A general hierarchical problem has been considered, and an explicit algorithm has been presented for solving this hierarchical problem. Also, it is shown that the suggested algorithm converges strongly to a solution of the hierarchical problem.

1. Introduction

Let H be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively. Let C be a nonempty closed convex subset of H. The hierarchical problem is of finding x̃∈Fix(T) such that〈Sx̃-x̃,x-x̃〉≤0,∀x∈Fix(T),
where S,T are two nonexpansive mappings and Fix(T) is the set of fixed points of T. Recently, this problem has been studied by many authors (see, e.g., [1–15]). The main reason is that this problem is closely associated with some monotone variational inequalities and convex programming problems (see [16–19]).

Now, we briefly recall some historic results which relate to the problem (1.1).

For solving the problem (1.1), in 2006, Moudafi and Mainge [1] first introduced an implicit iterative algorithm:xt,s=sQ(xt,s)+(1-s)[tS(xt,s)+(1-t)T(xt,s)]
and proved that the net {xt,s} defined by (1.2) strongly converges to xt as s→0, where xt satisfies xt=projFix(Pt)Q(xt), where Pt:C→C is a mapping defined by Pt(x)=tS(x)+(1-t)T(x),∀x∈C,t∈(0,1),
or, equivalently, xt is the unique solution of the quasivariational inequality 0∈(I-Q)xt+NFix(Pt)(xt),
where the normal cone to Fix(Pt), NFix(Pt), is defined as follows: NFix(Pt):x⟶{{u∈H:〈y-x,u〉≤0},ifx∈Fix(Pt),∅,otherwise.

Moreover, as t→0, the net {xt} in turn weakly converges to the unique solution x∞ of the fixed point equation x∞=projΩQ(x∞) or, equivalently, x∞ is the unique solution of the variational inequality 0∈(I-Q)x∞+NΩ(x∞).

Recently, Moudafi [2] constructed an explicit iterative algorithm:xn+1=(1-δn)xn+δn(σnSxn+(1-σn)Txn),∀n≥0,
where {δn} and {σn} are two real numbers in (0,1). By using this iterative algorithm, Moudafi [2] only proved a weak convergence theorem for solving the problem (1.1).

In order to obtain a strong convergence result, Mainge and Moudafi [3] further introduced the following iterative algorithm:xn+1=(1-δn)Qxn+δn[σnSxn+(1-σn)Txn],∀n≥0,
where {δn} and {σn} are two real numbers in (0,1), and proved that, under appropriate conditions, the iterative sequence {xn} generated by (1.8) has strong convergence.

Subsequently, some authors have studied some algorithms on hierarchical fixed problems (see, e.g., [4–15]).

Motivated and inspired by the results in the literature, in this paper, we consider a general hierarchical problem of finding x̃∈Fix(T) such that, for any n≥1,〈Wnx̃-x̃,x-x̃〉≤0,∀x∈Fix(T),
where Wn is the W-mapping defined by (2.3) below and T is a nonexpansive mapping, and introduce an explicit iterative algorithm which converges strongly to a solution x̃ of the hierarchical problem (1.9).

2. Preliminaries

Let C a nonempty closed convex subset of a real Hilbert space H. Recall that a mapping Q:C→C is said to be contractive if there exists a constant γ∈(0,1) such that ‖Qx-Qy‖≤γ‖x-y‖,∀x,y∈C.

A mapping T:C→C is called nonexpansive if‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.

Forward, we use Fix(T) to denote the fixed points set of T.

Let {Ti}i=1∞:C→C be an infinite family of nonexpansive mappings and {ξi}i=1∞ a real number sequence such that 0≤ξi≤1 for each i≥1.

For each n≥1, define a mapping Wn:C→C as follows:Un,n+1=I,Un,n=ξnTnUn,n+1+(1-ξn)I,Un,n-1=ξn-1Tn-1Un,n+(1-ξn-1)I,⋯Un,k=ξkTkUn,k+1+(1-ξk)I,Un,k-1=ξk-1Tk-1Un,k+(1-ξk-1)I,⋯Un,2=ξ2T2Un,3+(1-ξ2)I,Wn=Un,1=ξ1T1Un,2+(1-ξ1)I.

Such Wn is called the W-mapping generated by {Ti}i=1∞ and {ξi}i=1∞.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B18">20</xref>]).

Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Ti}i=1∞ be an infinite family of nonexpansive mappings of C into itself with ⋂n=1∞Fix(Tn)≠∅. Let ξ1,ξ2,… be real numbers such that 0<ξi≤b<1 for each i≥1. Then one has the following results:

for any x∈C and k≥1, the limit limn→∞Un,kx exists;

Fix(W)=⋂n=1∞Fix(Tn).

Using Lemma 3.1 in [21], we can define a mapping W of C into itself by Wx=limn→∞Wnx=limn→∞Un,1x for all x∈C. Thus we have the following.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B19">21</xref>]).

If {xn} is a bounded sequence in C, then one has
limn→∞‖Wxn-Wnxn‖=0.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B20">22</xref>]).

Let C be a nonempty closed convex of a real Hilbert space H and T:C→C be nonexpansive mapping. Then T is demiclosed on C, that is, if xn⇀x∈C and xn-Txn→0, then x=Tx.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B21">23</xref>]).

Assume {an} is a sequence of nonnegative real numbers such that
an+1≤(1-γn)an+δnγn+ηn,∀n≥1,
where {γn} is a sequence in (0,1) and {δn},{ηn} are two sequences such that

∑n=1∞γn=∞;

limsupn→∞δn≤0 or ∑n=1∞|δnγn|<∞;

∑n=1∞|ηn|<∞.

Then limn→∞an=0.

3. Main Results

In this section, we introduce our algorithm and give its convergence analysis.

Algorithm 3.1.

Let C be a nonempty closed convex subset of a real Hilbert space H and {Tn}n=1∞ be infinite family of nonexpansive mappings of C into itself. Let Q:C→C be a contraction with coefficient γ∈[0,1). For any x0∈C, let {xn} the sequence generated iteratively by
xn+1=αnWnxn+(1-αn)T(βnQxn+(1-βn)xn),∀n≥0,
where {αn},{βn} are two real numbers in (0,1) and Wn is the W-mapping defined by (2.3).

Now, we give the convergence analysis of the algorithm.

Theorem 3.2.

Let C be a nonempty closed convex subset of a real Hilbert space H and {Tn}n=1∞ be an infinite family of nonexpansive mappings of C into itself. Let Q:C→C be a contraction with coefficient γ∈[0,1). Assume that the set Ω of solutions of the hierarchical problem (1.9) is nonempty. Let {αn},{βn} be two real numbers in (0,1) and {xn} the sequence generated by (3.1). Assume that the sequence {xn} is bounded and

limn→∞αn=0 and limn→∞(βn/αn)=0;

∑n=0∞βn=∞;

limn→∞(1/βn)|(1/αn)-(1/αn-1)|=0 and limn→∞(∏i=1n-1ξi/αnβn)=limn→∞(1/αn)|1-(βn-1/βn)|=0.

Then limn→∞(∥xn+1-xn∥/αn)=0 and every weak cluster point of the sequence {xn} solves the following variational inequalityx̃∈Ω,〈(I-Q)x̃,x-x̃〉≥0,∀x∈Ω.

Proof.

Set yn=βnQxn+(1-βn)xn for each n≥0. Then we have
yn-yn-1=βnQxn+(1-βn)xn-βn-1Qxn-1-(1-βn-1)xn-1=βn(Qxn-Qxn-1)+(βn-βn-1)Qxn-1+(1-βn)(xn-xn-1)+(βn-1-βn)xn-1.
It follows that
‖yn-yn-1‖≤γβn‖xn-xn-1‖+(1-βn)‖xn-xn-1‖+|βn-βn-1|(‖Qxn-1‖+‖xn-1‖)=[1-(1-γ)βn]‖xn-xn-1‖+|βn-βn-1|(‖Qxn-1‖+‖xn-1‖).
From (3.1), we have
xn+1-xn=αnWnxn+(1-αn)Tyn-αn-1Wn-1xn-1-(1-αn-1)Tyn-1=αn(Wnxn-Wnxn-1)+(αn-αn-1)Wnxn-1+αn-1(Wnxn-1-Wn-1xn-1)+(1-αn)(Tyn-Tyn-1)+(αn-1-αn)Tyn-1.
Then we obtain
‖xn+1-xn‖≤αn‖Wnxn-Wnxn-1‖+(1-αn)‖Tyn-Tyn-1‖+|αn-αn-1|(‖Wnxn-1‖+‖Tyn-1‖)+αn-1‖Wnxn-1-Wn-1xn-1‖≤αn‖xn-xn-1‖+(1-αn)‖yn-yn-1‖+|αn-αn-1|(‖Wnxn-1‖+‖Tyn-1‖)+αn-1‖Wnxn-1-Wn-1xn-1‖.
From (2.3), since Ti and Un,i are nonexpansive, we have
‖Wnxn-1-Wn-1xn-1‖=‖ξ1T1Un,2xn-1-ξ1T1Un-1,2xn-1‖≤ξ1‖Un,2xn-1-Un-1,2xn-1‖=ξ1‖ξ2T2Un,3xn-1-ξ2T2Un-1,3xn-1‖≤ξ1ξ2‖Un,3xn-1-Un-1,3xn-1‖≤⋯≤ξ1ξ2⋯ξn-1‖Un,nxn-1-Un-1,nxn-1‖≤M1∏i=1n-1ξi,
where M1 is a constant such that supn≥1{∥Un,nxn-1-Un-1,nxn-1∥}≤M1. Substituting (3.4) and (3.7) into (3.6), we get
‖xn+1-xn‖≤αn‖xn-xn-1‖+(1-αn)[1-(1-γ)βn]‖xn-xn-1‖+|βn-βn-1|(‖Qxn-1‖+‖xn-1‖)+|αn-αn-1|(‖Wnxn-1‖+‖Tyn-1‖)+αn-1M1∏i=1n-1ξi=[1-(1-γ)βn(1-αn)]‖xn-xn-1‖+|βn-βn-1|(‖Qxn-1‖+‖xn-1‖)+|αn-αn-1|(‖Wnxn-1‖+‖Tyn-1‖)+αn-1M1∏i=1n-1ξi.
Therefore, it follows that
‖xn+1-xn‖αn≤[1-(1-γ)βn(1-αn)]‖xn-xn-1‖αn+|βn-βn-1|αn(‖Qxn-1‖+‖xn-1‖)+|αn-αn-1|αn(‖Wnxn-1‖+‖Tyn-1‖)+αn-1M1∏i=1n-1ξiαn=[1-(1-γ)βn(1-αn)]‖xn-xn-1‖αn-1+[1-(1-γ)βn(1-αn)](‖xn-xn-1‖αn-‖xn-xn-1‖αn-1)+|βn-βn-1|αn(‖Qxn-1‖+‖xn-1‖)+|αn-αn-1|αn(‖Wnxn-1‖+‖Tyn-1‖)+αn-1M1∏i=1n-1ξiαn≤[1-(1-γ)βn(1-αn)]‖xn-xn-1‖αn-1+(|1αn-1αn-1|+|αn-αn-1|αn+|βn-βn-1|αn+∏i=1n-1ξiαn)M=[1-(1-γ)βn(1-αn)]‖xn-xn-1‖αn-1+(1-γ)βn(1-αn)×{M(1-γ)(1-αn)(1βn|1αn-1αn-1|+1βn|αn-αn-1|αn+1βn|βn-βn-1|αn+∏i=1n-1ξiαnβn)},
where M is a constant such that
supn≥1{M1,‖xn-xn-1‖,(‖Wnxn-1‖+‖Tyn-1‖),(‖Qxn-1‖+‖xn-1‖)}≤M.
From (iii), we note that limn→∞(1/αn-1)|αn-αn-1/βnαn|=0, which implies that
limn→∞1βn|αn-αn-1|αn=0.
Thus it follows from (iii) and (3.11) that
limn→∞(1βn|1αn-1αn-1|+1βn|αn-αn-1|αn+1βn|βn-βn-1|αn+∏i=1n-1ξiαnβn)=0.
Hence, applying Lemma 2.4 to (3.9), we immediately conclude that
limn→∞‖xn+1-xn‖αn=0.
This implies that
limn→∞‖xn+1-xn‖=0.
Thus, from (3.1) and (3.14), we have
limn→∞‖xn-Tyn‖=0.
At the same time, we note that
yn-xn=βn(Qxn-xn)⟶0.
Hence we get
limn→∞‖yn-Tyn‖=0.
Since the sequence {xn} is bounded, {yn} is also bounded. Thus there exists a subsequence of {yn}, which is still denoted by {yn} which converges weakly to a point x̃∈H. Therefore, x̃∈Fix(T) by (3.17) and Lemma 2.3. By (3.1), we observe that
xn+1-xn=αn(Wnxn-xn)+(1-αn)(Tyn-yn)+(1-αn)βn(Qxn-xn),
that is,
xn-xn+1αn=(I-Wn)xn+1-αnαn(I-T)yn+βn(1-αn)αn(I-Q)xn.
Set zn=(xn-xn+1)/αn for each n≥1, that is,
zn=(I-Wn)xn+1-αnαn(I-T)yn+βn(1-αn)αn(I-Q)xn.
Using monotonicity of I-T and I-Wn, we derive that, for all u∈Fix(T),
〈zn,xn-u〉=〈(I-Wn)xn,xn-u〉+1-αnαn〈(I-T)yn-(I-T)u,yn-u〉+1-αnαn〈(I-T)yn,xn-yn〉+βn(1-αn)αn〈(I-Q)xn,xn-u〉≥〈(I-Wn)u,xn-u〉+βn(1-αn)αn〈(I-Q)xn,xn-u〉+(1-αn)βnαn〈(I-T)yn,xn-Qxn〉=〈(I-W)u,xn-u〉+〈(W-Wn)u,xn-u〉+βn(1-αn)αn〈(I-Q)xn,xn-u〉+(1-αn)βnαn〈(I-T)yn,xn-Qxn〉.
But, since zn→0,βn/αn→0 and limn→∞∥Wnu-Wu∥=0 (by Lemma 2.2), it follows from the above inequality that
limsupn→∞〈(I-W)u,xn-u〉≤0,∀u∈Fix(T).
This suffices to guarantee that ωw(xn)⊂Ω. As a matter of fact, if we take any x*∈ωw(xn), then there exists a subsequence {xnj} of {xn} such that xnj⇀x*. Therefore, we have
〈(I-W)u,x*-u〉=limj→∞〈(I-W)u,xnj-u〉≤0,∀u∈Fix(T).
Note that x*∈Fix(T). Hence x* solves the following problem:
x*∈Fix(T),〈(I-W)u,x*-u〉≤0,∀u∈Fix(T).
It is obvious that this is equivalent to the problem (1.9) since Wn→W uniformly in any bounded set (by Lemma 2.2). Thus x*∈Ω.

Let x̃ be the unique solution of the variational inequality (3.2). Now, take a subsequence {xni} of {xn} such thatlimsupn→∞〈(I-Q)x̃,xn-x̃〉=limi→∞〈(I-Q)x̃,xni-x̃〉.
Without loss of generality, we may further assume that xni⇀x¯. Then x¯∈Ω. Therefore, we have
limsupn→∞〈(I-Q)x̃,xn-x̃〉=〈(I-Q)x̃,x¯-x̃〉≥0.
This completes the proof.

Theorem 3.3.

Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Tn}n=1∞ be infinite family of nonexpansive mappings of C into itself. Let Q:C→C be a contraction with coefficient γ∈[0,1). Assume that the set Ω of solutions of the hierarchical problem (1.9) is nonempty. Let {αn},{βn} be two real numbers in (0,1) and {xn} the sequence generated by (3.1). Assume that the sequence {xn} is bounded and

limn→∞αn=0, limn→∞βn/αn=0 and limn→∞αn2/βn=0;

∑n=0∞βn=∞;

limn→∞(1/βn)|(1/αn)-(1/αn-1)|=0 and limn→∞∏i=1n-1ξi/αnβn = limn→∞(1/αn)|1-(βn-1/βn)| = 0;

there exists a constant k>0 such that ∥x-Tx∥≥kDist(x,Fix(T)), where

Dist(x,Fix(T))=infy∈Fix(T)‖x-y‖.
Then the sequence {xn} defined by (3.1) converges strongly to a point x̃∈Fix(T), which solves the variational inequality problem (3.2).Proof.

From (3.1), we have
xn+1-x̃=αn(Wnxn-Wnx̃)+αn(Wnx̃-x̃)+(1-αn)(Tyn-x̃).
Thus we have
‖xn+1-x̃‖2≤‖αn(Wnxn-Wnx̃)+(1-αn)(Tyn-x̃)‖2+2αn〈Wnx̃-x̃,xn+1-x̃〉≤(1-αn)‖Tyn-x̃‖2+αn‖Wnxn-Wnx̃‖2+2αn〈Wnx̃-x̃,xn+1-x̃〉≤(1-αn)‖yn-x̃‖2+αn‖xn-x̃‖2+2αn〈Wnx̃-x̃,xn+1-x̃〉.
At the same time, we observe that
‖yn-x̃‖2=‖(1-βn)(xn-x̃)+βn(Qxn-Qx̃)+βn(Qx̃-x̃)‖2≤‖(1-βn)(xn-x̃)+βn(Qxn-Qx̃)‖2+2βn〈Qx̃-x̃,yn-x̃〉≤(1-βn)‖xn-x̃‖2+βn‖Qxn-Qx̃‖2+2βn〈Qx̃-x̃,yn-x̃〉≤(1-βn)‖xn-x̃‖2+βnγ2‖xn-x̃‖2+2βn〈Qx̃-x̃,yn-x̃〉=[1-(1-γ2)βn]‖xn-x̃‖2+2βn〈Qx̃-x̃,yn-x̃〉.
Substituting (3.30) into (3.29), we get
‖xn+1-x̃‖2≤αn‖xn-x̃‖2+(1-αn)[1-(1-γ2)βn]‖xn-x̃‖2+2βn(1-αn)〈Qx̃-x̃,yn-x̃〉+2αn〈Wnx̃-x̃,xn+1-x̃〉=[1-(1-γ2)βn(1-αn)]‖xn-x̃‖2+2βn(1-αn)〈Qx̃-x̃,yn-x̃〉+2αn〈Wnx̃-x̃,xn+1-x̃〉=[1-(1-γ2)βn(1-αn)]‖xn-x̃‖2+(1-γ2)βn(1-αn)×{21-γ2〈Qx̃-x̃,yn-x̃〉+2(1-γ2)(1-αn)×αnβn〈Wnx̃-x̃,xn+1-x̃〉}.
By Theorem 3.2, we note that every weak cluster point of the sequence {xn} is in Ω. Since yn-xn→0, then every weak cluster point of {yn} is also in Ω. Consequently, since x̃=projΩ(Qx̃), we easily have
limsupn→∞〈Qx̃-x̃,yn-x̃〉≤0.

On the other hand, we observe that〈Wnx̃-x̃,xn+1-x̃〉=〈Wnx̃-x̃,projFix(T)xn+1-x̃〉+〈Wnx̃-x̃,xn+1-projFix(T)xn+1〉.
Since x̃ is a solution of the problem (1.9) and projFix(T)xn+1∈Fix(T), we have
〈Wnx̃-x̃,projFix(T)xn+1-x̃〉≤0.
Thus it follows that
〈Wnx̃-x̃,xn+1-x̃〉≤〈Wnx̃-x̃,xn+1-projFix(T)xn+1〉≤‖Wnx̃-x̃‖‖xn+1-projFix(T)xn+1‖=‖Wnx̃-x̃‖×Dist(xn+1,Fix(T))≤1k‖Wnx̃-x̃‖‖xn+1-Txn+1‖.
We note that
‖xn+1-Txn+1‖≤‖xn+1-Txn‖+‖Txn-Txn+1‖≤αn‖Wnxn-Txn‖+(1-αn)‖Tyn-Txn‖+‖xn+1-xn‖≤αn‖Wnxn-Txn‖+‖yn-xn‖+‖xn+1-xn‖≤αn‖Wnxn-Txn‖+βn‖Qxn-xn‖+‖xn+1-xn‖.
Hence we have
αnβn〈Wnx̃-x̃,xn+1-x̃〉≤αn2βn(1k‖Wnx̃-x̃‖‖Wnxn-Txn‖)+αn(1k‖Wnx̃-x̃‖‖Qxn-xn‖)+αn2βn‖xn+1-xn‖αn(1k‖Wnx̃-x̃‖).
From Theorem 3.2, we have limn→∞∥xn+1-xn∥/αn=0. At the same time, we note that {(1/k)∥Wnx̃-x̃∥∥Wnxn-Txn∥}, {(1/k)∥Wnx̃-x̃∥∥Qxn-xn∥}, and {(1/k)∥Wnx̃-x̃∥} are all bounded. Hence it follows from (i) and the above inequality that
limsupn→∞αnβn〈Wnx̃-x̃,xn+1-x̃〉≤0.

Finally, by (3.31)–(3.38) and Lemma 2.4, we conclude that the sequence {xn} converges strongly to a point x̃∈Fix(T). This completes the proof.

Remark 3.4.

In the present paper, we consider the hierarchical problem (1.9) which includes the hierarchical problem (1.1) as a special case.

From the above discussion, we can easily deduce the following result.

Algorithm 3.5.

Let C be a nonempty closed convex subset of a real Hilbert space H and S a nonexpansive mapping of C into itself. Let Q:C→C be a contraction with coefficient γ∈[0,1). For any x0∈C, let{xn} the sequence generated iteratively by
xn+1=αnSxn+(1-αn)T(βnQxn+(1-βn)xn),∀n≥0,
where {αn},{βn} are two real numbers in (0,1).

Corollary 3.6.

Let C be a nonempty closed convex subset of a real Hilbert space H. Let S:C→C be a nonexpansive mapping. Let Q:C→C be a contraction with coefficient γ∈[0,1). Assume that the set Ω′ of solutions of the hierarchical problem (1.1) is nonempty. Let {αn},{βn} be two real numbers in (0,1) and {xn} the sequence generated by (3.1). Assume that the sequence {xn} is bounded and

limn→∞αn=0, limn→∞βn/αn=0 and limn→∞αn2/βn=0;

∑n=0∞βn=∞;

limn→∞(1/βn)|(1/αn)-(1/αn-1)|=0 and limn→∞(1/αn)|1-(βn-1/βn)|=0;

there exists a constant k>0 such that ∥x-Tx∥≥kDist(x,Fix(T)), where

Dist(x,Fix(T))=infy∈Fix(T)‖x-y‖.
Then the sequence {xn} defined by (3.39) converges strongly to a point x̃∈Fix(T), which solves the hierarchical problem (1.1).Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021821).

MoudafiA.MaingeP. E.Towards viscosity approximations of hierarchical fixed-point problemsMoudafiA.Krasnosel'skii-Mann iteration for hierarchical fixed-point problemsMaingeP. E.MoudafiA.Strong convergence of an iterative method for hierarchical fixed-point problemsLuX.XuH. K.YinX.Hybrid methods for a class of monotone variational inequalitiesMarinoG.XuH. K.Explicit hierarchical fixed point approach to variational inequalitiesXuH. K.Viscosity method for hierarchical fixed point approach to variational inequalitiesCianciarusoF.ColaoV.MugliaL.XuH. K.On an implicit hierarchical fixed point approach to variational inequalitiesYaoY.LiouY. C.An implicit extragradient method for hierarchical variational inequalitiesYaoY.LiouY. C.ChenC. P.Hierarchical convergence of a double-net algorithm for equilibrium problems and variational inequality problemsYaoY.ChoY. J.LiouY. C.Iterative algorithms for hierarchical fixed points problems and variational inequalitiesCianciarusoF.MarinoG.MugliaL.YaoY.On a two-step algorithm for hierarchical fixed point problems and variational inequalitiesMarinoG.ColaoV.MugliaL.YaoY.Krasnosel'skii-Mann iteration for hierarchical fixed points and equilibrium problemYaoY.LiouY. C.MarinoG.Two-step iterative algorithms for hierarchical fixed point problems and variational inequality problemsYaoY.ChoY. J.LiouY. C.Hierarchical convergence of an implicit double-net algorithm for nonexpanseive semigroups and variational inequality problemsGouG.WangS.ChoY. J.Strong convergence algorithms for hierarchical fixed point problems and variational inequalitiesYamadaI.OguraN.Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappingsLuoZ. Q.PangJ. S.RalphD.CabotA.Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimizationSolodovM.An explicit descent method for bilevel convex optimizationShimojiK.TakahashiW.Strong convergence to common fixed points of infinite nonexpansive mappings and applicationsYaoY.LiouY. C.YaoJ. C.Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappingsGoebelK.KirkW. A.XuH. K.Iterative algorithms for nonlinear operators