# A New General System of Generalized Nonlinear Mixed Composite-Type Equilibria and Fixed Point Problems with an Application to Minimization Problems

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

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## Abstract

We introduce a new general system of generalized nonlinear mixed composite-type equilibria and propose a new iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem, the set of solutions of a general system of generalized nonlinear mixed composite-type equilibria, and the set of fixed points of a countable family of strict pseudocontraction mappings. Furthermore, we prove the strong convergence theorem of the purposed iterative scheme in a real Hilbert space. As applications, we apply our results to solve a certain minimization problem related to a strongly positive bounded linear operator. Finally, we also give a numerical example which supports our results. The results obtained in this paper extend the recent ones announced by many others.


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## Body

## 1.  Introduction

LetH be a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let C be a nonempty closed and convex subset of H. Let ϕ:C→ℝ be a real-valued function, where ℝ is the set of real numbers. Let G,B:C→H be two nonlinear mappings and Θ:H×C×C→ℝ be an equilibrium-like function, that is, Θ(w,u,v)=Θ(w,v,u)=0 for all (w,u,v)∈H×C×C. We consider the following new generalized equilibrium problem: find x*∈C such that
(1.1)Θ(Gx*,x*,y)+ϕ(y)-ϕ(x*)+〈Bx*,y-x*〉≥0,∀y∈C.
The set of solutions of the problem (1.1) is denoted by GEP(C,G,Θ,ϕ,B). As special cases of the problem (1.1), we have the following results.  (1)
IfB=0, then problem (1.1) reduces to the following generalized equilibrium problem: find x*∈C such that
(1.2)Θ(Gx*,x*,y)+ϕ(y)-ϕ(x*)≥0,∀y∈C,
which was considered by Cho et al. [1] for more details. The set of solutions of the problem (1.1) is denoted by GEP(C,G,Θ,ϕ).(2)
IfB=0 and Θ(w,u,v)=F(u,v), where F:C×C→ℝ is an equilibrium bifunction, then problem (1.1) reduces to the following mixed equilibrium problem: find x*∈C such that
(1.3)F(x*,y)+ϕ(y)-ϕ(x*)≥0,∀y∈C,
which was considered by Ceng and Yao [2] for more details. The set of solutions of the problem (1.3) is denoted by MEP(F,ϕ).(3)
IfB=0, ϕ=0 and Θ(w,u,v)=F(u,v) where F:C×C→ℝ is an equilibrium bifunction, then problem (1.1) reduces to the following equilibrium problem: find x*∈C such that
(1.4)F(x*,y)≥0,∀y∈C.
The set of solutions of problem (1.4) is denoted by EP(F).(4)
IfΘ=0, ϕ=0, then problem (1.1) reduces to the following classical variational inequality problem: find x*∈C such that
(1.5)〈Bx*,y-x*〉≥0,∀y∈C.
The set of solutions of the problem (1.5) is denoted by VI(C,B).In brief, for an appropriate choice of the mappingG, the function ϕ, and the convex set C, one can obtain a number of the various classes of equilibrium problems as special cases. In particular, the equilibrium problems (1.4) which were introduced by Blum and Oettli [3] and Noor and Oettli [4] in 1994 have had a great impact and influence on the development of several branches of pure and applied sciences. It has been shown that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. In [3, 4], it has been shown that equilibrium problems include variational inequalities, fixed point, minimax problems, Nash equilibrium problems in noncooperative games, and others as special cases. This means that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. Hence collectively, equilibrium problems cover a vast range of applications. Related to the equilibrium problems, we also have the problems of finding the fixed points of the nonlinear mappings, which is the subject of current interest in functional analysis. It is natural to construct a unified approach for these problems. In this direction, several authors have introduced some iterative schemes for finding a common element of the set of solutions of the equilibrium problems and the set of fixed points of nonlinear mappings (e.g., see [5–18] and the references therein).Recall the following definitions.Definition 1.1.
The mappingS:C→C is said to be  (1)
nonexpansive if(1.6)∥Sx-Sy∥≤∥x-y∥,∀x,y∈C,(2)
L-Lipschitzian if there exists a constant L>0 such that
(1.7)∥Sx-Sy∥≤L∥x-y∥,∀x,y∈C,(3)
k-strict pseudocontraction [19] if there exists a constant k∈[0,1) such that
(1.8)∥Sx-Sy∥2≤∥x-y∥2+k∥(I-S)x-(I-S)y∥2,∀x,y∈C,(4)
pseudocontractive if
(1.9)∥Sx-Sy∥2≤∥x-y∥2+∥(I-S)x-(I-S)y∥2,∀x,y∈C.
Clearly, the class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. It is easy to see that (1.8) is equivalent to
(1.10)〈Sx-Sy,x-y〉≤∥x-y∥2-1-k2∥(I-S)x-(I-S)y∥2,∀x,y∈C,
that is, I-S is (1-k)/2-inverse-strongly monotone. From [19], we know that if S is a k-strictly pseudocontractive mapping, then S is Lipschitz continuous with constant (3-k)/(1-k), that is, ∥Sx-Sy∥≤(3-k)/(1-k)∥x-y∥, for all x,y∈C.
In this paper, we useFix(S)={x∈C:Sx=x} to denote the set of fixed points of S.Definition 1.2.
A countable family of mapping{Tn}n=1∞:C→C is called a family ofk-strict pseudocontraction mappings if there exists a constant k∈[0,1) such that
(1.11)∥Tnx-Tny∥2≤∥x-y∥2+k∥(I-Tn)x-(I-Tn)y∥2,∀x,y∈C,n≥1.
On the other hand, let C be a nonempty closed and convex subset of a real Hilbert space H. Let F1,F2,F3:C×C→ℝ be three bifunctions and let Ψ1,Ψ2,Ψ3,Φ1,Φ2,Φ3:C→H be six nonlinear mappings and let φ1,φ2,φ3:C→ℝ be three functions. We consider the following problem of finding (x*,y*,z*)∈C×C×C such that
(1.12)μ1F1(x*,x)+〈μ1(Ψ1+Φ1)y*+x*-y*,x-x*〉≥μ1φ1(x*)-μ1φ1(x),∀x∈C,μ2F2(y*,y)+〈μ2(Ψ2+Φ2)z*+y*-z*,y-y*〉≥μ2φ2(y*)-μ2φ2(y),∀y∈C,μ3F3(z*,z)+〈μ3(Ψ3+Φ3)x*+z*-x*,z-z*〉≥μ3φ3(z*)-μ3φ3(z),∀z∈C,
which is called a new general system of generalized nonlinear mixed composite-type equilibria, where μi>0 for all i=1,2,3. Next, we present some special cases of problem (1.12) as follows.  (1)
IfΨi=Ψ, Φi=Φ, Fi=F, and φi=φ for all i=1,2,3, then the problem (1.12) reduces to the following new general system of generalized nonlinear mixed composite-type equilibria: find (x*,y*,z*)∈C×C×C such that
(1.13)μ1F(x*,x)+〈μ1(Ψ+Φ)y*+x*-y*,x-x*〉≥μ1φ(x*)-μ1φ(x),∀x∈C,μ2F(y*,y)+〈μ2(Ψ+Φ)z*+y*-z*,y-y*〉≥μ2φ(y*)-μ2φ(y),∀y∈C,μ3F(z*,z)+〈μ3(Ψ+Φ)x*+z*-x*,z-z*〉≥μ3φ(z*)-μ3φ(z),∀z∈C,
where μi>0 for all i=1,2.(2)
IfF3=0, Ψ3=Φ3=0, μ3=0, and z*=x*, then the problem (1.12) reduces to the following general system of generalized nonlinear mixed composite-type equilibria: find (x*,y*)∈C×C such that
(1.14)μ1F1(x*,x)+〈μ1(Ψ1+Φ1)y*+x*-y*,x-x*〉≥μ1φ1(x*)-μ1φ1(x),∀x∈C,μ2F2(y*,y)+〈μ2(Ψ2+Φ2)x*+y*-x*,y-y*〉≥μ2φ2(y*)-μ2φ2(y),∀y∈C,
which was introduced and considered by Ceng et al. [20], where μi>0 for all i=1,2.(3)
IfF3=0, Ψ3=Φ3=0, μ3=0, and φi=0 for all i=1,2,3 and z*=x*, then the problem (1.12) reduces to the following a general system of generalized equilibria: find (x*,y*)∈C×C such that
(1.15)F1(x*,x)+〈Ψ1y*,x-x*〉+1μ1〈x*-y*,x-x*〉≥0,∀x∈C,F2(y*,y)+〈Ψ2x*,y-y*〉+1μ2〈y*-x*,y-y*〉≥0,∀y∈C,
which was introduced and considered by Ceng and Yao [21], where μi>0 for all i=1,2.(4)
IfFi=F, Ψi=Ψ, and Φi=Φ, φi=φ, for all i=1,2,3, then the problem (1.12) reduces to the following generalized mixed equilibrium problem with perturbed mapping: find x*∈C such that
(1.16)F(x*,y)+φ(y)-φ(x*)+〈(Ψ+Φ)x*,y-x*〉≥0,∀y∈C,
which was introduced and considered by Hu and Ceng [22].(5)
IfFi=0, Φi=0, and φi=0 for all i=1,2,3, then the problem (1.12) reduces to the following general system of variational inequalities: find (x*,y*,z*)∈C×C×C such that
(1.17)〈μ1Ψ1y*+x*-y*,x-x*〉≥0,∀x∈C,〈μ2Ψ2z*+y*-z*,y-y*〉≥0,∀y∈C,〈μ3Ψ3x*+z*-x*,z-z*〉≥0,∀z∈C,
which was introduced and considered by Kumam et al. [23], where μi>0 for all i=1,2,3.(6)
IfFi=0, Φi=0, φi=0 for all i=1,2,3, Ψ3=0 and z*=x*, then the problem (1.12) reduces to the following general system of variational inequalities: find (x*,y*)∈C×C such that
(1.18)〈μ1Ψ1y*+x*-y*,x-x*〉≥0,∀x∈C,〈μ2Ψ2x*+y*-x*,y-y*〉≥0,∀y∈C,
which was introduced and considered by Ceng et al. [24], where μi>0 for all i=1,2.In 2010, Cho et al. [1] introduced an iterative method for finding a common element of the set of solutions of generalized equilibrium problems (1.2), the set of solutions for a systems of nonlinear variational inequalities problems (1.18), and the set of fixed points of nonexpansive mappings in Hilbert spaces. Ceng and Yao [21] introduced and considered a relaxed extragradient-like method for finding a common element of the set of solutions of a system of generalized equilibria, the set of fixed points of a strictly pseudocontractive mapping, and the set of solutions of a equilibrium problem in a real Hilbert space and obtained a strong convergence theorem. The result of Ceng and Yao [21] included, as special cases, the corresponding ones of S. Takahashi and W. Takahashi [10], Ceng et al. [24], Peng and Yao [25], and Yao et al. [26].Motivated and inspired by the works in the literature, we introduce a new general system of generalized nonlinear mixed composite-type equilibria (1.12) and propose a new iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem, a general system of generalized nonlinear mixed composite-type equilibria, and the set of fixed points of a countable family of strict pseudocontraction mappings. Furthermore, we prove the strong convergence theorem of the purposed iterative scheme in a real Hilbert space. As applications, we apply our results to solve a certain minimization problem related to a strongly positive bounded linear operator. The results presented in this paper extend the recent results of Cho et al. [1], Ceng and Yao [21], Ceng et al.[20], and many authors.


## 2.   Preliminaries

A bounded linear operatorA is said to be strongly positive, if there exists a constant γ->0 such that
(2.1)〈Ax,x〉≥γ-∥x∥2,∀x∈H.
Recall that, a mapping f:C→C is said to be contractive if there exists a constant α∈(0,1) such that
(2.2)∥f(x)-f(y)∥≤α∥x-y∥,∀x,y∈C.
A mapping A:C→H is called α-inverse-strongly monotone if there exists a constant α>0 such that
(2.3)〈x-y,Ax-Ay〉≥α∥Ax-Ay∥2,∀x,y∈C.
Let C be a nonempty closed convex subset of a real Hilbert space H. For every point x∈H there exists a unique nearest point in C denoted by PCx, such that
(2.4)∥x-PCx∥≤∥x-y∥,∀y∈C.PC is called the metric projection of H onto C. It is well known that PC is nonexpansive (see [27]) and for x∈H,
(2.5)z=PCx⟺〈x-z,y-z〉≤0,∀y∈C.
Let ϕ:C→ℝ be a real-valued function, G:C→H be a mapping and Θ:H×C×C→ℝ be an equilibrium-like function. Let r be a positive real number. For all x∈C, we consider the following problem. Find y∈C such that
(2.6)Θ(Gx,y,z)+ϕ(z)-ϕ(y)+1r〈y-x,z-y〉≥0,∀z∈C,
which is known as the auxiliary generalized equilibrium problem.LetT(r):H→C be the mapping such that, for all x∈H, T(r) is the solution set of the auxiliary problem (2.6), that is,
(2.7)T(r)(x)={y∈C:Θ(Gx,y,z)+ϕ(z)-ϕ(y)+1r〈y-z,z-x〉≥0,∀z∈C}.
Then, we will assume the Condition(Δ) [28] as follows:  (a)
T(r) is single-valued;(b)
T(r) is nonexpansive;(c)
Fix(T(r))=GEP(C,G,Θ,ϕ).Notice that the examples of showing the sufficient conditions for the existence of the condition(Δ) can be found in [6].Throughout this paper, we assume that a bifunctionF:C×C→ℝ and φ:C→ℝ is a lower semicontinuous and convex function satisfy the following conditions:  (H1)
F(x,x)=0,  ∀x∈C;(H2)
F is monotone, that is, F(x,y)+F(y,x)≤0,  ∀x,y∈C;(H3)
for ally∈C,  x↦F(x,y) is weakly upper semicontinuous;(H4)
for allx∈C,  y↦F(x,y) is convex and lower semicontinuous;(A1)
for allx∈H and r>0, there exist a bounded subset Dx⊂C and yx∈C such that for all z∈C∖Dx,
(2.8)F(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0;(A2)
C is a bounded set.In order to prove our main results in the next section, we need the following lemmas.Lemma 2.1 (see [29]).
LetC be a nonempty closed and convex subset of a real Hilbert sapce H. Let F:C×C→ℝ be a bifunction satisfying condition (H1)–(H4) and let φ:C→ℝ be a lower semicontinuous and convex function. For r>0 and x∈H define a mapping Tr(F,φ):H→C follows
(2.9)Tr(F,φ)(x)={y∈C:F(y,z)+φ(z)-φ(y)+1r〈y-z,z-x〉≥0,∀z∈C}.
Assume that either (A1) or (A2) holds, then the following statements hold  (i)
Tr(F,φ)≠∅ for all x∈H and Tr(F,φ) is single-valued;(ii)
Tr(F,φ) is firmly nonexpansive, that is, for all x,y∈H,
(2.10)∥Tr(F,φ)x-Tr(F,φ)y∥2≤〈Tr(F,φ)x-Tr(F,φ)y,x-y〉;(iii)
Fix(Tr(F,φ))=MEP(F,φ);(iv)
MEP(F,φ) is closed and convex.Remark 2.2.
Ifφ=0, then Tr(F,φ) is rewritten as TrF (see [21, Lemma 2.1] for more details).Lemma 2.3 (see [30]).
Let{xn} and {ln} be bounded sequences in a Banach space X and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose xn+1=(1-βn)ln+βnxn for all integers n≥0 and limsupn→∞(∥ln+1-ln∥-∥xn+1-xn∥)≤0. Then, limn→∞∥ln-xn∥=0.Lemma 2.4 (see [31]).
LetH be a real Hilbert space. Then the following inequalities hold.  (i)
∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)∥x-y∥2,  ∀x,y∈Hand λ∈[0,1].(ii)
∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.Definition 2.5 (see [32]).
Let{Tn} be a sequence of mappings from a subset C of a real Hilbert space H into itself. We say that {Tn} satisfies the  PT-condition if
(2.11)limk,l→∞ρlk=0,
where ρlk=supz∈C{∥Tkz-Tlz∥}<∞, for all k,l∈ℕ.Lemma 2.6 (see [32]).
Suppose that{Tn} satisfies the PT-condition such that  (i)
for eachx∈C, {Tn} is converse strongly to some point in C(ii)
let the mappingT:C→C defined by Tx=limn→∞Tnx for all x∈C. Then, limn→∞supω∈C∥Tω-Tnω∥=0.Lemma 2.7 (see [33]).
LetC be a closed and convex subset of a strictly convex Banach space X. Let {Tn:n∈ℕ} be a sequence of nonexpansive mappings on C. Suppose ⋂n=1∞‍Fix(Tn) is nonempty. Let {γn} be a sequence of positive numbers with ∑n=1∞‍γn=1. Then a mapping S on C defined by Sx=∑n=1∞‍γnTnx for all x∈C is well defined, nonexpansive, and Fix(S)=⋂n=1∞‍Fix(Tn) holds.Lemma 2.8 (see [19]).
LetT:C→H be a k-strict pseudocontraction. Define S:C→H by Sx=δx+(1-δ)Tx for each x∈C. Then, as δ∈[k,1), S is nonexpansive such that Fix(S)=Fix(T).Lemma 2.9 (see [34]).
LetC be a closed and convex subset of a real Hilbert space H and let S:C→C be a nonexpansive mapping. then, the mapping I-S is demiclosed. That is, if {xn} is a sequence in C such that xn⇀z and (I-S)xn→y, then (I-S)z=y.Lemma 2.10 (see [35]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.12)an+1≤(1-σn)an+δn,
where {σn} is a sequence in (0,1) and {δn} is a sequence in ℝ such that  (i)
∑n=0∞‍σn=∞;(ii)
limsupn→∞(δn/σn)≤0 or ∑n=0∞‍|δn|<∞.
then,limn→∞an=0.Lemma 2.11.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let mappings Ψ,Φ:C→H be β~-inverse-strongly monotone and γ~-inverse-strongly monotone, respectively. Then, we have
(2.13)∥(I-μ(Ψ+Φ))x-(I-μ(Ψ+Φ))y∥2≤∥x-y∥2+2μ(μ-β~)∥Ψx-Ψy∥2+2μ(μ-γ~)∥Φx-Φy∥2,
where μ>0. In particular, if μ∈(0,min{β~,γ~}), then I-μ(Ψ+Φ) is nonexpansive.Proof.
From Lemma2.4(i), for all x,y∈C, we have
(2.14)∥(I-μ(Ψ+Φ))x-(I-μ(Ψ+Φ))y∥2=∥(x-y)-μ((Ψ+Φ)x-(Ψ+Φ)y)∥2=∥12((x-y)-2μ(Ψx-Ψy))+12((x-y)-2μ(Φx-Φy))∥2≤12∥((x-y)-2μ(Ψx-Ψy))∥2+12∥((x-y)-2μ(Φx-Φy))∥2=12(∥x-y∥2-4μ〈x-y,Ψx-Ψy〉+4μ2∥Ψx-Ψy∥2)+12(∥x-y∥2-4μ〈x-y,Φx-Φy〉+4μ2∥Φx-Φy∥2)≤12(∥x-y∥2+4μ(μ-β~)∥Ψx-Ψy∥2)+12(∥x-y∥2+4μ(μ-γ~)∥Φx-Φy∥2)=∥x-y∥2+2μ(μ-β~)∥Ψx-Ψy∥2+2μ(μ-γ~)∥Φx-Φy∥2.
It is clear that, if 0<μ≤min{β~,γ~}, then I-μ(Ψ+Φ) is nonexpansive. This completes the proof.Lemma 2.12.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let mappings Ψi,Φi:C→H(i=1,2,3) be β~i-inverse-strongly monotone and γ~i-inverse-strongly monotone, respectively. Let Q:C→C be the mapping defined by
(2.15)Qx=Tμ1(F1,φ1)[Tμ2(F2,φ2)[Tμ3(F3,φ3)(x-μ3(Ψ3+Φ3)x)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(x-μ3(Ψ3+Φ3)x)]-μ1(Ψ1+Φ1)Tμ2(F2,φ2)[Tμ3(F3,φ3)(x-μ3(Ψ3+Φ3)x)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(x-μ3(Ψ3+Φ3)x)]],∀x∈C.
If μi∈(0,min{β~i,γ~i})(i=1,2,3), then Q is nonexpansive.Proof.
From Lemma2.11, for all x,y∈C, we have
(2.16)∥Qx-Qy∥=∥Tμ1(F1,φ1)[Tμ2(F2,φ2)[Tμ3(F3,φ3)(x-μ3(Ψ3+Φ3)x)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(x-μ3(Ψ3+Φ3)x)]-μ1(Ψ1+Φ1)Tμ2(F2,φ2)[Tμ3(F3,φ3)(x-μ3(Ψ3+Φ3)x)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(x-μ3(Ψ3+Φ3)x)]]-Tμ1(F1,φ1)[Tμ2(F2,φ2)[Tμ3(F3,φ3)(y-μ3(Ψ3+Φ3)y)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(y-μ3(Ψ3+Φ3)y)]-μ1(Ψ1+Φ1)Tμ2(F2,φ2)[Tμ3(F3,φ3)(y-μ3(Ψ3+Φ3)y)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(y-μ3(Ψ3+Φ3)y)]]∥≤∥Tμ2(F2,φ2)[Tμ3(F3,φ3)(x-μ3(Ψ3+Φ3)x)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(x-μ3(Ψ3+Φ3)x)]-μ1(Ψ1+Φ1)Tμ2(F2,φ2)[Tμ3(F3,φ3)(x-μ3(Ψ3+Φ3)x)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(x-μ3(Ψ3+Φ3)x)]-[Tμ2(F2,φ2)[Tμ3(F3,φ3)(y-μ3(Ψ3+Φ3)y)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(y-μ3(Ψ3+Φ3)y)]-μ1(Ψ1+Φ1)Tμ2(F2,φ2)[Tμ3(F3,φ3)(y-μ3(Ψ3+Φ3)y)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(y-μ3(Ψ3+Φ3)y)]]∥=∥Tμ2(F2,φ2)(I-μ2(Ψ2+Φ2))Tμ3(F3,φ3)(I-μ3(Ψ3+Φ3))(I-μ1(Ψ1+Φ1))x-Tμ2(F2,φ2)(I-μ2(Ψ2+Φ2))Tμ3(F3,φ3)(I-μ3(Ψ3+Φ3))(I-μ1(Ψ1+Φ1))y∥≤∥(I-μ2(Ψ2+Φ2))Tμ3(F3,φ3)(I-μ3(Ψ3+Φ3))(I-μ1(Ψ1+Φ1))x-(I-μ2(Ψ2+Φ2))Tμ3(F3,φ3)(I-μ3(Ψ3+Φ3))(I-μ1(Ψ1+Φ1))y∥≤∥Tμ3(F3,φ3)(I-μ3(Ψ3+Φ3))(I-μ1(Ψ1+Φ1))x-Tμ3(F3,φ3)(I-μ3(Ψ3+Φ3))(I-μ1(Ψ1+Φ1))y∥≤∥(I-μ3(Ψ3+Φ3))(I-μ1(Ψ1+Φ1))x-(I-μ3(Ψ3+Φ3))(I-μ1(Ψ1+Φ1))y∥≤∥(I-μ1(Ψ1+Φ1))x-(I-μ1(Ψ1+Φ1))y∥≤∥x-y∥,
which implies that Q:C→C is nonexpansive. This completes the proof.Lemma 2.13.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let Fi:C×C→ℝ  (i=1,2,3) be a bifunction satisfying conditions (H1)–(H4) and let Ψi,Φi:C→H(i=1,2,3) be a nonlinear mapping. Suppose that μi(i=1,2,3) be a real positive number. Let φi:C→ℝ(i=1,2,3) be a lower semicontinuous and convex function. Assume that either condition (A1) or (A2) holds. Then, for (x*,y*,z*)∈C×C×C is a solution of the problem (1.12) if and only if x*∈Fix(Q), y*=Tμ2(F2,φ2)(z*-μ2(Ψ2+Φ2)z*) and z*=Tμ3(F3,φ3)(x*-μ3(Ψ3+Φ3)x*), where Q is the mapping defined as in Lemma 2.12.Proof.
Let(x*,y*,z*)∈C×C×C be a solution of the problem (1.12). Then, we have
(2.17)μ1F1(x*,x)+〈μ1(Ψ1+Φ1)y*+x*-y*,x-x*〉≥μ1φ1(x*)-μ1φ1(x),∀x∈C,μ2F2(y*,y)+〈μ2(Ψ2+Φ2)z*+y*-z*,y-y*〉≥μ2φ2(y*)-μ2φ2(y),∀y∈C,μ3F3(z*,z)+〈μ3(Ψ3+Φ3)x*+z*-x*,z-z*〉≥μ3φ3(z*)-μ3φ3(z),∀z∈C,⇔(2.18)x*=Tμ1(F1,φ1)(y*-μ1(Ψ1+Φ1)y*),y*=Tμ2(F2,φ2)(z*-μ2(Ψ2+Φ2)z*),z*=Tμ3(F3,φ3)(x*-μ3(Ψ3+Φ3)x*),⇔(2.19)x*=Tμ1(F1,φ1)[Tμ2(F2,φ2)[Tμ3(F3,φ3)(x*-μ3(Ψ3+Φ3)x*)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(x*-μ3(Ψ3+Φ3)x*)]-μ1(Ψ1+Φ1)Tμ2(F2,φ2)[Tμ3(F3,φ3)(x*-μ3(Ψ3+Φ3)x*)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(x*-μ3(Ψ3+Φ3)x*)]].
This completes the proof.Corollary 2.14 (see [20]).
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let Fi:C×C→ℝ(i=1,2) be a bifunction satisfying conditions (H1)–(H4) and let Ψi,Φi:C→H(i=1,2) be a nonlinear mapping. Suppose that μi(i=1,2) be a real positive number. Let φi:C→ℝ(i=1,2) be a lower semicontinuous and convex function. Assume that either condition (A1) or (A2) holds. Then, for (x*,y*)∈C×C is a solution of the problem (1.14) if and only if x*∈Fix(Q), y*=Tμ2(F2,φ2)(x*-μ2(Ψ2+Φ2)x*), where G is the mapping defined by
(2.20)Qx=Tμ1(F1,φ1)[Tμ2(F2,φ2)(x-μ2(Ψ2+Φ2)x)-μ1(Ψ1+Φ1)Tμ2(F2,φ2)(x-μ2(Ψ2+Φ2)x)],∀x∈C.Corollary 2.15 (see [21]).
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let Fi:C×C→ℝ(i=1,2) be a bifunction satisfying conditions (H1)–(H4) and let Ψi:C→H(i=1,2) be a nonlinear mapping. Suppose that μi(i=1,2) is a real positive number. Assume that either condition (A1) or (A2) holds. Then, for (x*,y*)∈C×C is a solution of the problem (1.15) if and only if x*∈Fix(Q), y*=Tμ2F2(x*-μ2Ψ2x*), where Q is the mapping defined by
(2.21)Qx=Tμ1F1[Tμ2F2(x-μ2Ψ2x)-μ1Ψ1Tμ2F2(x-μ2Ψ2x)],∀x∈C.Corollary 2.16 (see [23]).
LetC be a nonempty closed and convex subset of a real Hilbert space H. For given (x*,y*,z*)∈C×C×C is a solution of the problem (1.17) if and only if x*∈Fix(Q), y*=PC(z*-μ2Ψ2z*) and z*=PC(x*-μ3Ψ3x*), where Q is the mapping defined by
(2.22)Qx=PC[PC[PC(x-μ3Ψ3x)-μ2Ψ2PC(x-μ3Ψx)]-μ1Ψ1PC[PC(x-μ3Ψ3x)-μ2Ψ2PC(x-μ3Ψ3x)]],∀x∈C.Corollary 2.17 (see [24]).
LetC be a nonempty closed and convex subset of a real Hilbert space H. For given (x*,y*)∈C×C is a solution of the problem (1.18) if and only if x*∈Fix(Q), y*=PC(x*-μ2Ψ2x*), where Q is the mapping defined by
(2.23)Qx=PC[PC(x-μ2Ψ2x)-μ1Ψ1PC(x-μ2Ψ2x)],∀x∈C.


## 3. Main Results

We are now in a position to prove the main result of this paper.Theorem 3.1.
LetC be a nonempty closed and convex subset of a real Hilbert space H such that C±C⊂C. Let ϕ,φi:C→ℝ(i=1,2,3) be lower semicontinuous and convex functionals, Θ:H×C×C→ℝ be an equilibrium-like function, G:C→H be a mapping, and Fi:C×C→ℝ(i=1,2,3) be a bifunction satisfying conditions (H1)–(H4). Assume that either condition (A1) or (A2) holds. Let B:C→H be β-inverse-strongly monotone, Ψi,Φi:C→H(i=1,2,3) be β~i-inverse-strongly monotone and γ~i-inverse-strongly monotone, respectively. Let {Tn}n=1∞:C→C be a family of k-strict pseudocontraction mappings. Define a mapping Snx:=δx+(1-δ)Tnx, for all x∈C, δ∈[k,1) and n≥1. Assume that the condition (Δ) is satisfied and Ω:=⋂n=1∞‍Fix(Tn)∩Fix(Q)∩GEP(C,G,Θ,ϕ)≠∅, where Q is defined as in Lemma 2.13. Let μ>0, γ>0, and r>0 be three constants. Let f:C→C be a contraction mapping with a coefficient α∈(0,1) and let A be a strongly positive bounded linear operator on C with a coefficient γ-∈(0,1] such that 0<γ<((1+μ)γ-)/α. For x1∈C, let the sequence {xn} defined by
(3.1)Θ(Gxn,un,y)+ϕ(y)-ϕ(un)+1r〈y-un,un-(xn-rBxn)〉≥0,∀y∈C,zn=Tμ3(F3,φ3)(xn-μ3(Ψ3+Φ3)xn),yn=Tμ2(F2,φ2)(zn-μ2(Ψ2+Φ2)zn),vn=Tμ1(F1,φ1)(yn-μ1(Ψ1+Φ1)yn),xn+1=αnγf(xn)+βnxn+((1-βn)I-αn(I+μA))[γ1Snxn+γ2un+γ3vn],∀n≥1,
where γ1,γ2,γ3∈[0,1] such that γ1+γ2+γ3=1, μ1∈(0,min{β~1,γ~1}), μ2∈(0,min{β~2,γ~2}), μ3∈(0,min{β~3,γ~3}), r∈(0,2β) and {αn},{βn} are two sequences in [0,1]. Suppose that {Tn} satisfies the PT-condition. Let T:C→C be the mapping defined by Tx=limn→∞Tnx for all x∈C and suppose that Fix(T)=⋂n=1∞‍Fix(Tn). Assume the following conditions are satisfied:  (C1)
limn→∞αn=0 and ∑n=1∞‍αn=∞,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1.
Then the sequence{xn} defined by (3.1) converges strongly to x^∈Ω, where x^ is the unique solution of the variational inequality
(3.2)〈(γf-(I+μA))x^,v-x^〉≤0,∀v∈Ω,
or equivalently, x^=PΩ(γf-μA)x^, where PΩ is a metric projection mapping from C onto Ω, and (x^,y^,z^) is a solution of the problem (1.12), where y^=Tμ2(F2,φ2)(z^-μ2(Ψ2+Φ2)z^) and z^=Tμ3(F3,φ3)(x^-μ3(Ψ3+Φ3)x^).Proof.
Note that from the conditions(C1) and (C2), we may assume, without loss of generality, that αn≤(1-βn)(1+μ∥A∥)-1 for all n∈ℕ. Since A is a linear bounded self-adjoint operator on C, by (2.2), we have
(3.3)∥A∥=sup{|〈Au,u〉|:u∈C,∥u∥=1}.
Observe that
(3.4)〈((1-βn)I-αn(I+μA))u,u〉=1-βn-αn-αnμ〈Au,u〉≥1-βn-αn-αnμ∥A∥≥0.
This show that (1-βn)I-αn(I+μA) is positive. It follows that
(3.5)∥(1-βn)I-αn(I+μA)∥=sup{|〈((1-βn)I-αn(I+μA))u,u〉|:u∈C,∥u∥=1}=sup{1-βn-αn-αnμ〈Au,u〉:u∈C,∥u∥=1}≤1-βn-αn(1+μγ-)≤1-βn-αn(1+μ)γ-.
First, we show that {xn} is bounded. Taking x*∈Ω, it follows from Lemma 2.13 that
(3.6)x*=Tμ1(F1,φ1)[Tμ2(F2,φ2)[Tμ3(F3,φ3)(x*-μ3(Ψ3+Φ3)x*)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(x*-μ3(Ψ3+Φ3)x*)]-μ1(Ψ1+Φ1)Tμ2(F2,φ2)[Tμ3(F3,φ3)(x*-μ3(Ψ3+Φ3)x*)-μ2(Ψ2+Φ2)Tμ3(F3,φ3)(x*-μ3(Ψ3+Φ3)x*)]].
Putting y*=Tμ2(F2,φ2)(z*-μ2(Ψ2+Φ2)z*) and z*=Tμ3(F3,φ3)(x*-μ3(Ψ3+Φ3)x*), we obtain x*=Tμ1(F1,φ1)(y*-μ1(Ψ1+Φ1)y*). Notice that un=T(r)(I-rB)xn. Since T(r) is nonexpansive and B is β-inverse-strongly monotone, we have
(3.7)∥un-x*∥2=∥T(r)(I-rB)xn-T(r)(I-rB)x*∥2≤∥(I-rB)xn-(I-rB)x*∥2=∥(xn-x*)-r(Bxn-Bx*)∥2=∥xn-x*∥2-2r〈xn-x*,Bxn-Bx*〉+r2∥Bxn-Bx*∥2≤∥xn-x*∥2+r(r-2β)∥Bxn-Bx*∥2≤∥xn-x*∥2,
and hence
(3.8)∥un-x*∥≤∥xn-x*∥.
We observe that
(3.9)∥vn-x*∥=∥Qxn-Qx*∥≤∥xn-x*∥.
Setting θn:=γ1Snxn+γ2un+γ3vn. By Lemma 2.8, we have Sn is a nonexpansive mapping such that Fix(Sn)=Fix(Tn) for all n≥1. Then, we have
(3.10)∥θn-x*∥=∥γ1(Snxn-x*)+γ2(un-x*)+γ3(vn-x*)∥≤γ1∥Snxn-x*∥+γ2∥un-x*∥+γ3∥vn-x*∥≤γ1∥xn-x*∥+γ2∥xn-x*∥+γ3∥xn-x*∥=∥xn-x*∥.
It follows that
(3.11)∥xn+1-x*∥=∥αn(γf(xn)-(I+μA)x*)+βn(xn-x*)+((1-βn)I-αn(I+μA))(θn-x*)∥≤αn∥γf(xn)-αn(I+μA)x*∥+βn∥xn-x*∥+(1-βn-αn(1+μγ-))∥θn-x*∥≤αnγ∥f(xn)-f(x*)∥+αn∥γf(x*)-(I+μA)x*∥+(1-αn(1+μγ-))∥xn-x*∥≤αnγα∥xn-x*∥+αn∥γf(x*)-(I+μA)x*∥+(1-αn(1+μγ-))∥xn-x*∥≤(1-((1+μ)γ--γα)αn)∥xn-x*∥+αn∥γf(x*)-(I+μA)x*∥=(1-((1+μ)γ--γα)αn)∥xn-x*∥+((1+μ)γ--γα)αn∥γf(x*)-(I+μA)x*∥(1+μ)γ--γα.
By induction, we have
(3.12)∥xn-x*∥≤max{∥x1-x*∥,∥γf(x*)-(I+μA)x*∥(1+μ)γ--γα},∀n≥1.
Hence, {xn} is bounded, so are {un}, {vn}, {yn}, and {zn}. From definition of Sn and for all k,l∈ℕ, it follows that
(3.13)supω∈C∥Skω-Slω∥=γ1supω∈C∥Tkω-Tlω∥.
By our assumption, {Tn} satisfies the PT-condition, we obtain that
(3.14)limn→∞supω∈C∥Skω-Slω∥=0,
that is {Sn} satisfies the PT-condition.
Next, we show thatlimn→∞∥xn+1-xn∥=0. Since un=T(r)(I-rB)xn, we have
(3.15)∥un+1-un∥=∥T(r)(xn+1-rBxn+1)-T(r)(xn-rBxn)∥≤∥(xn+1-rBxn+1)-(xn-rBxn)∥≤∥xn+1-xn∥,∥vn+1-vn∥=∥Qxn+1-Qxn∥≤∥xn+1-xn∥.
It follows from (3.15) that
(3.16)∥θn+1-θn∥=∥(γ1Sn+1xn+1+γ2un+1+γ3vn+1)-(γ1Snxn+γ2un+γ3vn)∥=∥γ1(Sn+1xn+1-Snxn)+γ(un+1-un)+γ3(vn+1-vn)∥≤γ1∥Sn+1xn+1-Snxn∥+γ2∥un+1-un∥+γ3∥vn+1-vn∥≤γ1∥Sn+1xn+1-Sn+1xn∥+γ1∥Sn+1xn-Snxn∥+γ2∥un+1-un∥+γ3∥vn+1-vn∥≤γ1∥xn+1-xn∥+γ2∥xn+1-xn∥+γ3∥xn+1-xn∥+γ1∥Sn+1xn-Snxn∥=∥xn+1-xn∥+γ1∥Sn+1xn-Snxn∥.
Let xn+1=(1-βn)ln+βnxn for all n≥1. Then, we have
(3.17)ln+1-ln=xn+2-βn+1xn+11-βn+1-xn+1-βnxn1-βn=αn+1γf(xn+1)+((1-βn+1)I-αn+1(I+μA))θn+11-βn+1-αnγf(xn)+((1-βn)I-αn(I+μA))θn1-βn=αn+11-βn+1γf(xn+1)+(1-βn+1)I-αn+1(I+μA)θn+11-βn+1-αn1-βnγf(xn)-(1-βn)I-αn(I+μA)θn1-βn=αn+11-βn+1(γf(xn+1)-(I+μA)θn+1)-αn1-βn(γf(xn)-(I+μA)θn)+θn+1-θn.
Combining (3.16) and (3.17), we have
(3.18)∥ln+1-ln∥-∥xn+1-xn∥≤αn+11-βn+1∥γf(xn+1)-(I+μA)θn+1∥+αn1-βn∥γf(xn)-(I+μA)θn∥+γ1∥Sn+1xn-Snxn∥.
Since {Sn} satisfies the PT-condition, we can define a mapping S:C→C by
(3.19)Sx=limn→∞Snx=limn→∞[δx+(1-δ)Tnx]=δx+(1-δ)Tx,∀x∈C.
We observe that
(3.20)∥Sn+1xn-Snxn∥≤∥Sn+1xn-Sxn∥+∥Sxn-Snxn∥≤supω∈C∥Sn+1ω-Sω∥+supω∈C∥Sω-Snω∥.
Be Lemma 2.6, we have
(3.21)limn→∞∥Sn+1xn-Snxn∥=0.
Consequently, it follows from the conditions (C1), (C2), and (3.18) that
(3.22)limsupn→∞(∥ln+1-ln∥-∥xn+1-xn∥)≤0.
Hence, by Lemma 2.3, we obtain that
(3.23)limn→∞∥ln-xn∥=0.
Consequently, we have
(3.24)limn→∞∥xn+1-xn∥=limn→∞(1-βn)∥ln-xn∥=0.
On the other hand, we observe that
(3.25)xn+1-xn=αn(γf(xn)-(I+μA)xn)+((1-βn)I-αn(I+μA))(θn-xn).
It follows that
(3.26)(1-βn-αn(1+μ)γ-)∥θn-xn∥≤∥xn+1-xn∥+αn∥γf(xn)-(I+μA)xn∥.
From the conditions (C1), (C2), and (3.24), we obtain that(3.27)limn→∞∥θn-xn∥=0.
Next, we show that(3.28)limsupn→∞〈γf(x^)-(I+μA)x^,xn-x^〉≤0.
To show this, we take a subsequence {xnj} of {xn} such that
(3.29)limsupn→∞〈γf(x^)-(I+μA)x^,xn-x^〉=limj→∞〈γf(x^)-(I+μA)x^,xnj-x^〉.
Since {xnj} is bounded, without loss of generality, we can assume that xnj⇀v∈C. So, we get
(3.30)limsupn→∞〈γf(x^)-(I+μA)x^,xn-x^〉=limj→∞〈γf(x^)-(I+μA)x^,xnj-x^〉=〈γf(x^)-(I+μA)x^,v-x^〉≤0.
Next, we show that v∈Ω:=⋂n=1∞‍Fix(Tn)∩Fix(Q)∩GEP(C,G,Θ,ϕ). Define a mapping Kn:C→C by
(3.31)Knx=γ1Snx+γ2Qx+γ3T(r)(I-rB)x,∀x∈C.
From (3.27), we have
(3.32)limn→∞∥Knxn-Kxn∥=0.
For all k,l∈ℕ, it follows that
(3.33)supω∈C∥Kkω-Klω∥=γ1supω∈C∥Skω-Slω∥.
Since {Sn} satisfies the PT-condition, we obtain that
(3.34)limk,l→∞supω∈C∥Kkω-Klω∥=0,
that is {Kn} satisfies the PT-condition. Define a mapping K:C→C by
(3.35)Kx=limn→∞Knx=limn→∞[γ1Snx+γ2Qx+γ3T(r)(I-rB)x]=γ1Sx+γ2Qx+γ3T(r)(I-rB)x,∀x∈C.
By Lemma 2.6, we obtain that
(3.36)limn→∞supω∈C∥Knω-Kω∥=0.
From Lemma 2.7, we see that K is nonexpansive and
(3.37)Fix(K)=Fix(S)∩Fix(Q)∩Fix(T(r)(I-rB))=⋂n=1∞‍Fix(Sn)∩Fix(Q)∩Fix(T(r)(I-rB))=⋂n=1∞‍Fix(Tn)∩Fix(Q)∩GEP(C,G,Θ,φ)=⋂n=1∞‍Fix(Kn).
Notice that
(3.38)∥xn-Kxn∥≤∥xn-Knxn∥+∥Knxn-Kxn∥≤∥xn-Knxn∥+supω∈C∥Knω-Kω∥.
From (3.32) and (3.39), we obtain that
(3.39)limn→∞∥xn-Kxn∥=0.
Thus, by Lemma 2.9, we obtain that v∈Fix(K)=Ω.
Finally, we show thatxn→x^ as n→∞. From Lemma 2.4, we compute
(3.40)∥xn+1-x^∥2=∥αn(γf(xn)-(I+μA)x^)+βn(xn-x^)+((1-βn)I-αn(I+μA))(θn-x^)∥2≤∥((1-βn)I-αn(I+μA))(θn-x^)+βn(xn-x^)∥2+2αn〈γf(xn)-(I+μA)x^,xn+1-x^〉≤[(1-βn-αn(1+μγ-))∥θn-x^∥+βn∥xn-x^∥]2+2αnγ〈f(xn)-f(x^),xn+1-x^〉+2αn〈γf(x^)-(I+μA)x^,xn+1-x^〉≤[(1-βn-αn(1+μ)γ-)∥xn-x^∥+βn∥xn-x^∥]2+2αnγα∥xn-x^∥∥xn+1-x^∥+2αn〈γf(x^)-(I+μA)x^,xn+1-x^〉≤[(1-βn-αn(1+μ)γ-)∥xn-x^∥+βn∥xn-x^∥]2+2αnγα∥xn-x^∥∥xn+1-x^∥+2αn〈γf(x^)-(I+μA)x^,xn+1-x^〉≤(1-αn(1+μ)γ-)2∥xn-x^∥2+αnγα(∥xn-x^∥2+∥xn+1-x^∥2)+2αn〈γf(x^)-(I+μA)x^,xn+1-x^〉=(1-2αn(1+μ)γ-+αn2[(1+μ)γ-]2+αnγα)∥xn-x^∥2+αnγα∥xn+1-x^∥2+2αn〈γf(x^)-(I+μA)x^,xn+1-x^〉.
It follows that
(3.41)∥xn+1-x^∥2≤1-2αn(1+μ)γ-+αn2[(1+μ)γ-]2+αnγα1-αnγα∥xn-x^∥2+2αn1-αnγα〈γf(x^)-(I+μA)x^,xn+1-x^〉=1-2αn(1+μ)γ-+αnγα1-αnγα∥xn-x^∥2+αn2[(1+μ)γ-]21-αnγα∥xn-x^∥2+2αn1-αnγα〈γf(x^)-(I+μA)x^,xn+1-x^〉≤[1-2((1+μ)γ--γα)αn1-αnγα]∥xn-x^∥2+αn2[(1+μ)γ-]21-αnγαM+2αn1-αnγα〈γf(x^)-(I+μA)x^,xn+1-x^〉,
whereM=supn≥1{∥xn-x^∥2}. Put σn=(2((1+μ)γ--γα)αn)/(1-αnγα) and
(3.42)δn=αn2[(1+μ)γ-]21-αnγαM+2αn1-αnγα〈γf(x^)-(I+μA)x^,xn+1-x^〉.
Then, the (3.41) reduces to the formula
(3.43)∥xn+1-x^∥2≤(1-σn)∥xn-x^∥2+δn.
It is easily seen that ∑n=1∞‍σn=∞ and (using (3.30)), we get
(3.44)limsupn→∞δnσn=limsupn→∞12((1+μ)γ--γα)[αn[(1+μ)γ-]2M+2〈γf(x^)-(I+μA)x^,xn+1-x^〉]≤0.
Hence, by Lemma 2.10, we conclude that xn→x^ as n→∞. This completes the proof.Remark 3.2.
Theorem3.1 improves and generalizes [1, Theorem 2.1] in the following ways. (i)
From a one nonexpansive mapping to a countable family of strict pseudocontraction mappings.(ii)
From a general system of variational inequalities to a general system of generalized nonlinear mixed composite-type equilibria.(iii)
Theorem3.1 for finding an element x^∈⋂n=1∞‍Fix(Tn)∩Fix(Q)∩GEP(C,G,Θ,Φ) (Q is defined as in Lemma 2.13) is more general the one of finding elements of Fix(S)∩Fix(D)∩GEP(C,G,Θ,Φ) (D is defined as in Lemma 2.17) in [1, Theorem 2.1].
Furthermore, our method of the proof is very different from that in [1, Theorem 2.1] because (3.1) involves the countable family of strict pseudocontraction mappings and strongly positive bounded linear operator.


## 4. Application to Minimization Problems

LetC be a nonempty closed and convex subset of a real Hilbert space H and A:H→H be a strongly positive linear bounded operator with a constant γ->0. In this section, we will utilize the results presented in Section 3 to study the following minimization problem:
(4.1)minx∈Ωμ2〈Ax,x〉+12∥x∥2-h(x),
where Ω is a nonempty closed and convex subsets of C, μ≥0 is some constant and h:C→ℝ is a potential function for γf (i.e., h'(x)=γf(x) for all x∈C), where f:C→C is a contraction mapping with a constant α∈(0,1). Note that this kind of minimization problems has been studied extensively by many authors (e.g., see [18, 36–39]). We can apply Theorem 3.1 to solve the above minimization problem in the framework of Hilbert spaces as follows.Theorem 4.1.
LetC be a nonempty closed and convex subset of a real Hilbert space H such that C±C⊂C. Let S:C→C be a nonexpansive mappings such that Fix(S)≠∅. Let μ>0 and γ>0 be two constants. Let f:C→C be a contraction mapping with a coefficient α∈(0,1) and A be a strongly positive bounded linear operator on C with a coefficient γ-∈(0,1) such that 0<γ<((1+μ)γ-)/α. Let {xn} be a sequence defined by x1∈C and
(4.2)xn+1=αnγf(xn)+βnxn+((1-βn)I-αn(I+μA))Sxn,∀n≥1,
where {αn},{βn} are two sequences in [0,1]. Assume the following conditions are satisfied:  (C1)
limn→∞αn=0 and ∑n=1∞‍αn=∞,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1. Suppose that Fix(S) is a compact subset of C. Then the sequence {xn} defined by (4.2) converges strongly to x^∈Fix(S) which solves the minimization problem (4.1).Proof.
Takingγ1=1 and γ2=γ3=0 in Theorem 3.1. Hence, from Theorem 3.1, we know that the sequence {xn} defined by (4.2) converges strongly to x^∈Fix(S), where x^ is the unique solution of the variational inequality
(4.3)〈(γf-(I+μA))x^,v-x^〉≤0,∀v∈Fix(S).
Since S is nonexpansive, then Fix(S) is convex. Again by the assumption that Fix(S) is compact, then it is a compact and convex subset of C, and
(4.4)μ2〈Ax,x〉+12∥x∥2-h(x):C→ℝ
is a continuous mapping. By virtue of the well-known Weierstrass's theorem, there exists a point x~∈Fix(S) which is a minimal point of minimization problem (4.1). As is known to all, (4.3) is the optimality necessary condition [18] for the minimization problem (4.1). Therefore we also have
(4.5)〈(γf-(I+μA))x~,v-x~〉≤0,∀v∈Fix(S).
Since x^ is the unique solution of (4.3), we have x^=x~. This completes the proof.


## 5. A Numerical Example

In this section, we give a real example in which the conditions satisfy the ones of Theorem3.1 and some numerical experiment results to explain the main result Theorem 3.1 as follows.Example 5.1.
LetH=ℝ, C=[0,1], Θ(x,y,z)=ϕ(x)=Bx=0, ∀x,y,z∈C, r=1, Fi(x,y)=φi(x)=Ψix=Φix=0, ∀x,y∈C, μi=1(i=1,2,3), A=I, f(x)=(1/2)x, and ∀x∈C, with a constant α=1/2, αn=1/n,  βn=(n+1)/2n, ∀n∈ℕ,  γi=(1/3)(i=1,2,3), γ=1, and μ=1. For all n∈ℕ, let Tn:C→C define by Tnx=nx2/(2n+1), ∀x∈C, we see that, {Tn} is a family of 0-strictly pseudocontractive with ⋂n=1∞‍Fix(Tn)={0}. Then, {xn} is the sequence defined by
(5.1)xn+1=(56-23n)xn+16(n-52n+1)xn2,
and xn→0 as n→∞, where 0 is the unique solution of the minimization problem
(5.2)minx∈C34x2+q.Proof.
Step
1. We show that
(5.3)T(r)(x)=PCx,∀x∈H,
where
(5.4)PCx={x|x|,x∈H∖C,x,x∈C.
Since Θ(x,y,z)=ϕ(x)=Bx=0, due to the definition of T(r)(x), ∀x∈H in (2.7), we have
(5.5)T(r)(x)={y∈C:〈y-z,z-x〉≥0,∀z∈C}.
Also by the equivalent property (2.5) of the nearest projection PC from H to C, we obtain this conclusion. When we take x∈C, then T(r)(x)=PCx=x. By the condition (Δ)(c), we have GEP(C,G,Θ,ϕ)=C. In a similar way, for all i=1,2,3, we can get
(5.6)Tμi(Fi,φi)(x)=PCx=x,∀x∈C,
and Fix(Q)=C. Hence
(5.7)GEP(C,G,Θ,ϕ)∩Fix(Q)=C.
Step
2. We show that {Tn} is satisfies the PT-condition. Since Tnx=(nx2/(2n+1)), ∀x∈C, and n∈ℕ. For all k,l∈ℕ, we have
(5.8)limk,l→∞|kx22k+1-lx22l+1|≤limk,l→∞supω∈C|kω22k+1-lω22l+1|=limk,l→∞|k2k+1-l2l+1|supω∈C{ω2}=0,
that is {Tn} satisfies the PT-condition.Step
3. We show that
(5.9)xn+1=(56-23n)xn+16(n-52n+1)xn2,xn→0,asn→∞,
where 0 is the unique solution of the minimization problem:
(5.10)minx∈C34x2+q.
Due to (5.3) and (5.4), we can obtain a special sequence {xn} of (3.1) in Theorem 3.1 as follows:
(5.11)xn+1=(56-23n)xn+16(n-52n+1)xn2.
Since ⋂n=1∞‍Fix(Tn)={0}, combining with (5.7), we have
(5.12)Ω:=GEP(C,G,Θ,ϕ)∩Fix(Q)∩⋂n=1∞‍Fix(Tn)={0}.
By Lemma 2.10, we obtain that xn→0, where 0 is the unique solution of the minimization problem minx∈C(3/4)x2+q, where q is a constant number.


### 5.1. Numerical Experiment Results

Next, we show the numerical experiment results using software MATLAB 7.0 and we obtain the results shown in Tables1 and 2 and Figure 1, which show that the iteration process of the sequence {xn} as initial points x1=1 and x1=0.5, respectively.Table 1
This table shows the value of sequence{xn} on each iteration step (initial value x1=1).
n
x
n
1
1.000000000000000
2
0.400000000000000
3
0.236825396825397
4
0.156844963280237
5
0.109791474296166
6
0.079448383160412
7
0.058780740964773
8
0.044187178532649
9
0.033618033450111
10
0.000000000000000Table 2
This table shows the value of sequence{xn} on each iteration step (initial value x1=0.5).
n
x
n
1
0.500000000000000
2
0.225000000000000
3
0.135089285714286
4
0.089721577233324
5
0.062805104063327
6
0.045409812093980
7
0.033562589425687
8
0.025205072875411
9
0.019159476038106
10
0.000000000000000These figures show the iteration comparison chart of different initial values (a)x1=1 and (b)  x1=0.5, respectively.
(a)
(b)


## 5.1. Numerical Experiment Results

Next, we show the numerical experiment results using software MATLAB 7.0 and we obtain the results shown in Tables1 and 2 and Figure 1, which show that the iteration process of the sequence {xn} as initial points x1=1 and x1=0.5, respectively.Table 1
This table shows the value of sequence{xn} on each iteration step (initial value x1=1).
n
x
n
1
1.000000000000000
2
0.400000000000000
3
0.236825396825397
4
0.156844963280237
5
0.109791474296166
6
0.079448383160412
7
0.058780740964773
8
0.044187178532649
9
0.033618033450111
10
0.000000000000000Table 2
This table shows the value of sequence{xn} on each iteration step (initial value x1=0.5).
n
x
n
1
0.500000000000000
2
0.225000000000000
3
0.135089285714286
4
0.089721577233324
5
0.062805104063327
6
0.045409812093980
7
0.033562589425687
8
0.025205072875411
9
0.019159476038106
10
0.000000000000000These figures show the iteration comparison chart of different initial values (a)x1=1 and (b)  x1=0.5, respectively.
(a)
(b)



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*Source: 414718-2012-11-14.xml*