# Hybrid Algorithms for Minimization Problems over the Solutions of Generalized Mixed Equilibrium and Variational Inclusion Problems

Authors: Thanyarat Jitpeera; Poom Kumam
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

We introduce a new general hybrid iterative algorithm for finding a common
element of the set of solution of fixed point for a nonexpansive mapping, the set of solution of
generalized mixed equilibrium problem, and the set of solution of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges
strongly to a common element of the above three sets under some mild conditions. Our results improve
and extend the corresponding results of Marino and Xu (2006), Yao and Liou (2010), Tan and Chang
(2011), and other authors.


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## Body

## 1. Introduction

In the theory of variational inequalities, variational inclusions, and equilibrium problems, the development of an efficient and implementable iterative algorithm is interesting and important. The important generalization of variational inequalities called variational inclusions, have been extensively studied and generalized in different directions to study a wide class of problems arising in mechanics, optimization, nonlinear programming, economics, finance, and applied sciences.Equilibrium theory represents an important area of mathematical sciences such as optimization, operations research, game theory, complementarity problems, financial mathematics, and mechanics. Equilibrium problems include variational inequalities, optimization problems, Nash equilibria problems, saddle point problems, fixed point problems, and complementarity problems as special cases; for example, see the references herein. LetC be a closed convex subset of a real Hilbert space H with the inner product 〈·,·〉 and the norm ∥·∥. Let F be a bifunction of C×C into ℛ, where ℛ is the set of real numbers, Φ:C→H be a mapping and φ:C→ℛ be a real-valued function. The generalized mixed equilibrium problem for finding x∈C such that(1.1)F(x,y)+〈Φx,y-x〉+φ(y)-φ(x)≥0,∀y∈C.
The set of solutions of (1.1) is denoted by GMEP(F,φ,Φ), that is(1.2)GMEP⁡(F,φ,Φ)={x∈C:F(x,y)+〈Φx,y-x〉+φ(y)-φ(x)≥0,∀y∈C}.
If Φ≡0 and φ≡0, the problem (1.1) is reduced into the equilibrium problem [1] for finding x∈C such that(1.3)F(x,y)≥0,∀y∈C.
The set of solutions of (1.3) is denoted by EP(F). This problem contains fixed point problems, includes as special cases numerous problems in physics, optimization, and economics. Some methods have been proposed to solve the equilibrium problem, please consult [2–4].IfF≡0 and φ≡0, the problem (1.1) is reduced into the Hartmann-Stampacchia variational inequality [5] for finding x∈C such that(1.4)〈Φx,y-x〉≥0,∀y∈C.
The set of solutions of (1.4) is denoted by VI(C,Φ). The variational inequality has been extensively studied in the literature [6].IfF≡0 and Φ≡0, the problem (1.1) is reduced into the minimize problem for finding x∈C such that(1.5)φ(y)-φ(x)≥0,∀y∈C.
The set of solutions of (1.5) is denoted by Argmin(φ).A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH:(1.6)θ(x)=12〈Ax,x〉-〈x,y〉,∀x∈F(S),
where A is a linear bounded operator, F(S) is the fixed point set of a nonexpansive mapping S, and y is a given point in H [7].Recall, a mappingS:C→C is said to be nonexpansive if (1.7)‖Sx-Sy‖≤‖x-y‖,
for all x,y∈C. If C is bounded closed convex and S is a nonexpansive mapping of C into itself, then F(S) is nonempty [8]. We denote weak convergence and strongly convergence by notations ⇀ and →, respectively. A mapping A of C into H is called monotone if(1.8)〈Ax-Ay,x-y〉≥0,
for all x,y∈C. A mapping A of C into H is called α-inverse-strongly monotone if there exists a positive real number α such that(1.9)〈Ax-Ay,x-y〉≥α‖Ax-Ay‖2,
for all x,y∈C. It is obvious that any α-inverse-strongly monotone mappings A is monotone and Lipschitz continuous mapping. A linear bounded operator A is strongly positive if there exists a constant γ¯>0 with the property(1.10)〈Ax,x〉≥γ¯‖x‖2,
for all x∈H. A self-mapping f:C→C is a contraction on C if there exists a constant α∈(0,1) such that(1.11)‖f(x)-f(y)‖≤α‖x-y‖,
for all x,y∈C. We use ΠC to denote the collection of all contraction on C. Note that each f∈ΠC has a unique fixed point in C.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. LetB:H→H be a single-valued nonlinear mapping and M:H→2H be a set-valued mapping. The variational inclusion problem is to find x∈H such that(1.12)θ∈B(x)+M(x),
where θ is the zero vector in H. The set of solutions of problem (1.12) is denoted by I(B,M). The variational inclusion has been extensively studied in the literature. See, for example,  [9–12] and the reference therein.A set-valued mappingM:H→2H is called monotone if for all x,y∈H,f∈M(x) and g∈M(y) imply 〈x-y,f-g〉≥0. A monotone mapping M is maximal if its graph G(M)∶={(f,x)∈H×H:f∈M(x)} of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for (x,f)∈H×H,〈x-y,f-g〉≥0 for all  (y,g)∈G(M) imply f∈M(x).LetB be an inverse-strongly monotone mapping of C into H and let NCv be normal cone to C at v∈C, that is, NCv={w∈H:〈v-u,w〉≥0,∀u∈C}, and define(1.13)Mv={Bv+NCv,ifv∈C,∅,ifv∉C.
Then M is a maximal monotone and θ∈Mv if and only if v∈VI(C,B) [13].LetM:H→2H be a set-valued maximal monotone mapping, then the single-valued mapping JM,λ:H→H defined by(1.14)JM,λ(x)=(I+λM)-1(x),x∈H
is called the resolvent operator associated with M, where λ is any positive number and I is the identity mapping. It is worth mentioning that the resolvent operator is nonexpansive, 1-inverse-strongly monotone, and that a solution of problem (1.12) is a fixed point of the operator JM,λ(I-λB)  for all λ>0, see [14], that is,  I(B,M)=F(JM,λ(I-λB)),∀λ>0.In 2000, Moudafi [15] introduced the viscosity approximation method for nonexpansive mapping and proved that if H is a real Hilbert space, the sequence {xn} defined by the iterative method below, with the initial guess x0∈C chosen arbitrarily,(1.15)xn+1=αnf(xn)+(1-αn)Sxn,n≥0,
where {αn}⊂(0,1) satisfies certain conditions, converges strongly to a fixed point of S (say x¯∈C) which is the unique solution of the following variational inequality:(1.16)〈(I-f)x¯,x-x¯〉≥0,∀x∈F(S).In 2006, Marino and Xu [7] introduced a general iterative method for nonexpansive mapping. They defined the sequence {xn} generated by the algorithm x0∈C,(1.17)xn+1=αnγf(xn)+(I-αnA)Sxn,n≥0,
where {αn}⊂(0,1) and A is a strongly positive linear bounded operator. They prove that if C=H and the sequence {αn} satisfies appropriate conditions, then the sequence {xn} generated by (1.17) converges strongly to a fixed point of S (say x¯∈H) which is the unique solution of the following variational inequality:(1.18)〈(A-γf)x¯,x-x¯〉≥0,∀x∈F(S),
which is the optimality condition for the minimization problem(1.19)min⁡x∈F(S)∩EP⁡(F)12〈Ax,x〉-h(x),
where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H).In 2010, Yao and Liou [16] introduced the following composite iterative scheme in a real Hilbert space: x0∈C(1.20)xn+1=μnPC[αnf(xn)+(1-αn)Sxn]+(1-μn)Tr(xn-rAxn),
for all n∈ℕ, where {αn},{μn}⊂[0,1). Furthermore, they proved {xn} and {un} converge strongly to the same point z∈F(S)∩EP(F), where PC is the projection of H onto C.In 2011, Tan and Chang [11] introduced the following iterative process for {Tn:C→C} be a sequence of nonexpansive mappings. Let {xn} be the sequence defined by(1.21)xn+1=αnxn+(1-αn)(SPC((1-tn)JM,λ(I-λA)Tμ(I-μB))xn),∀n≥0,
where {αn}⊂(0,1), λ∈(0,2α] and μ∈(0,2β]. Then, the sequence {xn} defined by (1.21) converges strongly to a common element of the set of fixed points of nonexpansive mapping, the set of solution of the variational inequality and the generalized equilibrium problem.In this paper, we modify the iterative methods (1.17), (1.20), and (1.21) by purposing the following new general viscosity iterative method: x0∈C,(1.22)xn+1=ξnPC[αnγf(xn)+(I-αnA)SJM,λ(xn-λBxn)]+(1-ξn)Tr(xn-rΦxn),
for all n∈ℕ, where {αn},{ξn}⊂(0,1), r∈(0,2σ), and λ∈(0,2β) satisfy some appropriate conditions. Consequently, we show that under some control conditions the sequence {xn} strongly converge to a common element of the set of fixed points of nonexpansive mapping, the solution of the generalized mixed equilibrium problem, and the set of solution of the variational inclusion in a real Hilbert space.


## 2. Preliminaries

LetH be a real Hilbert space and C be a nonempty closed convex subset of H. Recall that the (nearest point) projection PC from H onto C assigns to each x∈H, the unique point in PCx∈C satisfying the property (2.1)‖x-PCx‖=min⁡y∈C⁡‖x-y‖.
The following characterizes the projection PC. We recall some lemmas which will be needed in the rest of this paper.Lemma 2.1.
The functionu∈C is a solution of the variational inequality (1.4) if and only if u∈C satisfies the relation u=PC(u-λΦu)  for all λ>0.Lemma 2.2.
For a givenz∈H, u∈C, u=PCz⇔〈u-z,v-u〉≥0,∀v∈C.
It is well known thatPC is a firmly nonexpansive mapping of H onto C and satisfies
(2.2)‖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,
(2.3)〈x-PCx,y-PCx〉≤0.Lemma 2.3 (see [17]).
LetM:H→2H be a maximal monotone mapping and let B:H→H be a monotone and Lipschitz continuous mapping. Then the mapping L=M+B:H→2H is a maximal monotone mapping.Lemma 2.4 (see [18]).
Each Hilbert spaceH satisfies Opial's condition, that is, for any sequence {xn}⊂H with xn⇀x, the inequality liminfn→∞∥xn-x∥<liminfn→∞∥xn-y∥, hold for each y∈H with y≠x.Lemma 2.5 (see [19]).
Assume{an} is a sequence of nonnegative real numbers such that
(2.4)an+1≤(1-γn)an+δnγn,∀n≥0,
where {γn}⊂(0,1) and {δn} is a sequence in ℛ such that(i)
∑n=1∞γn=∞.(ii)
limsupn→∞δn≤0 or ∑n=1∞|δnγn|<∞.Then limn→∞an=0.Lemma 2.6 (see [20]).
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,
(2.5)xn⇀x,xn-Txn⟶0
implies x=Tx.For solving the generalized mixed equilibrium problem, let us assume that the bifunctionF:C×C→ℛ, the nonlinear mapping Φ:C→H is continuous monotone and φ:C→ℛ satisfies the following conditions:  (A1)
F(x,x)=0 for all x∈C;(A2)
F is monotone, that is,  F(x,y)+F(y,x)≤0 for any x,y∈C;(A3)
for each fixedy∈C, x↦F(x,y) is weakly upper semicontinuous;(A4)
for each fixedx∈C, y↦F(x,y) is convex and lower semicontinuous;(B1)
for eachx∈C and r>0, there exist a bounded subset Dx⊆C and yx∈C such that for any z∈C∖Dx,
(2.6)F(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0,(B2)
C is a bounded set.Lemma 2.7 (see [21]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F:C×C→ℛ be a bifunction mapping satisfies (A1)–(A4) and let φ:C→ℛ is convex and lower semicontinuous such that C∩domφ≠∅. Assume that either (B1) or (B2) holds. For r>0 and x∈H, then there exists u∈C such that
(2.7)F(u,y)+φ(y)-φ(u)+1r〈y-u,u-x〉.
Define a mapping Tr:H→C as follows:
(2.8)Tr(x)={u∈C:F(u,y)+φ(y)-φ(u)+1r〈y-u,u-x〉≥0,∀y∈C},
for all x∈H. Then, the following hold:  (i)
Tr is single-valued;(ii)
Tr is firmly nonexpansive, that is, for any x,y∈H,∥Trx-Try∥2≤〈Trx-Try,x-y〉;(iii)
F(Tr)=MEP(F,φ);(iv)
MEP(F,φ) is closed and convex.Lemma 2.8 (see[7]).
AssumeA is a strongly positive linear bounded operator on a Hilbert space H with coefficient γ¯>0 and 0<ρ≤∥A∥-1, then ∥I-ρA∥≤1-ργ¯.


## 3. Strong Convergence Theorems

In this section, we show a strong convergence theorem which solves the problem of finding a common element ofF(S),GMEP(F,φ,Φ), and I(B,M) of inverse-strongly monotone mappings in a Hilbert space.Theorem 3.1.
LetH be a real Hilbert space, C be a closed convex subset of H. Let F be a bifunction of C×C into ℛ satisfying (A1)–(A4) and B,Φ:C→H be β,σ-inverse-strongly monotone mappings, φ:C→ℛ is convex and lower semicontinuous function, f:C→C be a contraction with coefficient α(0<α<1), M:H→2H be a maximal monotone mapping and A be a strongly positive linear bounded operator of H into itself with coefficient γ¯>0, assume that 0<γ<γ¯/α. Let S be a nonexpansive mapping of H into itself and assume that either (B1) or (B2) holds such that
(3.1)Θ∶=F(S)∩GMEP⁡(F,φ,Φ)∩I(B,M)≠∅.
Suppose {xn} is a sequences generated by the following algorithm x0∈C arbitrarily:
(3.2)xn+1=ξnPC[αnγf(xn)+(I-αnA)SJM,λ(xn-λBxn)]+(1-ξn)Tr(xn-rΦxn),
where {αn},{ξn}⊂(0,1),λ∈(0,2β) such that 0<a≤λ≤b<2β and r∈(0,2σ) with 0<c≤d≤1-σ satisfy the following conditions:(C1)
limn→∞αn=0, Σn=0∞αn=∞ and limn→∞(αn+1/αn)=1,(C2)
0<liminfn→∞ξn<limsupn→∞ξn<1 and limn→∞((ξn+1-ξn)/αn+1)=1.
Then{xn} converges strongly to q∈Θ, where q=PΘ(γf+I-A)(q) which solves the following variational inequality:
(3.3)〈(γf-A)q,p-q〉≤0,∀p∈Θ
which is the optimality condition for the minimization problem
(3.4)min⁡q∈Θ⁡12〈Aq,q〉-h(q),
where h is a potential function for γf (i.e., h′(q)=γf(q) for q∈H).Proof.
Because of condition (C1), we may assume without loss of generality, thenαn∈(0,∥A∥-1) for all n. By Lemma 2.8, we have ∥I-αnA∥≤1-αnγ¯. Next, we will assume that ∥I-A∥≤∥1-γ¯∥.
Step 1.
We will show{xn},{un} are bounded.
SinceB,Φ are β,σ-inverse-strongly monotone mappings, we have
(3.5)‖(I-λB)x-(I-λB)y‖2=‖(x-y)-λ(Bx-By)‖2=‖x-y‖2-2λ〈x-y,Bx-By〉+λ2‖Bx-By‖2≤‖x-y‖2+λ(λ-2β)‖Bx-By‖2≤‖x-y‖2.
In similar way, we can obtain
(3.6)‖(I-rΦ)x-(I-rΦ)y‖2≤‖x-y‖2.
It is clear that if 0<λ<2β,0<r<2σ, then I-λB,I-rΦ are all nonexpansive.
Putyn=JM,λ(xn-λBxn),n≥0. It follows that
(3.7)‖yn-q‖=‖JM,λ(xn-λBxn)-JM,λ(q-λBq)‖≤‖xn-q‖.
By Lemma 2.7, we have un=Tr(xn-rΦxn) for all n≥0. Then, we have
(3.8)‖un-q‖2=‖Tr(xn-rΦxn)-Tr(q-rΦq)‖2≤‖(xn-rΦxn)-(q-rΦq)‖2≤‖xn-q‖2+r(r-2σ)‖Φxn-Φq‖2≤‖xn-q‖2.
Put zn=PC[αnγf(xn)+(I-αnA)Syn]  for all n≥0. From (3.2), we deduce that
(3.9)∥xn+1-q∥=‖ξn(zn-q)+(1-ξn)(un-q)‖≤ξn‖PC[αnγf(xn)+(I-αnA)Syn]-PCq‖+(1-ξn)‖un-q‖≤ξn‖αnγf(xn)+(I-αnA)Syn-q‖+(1-ξn)‖un-q‖=ξn‖αn(γf(xn)-Aq)+(I-αnA)(Syn-q)‖+(1-ξn)‖un-q‖≤ξnαn‖γf(xn)-Aq‖+ξn(1-αnγ¯)‖yn-q‖+(1-ξn)‖un-q‖≤ξnαn‖γf(xn)-γf(q)‖+ξnαn‖γf(q)-Aq‖+ξn(1-αnγ¯)‖xn-q‖+(1-ξn)‖xn-q‖≤ξnαnγα‖xn-q‖+ξnαn‖γf(q)-Aq‖+ξn(1-αnγ¯)‖xn-q‖+(1-ξn)‖xn-q‖=(1-(γ¯-γα)ξnαn)‖xn-q‖+ξnαn‖γf(q)-Aq‖=(1-(γ¯-γα)ξnαn)‖xn-q‖+(γ¯-γα)ξnαn‖γf(q)-Aq‖γ¯-γα≤max⁡⁡{‖xn-q‖,‖γf(q)-Aq‖γ¯-γα}⁡.
It follows from induction that
(3.10)‖xn-q‖≤max⁡⁡{‖x0-q‖,‖γf(q)-Aq‖γ¯-γα},n≥0.
Therefore {xn} is bounded, so are {yn}, {Syn}, {Bxn}, {f(xn)}, and {ASyn}.Step 2.
We claim thatlimn→∞∥xn+2-xn+1∥=0. From (3.2), we have
(3.11)‖xn+2-xn+1‖=‖ξn+1zn+1+(1-ξn+1)un+1-ξnzn-(1-ξn)un‖=‖ξn+1(zn+1-zn)+(ξn+1-ξn)zn+(1-ξn+1)(un+1-un)+(ξn+1-ξn)un‖≤ξn+1‖zn+1-zn‖+(1-ξn+1)‖un+1-un‖+|ξn+1-ξn|(‖zn‖+‖un‖).
We estimate ∥un+1-un∥, so we have
(3.12)‖un+1-un‖=‖Tr(xn+1-rΦxn+1)-Tr(xn-rΦxn)‖≤‖(xn+1-rΦxn+1)-(xn-rΦxn)‖≤‖xn+1-xn‖.
Substituting (3.12) into (3.11) that
(3.13)‖xn+2-xn+1‖≤ξn+1‖zn+1-zn‖+(1-ξn+1)‖xn+1-xn‖+|ξn+1-ξn|(‖zn‖+‖un‖).
We note that
(3.14)‖zn+1-zn‖=‖PC[αn+1γf(xn+1)+(I-αn+1A)Syn+1]-PC[αnγf(xn)+(I-αnA)Syn]‖≤‖αn+1γf(xn+1)+(I-αn+1A)Syn+1-(αnγf(xn)+(I-αnA)Syn)‖=‖αn+1γ(f(xn+1)-f(xn))+(αn+1-αn)γf(xn)+(I-αn+1A)(Syn+1-Syn)+(αn-αn+1)ASyn‖≤αn+1γα‖xn+1-xn‖+|αn+1-αn|‖γf(xn)‖+(1-αn+1γ¯)‖yn+1-yn‖+|αn+1-αn|‖ASyn‖=αn+1γα‖xn+1-xn‖+|αn+1-αn|(‖γf(xn)‖+‖ASyn‖)+(1-αn+1γ¯)‖yn+1-yn‖.
Next, we estimate ∥yn+1-yn∥, then we get
(3.15)‖yn+1-yn‖=‖JM,λ(xn+1-λBxn+1)-JM,λ(xn-λBxn)‖≤‖(xn+1-λBxn+1)-(xn-λBxn)‖≤‖xn+1-xn‖.
Substituting (3.15) into (3.14), we obtain that
(3.16)‖zn+1-zn‖≤αn+1γα‖xn+1-xn‖+|αn+1-αn|(‖γf(xn)‖+‖ASyn‖)+(1-αn+1γ¯)‖xn+1-xn‖.
And substituting (3.12), (3.16) into (3.11), we get
(3.17)‖xn+2-xn+1‖≤ξn+1{αn+1γα‖xn+1-xn‖+|αn+1-αn|(‖γf(xn)‖+‖ASyn‖)+(1-αn+1γ¯)‖xn+1-xn‖}+(1-ξn+1)‖xn+1-xn‖+|ξn+1-ξn|(‖zn‖+‖un‖)≤[1-(γ¯-γα)ξn+1αn+1]‖xn+1-xn‖+(|αn+1-αn|+|ξn+1-ξn|)M,
where M>0 is a constant satisfying
(3.18)sup⁡n{‖γf(xn)‖+‖ASyn‖,‖zn‖+‖un‖}≤M.
This together with (C1), (C2), and Lemma 2.5, implies that
(3.19)lim⁡n→∞⁡‖xn+2-xn+1‖=0.
From (3.15), we also have ∥yn+1-yn∥→0 as n→∞.Step 3.
We show the following:(i)
limn→∞∥Bxn-Bq∥=0;(ii)
limn→∞∥Φxn-Φq∥=0. For q∈Ω and q=JM,λ(q-λBq), then we get
(3.20)‖yn-q‖2=‖JM,λ(xn-λBxn)-JM,λ(q-λBq)‖2≤‖(xn-λBxn)-(q-λBq)‖2≤‖xn-q‖2+λ(λ-2β)‖Bxn-Bq‖2.
It follows that
(3.21)‖zn-q‖2=‖PC(αnγf(xn)+(I-αnA)Syn)-PC(q)‖2≤‖αn(γf(xn)-Aq)+(I-αnA)(Syn-q)‖2≤αn‖γf(xn)-Aq‖2+(1-αnγ¯)‖yn-q‖2+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖≤αn‖γf(xn)-Aq‖2+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖+(1-αnγ¯)(‖xn-q‖2+λ(λ-2β)‖Bxn-Bq‖2)≤αn‖γf(xn)-Aq‖2+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖+‖xn-q‖2+(1-αnγ¯)λ(λ-2β)‖Bxn-Bq‖2.
By the convexity of the norm ∥·∥, we have
(3.22)‖xn+1-q‖2=‖ξnzn+(1-ξn)un-q‖2=‖ξn(zn-q)+(1-ξn)(un-q)‖2≤ξn‖zn-q‖2+(1-ξn)‖un-q‖2.
Substituting (3.8), (3.21) into (3.22), we obtain
(3.23)‖xn+1-q‖2≤ξn{αn‖γf(xn)-Aq‖2+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖+‖xn-q‖2+(1-αnγ¯)λ(λ-2β)‖Bxn-Bq‖2}+(1-ξn)‖xn-q‖2=ξnαn‖γf(xn)-Aq‖2+2ξnαn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖+ξn‖xn-q‖2+ξn(1-αnγ¯)λ(λ-2β)‖Bxn-Bq‖2+(1-ξn)‖xn-q‖2.
So, we obtain
(3.24)ξn(1-αnγ¯)λ(2β-λ)‖Bxn-Bq‖2≤ξnαn‖γf(xn)-Aq‖2+ϵn+‖xn-xn+1‖(‖xn-q‖+‖xn+1-q‖),
where ϵn=2ξnαn(1-αnγ¯)∥γf(xn)Aq∥∥yn-q∥. Since condition (C1), (C2) and limn→∞∥xn+1-xn∥=0 then we obtain that ∥Bxn-Bq∥→0 as n→∞. We consider this inequality in (3.21) that
(3.25)‖zn-q‖2≤αn‖γf(xn)-Aq‖2+(1-αnγ¯)‖yn-q‖2+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖.
Substituting (3.20) into (3.25), we have
(3.26)‖zn-q‖2≤αn‖γf(xn)-Aq‖2+(1-αnγ¯){‖xn-q‖2+λ(λ-2β)‖Bxn-Bq‖2}+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖=αn‖γf(xn)-Aq‖2+(1-αnγ¯)‖xn-q‖2+(1-αnγ¯)λ(λ-2β)‖Bxn-Bq‖2+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖≤αn‖γf(xn)-Aq‖2+‖xn-q‖2+(1-αnγ¯)λ(λ-2β)‖Bxn-Bq‖2+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖.
Substituting (3.8) and (3.26) into (3.22), we obtain
(3.27)‖xn+1-q‖2≤ξn{αn‖γf(xn)-Aq‖2+‖xn-q‖2+(1-αnγ¯)λ(λ-2β)‖Bxn-Bq‖2+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖αn‖γf(xn)-Aq‖2+‖xn-q‖2+(1-αnγ¯)}+(1-ξn){‖xn-q‖2+r(r-2σ)‖Φxn-Φq‖2}=ξnαn‖γf(xn)-Aq‖2+ξn‖xn-q‖2+ξn(1-αnγ¯)λ(λ-2β)‖Bxn-Bq‖2+2ξnαn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖+(1-ξn)‖xn-q‖2+(1-ξn)r(r-2σ)‖Φxn-Φq‖2.
So, we also have
(3.28)(1-ξn)r(2σ-r)‖Φxn-Φq‖2≤ξnαn‖γf(xn)-Aq‖2+ϵn+‖xn-xn+1‖(‖xn-q‖+‖xn+1-q‖)+ξn(1-αnγ¯)λ(λ-2β)‖Bxn-Bq‖2,
where ϵn=2ξnαn(1-αnγ¯)∥γf(xn)-Aq∥∥yn-q∥. Since condition (C1), (C2), limn→∞∥xn+1-xn∥=0 and limn→∞∥Bxn-Bq∥=0 then we obtain that ∥Φxn-Φq∥→0 as n→∞.Step 4.
We show the following:(i)
limn→∞∥xn-un∥=0;(ii)
limn→∞∥xn-yn∥=0;(iii)
limn→∞∥yn-Syn∥=0. Since Tr is firmly nonexpansive, we observe that
(3.29)‖un-q‖2=‖Tr(xn-rΦxn)-Tr(q-rΦq)‖2≤〈(xn-rΦxn)-(q-rΦq),un-q〉=12{‖(xn-rΦxn)-(q-rΦq)‖2+‖un-q‖2-‖(xn-rΦxn)-(q-rΦq)-(un-q)‖2}≤12{‖xn-q‖2+‖un-q‖2-‖(xn-un)-r(Φxn-Φq)‖2}=12{+2r〈Φxn-Φq,xn-un〉-r2‖Φxn-Φq‖2‖xn-q‖2+‖un-q‖2-‖xn-un‖2+2r〈Φxn-Φq,xn-un〉-r2‖Φxn-Φq‖2}.
Hence, we have
(3.30)‖un-q‖2≤‖xn-q‖2-‖xn-un‖2+2r‖Φxn-Φq‖‖xn-un‖.
Since JM,λ is 1-inverse-strongly monotone, we have
(3.31)‖yn-q‖2=‖JM,λ(xn-λBxn)-JM,λ(q-λBq)‖2≤〈(xn-λBxn)-(q-λBq),yn-q〉=12{‖(xn-λBxn)-(q-λBq)‖2+‖yn-q‖2-‖(xn-λBxn)-(q-λBq)-(yn-q)‖2}≤12{‖xn-q‖2+‖yn-q‖2-‖(xn-yn)-λ(Bxn-Bq)‖2}=12{‖xn-q‖2+‖yn-q‖2-‖xn-yn‖2+2λ〈xn-yn,Bxn-Bq〉-λ2‖Bxn-Bq‖2},
which implies that
(3.32)‖yn-q‖2≤‖xn-q‖2-‖xn-yn‖2+2λ‖xn-yn‖‖Bxn-Bq‖.
Substituting (3.32) into (3.25), we have
(3.33)‖zn-q‖2≤αn‖γf(xn)-Aq‖2+(1-αnγ¯){‖xn-q‖2-‖xn-yn‖2+2λ‖xn-yn‖‖Bxn-Bq‖}+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖=αn‖γf(xn)-Aq‖2+(1-αnγ¯)‖xn-q‖2-(1-αnγ¯)‖xn-yn‖2+2λ(1-αnγ¯)‖xn-yn‖‖Bxn-Bq‖+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖.
Substituting (3.30) and (3.33) into (3.22), we obtain
(3.34)‖xn+1-q‖2≤ξn‖zn-q‖2+(1-ξn)‖un-q‖2≤ξn{αn‖γf(xn)-Aq‖2+(1-αnγ¯)‖xn-q‖2-(1-αnγ¯)‖xn-yn‖2+2λ(1-αnγ¯)‖xn-yn‖‖Bxn-Bq‖+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖(1-αnγ¯)‖xn-yn‖2}‖γf(xn)-Aq‖2+(1-ξn){‖xn-q‖2-‖xn-un‖2+2r‖Φxn-Φq‖‖xn-un‖‖γf(xn)-Aq‖2}≤ξn{αn‖γf(xn)-Aq‖2+‖xn-q‖2-‖xn-yn‖2+2λ(1-αnγ¯)‖xn-yn‖‖Bxn-Bq‖+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖‖γf(xn)-Aq‖2}+(1-ξn){‖xn-q‖2-‖xn-un‖2+2r‖Φxn-Φq‖‖xn-un‖}≤ξnαn‖γf(xn)-Aq‖2+ξn‖xn-q‖2-‖xn-yn‖2+2λξn(1-αnγ¯)‖xn-yn‖‖Bxn-Bq‖+2ξnαn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖+(1-ξn)‖xn-q‖2-‖xn-un‖2+2r(1-ξn)‖Φxn-Φq‖‖xn-un‖.
Then, we derive
(3.35)‖xn-un‖2+‖xn-yn‖2=ξnαn‖γf(xn)-Aq‖2+‖xn-q‖2-‖xn+1-q‖2+2ξnλ(1-αnγ¯)‖xn-yn‖‖Bxn-Bq‖+2ξnαn(1-αnγ¯)‖γf(xn)-Aq‖∥yn-q∥+2r(1-ξn)‖Φxn-Φq‖‖xn-un‖≤ξnαn‖γf(xn)-Aq‖2+‖xn+1-xn‖(‖xn-q‖+‖xn+1-q‖)+2ξnλ(1-αnγ¯)‖xn-yn‖‖Bxn-Bq‖+2ξnαn(1-αnγ¯)‖γf(xn)-Aq‖∥yn-q∥+2r(1-ξn)‖Φxn-Φq‖‖xn-un‖.
By condition (C1), (C2), limn→∞∥xn-xn+1∥=0,limn→∞∥Φxn-Φq∥=0, and limn→∞∥Bxn-Bq∥=0. So, we have ∥xn-un∥→0,∥xn-yn∥→0 as n→∞. It follows that
(3.36)‖un-yn‖≤‖xn-un‖+‖xn-yn‖⟶0,asn⟶∞.
We note that xn+1-xn=ξn(zn-xn)+(1-ξn)(un-xn). From limn→∞∥xn-un∥=0,limn→∞∥xn+1-xn∥=0, and hence
(3.37)lim⁡n→∞‖zn-xn‖=0.
Since
(3.38)‖zn-yn‖≤‖zn-xn‖+‖xn-yn‖.
So, by (3.37) and limn→∞∥xn-yn∥=0, we obtain
(3.39)lim⁡n→∞‖zn-yn‖=0.
Therefore, we observe that
(3.40)‖Syn-zn‖≤‖Syn-PC(αnγf(xn)+(I-αnA)Syn)‖≤‖Syn-αnγf(xn)-(I-αnA)Syn‖≤αn‖γf(xn)-ASyn‖.
By condition (C1), we have ∥Syn-zn∥→0 as n→∞. Next, we observe that
(3.41)‖Syn-yn‖≤‖Syn-zn‖+‖zn-yn‖.
By (3.39) and (3.40), we have ∥Syn-yn∥→0 as n→∞.Step 5.
We show thatq∈Θ∶=F(S)∩GMEP(F,φ,Φ)∩I(B,M) and limsupn→∞〈(γf-A)q,Syn-q〉≤0. It is easy to see that PΘ(γf+(I-A)) is a contraction of H into itself. Indeed, since 0<γ<γ¯/α we have
(3.42)‖PΘ(γf+(I-A))x-PΘ(γf+(I-A))y‖≤‖(γf+(I-A))x-(γf+(I-A))y‖≤γ‖f(x)-f(y)‖+|I-A|‖x-y‖≤γα‖x-y‖+(1-γ¯)‖x-y‖=(1-γ¯+γα)‖x-y‖.
Since H is complete, there exists a unique fixed point q∈H such that q=PΘ(γf+(I-A))(q). By Lemma 2.2, we obtain that 〈(γf-A)q,w-q〉≤0 for all w∈Θ.
Next, we show thatlimsupn→∞〈(γf-A)q,Syn-q〉≤0, where q=PΘ(γf+I-A)(q) is the unique solution of the variational inequality 〈(γf-A)q,p-q〉≥0,for allp∈Θ. We can choose a subsequence {yni} of {yn} such that
(3.43)limsup⁡n→∞〈(γf-A)q,Syn-q〉=lim⁡i→∞〈(γf-A)q,Syni-q〉.
Since {yni} is bounded, there exists a subsequence {ynij} of {yni} which converges weakly to w. We may assume without loss of generality that yni⇀w. We claim that w∈Θ, since limn→∞∥yn-Syn∥=0 and by Lemma 2.6, we have w∈F(S).
Next, we show thatw∈GMEP(F,φ,Φ). Since un=Tr(xn-rΦxn), we know that
(3.44)F(un,y)+φ(y)-φ(un)+〈Φxn,y-un〉+1r〈y-un,un-xn〉≥0,∀y∈C.
It follows by (A2) that
(3.45)φ(y)-φ(un)+〈Φxn,y-un〉+1r〈y-un,un-xn〉≥F(y,un),∀y∈C.
Hence,
(3.46)φ(y)-φ(uni)+〈Φxni,y-uni〉+1r〈y-uni,uni-xni〉≥F(y,uni),∀y∈C.
For t∈(0,1] and y∈H, let yt=ty+(1-t)w. From (3.46) we have
(3.47)〈yt-uni,Φyt〉≥〈yt-uni,Φyt〉-φ(yt)+φ(uni)-〈Φxni,yt-uni〉-1r〈yt-uni,uni-xni〉+F(yt,uni)=〈yt-uni,Φyt-Φuni〉+〈yt-uni,Φuni-Φxni〉-φ(yt)+φ(uni)-1r〈yt-uni,uni-xni〉+F(yt,uni).
From ∥uni-xni∥→0, we have ∥Φuni-Φxni∥→0. Further, from (A4) and the weakly lower semicontinuity of φ,(uni-xni)/r→0 and uni⇀w, we have
(3.48)〈yt-w,Φyt〉≥-φ(yt)+φ(w)+F(yt,w).
From (A1), (A4), and (3.48), we have
(3.49)0=F(yt,yt)-φ(yt)+φ(yt)≤tF(yt,y)+(1-t)F(yt,w)+tφ(y)+(1-t)φ(w)-φ(yt)=t[F(yt,y)+φ(y)-φ(yt)]+(1-t)[F(yt,w)+φ(w)-φ(yt)]≤t[F(yt,y)+φ(y)-φ(yt)]+(1-t)〈yt-w,Φyt〉=t[F(yt,y)+φ(y)-φ(yt)]+(1-t)t〈y-w,Φyt〉,
and hence
(3.50)0≤F(yt,y)+φ(y)-φ(yt)+(1-t)〈y-w,Φyt〉.
Letting t→0, we have, for each y∈C,
(3.51)F(w,y)+φ(y)-φ(w)+〈y-w,Φw〉≥0.
This implies that w∈GMEP(F,φ,Φ).
Lastly, we show thatw∈I(B,M). In fact, since B is a β-inverse-strongly monotone, B is monotone and Lipschitz continuous mapping. It follows from Lemma 2.3, that M+B is a maximal monotone. Let (v,g)∈G(M+B), since g-Bv∈M(v). Again since yni=JM,λ(xni-λBxni), we have xni-λBxni∈(I+λM)(yni), that is, (1/λ)(xni-yni-λBxni)∈M(yni). By virtue of the maximal monotonicity of M+B, we have
(3.52)〈v-yni,g-Bv-1λ(xni-yni-λBxni)〉≥0,
and hence
(3.53)〈v-yni,g〉≥〈v-yni,Bv+1λ(xni-yni-λBxni)〉=〈v-yni,Bv-Byni〉+〈v-yni,Byni-Bxni〉+〈v-yni,1λ(xni-yni)〉.
It follows from limn→∞∥xn-yn∥=0, we have limn→∞∥Bxn-Byn∥=0 and yni⇀w that
(3.54)limsup⁡n→∞〈v-yn,g〉=〈v-w,g〉≥0.
It follows from the maximal monotonicity of B+M that θ∈(M+B)(w), that is, w∈I(B,M). Therefore, w∈Θ. It follows that
(3.55)limsup⁡n→∞〈(γf-A)q,Syn-q〉=lim⁡k→∞〈(γf-A)q,Syni-q〉=〈(γf-A)q,w-q〉≤0.Step 6.
We provexn→q. By using (3.2) and together with Schwarz inequality, we have
(3.56)‖xn+1-q‖2=‖ξnPC[(αnγf(xn)+(I-αnA)Syn)-q]+(1-ξn)(un-q)‖2≤ξn‖PC[(αnγf(xn)+(I-αnA)Syn)-PC(q)]‖2+(1-ξn)‖un-q‖2≤ξn‖αn(γf(xn)-Aq)+(I-αnA)(Syn-q)‖2+(1-ξn)‖xn-q‖2≤ξn(I-αnA)2‖Syn-q‖2+ξnαn2‖γf(xn)-Aq‖2+2ξnαn〈(I-αnA)(Syn-q),γf(xn)-Aq〉+(1-ξn)‖xn-q‖2≤ξn(1-αnγ¯)2‖yn-q‖2+ξnαn2‖γf(xn)-Aq‖2+2ξnαn〈Syn-q,γf(xn)-Aq〉-2ξnαn2〈A(Syn-q),γf(xn)-Aq〉+(1-ξn)‖xn-q‖2=ξn(1-αnγ¯)2‖xn-q‖2+ξnαn2‖γf(xn)-Aq‖2+2ξnαn〈Syn-q,γf(xn)-γf(q)〉+2ξnαn〈Syn-q,γf(q)-Aq〉-2ξnαn2〈A(Syn-q),γf(xn)-Aq〉+(1-ξn)‖xn-q‖2≤ξn(1-αnγ¯)2‖xn-q‖2+ξnαn2‖γf(xn)-Aq‖2+2ξnαn‖Syn-q‖‖γf(xn)-γf(q)‖+2ξnαn〈Syn-q,γf(q)-Aq〉-2ξnαn2〈A(Syn-q),γf(xn)-Aq〉+(1-ξn)‖xn-q‖2≤ξn(1-αnγ¯)2‖xn-q‖2+ξnαn2‖γf(xn)-Aq‖2+2ξnγααn‖yn-q‖‖xn-q‖+2ξnαn〈Syn-q,γf(q)-Aq〉-2ξnαn2〈A(Syn-q),γf(xn)-Aq〉+(1-ξn)‖xn-q‖2≤(ξn-2ξnαnγ¯+ξnαn2γ¯2)‖xn-q‖2+ξnαn2‖γf(xn)-Aq‖2+2ξnγααn‖xn-q‖2+2ξnαn〈Syn-q,γf(q)-Aq〉-2ξnαn2〈A(Syn-q),γf(xn)-Aq〉+(1-ξn)‖xn-q‖2=(1-2ξnαnγ¯+2ξnγααn)‖xn-q‖2+ξnαn2‖γf(xn)-Aq‖2+2ξnαn〈Syn-q,γf(q)-Aq〉-2ξnαn2‖A(Syn-q)‖‖γf(xn)-Aq‖+ξnαn2γ¯2‖xn-q‖2=(1-2(γ¯-γα)ξnαn)‖xn-q‖2+ξnαn2‖γf(xn)-Aq‖2+2ξnαn〈Syn-q,γf(q)-Aq〉-2ξnαn2‖A(Syn-q)‖‖γf(xn)-Aq‖+ξnαn2γ¯2‖xn-q‖2=(1-2(γ¯-γα)ξnαn)‖xn-q‖2+2(γ¯-γα)ξnαn{αnγ¯22(γ¯-γα)αn2(γ¯-γα)‖γf(xn)-Aq‖2+1γ¯-γα〈Syn-q,γf(q)-Aq〉-αnγ¯-γα‖A(Syn-q)‖‖γf(xn)-Aq‖+αnγ¯22(γ¯-γα)‖xn-q‖2}=(1-γn)‖xn-q‖2+γnδn,
where γn=2(γ¯-γα) and δn=(αn/2(γ¯-γα))∥γf(xn)-Aq∥2+(1/(γ¯-γα))〈Syn-q,γf(q)-Aq〉-(αn/(γ¯-γα))∥A(Syn-q)∥∥γf(xn)-Aq∥+(αnγ¯2/2(γ¯-γα))∥xn-q∥2. It is clear that ∑n=0∞γn=∞ and limsupn→∞δn≤0. Hence, all conditions of Lemma 2.5, we can conclude that xn→q. This completes the proof.Corollary 3.2.
LetH be a real Hilbert space and C be a closed convex subset of H. Let F be a bifunction of C×C into ℛ satisfying (A1)–(A4) and B,Φ:C→H be β,σ-inverse-strongly monotone mappings, let φ:C→ℛ  be convex and lower semicontinuous function, f:C→C be a contraction with coefficient α(0<α<1), and M:H→2H be a maximal monotone mapping. Let S be a nonexpansive mapping of H into itself and assume that either (B1) or (B2) holds such that
(3.57)Θ∶=F(S)∩GMEP⁡(F,φ,Φ)∩I(B,M)≠∅.
Suppose {xn} is a sequence generated by the following algorithm x0∈C arbitrarily:
(3.58)xn+1=ξnPC[αnf(xn)+(I-αn)SJM,λ(xn-λBxn)]+(1-ξn)Tr(xn-rΦxn),
where {αn},{ξn}⊂(0,1),λ∈(0,2β) such that 0<a≤λ≤b<2β and r∈(0,2σ) with 0<c≤d≤1-σ satisfy the following conditions:(C1)
limn→∞αn=0, Σn=0∞αn=∞ and limn→∞(αn+1/αn)=1,(C2)
0<liminfn→∞ξn<limsupn→∞ξn<1 and limn→∞((ξn+1-ξn)/αn+1)=1.
Then{xn} converges strongly to q∈Θ, where q=PΘ(f+I)(q) which solves the following variational inequality:
(3.59)〈(f-I)q,p-q〉≤0,∀p∈Θ.Proof.
PuttingA≡I and γ≡1 in Theorem 3.1, we can obtain desired conclusion immediately.Corollary 3.3.
LetH be a real Hilbert space and C be a closed convex subset of H. Let F be a bifunction of C×C into ℛ satisfying (A1)–(A4) and B,Φ:C→H be β,σ-inverse-strongly monotone mappings, let φ:C→ℛ be convex and lower semicontinuous function, and M:H→2H be a maximal monotone mapping. Let S be a nonexpansive mapping of H into itself and assume that either (B1) or (B2) holds such that
(3.60)Θ∶=F(S)∩GMEP⁡(F,φ,Φ)∩I(B,M)≠∅.
Suppose {xn} is a sequence generated by the following algorithm x0∈C arbitrarily:
(3.61)xn+1=ξnPC[αnu+(I-αn)SJM,λ(xn-λBxn)]+(1-ξn)Tr(xn-rΦxn),
where {αn},{ξn}⊂(0,1),λ∈(0,2β) such that 0<a≤λ≤b<2β and r∈(0,2σ) with 0<c≤d≤1-σ satisfy the following conditions:(C1)
limn→∞αn=0, Σn=0∞αn=∞ and limn→∞(αn+1/αn)=1,(C2)
0<liminfn→∞ξn<limsupn→∞ξn<1 and limn→∞((ξn+1-ξn)/αn+1)=1.Then {xn} converges strongly to q∈Θ, where q=PΘ(q) which solves the following variational inequality:
(3.62)〈u-q,p-q〉≤0,∀p∈Θ.Proof.
Puttingf≡u∈C in Corollary 3.2, we can obtain desired conclusion immediately.Corollary 3.4.
LetH be a real Hilbert space, C be a closed convex subset of H. Let F be a bifunction of C×C into ℛ satisfying (A1)–(A4) and B,Φ:C→H be β,σ-inverse-strongly monotone mappings, φ:C→ℛ is convex and lower semicontinuous function, f:C→C be a contraction with coefficient α(0<α<1) and A be a strongly positive linear bounded operator of H into itself with coefficient γ¯>0. Assume that 0<γ<γ¯/α. Let S be a nonexpansive mapping of C into itself and assume that either (B1) or (B2) holds such that
(3.63)Θ:=F(S)∩GMEP⁡(F,φ,Φ)∩VI⁡(C,B)≠∅.
Suppose {xn} is a sequence generated by the following algorithm x0∈C arbitrarily:
(3.64)xn+1=ξnPC[αnγf(xn)+(I-αnA)SPC(xn-λBxn)]+(1-ξn)Tr(xn-rΦxn),
where {αn},{ξn}⊂(0,1), λ∈(0,2β) such that 0<a≤λ≤b<2β and r∈(0,2σ) with 0<c≤d≤1-σ satisfing the following conditions:(C1)
limn→∞αn=0, Σn=0∞αn=∞ and limn→∞(αn+1/αn)=1,(C2)
0<liminfn→∞ξn<limsupn→∞ξn<1 and limn→∞((ξn+1-ξn)/αn+1)=1.
Then{xn} converges strongly to q∈Θ, where q=PΘ(γf+I-A)(q) which solves the following variational inequality:
(3.65)〈(γf-)q,p-q〉≤0,∀p∈Θ.Proof.
TakingJM,λ=PC in Theorem 3.1, we can obtain desired conclusion immediately.Corollary 3.5.
LetH be a real Hilbert space, C be a closed convex subset of H. Let F be a bifunction of C×C into ℛ satisfying (A1)–(A4) and B,Φ:C→H be β,σ-inverse-strongly monotone mappings, φ:C→ℛ is convex and lower semicontinuous function, f:C→C be a contraction with coefficient α(0<α<1). Let S be a nonexpansive mapping of C into itself and assume that either (B1) or (B2) holds such that
(3.66)Θ:=F(S)∩GMEP⁡(F,φ,Φ)≠∅.
Suppose {xn} is a sequence generated by the following algorithm x0∈C arbitrarily:
(3.67)xn+1=ξnPC[αnf(xn)+(I-αn)Sxn]+(1-ξn)Tr(xn-rΦxn),
where {αn},{ξn}⊂(0,1) and r∈(0,2σ) with 0<c≤d≤1-σ satisfing the following conditions:(C1)
limn→∞αn=0, Σn=0∞αn=∞ and limn→∞(αn+1/αn)=1,(C2)
0<liminfn→∞ξn<limsupn→∞ξn<1 and limn→∞((ξn+1-ξn)/αn+1)=1.
Then{xn} converges strongly to q∈Θ, where q=PΘ(f+I)(q) which solves the following variational inequality:
(3.68)〈(f-I)q,p-q〉≤0,∀p∈Θ.Proof.
Takingγ≡1,A≡I,M=0, and B≡0 in Theorem 3.1, we can obtain desired conclusion immediately.Remark 3.6.
Corollary3.5 generalizes and improves the result of Yao and Liou [16].


## 4. Some Applications

In this section, we apply the iterative scheme (1.22) for finding a common fixed point of nonexpansive mapping and strictly pseudocontractive mapping and also apply Theorem 3.1 for finding a common fixed point of nonexpansive mappings and inverse-strongly monotone mappings.Definition 4.1.
A mappingT:C→C is called strictly pseudocontractionif there exists a constant 0≤κ<1 such that
(4.1)‖Tx-Ty‖2≤‖x-y‖2+κ‖(I-T)x-(I-T)y‖2,∀x,y∈C.
If κ=0, then S is nonexpansive. In this case, we say that T:C→C is a κ-strictly pseudocontraction. Putting B=I-T. Then, we have
(4.2)‖(I-B)x-(I-B)y‖2≤‖x-y‖2+κ‖Bx-By‖2,∀x,y∈C.
Observe that
(4.3)‖(I-B)x-(I-B)y‖2=‖x-y‖2+‖Bx-By‖2-2〈x-y,Bx-By〉,∀x,y∈C.
Hence, we obtain
(4.4)〈x-y,Bx-By〉≥1-κ2‖Bx-By‖2,∀x,y∈C.
Then, B is((1-κ)/2)-inverse-strongly monotone mapping.Using Theorem3.1, we first prove a strongly convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontraction.Theorem 4.2.
LetH be a real Hilbert space and C be a closed convex subset of H. Let F be a bifunction of C×C into ℛ satisfying (A1)–(A4) and B,Φ:C→H be β,σ-inverse-strongly monotone mapping, φ:C→ℛ be convex and lower semicontinuous function, f:C→C be a contraction with coefficient α(0<α<1), and A be a strongly positive linear bounded operator of H into itself with coefficient γ¯>0. Assume that 0<γ<γ¯/α. Let S be a nonexpansive mapping of C into itself and let T be a κ-strictly pseudocontraction of C into itself. Assume that either (B1) or (B2) holds such that
(4.5)Θ:=F(S)∩F(T)∩GMEP⁡(F,φ,Φ)≠∅.
Suppose {xn} is a sequence generated by the following algorithm x0∈C arbitrarily:
(4.6)xn+1=ξnPC[αnγf(xn)+(I-αnA)S((1-λ)un+λTun)]+(1-ξn)Tr(xn-rΦxn),
for all n=0,1,2,…, where {αn},{ξn}⊂(0,1), λ∈[0,1-κ) and r∈(0,2σ). If λ∈[a,b] for some a,b with 0<a<b<1-κ and {σn} is chosen so that r∈[c,d] for some c,d with 0<c<d<1-σ and satisfy the condition (C1)-(C2) in Theorem 3.1.
Then{xn} converges strongly to q∈Θ, where q=PΘ(γf+I-A)(q) which solves the following variational inequality:
(4.7)〈(γf-A)q,p-q〉≤0,∀p∈Θ,
which is the optimality condition for the minimization problem
(4.8)min⁡q∈Ω12〈Aq,q〉-h(q),
where h is a potential function for γf (i.e., h′(q)=γf(q) for q∈H).Proof.
PutB≡I-T, then B is ((1-κ)/2)-inverse-strongly monotone and F(T)=I(B,M)  and JM,λ(xn-λBxn)=(1-λ)xn+λTxn. So by Theorem 3.1, we obtain the desired result.Corollary 4.3.
LetH be a real Hilbert space and C be a closed convex subset of H. Let F be a bifunction of C×C into ℛ satisfying (A1)–(A4) and B,Φ:C→H be β,σ-inverse-strongly monotone mapping, φ:C→ℛ is convex and lower semicontinuous function. Letf:C→C be a contraction with coefficient α(0<α<1) and S be a nonexpansive mapping of C into itself and let T be a κ-strictly pseudocontraction of C into itself. Assume that either (B1) or (B2) holds such that
(4.9)Θ:=F(S)∩F(T)∩GMEP⁡(F,φ,Φ)≠∅.
Suppose {xn} is a sequence generated by the following algorithm x0∈C arbitrarily:
(4.10)xn+1=ξnPC[αnf(xn)+(I-αn)S((1-λ)un+λTun)]+(1-ξn)Tr(xn-rΦxn),
for all n=0,1,2,…, where {αn},{ξn}⊂(0,1), λ∈[0,1-κ) and r∈(0,2σ). If λ∈[a,b] is chosen for some a,b with 0<a<b<1-κ and {σn} is chosen so that r∈[c,d] for some c,d with 0<c<d<1-σ and satisfy the condition (C1)-(C2) in Theorem 3.1.Then{xn} converges strongly to q∈Ω, where q=PΩ(f+I)(q) which solves the following variational inequality:(4.11)〈(f-I)q,p-q〉,∀p∈Ω.Proof.
PutA≡I and γ≡1 in by Theorem 4.2, we obtain the desired result.



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*Source: 648617-2011-09-05.xml*