We consider a general variational inequality and fixed point problem, which is to find a point x* with the property that (GVF): x*∈GVI(C,A) and g(x*)∈Fix(S) where GVI(C,A) is the solution set of some variational inequality Fix(S) is the fixed points set of nonexpansive mapping S, and g is a nonlinear operator. Assume the solution set Ω of (GVF) is nonempty. For solving (GVF), we suggest the following method g(xn+1)=βg(xn)+(1-β)SPC[αnF(xn)+(1-αn)(g(xn)-λAxn)], n≥0. It is shown that the sequence {xn} converges strongly to x*∈Ω which is the unique solution of the variational inequality 〈F(x*)-g(x*),g(x)-g(x*)〉≤0, for all x∈Ω.

1. Introduction

Let A:C→H and g:C→C be two nonlinear mappings. We concern the following generalized variational inequality of finding u∈C,g(u)∈C such that〈g(v)-g(u),Au〉≥0,∀g(v)∈C.
The solution set of (1.1) is denoted by GVI(C,A,g). It has been shown that a large class of unrelated odd-order and nonsymmetric obstacle, unilateral, contact, free, moving, and equilibrium problems arising in regional, physical, mathematical, engineering, and applied sciences can be studied in the unified and general framework of the general variational inequalities (1.1), see [1–16] and the references therein. Noor [17] has introduced a new type of variational inequality involving two nonlinear operators, which is called the general variational inequality. It is worth mentioning that this general variational inequality is remarkably different from the so-called general variational inequality which was introduced by Noor [18] in 1988. Noor [17] proved that the general variational inequalities are equivalent to nonlinear projection equations and the Wiener-Hopf equations by using the projection technique. Using this equivalent formulation, Noor [17] suggested and analyzed some iterative algorithms for solving the special general variational inequalities and further proved that these algorithms have strong convergence.

For g=I, where I is the identity operator, problem (1.1) is equivalent to finding u∈C such that〈v-u,Au〉≥0,∀v∈C,
which is known as the classical variational inequality introduced and studied by Stampacchia [19] in 1964. This field has been extensively studied due to a wide range of applications in industry, finance, economics, social, pure and applied sciences. For related works, please see [20–35]. Our main purposes in the present paper is devoted to study this topic.

Motivated and inspired by the works in this field, in this paper, we consider a general variational inequality and fixed point problem, which is to find a point x* with the property thatx*∈GVI(C,A),g(x*)∈Fix(S),
where Fix(S) is the fixed points set of nonexpansive mapping S. Assume the solution set Ω of (GVF) is nonempty. For solving (GVF), we suggest the following methodg(xn+1)=βg(xn)+(1-β)SPC[αnF(xn)+(1-αn)(g(xn)-λAxn)],n≥0.
It is shown that the sequence {xn} converges strongly to x*∈Ω which is the unique solution of the following variational inequality〈F(x*)-g(x*),g(x)-g(x*)〉≤0,∀x∈Ω.
Our results contain some interesting results as special cases.

2. Preliminaries

Let H be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively. Let C be a nonempty closed convex subset of H. Recall that a mapping S:C→C is said to be nonexpansive if‖Sx-Sy‖≤‖x-y‖,
for all x,y∈C. We denote by Fix(S) the set of fixed points of S. A mapping F:C→H is said to be L-Lipschitz continuous, if there exists a constant L>0 such that ∥F(x)-F(y)∥≤L∥x-y∥ for all x,y∈C. A mapping A:C→H is said to be α-inverse strongly g-monotone if and only if〈Ax-Ay,g(x)-g(y)〉≥α‖Ax-Ay‖2,
for some α>0 and for all x,y∈C. A mapping g:C→C is said to be strongly monotone if there exists a constant γ>0 such that〈g(x)-g(y),x-y〉≥γ‖x-y‖2,
for all x,y∈C.

Let B be a mapping of H into 2H. The effective domain of B is denoted by dom(B), that is, dom(B)={x∈H:Bx≠∅}. A multivalued mapping B is said to be a monotone operator on H if and only if〈x-y,u-v〉≥0,
for all x,y∈dom(B),u∈Bx, and v∈By. A monotone operator B on H is said to be maximal if and only if its graph is not strictly contained in the graph of any other monotone operator on H. Let B be a maximal monotone operator on H and let B-10={x∈H:0∈Bx}.

It is well known that, for any u∈H, there exists a unique u0∈C such that‖u-u0‖=inf{‖u-x‖:x∈C}.
We denote u0 by PCu, where PC is called the metric projection of H onto C. The metric projection PC of H onto C has the following basic properties:

∥PCx-PCy∥≤∥x-y∥ for all x,y∈H;

〈x-y,PCx-PCy〉≥∥PCx-PCy∥2 for every x,y∈H;

〈x-PCx,y-PCx〉≤0 for all x∈H, y∈C.

It is easy to see that the following is true:u∈GVI(C,A,g)⟺g(u)=PC(g(u)-λA(u)),∀λ>0.

We use the following notation:

xn⇀x stands for the weak convergence of (xn) to x;

xn→x stands for the strong convergence of (xn) to x.

We need the following lemmas for the next section.

Lemma 2.1.

Let C be a nonempty closed convex subset of a real Hilbert space H. Let G:C→C be a nonlinear mapping and let the mapping A:C→H be α-inverse strongly g-monotone. Then, for any λ>0, one has
‖PC[g(x)-λAx]-PC[g(y)-λAy]‖2≤‖g(x)-g(y)‖2+λ(λ-2α)‖Ax-Ay‖2,x,y∈C.

Proof.

Consider the following:‖PC[g(x)-λAx]-PC[g(y)-λAy]‖2≤‖g(x)-g(y)-λ(Ax-Ay)‖2=‖g(x)-g(y)‖2-2λ〈Ax-Ay,g(x)-g(y)〉+λ2‖Ax-Ay‖2≤‖g(x)-g(y)‖2-2λα‖Ax-Ay‖2+λ2‖Ax-Ay‖2≤‖g(x)-g(y)‖2+λ(λ-2α)‖Ax-Ay‖2.
If λ∈[0,2α], we have
‖PC[g(x)-λAx]-PC[g(y)-λAy]‖≤‖g(x)-g(y)-λ(Ax-Ay)‖≤‖g(x)-g(y)‖.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B32">36</xref>]).

Let C be a closed convex subset of a Hilbert space H. Let S:C→C be a nonexpansive mapping. Then Fix(S) is a closed convex subset of C and the mapping I-S is demiclosed at 0, that is, whenever {xn}⊂C is such that xn⇀x and (I-S)xn→0, then (I-S)x=0.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B34">37</xref>]).

Let {xn} and {yn} 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)yn+βnxn for all n≥0 and limsupn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, limn→∞∥yn-xn∥=0.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B35">38</xref>]).

Assume {an} is a sequence of nonnegative real numbers such that
an+1≤(1-γn)an+δnγn,
where {γn} is a sequence in (0,1) and {δn} is a sequence such that

∑n=1∞γn=∞;

limsupn→∞δn≤0 or ∑n=1∞|δnγn|<∞.

Then limn→∞an=0.3. Main Results

In this section, we will prove our main results.

Theorem 3.1.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F:C→H be an L-Lipschitz continuous mapping, g:C→C be a weakly continuous and γ-strongly monotone mapping such that R(g)=C. Let A:C→H be an α-inverse strongly g-monotone mapping and let S:C→C be a nonexpansive mapping. Suppose that Ω≠∅. Let β∈(0,1) and γ∈(L,2α). For given x0∈C, let {xn}⊂C be a sequence generated by
g(xn+1)=βg(xn)+(1-β)SPC[αnF(xn)+(1-αn)(g(xn)-λAxn)],n≥0,
where {αn}⊂(0,1) satisfies (C1):limn→∞αn=0 and (C2):∑nαn=∞. Then the sequence {xn} generated by (3.1) converges strongly to x*∈Ω which is the unique solution of the following variational inequality:
〈F(x*)-g(x*),g(x)-g(x*)〉≤0,∀x∈Ω.

Proof.

First, we show the solution set of variational inequality (3.2) is singleton. Assume x̃∈Ω also solves (3.2). Then, we have
〈F(x*)-g(x*),g(x̃)-g(x*)〉≤0,〈F(x̃)-g(x̃),g(x*)-g(x̃)〉≤0.
It follows that
〈F(x̃)-g(x̃)-F(x*)+g(x*),g(x*)-g(x̃)〉≤0⟹‖g(x*)-g(x̃)‖2≤〈F(x*)-F(x̃),g(x*)-g(x̃)〉⟹‖g(x*)-g(x̃)‖2≤〈F(x*)-F(x̃),g(x*)-g(x̃)〉≤‖F(x*)-F(x̃)‖‖g(x*)-g(x̃)‖⟹‖g(x*)-g(x̃)‖≤‖F(x*)-F(x̃)‖.
Since g is γ-strongly monotone, we have
γ‖x-y‖2≤〈g(x)-g(y),x-y〉≤‖g(x)-g(y)‖‖x-y‖,∀x,y∈C.
Hence,
γ‖x-y‖≤‖g(x)-g(y)‖,∀x,y∈C.
In particular, γ∥x*-x̃∥≤∥g(x*)-g(x̃)∥. By (3.4), we deduce
γ‖x*-x̃‖≤‖g(x*)-g(x̃)‖≤‖F(x*)-F(x̃)‖≤L‖x*-x̃‖,
which implies that x̃=x* because of L<γ by the assumption. Therefore, the solution of variational inequality (3.2) is unique.

Pick up any u∈Ω. It is obvious that u∈GVI(C,A,g) and g(u)∈Fix(S). Set un=PC[αnF(xn)+(1-αn)(g(xn)-λAxn)],n≥0. From (2.6), we know g(u)=PC[g(u)-μAu] for any μ>0. Hence, we have
g(u)=PC[g(u)-(1-αn)λAu]=PC[αng(u)+(1-αn)(g(u)-λAu)],∀n≥0.
From (3.6), (3.8), and Lemma 2.1, we get
‖un-g(u)‖=‖PC[αnF(xn)+(1-αn)(g(xn)-λAxn)]-PC[αng(u)+(1-αn)(g(u)-λAu)]‖≤αn‖F(xn)-g(u)‖+(1-αn)‖(g(xn)-λAxn)-(g(u)-λAu)‖≤αn‖F(xn)-F(u)‖+αn‖F(u)-g(u)‖+(1-αn)‖g(xn)-g(u)‖≤αnL‖xn-u‖+αn‖F(u)-g(u)‖+(1-αn)‖g(xn)-g(u)‖≤αnLγ‖g(xn)-g(u)‖+αn‖F(u)-g(u)‖+(1-αn)‖g(xn)-g(u)‖=[1-(1-Lγ)αn]‖g(xn)-g(u)‖+αn‖F(u)-g(u)‖.
It follows from (3.1) that
‖g(xn+1)-g(u)‖≤β‖g(xn)-g(u)‖+(1-β)‖Sun-Sg(u)‖≤β‖g(xn)-g(u)‖+(1-β)‖un-g(u)‖≤β‖g(xn)-g(u)‖+(1-β)[1-(1-Lγ)αn]‖g(xn)-g(u)‖+(1-β)αn‖F(u)-g(u)‖=[1-(1-Lγ)(1-β)αn]‖g(xn)-g(u)‖+(1-Lγ)(1-β)αn‖F(u)-g(u)‖1-L/γ.
This indicates by induction that
‖g(xn+1)-g(u)‖≤max{‖g(xn)-g(u)‖,‖F(u)-g(u)‖1-L/γ}.
Hence, {g(xn)} is bounded. By (3.6), we have ∥xn-u∥≤(1/γ)∥g(xn)-g(u)∥. This implies that {xn} is bounded. Consequently, {F(xn)},{Axn},{un}, and {Sun} are all bounded.

Note that we can rewrite (3.1) as g(xn+1)=βg(xn)+(1-β)Sun for all n. Next, we will use Lemma 2.3 to prove that ∥xn+1-xn∥→0. In fact, we firstly have
‖Sun-Sun-1‖=‖SPC[αnF(xn)+(1-αn)(g(xn)-λAxn)]-SPC[αn-1F(xn-1)+(1-αn-1)(g(xn-1)-λAxn-1)]‖≤‖[αnF(xn)+(1-αn)(g(xn)-λAxn)]-[αn-1F(xn-1)+(1-αn-1)(g(xn-1)-λAxn-1)]‖≤αn‖F(xn)-F(xn-1)‖+|αn-αn-1|‖F(xn-1)‖+(1-αn)‖g(xn)-λAxn-(g(xn-1)-λAxn-1)‖+|αn-αn-1|‖g(xn-1)-λAxn-1‖≤αnL‖xn-xn-1‖+(1-αn)‖g(xn)-g(xn-1)‖+|αn-αn-1|(‖F(xn-1)‖+‖g(xn-1)-λAxn-1‖)≤αn(Lγ)‖g(xn)-g(xn-1)‖+(1-αn)‖g(xn)-g(xn-1)‖+|αn-αn-1|(‖F(xn-1)‖+‖g(xn-1)-λAxn-1‖)=[1-(1-Lγ)αn]‖g(xn)-g(xn-1)‖+|αn-αn-1|(‖F(xn-1)‖+‖g(xn-1)-λAxn-1‖).
It follows that
‖Sun-Sun-1‖-‖g(xn)-g(xn-1)‖≤|αn-αn-1|(‖F(xn-1)‖+‖g(xn-1)-λAxn-1‖).
Since αn→0 and the sequences {F(xn)},{g(xn)}, and {Axn} are bounded, we have
limsupn→∞(‖Sun-Sun-1‖-‖g(xn)-g(xn-1)‖)≤0.
By Lemma 2.3, we obtain
limn→∞‖Sun-g(xn)‖=0.
Hence,
limn→∞‖g(xn+1)-g(xn)‖=limn→∞(1-β)‖Sun-g(xn)‖=0.
This together with (3.6) imply that
limn→∞‖xn+1-xn‖=0.
By the convexity of the norm and (3.9), we have
‖g(xn+1)-g(u)‖2=‖β(g(xn)-g(u))+(1-β)(Sun-Sg(u))‖2≤β‖g(xn)-g(u)‖2+(1-β)‖Sun-Sg(u)‖2≤β‖g(xn)-g(u)‖2+(1-β)‖un-g(u)‖2≤β‖g(xn)-g(u)‖2+(1-β)[αn‖F(xn)-g(u)‖+(1-αn)‖(g(xn)-λAxn)-(g(u)-λAu)‖]2≤β‖g(xn)-g(u)‖2+(1-β)[αn‖F(xn)-g(u)‖2+(1-αn)‖(g(xn)-λAxn)-(g(u)-λAu)‖2].
From Lemma 2.1, we derive
‖(g(xn)-λAxn)-(g(u)-λAu)‖2≤‖g(xn)-g(u)‖2+λ(λ-2α)‖Axn-Au‖2.
Thus,
‖g(xn+1)-g(u)‖2≤β‖g(xn)-g(u)‖2+(1-β)[αn‖F(xn)-g(u)‖2+(1-αn)(‖g(xn)-g(u)‖2+λ(λ-2α)‖Axn-Au‖2)]=(1-β)αn‖F(xn)-g(u)‖2+[1-(1-β)αn]‖g(xn)-g(u)‖2+(1-β)(1-αn)λ(λ-2α)‖Axn-Au‖2.
So,
(1-β)(1-αn)λ(2α-λ)‖Axn-Au‖2≤(1-β)αn‖F(xn)-g(u)‖2+‖g(xn)-g(u)‖2-‖g(xn+1)-g(u)‖2≤(1-β)αn‖F(xn)-g(u)‖2+(‖g(xn)-g(u)‖+‖g(xn+1)-g(u)‖)‖g(xn+1)-g(xn)‖.
Since αn→0,∥g(xn+1)-g(xn)∥→0 and liminfn→∞(1-β)(1-αn)λ(2α-λ)>0, we obtain
limn→∞‖Axn-Au‖=0.
Set yn=g(xn)-λAxn-(g(u)-λAu) for all n. By using the property of projection, we get
‖un-g(u)‖2=‖PC[αnF(xn)+(1-αn)(g(xn)-λAxn)]-PC[αng(u)+(1-αn)(g(u)-λAu)]‖2≤〈αn(F(xn)-g(u))+(1-αn)yn,un-g(u)〉=12{‖αn(F(xn)-g(u))+(1-αn)yn‖2+‖un-g(u)‖2-‖αn(F(xn)-g(u))+(1-αn)yn-un+g(u)‖2}≤12{αn‖F(xn)-g(u)‖2+(1-αn)‖g(xn)-g(u)‖2+‖un-g(u)‖2-‖αn(F(xn)-g(u)-yn)+g(xn)-un-λ(Axn-Au)‖2}=12{αn‖F(xn)-g(u)‖2+(1-αn)‖g(xn)-g(u)‖2+‖un-g(u)‖2-‖g(xn)-un‖2-λ2‖Axn-Au‖-αn2‖F(xn)-g(u)-yn‖2+2λαn〈Axn-Au,F(xn)-g(u)-yn〉+2λ〈g(xn)-un,Axn-Au〉-2αn〈g(xn)-un,F(xn)-g(u)-yn〉(1-αn)‖g(xn)-g(u)‖2}.
It follows that
‖un-g(u)‖2≤αn‖F(xn)-g(u)‖2+(1-αn)‖g(xn)-g(u)‖2-‖g(xn)-un‖2+2λαn‖Axn-Au‖‖F(xn)-g(u)-yn‖+2λ‖g(xn)-un‖‖Axn-Au‖+2αn‖g(xn)-un‖‖F(xn)-g(u)-yn‖.
From (3.18) and (3.24), we have
‖g(xn+1)-g(u)‖2≤β‖g(xn)-g(u)‖2+(1-β)‖un-g(u)‖2≤β‖g(xn)-g(u)‖2+(1-β)αn‖F(xn)-g(u)‖2+(1-αn)(1-β)‖g(xn)-g(u)‖2-(1-β)‖g(xn)-un‖2+2λ(1-β)αn‖Axn-Au‖‖F(xn)-g(u)-yn‖+2λ(1-β)‖g(xn)-un‖‖Axn-Au‖+2(1-β)αn‖g(xn)-un‖‖F(xn)-g(u)-yn‖≤‖g(xn)-g(u)‖2+αn‖F(xn)-g(u)‖2-(1-β)‖g(xn)-un‖2+2λαn‖Axn-Au‖‖F(xn)-g(u)-yn‖+2λ‖g(xn)-un‖‖Axn-Au‖+2αn‖g(xn)-un‖‖F(xn)-g(u)-yn‖.
Then, we obtain
(1-β)‖g(xn)-un‖2≤(‖g(xn)-g(u)‖+‖g(xn+1)-g(u)‖)‖g(xn+1)-g(xn)‖+αn‖F(xn)-g(u)‖2+2λαn‖Axn-Au‖‖F(xn)-g(u)-yn‖+2λ‖g(xn)-un‖‖Axn-Au‖+2αn‖g(xn)-un‖‖F(xn)-g(u)-yn‖.
Since limn→∞αn=0, limn→∞∥g(xn+1)-g(xn)∥=0 and limn→∞∥Axn-Au∥=0, we have
limn→∞‖g(xn)-un‖=0.
Next, we prove limsupn→∞〈F(x*)-g(x*),un-g(x*)〉≤0 where x* is the unique solution of (3.2). We take a subsequence {uni} of {un} such that
limsupn→∞〈F(x*)-g(x*),un-g(x*)〉=limi→∞〈F(x*)-g(x*),uni-g(x*)〉=limi→∞〈F(x*)-g(x*),g(xni)-g(x*)〉.
Since {xni} is bounded, there exists a subsequence {xnij} of {xni} which converges weakly to some point z∈C. Without loss of generality, we may assume that xni⇀z. This implies that g(xni)⇀g(z) due to the weak continuity of g. Now, we show z∈Ω. First, we note that from (3.15) and (3.27) that ∥g(xn)-Sg(xn)∥→0. Hence, limi→∞∥g(xni)-Sg(xni)∥=0. By the demiclosedness principle of the nonexpansive mapping (see Lemma 2.2), we deduce g(z)∈Fix(S). Next, we only need to prove z∈GVI(C,A,g). Set
Tv={Av+NC(v),v∈C,∅,v∉C.
By [39], we know that T is maximal g-monotone. Let (v,w)∈G(T). Since w-Av∈NC(v) and xn∈C, we have
〈g(v)-g(xn),w-Av〉≥0.
From un=PC[αnF(xn)+(1-αn)(g(xn)-λAxn)], we get
〈g(v)-un,un-[αnF(xn)+(1-αn)(g(xn)-λAxn)]〉≥0.
It follows that
〈g(v)-un,un-g(xn)λ+Axn-αnλ(F(xn)-g(xn)+λAxn)〉≥0.
Then,
〈g(v)-g(xni),w〉≥〈g(v)-g(xni),Av〉≥〈g(v)-g(xni),Av〉-〈g(v)-uni,uni-g(xni)λ〉-〈g(v)-uni,Axni〉+αniλ〈g(v)-uni,F(xni)-g(xni)+λAxni〉=〈g(v)-g(xni),Av-Axni〉+〈g(v)-g(xni),-Axni〉-〈g(v)-uni,uni-g(xni)λ〉-〈g(v)-uni,Axni〉+αniλ〈g(v)-uni,F(xni)-g(xni)+λAxni〉≥-〈g(v)-uni,uni-g(xni)λ〉-〈g(xni)-uni,Axni〉+αniλ〈g(v)-uni,F(xni)-g(xni)+λAxni〉.
Since ∥g(xni)-uni∥→0 and g(xni)⇀g(z), we deduce that 〈g(v)-g(z),w〉≥0 by taking i→∞ in (3.33). Thus, z∈T-10 by the maximal g-monotonicity of T. Hence, z∈GVI(C,A,g). Therefore, z∈Ω. From (3.28), we obtain
limsupn→∞〈F(x*)-g(x*),un-g(x*)〉=limi→∞〈F(x*)-g(x*),g(xni)-g(x*)〉=〈F(x*)-g(x*),g(z)-g(x*)〉≤0.
We take u=x* in (3.23) to get
‖un-g(x*)‖2≤αn〈F(xn)-g(x*),un-g(x*)〉+(1-αn)〈g(xn)-λAxn-(g(x*)-λAx*),un-g(x*)〉≤αn〈F(xn)-F(x*),un-g(x*)〉+αn〈F(x*)-g(x*),un-g(x*)〉+(1-αn)‖g(xn)-λAxn-(g(x*)-λAx*)‖‖un-g(x*)‖≤αnL‖xn-x*‖‖un-g(x*)‖+αn〈F(x*)-g(x*),un-g(x*)〉+(1-αn)‖g(xn)-g(x*)‖‖un-g(x*)‖≤αn(Lγ)‖g(xn)-g(x*)‖‖un-g(x*)‖+αn〈F(x*)-g(x*),un-g(x*)〉+(1-αn)‖g(xn)-g(x*)‖‖un-g(x*)‖=[1-(1-Lγ)αn]‖g(xn)-g(x*)‖‖un-g(x*)‖+αn〈F(x*)-g(x*),un-g(x*)〉=1-(1-L/γ)αn2‖g(xn)-g(x*)‖2+12‖un-g(x*)‖2+αn〈F(x*)-g(x*),un-g(u)〉.
It follows that
‖un-g(x*)‖2≤[1-(1-Lγ)αn]‖g(xn)-g(x*)‖2+2αn〈F(x*)-g(x*),un-g(x*)〉.
Therefore,
‖g(xn+1)-g(x*)‖2≤β‖g(xn)-g(x*)‖2+(1-β)‖un-g(x*)‖2≤β‖g(xn)-g(x*)‖2+(1-β)[1-(1-Lγ)αn]‖g(xn)-g(x*)‖2+2(1-β)αn〈F(x*)-g(x*),un-g(x*)〉=[1-(1-Lγ)(1-β)αn]‖g(xn)-g(x*)‖2+2(1-β)αn〈F(x*)-g(x*),un-g(x*)〉=[1-(1-Lγ)(1-β)αn]‖g(xn)-g(x*)‖2+(1-Lγ)(1-β)αn(21-L/γ〈F(x*)-g(x*),un-g(x*)〉)=(1-γn)‖g(xn)-g(x*)‖2+δnγn,
where γn=(1-L/γ)(1-β)αn and δn=(2/(1-L/γ))〈F(x*)-g(x*),un-g(x*)〉. From condition (C2), we have ∑nγn=∞. By (3.34), we have limsupn→∞δn≤0. We can therefore apply Lemma 2.4 to conclude that g(xn)→g(x*) and xn→x*. This completes the proof.

Corollary 3.2.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F:C→H be an L-contraction. Let A:C→H be an α-inverse strongly monotone mapping and let S:C→C be a nonexpansive mapping. Suppose that Ω≠∅. Let β∈(0,1) and γ∈(L,2α). For given x0∈C, let {xn}⊂C be a sequence generated by
xn+1=βxn+(1-β)SPC[αnF(xn)+(1-αn)(xn-λAxn)],n≥0,
where {αn}⊂(0,1) satisfies (C1):limn→∞αn=0 and (C2):∑nαn=∞. Then the sequence {xn} generated by (3.38) converges strongly to x*∈Ω which is the unique solution of the following variational inequality:
〈F(x*)-x*,x-x*〉≤0,∀x∈Ω.

Acknowledgments

The authors are very grateful to the referees for their comments and suggestions which improved the presentation of this paper. The first author was partially supported by the Program TH-1-3, Optimization Lean Cycle, of Subprojects TH-1 of Spindle Plan Four in Excellence Teaching and Learning Plan of Cheng Shiu University and was supported in part by NSC 100-2221-E-230-012. The second author was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105.

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