖Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)-Vrn(Θ2,φ2)(x¯-rn(Ψ2+Φ1)x¯)‖2≤‖xn-x¯‖2+2rn(rn-β̃2)‖Ψ2xn-Ψ2x¯‖2+2rn(rn-γ̃2)‖Φ2xn-Φ2x¯‖2≤‖xn-x¯‖2.
It follows from (3.8), and (3.9) that
(3.10)‖vn-x¯‖≤‖un-x¯‖≤‖xn-x¯‖.
Setting zn:=Jsn(yn-snB1yn). Since 0<sn<2ξ̃1, we have
(3.11)‖zn-x¯‖2=‖Jsn(yn-snB1yn)-Jsn(x¯-snB1x¯)‖2≤‖(yn-snB1yn)-(x¯-snB1x¯)‖2=‖(yn-x¯)-sn(B1yn-B1x¯)‖2=‖yn-x¯‖2-2sn〈yn-x¯,B1yn-B1x¯〉+sn2‖B1yn-B1x¯‖2≤‖yn-x¯‖2+sn(sn-2ξ1)‖B1yn-B1x¯‖2≤‖yn-x¯‖2.
In a similar way, we can get
(3.12)‖yn-x¯‖2=‖Jtn(vn-tnB2vn)-Jtn(x¯-tnB2x¯)‖2≤‖vn-x¯‖2+tn(tn-2ξ2)‖B2vn-B2x¯‖2≤‖vn-x¯‖2.
It follows from (3.10), (3.11) and (3.12) that
(3.13)‖zn-x¯‖≤‖yn-x¯‖≤‖vn-x¯‖≤‖un-x¯‖≤‖xn-x¯‖.
Since xn+1=ϵn(u+γf(Snvn))+βnxn+((1-βn)I-ϵn(I+μA))Snzn, it follows from (3.13) that
(3.14)‖xn+1-x¯‖=‖ϵnu+ϵn(γf(Snvn)-(I+μA)x¯)+βn(xn-x¯)+((1-βn)I-ϵn(I+μA))(Snzn-x¯)‖≤ϵn‖u‖+ϵn‖γf(Snvn)-(I+μA)x¯‖+βn‖xn-x¯‖+(1-βn-ϵn(1+μγ¯))‖Snzn-x¯‖≤ϵn‖u‖+ϵnγ‖f(Snvn)-f(x¯)‖+ϵn‖γf(x¯)-(I+μA)x¯‖+βn‖xn-x¯‖+(1-βn-ϵn(1+μ)γ¯)‖zn-x¯‖≤ϵn‖u‖+ϵnγα‖Snvn-x¯‖+ϵn‖γf(x¯)-(I+μA)x¯‖+βn‖xn-x¯‖+(1-βn-ϵn(1+μ)γ¯)‖xn-x¯‖≤(1-((1+μ)γ¯-γα)ϵn)‖xn-x¯‖+ϵn‖γf(x¯)-(I+μA)x¯‖+ϵn‖u‖=(1-((1+μ)γ¯-γα)ϵn)‖xn-x¯‖+((1+μ)γ¯-γα)ϵn‖γf(x¯)-(I+μA)x¯‖+‖u‖(1+μ)γ¯-γα.
By induction, we have
(3.15)‖xn-x¯‖≤max⁡{‖x1-x¯‖,‖γf(x¯)-(I+μA)x¯‖+‖u‖(1+μ)γ¯-γα},∀n≥1.
Hence, {xn} is bounded, so are {vn}, {yn}, and {zn}.
Next, we show thatlimn→∞∥xn+1-xn∥. Since un=Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn) and un+1=Vrn+1(Θ2,φ2)(xn+1-rn+1(Ψ2+Φ2)xn+1), we have
(3.16)‖un+1-un‖=‖Vrn+1(Θ2,φ2)(xn+1-rn+1(Ψ2+Φ2)xn+1)-Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)‖≤‖Vrn+1(Θ2,φ2)(xn+1-rn+1(Ψ2+Φ2)xn+1)-Vrn+1(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)‖+‖Vrn+1(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)-Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)‖≤‖(xn+1-rn+1(Ψ2+Φ2)xn+1)-(xn-rn(Ψ2+Φ2)xn)‖+‖Vrn+1(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)-Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)‖=‖xn+1-xn-rn+1[(Ψ2+Φ2)xn+1-(Ψ2+Φ2)xn]+(rn-rn+1)(Ψ2+Φ2)xn‖+‖Vrn+1(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)-Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)‖=‖xn+1-rn+1(Ψ2+Φ2)xn+1-(xn-rn+1(Ψ2+Φ2)xn)+(rn-rn+1)(Ψ2+Φ2)xn‖+‖Vrn+1(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)-Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)‖≤‖xn+1-xn‖+rn+1|1-rnrn+1|‖(Ψ2+Φ2)xn‖+‖Vrn+1(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)-Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)‖.
In a similar way, we can get
(3.17)‖vn+1-vn‖=‖Vμn+1(Θ1,φ1)(un+1-μn+1(Ψ1+Φ1)un+1)-Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un)‖≤‖un+1-un‖+μn+1|1-μnμn+1|‖(Ψ1+Φ1)un‖+‖Vμn+1(Θ1,φ1)(un-μn(Ψ1+Φ1)un)-Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un)‖.
Substitution (3.16) into (3.17), we obtain
(3.18)‖vn+1-vn‖≤‖xn+1-xn‖+rn+1|1-rnrn+1|‖(Ψ2+Φ2)xn‖+μn+1|1-μnμn+1|‖(Ψ1+Φ1)un‖+‖Vrn+1(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)-Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)‖+‖Vμn+1(Θ1,φ1)(un-μn(Ψ1+Φ1)un)-Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un)‖≤‖xn+1-xn‖+(|1-rnrn+1|+|1-μnμn+1|)M1+‖Vrn+1(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)-Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)‖+‖Vμn+1(Θ1,φ1)(un-μn(Ψ1+Φ1)un)-Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un)‖,
where M1=supn≥1{μn+1∥(Ψ1+Φ1)un∥,rn+1∥(Ψ2+Φ2)xn∥}.
On the other hand, notice from Lemma2.6 that
(3.19)‖yn+1-yn‖=‖Jtn+1(vn+1-tn+1B2vn+1)-Jtn(vn-tnB2vn)‖≤‖Jtn+1(vn+1-tn+1B2vn+1)-Jtn+1(vn-tnB2vn)‖+‖Jtn+1(vn-tnB2vn)-Jsn(vn-tnB2vn)‖≤‖(vn+1-tn+1B2vn+1)-(vn-tnB2vn)‖+‖Jtn+1(vn-tnB2vn)-Jsn(vn-tnB2vn)‖=‖(vn+1-tn+1B2vn+1)-(vn-tn+1B2vn)+(tn-tn+1)B2vn‖+‖Jtn+1(vn-tnB2vn)-Jtn(vn-tnB2vn)‖≤‖vn+1-vn‖+tn+1|1-tntn+1|‖B2vn‖+|1-tntn+1|‖Jtn+1(vn-tnB2vn)-(vn-tnB2vn)‖≤‖vn+1-vn‖+2|1-tntn+1|M2,
where M2>0 is an appropriate constant such that M2=supn≥1{tn+1∥B2vn∥,∥Jtn+1(vn-tnB2vn)-(vn-tnB2vn)∥}.
In a similar way, we can get from Lemma2.6 that
(3.20)‖zn+1-zn‖=‖Jsn+1(yn+1-sn+1B1yn+1)-Jsn(yn-snB1yn)‖≤‖yn+1-yn‖+sn+1|1-snsn+1|‖B1yn‖+|1-snsn+1|‖Jsn+1(yn-snB1yn)-(yn-snB1yn)‖≤‖yn+1-yn‖+2|1-snsn+1|M3,
where M3>0 is an appropriate constant such that M3=supn≥1{sn+1∥B1yn∥,∥Jsn+1(yn-snB1yn)-(yn-snB1yn)∥}. It follows from (3.18) and (3.19) that
(3.21)‖zn+1-zn‖≤‖xn+1-xn‖+(|1-rnrn+1|+|1-μnμn+1|+2|1-tntn+1|+2|1-snsn+1|)L+‖Vrn+1(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)-Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)‖+‖Vμn+1(Θ1,φ1)(un-μn(Ψ1+Φ1)un)-Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un)‖,
where L=max{M1,M2,M3}.
Define the sequence{ln} by ln∶=(xn+1-βnxn)/(1-βn),foralln∈ℕ. Then, xn+1=βnxn+(1-βn)ln, foralln∈ℕ, we note that
(3.22)ln+1-ln=xn+2-βn+1xn+11-βn+1-xn+1-βnxn1-βn=ϵn+1(u+γf(Sn+1vn+1))+((1-βn+1)I-ϵn+1(I+μA))Sn+1zn+11-βn+1-ϵn(u+γf(Snvn))+((1-βn)I-ϵn(I+μA))Snzn1-βn=ϵn+11-βn+1(u+γf(Sn+1vn+1))-ϵn1-βn(u+γf(Snvn))+ϵn1-βn(I+μA)Snzn-ϵn+11-βn+1(I+μA)Sn+1zn+1+Sn+1zn+1-Snzn.
It follows that
(3.23)‖ln+1-ln‖≤ϵn+11-βn+1‖u+γf(Sn+1vn+1)-(I-μA)Sn+1zn+1‖+ϵn1-βn‖(I+μA)Snzn-u-γf(Snvn)‖+‖Sn+1zn+1-Snzn‖.
Now,  we estimate ∥Sn+1zn-Snzn∥. By definition of Sn, for all n∈ℕ and for all ω∈Ω, we have
(3.24)‖Sn+1zn-Snzn‖=‖Un+1,1zn-Un,1zn‖=‖α11T1Un+1,2zn+α21Un+1,2zn+α31zn-α11T1Un,2zn-α21Un,2zn-α31zn‖=(α11+α21)‖T1Un+1,2zn-T1Un,2zn‖≤(1-α31)‖Un+1,2zn-Un,2zn‖≤(1-α31)(1-α32)‖Un+1,3zn-Un,3zn‖=∏i=12(1-α3i)‖Un+1,3zn-Un,3zn‖⋮≤∏i=1n(1-α3i)‖Un+1,n+1zn-Un,n+1zn‖≤‖Un+1,n+1zn-zn‖=‖α1n+1Tn+1zn+(1-α1n+1)zn-zn‖=α1n+1‖Tn+1zn-zn‖≤α1n+1(‖Tn+1zn-Tn+1ω‖+‖Tn+1ω-zn‖)≤2α1n+1‖zn-ω‖.
By the condition (C3), we obtain that
(3.25)lim⁡n→∞‖Sn+1zn-Snzn‖=0.
It follows from (3.21) that
(3.26)‖Sn+1zn+1-Snzn‖≤‖Sn+1zn+1-Sn+1zn‖+‖Sn+1zn-Snzn‖≤‖zn+1-zn‖+‖Sn+1zn-Snzn‖≤‖xn+1-xn‖+(|1-rnrn+1|+|1-μnμn+1|+2|1-tntn+1|+2|1-snsn+1|)L+‖Vrn+1(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)-Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)‖+‖Vμn+1(Θ1,φ1)(un-μn(Ψ1+Φ1)un)-Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un)‖.
It follows from (3.23) and (3.26) that
(3.27)‖ln+1-ln‖-‖xn+1-xn‖≤ϵn+11-βn+1‖u+γf(Sn+1vn+1)-(I-μA)Sn+1zn+1‖+ϵn1-βn‖(I+μA)Snzn-u-γf(Snvn)‖+‖Sn+1zn+1-Snzn‖-‖xn+1-xn‖≤ϵn+11-βn+1‖u+γf(Sn+1vn+1)-(I-μA)Sn+1zn+1‖+ϵn1-βn‖(I+μA)Snzn-u-γf(Snvn)‖+(|1-rnrn+1|+|1-μnμn+1|+2|1-tntn+1|+2|1-snsn+1|)L+‖Vrn+1(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)-Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)‖+‖Vμn+1(Θ1,φ1)(un-μn(Ψ1+Φ1)un)-Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un)‖+‖Sn+1zn-Snzn‖.
Note that 0<liminfn→∞rn≤limsupn→∞rn<min{β̃2,γ̃2} and limn→∞rn/rn+1=1. Since K1′ is σ1-strongly monotone and ν1-Lipschitz continuous, then, from Lemma 2.10, we obtain that
(3.28)lim⁡n→∞‖Vrn+1(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)-Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn)‖=0.
In a similar way, we can get
(3.29)lim⁡n→∞‖Vμn+1(Θ1,φ1)(un-μn(Ψ1+Φ1)un)-Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un)‖=0.
Consequently, it follows from (3.25), (3.27), (3.28), (3.29), and the conditions (C4)–(C9) that
(3.30)limsup⁡n→∞(‖ln+1-ln‖-‖xn+1-xn‖)≤0.
Hence, by Lemma 2.3, we have
(3.31)lim⁡n→∞‖ln-xn‖=0.
Moreover, we have
(3.32)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞(1-βn)‖ln-xn‖=0.
Since
(3.33)xn+1-xn=ϵn(u+γf(Snvn)-(I+μA)xn)+((1-βn)I-ϵn(I+μA))(Snzn-xn)
it follows that
(3.34)(1-βn-ϵn(1+μ)γ¯)‖xn-Snzn‖≤‖xn+1-xn‖+ϵn‖u+γf(Snvn)-(I+μA)xn‖.
It follows from (3.32) and the conditions (C4) and (C5) that
(3.35)lim⁡n→∞‖xn-Snzn‖=0.
Next, we show that
(3.36)lim⁡n→∞‖Ψ1un-Ψ1x¯‖=lim⁡n→∞‖Φ1un-Φ1x¯‖=0,lim⁡n→∞‖Ψ2xn-Ψ2x¯‖=lim⁡n→∞‖Φ2xn-Φ2x¯‖=0,lim⁡n→∞‖B1yn-B1x¯‖=lim⁡n→∞‖B2vn-B1x¯‖=0.
From (3.8), (3.9), (3.11), and (3.12), we have
(3.37)‖zn-x¯‖2≤‖yn-x¯‖2+sn(sn-2ξ̃1)‖B1yn-B1x¯‖2≤‖vn-x¯‖2+tn(tn-2ξ̃2)‖B2vn-B2x¯‖2+sn(sn-2ξ̃1)‖B1yn-B1x¯‖2≤‖un-x¯‖2+2μn(μn-β̃1)‖Ψ1un-Ψ1x¯‖2+2μn(μn-γ̃1)‖Φ1un-Φ1x¯‖2+tn(tn-2ξ̃2)‖B2vn-B2x¯‖2+sn(sn-2ξ̃1)‖B1yn-B1x¯‖2≤‖xn-x¯‖2+2rn(rn-β̃2)‖Ψ2xn-Ψ2x¯‖2+2rn(rn-γ̃2)‖Φ2xn-Φ2x¯‖2+2μn(μn-β̃1)‖Ψ1un-Ψ1x¯‖2+2μn(μn-γ̃1)‖Φ1un-Φ1x¯‖2+tn(tn-2ξ̃2)‖B2vn-B2x¯‖2+sn(sn-2ξ̃1)‖B1yn-B1x¯‖2.
On the other hand, from Lemma 2.4   (ii), we have
(3.38)‖xn+1-x¯‖2=‖ϵn(u+γf(Snvn)-(I+μA)x¯)+βn(xn-Snzn)+(I-ϵn(I+μA))(Snzn-x¯)‖2≤‖(I+ϵn(I+μA))(Snzn-x¯)+βn(xn-Snzn)‖2+2ϵn〈u+γf(Snvn)-(I+μA)x¯,xn+1-x¯〉≤[‖(I-ϵn(I+μA))(Snzn-x¯)‖+βn‖xn-Snzn‖]2+2ϵn〈u+γf(Snvn)-(I+μA)x¯,xn+1-x¯〉≤[(1-ϵn-ϵnμγ¯)‖zn-x¯‖+βn‖xn-Snzn‖]2+2ϵn‖u+γf(Snvn)-(I+μA)x¯‖‖xn+1-x¯‖≤(1-ϵn-ϵnμγ¯)2‖zn-x¯‖2+cn,
where
(3.39)cn∶=βn2‖xn-Snzn‖2+2(1-ϵn-ϵnμγ¯)βn‖zn-x¯‖‖xn-Snzn‖+2ϵn‖u+γf(Snvn)-(I+μA)x¯‖‖xn+1-x¯‖.
It follows from the condition (C4) and (3.35) that
(3.40)lim⁡n→∞cn=0.
Substituting (3.37) into (3.38), we have
(3.41)‖xn+1-x¯‖2≤(1-ϵn-ϵnμγ¯)2×{‖xn-x¯‖2+2rn(rn-β̃2)‖Ψ2xn-Ψ2x¯‖2+2rn(rn-γ̃2)‖Φ2xn-Φ2x¯‖2+2μn(μn-β̃1)‖Ψ1un-Ψ1x¯‖2+2μn(μn-γ̃1)‖Φ1un-Φ1x¯‖2+tn(tn-2ξ̃2)‖B2vn-B2x¯‖2+sn(sn-2ξ̃1)‖B1yn-B1x¯‖2}+cn,
which in turn implies that
(3.42)(1-ϵn-ϵnμγ¯)2{2rn(β̃2-rn)‖Ψ2xn-Ψ2x¯‖2+2rn(γ̃2-rn)‖Φ2xn-Φ2x¯‖2+2μn(β̃1-μn)‖Ψ1un-Ψ1x¯‖2+2μn(γ̃1-μn)‖Φ1un-Φ1x¯‖2+tn(2ξ̃2-tn)‖B2vn-B2x¯‖2+sn(2ξ̃1-sn)‖B1yn-B1x¯‖2}≤(1-ϵn-ϵnμγ¯)2‖xn-x¯‖2-‖xn+1-x¯‖2+cn={1-2ϵn(1+μγ¯)+ϵn2(1+μγ¯)2}‖xn-x¯‖2-‖xn+1-x¯‖2+cn≤‖xn-x¯‖2-‖xn+1-x¯‖2+ϵn2(1+μγ¯)2‖xn-x¯‖2+cn≤(‖xn-x¯‖+‖xn+1-x¯‖)‖xn+1-xn‖+ϵn2(1+μγ¯)2‖xn-x¯‖2+cn.
It follows from the conditions (C4)–(C9), (3.32), and (3.40) that
(3.43)lim⁡n→∞‖Ψ1un-Ψ1x¯‖=lim⁡n→∞‖Φ1un-Φ1x¯‖=0,lim⁡n→∞‖Ψ2xn-Ψ2x¯‖=lim⁡n→∞‖Φ2xn-Φ2x¯‖=0,lim⁡n→∞‖B1yn-B1x¯‖=lim⁡n→∞‖B2vn-B1x¯‖=0.
Next, we show that limn→∞∥Szn-zn∥=0. Since Jsn is firmly nonexpansive, we have
(3.44)‖zn-x¯‖2=‖Jsn(yn-snB1yn)-Jsn(x¯-snB1x¯)‖2≤〈(yn-snB1yn)-(x¯-snB1x¯),zn-x¯〉=12(‖(yn-snB1yn)-(x¯-snB1x¯)‖2+‖zn-x¯‖2-‖(yn-snB1yn)-(x¯-snB1x¯)-(zn-x¯)‖2)≤12(‖yn-x¯‖2+‖zn-x¯‖2-‖yn-zn-sn(B1yn-B1x¯)‖2)=12(‖yn-x¯‖2+‖zn-x¯‖2-‖yn-zn‖2+2sn〈yn-zn,B1yn-B1x¯〉-sn2‖B1yn-B1x¯‖2),
which in turn implies that
(3.45)‖zn-x¯‖2≤‖yn-x¯‖2-‖yn-zn‖2+2sn〈yn-zn,B1yn-B1x¯〉-sn2‖B1yn-B1x¯‖2≤‖xn-x¯‖2-‖yn-zn‖2+2sn‖yn-zn‖‖B1yn-B1x¯‖.
Substituting (3.45) into (3.38), we have
(3.46)‖xn+1-x¯‖2≤(1-ϵn-ϵnμγ¯)2{‖xn-x¯‖2-‖yn-zn‖2+2sn‖yn-zn‖‖B1yn-B1x¯‖}+cn,
which in turn implies that
(3.47)(1-ϵn-ϵnμγ¯)2‖yn-zn‖2≤(1-ϵn-ϵnμγ¯)2‖xn-x¯‖2-‖xn+1-x¯‖2+2(1-ϵn-ϵnμγ¯)2sn‖yn-zn‖‖B1yn-B1x¯‖+cn≤‖xn-x¯‖2-‖xn+1-x¯‖2+ϵn2(1+μγ¯)2‖xn-x¯‖2+2sn(1-ϵn-ϵnμγ¯)2‖yn-zn‖‖B1yn-B1x¯‖+cn≤(‖xn-x¯‖+‖xn+1-x¯‖)‖xn+1-xn‖+ϵn2(1+μγ¯)2‖xn-x¯‖2+2sn(1-ϵn-ϵnμγ¯)2‖yn-zn‖‖B1yn-B1x¯‖+cn.
Since ϵn→0, ∥B1yn-B1x¯∥→0 and from (3.32), we obtain that
(3.48)lim⁡n→∞‖yn-zn‖=0.
In a similar way, we can get
(3.49)‖yn-x¯‖2≤‖xn-x¯‖2-‖vn-yn‖2+2tn‖vn-yn‖‖B2vn-B2x¯‖.
It follows from (3.38) and (3.49) that
(3.50)‖xn+1-x¯‖2≤(1-ϵn-ϵnμγ¯)2‖zn-x¯‖2+cn≤(1-ϵn-ϵnμγ¯)2‖yn-x¯‖2+cn≤(1-ϵn-ϵnμγ¯)2{‖xn-x¯‖2-‖vn-yn‖2+2tn‖vn-yn‖‖B2vn-B2x¯‖}+cn,
which in turn implies that
(3.51)(1-ϵn-ϵnμγ¯)2‖vn-yn‖2≤(1-ϵn-ϵnμγ¯)2‖xn-x¯‖2-‖xn+1-xn‖2+2(1-tnϵn-ϵnμγ¯)2‖vn-yn‖‖B2vn-B2x¯‖+cn≤‖xn-x¯‖2-‖xn+1-x¯‖2+ϵn2(1+μγ¯)2‖xn-x¯‖2+2tn(1-tnϵn-ϵnμγ¯)2‖vn-yn‖‖B2vn-B2x¯‖+cn≤(‖xn-x¯‖+‖xn+1-x¯‖)‖xn+1-xn‖+ϵn2(1+μγ¯)2‖xn-x¯‖2+2tn(1-ϵn-ϵnμγ¯)2‖vn-yn‖‖B2vn-B2x¯‖+cn.
Since ϵn→0,∥B2vn-B2x¯∥→0 and from (3.32), we obtain that
(3.52)lim⁡n→∞‖vn-yn‖=0.
In addition, from the firmly nonexpansivity of Vμn(Θ1,φ1), we have
(3.53)‖vn-x¯‖2=‖Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un)-Vμn(Θ1,φ1)(x¯-μn(Ψ1+Φ1)x¯)‖2≤〈(un-μn(Ψ1+Φ1)un)-(x¯-μn(Ψ1+Φ1)x¯),vn-x¯〉=12(‖(un-μn(Ψ1+Φ1)un)-(x¯-μn(Ψ1+Φ1)x¯)‖2+‖vn-x¯‖2-‖(un-μn(Ψ1+Φ1)un)-(x¯-μn(Ψ1+Φ1)x¯)-(vn-x¯)‖2)≤12(‖un-x¯‖2+‖vn-x¯‖2-‖(un-vn)-μn((Ψ1+Φ1)un-(Ψ1+Φ1)x¯)‖2)=12(‖un-x¯‖2+‖vn-x¯‖2-‖un-vn‖2+2μn〈un-vn,(Ψ1+Φ1)un-(Ψ1+Φ1)x¯〉-μn2‖(Ψ1+Φ1)un-(Ψ1+Φ1)x¯‖2),
which in turn implies that
(3.54)‖vn-x¯‖2≤‖un-x¯‖2-‖un-vn‖2+2μn〈un-vn,(Ψ1+Φ1)un-(Ψ1+Φ1)x¯〉-μn2‖(Ψ1+Φ1)un-(Ψ1+Φ1)x¯‖2≤‖xn-x¯‖2-‖un-vn‖2+2μn‖un-vn‖‖(Ψ1+Φ1)un-(Ψ1+Φ1)x¯‖.
It follows from (3.38) and (3.54) that
(3.55)‖xn+1-x¯‖2≤(1-ϵn-ϵnμγ¯)2‖zn-x¯‖2+cn≤(1-ϵn-ϵnμγ¯)2‖vn-x¯‖2+cn≤(1-ϵn-ϵnμγ¯)2{‖xn-x¯‖2-‖un-vn‖2+2μn‖un-vn‖‖(Ψ1+Φ1)un-(Ψ1+Φ1)x¯‖}+cn,
which in turn implies that
(3.56)(1-ϵn-ϵnμγ¯)2‖un-vn‖2≤(1-ϵn-ϵnμγ¯)2‖xn-x¯‖2-‖xn+1-x¯‖2+2μn(1-ϵn-ϵnμγ¯)2‖un-vn‖‖(Ψ1+Φ1)un-(Ψ1+Φ1)x¯‖+cn≤‖xn-x¯‖2-‖xn+1-x¯‖2+ϵn2(1+μγ¯)2‖xn-x¯‖2+2μn(1-ϵn-ϵnμγ¯)2‖un-vn‖‖(Ψ1+Φ1)un-(Ψ1+Φ1)x¯‖+cn≤(‖xn-x¯‖+‖xn+1-x¯‖)‖xn+1-xn‖+ϵn2(1+μγ¯)2‖xn-x¯‖2+2μn(1-ϵn-ϵnμγ¯)2‖un-vn‖‖(Ψ1+Φ1)un-(Ψ1+Φ1)x¯‖+cn.
Since ϵn→0, ∥Ψ1un-Ψ1x¯∥→0, ∥Φ1un-Φ1x¯∥→0 and from (3.32), we obtain that
(3.57)lim⁡n→∞‖un-vn‖=0.
In a similar way, we can get
(3.58)‖un-x¯‖2≤‖xn-x¯‖2-‖xn-un‖2+2rn‖xn-un‖‖(Ψ2+Φ2)xn-(Ψ2+Φ2)x¯‖.
It follows from (3.38) and (3.58) that
(3.59)‖xn+1-x¯‖2≤(1-ϵn-ϵnμγ¯)2‖zn-x¯‖2+cn≤(1-ϵn-ϵnμγ¯)2‖un-x¯‖2+cn≤(1-ϵn-ϵnμγ¯)2{‖xn-x¯‖2-‖xn-un‖2+2rn‖xn-un‖‖(Ψ2+Φ2)xn-(Ψ2+Φ2)x¯‖}+cn,
which in turn implies that
(3.60)(1-ϵn-ϵnμγ¯)2‖xn-un‖2≤(1-ϵn-ϵnμγ¯)2‖xn-x¯‖2-‖xn+1-x¯‖2+2rn(1-ϵn-ϵnμγ¯)2‖xn-un‖‖(Ψ2+Φ2)xn-(Ψ2+Φ2)x¯‖+cn≤‖xn-x¯‖2-‖xn+1-x¯‖2+ϵn2(1+μγ¯)2‖xn-x¯‖2+2rn(1-ϵn-ϵnμγ¯)2‖xn-un‖‖(Ψ2+Φ2)xn-(Ψ2+Φ2)x¯‖+cn≤(‖xn-x¯‖+‖xn+1-x¯‖)‖xn+1-xn‖+ϵn2(1+μγ¯)2‖xn-x¯‖2+2rn(1-ϵn-ϵnμγ¯)2‖xn-un‖‖(Ψ2+Φ2)xn-(Ψ2+Φ2)x¯‖+cn.
Since ϵn→0, ∥Ψ2xn-Ψ2x¯∥→0, ∥Φ2xn-Φ2x¯∥→0 and from (3.32), we obtain that
(3.61)lim⁡n→∞‖xn-un‖=0.
Notice that
(3.62)‖Snzn-zn‖≤‖Snzn-xn‖+‖xn-un‖+‖un-vn‖+‖vn-yn‖+‖yn-zn‖.
It follows from (3.35), (3.48), (3.52), (3.57), and (3.61) that
(3.63)lim⁡n→∞‖Snzn-zn‖=0.
Moreover, we note that
(3.64)‖Szn-zn‖≤‖Szn-Snzn‖+‖Snzn-zn‖≤sup⁡x∈K̃‖Sx-Snx‖+‖Snzn-zn‖,
where K̃ is any bounded subset of C. From Lemma 2.2 and (3.63), we obtain that
(3.65)lim⁡n→∞‖Szn-zn‖=0.
Next, we show that
(3.66)limsup⁡n→∞〈u+γf(x*)-(I+μA)x*,xn-x*〉≤0,
where x* is a solution of optimization problem (3.4). To show this, we choose a subsequence {znj} of {zn} such that
(3.67)limsup⁡n→∞〈u+γf(x*)-(I+μA)x*,zn-x*〉=lim⁡j→∞〈u+γf(x*)-(I+μA)x*,znj-x*〉.
Since {zn} is bounded, there exists a subsequence {znj} of {zn} such that znj⇀ω as j→∞. On the other hand, we note that
(3.68)‖xn-zn‖≤‖xn-un‖+‖un-vn‖+‖vn-yn‖+‖yn-zn‖.
It follows from (3.48), (3.52), (3.57), and (3.61) that
(3.69)lim⁡n→∞‖xn-zn‖=0.
Now, we show that ω∈Ω∶=⋂i=1∞F(Ti)∩GMEP(Θ1,φ1,Ψ1+Φ1)∩GMEP(Θ2,φ2,Ψ2+Φ2)∩(B1+W1)-1(0)∩(B2+W2)-1(0).
(i) First, we show thatω∈⋂i=1∞F(Ti). By Lemma 2.2, we have F(S)=⋂i=1∞F(Ti). Suppose the contrary, ω≠Sω, from (3.65) and Opial's condition (see [63]), we have
(3.70)liminf⁡n→∞‖znj-ω‖<liminf⁡n→∞‖znj-Sω‖≤liminf⁡n→∞{‖znj-Sznj‖+‖Sznj-Sω‖}≤liminf⁡n→∞‖znj-ω‖,
which is a contradiction. So, we obtain ω∈F(S)=⋂i=1∞F(Ti).
(ii) Now, we show thatω∈GMEP(Θ1,φ1,Ψ1+Φ1). From vn=Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un), we know that
(3.71)Θ1(vn,y)+φ1(y)-φ1(vn)+〈(Ψ1+Φ1)un,y-vn〉+1μn〈K1′(vn)-K1′(un),y-vn〉≥0,∀y∈C.
From (H2), we have
(3.72)φ1(y)-φ1(vn)+〈(Ψ1+Φ1)un,y-vn〉+1μn〈K1′(vn)-K1′(un),y-vn〉≥Θ1(y,vn),∀y∈C.
Now, replacing n by nj in (3.72), we have
(3.73)φ1(y)-φ1(vnj)+〈(Ψ1+Φ1)unj,y-vnj〉+1μnj〈K1′(vnj)-K1′(unj),y-vnj〉≥Θ1(y,vnj),∀y∈C.
Put vt=ty+(1-t)ω for all t∈(0,1] and y∈C. Then, from (3.73), we have
(3.74)〈vt-vnj,(Ψ1+Φ1)vt〉≥〈vt-vnj,(Ψ1+Φ1)vt〉-φ1(vt)+φ1(vnj)-〈vt-vnj,(Ψ1+Φ1)unj〉-〈K1′(vnj)-K1′(unj)μnj,vt-vnj〉+Θ1(vt,vnj)=〈vt-vnj,(Ψ1+Φ1)vt-(Ψ1+Φ1)vnj〉+〈vt-vnj,(Ψ1+Φ1)vnj-(Ψ1+Φ1)unj〉-φ1(vt)+φ1(vnj)-〈K1′(vnj)-K1′(unj)μnj,vt-vnj〉+Θ1(vt,vnj).
Since ∥vnj-unj∥→0, we have ∥(Ψ1+Φ1)vnj-(Ψ1+Φ1)unj∥→0 and from Lipschitz continuity of K1′, we have ∥K1′(vnj)-K1′(unj)∥→0. Further, from the monotonicity of Ψ1+Φ1, we have
(3.75)〈vt-vnj,(Ψ1+Φ1)vt-(Ψ1+Φ1)vnj〉≥0.
Since xnj⇀ω, ∥xnj-unj∥→0 and ∥vnj-unj∥→0, we have vnj⇀ω. Then, from (H4), (H5), the weakly lower semicontinuous of φ1, and (K1′(vnj)-K′(unj))/μnj→0, we obtain that
(3.76)〈vt-ω,(Ψ1+Φ1)vt〉≥-φ1(vt)+φ1(ω)+Θ1(vt,ω),asj⟶∞.
From (H1), (H4), and (3.76), we also have
(3.77)0=Θ1(vt,vt)+φ1(vt)-φ1(vt)≤tΘ1(vt,y)+(1-t)Θ1(vt,ω)+tφ1(y)+(1-t)φ1(ω)-φ1(vt)=t[Θ1(vt,y)+φ1(y)-φ1(vt)]+(1-t)[Θ1(vt,ω)+φ1(ω)-φ1(vt)]≤t[Θ1(vt,y)+φ1(y)-φ1(vt)]+(1-t)〈vt-ω,(Ψ1+Φ1)vt〉=t[Θ1(vt,y)+φ1(y)-φ1(vt)]+(1-t)t〈y-ω,(Ψ1+Φ1)vt〉,
and hence
(3.78)0≤Θ1(vt,y)+φ1(y)-φ1(vt)+(1-t)〈y-ω,(Ψ1+Φ1)vt〉.
Letting t→0, we have, for all y∈C,
(3.79)0≤Θ1(ω,y)+φ1(y)-φ1(ω)+〈(Ψ1+Φ1)ω,y-ω〉.
This implies that ω∈GMEP(Θ1,φ1,Ψ1+Φ1). In a similar way, we can get ω∈GMEP(Θ2,φ2,Ψ2+Φ2).
(iii) Now, we show thatω∈(B1+W1)-1(0). In fact, notice that
(3.80)yn-snB1yn∈zn+snW1zn.
Let ρ∈W1η. Since W1 is monotone, we have
(3.81)〈yn-znsn-B1yn-ρ,zn-η〉≥0.
It follows from the condition (C8) and (3.48) that
(3.82)〈-B1ω-ρ,ω-η〉≥0.
This implies that -B1ω∈W1ω, that is ω∈(B1+W1)-1(0). In a similar way, we can get ω∈(B2+W2)-1(0). Therefore ω∈Ω∶=⋂i=1∞F(Ti)∩GMEP(Θ1,φ1,Ψ1+Φ1)∩GMEP(Θ2,φ2,Ψ2+Φ2)∩(B1+W1)-1(0)∩(B2+W2)-1(0). Now, from Lemma 2.11, (3.67), and (3.69), we obtain that
(3.83)limsup⁡n→∞〈u+γf(x*)-(I+μA)x*,xn-x*〉=limsup⁡n→∞〈u+γf(x*)-(I+μA)x*,zn-x*〉=lim⁡j→∞〈u+γf(x*)-(I+μA)x*,znj-x*〉=〈u+γf(x*)-(I+μA)x*,ω-x*〉≤0.
Finally,  we show that xn→x* as n→∞. Again, from Lemma 2.4  (ii), we compute
(3.84)‖xn+1-x*‖2=‖ϵn(u+γf(Snvn)-(I+μA)x*)+βn(xn-x*)+((1-βn)I-ϵn(I+μA))(Snzn-x*)‖2≤‖((1-βn)I-ϵn(I+μA))(Snzn-x*)+βn(xn-x*)‖2+2ϵn〈u+γf(Snvn)-(I+μA)x*,xn+1-x*〉≤[‖((1-βn)I-ϵn(I+μA))(Snzn-x*)‖+βn‖xn-x*‖]2+2ϵnγ〈f(Snvn)-f(x*),xn+1-x*〉+2ϵn〈u+γf(x*)-(I+μA)x*,xn+1-x*〉≤[(1-βn-ϵn(1+μ)γ¯)‖zn-x*‖+βn‖xn-x*‖]2+2ϵnγα‖Snvn-x*‖‖xn+1-x*‖+2ϵn〈u+γf(x*)-(I+μA)x*,xn+1-x*〉≤[(1-βn-ϵn(1+μ)γ¯)‖xn-x*‖+βn‖xn-x*‖]2+2ϵnγα‖xn-x*‖‖xn+1-x*‖+2ϵn〈u+γf(x*)-(I+μA)x*,xn+1-x*〉≤(1-ϵn(1+μ)γ¯)2‖xn-x*‖2+ϵnγα(‖xn-x*‖2+‖xn+1-x*‖2)+2ϵn〈u+γf(x*)-(I+μA)x*,xn+1-x*〉=(1-2ϵn(1+μ)γ¯+ϵn2[(1+μ)γ¯]2+ϵnγα)‖xn-x*‖2+ϵnγα‖xn+1-x*‖2+2ϵn〈u+γf(x*)-(I+μA)x*,xn+1-x*〉.
It follows  that
(3.85)‖xn+1-x*‖2≤1-2ϵn(1+μ)γ¯+ϵn2[(1+μ)γ¯]2+ϵnγα1-ϵnγα‖xn-x*‖2+2ϵn1-ϵnγα〈u+γf(x*)-(I+μA)x*,xn+1-x*〉=1-2ϵn(1+μ)γ¯+ϵnγα1-ϵnγα‖xn-x*‖2+ϵn2[(1+μ)γ¯]21-ϵnγα‖xn-x*‖2+2ϵn1-ϵnγα〈u+γf(x*)-(I+μA)x*,xn+1-x*〉≤[1-2((1+μ)γ¯-γα)ϵn1-ϵnγα]‖xn-x*‖2+ϵn2[(1+μ)γ¯]21-ϵnγαM4+2ϵn1-ϵnγα〈u+γf(x*)-(I+μA)x*,xn+1-x*〉,
where M4=supn≥1{∥xn-x*∥2}. Put τn∶=2((1+μ)γ¯-γα)ϵn/(1-ϵnγα) and
(3.86)δn∶=ϵn2[(1+μ)γ¯]21-ϵnγαM+2ϵn1-ϵnγα〈u+γf(x*)-(I+μA)x*,xn+1-x*〉.
Then, (3.85) reduces to formula
(3.87)‖xn+1-x*‖2≤(1-τn)‖xn-x*‖2+δn.
It follows from the condition (C4) and (3.83) that ∑n=1∞τn=∞ and
(3.88)limsup⁡n→∞δnτn=limsup⁡n→∞12((1+μ)γ¯-γα)×[ϵn[(1+μ)γ¯]2M4+2〈u+γf(x*)-(I+μA)x*,xn+1-x*〉]≤0.
Hence, by Lemma 2.12, we obtain that xn→x* as n→∞. This completes the proof.Remark 3.2.
The control conditionlimn→∞|μn+1-μn|=0 and other on (C6)–(C9) are replaced by the strictly weaker conditions: limn→∞μn/μn+1=1 as shown in the next example.Example 3.3.
(a)  Iflimn→∞|μn+1-μn|=0, then limn→∞μn/μn+1=1.
(b)  The converse of(a) is not true.Proof.
(a) Since0<μn<min{β̃1,γ̃1} and liminfn→∞μn>0, there exists a constant μ̃ such that μn≥μ̃>0 for all n∈ℕ. We observe that
(3.89)lim⁡n→∞|μnμn+1-1|=lim⁡n→∞|μn-μn+1μn+1|≤lim⁡n→∞|μn+1-μn|μ̃=0.
Hence, we obtain that limn→∞μn/μn+1=1.
(b) Letμn=((n+1)/n)n, then μn+1=((n+2)/(n+1))n+1. We see that
(3.90)lim⁡n→∞μnμn+1=lim⁡n→∞(1+1n)n⋅(1+1/n)n+1(1+2/n)n+1=e⋅ee2=1,
but
(3.91)lim⁡n→∞|μn+1-μn|=lim⁡n→∞|(1+1/n)n+1(1+2/n)n+1-(1+1n)n|=e-1e≠0.
Then, converse of (a) is not true. Hence, (b) is proved.Remark 3.4.
Theorem3.1 improves and extends [45, Theorem 3.3] in the following respects. (1)
The problem of finding the common element⋂i=1∞F(Ti)∩GMEP(Θ1,φ1,Ψ1+Φ1)∩GMEP(Θ2,φ2,Ψ2+Φ2)∩(B1+W1)-1(0)∩(B2+W2)-1(0) is more general and more complex than the one of finding the common element F(T)∩GMEP(Θ1,φ,Ψ1)∩GMEP(Θ2,φ,Ψ2)∩VI(C,B) in [44, Theorem 3.3].(2)
In [44, Algorithm 3.13], the function Ki:C→ℝ is chosen as Ki(x)=∥x∥2/2 for i=1,2.(3)
The conditions (C3) and (C4) of Chantarangsi et al. [44, Theorem 3.3] are replaced by the strictly weaker conditions.Remark 3.5.
Theorem3.1 improves and extends [45, Theorem 2.1] in the following respects. (1)
Theorem3.1 extended [64, Theorem 2.1] to finding common element of the set of solutions of a system of generalized mixed equilibrium problem, and the set of infinite family of nonexpansive mapping involves strongly positive linear bounded operator and the optimization problem.(2)
The conditions (C3) and (C4) of Yu and Liang [45, Theorem 2.1] are replaced by the strictly weaker conditions.


## 4. Some Applications

LetH be a real Hilbert space and g:H→(-∞,+∞] a proper convex lower semicontinuous function. Then, the subdifferential∂g of g is defined as follows: (4.1)∂g(x)={y∈H:g(z)≥g(x)+〈z-x,y〉,∀z∈H},∀x∈H.
From Rockafellar [65], we know that ∂g is maximal monotone. Let C be a closed and convex subset of H, and let δC be the indicator function of C, that is,(4.2)δC(x)={0,x∈C,+∞,x∉C.
Since δC is a proper lower semicontinuous convex function on H, the subdifferential ∂δC of δC is a maximal monotone operator.Lemma 4.1 (see [57]).
LetC be a nonempty closed convex subset of a real Hilbert space H, let PC be the metric projection from H onto C, and let ∂δC be the subdifferential of δC, where δC is as defined in (4.2) and Jr=(I+r∂δC)-1. Then,
(4.3)y=Jrx⟺y=PCx,∀x∈H,y∈C.Now, we consider the existence of solution of the variational inequality (1.6).Theorem 4.2.
LetC be a nonempty closed convex subset of a Hilbert space H. Let φ1,φ2:C→ℝ be two lower semicontinuous and convex functionals, and let Θ1,Θ2:C×C→ℝ be two bifunctions satisfying conditions (H1)–(H5). Let Ψ1,Ψ2,Φ1,Φ2:C→H be β̃1-inverse-strongly monotone mapping, β̃2-inverse-strongly monotone mapping, γ̃1-inverse-strongly monotone mapping, and γ̃2-inverse-strongly monotone mapping, respectively, and let B1,B2:C→H be ξ̃1-inverse-strongly monotone mapping and ξ̃2-inverse-strongly monotone mapping, respectively. Let {Ti}i=1∞:C→C be an infinite family of nonexpansive mappings, and let αi=(α1j,α2j,α3j)∈I×I×I, where I=[0,1], α1j+α2j+α3j=1,α1j+α2j≤b<1, and α1j,α2j,α3j∈(0,1) for all j=1,2,…. For all n∈ℕ, let Sn and S be S-mappings generated by Tn,Tn-1,…,T1 and αn,αn-1,…,α1 and Tn,Tn-1,… and αn,αn-1,…, respectively. Assume that
(4.4)Ω∶=⋂i=1∞F(Ti)∩GMEP⁡(Θ1,φ1,Ψ1+Φ1)∩GMEP⁡(Θ2,φ2,Ψ2+Φ2)∩VI⁡(C,B1)∩VI⁡(C,B2)≠∅.
Let f:C→H be a contraction mapping with a coefficient α∈(0,1), and let A:C→H be a strongly positive linear bounded operator with a coefficient γ¯∈(0,1). Let μ>0 and γ>0 be two constants such that 0<γ<(1+μ)γ¯/α. Let {xn} be a sequence defined by u,x1∈H and
(4.5)un=Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn),vn=Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un),yn=PC(vn-tnB2vn),xn+1=ϵn(u+γf(Snvn))+βnxn+((1-βn)I-ϵn(I+μA))SnPC(yn-snB1yn),∀n≥1,
where {sn}⊂(0,2ξ̃1),{tn}⊂(0,2ξ̃2),{μn}⊂(0,min{β̃1,γ̃1}),{rn}⊂(0,min{β̃2,γ̃2}), and {ϵn} and {βn}⊂(0,1). Assume the following conditions are satisfied:  (C1)
for alli=1,2,Ki:C→ℝ is strongly convex with constant σi>0 and its derivative Ki′ is Lipschitz continuous with constant νi>0 such that the function x↦〈y-x,Ki′(x)〉 is weakly upper semicontinuous for each y∈C,(C2)
for alli=1,2 and for each x∈C, there exist a bounded subset Dx⊂C and zx∈C such that, for any y∉Dx,(4.6)Θi(y,zx)+φi(zx)-φi(y)+1ri〈Ki′(y)-Ki′(x),zx-y〉<0,(C3)
limn→∞αn1=0,(C4)
limn→∞ϵn=0 and ∑n=1∞ϵn=∞,(C5)
0<liminfn→∞βn≤limsupn→∞βn<1,(C6)
0<liminfn→∞μn≤limsupn→∞μn<min{β̃1,γ̃1} and limn→∞μn/μn+1=1,(C7)
0<liminfn→∞rn≤limsupn→∞rn<min{β̃2,γ̃2} and limn→∞rn/rn+1=1,(C8)
0<liminfn→∞sn≤limsupn→∞sn<2ξ̃1 and limn→∞sn/sn+1=1,(C9)
0<liminfn→∞tn≤limsupn→∞tn<2ξ̃2 and limn→∞tn/tn+1=1. Then, the sequence {xn} defined by (4.5) converges strongly to x*∈Ω, provided Vr,n(Θ,φ) is firmly nonexpansive, where x* solves the following optimization problem:
(4.7)min⁡x∈Ωμ2〈Ax,x〉+12‖x-u‖2-h(x).Proof.
Now, we show thatVI(C,B1)=(B1+∂δC)-1(0) and VI(C,B2)=(B2+∂δC)-1(0), respectively. Set W1=W2=∂δC in Theorem 3.1. Notice that
(4.8)x*∈(B1+∂δC)-1(0)⟺0∈B1x*+∂δCx*⟺-B1x*∈∂δCx*⟺〈B1x*,y-x*〉≥0⟺x*∈VI⁡(C,B1).
In a similar way, we can get
(4.9)x*∈(B2+∂δC)-1(0)⟺x*∈VI⁡(C,B2).
From Lemma 4.1, we can conclude the desired conclusion immediately. This completes the proof.LetT̂:C→C be k-strict pseudocontraction mapping. Setting B=I-T̂, we see that, for all x,y∈C, (4.10)‖(I-B)x-(I-B)y‖2≤‖x-y‖2+k‖Bx-By‖2.
On the other hand, we note that (4.11)‖(I-B)x-(I-B)y‖2≤‖x-y‖2-2〈Bx-By,x-y〉+‖Bx-By‖2.
For all x,y∈C, we obtain that (4.12)〈Bx-By,x-y〉≥1-k2‖Bx-By‖2.
Then, B is ((1-k)/2)-inverse-strongly monotone mapping.Theorem 4.3.
LetC be a nonempty closed convex subset of a Hilbert space H. Let φ1,φ2:C→ℝ be two lower semicontinuous and convex functionals, and  let Θ1,Θ2:C×C→ℝ be two bifunctions satisfying conditions (H1)–(H5). Let Ψ1,Ψ2,Φ1,Φ2:C→H be β̃1-inverse-strongly monotone mapping, β̃2-inverse-strongly monotone mapping, γ̃1-inverse-strongly monotone mapping, and γ̃2-inverse-strongly monotone mapping, respectively, and let B1,B2:C→H be ξ̃1-inverse-strongly monotone mapping and ξ̃2-inverse-strongly monotone mapping, respectively. Let {Ti}i=1∞:C→C be an infinite family of nonexpansive mappings, and let αi=(α1j,α2j,α3j)∈I×I×I, where I=[0,1], α1j+α2j+α3j=1,α1j+α2j≤b<1, and α1j,α2j,α3j∈(0,1) for all j=1,2,…. For all n∈ℕ, let Sn and S be S-mappings generated by Tn,Tn-1,…,T1 and αn,αn-1,…,α1 and Tn,Tn-1,… and αn,αn-1,…, respectively. Let T̂i:C→C be a ki-pseudocontraction mapping for all i=1,2. Assume that
(4.13)Ω∶=⋂i=1∞F(Ti)∩GMEP⁡(Θ1,φ1,Ψ1+Φ1)∩GMEP⁡(Θ2,φ2,Ψ2+Φ2)∩F(T̂1)∩F(T̂2)≠∅.
Let f:C→H be a contraction mapping with a coefficient α∈(0,1), and let A:C→H be a strongly positive linear bounded operator with a coefficient γ¯∈(0,1). Let μ>0 and γ>0 be two constants such that 0<γ<(1+μ)γ¯/α. Let {xn} be a sequence defined by u,x1∈H and
(4.14)un=Vrn(Θ2,φ2)(xn-rn(Ψ2+Φ2)xn),vn=Vμn(Θ1,φ1)(un-μn(Ψ1+Φ1)un),yn=(1-tn)vn+tnT̂2vn,xn+1=ϵn(u+γf(Snvn))+βnxn+((1-βn)I-ϵn(I+μA))×Sn[(1-sn)yn+snT̂1yn],∀n≥1,
where {sn}⊂(0,2ξ̃1), {tn}⊂(0,2ξ̃2), {μn}⊂(0,min{β̃1,γ̃1}), {rn}⊂(0,min{β̃2,γ̃2}), and {ϵn} and {βn}⊂(0,1). Assume the conditions (C1)– (C9) in Theorem 4.2. Then, the sequence {xn} defined by (4.14) converges strongly to x*∈Ω∶=⋂i=1∞F(Ti)∩GMEP(Θ1,φ1,Ψ1+Φ1)∩GMEP(Θ2,φ2,Ψ2+Φ2)∩F(T̂1)∩F(T̂2),  provided Vr,n(Θ,φ) is firmly nonexpansive, where x* solves the following optimization problem:
(4.15)min⁡x∈Ωμ2〈Ax,x〉+12‖x-u‖2-h(x).Proof.
TakingBi=I-T̂i:C→H, we see that Bi:C→H is λi-strict pseudocontraction mapping with λi=(1-ki)/2 and F(T̂i)=VI(C,Bi) for i=1,2. From Theorem 4.2, we can conclude the desired conclusion easily. This completes the proof.



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*Source: 316276-2011-12-25.xml*