# A New Modified Hybrid Steepest-Descent by Using a Viscosity Approximation Method with a Weakly Contractive Mapping for a System of Equilibrium Problems and Fixed Point Problems with Minimization Problems

Authors: Uamporn Witthayarat; Thanyarat Jitpeera; Poom Kumam
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The purpose of this paper is to consider a modified hybrid steepest-descent method by using a viscosity approximation method with a weakly contractive mapping for finding the common element of the set of a common fixed point for an infinite family of nonexpansive mappings and the set of solutions of a system of an equilibrium problem. The sequence is generated from an arbitrary initial point which converges in norm to the unique solution of the variational inequality under some suitable conditions in a real Hilbert space. The results presented in this paper generalize and improve the results of Moudafi (2000), Marino and Xu (2006), Tian (2010), Saeidi (2010), and some others. Finally, we give an application to minimization problems and a numerical example which support our main theorem in the last part.


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## Body

## 1. Introduction

Theconvex feasibility problem (CFP) is the problem for finding points in the intersection of a finite family of closed convex subsets Ci,i=1,2,…,N in the framework of Hilbert spaces, that is, to find a point x^ such that
(1.1)x^∈⋂i=1N‍Ci.
This problem plays an extremely important role in various fields, especially in applied mathematics and physical sciences; moreover, it has a great impact role on the real-world applications (see [1, 2]). The well-known applications are the theory of optimization [3, 4], image reconstruction by the projection method [5], signal processing problems [6], and model for the problem in sensor networks [7], as some powerful examples.We focus on the important subclass of convex feasibility problems, in which finitely many sets are given. Each set can be specified in various forms, such as the fixed point set of a nonexpansive mapping, the set of solutions of the variational inequality, and the set of solutions to an equilibrium problem. In a framework of Hilbert spaces, there are some applications of convex feasibility problems in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [8].Throughout this paper, we assume thatH is a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, and let C be a nonempty closed convex subset of H. Let ℑ={Fj}j∈Γ be bifunctions from C×C to ℝ, where ℝ is the set of real numbers, and Γ is an arbitrary index set. The system of equilibrium problems is to find x∈C such that
(1.2)Fj(x,y)≥0,∀y∈C,j∈Γ.
If Γ is a singleton, then problem (1.2), reduced to the equilibrium problems, is to find x∈C such that
(1.3)F(x,y)≥0,∀y∈C.
The set of solution of (1.3) is denoted by EP(F). The above formulation (1.3) was shown in [7] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In other words, the EP(F) is a unifying model for several problems arising in physics, engineering, science, optimization, economics, and so forth; Combettes and Hirstoaga [9] introduced an iterative scheme for finding a common element in the solution set of problem (1.3) in a Hilbert space.The equilibrium problems include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, and economics and the equilibrium problems; as special cases see, for example, [7, 10–14]. Some methods have been proposed to solve the equilibrium problem; see, for instance, [15–22].LetB:C→H be a mapping. The variational  inequality problem, denoted by VI(C,B), is to find x∈C such that
(1.4)〈Bx,y-x〉≥0,∀y∈C.
Existence and uniqueness of solutions are the most important problems of VI(C,B). The variational inequality problem has been extensively studied in the literature, see, for example, [23, 24] and the references therein. It is known that if B is a strong monotone and Lipschitzian mapping on C, then VI(C,B) has a unique solution. Variational inequalities are among the most interesting and important mathematical problems and have been studied intensively in the past years since they have wide applications in the optimization and control, economics and transportation equilibrium, and engineering science. For these reasons, many existence results and iterative algorithms for various variational inclusions have been studied extensively by many authors. For details, see [2, 7, 23–25] and references therein.On the other hand, iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences.A mappingT:C→C is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥, for all x,y∈C. We use F(T) to denote the set of fixed points of T, that is, F(T)={x∈C:Tx=x}. Recall that a self-mapping f:C→C is a contractive mapping on C if there exists a constant α∈[0,1) such that ∥f(x)-f(y)∥≤α∥x-y∥, for all x,y∈C. A mapping B:H→H is said to be a k-Lipschitzian if there exists a constant k>0 such that ∥Bx-By∥≤k∥x-y∥, for all x,y∈C. The concept of quasi-nonexpansive was introduced by Diaz and Metcalf [26]. The mapping T is said to be quasi-nonexpansive if ∥Tx-p∥≤∥x-p∥, for all x∈C and p∈F(T).In 2000, Moudafi [27] introduced the viscosity approximation method for a nonexpansive mapping T:C→C. Let f be a contraction on H, starting with an arbitrary initial x0∈H, defining a sequence {xn} recursively by
(1.5)xn+1=αnf(xn)+(1-αn)Txn,∀n≥0,
where αn is a sequence in (0,1). Xu [28] proved that under certain appropriate conditions on {αn}, the sequence {xn} generated by (1.5) converges strongly to the unique solution x*∈C of the variational inequality
(1.6)〈(I-f)x*,x-x*〉≥0,∀x∈F(T).
In 2006, Marino and Xu [29] introduced the following iterative scheme:
(1.7)xn+1=αnγf(xn)+(I-αnA)Txn,∀n≥0.
It was proved that if the sequence {αn} of parameters satisfies appropriate conditions, then the sequence {xn} generated by (1.7) converges strongly to the unique solution of the variational inequality
(1.8)〈(A-γf)x*,x-x*〉≥0,∀x∈F(T),
which is the optimality condition for the minimization problem
(1.9)minx∈F(T)12〈Ax,x〉-h(x),
where h is a potential function for γf (i.e.,h′(x)=γf(x), for x∈H). Assume A is strongly positive bounded linear operator. It can be referred that there is a constant γ->0 which satisfies the following property:
(1.10)〈Ax,x〉≥γ-∥x∥2,∀x∈H.
In 2007, Suzuki [30] extended Moudafi's viscosity approximations with MeirKeeler contractions and presented very simple proofs of Xu's theorems by concidering Moudafi's approximations.On the other hand, Yamada [31] introduced the following hybrid iterative scheme for finding the variational inequality:
(1.11)xn+1=Txn-μλnB(Txn),∀n≥0,
where B is k-Lipschitzian and η-strongly monotone operator with k>0,η>0,0<μ<2η/k2, then he proved that if {λn} satisfies some appropriate conditions, then {xn} generated by (1.11) converges strongly to the unique solution of variational inequality
(1.12)〈Bx*,x-x*〉≥0,x∈F(T).In 2010, Tian [32] combined (1.7) and (1.11) and considered the following general iterative method:
(1.13)xn+1=αnγf(xn)+(I-μαnB)Txn,∀n≥0.
If the sequence {αn} of parameters satisfies appropriate conditions, then the sequence {xn} generated by (1.13) converges strongly to the unique solution x*∈C of the variational inequality
(1.14)〈(γf-μB)x*,x-x*〉≤0,∀x∈F(T).Later, Saeidi [33] introduced the following modified hybrid steepest-descent iterative algorithm for finding a common element of the set of solutions of a system of equilibrium problems for a family ℱ={Fj:C×C→ℝ,j=1,2,…,M} and the set of common fixed points for a family of infinitely nonexpansive mappings 𝒮={Si:C→C}, with respect to W-mappings (see (2.14)). The proposed scheme was defined by
(1.15)yn=WnJrM,nFM⋯Jr2,nF2Jr1,nF1xn,xn+1=βxn+(1-β)(I-λnB)yn,∀n∈ℕ,
where B is a relaxed (γ,r)-cocoercive, k-Lipschitzian mapping such that r>γk2. Then, under weaker hypotheses on coefficients, he proved the strong convergence of the proposed iterative algorithm to the unique solution of the variational inequality. Zhang et al. [34] introduced a modified iterative algorithm by using a viscosity approximation method with a weakly contractive mapping with respect to W-mappings (see (2.14)). They defined
(1.16)xn+1=αnΦxn+(1-αn)WnJrM,nFM…Jr2,nF2Jr1,nF1xn,∀n∈ℕ,
where Φ is a π-weakly contractive self-mapping on C, and {αn} is a sequence in (0,1). They proved that under certain appropriate conditions imposed on {αn}, the proposed iterative algorithm converges strongly to the common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of a finite family of equilibrium problems.In this paper, motivated and inspired by the previously mentioned above results, we consider a modified hybrid steepest-descent method by using a viscosity approximation method with a weakly contractive mapping for finding the common element of the set of common fixed points for an infinite family of nonexpansive mappings with weakly contractive mappings and the set of solutions of a system of equilibrium problems. The sequence generated from an arbitrary initial pointx0∈H which will converge in norm to the unique solution of the variational inequality under some suitable conditions in a real Hilbert space. Furthermore, we give an application to minimization problems and a numerical example which support our main theorem in the last part.


## 2. Preliminaries

LetH be a real Hilbert space and C be a nonempty closed convex subset of H. We denote weak convergence and strong convergence by notations ⇀ and →, respectively. Recall that when the metric (nearest point) projection PC from H onto C assigns to each x∈H, the unique point in PCx∈C satisfies the property
(2.1)∥x-PCx∥=miny∈C∥x-y∥.
The following characterizes the projection PC.An important problem is how to find a solution ofVI(C,B). It is known that
(2.2)u∈VI(C,B)⟺u=PC(u-λBu),
where λ>0 is an arbitrarily fixed constant, and PC is the projection of H onto C.We recall some lemmas which will be needed in the rest of this paper.Lemma 2.1.
For a givenz∈H, u∈C,
(2.3)u=PCz⟺〈u-z,v-u〉≥0,∀v∈C.
It is well known that PC is a firmly nonexpansive mapping of H onto C and satisfies
(2.4)∥PCx-PCy∥2≤〈PCx-PCy,x-y〉,∀x,y∈H.
Moreover, PCx is characterized by the following properties: PCx∈C and for all x∈H,y∈C,
(2.5)〈x-PCx,y-PCx〉≤0.Definition 2.2.
A mappingΦ:C→H with domain D(Φ) and range R(Φ) in H, Alber and Guerre-Delabriere [35] defined a π-weakly contractive mapping by the following:
(2.6)∥Φx-Φy∥≤∥x-y∥-π(∥x-y∥),∀x,y∈D(Φ),
for some π:[0,+∞)→[0,+∞) which is a continuous and strictly increasing function such that π is positive on (0,+∞) and π(0)=0. If π(t)≡(1-k)t, then Φ is said to be contractive mapping with the contractive coefficient k. If π(t)=0, then Φ is said to be nonexpansive. If π(t)=0 and y=Φy, then Φ with a fixed point y is said to be qusi-nonexpansive.Definition 2.3.
A mappingB:C→H is said to be an η-strongly monotone if there exists a constant η>0 with the following property:
(2.7)〈Bx-By,x-y〉≥η∥x-y∥2,∀x,y∈C.Definition 2.4.
A mappingB:C→H is said to be relaxed (γ,r)-cocoercive if there exist two constants γ>0 and r>0 which satisfies the following property:
(2.8)〈Bx-By,x-y〉≥-γ∥Bx-By∥2+r∥x-y∥2,∀x,y∈C.Lemma 2.5 (see [28]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.9)an+1≤(1-ln)an+σn,n≥0,
where {ln} is a sequence in (0,1), and {σn} is a sequence in ℝ such that  (1)
∑n=1∞ln=∞,(2)
limsupn→∞σn/ln≤0 or ∑n=1∞|σn|<∞. Then limn→∞an=0.Lemma 2.6 (see [36]).
LetC be a closed convex subset of a real Hilbert space H and let T:C→C be a nonexpansive mapping. Then I-T is demiclosed at zero, that is, xn⇀x,xn-Txn→0 implies x=Tx.Lemma 2.7 (see [37]).
LetC be a closed convex subset of H. Let {xn} be a bounded sequence in H. Assume that  (1)
the weakω-limit set ωw(xn)⊂C,(2)
for eachz∈C, limn→∞∥xn-z∥ exists. Then {xn} is weakly convergent to a point in C.Lemma 2.8 (see [38]).
Each Hilbert spaceH satisfies Opial's condition, that is, for any sequence {xn}⊂H with xn⇀x, the inequality,
(2.10)liminfn→∞∥xn-x∥<liminfn→∞∥xn-y∥,
holds for each y∈H with y≠x.Lemma 2.9 (see [39]).
Each Hilbert spaceH satisfies the Kadec-Klee property, that is, for any sequence {xn} with xn⇀x and together with ∥xn∥→∥x∥ implies ∥xn-x∥→0.For solving the equilibrium problem, let us give the following assumptions for a bifunctionF of C×C into ℝ which were imposed in [9, 40]:  (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.Lemma 2.10 (see [9, 40]).
LetC be a nonempty closed convex subset of H, and let F be a bifunction of C×C into ℝ satisfying (A1)–(A4). If r>0 and x∈H, then there exists z∈C such that
(2.11)F(z,y)+1r〈y-z,z-x〉≥0.Lemma 2.11 (see [9]).
LetC be a nonempty closed convex subset of H, and let F be a bifunction of C×C into ℝ satisfying (A1)–(A4). For r>0 and x∈H, define a mapping JrF:H→C as follows:
(2.12)JrF(x)={z∈C:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C},
for all x∈H. Then, the following conclusions hold that  (1)
JrF is single-valued;(2)
JrF is firmly nonexpansive, that is, for any x,y∈H,
(2.13)∥JrF(x)-JrF(y)∥2≤〈JrF(x)-JrF(y),x-y〉;(3)
F(JrF)=EP(F);(4)
EP(F) is closed and convex.A family of nonexpansive mappings has been considered by many authors (see [41–52] and references therein). Recently, Shang et al. [47] improved the results of Kim and Xu [53] from a single mapping to a finite family of mappings in the framework of Hilbert spaces.Now, we consider the mappingWn defined, as in Shimoji and Takahashi [48], by
(2.14)Un,n+1≔I,Un,n≔γnSnUn,n+1+(1-γn)I,Un,n-1≔γn-1Sn-1Un,n+(1-γn-1)I,⋮Un,k≔γkSkUn,k+1+(1-γk)I,Un,k-1≔γk-1Sk-1Un,k+(1-γk-1)I,⋮Un,2:=γ2S2Un,3+(1-γ2)I,Wn≔Un,1=γ1S1Un,2+(1-γ1)I,
where γn,γn-1,…,γ1 are real numbers such that 0≤γn≤1 and S1,S2,… are an infinite family of mappings of H into itself. Nonexpansivity of each Si ensures the nonexpansivity of Wn.Lemma 2.12 (see [48]).
LetH be a real Hilbert space H. Let S1,S2,… be nonexpansive mappings from H into itself such that ⋂n=1∞F(Sn)≠∅ and {γ1,γ2,…} are real numbers such that 0<γn≤b<1, for all n≥1. Then, for every x∈H and k∈N, the limit limn→∞Un,kx exists.Using Lemma2.12, one can define the mapping W from H into itself as follows:
(2.15)limn→∞Wnx=limn→∞Un,1x,∀x∈H.SuchW is called the W-mapping generated by S1,S2,… and γ1,γ2,….Lemma 2.13 (see [48]).
LetH be a real Hilbert space H. Let S1,S2,… be nonexpansive mappings from H into itself such that ⋂n=1∞‍F(Sn)≠∅ and {γ1,γ2,…} are real numbers such that 0<γn≤b<1, for all n≥1. Then, F(W)=⋂n=1∞‍F(Sn).


## 3. Main Results

In this section, we will introduce an iterative scheme by using a modified hybrid steepest-descent method for finding the common element of the set of common fixed points for an infinite family of nonexpansive mappings with weakly contractive mappings and the set of solutions of a system of equilibrium problems in a real Hilbert space.Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H, such that C±C⊂C. Let 𝒮={Si:C→C} be a family of infinitely nonexpansive mappings, and let ℱ={Fj:j=1,2,3,…,M} be a finite family of bifunctions C×C to ℝ satisfying (A1)–(A4). Assume that Θ≔(⋂i=1∞F(Si))∩(⋂j=1MEP(Fj))≠∅. Let B be a k-Lipschitzian and η-strongly monotone mapping on C with k>0,η>0. Let Φ be a π-weakly contractive self-mapping on C with α∈[0,1). Denote the collection of all weakly contractive Φ on C by 𝒞. Let 0<μ<2η/k2 and 0<γ<μ(η-μk2/2)=τ. Let the mapping Wn be defined by (2.14) and {rj,n}j=1M be a sequence in (0,∞). If {xn} is the sequence generated by x1∈C and
(3.1)wn=WnJrM,nFM⋯Jr2,nF2Jr1,nF1xn,xn+1=αnγΦ(xn)+(I-αnμB)wn,∀n∈ℕ,
where {αn} is a sequence in (0,1) and satisfies the following conditions:  (C1)
limn→∞αn=0  and ∑n=1∞αn=∞;(C2)
limn→∞|αn+1-αn|=0;(C3)
limn→∞|γn+1-γn|=0;(C4)
liminfn→∞rj,n>0, for all j∈{1,2,…,M}. Then, the sequence {xn} converges strongly to x*=PΘ(I-μB+γΦ)x* which is the unique solution of the variational inequality
(3.2)〈(μB-γΦ)x*,x*-x〉≤0,∀x∈Θ,
which is the optimality condition for the minimization problem
(3.3)minx∈F(T)12〈Bx,x〉-h(x),
where h is a potential function for γΦ (i.e.,  h′(x)=γΦ(x), for x∈H).Proof.
We will divide the proof of Theorem3.1 into several steps.
Step
1. We will show that {xn} is bounded. Let p∈Θ. By taking ℑnk=Jrk,nFkJrk-1,nFk-1Jrk-2,nFk-2⋯Jr2,nF2Jr1,nF1 for k∈{1,2,3,…,M} and ℑn0=I, for all n∈ℕ. Since Jrk,nFk is nonexpansive for each k=1,2,3,…,M, then, we have(3.4)∥ℑnMxn-p∥=∥ℑnMxn-ℑnMp∥≤∥xn-p∥.
From Lemmas 2.11 and 2.12, it follows that
(3.5)∥xn+1-p∥=∥αnγΦ(xn)+(I-αnμB)wn-p∥=∥αnγΦ(xn)+(I-αnμB)WnℑnMxn-p∥=∥αn(γΦ(xn)-μBWnℑnM(p))+(I-αnμB)WnℑnMxn-(I-αnμB)WnℑnMp∥≤αn∥γΦ(xn)-γΦ(p)∥+αn∥γΦ(p)-μBWnℑnM(p)∥+∥(I-αnμB)WnℑnMxn-(I-αnμB)WnℑnMp∥≤αnγ{∥xn-p∥-π(∥xn-p∥)}+αn∥γΦ(p)-μB(p)∥+(1-αnτ)∥xn-p∥≤(1-αn(τ-γ))∥xn-p∥+αn(τ-γ)∥γΦ(p)-μB(p)∥τ-γ≤max{∥xn-p∥,∥γΦ(p)-μB(p)∥τ-γ}.
By mathematical induction, it becomes
(3.6)∥xn-p∥≤max{∥x1-p∥,∥γΦ(p)+μB(p)∥τ-γ},∀n≥1,
and we obtain that {xn} is bounded. So are {WnℑnM(xn)} and {Φ(xn)}.
Step
2. We claim that(3.7)limn→∞∥ℑnkxn-ℑn+1kxn∥=0,
for every k∈{1,2,3,…,M}. From Step 2 of the proof in [54, Theorem 3.1], we have for k∈{1,2,3,…,M},
(3.8)limn→∞∥Jrk,n+1Fkxn-Jrk,nFkxn∥=0.
Note that for every k∈{1,2,3,…,M}, we obtain
(3.9)ℑnk=Jrk,nFkJrk-1,nFk-1Jrk-2,nFk-2⋯Jr2,nF2Jr1,nF1=Jrk,nFkℑnk-1.
So, we have
(3.10)∥ℑnkxn-ℑn+1kxn∥=∥Jrk,nFkℑnk-1xn-Jrk,n+1Fkℑn+1k-1xn∥≤∥Jrk,nFkℑnk-1xn-Jrk,n+1Fkℑnk-1xn∥+∥Jrk,n+1Fkℑnk-1xn-Jrk,n+1Fkℑn+1k-1xn∥≤∥Jrk,nFkℑnk-1xn-Jrk,n+1Fkℑnk-1xn∥+∥ℑnk-1xn-ℑn+1k-1xn∥≤∥Jrk,nFkℑnk-1xn-Jrk,n+1Fkℑnk-1xn∥+∥Jrk-1,nFk-1ℑnk-2xn-Jrk-1,n+1Fk-1ℑnk-2xn∥+∥ℑnk-2xn-ℑn+1k-2xn∥≤∥Jrk,nFkℑnk-1xn-Jrk,n+1Fkℑnk-1xn∥+∥Jrk-1,nFk-1ℑnk-2xn-Jrk-1,n+1Fk-1ℑnk-2xn∥+⋯+∥Jr2,nF2ℑn1xn-Jr2,n+1F2ℑn1xn∥+∥Jr1,nF1xn-Jr1,n+1F1xn∥.
Now, apply (3.8) to (3.10), we conclude (3.7).
Step
3. We may assume that Bn=(I-αnμB). Let {wn} be a bounded sequence in C. Then, we show that limn→∞∥Bn+1wn-Bnwn∥=0. Indeed, since {wn} is bounded and B is a Lipschitzian mapping, now, from condition (C2), we have
(3.11)∥Bn+1wn-Bnwn∥=∥(I-αn+1μB)wn-(I-αnμB)wn∥=∥αn+1μBwn-αnμBwn∥=μ|αn+1-αn|∥Bwn∥≤M1|αn+1-αn|,
where M1 is an approximate constant such that M1≥max{supn≥1∥Bwn∥}. Hence ∥Bn+1wn-Bnwn∥→0 as n→∞.
Step
4. We show that limn→∞∥Wn+1wn-Wnwn∥=0. By the definition of Wn, it follows that
(3.12)∥Wn+1wn-Wnwn∥=∥γn+1,NSNUn+1,N-1wn+(1-γn+1,N)Un+1,N-1wn-γn,NSNUn,N-1wn-(1-γn,N)Un,N-1wn∥≤∥γn+1,NSNUn+1,N-1wn-γn+1,NSNUn,N-1wn∥+∥γn+1,NSNUn,N-1wn-γn,NSNUn,N-1wn∥+∥Un+1,N-1wn-γn+1,NUn+1,N-1wn-Un,N-1wn+γn,NUn,N-1wn∥≤γn+1,N∥SNUn+1,N-1wn-SNUn,N-1wn∥+|γn+1,N-γn,N|∥SNUn,N-1wn∥+∥Un+1,N-1wn-Un,N-1wn∥+∥γn+1,NUn+1,N-1wn-γn,NUn,N-1wn∥≤γn+1,N∥Un+1,N-1wn-Un,N-1wn∥+|γn+1,N-γn,N|∥SNUn,N-1wn∥+∥Un+1,N-1wn-Un,N-1wn∥+∥γn+1,NUn+1,N-1wn-γn,NUn+1,N-1wn∥+∥γn,NUn+1,N-1wn-γn,NUn,N-1wn∥≤γn+1,N∥Un+1,N-1wn-Un,N-1wn∥+|γn+1,N-γn,N|∥SNUn,N-1wn∥+∥Un+1,N-1wn-Un,N-1wn∥+|γn+1,N-γn,N|∥Un+1,N-1wn∥+γn,N∥Un+1,N-1wn-Un,N-1wn∥≤|γn+1,N+1+γn,N|∥Un+1,N-1wn-Un,N-1wn∥+2M2|γn+1,N-γn,N|,
where M2 is an approximate constant such that M2≥max{supn≥1{∥SiUn,i-1wn∥},supn≥0{∥Un+1,i-1wn∥}∣i=1,2,…}. Since 0<γn,i≤1 for all n≥1 and i=1,2,…,N, we compute
(3.13)∥Un+1,N-1wn-Un,N-1wn∥=∥γn+1,N-1SN-1Un+1,N-2wn+(1-γn+1,N-1)Un+1,N-2wn-γn,N-1SN-1Un,N-2wn-(1-γn,N-1)Un,N-2wn∥≤∥γn+1,N-1SN-1Un+1,N-2wn-γn,N-1SN-1Un,N-2wn∥+∥Un+1,N-2wn-γn+1,N-1Un+1,N-2wn-Un,N-2wn+γn,N-1Un,N-2wn∥≤∥γn+1,N-1SN-1Un+1,N-2wn-γn+1,N-1SN-1Un,N-2wn∥+∥γn+1,N-1SN-1Un,N-2wn-γn,N-1SN-1Un,N-2wn∥+∥Un+1,N-2wn-Un,N-2wn∥+∥γn+1,N-1Un+1,N-2wn-γn,N-1Un,N-2wn∥≤γn+1,N-1∥SN-1Un+1,N-2wn-SN-1Un,N-2wn∥+|γn+1,N-1-γn,N-1|∥SN-1Un,N-2wn∥+∥Un+1,N-2wn-Un,N-2wn∥+∥γn+1,N-1Un+1,N-2wn-γn+1,N-1Un,N-2wn∥+∥γn+1,N-1Un,N-2wn-γn,N-1Un,N-2wn∥≤γn+1,N-1∥Un+1,N-2wn-Un,N-2wn∥+|γn+1,N-1-γn,N-1|∥SN-1Un,N-2wn∥+∥Un+1,N-2wn-Un,N-2wn∥+γn+1,N-1∥Un+1,N-2wn-Un,N-2wn∥+|γn+1,N-1-γn,N-1|∥Un,N-2wn∥≤|γn+1,N-1+1+γn+1,N-1|∥Un+1,N-2wn-Un,N-2wn∥+2M3|γn+1,N-1-γn,N-1|,
where M3 is an approximate constant such that M3≥max{supn≥1{∥SiUn,i-1wn∥},supn≥0{∥Un+1,i-1wn∥}∣i=2,3,…}. It follows that
(3.14)∥Un+1,N-1wn-Un,N-1wn∥≤2M3|γn+1,N-1-γn,N-1|+2M3|γn+1,N-2-γn,N-2|+∥Un+1,N-3wn-Un,N-3wn∥≤2M3∑i=2N-1‍|γn+1,i-γn,i|+∥Un+1,1wn-Un,1wn∥=2M3∑i=2N-1‍|γn+1,i-γn,i|+∥γn+1,1S1wn+(1-γn+1,1)wn-γn,1S1wn-(1-γn,1)wn∥≤2M3∑i=1N-1‍|γn+1,i-γn,i|.
Substituting (3.14) into (3.12), it yields that
(3.15)∥Wn+1wn-Wnwn∥≤2M2|γn+1,N-γn,N|+2γn+1,NM3∑i=1N-1‍|γn+1,i-γn,i|≤2M∑i=1N‍|γn+1,i-γn,i|,
where M is an approximate constant such that M≥max{M2,M3}. By condition (C3), we obtain that ∥Wn+1wn-Wnwn∥→0 as n→∞.
Step
5. We will show that
(3.16)limn→∞∥xn+1-xn∥=0.
We observe that
(3.17)∥xn+1-xn∥=∥αnγΦ(xn)+(I-αnμB)wn-αn-1γΦ(xn-1)-(I-αn-1μB)wn-1∥≤∥αnγ(Φ(xn)-Φ(xn-1))+(I-αnμB)wn-(I-αnμB)wn-1∥+∥γ(αn-αn-1)Φ(xn-1)+μ(αn-αn-1)Bwn-1∥≤αnγ{∥xn-xn-1∥-π(∥xn-xn+1∥)}+(1-αnτ)∥wn-wn-1∥+2M4|αn-αn-1|≤αnγ∥xn-xn-1∥+(1-αnτ)∥wn-wn-1∥+2M4|αn-αn-1|,
where M4 is an approximate constant such that M4≥max{supn≥2{∥Φ(xn-1)∥},supn≥2{∥Bwn-1∥}}. We compute
(3.18)∥wn-wn-1∥=∥WnℑnMxn-Wn-1ℑn-1Mxn-1∥≤∥WnℑnMxn-WnℑnMxn-1∥+∥WnℑnMxn-1-Wnℑn-1Mxn-1∥+∥Wnℑn-1Mxn-1-Wn-1ℑn-1Mxn-1∥≤∥xn-xn-1∥+∥ℑnMxn-1-ℑn-1Mxn-1∥+∥Wnℑn-1Mxn-1-Wn-1ℑn-1Mxn-1∥.
By Step 2 and Step 4, we have immediately concluded from (3.17) that
(3.19)∥xn+1-xn∥≤αnγ∥xn-xn-1∥+(1-αnτ)∥wn-wn-1∥+2M4|αn-αn-1|≤αnγ∥xn-xn-1∥+(1-αnτ)∥xn-xn-1∥+2M4|αn-αn-1|≤(1-αn(τ-γ))∥xn-xn-1∥+2M4|αn-αn-1|.
By Lemma 2.5, we have limn→∞∥xn+1-xn∥=0.
Step
6. We will show that
(3.20)limn→∞∥ℑnkxn-ℑnk+1xn∥=0,∀k=1,2,…,M-1.
For any p∈Θ and for all k=1,2,…,M-1, note that Jrk+1,nFk+1 is firmly nonexpansive. Then by Lemma 2.11, we get
(3.21)∥ℑnk+1xn-p∥2=∥Jrk+1,nFk+1ℑnkxn-Jrk+1,nFk+1p∥2≤〈Jrk+1,nFk+1ℑnkxn-Jrk+1,nFk+1p,ℑnkxn-p〉=〈ℑnk+1xn-p,ℑnkxn-p〉=12(∥ℑnk+1xn-p∥2+∥ℑnkxn-p∥2-∥ℑnk+1xn-ℑnkxn∥2),
and, hence,
(3.22)∥ℑnk+1xn-p∥2≤∥ℑnkxn-p∥2-∥ℑnk+1xn-ℑnkxn∥2≤∥xn-p∥2-∥ℑnk+1xn-ℑnkxn∥2.
By (3.22), we compute
(3.23)∥xn+1-p∥2=∥αnγΦ(xn)+(I-αnμB)wn-p∥2=∥αnγΦ(xn)+(I-αnμB)WnℑnMxn-p∥2=∥αn(γΦ(xn)-μBp)+(I-αnμB)WnℑnMxn-(I-αnμB)p∥2=αn2∥γΦ(xn)-μBp∥2+∥(I-αnμB)WnℑnMxn-(I-αnμB)WnℑnMp∥2+2αn〈γΦ(xn)-μBp,(I-αnμB)WnℑnMxn-(I-αnμB)WnℑnMp〉≤αn2∥γΦ(xn)-μBp∥2+(1-αnτ)2∥ℑnMxn-ℑnMp∥2+2αn〈γΦ(xn)-μBp,WnℑnMxn-WnℑnMp〉+2αn2μ〈γΦ(xn)-μBp,BWnℑnMxn-BWnℑnMp〉≤αn2∥γΦ(xn)-μBp∥2+(1-αnτ)2∥ℑnMxn-p∥2+2αn〈γΦ(xn)-γΦ(p),WnℑnMxn-WnℑnMp〉+2αn〈γΦ(p)-μB(p),WnℑnMxn-WnℑnMp〉+2αn2μ∥γΦ(xn)-μBp∥∥BWnℑnMxn-BWnℑnMp∥≤αn2∥γΦ(xn)-μBp∥2+(1-αnτ)2∥ℑnMxn-p∥2+2αnγ{∥xn-p∥-π(∥xn-p∥)}∥xn-p∥+2αn〈γΦ(p)-μB(p),WnℑnMxn-WnℑnMp〉+2αn2μ∥γΦ(xn)-μBp∥∥BWnℑnMxn-BWnℑnMp∥≤αn2∥γΦ(xn)-μBp∥2+(1-2αnτ+αn2τ2+2αnγ)∥ℑnMxn-p∥2+2αn〈γΦ(p)-μB(p),WnℑnMxn-WnℑnMp〉+2αn2μ∥γΦ(xn)-μBp∥∥BWnℑnMxn-BWnℑnMp∥≤(1-2αn(τ-γ))∥ℑnk+1xn-p∥2+αn(αnτ2∥ℑnMxn-p∥2+αn∥γΦ(xn)-μBp∥2+αn+2〈γΦ(p)-μB(p),WnℑnMxn-WnℑnMp〉+αn+2αnμ∥γΦ(xn)-μBp∥∥BWnℑnMxn-BWnℑnMp∥∥ℑnk+1xn-p∥2)≤(1-2αn(τ-γ))(∥xn-p∥2-∥ℑnkxn-ℑnk+1xn∥2)+αn(αnτ2∥xn-p∥2+αn∥γΦ(xn)-μBp∥2+αn+2〈γΦ(p)-μB(p),WnℑnMxn-WnℑnMp〉+αn+2αnμ∥γΦ(xn)-μBp∥∥BWnℑnMxn-BWnℑnMp∥).
So, we obtain
(3.24)(1-2αn(τ-γ))∥ℑnkxn-ℑnk+1xn∥2≤(1-2αn(τ-γ))∥xn-p∥2-∥xn+1-p∥2+αn(αnτ2∥xn-p∥2+αn∥γΦ(xn)-μBp∥2+2〈γΦ(p)-μB(p),WnℑnMxn-WnℑnMp〉+2αnμ∥γΦ(xn)-μBp∥∥BWnℑnMxn-BWnℑnMp∥)≤∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥)+αn(αnτ2∥xn-p∥2+αn∥γΦ(xn)-μBp∥2+αn+2〈γΦ(p)-μB(p),WnℑnMxn-WnℑnMp〉+αn+2αnμ∥γΦ(xn)-μBp∥∥BWnℑnMxn-BWnℑnMp∥).
Using condition (C1) and (3.16), we obtain
(3.25)limn→∞∥ℑnkxn-ℑnk+1xn∥=0,∀k=1,2,…,M-1.
Step
7. Next, we show that
(3.26)limn→∞∥xn-WnℑnMxn∥=0.
Since
(3.27)∥xn-WnℑnMxn∥≤∥xn-xn+1∥+∥xn+1-WnℑnMxn∥=∥xn-xn+1∥+αn∥γf(xn)-μBWnℑnMxn∥≤∥xn-xn+1∥+αn∥γf(xn)-μBxn∥,
by condition (C1) and (3.16), we get ∥xn-WnℑnMxn∥→0 as n→∞.
Step
8. We show that z∈Θ. The weak w-limit set of {xn},ww(xn) is a subset of Θ. Let z∈ww(xn), and let {xnm} be a subsequence of {xn} which converges weakly to z. By Step 6, without loss of generality, we may assume that
(3.28)ℑnmkxnm⇀z,∀k∈{1,2,…,M-1}.
We need to show that z∈Θ. At first, note that by (A2) and given y∈C and k∈{1,2,…,M-1}, we have
(3.29)1rk+1,n〈y-ℑnk+1xn,ℑnk+1xn-ℑnkxn〉≥Fk+1(y,ℑnk+1xn).
Thus,
(3.30)〈y-ℑnmk+1xnm,ℑnmk+1xnm-ℑnmkxnmrk+1,nm〉≥Fk+1(y,ℑnmk+1xnm).
By (A4), F(y,·) is a lower semicontinuous and convex, thus, weakly semicontinuous. By condition (C3) and (3.20), imply that
(3.31)ℑnmk+1xnm-ℑnmkxnmrk+1,nm→0,
in norm. Therefore, letting m→∞ in (3.30) yields
(3.32)Fk+1(y,z)≤limm→∞Fk+1(y,ℑnmk+1xnm)≤0,
for all y∈H and k∈{1,2,…,M-1}. Replacing y with yt=ty+(1-t)z with t∈(0,1) and using (A1) and (A4), we obtain
(3.33)0=Fk+1(yt,yt)≤tFk+1(yt,y)+(1-t)Fk+1(yt,z)≤tFk+1(yt,y).
Hence, Fk+1(ty+(1-t)z,y)≥0, for all t∈(0,1) and y∈H. Letting t→0+ and using (A3), we conclude Fk+1(z,y)≥0, for all y∈H and k∈{1,2,…,M}. Therefore,
(3.34)z∈⋂j=1MEP(Fj)=EP(ℱ).
Next, we show that z∈⋂i=1∞EP(Si). By Lemma 2.12, we have for all z∈C,
(3.35)Wnmz→Wz,
and F(W)=⋂i=1∞F(Si). Assume that z∉F(W), then z≠Wz. Therefore, from Opial's property of Hilbert space, (3.26), (3.34), and (3.35), we have
(3.36)liminfm→∞∥xnm-z∥<liminfm→∞∥xnm-Wz∥≤liminfm→∞{∥xnm-WnmℑnmMxnm∥+∥WnmℑnmMxnm-Wnmz∥+∥Wnmz-Wz∥}≤liminfm→∞{∥xnm-z∥+∥Wnmz-Wz∥}≤liminfm→∞∥xnm-z∥.
This is a contradiction. Therefore, z must belong to F(W)=⋂i=1∞F(Si).
Step
9. We show that limsupn→∞〈(μB-γΦ)x*,x*-xn〉≤0, where x*=PΘ(I-μB+γΦ)x*. By Banach's contraction mapping principle, it guarantees that PΘ(I-μB+γΦ) has a unique fixed point x* which is the unique solution of (3.2). Let {xnk} be a subsequence of {xn} such that
(3.37)limk→∞〈(μB-γΦ)x*,x*-xnk〉=limsupn→∞〈(μB-γΦ)x*,x*-xn〉.
Without loss of generality, we can assume that {xnk} converges weakly to some z∈C. It follows that from Lemma 2.6 and limn→∞∥xn+1-xn∥=0 that z∈Θ. Hence by (3.2), we obtain
(3.38)limsupn→∞〈(μB-γΦ)x*,x*-xn〉=〈(μB-γΦ)x*,x*-z〉≤0.Step10. Finally, we show that xn→x*. As a matter of fact, we have
(3.39)∥xn+1-x*∥2=∥αnγΦ(xn)+(I-αnμB)wn-x*∥2=∥αnγΦ(xn)+(I-αnμB)WnℑnMxn-x*∥2=∥αn(γΦ(xn)-μBx*)+(I-αnμB)WnℑnMxn-(I-αnμB)x*∥2=αn2∥γΦ(xn)-μBx*∥2+∥(I-αnμB)WnℑnMxn-(I-αnμB)WnℑnMx*∥2+2αn〈γΦ(xn)-μBx*,(I-αnμB)WnℑnMxn-(I-αnμB)WnℑnMx*〉≤αn2∥γΦ(xn)-μBx*∥2+(1-αnτ)2∥xn-x*∥2+2αn〈γΦ(xn)-μBx*,WnℑnMxn-WnℑnMx*〉-2αn2μ〈γΦ(xn)-μBx*,BWnℑnMxn-BWnℑnMx*〉≤αn2∥γΦ(xn)-μBx*∥2+(1-αnτ)2∥xn-x*∥2+2αn〈γΦ(xn)-γΦ(x*),WnℑnMxn-WnℑnMx*〉+2αn〈γΦ(x*)-μB(x*),WnℑnMxn-WnℑnMx*〉-2αn2μ∥γΦ(xn)-μBx*∥∥BWnℑnMxn-BWnℑnMx*∥≤αn2∥γΦ(xn)-μBx*∥2+(1-αnτ)2∥xn-x*∥2+2αnγ{∥xn-x*∥-π(∥xn-x*∥)}∥xn-x*∥+2αn〈γΦ(x*)-μB(x*),WnℑnMxn-WnℑnMx*〉+2αn2μ∥γΦ(xn)-μBx*∥∥BWnℑnMxn-BWnℑnMx*∥≤αn2∥γΦ(xn)-μBx*∥2+(1-2αnτ+αn2τ2+2αnγ)∥xn-x*∥2+2αn〈γΦ(x*)-μB(x*),WnℑnMxn-WnℑnMx*〉+2αn2μ∥γΦ(xn)-μBx*∥∥BWnℑnMxn-BWnℑnMx*∥≤(1-2αn(τ-γ))∥xn-x*∥2+αn(αnτ2∥xn-x*∥2+αn∥γΦ(xn)-μBx*∥2+αn+2〈γΦ(x*)-μB(x*),WnℑnMxn-WnℑnMx*〉+αn+2αnμ∥γΦ(xn)-μBx*∥∥BWnℑnMxn-BWnℑnMx*∥),
where
(3.40)ln=2αn(τ-γ),δn=12(τ-γ)(αnτ2∥xn-x*∥2+αn∥γΦ(xn)-μBx*∥2=======+2〈γΦ(x*)-μB(x*),WnℑnMxn-WnℑnMx*〉=======+2αnμ∥γΦ(xn)-μBx*∥∥BWnℑnMxn-BWnℑnMx*∥).
It is easily to see that limn→∞ln=0,∑n=1∞ln=∞ and limsupn→∞δn≤0. By Lemma 2.5, we conclude that xn→x*; this completes the proof.Corollary 3.2.
LetC be a nonempty closed convex subset of a real Hilbert space H such that C±C⊂C. Let 𝒮={Si:C→C} a family of infinitely nonexpansive mappings, and let ℱ={Fj:j=1,2,3,…,M} be a finite family of bifunctions C×C to ℝ satisfying (A1)–(A4). Assume that Θ:=(⋂i=1∞F(Si))∩(⋂j=1MEP(Fj))≠∅. Let B be a k-Lipschitzian and η-strongly monotone mapping on C with k>0,η>0. Let Φ be a π-weakly contractive self-mapping on C with α∈[0,1). Denote by 𝒞 the collection of all weakly contractive mapping Φ on C. Let 0<μ<2η/k2 and 0<γ<μ(η-μk2/2)=τ. Let the mapping Wn be defined by (2.14) and {rj,n}j=1M be a sequence in (0,∞). If {xn} is the sequence generated by x1∈C and
(3.41)wn=WnJrM,nFM⋯Jr2,nF2Jr1,nF1xn,xn+1=αnΦ(xn)+(I-αnμB)wn,∀n∈ℕ,
where {αn} is a sequence in (0,1) which satisfies the following conditions (C1)–(C4), then the sequence {xn} converges strongly to x*=PΘ(I-μB+Φ)(x*) which is the unique solution of the variational inequality
(3.42)〈(μB-Φ)x*,x*-x〉≤0,∀x∈Θ,
which is the optimality condition for the minimization problem
(3.43)minx∈F(T)12〈Bx,x〉-h(x),
where h is a potential function for γΦ (i.e.,  h′(x)=γΦ(x), for x∈H).Proof.
Takingγ≡1, in Theorem 3.1, it is easy to obtain the desired conclusion.Corollary 3.3.
LetC be a nonempty closed convex subset of a real Hilbert space H such that C±C⊂C. Let 𝒮={Si:C→C} a family of infinitely nonexpansive mappings, and let ℱ={Fj:j=1,2,3,…,M} be a finite family of bifunctions C×C to ℝ satisfying (A1)–(A4). Assume that Θ:=(⋂i=1∞F(Si))∩(⋂j=1MEP(Fj))≠∅. Let B be a k-Lipschitzian and η-strongly monotone mapping on C, and let f be a contraction self-mapping on C with α∈[0,1). Denote by 𝒞 the collection of all contraction f on C. Let 0<μ<2η/k2 and 0<γ<μ(η-μk2/2)/α=τ/α. Let the mapping Wn be defined by (2.14) and {rj,n}j=1M be a sequence in (0,∞). If {xn} is the sequence generated by x1∈C and
(3.44)wn=WnJrM,nFM⋯Jr2,nF2Jr1,nF1xn,xn+1=αnγf(xn)+(I-αnμB)wn,∀n∈ℕ,
where {αn} is a sequence in (0,1) which satisfies the following conditions (C1)–(C4), then the sequence {xn} converges strongly to x*=PΘ(I-μB+γf)(x*), which is the unique solution of the variational inequality
(3.45)〈(μB-γf)x*,x*-x〉≤0,∀x∈Θ,
which is the optimality condition for the minimization problem
(3.46)minx∈F(T)12〈Bx,x〉-h(x),
where h is a potential function for γf (i.e.,h′(x)=γf(x), for x∈H).Proof.
TakingΦ≡f in Theorem 3.1, it is easy to obtain the desired conclusion.Corollary 3.4.
LetC be a nonempty closed convex subset of a real Hilbert space H such that C±C⊂C, and let ℱ={Fj:j=1,2,3,…,M} be a finite family of bifunctions C×C to ℝ satisfying (A1)–(A4). Assume that Θ:=⋂j=1MEP(Fj)≠∅. Let B be a k-Lipschitzian and η-strongly monotone mapping on C with k>0,η>0. Let Φ be a π-weakly contractive self-mapping on C with α∈[0,1). Denote by 𝒞 the collection of all weakly contractivemappingon C, and let f∈𝒞 with α=1. Let 0<μ<2η/k2 and 0<γ<μ(η-μk2/2)=τ. Let {rj,n}j=1M be a sequence in (0,∞). If {xn} is the sequence generated by x1∈C and
(3.47)wn=JrM,nFM⋯Jr2,nF2Jr1,nF1xn,xn+1=αnγΦ(xn)+(I-αnμB)wn,∀n∈ℕ,
where {αn} is a sequence in (0,1) which satisfies the following conditions (C1), (C2), and (C4) in Theorem 3.1, then the sequence {xn} converges strongly to x*=PΘ(I-μB+γΦ)(x*), which is the unique solution of the variational inequality
(3.48)〈(μB-γΦ)x*,x*-x〉≤0,∀x∈Θ,
which is the optimality condition for the minimization problem
(3.49)minx∈F(T)12〈Bx,x〉-h(x),
where h is a potential function for γΦ (i.e.,h′(x)=γΦ(x), for x∈H).Proof.
TakingWn≡0 in Theorem 3.1, it is easy to obtain the desired conclusion.Corollary 3.5.
LetC be a nonempty closed convex subset of a real Hilbert space H such that C±C⊂C, and let 𝒮={Si:C→C} a family of infinitely nonexpansive mappings. Assume that Θ:=⋂i=1∞F(Si)≠∅. Let B be a k-Lipschitzian and η-strongly monotone mapping on C with k>0,η>0. Let Φ be a π-weakly contractive self-mapping on C with α∈[0,1). Denote by 𝒞 the collection of all weakly contractive mapping Φ on C. Let 0<μ<2η/k2 and 0<γ<μ(η-μk2/2)=τ. Let the mapping Wn be defined by (2.14). If {xn} is the sequence generated by x1∈C and
(3.50)wn=Wnxn,xn+1=αnγΦ(xn)+(I-αnμB)wn,∀n∈ℕ,
where {αn} is a sequence in (0,1) which satisfies the following conditions (C1)–(C3), then the sequence {xn} converges strongly to x*=PΘ(I-μB+γΦ)(x*), which is the unique solution of the variational inequality
(3.51)〈(μB-γΦ)x*,x*-x〉≤0,∀x∈Θ,
which is the optimality condition for the minimization problem
(3.52)minx∈F(T)12〈Bx,x〉-h(x),
where h is a potential function for γΦ (i.e.,h′(x)=γΦ(x), for x∈H).Proof.
TakingFj≡0, for each j=1,2,…,M in Theorem 3.1, it is easy to obtain the desired conclusion.Corollary 3.6.
LetC be a nonempty closed convex subset of a real Hilbert space H such that C±C⊂C, and let S:C→C be a nonexpansive mapping with Θ:=F(S)≠∅. Let B be a k-Lipschitzian and η-strongly monotone mapping on C with k>0,η>0. Let Φ be a π-weakly contractive self-mapping on C with α∈[0,1). Denote by 𝒞 the collection of all weakly contractive Φ on C. Let 0<μ<2η/k2 and 0<γ<μ(η-μk2/2)=τ. If {xn} is the sequence generated by x1∈C and
(3.53)xn+1=αnγΦ(xn)+(I-αnμB)Sxn,∀n∈ℕ,
where {αn} is a sequence in (0,1) which satisfies the following conditions (C1)–(C3), then the sequence {xn} converges strongly to x*=PΘ(I-μB+γΦ)(x*), which is the unique solution of the variational inequality
(3.54)〈(μB-γΦ)x*,x*-x〉≤0,∀x∈Θ,
which is the optimality condition for the minimization problem
(3.55)minx∈F(T)12〈Bx,x〉-h(x),
where h is a potential function for γΦ (i.e.,h′(x)=γΦ(x), for x∈H).Proof.
TakingFj≡0, for each j=1,2,…,M and replacing Wn by nonexpansive mapping S in Theorem 3.1, it is easy to obtain the desired conclusion.


## 4. An Example and Numerical Result

In this section, we give a real simple numerical example of Theorem3.1 as follows:Example 4.1.
For simplicity, letH=ℝ,C=[0,1],Sn=I.Fk(x,y)=0, for all x,y∈H,rj,n=1,j∈1,2,3,…,M,B=I,f(x)=x2/(1+x),αn=1/n for every n∈ℕ and μ=1. Then {xn} is the sequence generated by
(4.1)xn+1=xn22n(1+xn)+(1-12n)xn,
and z→0 as n→∞, where 0 is the unique solution of the minimization problem
(4.2)minx∈Cx2-x+ln|x+1|+K,
where K is a constant.Proof.
We divide the proof into 4 steps.
Step
1. Using the idea in [55], we can show that
(4.3)Jrj,nFjx=PCx,∀x∈H,j∈{1,2,…,M},
where
(4.4)PCx={x|x|,x∈H-Cx,x∈C.
Since Fj(x,y)=0,for all x,y∈C,j∈{1,2,…,M}, with the definition of Jr(x),for all x∈H in Lemma 2.13, we have
(4.5)JrF(x)={z∈C:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}.
By the equivalent property of the nearest projection PC from H to C, we can conclude that if we take x∈C,Jrj,nFjx=PCx=Ix. By (3) in Lemma 2.11, we have
(4.6)⋂j=1MEP(Fj)=C.Step2. We show that
(4.7)Wn=I.Sn is nonexpansive mapping. By (2.14) we have
(4.8)W1=U1,1=γ1S1U1,2+(1-γ1)I,W2=U2,1=γ1S1U2,2+(1-γ1)I=γ1S1(γ2S2U2,3+(1-γ2)I)+(1-γ1)IW2=U2,1=γ1γ2S1S2+γ1(1-γ2)S1+(1-γ1)I,W3=U3,1=γ1S1U3,2+(1-γ1)I=γ1S1(γ2S2U3,3+(1-γ2)I)+(1-γ1)IW2=U2,1=γ1γ2S1S2U3,3+γ1(1-γ2)S1+(1-γ1)IW2=U2,1=γ1γ2S1S2(γ3S3U3,4+(1-γ3)I)+γ1(1-γ2)S1+(1-γ1)IW2=U2,1=γ1γ2γ3S1S2S3+γ1γ2(1-γ3)S1S2+γ1(1-γ2)S1+(1-γ1)I,
and we compute (2.14) in the same way as above, so we obtain
(4.9)Wn=Un,1=γ1γ2⋯γnS1S2n+γ1γ2⋯γn-1(1-γn)S1S2n-1+γ1γ2⋯γn-2(1-γn-1)S1S2n-2+⋯+γ1(1-γ2)S1+(1-γ1)I.
Since Sn=I,γn=β,n∈ℕ, hence,
(4.10)Wn=[βn+βn-1(1-β)+⋯+β(1-β)+(1-β)]I=I.
Step
3. We prove
(4.11)xn+1=xn2n(1+xn)+(1-1n)xn,xn→0asn→∞,
where 0 is the unique solution of the minimization problem
(4.12)minx∈Cx2-x+ln|x+1|+K.
Since we let B=I, γ be a real number, we choose γ=1. From (4.3), (4.4), and (4.7), we can obtain a special sequence {xn} of Theorem 3.1 as follows:
(4.13)xn+1=xn2n(1+xn)+(1-1n)xn.
Since Sn=I,n∈ℕ, we have
(4.14)⋂n∈ℕF(Sn)=H.
Combining with (4.6), we obtain
(4.15)Θ:=(⋂j=1MEP(Fj))∩(⋂i=1∞F(Si))=C=[0,1].
It is obvious that {xn}→0,and0 is the unique solution of the minimization problem
(4.16)minx∈Cx2-x+ln|x+1|+K,
where K is a constant number.


## 5. Numerical Result

In this step, we give the numerical results (see Table1) that support our main theorem as shown by the plotting graph using MATLAB 7.11.0. We choose the initial values as x=0.15 in Figure 1. From the example, we can see that {xn} converges to 0.Table 1
The sequence values on each different iteration steps.
Iteration step (n)
x
(
1
)
=
0.15
Iteration step (n)
x
(
1
)
=
0.15
0
0.1500
7
0.0029
1
0.0196
8
0.0025
2
0.0100
9
0.0022
3
0.0067
⋮
⋮
4
0.0050
403
0.0001
5
0.0040
404
0.0001
6
0.0034
405
0.0000Figure 1
The initial valuex(1)=0.15 and iteration steps n=500.



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*Source: 206345-2012-10-09.xml*