We study the existence and approximation of a solution for a system of hierarchical variational inclusion problems in Hilbert spaces. In this study, we use Maingé’s approach for finding the solutions of the system of hierarchical variational inclusion problems. Our result in this paper improves and generalizes some known corresponding results in the literature.

1. Introduction

Let H be a real Hilbert space with inner product and norm being 〈·,·〉 and ∥·∥, respectively, and let C be a nonempty closed convex subset of H. A mapping T:H→H is called nonexpansive if
(1)Tx-Ty≤x-y,∀x,y∈H.
We use F(T) to denote the set of fixed points of T; that is, F(T)={x∈H:Tx=x}. It is well known that F(T) is a closed convex set, if T is nonexpansive mappings.

A variational inclusion problem [1–3] is the problem of finding a point u∈H such that
(2)θ∈A(u)+M(u),
where A:H→H is a single-valued nonlinear mapping and M:H→2H is a multivalued mapping. We use Ω to denote the set of solutions of the variational inclusion (2).

On the other hand, a hierarchical fixed point problem [4–11] is the problem of finding a point x*∈F(T) such that
(3)Ax*,x-x*≥0,∀x∈F(T).
If the set F(T) is replaced by the solution set of the variational inequality, then the hierarchical fixed point problems are called hierarchical variational inequality problems or hierarchical optimization problems. Many problems in mathematics, for example, the signal recovery [12], the power control problem [13], and the beamforming problem [14], can be considered in the framework of this kind of the hierarchical variational inequality problems.

Recently, Chang et al. [15] introduced bilevel hierarchical variational inclusion problems; that is, find (x*,y*)∈Ω1×Ω2 such that, for given positive real numbers ρ and η, the following inequalities hold:
(4)ρFy*+x*-y*,x-x*≥0,∀x∈Ω1,ηFx*+y*-x*,y-y*≥0,∀y∈Ω2,
where F,A1,A2:H→H are mappings, M1,M2:H→2H are multivalued mappings, and Ωi is the set of solutions to variational inclusion problem (2) with A=Ai, M=Mi for i=1,2. They solved the convex programming problems and quadratic minimization problems by using Maingés scheme.

In this paper, we consider the following system of hierarchical variational inclusion problem: find (x*,y*,z*)∈Ω1×Ω2×Ω3, such that, for given positive real numbers ρ,η, and ξ, the following inequalities hold:
(5)ρFy*+x*-y*,x-x*≥0,∀x∈Ω1,ηFz*+y*-z*,y-y*≥0,∀y∈Ω2,ξFx*+z*-x*,z-z*≥0,∀z∈Ω3.
Some special cases of the system of hierarchical variational inclusion problem (5) are as follows.

If Mi=0, Ai=I-Ti, where Ti:H→H is a nonlinear mapping for each i=1,2,3, in (5), then Ωi=F(Ti) and the system of hierarchical variational inclusion problem (5) reduces to the following system of hierarchical optimization problem: find (x*,y*,z*)∈F(T1)×F(T2)×F(T3), such that
(6)ρFy*+x*-y*,x-x*≥0,∀x∈F(T1),ηFz*+y*-z*,y-y*≥0,∀y∈F(T2),ξFx*+z*-x*,z-z*≥0,∀z∈F(T3),

which was studied by Li [16].

If Ti=PKi for each i=1,2,3, where PKi is the metric projection from H onto a nonempty closed convex subset Ki in (6), then it is clear that the Ωi=F(Ti)=Ki and the system of hierarchical optimization problem (6) reduces to the following system of optimization problem: find (x*,y*,z*)∈K1×K2×K3 such that
(7)ρFy*+x*-y*,x-x*≥0,∀x∈K1,ηFz*+y*-z*,y-y*≥0,∀y∈K2,ξFx*+z*-x*,z-z*≥0,∀z∈K3.

If K1=K2=K3, then the system of optimization problem (7) reduces to the following system of variational inequality problem: find (x*,y*,z*)∈K1×K1×K1 such that
(8)ρFy*+x*-y*,x-x*≥0,∀x∈K1,ηFz*+y*-z*,y-y*≥0,∀y∈K1,ξFx*+z*-x*,z-z*≥0,∀z∈K1.

If ξ=0,ρ,η>0, Ω1=Ω3, and x*=z* in (5) then the system of hierarchical variational inclusion problem (5) reduces to the following bilevel hierarchical variational inclusion problem: find (x*,y*)∈Ω1×Ω2 such that
(9)ρFy*+x*-y*,x-x*≥0,∀x∈Ω1,ηFx*+y*-x*,y-y*≥0,∀y∈Ω2,

which was studied by Chang et al. [15].

In (9), if Mi=0, Ai=I-Ti, for each i=1,2, then bilevel hierarchical variational inclusion problem (9) reduces to the following bilevel hierarchical optimization problem: find (x*,y*)∈F(T1)×F(T2) such that
(10)ρFy*+x*-y*,x-x*≥0,∀x∈F(T1),ηFx*+y*-x*,y-y*≥0,∀y∈F(T2),

which was studied by Maingé [17] and Kraikaew and Saejung [18].

In (10), if Ti=PKi for each i=1,2, then bilevel hierarchical optimization problem (10) reduces to the following problem [19–21]: find (x*,y*)∈K1×K2 such that
(11)ρFy*+x*-y*,x-x*≥0,∀x∈K1,ηFx*+y*-x*,y-y*≥0,∀y∈K2.

In (11), if K1=K2 then the problem (11) reduces to the following problem: find (x*,y*)∈K1×K1 such that
(12)ρFy*+x*-y*,x-x*≥0,∀x∈K1,ηFx*+y*-x*,y-y*≥0,∀y∈K1.

In (5), if ξ=η=0, ρ>0, Ω1=Ω2=Ω3, and x*=y*=z* then the system of hierarchical variational inclusion problem (5) reduces to the following hierarchical variational inclusion problem: find x*∈Ω1 such that
(13)Fy*,x-x*≥0,∀x∈Ω1.

In (13), if M1=0, Ai=I-T1 then the hierarchical variational inclusion problem (13) reduces to the following hierarchical fixed point problem: find x*∈F(T1) such that
(14)Fy*,x-x*≥0,∀x∈F(T1).

In (15), if T1=PK1 then the hierarchical fixed point problem (15) reduces to the following classic variational inequality problem: find x*∈K1 such that
(15)Fy*,x-x*≥0,∀x∈K1.

Motivated and inspired by Chang et al. [15], we introduce the system of a hierarchical variational inclusion problem (5) and investigate a more general variant of the scheme proposed by Chang et al. [15] to solve the system of a hierarchical variational inclusion problem. Our analysis and method allow us to prove the existence and approximation of solutions to the system of a hierarchical variational inclusion problem (5). The results presented in this paper extend and improve the results of Chang et al. [15], Maingé [17], Kraikaew and Saejung [18], and some authors.

2. Preliminaries

This section collects some definitions and lemmas which can be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive. We use → for strong convergence and ⇀ for weak convergence.

Definition 1.

Let A,T,F:H→H be a mapping and let M:H→2H be a multivalued mapping.

A mapping T is called nonexpansive if
(16)Tx-Ty≤x-y,∀x,y∈H.

A mapping T is called quasinonexpansive if F(T)≠∅ and
(17)Tx-p≤x-p,∀x∈H,p∈F(T).

It should be noted that T is quasinonexpansive if and only if for all x∈H, p∈F(T)(18)x-Tx,x-p≥12x-Tx2.

A mapping T is called strongly quasinonexpansive if T is quasinonexpansive and xn-Txn→0, whenever {xn} is a bounded sequence in H and ∥xn-p∥-∥Txn-p∥→0 for some p∈F(T).

A mapping F is called μ-Lipschitzian if there exists α>0 such that
(19)Fx-Fy≤μx-y,∀x,y∈H.

A mapping F is called r-strongly monotone if there exists r>0 such that
(20)Fx-Fy,x-y≥rx-y2,∀x,y∈H.

It is easy to prove that if F:H→H is a μ-Lipschitzian and r-strongly monotone mapping and if ρ∈(0,2r/μ2), then the mapping I-ρF is a contraction.

A mapping A is called α-inverse-strongly monotone if there exists μ>0 such that
(21)Ax-Ay,x-y≥αAx-Ay2,∀x,y∈H.

A multivalued mapping M is called monotone if for all x,y∈H, u∈Mx and v∈My imply that
(22)u-v,x-y≥0.

A multivalued mapping M is called maximal monotone if it is monotone and for any (x,u)∈H×H,
(23)u-v,x-y≥0

for every (y,v)∈Graph(M) (the graph of mapping M) implies that u∈Mx.

Lemma 2 (see [<xref ref-type="bibr" rid="B22">22</xref>]).

Let A:H→H be an α-inverse-strongly monotone mapping. Then

A is an 1/α-Lipschitz continuous and monotone mapping;

for any constant λ>0, one has
(24)(I-λA)x-(I-λA)y2≤x-y2+λ(λ-2α)Ax-Ay2;

if λ∈(0,2α], then I-λA is a nonexpansive mapping, where I is the identity mapping on H.

Lemma 3.

Let x∈H and z∈C be any points. Then one has the following.

That z=PC[x] if and only if there holds the relation:
(25)x-z,y-z≤0,∀y∈C.

That z=PC[x] if and only if there holds the relation:
(26)x-z2≤x-y2-y-z2,∀y∈C.

There holds the relation:
(27)PCx-PCy,x-y≥PCx-PCy2,∀x,y∈H.

Consequently, PC is nonexpansive and monotone.

Definition 4.

Let M:H→2H be a multivalued maximal monotone mapping. Then the mapping JM,λ:H→H defined by
(28)JM,λ(u)=I+λM-1(u),u∈H
is called the resolvent operator associated with M, where λ is any positive number and I is the identity mapping.

Proposition 5 (see [<xref ref-type="bibr" rid="B22">22</xref>]).

Let M:H→2H be a multivalued maximal monotone mapping, and let A:H→H be an α-inverse-strongly monotone mapping. Then the following conclusions hold.

The resolvent operator JM,λ associated with M is single-valued and nonexpansive for all λ>0.

The resolvent operator JM,λ is 1-inverse-strongly monotone; that is,
(29)JM,λ(x)-JM,λ(y)2≤x-y,JM,λx-JM,λy,hhhhhhhhhhhhhhhhh∀x,y∈H.

u∈H is a solution of the variational inclusion (2) if and only if u=JM,λ(u-λAu), for all λ>0; that is, u is a fixed point of the mapping JM,λ(I-λA). Therefore one has
(30)Ω=FJM,λI-λA,∀λ>0,

where Ω is the set of solutions of variational inclusion problem (2).

If λ∈(0,2α], then Ω is a closed convex subset in H.

Lemma 6 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

For x,y∈H and ω∈(0,1), the following statements hold:

∥x+y∥2≤∥x∥2+2〈y,x+y〉;

∥(1-ω)x+ωy∥2=(1-ω)∥x∥2+ω∥y∥2-ω(1-ω)∥x-y∥2.

Lemma 7 (see [<xref ref-type="bibr" rid="B24">24</xref>]).

Let {an} be a sequence of real numbers, and there exists a subsequence {amj} of {an} such that amj<amj+1 for all j∈N, where N is the set of all positive integers. Then there exists a nondecreasing sequence {nk} of N such that limk→∞nk=∞ and the following properties are satisfied by all (sufficiently large) number k∈N:
(31)ank≤ank+1,ak≤ank+1.
In fact, nk is the largest number n in the set {1,2,…,k} such that an<an+1 holds.

Lemma 8 (see [<xref ref-type="bibr" rid="B18">18</xref>]).

Let {an}⊂[0,∞), {αn}⊂[0,1), {bn}⊂(-∞,+∞), and h∈[0,1) be such that

{an} is a bounded sequence;

an+1≤(1-αn)2an+2αnhanan+1+αnbn, for all n≥1;

whenever {ank} is a subsequence of {an} satisfying
(32)liminfk→∞(ank+1-ank)≥0,

it follows that limsupk→∞bnk≤0;

limn→∞αn=0 and ∑n=1∞αn=∞.

Then limn→∞an=0.

Lemma 9 (see [<xref ref-type="bibr" rid="B15">15</xref>]).

Let M:H→2H be a multivalued maximal monotone mapping, let A:H→H be an α-inverse-strongly monotone mapping, and let Ω be the set of solutions of variational inclusion problem (2) and Ω≠∅. Then the following statements hold.

If λ∈(0,2α], then the mapping K:H→H defined by
(33)K∶=JM,λ(I-λA)

is quasinonexpansive, where I is the identity mapping and JM,λ is the resolvent operator associated with M.

The mapping I-K:H→H is demiclosed at zero; that is, for any sequence {xn}⊂H, if xn⇀x and (I-K)xn→0, then x=Kx.

For any β∈(0,1), the mapping Kβ defined by
(34)Kβ=(1-β)I+βK

is a strongly quasinonexpansive mapping and F(Kβ)=F(K).

I-Kβ,β∈(0,1) is demiclosed at zero.

3. Main Results

Throughout this section, we always assume that the following conditions are satisfied:

Mi:H→2H is a multivalued maximal monotone mapping, Ai:H→H is an αi-inverse-strongly monotone mapping, and Ωi is the set of solutions to variational inclusion problem (2) with A=Ai, M=Mi, and Ωi≠∅, for all i=1,2,3;

Ki and Ki,β,β∈(0,1), i=1,2,3, are the mappings defined by
(35)Ki∶=JMi,λ(I-λAi),λ∈(0,2αi],Ki,β∶=(1-β)I+βKi,β∈(0,1),

respectively.

Next, there are our main results.

3.1. An Existence TheoremTheorem 10.

Let Ai, Mi, Ωi, Ki, and Ki,β satisfy conditions (C1) and (C2), and let fi:H→H be contractions with a contractive constant hi∈(0,1), for all i=1,2,3. Then there exists a unique element (x*,y*,z*)∈Ω1×Ω2×Ω3 such that the following three inequalities are satisfied:
(36)x*-f1y*,x-x*≥0,∀x∈Ω1,y*-f2z*,y-y*≥0,∀y∈Ω2,z*-f3x*,z-z*≥0,∀z∈Ω3.

Proof.

The proof is a consequence of Banach’s contraction principle but it is given here for the sake of completeness. By Proposition 5 and Lemma 9, Ω1, Ω2, and Ω3 are nonempty closed and convex. Therefore the metric projection PΩi is well defined for each i=1,2,3.

Since fi is a contraction mapping for each i=1,2,3, then we have PΩifi which is a contraction and also have
(37)PΩ1f1∘PΩ2f2∘PΩ3f3
which is a contraction. Hence there exists a unique element x*∈H such that
(38)x*=(PΩ1f1∘PΩ2f2∘PΩ3f3)x*.
Putting z*=PΩ3f3(x*) and y*=PΩ2f2(z*), then z*∈Ω3, y*∈Ω2, and x*=PΩ1f1(y*).

Suppose that there is an element (x^,y^,z^)∈Ω1×Ω2×Ω3 such that the following three inequalities are satisfied:
(39)x^-f1y^,x-x^≥0,∀x∈Ω1,y^-f2z^,y-y^≥0,∀y∈Ω2,z^-f3x^,z-z^≥0,∀z∈Ω3.
Then
(40)x^=PΩ1f1(y^),y^=PΩ2f2(z^),z^=PΩ3f3(x^).
Therefore
(41)x^=PΩ1f1∘PΩ2f2∘PΩ3f3x^.
This implies that x^=x*, y^=y*, and z^=z*. This completes the proof.

3.2. A Convergence TheoremTheorem 11.

Let Ai, Mi, Ωi, Ki, and Ki,β satisfy conditions (C1) and (C2), and let fi:H→H be contractions with a contractive constant hi∈(0,1), for all i=1,2,3. Let {xn}, {yn}, and {zn} be three sequences defined by
(42)x0,y0,z0∈H,xn+1=1-αnK1,βxn+αnf1K2,βyn,yn+1=1-αnK2,βyn+αnf2K3,βzn,zn+1=(1-αn)K3,βzn+αnf3(K1,βxn),hhhhhhhhhhhhhhhhhhhn=0,1,2,…,
where {αn} is a sequence in (0,1) satisfying αn→0 and ∑n=0∞αn=∞. Then the sequences {xn}, {yn}, and {zn} generated to be (42) converge to x*,y*, and z*, respectively, where (x*,y*,z*) is the unique element in Ω1×Ω2×Ω3 verifying (36).

Proof.

(i) First we prove that sequences {xn}, {yn}, and {zn} are bounded.

From Lemma 9, it follow that Ki,β is strongly quasinonexpansive and F(Ki,β)=F(Ki)=Ωi for each i=1,2,3. Since fi is contraction with the coefficient hi for each i=1,2,3 and x*∈F(K1,β), y*∈F(K2,β), and z*∈F(K3,β), it follows that
(43)xn+1-x*≤(1-αn)K1,βxn-x*+αnf1(K2,βyn)-x*≤(1-αn)xn-x*+αnf1(K2,βyn)-f1(y*)+αnf1(y*)-x*≤(1-αn)xn-x*+αnh1K2,βyn-y*+αnf1(y*)-x*≤(1-αn)xn-x*+αnh1yn-y*+αnf1(y*)-x*≤(1-αn)xn-x*+αnhyn-y*+αnf1(y*)-x*,
where h=max{h1,h2,h3}. Similarly, we can also compute that
(44)yn+1-y*≤(1-αn)yn-y*+αnhzn-z*+αnf2z*-y*,zn+1-z*≤(1-αn)zn-z*+αnhxn-x*+αnf3(x*)-z*.
This implies that
(45)xn+1-x*+yn+1-y*+zn+1-z*≤1-αn1-hxn-x*+yn-y*+zn-z*+αn(1-h)×f1(y*)-x*+f2(z*)-y*+f3(x*)-z*1-h≤maxxn-x*+yn-y*+zn-z*,1-h-1f1y*-x*+f2z*-y*+f3x*-z*×1-h-1.
By induction, we have
(46)xn+1-x*+yn+1-y*+zn+1-z*≤maxx0-x*+y0-y*+z0-z*,1-h-1f1y*-x*+f2z*-y*+f3(x*)-z*×1-h-1,
for all n≥1.

Hence {xn}, {yn}, and {zn} are bounded. Consequently, {K1,βxn}, {K2,βyn}, and {K3,βzn} are bounded.

(ii) Next we prove that for each n≥1 the following inequality holds:
(47)xn+1-x*2+yn+1-y*2+zn+1-z*2≤1-αn2(xn-x*2+yn-y*2+zn-z*2)+2αnhxn+1-x*yn-y*+yn+1-y*×zn-z*+zn+1-z*xn-x*+2αnf1y*-x*,xn+1-x*+f2z*-y*,yn+1-y*+f3x*-z*,zn+1-z*.
From (42) and Lemma 6, we have
(48)xn+1-x*2=(1-αn)(K1,βxn-x*)+αn(f1(K2,βyn)-x*)2≤1-αnK1,βxn-x*2+2αnf1K2,βyn-x*,xn+1-x*=1-αn2K1,βxn-x*2+2αnf1K2,βyn-f1y*,xn+1-x*+2αnf1y*-x*,xn+1-x*≤1-αn2xn-x*2+2αnf1(K2,βyn)-f1(y*)×xn+1-x*+2αnf1y*-x*,xn+1-x*≤1-αn2xn-x*2+2αnh1K2,βyn-y*×xn+1-x*+2αnf1y*-x*,xn+1-x*≤1-αn2xn-x*2+2αnhyn-y*xn+1-x*+2αnf1y*-x*,xn+1-x*.
Similarly, we can also prove that
(49)yn+1-y*2≤1-αn2yn-y*2+2αnhzn-z*yn+1-y*+2αnf2z*-y*,yn+1-y*,zn+1-z*2≤1-αn2zn-z*2+2αnhxn-x*zn+1-z*+2αnf3x*-z*,zn+1-z*.
Adding up inequalities (48) and (49), inequality (47) is proved.

(iii) Next, we prove that if there exists a subsequence {nk}⊂{n} such that
(50)liminfk→∞xnk+1-x*2+ynk+1-y*2+znk+1-z*2-xnk-x*2+ynk-y*2+znk-z*2≥0,
then
(51)limsupk→∞f1y*-x*,xnk+1-x*+f2z*-y*,ynk+1-y*f2z*-y*-y*+f3x*-z*,znk+1-z*≤0.
Since the norm ∥·∥2 is convex and limn→∞αn=0, by (42), we have
(52)0≤liminfk→∞xnk+1-x*2+ynk+1-y*2+znk+1-z*2-xnk-x*2+ynk-y*2+znk-z*2≤liminfk→∞(1-αnk)K1,βxnk-x*2+αnkf1(K2,βynk)-x*2+1-αnkK2,βynk-y*2+αnkf2(K3,βznk)-y*2+(1-αnk)K3,βznk-z*2+αnkf3(K1,βxnk)-z*2-xnk-x*2+ynk-y*2+znk-z*2=liminfk→∞K1,βxnk-x*2-xnk-x*2+(K2,βynk-y*2-ynk-y*2)+K3,βznk-z*2-znk-z*2≤limsupk→∞K1,βxnk-x*2-xnk-x*2+K2,βynk-y*2-ynk-y*2+K3,βznk-z*2-znk-z*2≤0.
This implies that
(53)limk→∞(K1,βxnk-x*2-xnk-x*2)=limk→∞(K2,βynk-y*2-ynk-y*2)=limk→∞(K3,βznk-z*2-znk-z*2)=0.
Since the sequences {∥K1,βxnk-x*∥+∥xnk-x*∥}, {∥K2,βynk-y*∥+∥ynk-y*∥}, and {∥K3,βznk-z*∥+∥znk-z*∥} are bounded, we have
(54)limk→∞K1,βxnk-x*-xnk-x*=limk→∞(K2,βynk-y*-ynk-y*)=limk→∞K3,βznk-z*-znk-z*=0.
By Lemma 9, K1,β, K2,β, and K3,β are strongly quasinonexpansive. We have
(55)K1,βxnk-xnk⟶0,K2,βynk-ynk⟶0,K3,βznk-znk⟶0.
Consequently, we obtain that
(56)xnk-xnk+1⟶0,ynk-ynk+1⟶0,znk-znk+1⟶0.
It follows from the boundedness of {xnk} and H which is reflexive that there exists a subsequence {xnkl} of {xnk} such that xnkl⇀p and
(57)liml→∞f1y*-x*,xnkl-x*=limsupk→∞f1y*-x*,xnk-x*=limsupk→∞f1y*-x*,xnk+1-x*.
By Lemma 9, I-K1,β is demiclosed at zero, and so p∈F(K1,β)=Ω1. Hence from (36) we have
(58)liml→∞f1y*-x*,xnkl-x*=f1y*-x*,p-x*≤0.
Therefore
(59)limsupk→∞f1y*-x*,xnk+1-x*=liml→∞f1y*-x*,xnkl-x*≤0.
Similarly, we can also prove that
(60)limsupk→∞f2z*-y*,ynk+1-y*≤0,limsupk→∞f3x*-z*,znk+1-z*≤0.
Hence, we have the desired inequality.

(iv) Finally, we prove that the sequences {xn}, {yn}, and {zn} generated to be (42) converge to x*,y*, and z*, respectively.

It is clear that
(61)xn+1-x*yn-y*+yn+1-y*zn-z*+zn+1-z*xn-x*≤xn-x*2+yn-y*2+zn-z*21/2×xn+1-x*2+yn+1-y*2+zn+1-z*21/2.
Substituting (61) into (47), we have
(62)xn+1-x*2+yn+1-y*2+zn+1-z*2≤1-αn2xn-x*2+yn-y*2+zn-z*2+2αnhxn-x*2+yn-y*2+zn-z*21/2+zn-z*21/2×xn+1-x*2+yn+1-y*2+zn+1-z*21/2+2αnf1y*-x*,xn+1-x*+f2z*-y*,yn+1-y*+f3x*-z*,zn+1-z*.
Set
(63)an∶=xn-x*2+yn-y*2+zn-z*2,bn∶=2f1y*-x*,xn+1-x*+f2z*-y*,yn+1-y*+f3x*-z*,zn+1-z*.
Then, we have the following statements.

From (i), {an} is bounded sequence.

From (62), an+1≤(1-αn)2an+2αnhanan+1+αnbn, for all n≥1.

From (iii), whenever {ank} is a subsequence of {an} satisfying
(64)liminfk→∞(ank+1-ank)≥0,

it follows that limsupk→∞bnk≤0.

By Lemma 8, we have
(65)limn→∞xn-x*2+yn-y*2+zn-z*2=0.
Hence, we obtain that
(66)limn→∞xn-x*=limn→∞yn-y*=limn→∞zn-z*=0.
This completes the proof.

3.3. Consequence Results

Using Theorem 11, we can prove the following results.

Theorem 12.

Let Ai, Mi, Ωi, Ki, and Ki,β satisfy conditions (C1) and (C2), and let F:H→H be a μ-Lipschitzian and r-strongly monotone mapping. Let {xn}, {yn}, and {zn} be three sequences defined by
(67)x0,y0,z0∈H,xn+1=(1-αn)K1,βxn+αnf1(K2,βyn),yn+1=(1-αn)K2,βyn+αnf2(K3,βzn),zn+1=(1-αn)K3,βzn+αnf3(K1,βxn),hhhhhhhhhhhhhhhhhhhhn=0,1,2,…,
where f1∶=I-ρF, f2∶=I-ηF, f3∶=I-ξF with ρ,η,ξ∈(0,2r/μ2), and {αn} is a sequence in (0,1) satisfying αn→0 and ∑n=0∞αn=∞. Then the sequences {xn}, {yn}, and {zn} converge to x*,y*, and z*, respectively, where (x*,y*,z*) is the unique element in Ω1×Ω2×Ω3 such that the following three inequalities are satisfied:
(68)ρFy*+x*-y*,x-x*≥0,∀x∈Ω1,ηFz*+y*-z*,y-y*≥0,∀y∈Ω2,ξFx*+z*-x*,z-z*≥0,∀z∈Ω3.

Proof.

It is easy to see that f1, f2, and f3 are contraction mappings and all the conditions in Theorem 11 are satisfied. By Theorem 11, we have the sequences {xn}, {yn}, and {zn} which converge to (x*,y*,z*)∈Ω1×Ω2×Ω3 such that the following three inequalities are satisfied:
(69)x*-f1y*,x-x*≥0,∀x∈Ω1,y*-f2z*,y-y*≥0,∀y∈Ω2,z*-f3x*,z-z*≥0,∀z∈Ω3.
Substituting f1∶=I-ρF, f2∶=I-ηF, and f3∶=I-ξF into (69), we obtain that the sequences {xn}, {yn}, and {zn} converge to (x*,y*,z*)∈Ω1×Ω2×Ω3 such that the following three inequalities are satisfied:
(70)ρFy*+x*-y*,x-x*≥0,∀x∈Ω1,ηFz*+y*-z*,y-y*≥0,∀y∈Ω2,ξFx*+z*-x*,z-z*≥0,∀z∈Ω3.
This completes the proof

Setting A1=A2=A3 in Theorem 11, we obtain the following corollary.

Corollary 13.

Let A1, M1, Ω1, K1, and K1,β satisfy conditions (C1) and (C2), and let fi:H→H be contractions with a contractive constant hi∈(0,1), for all i=1,2,3. Let {xn}, {yn}, and {zn} be three sequences defined by
(71)x0,y0,z0∈H,xn+1=(1-αn)K1,βxn+αnf1(K1,βyn),yn+1=1-αnK1,βyn+αnf2K1,βzn,zn+1=(1-αn)K1,βzn+αnf3(K1,βxn),hhhhhhhhhhhhhhhhhhhn=0,1,2,…,
where {αn} is a sequence in (0,1) satisfying αn→0 and ∑n=0∞αn=∞. Then the sequences {xn}, {yn}, and {zn} generated to be (42) converge to x*, y*, and z*, respectively, where (x*,y*,z*) is the unique element in Ω1×Ω1×Ω1 such that the following three inequalities are satisfied:
(72)x*-f1y*,x-x*≥0,∀x∈Ω1,y*-f2z*,x-y*≥0,∀x∈Ω1,z*-f3x*,x-z*≥0,∀x∈Ω1.

Corollary 14.

Let A1, M1, Ω, K1, and K1,β satisfy conditions (C1) and (C2), and let F:H→H be μ-Lipschitzian and r-strongly monotone mapping. Let {xn}, {yn}, and {zn} be three sequences defined by
(73)x0,y0,z0∈H,xn+1=(1-αn)K1,βxn+αnf1(K1,βyn),yn+1=(1-αn)K1,βyn+αnf2(K1,βzn),zn+1=1-αnK1,βzn+αnf3K1,βxn,hhhhhhhhhhhhhhhhhhhn=0,1,2,…,
where f1∶=I-ρF, f2∶=I-ηF, f3∶=I-ξF with ρ,η,ξ∈(0,2r/μ2), and {αn} is a sequence in (0,1) satisfying αn→0 and ∑n=0∞αn=∞. Then the sequences {xn}, {yn}, and {zn} converge to x*, y*, and z*, respectively, where (x*,y*,z*) is the unique element in Ω1×Ω1×Ω1 such that the following three inequalities are satisfied:
(74)ρFy*+x*-y*,x-x*≥0,∀x∈Ω1,ηFz*+y*-z*,x-y*≥0,∀x∈Ω1,ξFx*+z*-x*,x-z*≥0,∀x∈Ω1.

Setting A1=A2=A3, f1=f2=f3, and x0=y0=z0 in Theorem 11, we obtain the following corollary.

Corollary 15.

Let A1, M1, Ω1, K1, and K1,β satisfy conditions (C1) and (C2), and let f:H→H be contractions with a contractive constant h∈(0,1). Let {xn} be the sequences defined by
(75)x0∈H,xn+1=(1-αn)K1,βxn+αnf(K1,βxn),hhhhhhhhhhhhhhhhhhhn=0,1,2,…,
where {αn} is a sequence in (0,1) satisfying αn→0 and ∑n=0∞αn=∞. Then the sequences {xn} converge to x*∈Ω1 such that the following three inequalities are satisfied:
(76)x*-f1x*,x-x*≥0,∀x∈Ω1.

Corollary 16.

Let A1, M1, Ω1, K1, and K1,β satisfy conditions (C1) and (C2), and let F:H→H be μ-Lipschitzian and r-strongly monotone mapping. Let {xn} be the sequences defined by
(77)x0∈H,xn+1=(1-αn)K1,βxn+αn(I-ρF)(K1,βxn),hhhhhhhhhhhhhhhhhhhhhhhhhn=0,1,2,…,
where ρ∈(0,2r/μ2) and {αn} is a sequence in (0,1) satisfying αn→0 and ∑n=0∞αn=∞. Then the sequences {xn} converge to x*∈Ω1 such that the following three inequalities are satisfied:
(78)Fx*,x-x*≥0,∀x∈Ω1.

Conflict of Interests

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

Acknowledgment

The authors would like to thank the National Research University Project of Thailand’s Office of the Higher Education Commission for financial support (NRU no. 57000621).

NoorM. A.NoorK. I.Sensitivity analysis for quasi-variational inclusionsChangS. S.Set-valued variational inclusions in Banach spacesChangS. S.Existence and approximation of solutions of set-valued variational inclusions in Banach spacesYaoY.ChoY. J.LiouY. C.Iterative algorithms for hierarchical fixed points problems and variational inequalitiesMoudafiA.MaingéP.Towards viscosity approximations of hierarchical fixed-point problemsXuH.-K.Viscosity method for hierarchical fixed point approach to variational inequalitiesCianciarusoF.ColaoV.MugliaL.XuH.-K.On an implicit hierarchical fixed point approach to variational inequalitiesMaingéP. E.MoudafiA.Strong convergence of an iterative method for hierarchical fixed-point problemsMoudafiA.Krasnoselski-Mann iteration for hierarchical fixed-point problemsYaoY.LiouY.-C.Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problemsMarinoG.ColaoV.MugliaL.YaoY.Krasnoselski-Mann iteration for hierarchical fixed points and equilibrium problemCombettesP. L.A block-iterative surrogate constraint splitting method for quadratic signal recoveryIidukaH.Fixed point optimization algorithm and its application to power control in CDMA data networksSlavakisK.YamadaI.Robust wideband beamforming by the hybrid steepest descent methodChangS. S.KimJ. K.LeeH. W.ChunC. K.On the hierarchical variational inclusion problems in Hilbert spacesLiY.Improving strong convergence results for hierarchical optimizationMaingéP. E.New approach to solving a system of variational inequalities and hierarchical problemsKraikaewR.SaejungS.On Maingé approach for hierarchical optimization problemsKassayG.KolumbánJ.PálesZ.Factorization of Minty and Stampacchia variational inequality systemsKassayG.KolumbnJ.System of multi-valued variational inequalitiesVermaR. U.Projection methods, algorithms, and a new system of nonlinear variational inequalitiesZhangS.LeeJ. H. W.ChanC. K.Algorithms of common solutions to quasi variational inclusion and fixed point problemsTakahashiW.MaingéP. E.The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces