# A General Iterative Algorithm for Generalized Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities

Authors: Thanyarat  Jitpeera; Poom Kumam
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We introduce a new general iterative method for finding a common element
of the set of solutions of fixed point for nonexpansive mappings, the set of solution of generalized
mixed equilibrium problems, and the set of solutions 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), Su et al. (2008), Klin-eam and Suantai
(2009), Tan and Chang (2011), and some other authors.


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## Body

## 1. Introduction

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, Ψ:C→H a mapping, and φ:C→ℛ a real-valued function. The generalized mixed equilibrium problem is for finding x∈C such that(1.1)F(x,y)+〈Ψx,y-x〉+φ(y)-φ(x)≥0,∀y∈C.
The set of solutions of (1.1) is denoted by GMEP(F,φ,Ψ), that is, (1.2)GMEP(F,φ,Ψ)={x∈C:F(x,y)+〈Ψx,y-x〉+φ(y)-φ(x)≥0,∀y∈C}.
If F≡0, the problem (1.1) is reduced into the mixed variational inequality of Browder type [1] for finding x∈C such that(1.3)〈Ψx,y-x〉+φ(y)-φ(x)≥0,∀y∈C.
The set of solutions of (1.3) is denoted by MVI(C,φ,Ψ).IfΨ≡0 and φ≡0, the problem (1.1) is reduced into the equilibrium problem [2] for finding x∈C such that(1.4)F(x,y)≥0,∀y∈C.
The set of solutions of (1.4) is denoted by EP(F). This problem contains fixed point problems and includes as special cases numerous problems in physics, optimization, and economics. Some methods have been proposed to solve the equilibrium problem; see [3–5].IfF≡0 and φ≡0, the problem (1.1) is reduced into the Hartmann-Stampacchia variational inequality [6] for finding x∈C such that(1.5)〈Ψx,y-x〉≥0,∀y∈C.
The set of solutions of (1.5) is denoted by VI(C,Ψ). The variational inequality has been extensively studied in the literature [7].IfF≡0 and Ψ≡0, the problem (1.1) is reduced into the minimize problem for finding x∈C such that(1.6)φ(y)-φ(x)≥0,∀y∈C.
The set of solutions of (1.6) is denoted by Argmin(φ).Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence on the development of almost all branches of pure and applied sciences. 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:(1.7)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 [8].Recall that a mappingS:C→C is said to be nonexpansive if (1.8)‖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 [9]. We denote weak convergence and strong convergence by notations ⇀ and →, respectively. A mapping A of C into H is called monotone if (1.9)〈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 (1.10)〈Ax-Ay,x-y〉≥α‖Ax-Ay‖2,
for all x,y∈C. It is obvious that any α-inverse-strongly monotone mapping 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 (1.11)〈Ax,x〉≥γ¯‖x‖2,
for all x∈H. A self mapping f:C→C is a contractions on C if there exists a constant α∈(0,1) such that (1.12)‖f(x)-f(y)‖≤α‖x-y‖,
for all x,y∈C. We use ∏C to denote the collection of all contraction on C. Note that each f∈∏C has a unique fixed point in C.LetB:H→H be a single-valued nonlinear mapping and M:H→2H a set-valued mapping. The variational inclusion problem is to find x∈H such that(1.13)θ∈B(x)+M(x),
where θ is the zero vector in H. The set of solutions of problem (1.13) is denoted by I(B,M). The variational inclusion has been extensively studied in the literature, see, for example, [10–13] 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(1.14)Tv={Bv+NCv,ifv∈C,∅,ifv∉C.
Then, T is a maximal monotone and θ∈Tv if and only if v∈VI(C,B) [14].LetM:H→2H be a set-valued maximal monotone mapping; then the single-valued mapping JM,λ:H→H defined by(1.15)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 (1.13) is a fixed point of the operator JM,λ(I-λB) for all λ>0 [15].In 2000, Moudafi [16] 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, is chosen arbitrarily,(1.16)xn+1=αnf(xn)+(1-αn)Sxn,n≥0,
where {αn}⊂(0,1) satisfies certain conditions and converges strongly to a fixed point of S (say x¯∈C) which is the unique solution of the following variational inequality:(1.17)〈(I-f)x¯,x-x¯〉≥0,∀x∈F(S).In 2006, Marino and Xu [8] introduced a general iterative method for nonexpansive mapping. They defined the sequence {xn} generated by the algorithm x0∈C:(1.18)xn+1=αnγf(xn)+(I-αnA)Sxn,n≥0,
where {αn}⊂(0,1) and A is a strongly positive linear bounded operator. They proved that, if C=H and the sequence {αn} satisfies appropriate conditions, then the sequence {xn} generated by (1.18) converges strongly to a fixed point of S (say x¯∈H) which is the unique solution of the following variational inequality:(1.19)〈(A-γf)x¯,x-x¯〉≥0,∀x∈F(S),
which is the optimality condition for the minimization problem(1.20)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).For finding a common element of the set of fixed points of nonexpansive mappings and the set of solution of the variational inequalities, letPC be the projection of H onto C. In 2005, Iiduka and Takahashi [17] introduced following iterative process for x0∈C:(1.21)xn+1=αnu+(1-αn)SPC(xn-λnAxn),∀n≥0,
where u∈C, {αn}⊂(0,1), and {λn}⊂[a,b] for some a,b with 0<a<b<2β. They proved that under certain appropriate conditions imposed on {αn} and {λn}, the sequence {xn} generated by (1.21) converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping (say x¯∈C) which solve some variational inequality(1.22)〈x¯-u,x-x¯〉≥0,∀x∈F(S)∩VI(C,A).In 2008, Su et al. [18] introduced the following iterative scheme by the viscosity approximation method in a real Hilbert space: x1,un∈H(1.23)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnf(xn)+(1-αn)SPC(un-λnAun),
for all n∈ℕ, where {αn}⊂[0,1) and {rn}⊂(0,∞) satisfy some appropriate conditions. Furthermore, they proved that {xn} and {un} converge strongly to the same point z∈F(S)∩VI(C,A)∩EP(F), where z=PF(S)∩VI(C,A)∩EP(F)f(z).In 2011, Tan and Chang [12] introduced following iterative process for {Tn:C→C} which is a sequence of nonexpansive mappings. Let {xn} be the sequence defined by(1.24)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]. The sequence {xn} defined by (1.24) converges strongly to a common element of the set of fixed points of nonexpansive mapping, the set of solutions of the variational inequality, and the generalized equilibrium problem.In this paper, we modify the iterative methods (1.18), (1.23), and (1.24) by proposing the following new general viscosity iterative method: x0,un∈C,(1.25)F(un,y)+φ(y)-φ(un)+〈Ψxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=PC[αnγf(xn)+(I-αnA)SJM,λ(un-λBun)],
for all n∈ℕ, where {αn}⊂(0,1), {rn}⊂(0,2σ), and λ∈(0,2β) satisfy some appropriate conditions. The purpose of this paper is to show that under some control conditions the sequence {xn} strongly converges to a common element of the set of fixed points of nonexpansive mapping, the solution of the generalized mixed equilibrium problems, and the set of solutions of the variational inclusion in a real Hilbert space.


## 2. Preliminaries

LetH be a real Hilbert space and C 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 (1.5) if and only if u∈C satisfies the relation u=PC(u-λΨu) for all λ>0.Lemma 2.2.
For a givenz∈H, u∈C, u=PCz⇔〈u-z,v-u〉≥0, for all 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 [19]).
LetM:H→2H be a maximal monotone mapping, and let B:H→H be a monotone and Lipshitz continuous mapping. Then the mapping L=M+B:H→2H is a maximal monotone mapping.Lemma 2.4 (see [20]).
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∥ holds for each y∈H with y≠x.Lemma 2.5 (see [21]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.4)an+1≤(1-γn)an+δn,∀n≥0,
where {γn}⊂(0,1) and {δn} is a sequence in ℛ such that(i)
∑n=1∞γn=∞,(ii)
limsupn→∞(δn/γn)≤0 or ∑n=1∞|δn|<∞.Then limn→∞an=0.Lemma 2.6 (see [22]).
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 Ψ: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 [23]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F:C×C→ℛ be a bifunction mapping satisfying (A1)–(A4), and let φ:C→ℛ be convex and lower semicontinuous such that C∩domφ≠∅. Assume that either (B1) or (B2) holds. For r>0 and x∈H, there exists u∈C such that
(2.7)F(u,y)+φ(y)-φ(u)+1r〈y-u,u-x〉.
Define a mapping Kr:H→C as follows:
(2.8)Kr(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)
Kr is single valued;(ii)
Kr is firmly nonexpansive, that is, for any x,y∈H,∥Krx-Kry∥2≤〈Krx-Kry,x-y〉;(iii)
F(Kr)=MEP(F,φ);(iv)
MEP(F,φ) is closed and convex.Lemma 2.8 (see [8]).
Assume thatA 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,Ψ), and I(B,M) of an inverse-strongly monotone mappings in a Hilbert space.Theorem 3.1.
LetH be a real Hilbert space, C a closed convex subset of H, B, Ψ:C→H be β, σ-inverse-strongly monotone mappings, respectively. Let φ:C→ℛ be a convex and lower semicontinuous function, f:C→C a contraction with coefficient α(0<α<1), M:H→2H a maximal monotone mapping, and A 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 such that
(3.1)Ω:=F(S)∩GMEP(F,φ,Ψ)∩I(B,M)≠∅.
Suppose that {xn} is a sequences generated by the following algorithm for x0∈C arbitrarily:
(3.2)F(un,y)+φ(y)-φ(un)+〈Ψxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=PC[αnγf(xn)+(I-αnA)SJM,λ(un-λBun)]
for all n=0,1,2,…, where(C1)
{αn}⊂(0,1), limn→0αn=0, ∑n=1∞αn=∞, and ∑n=1∞|αn+1-αn|<∞,(C2)
{rn}⊂[c,d] with c,d∈(0,2σ) and ∑n=1∞|rn+1-rn|<∞,(C3)
λ∈(0,2β).
Then{xn} converges strongly to q∈Ω, where q=PΩ(γf+I-A)(q) which solves the following variational inequality:
(3.3)〈(γf-A)q,p-q〉≤0,∀p∈Ω
which is the optimality condition for the minimization problem
(3.4)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.
Due to condition (C1), we may assume without loss of generality, then, thatαn∈(0,∥A∥-1) for all n. By Lemma 2.8, we have that ∥I-αnA∥≤1-αnγ¯. Next, we will assume that ∥I-A∥≤∥1-γ¯∥.
Next, we will divide the proof into six steps.
Step 1.
We will show that{xn},{un} are bounded.
SinceB, Ψ are β, σ-inverse-strongly monotone mappings, we have that
(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 a similar way, we can obtain
(3.6)‖(I-rnΨ)x-(I-rnΨ)y‖2≤‖x-y‖2.
It is clear that if 0<λ<2β, 0<rn<2σ, then I-λB, I-rnΨ are all nonexpansive.
Putyn=JM,λ(un-λBun), n≥0. It follows that
(3.7)‖yn-q‖=‖JM,λ(un-λBun)-JM,λ(q-λBq)‖≤‖un-q‖.
By Lemma 2.7, we have that un=Krn(xn-rnΨxn) for all n≥0. Then, we have that
(3.8)‖un-q‖2=‖Krn(xn-rnΨxn)-Krn(q-rnΨq)‖2≤‖(xn-rnΨxn)-(q-rnΨq)‖2≤‖xn-q‖2+rn(rn-2σ)‖Ψxn-Ψq‖2=‖xn-q‖2.
Hence, we have that
(3.9)‖yn-q‖≤‖xn-q‖.
From (3.2), we deduce that
(3.10)‖xn+1-q‖=‖PC(αnγf(xn)+(I-αnA)Syn)-PC(q)‖≤‖αn(γf(xn)-Aq)+(I-αnA)(Syn-q)‖≤αn‖γf(xn)-Aq‖+(1-αnγ¯)‖yn-q‖≤αn‖γf(xn)-γf(q)‖+αn‖γf(q)-Aq‖+(1-αnγ¯)‖yn-q‖≤αγαn‖xn-q‖+αn‖γf(q)-Aq‖+(1-αnγ¯)‖xn-q‖=(1-(γ¯-γα)αn)‖xn-q‖+αn‖γf(q)-Aq‖=(1-(γ¯-γα)αn)‖xn-q‖+(γ¯-γα)αn‖γf(q)-Aq‖γ¯-γα≤max⁡{‖xn-q‖,‖γf(q)-Aq‖γ¯-γα}.
It follows by induction that
(3.11)‖xn-q‖≤max⁡{‖x0-q‖,‖γf(q)-Aq‖γ¯-γα},n≥0.
Therefore {xn} is bounded, so are {yn}, {Syn}, {Bun}, {f(xn)}, and {ASyn}.
Step 2.
We claim thatlimn→∞∥xn+1-xn∥=0. From (3.2), we have that
(3.12)‖xn+1-xn‖=‖PC(αnγf(xn)+(I-αnA)Syn)-PC(αn-1γf(xn-1)+(I-αn-1A)Syn-1)‖≤‖(I-αnA)(Syn-Syn-1)-(αn-αn-1)ASyn-1+γαn(f(xn)-f(xn-1))+γ(αn-αn-1)f(xn-1)‖≤(1-αnγ¯)‖yn-yn-1‖+|αn-αn-1|‖ASyn-1‖+γααn‖xn-xn-1‖+γ|αn-αn-1|‖f(xn-1)‖.
SinceI-λB are nonexpansive, we also have that
(3.13)∥yn-yn-1∥=‖JM,λ(un-λBun)-JM,λ(un-1-λBun-1)‖≤‖(un-λBun)-(un-1-λBun-1)‖≤‖un-un-1‖.
On the other hand, from un-1=Krn-1(xn-1-rn-1Ψxn-1) and un=Krn(xn-rnΨxn), it follows that
(3.14)F(un-1,y)+〈Ψxn-1,y-un-1〉+φ(y)-φ(un-1)+1rn-1〈y-un-1,un-1-xn-1〉≥0,∀y∈C,(3.15)F(un,y)+〈Ψxn,y-un〉+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C.
Substituting y=un into (3.14) and y=un-1 into (3.15), we get
(3.16)F(un-1,un)+〈Ψxn-1,un-un-1〉+φ(un)-φ(un-1)+1rn-1〈un-un-1,un-1-xn-1〉≥0,F(un,un-1)+〈Ψxn,un-1-un〉+φ(un-1)-φ(un)+1rn〈un-1-un,un-xn〉≥0.
From (A2), we obtain
(3.17)〈un-un-1,Ψxn-1-Ψxn+un-1-xn-1rn-1-un-xnrn〉≥0,
and then
(3.18)〈un-un-1,rn-1(Ψxn-1-Ψxn)+un-1-xn-1-rn-1rn(un-xn)〉≥0.
So
(3.19)〈un-un-1,rn-1Ψxn-1-rn-1Ψxn+un-1-un+un-xn-1-rn-1rn(un-xn)〉≥0.
It follows that
(3.20)〈un-un-1,(I-rn-1Ψ)xn-(I-rn-1Ψ)xn-1+un-1-un+un-xn-rn-1rn(un-xn)〉≥0,〈un-un-1,un-1-un〉+〈un-un-1,xn-xn-1+(1-rn-1rn)(un-xn)〉≥0.
Without loss of generality, let us assume that there exists a real number c such that rn-1>c>0, for all n∈ℕ. Then, we have that
(3.21)‖un-un-1‖2≤〈un-un-1,xn-xn-1+(1-rn-1rn)(un-xn)〉≤‖un-un-1‖{‖xn-xn-1‖+|1-rn-1rn|‖un-xn‖},
and hence
(3.22)‖un-un-1‖≤‖xn-xn-1‖+1rn|rn-rn-1|‖un-xn‖≤‖xn-xn-1‖+M1c|rn-rn-1|,
where M1=sup{∥un-xn∥:n∈ℕ}. Substituting (3.22) into (3.13), we have that
(3.23)‖yn-yn-1‖≤‖xn-xn-1‖+M1c|rn-rn-1|.
Substituting (3.23) into (3.12), we get
(3.24)‖xn+1-xn‖≤(1-αnγ¯)(‖xn-xn-1‖+M1c|rn-rn-1|)+|αn-αn-1|‖ASyn-1‖+γααn‖xn-xn-1‖+γ|αn-αn-1|‖f(xn-1)‖=(1-αnγ¯)‖xn-xn-1‖+(1-αnγ¯)M1c|rn-rn-1|+|αn-αn-1|‖ASyn-1‖+γααn‖xn-xn-1‖+γ|αn-αn-1|‖f(xn-1)‖≤(1-(γ¯-γα)αn)∥xn-xn-1∥+M1c|rn-rn-1|+|αn-αn-1|‖ASyn-1‖+γ|αn-αn-1|‖f(xn-1)‖≤(1-(γ¯-γα)αn)‖xn-xn-1‖+M1c|rn-rn-1|+M2|αn-αn-1|,
where M2=sup{max{∥ASyn-1∥,∥f(xn-1)∥:n∈ℕ}}. By conditions (C1)-(C2) and Lemma 2.5, we have that ∥xn+1-xn∥→0 as n→∞. From (3.23), we also have that ∥yn+1-yn∥→0 as n→∞.
Step 3.
We show the following:(i)
limn→∞∥Bun-Bq∥=0;(ii)
limn→∞∥Ψxn-Ψq∥=0.For q∈Ω and q=JM,λ(q-λBq), by (3.5) and (3.8), we get
(3.25)‖yn-q‖2=‖JM,λ(un-λBun)-JM,λ(q-λBq)‖2≤‖(un-λBun)-(q-λBq)‖2≤‖un-q‖2+λ(λ-2β)‖Bun-Bq‖2≤‖xn-q‖2+λ(λ-2β)‖Bun-Bq‖2.
It follows that
(3.26)‖xn+1-q‖2=‖PC(αnγf(xn)+(I-αnA)Syn)-PC(q)‖2≤‖αn(γf(xn)-Aq)+(I-αnA)(Syn-q)‖2≤(αn‖γf(xn)-Aq‖+(1-αnγ¯)‖yn-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)‖Bun-Bq‖2)≤αn‖γf(xn)-Aq‖2+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖+∥xn-q∥2+(1-αnγ¯)λ(λ-2β)∥Bun-Bq∥2.
So, we obtain
(3.27)(1-αnγ¯)λ(2β-λ)‖Bun-Bq‖2≤αn‖γf(xn)-Aq‖2+‖xn-xn+1‖(‖xn-q‖+‖xn+1-q‖)+ϵn,
where ϵn=2αn(1-αnγ¯)∥γf(xn)-Aq∥∥yn-q∥. By conditions (C1) and (C3) and limn→∞∥xn+1-xn∥=0, we obtain that ∥Bun-Bq∥→0 as n→∞.
Substituting (3.8) into (3.25), we get
(3.28)‖yn-q‖2≤‖un-q‖2+λ(λ-2β)‖Bun-Bq‖2≤(‖xn-q‖2+rn(rn-2σ)‖Ψxn-Ψq‖2)+λ(λ-2β)‖Bun-Bq‖2.
From (3.26), we have that
(3.29)‖xn+1-q‖2≤αn‖γf(xn)-Aq‖2+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖+(1-αnγ¯)(‖xn-q‖2+rn(rn-2σ)‖Ψxn-Ψq‖2+λ(λ-2β)‖Bun-Bq‖2)≤αn‖γf(xn)-Aq‖2+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖+‖xn-q‖2+(1-αnγ¯)rn(rn-2σ)‖Ψxn-Ψq‖2+(1-αnγ¯)λ(λ-2β)‖Bun-Bq‖2.
So, we also have that
(3.30)(1-αnγ¯)rn(2σ-rn)∥Ψxn-Ψq∥2≤αn‖γf(xn)-Aq‖2+‖xn-xn+1‖(‖xn-q‖+‖xn+1-q‖)+ϵn+(1-αnγ¯)λ(λ-2β)‖Bun-Bq‖2,
where ϵn=2αn(1-αnγ¯)∥γf(xn)-Aq∥∥yn-q∥. By conditions (C1)–(C3), limn→∞∥xn+1-xn∥=0 and limn→∞∥Bun-Bq∥=0, we obtain that ∥Ψxn-Ψq∥→0 as n→∞.
Step 4.
We show the following:(i)
limn→∞∥xn-un∥=0;(ii)
limn→∞∥un-yn∥=0;(iii)
limn→∞∥yn-Syn∥=0.Since Krn is firmly nonexpansive and by (2.2), we observe that
(3.31)‖un-q‖2=‖Krn(xn-rnΨxn)-Krn(q-rnΨq)‖2≤〈(xn-rnΨxn)-(q-rnΨq),un-q〉=12(‖(xn-rnΨxn)-(q-rnΨq)‖2+‖un-q‖2-‖(xn-rnΨxn)-(q-rnΨq)-(un-q)‖2)≤12(‖xn-q‖2+‖un-q‖2-‖(xn-un)-rn(Ψxn-Ψq)‖2)=12(‖xn-q‖2+‖un-q‖2-‖xn-un‖2+2rn〈Ψxn-Ψq,xn-un〉-rn2‖Ψxn-Ψq‖2).
Hence, we have that
(3.32)‖un-q‖2≤‖xn-q‖2-‖xn-un‖2+2rn‖Ψxn-Ψq‖‖xn-un‖.
Since JM,λ is 1-inverse-strongly monotone and by (2.2), we compute
(3.33)‖yn-q‖2=‖JM,λ(un-λBun)-JM,λ(q-λBq)‖2≤〈(un-λBun)-(q-λBq),yn-q〉=12(‖(un-λBun)-(q-λBq)‖2+‖yn-q‖2-‖(un-λBun)-(q-λBq)-(yn-q)‖2)≤12(‖un-q‖2+‖yn-q‖2-‖(un-yn)-λ(Bun-Bq)‖2)=12(‖un-q‖2+‖yn-q‖2-‖un-yn‖2+2λ〈un-yn,Bun-Bq〉-λ2‖Bun-Bq‖2),
which implies that
(3.34)‖yn-q‖2≤‖un-q‖2-‖un-yn‖2+2λ‖un-yn‖‖Bun-Bq‖.
Substituting (3.32) into (3.34), we have that
(3.35)‖yn-q‖2≤(‖xn-q‖2-‖xn-un‖2+2rn‖Ψxn-Ψq‖‖xn-un‖)-‖un-yn‖2+2λ‖un-yn‖‖Bun-Bq‖.
Substituting (3.35) into (3.26), we get
(3.36)‖xn+1-q‖2≤αn‖γf(xn)-Aq‖2+‖yn-q‖2+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖≤αn‖γf(xn)-Aq‖2+(‖xn-q‖2-‖xn-un‖2+2rn‖Ψxn-Ψq‖‖xn-un‖-‖un-yn‖2+2λn‖un-yn‖‖Bun-Bq‖)+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖.
Then, we derive
(3.37)‖xn-un‖2+‖un-yn‖2≤αn‖γf(xn)-Aq‖2+‖xn-q‖2-‖xn+1-q‖2+2rn‖Ψxn-Ψq‖‖xn-un‖+2λ‖un-yn‖‖Bun-Bq‖+2αn(1-αnγ¯)‖γf(xn)-Aq‖‖yn-q‖=αn‖γf(xn)-Aq‖2+‖xn-xn+1‖(‖xn-q‖+‖xn+1-q‖)+2rn‖Ψxn-Ψq‖‖xn-un‖+2λ‖un-yn‖‖Bun-Bq‖+2αn(1-αnγ¯)∥γf(xn)-Aq∥∥yn-q∥.
By condition (C1), limn→∞∥xn-xn+1∥=0, limn→∞∥Ψxn-Ψq∥=0, and limn→∞∥Bun-Bq∥=0. So, we have that ∥xn-un∥→0, ∥un-yn∥→0 as n→∞. It follows that
(3.38)‖xn-yn‖≤‖xn-un‖+‖un-yn‖⟶0,asn⟶∞.
From (3.2), we have that
(3.39)‖xn-Syn‖≤‖xn-Syn-1‖+‖Syn-1-Syn‖≤‖PC(αn-1γf(xn-1)+(I-αn-1A)Syn-1)-PC(Syn-1)‖+‖yn-1-yn‖≤αn-1‖γfxn-1-ASyn-1‖+‖yn-1-yn‖.
By condition (C1) and limn→∞∥yn-1-yn∥=0, we obtain that ∥xn-Syn∥→0 as n→∞. Next, we observe that
(3.40)‖xn+1-Syn‖=‖PC(αnγf(xn)+(I-αnA)Syn)-PC(Syn)‖≤‖αnγf(xn)+(I-αnA)Syn-Syn‖=αn‖γf(xn)-ASyn‖.
Since {ASyn} is bounded and by condition (C1), we have that ∥xn+1-Syn∥→0 as n→∞, and
(3.41)‖xn-Syn‖≤‖xn-xn+1‖+‖xn+1-Syn‖.
Since limn→∞∥xn-xn+1∥=0 and limn→∞∥xn+1-Syn∥=0, it implies that ∥xn-Syn∥→0 as n→∞. Hence, we have that
(3.42)‖xn-Sxn‖≤‖xn-Syn‖+‖Syn-Sxn‖≤‖xn-Syn‖+‖yn-xn‖.
By (3.38) and limn→∞∥xn-Syn∥=0, we obtain ∥xn-Sxn∥→0 as n→∞. Moreover, we also have that
(3.43)‖yn-Syn‖≤‖yn-xn‖+‖xn-Syn‖.
By (3.38) and limn→∞∥xn-Syn∥=0, we obtain ∥yn-Syn∥→0 as n→∞.
Step 5.
We show thatq∈Ω:=F(S)∩GMEP(F,φ,Ψ)∩I(B,M) and limsupn→∞〈(γf-A)q,Syn-q〉≤0. It is easy to see that PΩ(γf+(I-A)) is a contraction of H into itself. Indeed, since 0<γ<γ¯/α, we have that
(3.44)‖PΩ(γf+(I-A))x-PΩ(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‖.
Hence H is complete, and there exists a unique fixed point q∈H such that q=PΩ(γf+(I-A))(q). By Lemma 2.2, we obtain that 〈(γf-A)q,w-q〉≤0 for all w∈Ω.
Next, we show thatlimsupn→∞〈(γf-A)q,Syn-q〉≤0, where q=PΩ(γf+I-A)(q) is the unique solution of the variational inequality 〈(γf-A)q,p-q〉≥0, for all p∈Ω. We can choose a subsequence {yni} of {yn} such that
(3.45)limsup⁡n→∞〈(γf-A)q,Syn-q〉=lim⁡i→∞〈(γf-A)q,Syni-q〉.
As {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 thatw∈Ω. Since ∥yn-Syn∥→0, ∥xn-Sxn∥→0, and ∥xn-yn∥→0 and by Lemma 2.6, we have that w∈F(S).
Next, we show thatw∈GMEP(F,φ,Ψ). Since un=Krn(xn-rnΨxn), we know that
(3.46)F(un,y)+φ(y)-φ(un)+〈Ψxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C.
It follows by (A2) that
(3.47)φ(y)-φ(un)+〈Ψxn,y-un〉+1rn〈y-un,un-xn〉≥F(y,un),∀y∈C.
Hence,
(3.48)φ(y)-φ(uni)+〈Ψxni,y-uni〉+1rni〈y-uni,uni-xni〉≥F(y,uni),∀y∈C.
For t∈(0,1] and y∈H, let yt=ty+(1-t)w. From (3.48), we have that
(3.49)〈yt-uni,Ψyt〉≥〈yt-uni,Ψyt〉-φ(yt)+φ(uni)-〈Ψxni,yt-uni〉-1rni〈yt-uni,uni-xni〉+F(yt,uni)=〈yt-uni,Ψyt-Ψuni〉+〈yt-uni,Ψuni-Ψxni〉-φ(yt)+φ(uni)-1rni〈yt-uni,uni-xni〉+F(yt,uni).
From ∥uni-xni∥→0, we have that ∥Ψuni-Ψxni∥→0. Further, from (A4) and the weakly lower semicontinuity of φ,(uni-xni)/rni→0 and uni⇀w, we have that
(3.50)〈yt-w,Ψyt〉≥-φ(yt)+φ(w)+F(yt,w).
From (A1), (A4), and (3.50), we have that
(3.51)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,Ψyt〉=t[F(yt,y)+φ(y)-φ(yt)]+(1-t)t〈y-w,Ψyt〉,
and hence
(3.52)0≤F(yt,y)+φ(y)-φ(yt)+(1-t)〈y-w,Ψyt〉.
Letting t→0, we have, for each y∈C, that
(3.53)F(w,y)+φ(y)-φ(w)+〈y-w,Ψw〉≥0.
This implies that w∈GMEP(F,φ,Ψ).
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(uni-λBuni), we have that uni-λBuni∈(I+λM)(yni), that is, (1/λ)(uni-yni-λBuni)∈M(yni). By virtue of the maximal monotonicity of M+B, we have that
(3.54)〈v-yni,g-Bv-1λ(uni-yni-λBuni)〉≥0,
and hence
(3.55)〈v-yni,g〉≥〈v-yni,Bv+1λ(uni-yni-λBuni)〉=〈v-yni,Bv-Byni〉+〈v-yni,Byni-Buni〉+〈v-yni,1λ(uni-yni)〉.
It follows from limn→∞∥un-yn∥=0, limn→∞∥Bun-Byn∥=0, and yni⇀w that
(3.56)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∈Ω. It follows that
(3.57)limsup⁡n→∞〈(γf-A)q,Syn-q〉=lim⁡i→∞〈(γf-A)q,Syni-q〉=〈(γf-A)q,w-q〉≤0.
Step 6.
We prove thatxn→q. By using (3.2) and together with Schwarz inequality, we have that
(3.58)‖xn+1-q‖2=‖PC(αnγf(xn)+(I-αnA)Syn)-PC(q)‖2≤‖αn(γf(xn)-Aq)+(I-αnA)(Syn-q)‖2≤(I-αnA)2‖(Syn-q)‖2+αn2‖γf(xn)-Aq‖2+2αn〈(I-αnA)(Syn-q),γf(xn)-Aq〉≤(1-αnγ¯)2‖yn-q‖2+αn2‖γf(xn)-Aq‖2+2αn〈Syn-q,γf(xn)-Aq〉-2αn2〈A(Syn-q),γf(xn)-Aq〉≤(1-αnγ¯)2‖xn-q‖2+αn2‖γf(xn)-Aq‖2+2αn〈Syn-q,γf(xn)-γf(q)〉+2αn〈Syn-q,γf(q)-Aq〉-2αn2〈A(Syn-q),γf(xn)-Aq〉≤(1-αnγ¯)2‖xn-q‖2+αn2‖γf(xn)-Aq‖2+2αn‖Syn-q‖‖γf(xn)-γf(q)‖+2αn〈Syn-q,γf(q)-Aq〉-2αn2〈A(Syn-q),γf(xn)-Aq〉≤(1-αnγ¯)2‖xn-q‖2+αn2‖γf(xn)-Aq‖2+2γααn‖yn-q‖‖xn-q‖+2αn〈Syn-q,γf(q)-Aq〉-2αn2〈A(Syn-q),γf(xn)-Aq〉≤(1-αnγ¯)2‖xn-q‖2+αn2‖γf(xn)-Aq‖2+2γααn‖xn-q‖2+2αn〈Syn-q,γf(q)-Aq〉-2αn2〈A(Syn-q),γf(xn)-Aq〉≤((1-αnγ¯)2+2γααn)‖xn-q‖2+αn{αn‖γf(xn)-Aq‖2+2〈Syn-q,γf(q)-Aq〉-2αn‖A(Syn-q)‖‖γf(xn)-Aq‖‖γf(xn)-Aq‖2}=(1-2(γ¯-γα)αn)‖xn-q‖2+αn{αn‖γf(xn)-Aq‖2+2〈Syn-q,γf(q)-Aq〉-2αn‖A(Syn-q)‖‖γf(xn)-Aq‖+αnγ¯2‖xn-q‖2}.
Since{xn} is bounded, where η≥∥γf(xn)-Aq∥2-2∥A(Syn-q)∥∥γf(xn)-Aq∥+γ¯2∥xn-q∥2 for all n≥0, it follows that
(3.59)‖xn+1-q‖2≤(1-2(γ¯-γα)αn)‖xn-q‖2+αnςn,
where ςn=2〈Syn-q,γf(q)-Aq〉+ηαn. By limsupn→∞〈(γf-A)q,Syn-q〉≤0, we get limsupn→∞ςn≤0. Applying Lemma 2.5, we can conclude that xn→q. This completes the proof.Corollary 3.2.
LetH be a real Hilbert space and C a closed convex subset of H. Let B,Ψ:C→H be β,σ-inverse-strongly monotone mappings and φ:C→ℛ a convex and lower semicontinuous function. Let f:C→C be a contraction with coefficient α(0<α<1), M:H→2H a maximal monotone mapping, and S a nonexpansive mapping of C into itself such that
(3.60)Ω:=F(S)∩GMEP(F,φ,Ψ)∩I(B,M)≠∅.
Suppose that {xn} is a sequence generated by the following algorithm for x0,un∈C arbitrarily:
(3.61)F(un,y)+φ(y)-φ(un)+〈Ψxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=PC[αnf(xn)+(1-αn)SJM,λ(un-λBun)]
for all n=0,1,2,…, by (C1)–(C3) in Theorem 3.1.
Then{xn} converges strongly to q∈Ω, where q=PΩ(f+I)(q) which solves the following variational inequality:
(3.62)〈(f-I)q,p-q〉≤0,∀p∈Ω.Proof.
PuttingA≡I and γ≡1 in Theorem 3.1, we can obtain the desired conclusion immediately.Corollary 3.3.
LetH be a real Hilbert space and C a closed convex subset of H. Let B, Ψ:C→H be β, σ-inverse-strongly monotone mappings, φ:C→ℛ a convex and lower semicontinuous function, and M:H→2H a maximal monotone mapping. Let S be a nonexpansive mapping of C into itself such that
(3.63)Ω:=F(S)∩GMEP(F,φ,Ψ)∩I(B,M)≠∅.
Suppose that {xn} is a sequence generated by the following algorithm for x0,u∈C and un∈C:
(3.64)F(un,y)+φ(y)-φ(un)+〈Ψxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=PC[αnu+(1-αn)SJM,λ(un-λBun)]
for all n=0,1,2,…, by (C1)–(C3) in Theorem 3.1.
Then{xn} converges strongly to q∈Ω, where q=PΩ(q) which solves the following variational inequality:
(3.65)〈u-q,p-q〉≤0,∀p∈Ω.Proof.
Puttingf(x)≡u, for all x∈C, in Corollary 3.2, we can obtain the desired conclusion immediately.Corollary 3.4.
LetH be a real Hilbert space, C a closed convex subset of H, B:C→H be β-inverse-strongly monotone mappings, and A a strongly positive linear bounded operator of H into itself with coefficient γ¯>0. Assume that 0<γ<γ¯/α. Let f:C→C be a contraction with coefficient α(0<α<1) and S a nonexpansive mapping of C into itself such that
(3.66)Ω:=F(S)∩VI(C,B)≠∅.
Suppose that {xn} is a sequence generated by the following algorithm for x0∈C arbitrarily:
(3.67)xn+1=PC[αnγf(xn)+(I-αnA)SPC(xn-λBxn)]
for all n=0,1,2,…, by (C1)–(C3) in Theorem 3.1.
Then{xn} converges strongly to q∈Ω, where q=PΩ(γf+I-A)(q) which solves the following variational inequality:
(3.68)〈(γf-A)q,p-q〉≤0,∀p∈Ω.Proof.
TakingF≡0,Ψ≡0,φ≡0,un=xn, and JM,λ=PC in Theorem 3.1, we can obtain the desired conclusion immediately.Remark 3.5.
In Corollary3.4 we generalize and improve the result of Klin-eam and Suantai [24].


## 4. Applications

In this section, we apply the iterative scheme (1.25) 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 that
(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 strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strict pseudocontraction.Theorem 4.2.
LetH be a real Hilbert space, C a closed convex subset of H, B, Ψ:C→H be β, σ-inverse-strongly monotone mappings, φ:C→ℛ a convex and lower semicontinuous function, f:C→C a contraction with coefficient α(0<α<1), and A 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 such that
(4.5)Ω:=F(S)∩F(T)∩GMEP(F,φ,Ψ)≠∅.
Suppose that {xn} is a sequence generated by the following algorithm for x0,un∈C arbitrarily:
(4.6)F(un,y)+φ(y)-φ(un)+〈Ψxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=PC[αnγf(xn)+(I-αnA)S((1-λ)un+λTun)]
for all n=0,1,2,…, by (C1)–(C3) in Theorem 3.1.
Then{xn} converges strongly to q∈Ω, where q=PΩ(γf+I-A)(q) which solves the following variational inequality:
(4.7)〈(γf-A)q,p-q〉≤0,∀p∈Ω
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, 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, C a closed convex subset of H, B, Ψ:C→H be β, σ-inverse-strongly monotone mappings, and φ:C→ℛ a convex and lower semicontinuous function. Let f:C→C be a contraction with coefficient α(0<α<1) and S a nonexpansive mapping of C into itself, and let T be a κ-strictly pseudocontraction of C into itself such that
(4.9)Ω:=F(S)∩F(T)∩GMEP(F,φ,Ψ)≠∅.
Suppose that {xn} is a sequence generated by the following algorithm for x0∈C arbitrarily:
(4.10)F(un,y)+φ(y)-φ(un)+〈Ψxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=PC[αnf(xn)+(I-αn)S((1-λ)un+λTun)]
for all n=0,1,2,…, by (C1)–(C3) 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〉≤0,∀p∈Ω
which is the optimality condition for the minimization problem
(4.12)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.
PuttingA≡I and γ≡1 in Theorem 4.2, we obtain the desired result.



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*Source: 619813-2011-04-13.xml*