We introduce the triple hierarchical problem over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence of the algorithm is proved under some mild conditions. Our results extend those of Yao et al., Iiduka, Ceng et al., and other authors.

1. Introduction

Let C be a closed convex subset of a real Hilbert space H with inner product 〈·,·〉 and norm ∥·∥. We denote weak convergence and strong convergence by notations ⇀ and →, respectively. Let A be a nonlinear mapping. The Hartman-Stampacchia variational inequality [1] is to find x∈C such that 〈Ax,y-x〉≥0,∀y∈C. The set of solutions is denoted by VI(C,A). f:C→C is said to be a ρ-contraction if there exists a constant ρ∈[0,1) such that ∥f(x)-f(y)∥≤ρ∥x-y∥,∀x,y∈C. A mapping A:H→H is said to be monotone if 〈Ax-Ay,x-y〉≥0,∀x,y∈H. A mapping A:H→H is said to be α- strongly monotone if there exists a positive real number α such that 〈Ax-Ay,x-y〉≥α∥x-y∥2,∀x,y∈H. A mapping A:H→H is said to be β-inverse-strongly monotone if there exists a positive real number β such that 〈Ax-Ay,x-y〉≥β∥Ax-Ay∥2,∀x,y∈H. A mapping A:H→H is said to be L-Lipschitz continuous if there exists a positive real number L such that ∥Ax-Ay∥≤L∥x-y∥,∀x,y∈H. A linear bounded operator A is said to be strongly positive on H if there exists a constant γ¯>0 with the property 〈Ax,x〉≥γ¯∥x∥2,∀x∈H. A mapping T:C→C is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.

A point x∈C is a fixed point of T provided Tx=x. Denote by F(T) the set of fixed points of T; that is, F(T)={x∈C:Tx=x}. If C is bounded closed convex and T is a nonexpansive mapping of C into itself, then F(T) is nonempty (see [2]).

We discuss the following variational inequality problem over the fixed point set of a nonexpansive mapping (see [3–16]), which is said to be the hierarchical problem. Let a monotone, continuous mapping A:H→H and a nonexpansive mapping T:H→H. Find x∈VI(F(T),A)={x∈F(T):〈Ax,y-x〉≥0,∀y∈F(T)}, where F(T)≠∅. This solution set is denoted by Ξ.

We introduce the following variational inequality problem over the solution set of variational inequality problem and the fixed point set of a nonexpansive mapping (see [17, 18]), which is said to be the triple hierarchical problem. Let an inverse-strongly monotone A:H→H, a strongly monotone and Lipschitz continuous B:H→H, and a nonexpansive mapping T:H→H. Find x∈VI(Ξ,B)={x∈Ξ:〈Bx,y-x〉≥0,∀y∈Ξ}, where Ξ:=VI(F(T),A)≠∅.

In 2009, Yao et al. [19] considered the following two-step iterative algorithm with the initial guess x0∈C which is chosen arbitrarily:
(1)xn+1=αnf(xn)+(1-αn)Tyn,yn=βnSxn+(1-βn)xn,∀n≥0,
where αn,βn∈(0,1) satisfies certain assumptions. Let S,T be two nonexpansive mappings and let f:C→C be a contraction mapping. Then, they proved that the above iterative sequence {xn} converges strongly to fixed point.

Next, Iiduka [17] introduced a monotone variational inequality with variational inequality constraint over the fixed point set of a nonexpansive mapping; the sequence {xn} defined by the iterative method below, with the initial guess x1∈H, is chosen arbitrarily:
(2)yn=T(xn-λnA1xn),xn+1=yn-μαnA2yn,∀n≥0,
where αn∈(0,1] and λn∈(0,2α] satisfy certain conditions, A1:H→H is an inverse-strongly monotone, A2:H→H is a strongly monotone and Lipschitz continuous, and T:H→H is a nonexpansive mapping; then the strongly convergence analysis of the sequence generated by (2) is proved under some appropriate conditions.

In 2011, Yao et al. [20] studied the hierarchical problem over the fixed point set. Let the sequences {xn} be generated by these two following algorithms:

They illustrated that these two algorithms converge strongly to the unique solution of the variational inequality which is to find x*∈F(T) such that
(3)〈(A-γf)x*,x-x*〉≥0,∀x∈F(T),
where A:C→H is a strongly positive linear bounded operator, f:C→H is a ρ-contraction, and T:C→C is a nonexpansive mapping satisfying some conditions.

Very recently, Ceng et al. [21] studied the following new algorithms. For x0∈C is chosen arbitrarily, they defined a sequence {xn} by
(4)xn+1=PC[λnγ(αnf(xn)+(1-αn)Sxn)+(I-λnμF)Txn],wwwwwwwwwwwwwwwwwwwwwwwwwwwi∀n≥0,
where the mappings S, T are nonexpansive mappings with F(T)≠∅. Let F:C→H be a Lipschitzian and strongly monotone operator and let f:C→H be a contraction mapping satisfying some appropriate conditions. They proved that the proposed algorithms strongly converge to the minimum norm fixed point of T.

In this paper, we consider a new iterative algorithm for solving the triple hierarchical problem over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping which contain algorithms (1) and (4) as follows:
(5)yn=PC[βnSxn+(1-βn)xn],xn+1=γλnϕ(xn)+(I-λnμF)Tyn,∀n≥0,
where the mappings S, T are nonexpansive mappings with F(T)≠∅. Let F:C→H be a Lipschitzian and strongly monotone operator, and let ϕ:H→H be a contraction mapping satisfying some mild conditions. Find a point x*∈F(T) such that
(6)〈(I-S)x*,x-x*〉≥0,∀x∈F(T).
This solution set of (6) is denoted by Ω:=VI(F(T),S). The strong convergence for the proposed algorithms to the solution is solved under some appropriate assumptions. Our results improve the results of Ceng et al. [21], Iiduka [17], Yao et al. [19], Yao et al. [20], and some authors.

2. Preliminaries

Let C be a nonempty closed convex subset of H. There holds the following inequality in an inner product space ∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H. For every point x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(7)∥x-PCx∥≤∥x-y∥,∀y∈C.PC is called the metric projection of H onto C. It is well known that PC is a nonexpansive mapping of H onto C and satisfies
(8)〈x-y,PCx-PCy〉≥∥PCx-PCy∥2,
for every x,y∈H. Moreover, PCx is characterized by the following properties: PCx∈C and
(9)〈x-PCx,y-PCx〉≤0,(10)∥x-y∥2≥∥x-PCx∥2+∥y-PCx∥2,
for all x∈H,y∈C. Let B be a monotone mapping of C into H. In the context of the variational inequality problem the characterization of projection (9) implies the following:
(11)u∈VI(C,B)⟺u=PC(u-λBu),λ>0.
It is also known that H satisfies the Opial’s condition [22]; that is, for any sequence {xn}⊂H with xn⇀x, the inequality liminfn→∞∥xn-x∥<liminfn→∞∥xn-y∥ holds for every y∈H with x≠y.

Lemma 1 (see [<xref ref-type="bibr" rid="B1">23</xref>]).

Let C be a closed convex subset of a real Hilbert space H and let T:C→C be a nonexpansive mapping. Then I-T is demiclosed at zero; that is, xn⇀x and xn-Txn→0 imply x=Tx.

Lemma 2 (see [<xref ref-type="bibr" rid="B13">24</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 integers n≥0 and limsupn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, limn→∞∥yn-xn∥=0.

Lemma 3 (see [<xref ref-type="bibr" rid="B16">10</xref>]).

Let B:H→H be β-strongly monotone and L-Lipschitz continuous and let μ∈(0,2β/L2). For λ∈[0,1], define Tλ:H→H by Tλ(x):=x-λμB(x) for all x∈H. Then, for all x,y∈H, ∥Tλ(x)-Tλ(y)∥≤(1-λτ)∥x-y∥ hold, where τ:=1-1-μ(2β-μL2)∈(0,1].

Lemma 4 (see [<xref ref-type="bibr" rid="B25">25</xref>]).

Assume that {an} is a sequence of nonnegative real numbers such that
(12)an+1≤(1-γn)an+δn,∀n≥0,
where {γn}⊂(0,1) and {δn} is a sequence in R such that

∑n=1∞γn=∞;

limsupn→∞(δn/γn)≤0 or ∑n=1∞|δn|<∞.

Then limn→∞an=0.3. Strong Convergence Theorem

In this section, we introduce an iterative algorithm of triple hierarchical for solving monotone variational inequality problems for κ-Lipschitzian and η-strongly monotone operators over the solution set of variational inequality problems and the fixed point set of a nonexpansive mapping.

Theorem 5.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F:C→C be κ-Lipschitzian and η-strongly monotone operators with constant κ and η>0, respectively, and let ϕ:C→C be a ρ-contraction with coefficient ρ∈[0,1). Let T:C→C be a nonexpansive mapping with F(T)≠∅, and let S:H→H be a nonexpansive mapping. Let 0<μ<2η/κ2 and 0<γ<τ, where τ=1-1-μ(2η-μκ2). Suppose that {xn} is a sequence generated by the following algorithm where x0∈C is chosen arbitrarily:
(13)yn=PC[βnSxn+(1-βn)xn],xn+1=γλnϕ(xn)+(I-λnμF)Tyn,∀n≥0,
where {βn},{λn},⊂(0,1) satisfy the following conditions:

βn≤kλn;

limn→∞λn=0, limn→∞((λn-λn-1)/λn)=0, ∑n=0∞λn=∞;

limn→∞((βn-βn-1)/βn)=0.

Then {xn} converges strongly to x*∈Ω, which is the unique solution of another variational inequality:
(14)〈(μF-γϕ)x*,x-x*〉≥0,∀x∈Ω,
where Ω:=VI(F(T),S)≠∅.Proof.

We will divide the proof into four steps.

Step 1. We will show that {xn} is bounded. Indeed, for any x*∈F(T), we have
(15)∥yn-x*∥=∥PC[βnSxn+(1-βn)xn]-PCx*∥≤∥βnSxn+(1-βn)xn-x*∥=∥βn(Sxn-Sx*)+(1-βn)(xn-x*)+βn(Sx*-x*)∥≤βn∥xn-x*∥+(1-βn)∥xn-x*∥+βn∥Sx*-x*∥≤∥xn-x*∥+βn∥Sx*-x*∥.
From (13), we deduce that
(16)∥xn+1-x*∥=∥γλnϕ(xn)+(I-λnμF)Tyn-x*∥=∥γλn(ϕ(xn)-ϕ(x*))+(I-λnμF)(Tyn-x*)+λn(γϕ(x*)-μFx*)(ϕ(xn)-ϕ(x*))∥≤γλn∥ϕ(xn)-ϕ(x*)∥+(I-λnμF)∥Tyn-x*∥+λn∥γϕ(x*)-μFx*∥≤γρλn∥xn-x*∥+(1-λnτ)∥yn-x*∥+λn∥γϕ(x*)-μFx*∥.
Substituting (15) into (16), we obtain
(17)∥xn+1-x*∥≤γρλn∥xn-x*∥+(1-λnτ){∥xn-x*∥+βn∥Sx*-x*∥}+λn∥γϕ(x*)-μFx*∥≤γρλn∥xn-x*∥+(1-λnτ)∥xn-x*∥+βn∥Sx*-x*∥+λn∥γϕ(x*)-μFx*∥≤[1-λn(τ-γρ)]∥xn-x*∥+kλn∥Sx*-x*∥+λn∥γϕ(x*)-μFx*∥≤[1-λn(τ-γρ)]∥xn-x*∥+λn(k∥Sx*-x*∥+∥γϕ(x*)-μFx*∥)≤max{∥xn-x*∥+1τ-γρ×(k∥Sx*-x*∥+∥γϕ(x*)-μFx*∥)1τ-γρ}.
By induction, it follows that
(18)∥xn-x*∥≤max{∥x0-x*∥+1τ-γρwwwwww×(k∥Sx*-x*∥+∥γϕ(x*)-μFx*∥)1τ-γρ},wwwwwwwwwwwwwwwwwwwwwwwwwn≥0.
Therefore, {xn} is bounded and so are {yn}, {Tyn}, {Sxn}, {ϕ(xn)}, and {FT(yn)}.

Step 2. We will show that limn→∞∥xn-Txn∥=0. Setting vn:=βnSxn+(1-βn)xn, we obtain
(19)∥vn-vn-1∥=∥βnSxn+(1-βn)xn-βn-1Sxn-1-(1-βn-1)xn-1∥=∥βn(Sxn-Sxn-1)+(βn-βn-1)Sxn-1+(1-βn)(xn-xn-1)+(βn-1-βn)xn-1∥≤βn∥xn-xn-1∥+|βn-βn-1|(∥Sxn-1∥+∥xn-1∥)+(1-βn)∥xn-xn-1∥≤∥xn-xn-1∥+|βn-βn-1|(∥Sxn-1∥+∥xn-1∥),
which implies that
(20)∥yn-yn-1∥=∥PCvn-PCvn-1∥≤∥vn-vn-1∥≤∥xn-xn-1∥+|βn-βn-1|(∥Sxn-1∥+∥xn-1∥).
It follows from (13) that
(21)∥xn+1-xn∥=∥γλnϕ(xn)+(I-λnμF)Tyn-γλn-1ϕ(xn-1)-(I-λn-1μF)Tyn-1∥=∥γλn(ϕ(xn)-ϕ(xn-1))+(λn-λn-1)γϕ(xn-1)+(I-λnμF)Tyn-(I-λn-1μF)Tyn-1∥≤γρλn∥xn-xn-1∥+|λn-λn-1|γ∥ϕ(xn-1)∥+∥(I-λnμF)Tyn-(I-λnμF)Tyn-1+(I-λnμF)Tyn-1-(I-λn-1μF)Tyn-1∥≤γρλn∥xn-xn-1∥+|λn-λn-1|γ∥ϕ(xn-1)∥+(1-λnτ)∥yn-yn-1∥+|λn-λn-1|μ∥FTyn-1∥≤γρλn∥xn-xn-1∥+|λn-λn-1|×(γ∥ϕ(xn-1)∥+μ∥FTyn-1∥)+(1-λnτ){∥xn-xn-1∥+|βn-βn-1|×(∥Sxn-1∥+∥xn-1∥)}≤[1-λn(τ-γρ)]∥xn-xn-1∥+|λn-λn-1|(γ∥ϕ(xn-1)∥+μ∥FTyn-1∥)+|βn-βn-1|(∥Sxn-1∥+∥xn-1∥)=[1-λn(τ-γρ)]∥xn-xn-1∥+(|λn-λn-1|λn+|βn-βn-1|λn)λnM1≤[1-λn(τ-γρ)]∥xn-xn-1∥+(|λn-λn-1|λn+k|βn-βn-1|βn)λnM1,
where M1 is a constant such that
(22)supn≥0{(γ∥ϕ(xn)∥+μ∥FTyn∥),(∥Sxn∥+∥xn∥)}≤M1.
Hence, conditions (C2) and (C3) allow us to apply Lemma 4; then we get
(23)limn→∞∥xn+1-xn∥=0.
By (21), we get
(24)∥xn+1-xn∥λn≤[1-λn(τ-γρ)]∥xn-xn-1∥λn+|λn-λn-1|+|βn-βn-1|λnM1=[1-λn(τ-γρ)]∥xn-xn-1∥λn-1+[1-λn(τ-γρ)](∥xn-xn-1∥λn-∥xn-xn-1∥λn-1)+|λn-λn-1|+|βn-βn-1|λnM1≤[1-λn(τ-γρ)]∥xn-xn-1∥λn-1+λn∥xn-xn-1∥1λn|1λn-1λn-1|+M1λn|λn-λn-1|+|βn-βn-1|λn2.
Using the conditions (C2) and (C3), we can apply Lemma 4 to conclude that
(25)limn→∞∥xn+1-xn∥λn=0.
By (13), we compute
(26)∥xn+1-Tyn∥=∥γλnϕ(xn)+(I-λnμF)Tyn-Tyn∥=∥γλnϕ(xn)+Tyn-λnμFTyn-Tyn∥≤λn∥γϕ(xn)-μFTyn∥.
From the condition (C2), we note that limn→∞∥xn+1-Tyn∥=0. At the same time, from (13), we also have
(27)∥yn-xn∥=∥PC[βnSxn+(1-βn)xn]-PCxn∥≤∥βnSxn+(1-βn)xn-xn∥≤βn∥Sxn-xn∥.
By the conditions (C1) and (C2), we note that limn→∞∥yn-xn∥=0. Consider
(28)∥yn-Tyn∥≤∥yn-xn∥+∥xn-xn+1∥+∥xn+1-Tyn∥⟶0.
From (23), (26), and (27), we obtain
(29)limn→∞∥yn-Tyn∥=0.
We set vn=βnSxn+(1-βn)xn; then we get
(30)∥yn-vn∥=∥PCvn-vn∥≤∥vn-vn∥⟶0,asn⟶∞.
From (13), we have
(31)∥Tyn-Txn∥=∥TPC[βnSxn+(1-βn)xn]-TPCxn∥≤∥βnSxn+(1-βn)xn-xn∥≤βn∥Sxn-xn∥.
By the conditions (C1) and (C2) again, we note that limn→∞∥Tyn-Txn∥=0. Consider(32)∥xn-Txn∥≤∥xn-yn∥+∥yn-Tyn∥+∥Tyn-Txn∥⟶0.
From (29), limn→∞∥xn-yn∥=0, and limn→∞∥Tyn-Txn∥=0, we obtain
(33)limn→∞∥xn-Txn∥=0.

Step 3. We will show that limsupn→∞〈μFx*-γϕ(x*),xn-x*〉≤0. Rewrite (13) as
(34)xn+1=γλnϕ(xn)+(I-μλnF)Tyn-vn+βnSxn+(1-βn)xn.
We observe that
(35)xn-xn+1=xn-γλnϕ(xn)-(I-μλnF)Tyn+vn-βnSxn-xn+βnxn=λn(μF-γϕ)xn-λnμFxn-(I-μλnF)Tyn+(I-μλnF)yn-(I-μλnF)yn+vn+βn(I-S)xn=λn(μF-γϕ)xn+λnμ(Fyn-Fxn)+(yn-Tyn)-μλnF(yn-Tyn)+(vn-yn)+βn(I-S)xn=λn(μF-γϕ)xn+λnμ(Fyn-Fxn)+(yn-Tyn)-μλnF(yn-Tyn)+λn(yn-Tyn)-λn(yn-Tyn)+(vn-yn)+βn(I-S)xn=λn(μF-γϕ)xn+λnμ(Fyn-Fxn)+λn(I-μF)(yn-Tyn)+(1-λn)(yn-Tyn)+(vn-yn)+βn(I-S)xn.
Set
(36)zn=xn-xn+1λn,∀n≥0.
We note from (35) that
(37)zn=(μF-γϕ)xn+μ(Fyn-Fxn)+(I-μF)(yn-Tyn)+1-λnλn(yn-Tyn)+1λn(vn-yn)+βnλn(I-S)xn.
This yields that, for each x*∈F(T),
(38)〈zn,xn-x*〉=〈(μF-γϕ)xn,xn-x*〉+μ〈(Fyn-Fxn),xn-x*〉+〈(I-μF)yn-(I-μF)Tyn,xn-x*〉+1-λnλn〈yn-Tyn,xn-x*〉+1λn〈vn-yn,xn-x*〉+βnλn〈(I-S)xn,xn-x*〉=〈(μF-γϕ)x*,xn-x*〉+〈(μF-γϕ)xn-(μF-γϕ)x*,xn-x*〉+μ〈(Fyn-Fxn),xn-x*〉+〈(I-μF)yn-(I-μF)Tyn,xn-x*〉+1-λnλn〈yn-Tyn,xn-x*〉+1λn〈vn-yn,xn-x*〉+βnλn〈(I-S)xn,xn-x*〉.
In view of (38), 〈(μF-γϕ)xn-(μF-γϕ)x*,xn-x*〉 is nonnegative due to the monotonicity of μF-γϕ. From (38), we derive that
(39)〈zn,xn-x*〉≥〈(μF-γϕ)x*,xn-x*〉+μ〈(Fyn-Fxn),xn-x*〉+〈(I-μF)yn-(I-μF)Tyn,xn-x*〉+1-λnλn〈yn-Tyn,xn-x*〉+1λn〈vn-yn,xn-x*〉+βnλn〈(I-S)xn,xn-x*〉.
Since (29) implies ∥(I-μF)yn-(I-μF)Tyn∥→0, as n→∞, from (25), then we get zn→0. Using (C1) and (30), ∥yn-xn∥→0, as n→∞ and {xn} is bounded. We obtain from (39) that
(40)limsupn→∞〈(μF-γϕ)x*,xn-x*〉≤0,∀x*∈F(T).
Since the sequence {xn} is bounded, we can take a subsequence {xnj} of {xn} such that
(41)limsupn→∞〈(μF-γϕ)x*,xn-x*〉=limsupj→∞〈(μF-γϕ)x*,xnj-x*〉
and xnj⇀x~. From (33), by the demiclosed principle of the nonexpansive mapping, it follows that x~∈F(T). Then
(42)limsupj→∞〈(μF-γϕ)x*,xnj-x*〉=〈(μF-γϕ)x*,x~-x*〉≤0.

Step 4. Finally, we will prove xn+1→x*. From (13), we note that
(43)∥yn-x*∥2=∥PC[βnSxn+(1-βn)xn]-PCx*∥2≤∥[βnSxn+(1-βn)xn]-x*∥2≤∥βn(Sxn-Sx*)+(1-βn)(xn-x*)+βn(Sx*-x*)∥2≤∥βn(Sxn-Sx*)+(1-βn)(xn-x*)∥2+2βn〈Sx*-x*,yn-x*〉≤βn∥xn-x*∥2+(1-βn)∥xn-x*∥2+2βn〈Sx*-x*,yn-x*〉≤∥xn-x*∥2+2βn∥Sx*-x*∥∥yn-x*∥.
Using (43), we compute
(44)∥xn+1-x*∥2=∥γλnϕ(xn)+(I-λnμF)Tyn-x*∥2=∥γλn(ϕ(xn)-ϕ(x*))+(I-λnμF)Tyn-(I-λnμF)x*+(I-λnμF)x*-x*+γλnϕ(x*)∥2=∥γλn(ϕ(xn)-ϕ(x*))+(I-λnμF)(Tyn-x*)+λn(γϕ(x*)-μFx*)∥2≤∥γλn(ϕ(xn)-ϕ(x*))+(I-λnμF)(Tyn-x*)∥2+2λn〈γϕ(x*)-μFx*,xn+1-x*〉≤γ2λn2∥ϕ(xn)-ϕ(x*)∥2+(1-λnτ)2∥Tyn-x*∥2+2λn〈γϕ(x*)-μFx*,xn+1-x*〉+2〈γλn(ϕ(xn)-ϕ(x*)),(I-μλnF)(Tyn-x*)〉≤γ2ρ2λn2∥xn-x*∥2+(1-2λnτ+λn2τ2)∥yn-x*∥2+2λn〈γϕ(x*)-μFx*,xn+1-x*〉+2γλn〈ϕ(xn)-ϕ(x*),(I-μλnF)Tyn-(I-μλnF)x*〉=γ2ρ2λn2∥xn-x*∥2+(1-2λnτ+λn2τ2)∥yn-x*∥2+2λn〈γϕ(x*)-μFx*,xn+1-x*〉+2γλn〈ϕ(xn)-ϕ(x*),(Tyn-x*)-μλnF(Tyn-x*)〉=γ2ρ2λn2∥xn-x*∥2+(1-2λnτ+λn2τ2)∥yn-x*∥2+2λn〈γϕ(x*)-μFx*,xn+1-x*〉+2γλn〈ϕ(xn)-ϕ(x*),Tyn-x*〉-2γλn〈ϕ(xn)-ϕ(x*),μλnF(Tyn-x*)〉≤γ2ρ2λn2∥xn-x*∥2+(1-2λnτ+λn2τ2)×{∥xn-x*∥2+2βn∥Sx*-x*∥∥yn-x*∥}+2λn〈γϕ(x*)-μFx*,xn+1-x*〉+2γρλn∥xn-x*∥∥Tyn-x*∥-2γρμλn2∥xn-x*∥∥F(Tyn-x*)∥≤[1-λn(2τ-λnτ2-λnγ2ρ2)]∥xn-x*∥2+2εnλn∥Sx*-x*∥∥yn-x*∥+2λn〈γϕ(x*)-μFx*,xn+1-x*〉+2γρλn∥xn-x*∥∥Tyn-x*∥-2γρμλn2∥xn-x*∥∥F(Tyn-x*)∥.
Since {xn}, {Tyn}, and {FTyn} are all bounded, we can choose a constant M2>0 such that
(45)supn≥012τ-λnτ2-λnγ2ρ2×{2γρμ∥xn-x*∥∥F(Tyn-x*)∥}≤M2.
It follows that
(46)∥xn+1-x*∥2≤[1-λn(2τ-λnτ2-λnγ2ρ2)]∥xn-x*∥2+λn(2τ-λnτ2-λnγ2ρ2)δn,
where
(47)δn=2εn2τ-λnτ2-λnγ2ρ2∥Sx*-x*∥∥yn-x*∥+22τ-λnτ2-λnγ2ρ2〈γϕ(x*)-μFx*,xn+1-x*〉+22τ-λnτ2-λnγ2ρ2γρ∥xn-x*∥∥Tyn-x*∥-λnM2.
Now, applying Lemma 4 and (35), we conclude that xn→x*. This completes the proof.

Corollary 6.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F:C→C be κ-Lipschitzian and η-strongly monotone operators with constant κ and η>0, respectively. Let T:C→C be a nonexpansive mapping with F(T)≠∅, and let S:H→H be a nonexpansive mapping. Let 0<μ<2η/κ2 and 0<γ<τ, where τ=1-1-μ(2η-μκ2). Suppose {xn} is a sequence generated by the following algorithm x0∈C arbitrarily:
(48)xn+1=(I-λnμF)TPC[βnSxn+(1-βn)xn],∀n≥0,
where {βn},{λn}⊂(0,1) satisfy the following conditions (C1)–(C3). Then {xn} converges strongly to x*∈Ω, which is the unique solution of variational inequality:
(49)〈(I-μF)x*,x-x*〉≥0,∀x∈Ω,
where Ω:=VI(F(T),S)≠∅.

Proof.

Putting ϕ≡0 in Theorem 5, we can obtain the desired conclusion immediately.

Corollary 7.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let ϕ:H→H be a ρ-contraction with coefficient ρ∈[0,1), and let T:C→C be a nonexpansive mapping with F(T)≠∅ and S:H→H a nonexpansive mapping. Suppose {xn} is a sequence generated by the following algorithm, x0∈C, arbitrarily:
(50)yn=PC[βnSxn+(1-βn)xn],xn+1=λnϕ(xn)+(1-λn)Tyn,∀n≥0,
where {βn},{λn}⊂(0,1) satisfy the following conditions (C1)–(C3). Then {xn} converges strongly to x*∈Ω, which is the unique solution of variational inequality:
(51)〈(I-ϕ)x*,x-x*〉≥0,∀x∈Ω,
where Ω:=VI(F(T),S)≠∅.

Proof.

Putting γ=1, μ=2, and F≡I/2 in Theorem 5, we can obtain the desired conclusion immediately.

Corollary 8.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let T:C→C be a nonexpansive mapping with F(T)≠∅ and let S:H→H be a nonexpansive mapping. Suppose {xn} is a sequence generated by the following algorithm, x0∈C, arbitrarily:
(52)xn+1=(1-λn)TPC[βnSxn+(1-βn)xn],∀n≥0,
where {βn},{λn}⊂(0,1) satisfy the following conditions (C1)–(C3). Then {xn} converges strongly to x*∈F(T), which is the unique solution of variational inequality:
(53)〈(I-S)x*,x-x*〉≥0,∀x∈F(T).

Proof.

Putting ϕ≡0 in Corollary 7, we can obtain the desired conclusion immediately.

Corollary 9.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let ϕ:H→H be a ρ-contraction with coefficient ρ∈[0,1), and let T:C→C be a nonexpansive mapping with F(T)≠∅ and S:C→C a nonexpansive mapping. Suppose {xn} is a sequence generated by the following algorithm, x0∈C, arbitrarily:
(54)xn+1=λnxn+(1-λn)T[βnSxn+(1-βn)xn],wwwwwwwwwwwwwwwwwwwwwwii∀n≥0,
where {βn},{λn}⊂(0,1) satisfy the following conditions (C1)–(C3). Then {xn} converges strongly to x*∈F(T), which is the unique solution of variational inequality:
(55)〈(I-S)x*,x-x*〉≥0,∀x∈F(T).

Proof.

Putting PC≡I in Corollary 7, we can obtain the desired conclusion immediately.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author was supported by the Thailand Research Fund and the King Mongkut’s University of Technology Thonburi (Grant no. RSA5780059). The second author was supported by the Commission on Higher Education, the Thailand Research Fund, and Rajamangala University of Technology Lanna Chiangrai under Grant no. MRG5680157 during the preparation of this paper.

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