# Common Solutions of Generalized Mixed Equilibrium Problems, Variational Inclusions, and Common Fixed Points for Nonexpansive Semigroups and Strictly Pseudocontractive Mappings

Authors: Poom Kumam; Usa Hamphries; Phayap Katchang
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

We introduce a new iterative scheme by shrinking projection method for finding a common
element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of
variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone
mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed
points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. We prove that
the sequence converges strongly to a common element of the above four sets under some mind conditions.
Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was
obtained. Our results improve and extend the corresponding results of Martinez-Yanes and Xu  (2006), Shehu
(2011), Zhang et al. (2008), and many authors.


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## Body

## 1. Introduction

Throughout this paper, we assume thatH is a real Hilbert space with inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let 2H denote the family of all subsets of H, and let C be a closed-convex subset of H. Recall that a mapping T:C→C is said to be a k-strict pseudocontraction [1] if there exists 0≤k<1 such that(1.1)‖Tx-Ty‖2≤‖x-y‖2+k‖(I-T)x-(I-T)y‖2,∀x,y∈C,
where I denotes the identity operator on C. When k=0, T:C→C is said to be nonexpansive [2] if (1.2)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.
And when k=1, T:C→C is said to be pseudocontraction if (1.3)‖Tx-Ty‖2≤‖x-y‖2+‖(I-T)x-(I-T)y‖2,∀x,y∈C.
Clearly, the class of k-strict pseudocontraction falls into the one between classes of nonexpansive mappings and pseudocontraction mapping. We denote the set of fixed points of T by F(T).A family𝒮={S(s):0≤s<∞} of mappings of C into itself is called a nonexpansive semigroup on C if it satisfies the following conditions: (i)
S(0)x=x for all x∈C,(ii)
S(s+t)=S(s)S(t) for all s,t≥0,(iii)
∥S(s)x-S(s)y∥≤∥x-y∥ for all x,y∈C and s≥0,(iv)
for allx∈C,s↦S(s)x is continuous.We denote byF(𝒮) the set of all common fixed points of 𝒮={S(s):s≥0}, that is, F(S)=⋂s≥0F(S(s)). It is known that F(𝒮) is closed and convex.LetA:H→H be a single-valued nonlinear mapping, and let M:H→2H be a set-valued mapping. We consider the following variational inclusion problem, which is to find a point u∈H such that(1.4)θ∈A(u)+M(u),
where θ is the zero vector in H. The set of solutions of problem (1.4) is denoted by I(A,M).Let the set-valued mappingM:H→2H be a maximal monotone. We define the resolvent operatorJM,λ associate with M and λ as follows:(1.5)JM,λ(u)=(I+λM)-1(u),u∈H,
where λ is a positive number. It is worth mentioning that the resolvent operator JM,λ is single-valued, nonexpansive, and 1-inverse-strongly monotone ([3, 4]).LetF be a bifunction of C×C into ℝ, where ℝ is the set of real numbers, let A:C→H be a mapping, and let φ:C→ℝ be a real-valued function. The generalized mixed equilibrium problem is for finding x∈C such that(1.6)F(x,y)+〈Ax,y-x〉+φ(y)-φ(x)≥0,∀y∈C.
The set of solutions of (1.6) is denoted by GMEP(F,φ,A), that is,(1.7)GMEP⁡(F,φ,A)={x∈C:F(x,y)+〈Ax,y-x〉+φ(y)-φ(x)≥0,∀y∈C}.
If A≡0, then the problem (1.6) is reduced into the mixed equilibrium problem for finding x∈C such that(1.8)F(x,y)+φ(y)-φ(x)≥0,∀y∈C.
The set of solutions of (1.8) is denoted by MEP(F,φ). The (generalized) mixed equilibrium problems include fixed-point problems, variational inequality problems, optimization problems, Nash equilibrium problems, noncooperative games, economics, and the equilibrium problem as special cases ([5–15]). In the last two decades, many papers have appeared in the literature on the existence of solutions of equilibrium problems; see, for example, [9] and references therein. Some solution methods have been proposed to solve the mixed equilibrium problems; see, for example, ([7–10, 12–20]) and references therein.In 2006, Martinez-Yanes and Xu [21] introduced the following iterative:
(1.9)x0∈C,yn=αnx0+(1-αn)Txn,Cn={z∈C:‖yn-z‖2≤‖xn-z‖2+αn(‖x0‖2+2〈xn-x0,z〉)},Qn={z∈C:〈xn-z,xn-x0〉≤0},xn+1=PCn∩Qnx0,
where T is a nonexpansive mapping in a Hilbert space H, and PC is metric projection of H onto a closed and convex subset C of H. They proved that if the sequence {αn} of parameters satisfies appropriate conditions, then the sequence {xn} converges strongly to PF(T)x0.In 2008, Zhang et al. [4] introduced an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with a multivalued maximal monotone mapping and an inverse-strongly monotone mapping and the set of fixed points of nonexpansive mapping in Hilbert spaces. The following iterative scheme x0=x∈H and
(1.10)yn=JM,λ(xn-λAxn),xn+1=αnx+(1-αn)Syn,
for all n≥0. They proved the strong convergence theorem under some mind conditions.Recently, Shehu [19] introduced a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of k-strictly pseudocontractive mappings, the set of common solutions to a system of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in Hilbert spaces. Starting with an arbitrary x0∈C,C1,i=C,C1=⋂i=1∞C1,i, and x1=PC1x0 define sequence {xn},{wn},{un},{zn}, and {yn,i} as follows:
(1.11)zn=Trn(F1,φ1)(xn-rnAxn),yn=Tλn(F2,φ2)(zn-λnBzn),wn=PC(un-snDun),yn,i=αn,iwn+(1-αn,i)Tiwn,n≥1,Cn+1,i={z∈Cn,i:‖yn,i-z‖≤‖xn-z‖},n≥1,Cn+1=∩i=1∞Cn+1,i,xn+1=PCn+1x0,n≥1,
where Ti is a ki-strictly pseudocontractive mapping and for some 0≤ki<1, A,B is α,β-inverse-strongly monotone mapping of C into H. He proved that if the sequence {αn,i},{rn},{sn}, and {λn} of parameters satisfies appropriate conditions, then {xn} generated by (1.11) converges strongly to PΩx0.In this paper, motivated by the above results, we present a new general iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroup mappings, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. Then, we prove strong convergence theorem under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. The results presented in this paper extend and improve the results of Martinez-Yanes and Xu [21], Shehu [19], Zhang et al. [4], and many authors.


## 2. Preliminaries

LetH be a real Hilbert space with norm ∥·∥ and inner product 〈·,·〉, and let C be a closed-convex subset of H. When {xn} is a sequence in H, xn⇀x means that {xn} converges weakly to x, and xn→x means that {xn} converges strongly to x. In a real Hilbert space H, we have
(2.1)‖x-y‖2=‖x‖2-‖y‖2-2〈x-y,y〉,(2.2)‖λx+(1-λ)y‖2=λ‖x‖2+(1-λ)‖y‖2-λ(1-λ)‖x-y‖2,∀x,y∈H,
and λ∈ℝ. For every point x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(2.3)‖x-PCx‖≤‖x-y‖,∀y∈C.PC is called the metric projection of H onto C. It is well known that PC is a nonexpansive mapping of H onto C and satisfies(2.4)〈x-y,PCx-PCy〉≥‖PCx-PCy‖2,∀x,y∈H.
Moreover, PCx is characterized by the following properties: PCx∈C and
(2.5)〈x-PCx,y-PCx〉≤0,(2.6)‖x-y‖2≥‖x-PCx‖2+‖y-PCx‖2,∀x∈H,y∈C.
Recall that a mapping A of H into itself is called α-inverse-strongly monotone if there exists a positive real number α such that(2.7)〈Ax-Ay,x-y〉≥α‖Ax-Ay‖2,∀,y∈H.
It is obvious that any α-inverse-strongly monotone mapping A is (1/α)-Lipschitz monotone and continuous mapping.In order to prove our main results, we need the following Lemmas.Lemma 2.1 (see [22]).
LetV:C→H be a k-strict pseudocontraction, then
(1)
the fixed-point setF(V) of V is closed convex, so that the projection PF(V) is well defined;(2)
define a mappingT:C→H by(2.8)Tx=tx+(1-t)Vx,∀x∈C.
If t∈[k,1), then T is a nonexpansive mapping such that F(V)=F(T).A family of mappings{Vi:C→H}i=1∞ is called a family of uniformlyk-strict pseudocontractions if there exists a constant k∈[0,1) such that
(2.9)‖Vix-Viy‖2≤‖x-y‖2+k‖(I-Vi)x-(I-Vi)y‖2,∀x,y∈C,∀i≥1.
Let {Vi:C→C}i=1∞ be a countable family of uniformly k-strict pseudocontractions. Let {Ti:C→C}i=1∞ be the sequence of nonexpansive mappings defined by (2.8), that is,(2.10)Tix=tx+(1-t)Vix,∀x∈C,∀i≥1,t∈[k,1).Let{Ti} be a sequence of nonexpansive mappings of C into itself defined by (2.10), and let {μi} be a sequence of nonnegative numbers in [0,1]. For each n≥1, define a mapping Wn of C into itself as follows:(2.11)Un,n+1=I,Un,n=μnTnUn,n+1+(1-μn)I,Un,n-1=μn-1Tn-1Un,n+(1-μn-1)I,⋮Un,k=μkTkUn,k+1+(1-μk)I,Un,k-1=μk-1Tk-1Un,k+(1-μk-1)I,⋮Un,2=μ2T2Un,3+(1-μ2)I,Wn=Un,1=μ1T1Un,2+(1-μ1)I.
Such a mapping Wn is nonexpansive from C to C and it is called the W-mapping generated by T1,T2,…,Tn and μ1,μ2,…,μn.For eachn,k∈ℕ, let the mapping Un,k be defined by (2.11), then we can have the following crucial conclusions concerning Wn. You can find them in [23]. Now, we only need the following similar version in Hilbert spaces.Lemma 2.2 (see [23]).
LetC be a nonempty closed-convex subset of a real Hilbert space H. Let T1,T2,… be nonexpansive mappings of C into itself such that ⋂n=1∞F(Tn) is nonempty, and let μ1,μ2,… be real numbers such that 0≤μn≤b<1 for every n≥1, then
(1)
Wn is nonexpansive and F(Wn)=⋂i=1nF(Ti), ∀n≥1,(2)
for everyx∈C and k∈ℕ, the limit limn→∞Un,kx exists,(3)
a mappingW:C→C defined by(2.12)Wx:=lim⁡n→∞Wnx=lim⁡n→∞Un,1x,∀x∈C
is a nonexpansive mapping satisfying F(W)=⋂i=1∞F(Ti), and it is called the W-mapping generated by T1,T2,… and μ1,μ2,….Lemma 2.3 (see [24]).
LetC be a nonempty closed-convex subset of a Hilbert space H, let {Ti:C→C} be a countable family of nonexpansive mappings with ⋂i=1∞F(Ti)≠∅, and let {μi} be a real sequence such that 0<μi≤b<1,∀i≥1. If D is any bounded subset of C, then
(2.13)lim⁡n→∞sup⁡x∈D‖Wx-Wnx‖=0.Lemma 2.4 (see [25]).
Each Hilbert spaceH satisfies Opials condition, that is, for any sequence {xn}⊂H with xn⇀x, the inequality
(2.14)liminf⁡n→∞‖xn-x‖<liminf⁡n→∞‖xn-y‖
holds for each y∈H with y≠x.Lemma 2.5 (see [3]).
LetM:H→2H be a maximal monotone mapping, and let A:H→H be a monotone mapping, then the mapping S=M+A:H→2H is a maximal monotone mapping.Remark 2.6.
Lemma2.5 implies that I(A,M) is closed and convex if M:H→2H is a maximal monotone mapping and A:H→H is a monotone mapping.Lemma 2.7 (see [4]).
Letu∈H be a solution of variational inclusion (1.4) if and only if u=JM,λ(u-λAu),∀λ>0, that is,
(2.15)I(A,M)=F(JM,λ(I-λA)),∀λ>0.Lemma 2.8 (see [20]).
LetC be a nonempty bounded closed-convex subset of a Hilbert space H, and let 𝒮={S(s):0≤s<∞} be a nonexpansive semigroup on C, then for any h≥0,
(2.16)lim⁡t→∞sup⁡x∈C‖1t∫0tS(s)xds-S(h)(1t∫0tS(s)xds)‖=0.Lemma 2.9 (see [26]).
Let C be a nonempty bounded closed-convex subset of H, let{xn} be a sequence in C, and let 𝒮={S(s):0≤s<∞} be a nonexpansive semigroup on C. If the following conditions are satisfied:
(1)
xn⇀z,(2)
limsups→∞limsupn→∞∥S(s)xn-xn∥=0,thenz∈F(S).
For solving the generalized mixed equilibrium problem forF:C×C→ℝ, one gives the following assumptions for the bifunction F, φ and the set C:
(A1)
F(x,x)=0 for all x∈C,(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0 for all x,y∈C,(A3)
for eachx,y,z∈C,limt→0F(tz+(1-t)x,y)≤F(x,y),(A4)
for eachx∈C,y↦F(x,y) is convex and lower semicontinuous,(A5)
for eachy∈C,x↦F(x,y) is weakly upper semicontinuous,(B1)
for eachx∈H and r>0, there exist a bounded subset Dx⊆C and yx∈C such that for any z∈C∖Dx,
(2.17)F(z,yx)+φ(yx)+1r〈yx-z,z-x〉<φ(z),(B2)
C is a bounded set,
then one has the following lemma.Lemma 2.10 (see [18]).
Let C  be a nonempty closed-convex subset of H. LetF:C×C→ℝ be a bifunction that satisfies (A1)–(A5), and let φ:C→ℝ∪{+∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r>0 and x∈H, define a mapping Tr:H→C as follows:
(2.18)Tr(F,φ)(x)={z∈C:F(z,y)+φ(y)+1r〈y-z,z-x〉≥φ(z),∀y∈C},
for all z∈H, then the following hold:(1)
for eachx∈H,Tr(F,φ)(x)≠∅,(2)
Tr(F,φ) is single valued,(3)
Tr(F,φ) is firmly nonexpansive, that is, for any x,y∈H,∥Tr(F,φ)x-Tr(F,φ)y∥2≤〈Tr(F,φ)x-Tr(F,φ)y,x-y〉,(4)
F(Tr(F,φ))=MEP(F,φ),(5)
MEP(F,φ) is closed and convex.


## 3. Main Result

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroup mappings, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space.Theorem 3.1.
LetC be a nonempty closed-convex subset of a real Hilbert Space H. Let F1,F2 be bifunctions of C×C into real numbers ℝ satisfying (A1)–(A5), and let φ1,φ2:C→ℝ∪{+∞} be proper lower semicontinuous and convex functions with assumption (B1) or (B2). Let A,B,E1,E2 be α,β,η1,η2-inverse-strongly monotone mappings of C into H, respectively, and let M1,M2:H→2H be maximal monotone mappings. Let 𝒮={S(s):0≤s<∞} be a nonexpansive semigroup on C, and let {tn} be a positive real divergent sequence. Let {Vi:C→C}i=1∞ be a countable family of uniformly k-strict pseudocontractions, let {Ti:C→C}i=1∞ be a countable family of nonexpansive mappings defined by Tix=tx+(1-t)Vix,∀x∈C,∀i≥1,t∈[k,1), and let Wn be the W-mapping defined by (2.11) and W a mapping defined by (2.12) with F(W)≠∅. Suppose that Θ:=F(S)∩F(W)∩GMEP(F1,φ1,A)∩GMEP(F2,φ2,B)∩I(E1,M1)∩I(E2,M2)≠∅. Let {xn} be a sequence generated by x0∈C,C1,i=C,C1=⋂i=1∞C1,i,x1=PC1x0, and
(3.1)tn=Trn(F1,φ1)(xn-rnAxn),un=Tqn(F2,φ2)(tn-qnBtn),vn=JM1,λ1(un-λ1E1un),wn=JM2,λ2(vn-λ2E2vn),yn,i=αn,ix0+(1-αn,i)1tn∫0tnS(s)Wnwnds,Cn+1,i={z∈Cn,i:‖yn,i-z‖2≤‖xn-z‖2+αn,i(‖x0‖2+2〈xn-x0,z〉)},Cn+1=⋂i=1∞Cn+1,i,xn+1=PCn+1x0,
for every n≥0, where {αn,i}n=1∞⊂(0,1), {rn},{qn}⊂(0,∞),λ1∈(0,2η1), and λ2∈(0,2η2) satisfy the following conditions:(i)
0<a≤rn≤b<2α,(ii)
0<c≤qn≤d<2β,(iii)
limn→∞αn,i=0,(iv)
0<e≤λ1≤f<2η1,(v)
0<g≤λ2≤j<2η2,
then{xn} converges strongly to PΘx0.Proof.
First, we show thatI-λ1E1 and I-λ2E2 are nonexpansive. Indeed, for all x,y∈C and λ1∈(0,2η1), we obtain
(3.2)‖(I-λ1E1)x-(I-λ1E1)y‖2=‖x-y-λ1(E1x-E1y)‖2=‖x-y‖2-2λ1〈x-y,E1x-E1y〉+λ12‖E1x-E1y‖2≤‖x-y‖2-2λ1η1‖E1x-E1y‖2+λ12‖E1x-E1y‖2≤‖x-y‖2+λ1(λ1-2η1)‖E1x-E1y‖2≤‖x-y‖2,
which implies that the mapping I-λ1E1 is nonexpansive, so is I-λ2E2. Let p∈Θ. We observe that
(3.3)‖wn-p‖2=‖JM2,λ2(vn-λ2E2vn)-JM2,λ2(p-λ2E2p)‖2≤‖(vn-λ2E2vn)-(p-λ2E2p)‖2≤‖vn-p‖2=‖JM1,λ1(un-λ1E1un)-JM1,λ1(p-λ1E1p)‖2≤‖(un-λ1E1un)-(p-λ1E1p)‖2=‖un-p‖2.
Since both I-rnA and I-qnB are nonexpansive for each n≥1, let p∈Θ, then p=TrnF1,φ1(p-rnAp) and p=TqnF2,φ2(p-qnBp); by conditions (i) and (ii), we have
(3.4)‖un-p‖2=‖Tqn(F2,φ2)(I-qnB)tn-Tqn(F2,φ2)(I-qnB)p‖2≤‖(I-qnB)tn-(I-qnB)p‖2≤‖tn-p‖2+qn(qn-2β)‖Btn-Bp‖2≤‖tn-p‖2,‖tn-p‖2=‖Trn(F1,φ1)(I-rnA)xn-Trn(F1,φ1)(I-rnA)p‖2≤‖(I-rnA)xn-(I-rnA)p‖2≤‖xn-p‖2+rn(rn-2α)‖Axn-Ap‖2≤‖xn-p‖2.
Therefore, we get
(3.5)‖un-p‖≤‖xn-p‖.
Next, we will divide the proof into five steps.
Step 1.
We show that{xn} is well defined. Let n=1, then C1,i=C is closed and convex for each i≥1. Suppose that Cn,i is closed convex for some n>1, then, from the definition of Cn+1,i, we know that Cn+1,i is closed convex for the same n≥1. Hence, Cn,i is closed convex for n≥1 and for each i≥1. This implies that Cn is closed convex for n≥1. Furthermore, we show that Θ⊂Cn. For n=1,Θ⊂C=C1,i. For n≥2, let p∈Θ, then
(3.6)‖yn,i-p‖2=‖αn,i(x0-p)+(1-αn,i)(1tn∫0tnS(s)Wnwnds-p)‖2≤αn,i‖x0-p‖2+(1-αn,i)‖1tn∫0tnS(s)Wnwnds-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖wn-p‖2=‖wn-p‖2+αn,i(‖x0-p‖2-‖wn-p‖2)≤‖xn-p‖2+αn,i(‖x0‖2+2〈xn-x0,p〉),
which shows that p∈Cn,i,∀n≥2,∀i≥1. Thus, Θ⊂Cn,i,∀n≥1,∀i≥1. Hence, it follows that ∅≠Θ⊂Cn,∀n≥1. This implies that {xn} is well defined.Step 2.
We claim thatlimn→∞∥xn+1-xn∥=0 and limn→∞∥yn,i-xn∥=0, for i≥1. Since xn=PCnx0 and xn+1=PCn+1x0∈Cn+1⊂Cn,∀n≥1, we have
(3.7)‖xn-x0‖≤‖xn+1-x0‖,∀n≥1.
Also, as Θ⊂Cn by (2.1), it follows that
(3.8)‖xn-x0‖≤‖z-x0‖,z∈Θ,∀n≥1.
Form (3.7) and (3.8), we have that limn→∞∥xn-x0∥ exists. Hence, {xn} is bounded and so are {yn,i},∀i≥1, {wn}, {vn}, {un}, {tn}, {Axn}, {Btn}, {E1un}, {E2vn}, {Wnwn}, and {(1/tn)∫0tnS(s)Wnwnds}. For m>n≥1, we have that xm=PCmx0∈Cm⊂Cn. By (2.5), we obtain
(3.9)‖xm-xn‖2≤‖xn-x0‖2-‖xm-x0‖2.
Letting m,n→∞ and taking the limit in (3.9), we have ∥xm-xn∥→0, which shows that {xn} is Cauchy. In particular,
(3.10)lim⁡n→∞‖xn+1-xn‖=0.
Since {xn} is Cauchy, we assume that xn→z∈C. Since xn+1=PCn+1x0∈Cn+1, then
(3.11)‖yn,i-xn+1‖2≤‖xn-xn+1‖2+αn,i(‖x0‖2+2〈xn-x0,xn+1〉)⟶0,
and it follows that
(3.12)‖yn,i-xn‖≤‖yn,i-xn+1‖+‖xn+1-xn‖.
Therefore,
(3.13)lim⁡n→∞‖yn,i-xn‖=0,∀i≥1.Step 3.
We claim that the following statements hold:(1)
limn→∞∥un-tn∥=0,(2)
limn→∞∥tn-xn∥=0,(3)
limn→∞∥wn-vn∥=0,(4)
limn→∞∥vn-un∥=0.
Forp∈Θ, from (3.4), and (3.6), we obtain(3.14)‖yn,i-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖wn-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖un-p‖2≤αn,i‖x0-p‖2+(1-αn,i)(‖tn-p‖2+qn(qn-2β)‖Btn-Bp‖2)≤αn,i‖x0-p‖2+‖xn-p‖2+(1-αn,i)qn(qn-2β)‖Btn-Bp‖2.
Since 0<c≤qn≤d<2β, we have
(3.15)(1-αn,i)qn(2β-qn)‖Btn-Bp‖2≤αn,i‖x0-p‖2+‖xn-p‖2-‖yn,i-p‖2≤αn,i‖x0-p‖2+‖xn-yn,i‖(‖xn-p‖+‖yn,i-p‖).
Hence, by condition (iii) and (3.13), we have
(3.16)lim⁡n→∞‖Btn-Bp‖=0.
From (3.6), we have
(3.17)‖yn,i-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖wn-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖un-p‖2.
On the other hand,
(3.18)‖un-p‖2=‖Tqn(F2,φ2)(tn-qnBtn)-Tqn(F2,φ2)(p-qnBp)‖2≤〈(tn-qnBtn)-(p-qnBp),un-p〉=12{‖(tn-qnBtn)-(p-qnBp)‖2+‖un-p‖2-‖(tn-qnBtn)-(p-qnBp)-(un-p)‖2}≤12{‖tn-p‖2+‖un-p‖2-‖(tn-qnBtn)-(p-qnBp)-(un-p)‖2}≤12{‖xn-p‖2+‖un-p‖2-‖un-tn‖2+2qn〈tn-un,Btn-Bp〉-sn2‖Btn-Bp‖2},
and hence,
(3.19)‖un-p‖2≤‖xn-p‖2-‖un-tn‖2+2qn‖tn-un‖‖Btn-Bp‖.
Putting (3.19) into (3.17), for i≥1, we have
(3.20)‖yn,i-p‖2≤αn,i‖x0-p‖2+(1-αn,i)(‖xn-p‖2-‖un-tn‖2+2qn‖tn-un‖‖Btn-Bp‖)≤αn,i‖x0-p‖2+‖xn-p‖2-(1-αn,i)‖un-tn‖2+2qn‖tn-un‖‖Btn-Bp‖.
It follows that
(3.21)(1-αn,i)‖un-tn‖2≤αn,i‖x0-p‖2+‖xn-p‖2-‖yn,i-p‖2+2qn‖tn-un‖‖Btn-Bp‖≤αn,i‖x0-p‖2+‖xn-yn,i‖(‖xn-p‖+‖yn,i-p‖)+2qn‖tn-un‖‖Btn-Bp‖.
Therefore, from condition (iii), (3.13), and (3.16), we have
(3.22)lim⁡n→∞‖un-tn‖=0.
Furthermore, from (3.4), and (3.6), we get
(3.23)‖yn,i-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖wn-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖un-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖tn-p‖2≤αn,i‖x0-p‖2+(1-αn,i){‖xn-p‖2+rn(rn-2α)‖Axn-Ap‖2}≤αn,i‖x0-p‖2+‖xn-p‖2+(1-αn,i)rn(rn-2α)‖Axn-Ap‖2.
Since 0<a≤rn≤b<2α, we have
(3.24)(1-αn,i)rn(2α-rn)‖Axn-Ap‖2≤αn,i‖x0-p‖2+‖xn-p‖2-‖yn,i-p‖2≤αn,i‖x0-p‖2+‖xn-yn,i‖(‖xn-p‖+‖yn,i-p‖).
Then, by condition (iii) and (3.13), we obtain that
(3.25)lim⁡n→∞‖Axn-Ap‖=0.
From (3.6), we have
(3.26)‖yn,i-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖wn-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖un-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖tn-p‖2.
On the other hand, we note that
(3.27)‖tn-p‖2≤‖Trn(F1,φ1)(xn-rnAxn)-Trn(F1,φ1)(p-rnAp)‖2≤〈(xn-rnAxn)-(p-rnAp),tn-p〉=12{‖(xn-rnAxn)-(p-rnAp)‖2+‖tn-p‖2-‖(xn-rnAxn)-(p-rnAp)-(tn-p)‖2}≤12{‖xn-p‖2+‖tn-p‖2-‖(xn-rnAxn)-(p-rnAp)-(tn-p)‖2}=12{‖xn-p‖2+‖tn-p‖2-‖tn-xn‖2+2rn〈xn-tn,Axn-Ap〉-rn2‖Axn-Ap‖2},
and hence,
(3.28)‖tn-p‖2≤‖xn-p‖2-‖tn-xn‖2+2rn‖xn-tn‖‖Axn-Ap‖.
Putting (3.28) into (3.26), we have
(3.29)‖yn,i-p‖2≤αn,i‖x0-p‖2+(1-αn,i)(‖xn-p‖2-‖tn-xn‖2+2rn‖xn-tn‖‖Axn-Ap‖)≤αn,i‖x0-p‖2+‖xn-p‖2-(1-αn,i)‖tn-xn‖2+2rn‖xn-tn‖‖Axn-Ap‖.
It follows that
(3.30)(1-αn,i)‖xn-tn‖2≤αn,i‖x0-p‖2+‖xn-p‖2-‖yn,i-p‖2+2rn‖xn-zn‖‖Axn-Ap‖≤αn,i‖x0-p‖2+‖xn-yn,i‖(‖xn-p‖+‖yn,i-p‖)+2rn‖xn-zn‖‖Axn-Ap‖.
Therefore, by condition (iii), (3.13), and (3.25), we have
(3.31)lim⁡n→∞‖xn-tn‖=0.
Condition (iii) implies that
(3.32)‖yn,i-1tn∫0tnS(s)Wnwnds‖2=αn,i‖x0-1tn∫0tnS(s)Wnwnds‖⟶0.
It follows that
(3.33)‖xn-1tn∫0tnS(s)Wnwnds‖≤‖xn-yn,i‖+‖yn,i+1tn∫0tnS(s)Wnwnds‖⟶0.
From (3.6), we have
(3.34)‖yn,i-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖wn-p‖2=αn,i‖x0-p‖2+(1-αn,i)‖JM2,λ2(vn-λ2E2vn)-JM2,λ2(p-λ2E2p)‖2≤αn,i‖x0-p‖2+(1-αn,i)‖(vn-λ2E2vn)-(p-λ2E2p)‖2≤αn,i‖x0-p‖2+(1-αn,i)(‖vn-p‖2+λ2(λ2-2η2)‖E2vn-E2p‖2)≤αn,i‖x0-p‖2+‖xn-p‖2+(1-αn,i)λ2(λ2-2η2)‖E2vn-E2p‖2.
Since 0<g≤λ2≤j<2η2, we have
(3.35)(1-αn,i)λ2(2η2-λ2)‖E2vn-E2p‖2≤αn,i‖x0-p‖2+‖xn-p‖2-‖yn,i-p‖2≤αn,i‖x0-p‖2+‖xn-yn,i‖(‖xn-p‖+‖yn,i-p‖).
Then, by condition (iii) and (3.13), we obtain that
(3.36)lim⁡n→∞‖E2vn-E2p‖=0.
From (3.6), we have
(3.37)‖yn,i-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖wn-p‖2.
On the other hand, we note that
(3.38)‖wn-p‖2≤‖JM2,λ2(vn-λ2E2vn)-JM2,λ2(p-λ2E2p)‖2≤〈(vn-λ2E2vn)-(p-λ2E2p),wn-p〉=12{‖(vn-λ2E2vn)-(p-λ2E2p)‖2+‖wn-p‖2-‖(vn-λ2E2vn)-(p-λ2E2p)-(wn-p)‖2}≤12{‖vn-p‖2+‖wn-p‖2-‖(vn-λ2E2vn)-(p-λ2E2p)-(wn-p)‖2}≤12{‖xn-p‖2+‖wn-p‖2-‖wn-vn‖2+2λ2〈vn-wn,E2vn-E2p〉-λ22‖E2vn-E2p‖2},
and hence,
(3.39)‖wn-p‖2≤‖xn-p‖2-‖wn-vn‖2+2λ2‖vn-wn‖‖E2vn-E2p‖.
Putting (3.39) into (3.37),
(3.40)‖yn,i-p‖2≤αn,i‖x0-p‖2+(1-αn,i)(‖xn-p‖2-‖wn-vn‖2+2λ2‖vn-wn‖‖E2vn-E2p‖)≤αn,i‖x0-p‖2+‖xn-p‖2-(1-αn,i)‖wn-vn‖2+2λ2‖vn-wn‖‖E2vn-E2p‖,
this implies that
(3.41)(1-αn,i)‖wn-vn‖2≤αn,i‖x0-p‖2+‖xn-p‖2-‖yn,i-p‖2+2λ2‖wn-vn‖‖E2vn-E2p‖≤αn,i‖x0-p‖2+‖xn-yn,i‖(‖xn-p‖+‖yn,i-p‖)+2λ2‖wn-vn‖‖E2vn-E2p‖.
Therefore, by condition (iii), (3.13), and (3.36), we have
(3.42)lim⁡n→∞‖wn-vn‖=0.
Furthermore, from (3.6), we have
(3.43)‖yn,i-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖wn-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖vn-p‖2=αn,i‖x0-p‖2+(1-αn,i)‖JM1,λ1(un-λ1E1un)-JM1,λ1(p-λ1E1p)‖2≤αn,i‖x0-p‖2+(1-αn,i)‖(un-λ1E1un)-(p-λ1E1p)‖2≤αn,i‖x0-p‖2+(1-αn,i)(‖un-p‖2+λ1(λ1-2η1)‖E1un-E1p‖2)≤αn,i‖x0-p‖2+‖xn-p‖2+(1-αn,i)λ1(λ1-2η1)‖E1un-E1p‖2.
Since 0<e≤λ1≤f<2η1, we have
(3.44)(1-αn,i)λ1(2η1-λ1)‖E1un-E1p‖2≤αn,i‖x0-p‖2+‖xn-p‖2-‖yn,i-p‖2≤αn,i‖x0-p‖2+‖xn-yn,i‖(‖xn-p‖+‖yn,i-p‖).
Then, by condition (iii) and (3.13), we obtain that
(3.45)lim⁡n→∞‖E1un-E1p‖=0.
From (3.6), we have
(3.46)‖yn,i-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖wn-p‖2≤αn,i‖x0-p‖2+(1-αn,i)‖vn-p‖2.
On the other hand, we note that
(3.47)‖vn-p‖2≤‖JM1,λ1(un-λ1E1un)-JM1,λ1(p-λ1E1p)‖2≤〈(un-λ1E1un)-(p-λ1E1p),vn-p〉=12{‖(un-λ1E1un)-(p-λ1E1p)‖2+‖vn-p‖2-‖(un-λ1E1un)-(p-λ1E1p)-(vn-p)‖2}≤12{‖un-p‖2+‖vn-p‖2-‖(un-λ1E1un)-(p-λ1E1p)-(vn-p)‖2}≤12{‖xn-p‖2+‖vn-p‖2-‖vn-un‖2+2λ1〈un-vn,E1un-E1p〉-λ12‖E1un-E1p‖2},
and hence,
(3.48)‖vn-p‖2≤‖xn-p‖2-‖vn-un‖2+2λ1‖un-vn‖‖E1un-E1p‖.
Putting (3.48) into (3.46),
(3.49)‖yn,i-p‖2≤αn,i‖x0-p‖2+(1-αn,i)(‖xn-p‖2-‖vn-un‖2+2λ1‖un-vn‖‖E1un-E1p‖)≤αn,i‖x0-p‖2+‖xn-p‖2-(1-αn,i)‖vn-un‖2+2λ1‖un-vn‖‖E1un-E1p‖,
this implies that
(3.50)(1-αn,i)‖vn-un‖2≤αn,i‖x0-p‖2+‖xn-p‖2-‖yn,i-p‖2+2λ1‖vn-un‖‖E1un-E1p‖≤αn,i‖x0-p‖2+‖xn-yn,i‖(‖xn-p‖+‖yn,i-p‖)+2λ1‖vn-un‖‖E1un-E1p‖.
Therefore, by condition (iii), (3.13), and (3.45), we have
(3.51)lim⁡n→∞‖vn-un‖=0.Step 4.
We show thatz∈Θ:=F(𝒮)∩F(W)∩GMEP(F1,φ1,A)∩GMEP(F2,φ2,B)∩I(E1,M1)∩I(E2,M2). Since {wni} is bounded, there exists a subsequence {wnij} of {wni} which converges weakly to z∈C. Without loss of generality, we can assume that wni⇀z.
(1) First, we prove thatz∈F(𝒮). From (3.22), (3.31), (3.33), (3.42), and (3.51), we get(3.52)lim⁡n→∞‖wn-1tn∫0tnS(s)Wnwnds‖=0.
Since {Wnwn} is bounded and from Lemma 2.8 for all s≥0, we have
(3.53)lim⁡n→∞‖1tn∫0tnS(s)Wnwnds-S(s)(1tn∫0tnS(s)Wnwnds)‖=0,
and since
(3.54)‖wn-S(s)wn‖≤‖wn-1tn∫0tnS(s)Wnwnds‖+‖1tn∫0tnS(s)Wnwnds-S(s)1tn∫0tnS(s)Wnwnds‖+‖S(s)1tn∫0tnS(s)Wnwnds-S(s)wn‖≤2‖wn-1tn∫0tnS(s)Wnwnds‖+‖1tn∫0tnS(s)Wnwnds-S(s)1tn∫0tnS(s)Wnwnds‖.
It follows from (3.52) and (3.53) that
(3.55)lim⁡n→∞‖wn-S(s)wn‖=0.
Indeed, from Lemma 2.9 and (3.55), we get z∈F(𝒮), that is, z=S(s)z,∀s≥0.
(2) Next, we show thatz∈F(W)=⋂n=1∞F(Wn), where F(Wn)=⋂i=1nF(Ti),∀n≥1, and F(Wn+1)⊂F(Wn). Assume that z∉F(W), then there exists a positive integer m such that z∉F(Tm), and so z∉⋂i=1mF(Ti). Hence, for any n≥m,z∉⋂i=1nF(Ti)=F(Wn), that is, z≠Wnz. This together with z=S(s)z,∀s≥0 shows that z=S(s)z≠S(s)Wnz,∀s≥0; therefore, we have z≠(1/tn)∫0tnS(s)Wnzds,∀n≥m. It follows from the Opial's condition and (3.52) that
(3.56)liminf⁡i→∞‖wni-z‖<liminf⁡i→∞‖wni-1tni∫0tniS(s)Wnizds‖≤liminf⁡i→∞(‖wni-1tni∫0tniS(s)Wniwnids‖+‖1tni∫0tniS(s)Wniwnids-1tni∫0tniS(s)Wnizds‖)≤liminf⁡i→∞‖wni-z‖,
which is a contradiction. Thus, we get z∈F(W).
(3) Now, we prove thatz∈GMEP(F1,φ,A). Since tn=Trn(F1,φ1)(xn-rnAxn),n≥1, we have for any y∈C that
(3.57)F1(tn,y)+φ1(y)-φ1(tn)+〈Axn,y-tn〉+1rn〈y-tn,tn-xn〉≥0,∀y∈C.
From (A2), we also have
(3.58)φ1(y)-φ1(tn)+〈Axn,y-tn〉+1rn〈y-tn,tn-xn〉≥F1(y,tn),∀y∈C.
For t with 0<t≤1 and y∈C, let yt=ty+(1-t)z. Since y∈C and z∈C, we have yt∈C. Then, we have
(3.59)〈yt-tni,Ayt〉≥〈yt-tni,Ayt〉-φ1(yt)+φ1(tni)-〈yt-tni,Axni〉-〈yt-tni,tni-xnirnitni-xnirni〉+F1(yt,tni)=〈yt-tni,Ayt-Atni〉+〈yt-tni,Atni-Axni〉-φ1(yt)+φ1(tni)-〈yt-tni,tni-xnirnitni-xnirni〉+F1(yt,tni).
Since ∥tni-xni∥→0, we have ∥Atni-Axni∥→0. Furthermore, from the inverse-strongly monotonicity of A, we have 〈yt-tni,Ayt-Atni〉≥0. So, from (A4), (A5), and the weak lower semicontinuity of φ1,(tni-xni)/rni→0 and tni⇀z, we have at the limit
(3.60)〈yt-z,Ayt〉≥-φ1(yt)+φ1(z)+F1(yt,z),
as i→∞. From (A1), (A4), and (3.60), we also get
(3.61)0=F1(yt,yt)+φ1(yt)-φ1(yt)≤tF1(yt,y)+(1-t)F1(yt,z)+tφ1(y)-(1-t)φ1(z)-φ(yt)=t[F1(yt,y)+φ1(y)-φ1(yt)]+(1-t)[F1(yt,z)+φ1(z)-φ1(yt)]≤t[F1(yt,y)+φ1(y)-φ1(yt)]+(1-t)〈yt-z,Ayt〉=t[F1(yt,y)+φ1(y)-φ1(yt)]+(1-t)t〈y-z,Ayt〉,
and hence,
(3.62)0≤F1(yt,y)+φ1(y)-φ1(yt)+(1-t)〈y-z,Ayt〉.
Letting t→0, we have, for each y∈C,
(3.63)F1(z,y)+φ1(y)-φ1(z)+〈y-z,Az〉≥0.
This implies that z∈GMEP(F1,φ,A). By following the same arguments, we can show that z∈GMEP(F2,φ,B).
(4) At last, we show thatz∈I(E2,M2). Infact, since E2 is η2-inverse-strongly monotone, this implies that E2 is(1/η2)-Lipschitz continuous monotone mapping and domain of E2 equal to H. It follows from Lemma 2.5 that M2+E2 is a maximal monotone. Let (y,g)∈G(M2+E2), that is, g-E2y∈M2(y). Since wni=JM2,λ2(vni-λ2E2vni), we have vni-λ2E2vni∈(I+λ2M2)(wni), that is,
(3.64)1λ2(vni-wni-λ2E2vni)∈M2(wni).
Since M2+E2 is a maximal monotone, we have
(3.65)〈y-wni,g-By-1λ2(vni-wni-λ2E2vni)〉≥0,
and so
(3.66)〈y-wni,g〉≥〈y-wni,E2y+1λ2(vni-wni-λ2E2vni)〉=〈y-wni,E2y-E2wni+E2wni-E2vni+1λ2(vni-wni)〉≥0+〈y-wni,E2wni-E2vni〉+〈y-wni,1λ2(vni-wni)〉.
It follows from ∥vn-wn∥→0, ∥E2vn-E2wn∥→0, and wni⇀z that
(3.67)lim⁡i→∞〈y-wni,g〉=〈y-w,g〉≥0.
It follows from the maximal monotonicity of M2+E2 that 0∈(M2+E2)(z), that is, z∈I(E2,M2). By following the same arguments, we can show that z∈I(E1,M1). Hence, by (1)–(4), we have z∈Θ.Step 5.
Noting thatxn=PCnx0, by (2.5), we have
(3.68)〈x0-xn,y-xn〉≤0,∀y∈Cn.
Since Θ⊂Cn and by the continuity of inner product, we obtain from the above inequality that
(3.69)〈x0-z,y-z〉≤0,∀y∈C.
By (2.5) again, we conclude that z=PΘx0. This completes the proof.Using Theorem3.1, we obtain the following corollaries.Corollary 3.2.
LetC be a nonempty closed-convex subset of a real Hilbert Space H. Let F1,F2 be bifunctions of C×C into real numbers ℝ satisfying (A1)–(A5), and let φ1,φ2:C→ℝ∪{+∞} be proper lower semicontinuous and convex functions with assumption (B1) or (B2). Let A,B,E1,E2 be α,β,η1,η2-inverse-strongly monotone mappings of C into H, respectively. Let 𝒮={S(s):0≤s<∞} be a nonexpansive semigroup on C, and let {tn} be a positive real divergent sequence. Let {Vi:C→C}i=1∞ be a countable family of uniformly k-strict pseudocontractions, let {Ti:C→C}i=1∞ be a countable family of nonexpansive mappings defined by Tix=tx+(1-t)Vix,∀x∈C,∀i≥1,t∈[k,1), and let Wn be the W-mapping defined by (2.11) and W a mapping defined by (2.12) with F(W)≠∅. Suppose that Θ:=F(𝒮)∩F(W)∩GMEP(F1,φ1,A)∩GMEP(F2,φ2,B)∩VI(C,E1)∩VI(C,E2)≠∅. Let {xn} be a sequence generated by x0∈C,C1,i=C,C1=⋂i=1∞C1,i,x1=PC1x0, and
(3.70)tn=Trn(F1,φ1)(xn-rnAxn),un=Tqn(F2,φ2)(tn-qnBtn),vn=PC(un-λ1E1un),wn=PC(vn-λ2E2vn),yn,i=αn,ix0+(1-αn,i)1tn∫0tnS(s)Wnwnds,Cn+1,i={z∈Cn,i:‖yn,i-z‖2≤‖xn-z‖2+αn,i(‖x0‖2+2〈xn-x0,z〉)},Cn+1=⋂i=1∞Cn+1,i,xn+1=PCn+1x0,
for every n≥0, where {αn,i}n=1∞⊂(0,1), {rn},{qn}⊂(0,∞),λ1∈(0,2η1), and λ2∈(0,2η2) satisfy the following conditions:(i)
0<a≤rn≤b<2α,(ii)
0<c≤qn≤d<2β,(iii)
limn→∞αn,i=0,(iv)
0<e≤λ1≤f<2η1,(v)
0<g≤λ2≤j<2η2,
then{xn} converges strongly to PΘx0.Proof.
From Theorem3.1, put M=∂δC, then JM,λ1=PC and JM,λ2=PC. So we have vn=PC(un-λ1E1un) and wn=PC(vn-λ2E2vn). The conclusion of Corollary 3.2 can be obtained from Theorem 3.1 immediately.


## 4. Applications

In this section, we study a kind of multiobjective optimization problem by using the result of this paper. We will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions:(4.1)min⁡⁡h1(x)min⁡⁡h2(x)x∈C,
where h(x) is a convex and lower semicontinuous functional, and define C as a closed-convex subset of a real Hilbert space H. We denote the set of solutions of (4.1) by M(h1) and M(h2). Let Fi:C×C→ℝ be a bifunction defined by Fi(x,y)=hi(y)-hi(x). We consider the equilibrium problem, and it is obvious that EP(Fi)=M(hi),i=1,2. Therefore, from Theorem 3.1, we obtain the following Corollaries.Corollary 4.1.
LetC be a nonempty closed-convex subset of a real Hilbert Space H. Let h1,h2:C→ℝ∪{+∞} be proper lower semicontinuous and convex functions. Let E1,E2 be η1,η2-inverse-strongly monotone mappings of C into H, respectively, and let M1,M2:H→2H be maximal monotone mappings. Let 𝒮={S(s):0≤s<∞} be a nonexpansive semigroup on C, and let {tn} be a positive real divergent sequence. Let {Vi:C→C}i=1∞ be a countable family of uniformly k-strict pseudocontractions, let {Ti:C→C}i=1∞ be a countable family of nonexpansive mappings defined by Tix=tx+(1-t)Vix,∀x∈C,∀i≥1,t∈[k,1), and let Wn be the W-mapping defined by (2.11) and W a mapping defined by (2.12) with F(W)≠∅. Suppose that Θ:=F(𝒮)∩F(W)∩M(h1)∩M(h2)∩I(E1,M1)∩I(E2,M2)≠∅. Let {xn} be a sequence generated by x0∈C,C1,i=C,C1=⋂i=1∞C1,i,x1=PC1x0, and
(4.2)h1(t)-h1(tn)+1rn〈t-tn,tn-xn〉≥0,∀t∈C,h2(u)-h2(un)+1qn〈u-un,un-tn〉≥0,∀u∈C,vn=JM1,λ1(un-λ1E1un),wn=JM2,λ2(vn-λ2E2vn),yn,i=αn,ix0+(1-αn,i)1tn∫0tnS(s)Wnwnds,Cn+1,i={z∈Cn,i:‖yn,i-z‖2≤‖xn-z‖2+αn,i(‖x0‖2+2〈xn-x0,z〉)},Cn+1=⋂i=1∞Cn+1,i,xn+1=PCn+1x0,
for every n≥0, where {αn,i}n=1∞⊂(0,1), {rn},{qn}⊂(0,∞),λ1∈(0,2η1), and λ2∈(0,2η2) satisfy the following conditions:(i)
liminfn→∞rn>0,(ii)
liminfn→∞qn>0,(iii)
limn→∞αn,i=0,(iv)
0<e≤λ1≤f<2η1,(v)
0<g≤λ2≤j<2η2,then {xn} converges strongly to PΘx0.Proof.
From Theorem3.1, put F1(tn,t)=h1(t)-h1(tn),F2(un,u)=h2(u)-h2(un), and A,B,φ1,φ2≡0. The conclusion of Corollary 4.1 can be obtained from Theorem 3.1 immediately.Corollary 4.2.
LetC be a nonempty closed-convex subset of a real Hilbert Space H. Let h1,h2:C→ℝ∪{+∞} be proper lower semicontinuous and convex functions. Let E1,E2 be η1,η2-inverse-strongly monotone mappings of C into H, respectively, and let M1,M2:H→2H be maximal monotone mappings. Let 𝒮={S(s):0≤s<∞} be a nonexpansive semigroup on C, and let {tn} be a positive real divergent sequence. Let {Vi:C→C}i=1∞ be a countable family of uniformly k-strict pseudocontractions, let {Ti:C→C}i=1∞ be a countable family of nonexpansive mappings defined by Tix=tx+(1-t)Vix,∀x∈C,∀i≥1,t∈[k,1), and let Wn be the W-mapping defined by (2.11) and W a mapping defined by (2.12) with F(W)≠∅. Suppose that Θ:=F(𝒮)∩F(W)∩M(h1)∩M(h2)∩VI(C,E1)∩VI(C,E2)≠∅. Let {xn} be a sequence generated by x0∈C,C1,i=C,C1=⋂i=1∞C1,i,x1=PC1x0, and
(4.3)h1(t)-h1(tn)+1rn〈t-tn,tn-xn〉≥0,∀t∈C,h2(u)-h2(un)+1qn〈u-un,un-tn〉≥0,∀u∈C,vn=PC(un-λ1E1un),wn=PC(vn-λ2E2vn),yn,i=αn,ix0+(1-αn,i)1tn∫0tnS(s)Wnwnds,Cn+1,i={z∈Cn,i:‖yn,i-z‖2≤‖xn-z‖2+αn,i(‖x0‖2+2〈xn-x0,z〉)},Cn+1=⋂i=1∞Cn+1,i,xn+1=PCn+1x0,
for every n≥0, where {αn,i}n=1∞⊂(0,1), {rn},{qn}⊂(0,∞),λ1∈(0,2η1), and λ2∈(0,2η2) satisfy the following conditions:
(i)
liminfn→∞rn>0,(ii)
liminfn→∞qn>0,(iii)
limn→∞αn,i=0,(iv)
0<e≤λ1≤f<2η1,(v)
0<g≤λ2≤j<2η2,
then {xn} converges strongly to PΘx0.



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*Source: 953903-2011-09-11.xml*