# Convergence Theorems for Equilibrium Problems and Fixed-Point Problems of an Infinite Family ofki-Strictly Pseudocontractive Mapping in Hilbert Spaces

Authors: Haitao Che; Meixia Li; Xintian Pan
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We first extend the definition of Wn from an infinite family of nonexpansive mappings to an infinite family of strictly pseudocontractive mappings, and then propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of an infinite family of ki-strictly pseudocontractive mappings in Hilbert spaces. The results obtained in this paper extend and improve the recent ones announced by many others. Furthermore, a numerical example is presented to illustrate the effectiveness of the proposed scheme.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and induced norm ∥·∥. Let C be a nonempty closed convex subset of H and let F:C×C→R be a bifunction. We consider the following equilibrium problem (EP) which is to find z∈C such that
(1.1)EP⁡:F(z,y)≥0,∀y∈C.
Denote the set of solutions of EP by EP(F). Given a mapping T:C→H, let F(x,y)=〈Tx,y-x〉 for all x,y∈C. Then, z∈EP(F) if and only if 〈Tx,y-x〉≥0 for all y∈C, that is, z is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem [1–13].A mappingB:C→C is called θ-Lipschitzian if there exists a positive constant θ such that
(1.2)‖Bx-By‖≤θ‖x-y‖,∀x,y∈C.B is said to be η-strongly monotone if there exists a positive constant η such that
(1.3)〈Bx-By,x-y〉≥η‖x-y‖2,∀x,y∈C.
A mapping S:C→C is said to be k-strictly pseudocontractive mapping if there exists a constant 0≤k<1 such that
(1.4)‖Sx-Sy‖2≤‖x-y‖2+k‖(I-S)x-(I-S)y‖2,
for all x,y∈C and F(S) denotes the set of fixed point of the mapping S, that is F(S)={x∈C:Sx=x}.Ifk=1, then S is said to a pseudocontractive mapping, that is,
(1.5)‖Sx-Sy‖2≤‖x-y‖2+‖(I-S)x-(I-S)y‖2,
is equivalent to
(1.6)〈(I-S)x-(I-S)y,x-y〉≥0,
for all x,y∈C.The class ofk-strict pseudo-contractive mappings extends the class of nonexpansive mappings (A mapping T is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥, for all x,y∈C). That is, S is nonexpansive if and only if S is a 0-strict pseudocontractive mapping. Clearly, the class of k-strictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudo-contractive mapping.In 2006, Marino and Xu [14] introduced the general iterative method and proved that for a given x0∈H, the sequence {xn} generated by the algorithm
(1.7)xn+1=αnγf(xn)+(I-αnB)Txn,n∈N,
where T is a self-nonexpansive mapping on H, f is an α-contraction of H into itself (i.e., ∥f(x)-f(y)∥≤α∥x-y∥, for all x,y∈H and α∈(0,1)), {αn}⊂(0,1) satisfies certain conditions, B is strongly positive bounded linear operator on H, and converges strongly to fixed point x* of T which is the unique solution to the following variational inequality:
(1.8)〈(γf-B)x*,x*-x〉≤0,∀x∈F(T).Tian [15] considered the following iterative method, for a nonexpansive mapping T:H→H with F(T)≠∅,
(1.9)xn+1=αnγf(xn)+(I-μαnF)Txn,n∈N,
where F is k-Lipschitzian and η-strongly monotone operator. The sequence {xn} converges strongly to fixed-point q in F(T) which is the unique solution to the following variational inequality:
(1.10)〈(γf-μF)q,p-q〉≤0,p∈F(T).For finding a common element ofEP(F)∩F(S), S. Takahashi and W. Takahashi [16] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let S:C→H be a nonexpansive mapping. Starting with arbitrary initial point x1∈H, define sequences {xn} and {un} recursively by
(1.11)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnf(xn)+(1-αn)Sun,∀n∈N.
They proved that under certain appropriate conditions imposed on {αn} and {rn}, the sequences {xn} and {un} converge strongly to z∈F(S)∩EP(F), where z=PF(S)∩EP(F)f(z).Liu [17] introduced the following scheme: x1∈H and
(1.12)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=βnun+(1-βn)Sun,xn+1=αnγf(xn)+(I-αnB)yn,∀n∈N,
where S is a k-strict pseudo-contractive mapping and B is a strongly positive bounded linear operator. They proved that under certain appropriate conditions imposed on {αn},{βn}, and {rn}, the sequence {xn} converges strongly to z∈F(S)∩EP(F), where z=PF(S)∩EP(F)(I-B+γf)(z).In [18], the concept of W mapping had been modified for a countable family {Tn}n∈N of nonexpansive mappings by defining the sequence {Wn}n∈N of W-mappings generated by {Tn}n∈N and {λn}⊂(0,1), proceeding backward
(1.13)Un,n+1∶=I,Un,n∶=λnTnUn,n+1+(1-λn)I,⋅⋅⋅Un,k∶=λkTkUn,k+1+(1-λk)I,⋅⋅⋅Un,2∶=λ2T2Un,3+(1-λ2)I,Wn=Un,1∶=λ1T1Un,2+(1-λ1)I.
Yao et al. [19] using this concept, introduced the following algorithm: x1∈H and
(1.14)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnf(xn)+βnxn+(1-αn-βn)Wnun,∀n∈N.
They proved that under certain appropriate conditions imposed on {αn} and {rn}, the sequences {xn} and {un} converge strongly to z∈⋂i=1∞F(Ti)∩EP(F).Colao and Marino [20] considered the following explicit viscosity scheme
(1.15)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnrf(xn)+βnxn+((1-βn)I-αnA)Wnun,∀n∈N,
where A is a strongly positive operator on H. Under certain appropriate conditions, the sequences {xn} and {un} converge strongly to z∈⋂i=1∞F(Ti)∩EP(F).Motivated and inspired by these facts, in this paper, we first extend the definition ofWn from an infinite family of nonexpansive mappings to an infinite family of strictly pseudo-contractive mappings, and then propose the iteration scheme (3.2) for finding an element of EP(F)⋂i=1∞F(Si), where {Si} is an infinite family of ki-strictly pseudo-contractive mappings of C into itself. Finally, the convergence theorem of the iteration scheme is obtained. Our results include Yao et al. [19], Colao and Marino [20] as some special cases.


## 2.  Preliminaries

Throughout this paper, we always assume thatC is a nonempty closed convex subset of a Hilbert space H. We write xn⇀x to indicate that the sequence {xn} converges weakly to x. xn→x implies that {xn} converges strongly to x. We denote by N and R the sets of positive integers and real numbers, respectively. For any x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(2.1)‖x-PCx‖≤‖x-y‖,∀y∈C.
Such a PC is called the metric projection of H onto C. It is known that PC is nonexpansive. Furthermore, for x∈H and u∈C,
(2.2)u=PCx⟺〈x-u,u-y〉≥0,∀y∈C.
It is widely known that H satisfies Opial′s condition [21], that is, for any sequence {xn} with xn⇀x, the inequality
(2.3)lim⁡n→∞inf⁡‖xn-x‖<lim⁡n→∞inf⁡‖xn-y‖
holds for every y∈H with y≠x.In order to solve the equilibrium problem for a bifunctionF:C×C→R, we assume that F satisfies the following conditions:(A1)
F(x,x)=0,for all x∈C.(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0,  for all x,y∈C.(A3)
limt↓0F(tz+(1-t)x,y)≤F(x,y), for all x,y,z∈C.(A4)
For eachx∈C,y↦F(x,y) is convex and lower semicontinuous.Let us recall the following lemmas which will be useful for our paper.Lemma 2.12.1 (see [22]).
LetF be a bifunction from C×C into R satisfying (A1), (A2), (A3), and (A4). Then, for any r>0 and x∈H, there exists z∈C such that
(2.4)F(z,y)+1r(y-z,z-x)≥0,∀y∈C.
Furthermore, if Trx={z∈C:F(z,y)+(1/r)(y-z,z-x)≥0,∀y∈C}, then the following hold:(1)
Tr is single-valued.(2)
Tr is firmly nonexpansive, that is,
(2.5)‖Trx-Try‖2≤〈Trx-Try,x-y〉,∀x,y∈H.(3)
F(Tr)=EP(F).(4)
EP(F) is closed and convex.Lemma 2.22.2 (see [23]).
LetS:C→H be a k-strictly pseudo-contractive mapping. Define T:C→H by Tx=λx+(1-λ)Sx for each x∈C. Then, as λ∈[k,1), T is nonexpansive mapping such that F(T)=F(S).Lemma 2.32.3 (see [24]).
In a Hilbert spaceH, there holds the inequality
(2.6)‖x+y‖2≤‖x‖2+2〈y,x+y〉,∀x,y∈H.Lemma 2.4 (see [25]).
LetH be a Hilbert space and C be a closed convex subset of H, and T:C→C a nonexpansive mapping with F(T)≠∅. If {xn} is a sequence in C weakly converging to x and if {(I-T)xn} converges strongly to y, then (I-T)x=y.Lemma 2.5 (see [26]).
Let{xn} and {zn} be bounded sequences in a Banach space E and {γn} be a sequence in [0,1] satisfying the following condition
(2.7)0<lim⁡n→∞inf⁡γn≤lim⁡n→∞sup⁡γn<1.
Suppose that xn+1=γnxn+(1-γn)zn,n≥0 and limn→∞sup(∥zn+1-zn∥-∥xn+1-xn∥)≤0. Then limn→∞∥zn-xn∥=0.Lemma 2.6 (see [27]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.8)an+1≤(1-bn)an+bnδn,n≥0,
where {bn} is a sequence in (0,1) and {δn} is a sequence in R, such that(i)
∑i=1∞bi=∞.(ii)
limn→∞supδn≤0 or ∑i=1∞|bnδn|<∞.
Then,limn→∞an=0.Let{Si} be an infinite family of ki-strictly pseudo-contractive mappings of C into itself, we define a mapping Wn of C into itself as follows,
(2.9)Un,n+1:=I,Un,n:=τnSn′Un,n+1+(1-τn)I,⋅⋅⋅Un,k:=τkSk′Un,k+1+(1-τk)I,⋅⋅⋅Un,2:=τ2S2′Un,3+(1-τ2)I,Wn=Un,1:=τ1S1′Un,2+(1-τ1)I,
where 0≤τi≤1, Si′=σiI+(1-σi)Si and σi∈[ki,1) for i∈N. We can obtain Si′ is a nonexpansive mapping and F(Si)=F(Si′) by Lemma 2.2. Furthermore, we obtain that Wn is a nonexpansive mapping.Remark 2.7.
Ifki=0, and σi=0 for i∈N, then the definition of Wn in (2.9) reduces to the definition of Wn in (1.13).To establish our results, we need the following technical lemmas.Lemma 2.8 (see [18]).
LetC be a nonempty closed convex subset of a strictly convex Banach space. Let {Si′} be an infinite family of nonexpansive mappings of C into itself and let {τi} be a real sequence such that 0<τi≤b<1 for every i∈N. Then, for every x∈C and k∈N, the limit limn→∞Un,kx exists.In view of the previous lemma, we will define(2.10)Wx∶=lim⁡n→∞Wnx=lim⁡n→∞Un,1x,x∈C.Lemma 2.9 (see [18]).
LetC be a nonempty closed convex subset of a strictly convex Banach space. Let {Si′} be an infinite family of nonexpansive mappings of C into itself such that ⋂i=1∞F(Si′)≠∅ and let {τi} be a real sequence such that 0<τi≤b<1 for every i∈N. Then, F(W)=⋂i=1∞F(Si′)≠∅.The following lemmas follow from Lemmas2.2, 2.8, and 2.9.Lemma 2.10.
LetC be a nonempty closed convex subset of a strictly convex Banach space. Let {Si} be an infinite family of ki-strictly pseudo-contractive mappings of C into itself such that ⋂i=1∞F(Si)≠∅. Define Si′=σiI+(1-σi)Si and σi∈[ki,1) and let {τi} be a real sequence such that 0<τi≤b<1 for every i∈N. Then, F(W)=⋂i=1∞F(Si)=⋂i=1∞F(Si′)≠∅.Lemma 2.11 (see [28]).
LetC be a nonempty closed convex subset of a Hilbert space. Let {Si′} be an infinite family of nonexpansive mappings of C into itself such that ⋂i=1∞F(Si′)≠∅ and let {τi} be a real sequence such that 0<τi≤b<1 for every i∈N. If K is any bounded subset of C, then
(2.11)lim⁡n→∞sup⁡x∈K‖Wx-Wnx‖=0.


## 3. Main Results

LetH be a real Hilbert space and F be a k-Lipschitzian and η-strongly monotone operator with k>0, η>0, 0<μ<2η/k2 and 0<t<1. Then, for t∈min{0,{1,1/τ}}, S=(I-tμF):H→H is a contraction with contractive coefficient 1-tτ and τ=(1/2)μ(2η-μk2).In fact, from (1.2) and (1.3), we obtain
(3.1)‖Sx-Sy‖2=‖x-y-tμ(Fx-Fy)‖2=‖x-y‖2+t2μ2‖Fx-Fy‖2-2tμ〈Fx-Fy,x-y〉≤‖x-y‖2+k2t2μ2‖x-y‖2-2tημ‖x-y‖2≤(1-tμ(2η-μk2))‖x-y‖2≤(1-tτ)2‖x-y‖2.
Thus, S=(1-tμF) is a contraction with contractive coefficient 1-tτ∈(0,1).Now, we show the strong convergence results for an infinite familyki-strictly pseudo-contractive mappings in Hilbert space.Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H and F be a bifunction from C×C→R satisfying (A1)–(A4). Let Si:C→C be a ki-strictly pseudo-contractive mapping with ⋂i=1∞F(Si)∩EP≠∅ and {τi} be a real sequence such that 0<τi≤b<1, i∈N. Let f be a contraction of H into itself with β∈(0,1) and B be k-Lipschitzian and η-strongly monotone operator on H with coefficients k,η>0, 0<μ<2η/k2, 0<r<(1/2)μ(2η-μk2)/β=(τ/β) and τ<1. Let {xn} be a sequence generated by
(3.2)F(un,y)+1λn〈y-un,un-xn〉≥0,∀y∈C,yn=δnun+(1-δn)Wnun,xn+1=αnrf(xn)+βnxn+((1-βn)I-μαnB)yn,∀n∈N,
where un=Tλnxn and {Wn:C→C} is the sequence defined by (2.9). If {αn}, {βn}, {δn}, and {λn} satisfy the following conditions:(i)
{αn}⊂(0,1), limn→∞αn=0,   ∑i=1∞αn=∞,(ii)
0<limn→∞infβn≤limn→∞supβn<1,(iii)
0<limn→∞infδn≤limn→∞supδn<1,
limn→∞|δn+1-δn|=0,(iv)
{λn}⊂(0,∞), limn→∞λn>0, limn→∞|λn+1-λn|=0.
Then{xn} converges strongly to z∈⋂i=1∞F(Si)∩EP≠∅, where z is the unique solution of variational inequality
(3.3)lim⁡n→∞sup⁡〈(rf-μB)z,p-z〉≤0,∀p∈⋂i=1∞F(Si)∩EP⁡≠∅,
that is, z=PF(W)∩EP(F)(I-μB+rf)z, which is the optimality condition for the minimization problem
(3.4)min⁡z∈⋂i=1∞F(Si)∩EP12〈μBz,z〉-h(z),
where h is a potential function for rf (i.e., h′(z)=rf(z) for z∈H).Proof .
We divide the proof into five steps.
Step 1. We prove that {xn} is bounded.
Noting the conditions (i) and (ii), we may assume, without loss of generality, thatαn/(1-βn)≤min{1,1/τ}. For x,y∈C, we obtain
(3.5)‖((1-βn)I-αnμB)x-((1-βn)I-αnμB)y‖≤(1-βn)‖(I-αn1-βnμB)x-(I-αn1-βnμB)y‖≤(1-βn)(1-αn1-βnτ)‖x-y‖=(1-βn-αnτ)‖x-y‖.
Take p∈⋂i=1∞F(Si)∩EP≠∅. Since un=Tλnxn and p=Tλnp, then from Lemma 2.1, we know that, for any n∈N,
(3.6)‖un-p‖=‖Tλnxn-Tλnp‖≤‖xn-p‖.
Furthermore, since Wnp=p and (3.6), we have
(3.7)‖yn-p‖=‖δnun+(1-δn)Wnun-p‖=‖δn(un-p)+(1-δn)(Wnun-p)‖≤δn‖un-p‖+(1-δn)‖Wnun-p‖≤‖un-p‖≤‖xn-p‖.
Thus, it follows from (3.7) that
(3.8)‖xn+1-p‖=‖αnrf(xn)+βnxn+((1-βn)I-μαnB)yn-p‖=‖αnr(f(xn)-f(p))+αn(rf(p)-μBp)+βn(xn-p)+((1-βn)I-μαnB)(yn-p)‖≤αnrβ‖xn-p‖+αn‖rf(p)-μBp‖+βn‖xn-p‖+(1-βn-ταn)‖yn-p‖≤(1-αn(τ-rβ))‖xn-p‖+αn‖rf(p)-μBp‖≤max⁡{‖xn-p‖,‖rf(p)-μBp‖τ-rβ}.
By induction, we have
(3.9)‖xn-p‖≤max⁡{‖x1-p‖,‖rf(p)-μBp‖τ-rβ},n≥1.
Hence, {xn} is bounded and we also obtain that {un}, {Wnun}, {yn}, {Byn}, and {f(xn)} are all bounded. Without loss of generality, we can assume that there exists a bounded set K⊂C such that {un}, {Wnun}, {yn}, {Byn}, {f(xn)}∈K, for all n∈N.
Step 2. We show that limn→∞∥xn-xn+1∥=0.
Letxn+1=(1-βn)zn+βnxn. We note that
(3.10)zn=xn+1-βnxn1-βn=αnrf(xn)+((1-βn)I-μαnB)yn1-βn,
and then
(3.11)zn+1-zn=αn+1rf(xn+1)+((1-βn+1)I-μαn+1B)yn+11-βn+1-αnrf(xn)+((1-βn)I-μαnB)yn1-βn=αn+11-βn+1(rf(xn+1)-μByn+1)-αn1-βn(rf(xn)-μByn)+yn+1-yn.
Therefore,
(3.12)‖zn+1-zn‖≤αn+11-βn+1(‖rf(xn+1)‖+‖μByn+1‖)+αn1-βn(‖rf(xn)‖+‖μByn‖)+‖yn+1-yn‖.
It follows from (3.2) that
(3.13)‖yn+1-yn‖=‖δn+1un+1+(1-δn+1)Wn+1un+1-(δnun+(1-δn)Wnun)‖≤|δn+1-δn|‖un‖+δn+1‖un+1-un‖+(1-δn+1)‖Wn+1un+1-Wnun‖+|δn+1-δn|‖Wnun‖.
We will estimate∥un+1-un∥. From un+1=Tλn+1xn+1 and un=Tλnxn, we obtain
(3.14)F(un+1,y)+1λn+1〈y-un+1,un+1-yn+1〉≥0,∀y∈C,(3.15)F(un,y)+1λn〈y-un,un-yn〉≥0,∀y∈C.
Takingy=un in (3.14) and y=un+1 in (3.15), we have
(3.16)F(un+1,un)+1λn+1〈un-un+1,un+1-xn+1〉≥0,F(un,un+1)+1λn〈un+1-un,un-xn〉≥0.
So, from (A2), one has(3.17)〈un+1-un,un-xnλn-un+1-xn+1λn+1〉≥0,
furthermore,
(3.18)〈un+1-un,un-un+1-xn-λnλn+1(un+1-xn+1)〉≥0.
Since limn→∞λn>0, we assume that there exists a real number such that λn>a>0 for all n∈N. Thus, we obtain
(3.19)‖un+1-un‖2≤〈un+1-un,xn+1-xn+(1-λnλn+1)(un+1-xn+1)〉≤‖un+1-un‖{‖xn+1-xn‖+|1-λnλn+1|‖un+1-xn+1‖},
which means
(3.20)‖un+1-un‖≤‖xn+1-xn‖+|1-λnλn+1|‖un+1-xn+1‖≤‖xn+1-xn‖+1a|λn+1-λn|‖un+1-xn+1‖≤‖xn+1-xn‖+L1|λn+1-λn|,
where L1=sup{∥un+1-xn+1∥:n∈N}.
Next, we estimate∥Wn+1un+1-Wnun∥. Notice that
(3.21)‖Wn+1un+1-Wnun‖=‖Wn+1un+1-Wn+1un+Wn+1un-Wnun‖≤‖un+1-un‖+‖Wn+1un-Wnun‖.
From (2.9), we obtain
(3.22)‖Wn+1un-Wnun‖=‖τ1S1′Un+1,2un-τ1S1′Un,2un‖≤τ1‖Un+1,2un-Un,2un‖=τ1‖τ2S2′Un+1,3un-τ2S2′Un,3un‖≤τ1τ2‖Un+1,3un-Un,3un‖≤⋯≤τ1τ2⋅⋅⋅τn‖Un+1,n+1un-Un,n+1un‖≤L2∏i=1n‍τi,
where L2≥0 is a constant such that ∥Un+1,n+1un-Un,n+1un∥≤L2, for all n∈N.
Substituting (3.20) and (3.22) into (3.21), we obtain
(3.23)‖Wn+1un+1-Wnun‖≤‖xn+1-xn‖+L1|λn+1-λn|+L2∏i=1n‍τi.
Hence, we have
(3.24)‖yn+1-yn‖≤|δn+1-δn|(‖un‖+‖Wnun‖)+‖xn+1-xn‖+(1-δn+1)L2∏i=1n‍τi+L1|λn+1-λn|≤L3|δn+1-δn|+‖xn+1-xn‖+(1-δn+1)L2∏i=1n‍τi+L1|λn+1-λn|,
where L3=sup{∥un∥+∥Wnun∥:n∈N}.
Furthermore,(3.25)‖zn+1-zn‖≤αn+11-βn+1(‖rf(xn+1)‖+‖μByn+1‖)+αn1-βn(‖rf(xn)‖+‖μByn‖)+‖xn+1-xn‖+L1|λn+1-λn|+L2(1-δn+1)∏i=1n‍τi+L3|δn+1-δn|.
It follows from (3.25) that
(3.26)‖zn+1-zn‖-‖xn+1-xn‖≤αn+11-βn+1(‖rf(xn+1)‖+‖μByn+1‖)+αn1-βn(‖rf(xn)‖+‖μByn‖)+L1|λn+1-λn|+L2(1-δn+1)∏i=1n‍τi+L3|δn+1-δn|.
By the conditions (i), (iii), and (iv), we obtain
(3.27)lim⁡n→∞sup⁡(‖zn+1-zn‖-‖xn+1-xn‖)≤0.
Hence, by Lemma 2.5, one has
(3.28)lim⁡n→∞‖zn-xn‖=0,
which implies
(3.29)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞(1-βn)‖zn-xn‖=0.Step 3. We claim that limn→∞∥Wun-un∥=0.
Notice that(3.30)‖Wun-un‖=‖Wun-Wnun+Wnun-un‖≤‖Wun-Wnun‖+‖Wnun-un‖≤sup⁡u∈K‖Wu-Wnu‖+‖Wnun-un‖.
It follows from (3.2) that
(3.31)‖Wnun-un‖=‖Wnun-yn+yn-un‖≤‖yn-un‖+‖Wnun-yn‖=‖yn-un‖+δn‖Wnun-un‖≤‖xn-un‖+‖yn-xn‖+δn‖Wnun-un‖.
By the condition (iii), we obtain
(3.32)‖Wnun-un‖≤11-δn(‖xn-un‖+‖yn-xn‖).
First, we showlimn→∞∥xn-un∥=0. From (3.2), for all p∈⋂i=1∞F(Si)∩EP(F), applying Lemma 2.3 and noting that ∥·∥ is convex, we obtain
(3.33)‖xn+1-p‖2=‖αnrf(xn)+βnxn+((1-βn)I-μαnB)yn-p‖2=‖αn(rf(xn)+μByn)+βn(xn-p)+(1-βn)(yn-p)‖2≤‖βn(xn-p)+(1-βn)(yn-p)‖2+2αn〈rf(xn)+μByn,xn+1-p〉≤βn‖xn-p‖2+(1-βn)‖yn-p‖2+2αn‖rf(xn)+μByn‖‖xn+1-p‖≤βn‖xn-p‖2+(1-βn)‖un-p‖2+2αn‖rf(xn)+μByn‖‖xn+1-p‖.
Since un=Tλnxn, p=Tλnp, we have
(3.34)‖un-p‖2=‖Tλnxn-Tλnp‖2≤〈xn-p,un-p〉=12(‖xn-p‖2+‖un-p‖2-‖xn-un‖2),
which implies
(3.35)‖un-p‖2≤‖xn-p‖2-‖xn-un‖2.
Substituting (3.35) into (3.33), we have
(3.36)‖xn+1-p‖2≤‖xn-p‖2-(1-βn)‖xn-un‖2+2αn‖rf(xn)+μByn‖‖xn+1-p‖,
which means
(3.37)(1-βn)‖xn-un‖2≤‖xn-p‖2-‖xn+1-p‖2+2αn‖rf(xn)+μByn‖‖xn+1-p‖≤‖xn+1-xn‖(‖xn-p‖+‖xn+1-p‖)+2αn‖rf(xn)+μByn‖‖xn+1-p‖.
Noticing limn→∞αn=0 and limn→∞inf(1-βn)>0, we have
(3.38)lim⁡n→∞‖xn-un‖=0.
Second, we showlimn→∞∥yn-xn∥=0. It follows from (3.2) that
(3.39)‖yn-xn‖≤‖yn-xn+1‖+‖xn+1-xn‖=‖αnrf(xn)+βnxn+((1-βn)I-μαnB)yn-yn‖+‖xn+1-xn‖≤αn‖rf(xn)+μByn‖+βn‖xn-yn‖+‖xn+1-xn‖.
This implies that
(3.40)(1-βn)‖yn-xn‖≤αn‖rf(xn)+μByn‖+‖xn+1-xn‖.
Noticing limn→∞αn=0, limn→∞inf(1-βn)>0 and (3.30), we have
(3.41)lim⁡n→∞‖yn-xn‖=0.
Thus, substituting (3.41) and (3.38) into (3.32), we obtain
(3.42)lim⁡n→∞‖Wnun-un‖=0.
Furthermore, (3.42), (3.30), and Lemma 2.11 lead to
(3.43)lim⁡n→∞‖Wun-un‖=0.
Step 4. Letting z=PF(W)∩EP(F)(I-μB+rf)z, we show
(3.44)lim⁡n→∞sup⁡〈(rf-μB)z,xn-z〉≤0.
We know that PF(W)∩EP(F)(I-μB+rf) is a contraction. Indeed, for any x,y∈H, we have
(3.45)‖PF(W)∩EP⁡(F)(I-μB+rf)x-PF(W)∩EP⁡(F)(I-μB+rf)y‖≤‖(I-μB+rf)x-(I-μB+rf)y‖≤(1-τ+rβ)‖x-y‖,
and hence PF(W)∩EP(F)(I-μB+rf) is a contraction due to (1-τ+rβ)∈(0,1). Thus, Banach’s Contraction Mapping Principle guarantees that PF(W)∩EP(F)(I-μB+rf) has a unique fixed point, which implies z=PF(W)∩EP(F)(I-μB+rf)z.
Since{uni}⊂{un} is bounded in C, without loss of generality, we can assume that {uni}⇀ω, it follows from (3.43) that Wuni⇀ω. Since C is closed and convex, C is weakly closed. Thus we have ω∈C.
Let us showω∈F(W). For the sake of contradiction, suppose that ω∉F(W), that is, Wω≠ω. Since {uni}⇀ω, by the Opial condition, we have
(3.46)lim⁡n→∞inf⁡‖uni-ω‖<lim⁡n→∞inf⁡‖uni-Wω‖≤lim⁡n→∞inf⁡{‖uni-Wuni‖+‖Wuni-Wω‖}≤lim⁡n→∞inf⁡{‖uni-Wuni‖+‖uni-ω‖}.
It follows (3.43) that
(3.47)lim⁡n→∞inf⁡‖uni-ω‖<lim⁡n→∞inf⁡‖uni-ω‖.
This is a contradiction, which shows that ω∈F(W).
Next, we prove thatω∈EP(F). By (3.2), we obtain
(3.48)F(un,y)+1λn〈y-un,un-xn〉≥0.
It follows from (A2) that
(3.49)1λn〈y-un,un-xn〉≥F(y,un).
Replacing n by ni, we have
(3.50)〈y-uni,1λni(uni-xni)〉≥F(y,uni).
Since (1/λni)(uni-xni)→0 and {uni}⇀ω, it follows from (A4) that F(y,ω)≥0 for all y∈C. Put zt=ty+(1-t)ω for all t∈(0,1] and y∈C. Then, we have zt∈C and then F(zt,ω)≥0. Hence, from (A1) and (A4), we have
(3.51)0=F(zt,zt)≤tF(zt,y)+(1-t)F(zt,y)≤tF(zt,y),
which means F(zt,y)≥0. From (A3), we obtain F(ω,y)≥0 for y∈C and then ω∈EP(F). Therefore, ω∈F(W)∩EP(F).
Sincez=PF(W)∩EP(F)(I-μB+rf)z, it follows from (3.38), (3.42), and Lemma 2.11 that
(3.52)lim⁡i→∞sup⁡〈(rf-μB)z,xn-z〉≤lim⁡n→∞〈(rf-μB)z,xni-z〉=lim⁡i→∞〈(rf-μB)z,xni-uni〉+lim⁡i→∞〈(rf-μB)z,uni-Wniuni〉+lim⁡i→∞〈(rf-μB)z,Wniuni-Wuni〉+lim⁡i→∞〈(rf-μB)z,Wuni-z〉=〈(rf-μB)z,ω-z〉≤0.
Step 5. Finally we prove that xn→ω as n→∞. In fact, from (3.2) and (3.7), we obtain
(3.53)‖xn+1-ω‖2=‖αnrf(xn)+βnxn+((1-βn)I-μαnB)yn-ω‖2=‖αnr(f(xn)-f(ω))+αn(rf(ω)-μBω)+βn(xn-ω)+((1-βn)I-μαnB)(yn-ω)‖2=αnr〈f(xn)-f(ω),xn+1-ω〉+αn〈rf(ω)-μBω,xn+1-ω〉+βn〈xn-ω,xn+1-ω〉+〈((1-βn)I-μαnB)(yn-ω),xn+1-ω〉≤αnrβ‖xn-ω‖2+‖xn+1-ω‖22+αn〈rf(ω)-μBω,xn+1-ω〉+βn‖xn-ω‖2+‖xn+1-ω‖22+(1-βn-αnτ)‖yn-ω‖2+‖xn+1-ω‖22≤1-αn(τ-rβ)2(‖xn-ω‖2+‖xn+1-ω‖2)+αn〈rf(ω)-μBω,xn+1-ω〉,
which implies
(3.54)‖xn+1-ω‖2≤1-αn(τ-rβ)1+αn(τ-rβ)‖xn-ω‖2+2αn(τ-rβ)(1+αn(τ-rβ))(τ-rβ)〈rf(ω)-μBω,xn+1-ω〉≤(1-αn(τ-rβ))‖xn-ω‖2+2αn(τ-rβ)(1+αn(τ-rβ))(τ-rβ)〈rf(ω)-μBω,xn+1-ω〉.
From condition (i) and (3.7), we know that ∑i=1nαn(τ-rβ)=∞ and limi→∞sup(2/(1+αn(τ-rβ))(τ-rβ))〈rf(ω)-μBω,xn+1-ω〉≤0. we can conclude from Lemma 2.6 that xn→ω as n→∞. This completes the proof of Theorem 3.1.Remark 3.2.
Ifr=1, μ=1, B=I and δi=0, ki=0, σi=0 for i∈N, then Theorem 3.1 reduces to Theorem 3.5 of Yao et al. [19]. Furthermore, we extend the corresponding results of Yao et al. [19] from one infinite family of nonexpansive mapping to an infinite family of strictly pseudo-contractive mappings.Remark 3.3.
Ifμ=1 and δi=0, ki=0, σi=0 for i∈N, then Theorem 3.1 reduces to Theorem 10 of Colao and Marino [20]. Furthermore, we extend the corresponding results of Colao and Marino [20] from one infinite family of nonexpansive mapping to an infinite family of strictly pseudo-contractive mappings, and from a strongly positive bounded linear operator A to a k-Lipschitzian and η-strongly monotone operator B.Theorem 3.4.
LetC be a nonempty closed convex subset of a real Hilbert space H and F be a bifunction from C×C→R satisfying (A1)–(A4). Let S:C→C be a nonexpansive mapping with F(S)∩EP≠∅. Let f be a contraction of H into itself with β∈(0,1) and B be k-Lipschitzian and η-strongly monotone operator on H with coefficients k,η>0, 0<μ<2η/k2, 0<r<(1/2)μ(2η-μk2)/β=τ/β and τ<1. Let {xn} be sequence generated by
(3.55)F(un,y)+1λn〈y-un,un-xn〉≥0,∀y∈C,yn=δnun+(1-δn)Snun,xn+1=αnrf(xn)+βnxn+((1-βn)I-μαnB)yn,∀n∈N,
where un=Tλnxn. If {αn},{βn},{δn}, and {λn} satisfy the following conditions:(i)
{αn}⊂(0,1), limn→∞αn=0,   ∑i=1∞αn=∞,(ii)
0<limn→∞infβn≤limn→∞supβn<1,(iii)
0<limn→∞infδn≤limn→∞supδn<1,
limn→∞|δn+1-δn|=0,(iv)
{λn}⊂(0,∞), limn→∞λn>0, limn→∞|λn+1-λn|=0.
Then{xn} converges strongly to z∈F(S)∩EP≠∅, where z is the unique solution of variational inequality
(3.56)lim⁡n→∞sup⁡〈(rf-μB)z,p-z〉≤0,∀p∈F(S)∩EP≠∅,
that is, z=PF(S)∩EP(F)(I-μB+rf)z.Proof .
By Theorem3.1, letting ki=0, σi=0, τi=1 and Si=S for i∈N, we can obtain Theorem 3.4.


## 4. Numerical Example

Now, we present a numerical example to illustrate our theoretical analysis results obtained in Section3.Example 4.1.
LetH=R, C=[-1,1],   Sn=I, τn=τ∈(0,1),  λn=1, n∈N, F(x,y)=0, for all x,y∈C, B=I, r=μ=1, f(x)=(1/10)x, for all x, with contraction coefficient β=1/5, δn=1/2, αn=1/n, βn=1/4+1/2n for every n∈N. Then {xn} is the sequence generated by
(4.1)xn+1=(1-910n)xn,
and {xn}→0, as n→∞, where 0 is the unique solution of the minimization problem
(4.2)min⁡x∈C920x2+c.Proof.
We divide the proof into four steps.
Step 1. We show
(4.3)Tλnx=PCx,∀x∈H,
where
(4.4)PCx={x|x|,x∉C,x,x∈C.
SinceF(x,y)=0, for all x,y∈C, due to the definition of Tλn(x), for all x∈H, by Lemma 2.1, we obtain
(4.5)Tλnx={z∈C:(y-z,z-x)≥0,∀y∈C}.
By the property ofPC, for x∈C, we have Tλnx=PCx=Ix. Furthermore, it follows from (3) in Lemma 2.1 that
(4.6)EP⁡(F)=C.
Step 2. We show that
(4.7)Wn=I.
It follows from (2.9) that
(4.8)W1=U1,1=τ1S1′U1,2+(1-τ1)I=τ1S1′+(1-τ1)I,W2=U2,1=τ1S1′U2,2+(1-τ1)I=τ1S1′(τ2S2′U2,3+(1-τ2)I)+(1-τ1)I=τ1τ2S1′S2′+τ1(1-τ2)S1′+(1-τ1)I,W3=U3,1=τ1S1′U3,2+(1-τ1)I=τ1S1′(τ2S2′U3,3+(1-τ2)I)+(1-τ1)I=τ1τ2S1′S2′U3,3+τ1(1-τ2)S1′+(1-τ1)I=τ1τ2S1′S2′(τ3S3′U3,4+(1-τ3)I)+τ1(1-τ2)S1′+(1-τ1)I=τ1τ2τ3S1′S2′S3′+τ1τ2(1-τ3)S1′S2′+τ1(1-τ2)S1′+(1-τ1)I.
Furthermore, we obtain
(4.9)Wn=Un,1=τ1τ2τ3⋅⋅⋅τnS1′S2′S3′⋅⋅⋅Sn′+τ1τ2⋅⋅⋅τn-1(1-τn)S1′S2′⋅⋅⋅Sn-1′+τ1τ2⋅⋅⋅τn-2(1-τn-1)S1′S2′⋅⋅⋅Sn-2′+⋅⋅⋅+τ1(1-τ2)S1′+(1-τ1)I.
Since Si′=I, τi=τ for i∈N, one has
(4.10)Wn=[τn+τn-1(1-τ)+⋅⋅⋅+τ(1-τ)+(1-τ)]I=I.
Step 3. We show that
(4.11)xn+1=(1-910n)xn,{xn}→0, as n→∞, where 0 is the unique solution of the minimization problem
(4.12)min⁡x∈C920x2+c.
In fact, we can see thatB=I is k-Lipschitzian and η-strongly monotone operator on H with coefficient k=1, η=3/4 such that 0<μ<2η/k2, 0<r<(1/2)μ(2η-μk2)/β=τ/β, so we take r=μ=1. Since Sn′=I, n∈N, we have
(4.13)⋂i=1∞F(Si)=H.
Furthermore, we obtain
(4.14)⋂i=1∞F(Si)∩EP⁡(F)=C=[-1,1].
Next, we need prove{xn}→0, as n→∞. Since yn=un for all n∈N, we have
(4.15)xn+1=αnrf(xn)+βnxn+((1-βn)I-μαnB)yn=(1-910n)xn,
for all n∈N.
Thus, we obtain a special sequence{xn} of (3.2) in Theorem 3.1 as follows
(4.16)xn+1=(1-910n)xn.
By Lemma 2.6, it is obviously that xn→0, 0 is the unique solution of the minimization problem
(4.17)min⁡x∈C920x2+c,
where c is a constant number.
Step 4. Finally, we use software Matlab 7.0 to give the numerical experiment results and then obtain Table 1 which show that the iteration process of the sequence {xn} is a monotonedecreasing sequence and converges to 0. From Table 1 and the corresponding graph Figure 1, we show that the more the iteration steps are, the more slowly the sequence {xn} converges to 0.Table 1
This  table  shows  the  value  of  sequence{xn}  on  each  iteration  step  (initial  value  x1=0.2).
n
x
n
n
x
n
1
0.2000
17
0.0017
2
0.0200
18
0.0016
3
0.0110
19
0.0016
4
0.0077
20
0.0015
5
0.0060
21
0.0014
⋮
⋮
⋮
⋮
9
0.0032
26
0.0012
10
0.0029
27
0.0011
⋮
⋮
⋮
⋮
14
0.0021
30
0.0010
15
0.0019
31
0.0009
16
0.0018
32
0.0009Figure 1
The corresponding graph atx=0.2.



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*Source: 416476-2012-07-31.xml*