# A New Iterative Method for Equilibrium Problems and Fixed Point Problems

Authors: Abdul Latif; Mohammad Eslamian
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative method for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh (2012), Cianciaruso et al. (2010), and many others.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let C be a nonempty closed convex subset of H, and let ProjC be a nearest point projection of H into C; that is, for x∈H, ProjCx is the unique point in C with the property ∥x-ProjCx∥:=inf{∥x-y∥:y∈C}. It is well known that y=ProjCx iff 〈x-y,y-z〉≥0 for all z∈C.Letf be a bifunction from C×C into ℝ, such that f(x,x)=0 for all x∈C. Consider the Fan inequality [1]: find a point x⋆∈C such that
(1)f(x⋆,y)≥0,∀y∈C,
where f(x,·) is convex and subdifferentiable on C for every x∈C. The set of solutions of this problem is denoted by Sol(f,C). In fact, the Fan inequality can be formulated as an equilibrium problem. Such problems arise frequently in mathematics, physics, engineering, game theory, transportation, economics, and network. Due to importance of the solutions of such problems, many researchers are working in this area and studying the existence of the solutions of such problems; for example, see, [2–4]. Further, if f(x,y)=〈Fx,y-x〉 for every x,y∈C, where F is a mapping from C into H, then the Fan inequality problem (equilibrium problem) becomes the classical variational inequality problem which is formulated as finding a point x⋆∈C such that
(2)〈Fx⋆,y-x⋆〉≥∀y∈C.Variational inequalities were introduced and studied by Stampacchia [5]. It is well known that this area covers many branches of mathematics, such as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance; see [6–11].Here we recall some useful notions.A mappingT:C→H is said to be L-Lipschitz on C if there exists a constant L≥0 such that for each x,y∈C,
(3)∥Tx-Ty∥≤L∥x-y∥.
In particular, if L∈[0,1[, then T is called a contraction on C; if L=1, then T is called a nonexpansive mapping on C. The set of fixed points of T is denoted by F(T).A family𝒯:={T(s):0≤s<∞} of mappings on a closed convex subset C of a Hilbert space H is called a nonexpansive semigroup if it satisfies the following: (i) T(0)x=x for all x∈C; (ii) T(s+t)=T(s)T(t) for all s,t≥0; (iii) for all x∈C, s→T(s)x is continuous; (iv) ∥T(s)x-T(s)y∥≤∥x-y∥ for all x,y∈C and s≥0.We useF(𝒯) to denote the common fixed point set of the semigroup 𝒯; that is, F(𝒯)={x∈C:T(s)x=x,∀s≥0}. It is well known that F(𝒯) is closed and convex [12]. A nonexpansive semigroup 𝒯 on C is said to be uniformly asymptotically regular (in short, u.a.r.) on C if for all h≥0 and any bounded subset B of C,
(4)limt→∞supx∈B∥T(h)(T(t)x)-T(t)x∥=0.
For each h≥0, define σt(x)=(1/t)∫0t‍T(s)xds. Then
(5)limt→∞supx∈B∥T(h)(σt(x))-σt(x)∥=0,
Provided that B is closed bounded convex subset of C. It is known that the set {σt(x):t>0} is a u.a.r. nonexpansive semigroup; see [13]. The other examples of u.a.r. operator semigroup can be found in [14].A bifunctionf:C×C→ℝ is said to be (i) strongly monotone on C with α>0 if f(x,y)+f(y,x)≤-α∥x-y∥2, ∀x,y∈C; (ii) monotone on C if f(x,y)+f(y,x)≤0, ∀x,y∈C; (iii) pseudomonotone on C if f(x,y)≥0⇒f(y,x)≤0, ∀x,y∈C; (iv) Lipschitz-type continuous on C with constants c1>0 and c2>0 if f(x,y)+f(y,z)≥f(x,z)-c1∥x-y∥2-c2∥y-z∥2, ∀x,y,z∈C.Note that ifT is L-Lipschitz on C, then for each x,y∈C, the function f(x,y)=〈Fx,y-x〉 is a Lipschitz-type continuous with constants c1=c2=L/2.An operatorA on H is called strongly positive if there is a constant γ¯>0 such that
(6)〈Ax,x〉≥γ¯∥x∥2,∀x∈H.Recently, iterative methods for nonexpansive mappings have been applied to solve convex minimization problems. In [15], Xu defined an iterative sequence {xn} in C which converges strongly to the unique solution of the minimization problem under some suitable conditions. A well-known typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping T on a real Hilbert space H:
(7)minx∈F(T)12〈Ax,x〉-〈x,b〉,
where b is a given point in H and A is strongly positive operator.For solving the variational inequality problem, Marino and Xu [16] introduced the following general iterative process for nonexpansive mapping T based on the viscosity approximation method (see [17]):
(8)xn+1=anγh(xn)+(I-anA)Txn,∀n≥0,
where A is strongly positive bounded linear operator on H, h is contraction on H, and {an}⊂]0,1[. They proved that, under some appropriate conditions on the parameters, the sequence {xn} generated by (8) converges strongly to the unique solution x⋆∈F(T) of the variational inequality
(9)〈(A-γh)x⋆,x-x⋆〉≥0,∀x∈F(T),
which is the optimality condition for the minimization problem
(10)minx∈F(T)12〈Ax,x〉-g(x),
where g is a potential function for γh (i.e., g′(x)=γh(x), ∀x∈H).Iterative process for approximating common fixed points of a nonexpansive semigroup has been investigated by various authors (see [13, 14, 18–21]). Recently, Li et al. [19] introduced the following iterative procedure for the approximation of common fixed points of a nonexpansive semigroup 𝒯={T(s):0≤s<∞} on a closed convex subset C of a Hilbert space H:
(11)xn=αnγh(xn)+(I-αnA)1sn∫0sn‍T(s)xnds,n≥1,
where A is a strongly positive bounded linear operator on H and h is a contraction on C. Imposing some appropriate conditions on the parameters, they proved that the iterative sequence {xn} generated by (11) converges strongly to the unique solution x⋆∈F(𝒯) of the variational inequality 〈(γh-A)x⋆,z-x⋆〉≤0, ∀z∈F(𝒯).For obtaining a common element ofSol(f,C) and the set of fixed points of a nonexpansive mapping T, S. Takahashi and W. Takahashi [9] first introduced an iterative scheme by the viscosity approximation method. They proved that under certain conditions the iterative sequences converge strongly to z=ProjF(T)∩Sol(f,C)(h(z)).During last few years, iterative algorithms for finding a common element of the set of solutions of Fan inequality and the set of fixed points of nonexpansive mappings in a real Hilbert space have been studied by many authors (see, e.g., [2, 4, 22–28]). Recently, Anh [22] studied the existence of a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of Fan inequality for monotone and Lipschitz-type continuous bifunctions. He introduced the following new iterative process:
(12)wn=argmin{λnf(xn,w)+12∥w-xn∥2:w∈C},zn=argmin{λnf(wn,z)+12∥z-xn∥2:z∈C},xn+1=αnh(xn)+βnxn+γn(μS(xn)+(1-μ)zn),∀n≥0,
where μ∈]0,1[, C is nonempty, closed convex subset of a real Hilbert space H, f is monotone, continuous, and Lipschitz-type continuous bifunction, h is self-contraction on C with constant k∈]0,1[, and S is self nonexpansive mapping on C. He proved that, under some appropriate conditions over positive sequences {αn}, {βn}, {γn}, and {λn}, the sequences {xn}, {wn}, and {zn} converge strongly to q∈F(S)∩Sol(f,C) which is a solution of the variational inequality 〈(I-h)q,x-q〉≥0, ∀x∈F(S)∩Sol(f,C).In this paper, we introduce a new iterative scheme based on the viscosity method and study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative scheme for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh [22], Cianciaruso et al. [18], and many others.


## 2. Preliminaries

In this section we collect some lemmas which are crucial for the proofs of our results.Let{xn} be a sequence in H and x∈H. In the sequel, xn⇀x denotes that {xn} weakly converges to x and xn→x denotes that {xn} weakly converges to x.Lemma 1.
LetH be a real Hilbert space. Then the following inequality holds:
(13)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.Lemma 2 (see [15]).
Assume that{an} is a sequence of nonnegative real numbers such that
(14)an+1≤(1-ηn)an+ηnδn,n≥0,
where {ηn} is a sequence in ]0,1[ and δn is a sequence in ℝ such that (i)
∑n=1∞‍ηn=∞,(ii)
limsupn→∞δn≤0 or ∑n=1∞‍|ηnδn|<∞.Then limn→∞an=0.Lemma 3 (see [16]).
LetA be a strongly positive linear bounded self-adjoint operator on H with coefficient γ¯>0 and 0<ρ≤∥A∥-1. Then ∥I-ρA∥≤1-ργ¯.Lemma 4 (see [29]).
LetH be a Hilbert space and xi∈H, (1≤i≤m). Then for any given {λi}i=1m⊂]0,1[ with ∑i=1m‍λi=1 and for any positive integer k,j with 1≤k<j≤m,
(15)∥∑i=1m‍λixi∥2≤∑i=1m‍λi∥xi∥2-λkλj∥xk-xj∥2.Lemma 5 (see [30]).
Let{tn} be a sequence of real numbers such that there exists a subsequence {ni} of {n} such that tni<tni+1 for all i∈ℕ. Then there exists a nondecreasing sequence {τ(n)}⊂ℕ such that τ(n)→∞ and the following properties are satisfied by all (sufficiently large) numbers n∈ℕ:
(16)tτ(n)≤tτ(n)+1,tn≤tτ(n)+1.
In fact
(17)τ(n)=max{k≤n:tk<tk+1}.Lemma 6 (see [2]).
LetC be a nonempty closed convex subset of a real Hilbert space H and let f:C×C→ℝ be a pseudomonotone and Lipschitz-type continuous bifunction with constants c1,c2≥0. For each x∈C, let f(x,·) be convex and subdifferentiable on C. Let x0∈C and let {xn}, {zn}, and {wn} be sequences generated by
(18)wn=argmin{λnf(xn,w)+12∥w-xn∥2:w∈C},zn=argmin{λnf(wn,z)+12∥z-xn∥2:z∈C}.
Then for each x⋆∈Sol(f,C),
(19)∥zn-x⋆∥2≤∥xn-x⋆∥2-(1-2λnc1)∥xn-wn∥2-(1-2λnc2)∥wn-zn∥2,∀n≥0.


## 3. Main Results

In this section, we prove the main strong convergence result which solves the problem of finding a common element of three setsF(𝒯), F(𝒮), and Sol(fi,C) for finite family of monotone, continuous, and Lipschitz-type continuous bifunctions fi in a real Hilbert space H.Theorem 7.
LetC be a nonempty closed convex subset of a real Hilbert space H and let fi:C×C→ℝ(i=1,2,…,m) be a finite family of monotone, continuous, and Lipschitz-type continuous bifunctions with constants c1,i and c2,i. Let 𝒯={T(u):u≥0} and 𝒮={S(u):u≥0} be two u.a.r. nonexpansive self-mapping semigroups on C such that Ω=⋂i=1m‍Sol(fi,C)⋂‍F(𝒯)⋂‍F(𝒮)≠∅. Assume that h is a k-contraction self-mapping of C and A is a strongly positive bounded linear self-adjoint operator on H with coefficient γ¯<1 and 0<γ<γ¯/k. Let x0∈C and let {xn}, {wn,i}, and {zn,i} be sequences generated by
(20)wn,i=argmin{λn,ifi(xn,w)+12∥w-xn∥2:w∈C},i∈{1,2,…,m},zn,i=argmin{λn,ifi(wn,i,z)+12∥z-xn∥2:z∈C},i∈{1,2,…,m},yn=αnxn+∑i=1m‍βn,izn,i+γnT(tn)xn+ηnS(tn)xn,xn+1=θnγh(xn)+(I-θnA)yn,∀n≥0,
where αn+∑i=1m‍βn,i+γn+ηn=1 and {tn}, {αn}, {βn,i}, {γn}, {ηn}, {λn,i}, and {θn} satisfy the following conditions: (i)
limn→∞tn=∞,(ii)
{θn}⊂]0,1[, limn→∞θn=0, and ∑n=1∞‍θn=∞,(iii)
{λn,i}⊂[a,b]⊂]0,1/L[, where L=max{2c1,i,2c2,i,1≤i≤m},(iv)
{αn},{βn,i},{γn},{ηn}⊂]0,1[, liminfnαnβn,i>0, liminfnαnγn>0, liminfnαnηn>0, and liminfnβn,i(1-2λn,ic1,i)>0 for each 1≤i≤m.Then, the sequence {xn} converges strongly to x⋆∈⋂i=1m‍Sol(fi,C)⋂‍F(𝒯)⋂‍F(𝒮) which solves the variational inequality:
(21)〈(A-γh)x⋆,x-x⋆〉≥0,∀x∈Ω.Proof.
SinceF(𝒯), F(𝒮), and Sol(fi,C) are closed and convex, ProjΩ is well-defined. We claim that ProjΩ(I-A+γh) is a contraction from C into itself. Indeed, for each x,y∈C, we have
(22)∥ProjΩ(I-A+γh)(x)-ProjΩ(I-A+γh)(y)∥≤∥(I-A+γh)(x)-(I-A+γh)(y)∥≤∥(I-A)x-(I-A)y∥+γ∥hx-hy∥≤(1-γ¯)∥x-y∥+γk∥x-y∥≤(1-(γ¯-γk))∥x-y∥.
Therefore, by the Banach contraction principle, there exists a unique elementx⋆∈C such that x⋆=ProjΩ(I-A+γh)x⋆. We show that {xn} is bounded. Since limn→∞θn=0, we can assume, with no loss of generality, that θn∈(0,∥A∥-1), for all n≥0. By Lemma 6, for each 1≤i≤m, we have
(23)∥zn,i-x⋆∥≤∥xn-x⋆∥.
This implies that(24)∥yn-x⋆∥≤∥αnxn+∑i=1mβn,izn,i+γnT(tn)xn+ηnS(tn)xn-x⋆∥≤αn∥xn-x⋆∥+∑i=1mβn,i∥zn,i-x⋆∥+γn∥T(tn)xn-x⋆∥+ηn∥S(tn)xn-x⋆∥≤αn∥xn-x⋆∥+∑i=1mβn,i∥xn-x⋆∥+γn∥xn-x⋆∥+ηn∥xn-x⋆∥=∥xn-x⋆∥.
It follows from Lemma 3 that
(25)∥xn+1-x⋆∥=∥θn(γh(xn)-Ax⋆)+(I-θnA)(yn-x⋆)∥≤θn∥γh(xn)-Ax⋆∥+∥I-θnA∥∥yn-x⋆∥≤θn∥γh(xn)-Ax⋆∥+(1-θnγ¯)∥xn-x⋆∥≤θnγ∥h(xn)-h(x⋆)∥+θn∥γh(x⋆)-Ax⋆∥+(1-θnγ¯)∥xn-x⋆∥≤θnγk∥xn-x⋆∥+θn∥γh(x⋆)-Ax⋆∥+(1-θnγ¯)∥xn-x⋆∥≤(1-θn(γ¯-γk))∥xn-x⋆∥+θn∥γh(x⋆)-Ax⋆∥=(1-θn(γ¯-γk))∥xn-x⋆∥+θn(γ¯-γk)∥γh(x⋆)-Ax⋆∥γ¯-γk≤max{∥xn-x⋆∥,∥γhx⋆-𝒜x⋆∥γ¯-γk}.
This implies that {xn} is bounded and so are {zn,i}, {h(xn)}, {T(tn)xn}, and {S(tn)xn}. Next, we show that limn→∞∥xn-T(tn)xn∥=limn→∞∥xn-S(tn)xn∥=0. Indeed, by Lemma 6, for each 1≤i≤m, we have
(26)∥zn,i-x⋆∥2≤∥xn-x⋆∥2-(1-2λn,ic1,i)∥xn-wn,i∥2-(1-2λn,ic2,i)∥wn,i-zn,i∥2.
Applying Lemma 4 and inequality (26) for k∈{1,2,…,m} we have that
(27)∥yn-x⋆∥2=∥αnxn+∑i=1m‍βn,izn,i+γnT(tn)xn+ηnS(tn)xn-x⋆∥2≤αn∥xn-x⋆∥2+∑i=1m‍βn,i∥zn,i-x⋆∥2+γn∥T(tn)xn-x⋆∥2+ηn∥S(tn)xn-x⋆∥2-αnβn,k∥xn-zn,k∥2-αnγn∥xn-T(tn)xn∥2-αnηn∥xn-S(tn)xn∥2≤αn∥xn-x⋆∥2+∑i=1m‍βn,i∥xn-x⋆∥2+γn∥xn-x⋆∥2+ηn∥xn-x⋆∥2-αnβn,k∥xn-zn,k∥2-βn,k(1-2λn,kc1,k)∥xn-wn,k∥2-αnγn∥xn-T(tn)xn∥2-αnηn∥xn-S(tn)xn∥2=∥xn-x⋆∥2-βn,k(1-2λn,kc1,k)∥xn-wn,k∥2-αnβn,k∥xn-zn,k∥2-αnγn∥xn-T(tn)xn∥2-αnηn∥xn-S(tn)xn∥2.
We now compute
(28)∥xn+1-x⋆∥2=∥θn(γh(xn)-Ax⋆)+(I-θnA)(yn-x⋆)∥2≤θn2∥γh(xn)-Ax⋆∥2+(1-θnγ¯)2∥yn-x⋆∥2+2θn(1-θnγ¯)∥γh(xn)-Ax⋆∥∥yn-x⋆∥≤θn2∥γh(xn)-Ax⋆∥2+(1-θnγ¯)2∥xn-x⋆∥2+2θn(1-θnγ¯)∥γh(xn)-Ax⋆∥∥xn-x⋆∥-(1-θnγ¯)2βn,k(1-2λn,kc1,k)∥xn-wn,k∥2-(1-θnγ¯)2αnβn,k∥xn-zn,k∥2-(1-θnγ¯)2αnγn∥xn-T(tn)xn∥2-(1-θnγ¯)2αnηn∥xn-S(tn)xn∥2.
Therefore,
(29)(1-θnγ¯)2αnβn,k∥xn-zn,k∥2≤∥xn-x⋆∥2-∥xn+1-x⋆∥2+2θn(1-θnγ¯)×∥γh(xn)-Ax⋆∥∥xn-x⋆∥+θn2∥γh(xn)-Ax⋆∥2.
In order to prove that xn→x⋆ as n→∞, we consider the following two cases.Case 1.
Assume that{∥xn-x⋆∥} is a monotone sequence. In other words, for large enough n0, {∥xn-x⋆∥}n≥n0 is either nondecreasing or nonincreasing. Since {∥xn-x⋆∥} is bounded, it is convergent. Since limn→∞θn=0 and {h(xn)} and {xn} are bounded, from (29), we have
(30)limn→∞(1-θnγ¯)2αnβn,k∥xn-zn,k∥2=0,
and by assumption we get
(31)limn→∞∥xn-zn,k∥=0,1≤k≤m.
By similar argument we can obtain that
(32)limn→∞∥xn-wn,k∥=limn→∞∥xn-T(tn)xn∥=limn→∞∥xn-S(tn)xn∥=0.
Further, for all h≥0 and n≥0, we see that
(33)∥xn-T(h)xn∥≤∥xn-T(tn)xn∥+∥T(tn)xn-T(h)T(tn)xn∥+∥T(h)T(tn)xn-T(h)xn∥≤2∥xn-T(tn)xn∥+supx∈{xn}∥T(tn)xn-T(h)T(tn)xn∥.
Since {T(t)} is u.a.r. nonexpansive semigroup and limn→∞tn=∞, we have
(34)limn→∞∥xn-T(h)xn∥=0.
Similarly, for all h≥0 and n≥0, we obtain that
(35)limn→∞∥xn-S(h)xn∥=0.
Next, we show that
(36)limsupn→∞〈(A-γh)x⋆,x⋆-xn〉≤0.
To show this inequality, we choose a subsequence {xni} of {xn} such that
(37)limi→∞〈(A-γh)x⋆,x⋆-xni〉=limsupn→∞〈(A-γh)x⋆,x⋆-xn〉.
Since {xni} is bounded, there exists a subsequence {xnij} of {xni} which converges weakly to x~. Without loss of generality, we can assume that xnj⇀x~. Consider
(38)∥xnj-T(t)x~∥≤∥xnj-T(t)xnj∥+∥T(t)xnj-T(t)x~∥≤∥xnj-T(t)xnj∥+∥xnj-x~∥.
Thus, we have
(39)limsupn→∞∥xni-T(t)x~∥≤limsupn→∞∥xni-x~∥.
By the Opial property of the Hilbert space H we obtain T(t)x~=x~ for all t≥0. Similarly we have that S(t)x~=x~ for all t≥0. This implies that x~∈F(𝒯)⋂‍F(𝒮). Now we show that x~∈Sol(fi,C). For each 1≤i≤m, since fi(x,·) is convex on C for each x∈C, we see that
(40)wn,i=argmin{λn,ifi(xn,y)+12∥y-xn∥2:y∈C}
if and only if
(41)0∈∂2(fi(xn,y)+12∥y-xn∥2)(wn,i)+NC(wn,i),
where NC(x) is the (outward) normal cone of C at x∈C. This follows that
(42)0=λn,iv+wn,i-xn+un,
where v∈∂2fi(xn,wn,i) and un∈NC(wn,i). By the definition of the normal cone NC we have
(43)〈wn,i-xn,y-wn,i〉≥λn,i〈v,wn,i-y〉,∀y∈C.
Since fi(xn,·) is subdifferentiable on C, by the well-known Moreau-Rockafellar theorem [31] (also see [6]), for v∈∂2fi(xn,wn,i), we have
(44)fi(xn,y)-fi(xn,wn,i)≥〈v,y-wn,i〉,∀y∈C.
Combining this with (43), we have
(45)λn,i(fi(xn,y)-fi(xn,wn,i))≥〈wn,i-xn,wn,i-y〉,∀y∈C.
In particular, we have
(46)λnj,i(fi(xnj,y)-fi(xnj,wn,i))≥〈wnj,i-xnj,wnj,i-y〉,∀y∈C.
Since xnj⇀x~, it follows from (32) that wnj,i⇀x~. And thus we have
(47)fi(x~,y)≥0,∀y∈C.
This implies that x~∈Sol(fi,C) and hence x~∈Ω. Since x⋆=ProjΩ(I-A+γh)x⋆ and x~∈Ω, we have
(48)limsupn→∞〈(A-γh)x⋆,x⋆-xn〉=limi→∞〈(A-γh)x⋆,x⋆-xni〉=〈(A-γh)x⋆,x⋆-x~〉≤0.
From Lemma 1, it follows that
(49)∥xn+1-x⋆∥2≤∥(I-θnA)(yn-x⋆)∥2+2θn〈γh(xn)-Ax⋆,xn+1-x⋆〉≤(1-θnγ¯)2∥xn-x⋆∥2+2θnγ〈h(xn)-h(x⋆),xn+1-x⋆〉+2θn〈γh(x⋆)-Ax⋆,xn+1-x⋆〉≤(1-θnγ¯)2∥xn-x⋆∥2+2θnkγ∥xn-x⋆∥∥xn+1-x⋆∥+2θn〈γh(x⋆)-Ax⋆,xn+1-x⋆〉≤(1-θnγ¯)2∥xn-x⋆∥2+θnkγ(∥xn-x⋆∥2+∥xn+1-x⋆∥2)+2θn〈γh(x⋆)-Ax⋆,xn+1-x⋆〉≤((1-θnγ¯)2+θnkγ)∥xn-x⋆∥2+θnγk∥xn+1-x⋆∥2+2θn〈γh(x⋆)-Ax⋆,xn+1-x⋆〉.
This implies that
(50)∥xn+1-x⋆∥2≤1-2θnγ¯+(θnγ¯)2+θnγk1-θnγk∥xn-x⋆∥2+2θn1-θnγk〈γh(x⋆)-Ax⋆,xn+1-x⋆〉=(1-2(γ¯-γk)θn1-θnγk)∥xn-x⋆∥2+(θnγ¯)21-θnγk∥xn-x⋆∥2+2θn1-θnγk〈γh(x⋆)-Ax⋆,xn+1-x⋆〉≤(1-2(γ¯-γk)θn1-θnγk)∥xn-x⋆∥2+2(γ¯-γk)θn1-θnγk×((θnγ¯2)M2(γ¯-γk)+1γ¯-γk〈γh(x⋆)-Ax⋆,xn+1-x⋆〉)=(1-ηn)∥xn-x⋆∥2+ηnδn,
where
(51)M=sup{∥xn-x⋆∥2:n≥0},ηn=2(γ¯-γk)θn1-θnγk,δn=(θnγ¯2)M2(γ¯-γk)+1γ¯-γk〈γhx⋆-Ax⋆,xn+1-x⋆〉.
It is easy to see that ηn→0, ∑n=1∞‍ηn=∞, and limsupn→∞δn≤0. Hence, by Lemma 2 the sequence {xn} converges strongly to x⋆. From (32) we have that {wn,i} and {zn,i} converge strongly to x⋆.Case 2.
Assume that{∥xn-x⋆∥} is not a monotone sequence. Then, we can define an integer sequence {τ(n)} for all n≥n0 (for some large enough n0) by
(52)τ(n):=max{k∈ℕ,k≤n:∥xk-x⋆∥<∥xk+1-x⋆∥}.
Clearly, τ is a nondecreasing sequence such that τ(n)→∞ as n→∞, and for all n≥n0,
(53)∥xτ(n)-x⋆∥<∥xτ(n)+1-x⋆∥.
From (33) we obtain that
(54)limn→∞∥xτ(n)-zτ(n),i∥=limn→∞∥xτ(n)-wτ(n),i∥=limn→∞∥xτ(n)-T(tτn)xτ(n)∥=0.
Following an argument similar to that in Case 1 we have
(55)∥xτ(n)+1-x⋆∥2≤(1-ητ(n))∥xτ(n)-x⋆∥2+ητ(n)δτ(n),
where ητ(n)→0, ∑n=1∞‍ητ(n)=∞, and limsupn→∞δτ(n)≤0. Hence, by Lemma 2, we obtain limn→∞∥xτ(n)-x⋆∥=0 and limn→∞∥xτ(n)+1-x⋆∥=0. Now Lemma 5 implies that
(56)0≤∥xn-x⋆∥≤max{∥xτ(n)-x⋆∥,∥xn-x⋆∥}≤∥xτ(n)+1-x⋆∥.
Therefore, {xn} converges strongly to x⋆=ProjΩ(I-A+γh)x⋆. This completes the proof.


## 4. Application

In this section, we consider a particular Fan inequality corresponding to the functionf defined by the following: for every x,y∈C(57)f(x,y)=〈F(x),y-x〉,
where F:C→H. Then, we obtain the classical variational inequality as follows.
(58)Findz∈Csuchthat〈F(z),y-z〉≥0,∀y∈C.
The set of solutions of this problem is denoted by VI(F,C). In that particular case, the solution yn of the minimization problem
(59)argmin{λnf(xn,y)+12∥y-xn∥2:y∈C}
can be expressed as
(60)yn=PC(xn-λnF(xn)).
Let F be L-Lipschitz continuous on C. Then
(61)f(x,y)+f(y,z)-f(x,z)=〈F(x)-F(y),y-z〉,x,y,z∈C.
Therefore,
(62)|〈F(x)-F(y),y-z〉|≤L∥x-y∥∥y-z∥≤L2(∥x-y∥2+∥y-z∥2)
hence, f satisfies Lipschitz-type continuous condition with c1=c2=L/2.Using Theorem7 we obtain the following convergence theorem.Theorem 8.
LetC be a nonempty closed convex subset of a real Hilbert space H and let Fi:C→H  (i=1,2,…,m) be functions such that, for each 1≤i≤m, Fi is monotone and L-Lipschitz continuous on C. Let 𝒯={T(u):u≥0} and 𝒮={S(u):u≥0} be two u.a.r. nonexpansive self-mapping semigroups on C such that Ω=⋂i=1m‍VI(Fi,C)⋂‍F(𝒯)⋂‍F(𝒮)≠∅. Assume that h is a k-contraction of C into itself and A is a strongly positive bounded linear self-adjoint operator on H with coefficient γ¯<1 and 0<γ<γ¯/k. Let x0∈C and let {xn}, {wn,i}, and {zn,i} be sequences generated by
(63)wn,i=PC(xn-λn,iFi(xn)),i∈{1,2,…,m},zn,i=PC(xn-λn,iFi(wn,i)),i∈{1,2,…,m},yn=αnxn+∑i=1m‍βn,izn,i+γnT(tn)xn+ηnS(tn)xn,xn+1=θnγh(xn)+(I-θnA)yn,∀n≥0,
where αn+∑i=1m‍βn,i+γn+ηn=1 and {αn}, {βn,i}, {γn}, {ηn}, {λn,i}, and {θn} satisfy the following conditions:  (i)
limn→∞tn=∞,(ii)
{θn}⊂]0,1[, limn→∞θn=0, and ∑n=1∞‍θn=∞,(iii)
{λn,i}⊂[a,b]⊂]0,1/L[,(iv)
{αn},{βn,i},{γn},{ηn}⊂]0,1[, liminfnαnβn,i>0, liminfnαnγn>0, liminfnαnηn>0, and liminfnβn,i(1-2λn,ic1,i)>0 for each 1≤i≤m.Then, the sequence {xn} converges strongly to x⋆∈⋂i=1m‍VI(Fi,C)⋂‍F(𝒯)⋂‍F(𝒮) which solves the variational inequality
(64)〈(A-γh)x⋆,x-x⋆〉≥0,∀x∈Ω.In [32], Baillon proved a strong mean convergence theorem for nonexpansive mappings, and it was generalized in [33]. It follows from the above proof that Theorems 7 is valid for nonexpansive mappings. Thus, we have the following mean ergodic theorems for nonexpansive mappings in a Hilbert space.Theorem 9.
LetC be a nonempty closed convex subset of a real Hilbert space H and let Fi:C→H  (i=1,2,…,m) be functions such that for each 1≤i≤m, Fi is monotone and L-Lipschitz continuous on C. Let T and S be two nonexpansive mappings on C such that Ω=⋂i=1m‍VI(Fi,C)⋂‍F(T)⋂‍F(S)≠∅. Assume that h is a k-contraction of C into itself and A is a strongly positive bounded linear self-adjoint operator on H with coefficient γ¯<1 and 0<γ<γ¯/k. Let {xn}, {wn,i}, and {zn,i} be sequences generated by x0∈C and by
(65)wn,i=PC(xn-λn,iFi(xn)),i∈{1,2,…,m},zn,i=PC(xn-λn,iFi(wn,i)),i∈{1,2,…,m},yn=αnxn+∑i=1m‍βn,izn,i+γnTnxn+ηnSnxn,xn+1=θnγh(xn)+(I-θnA)yn,∀n≥0,
where αn+∑i=1m‍βn,i+γn+ηn=1 and {αn}, {βn,i}, {γn}, {ηn}, {λn,i}, and {θn} satisfy the following conditions:(i)
{θn}⊂]0,1[, limn→∞θn=0, and ∑n=1∞‍θn=∞,(ii)
{λn,i}⊂[a,b]⊂]0,1/L[, where L=max{2c1,i,2c2,i,1≤i≤m},(iii)
{αn},{βn,i},{γn},{ηn}⊂]0,1[, liminfnαnβn,i>0, liminfnαnγn>0, liminfnαnηn>0, and liminfnβn,i(1-2λn,ic1,i)>0 for each 1≤i≤m.Then, the sequence {xn} converges strongly to x⋆∈⋂i=1m‍VI(Fi,C)⋂‍F(𝒯)⋂‍F(𝒮) which solves the variational inequality
(66)〈(A-γh)x⋆,x-x⋆〉≥0,∀x∈Ω.



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*Source: 178053-2013-12-31.xml*