General Iterative Method for Convex Feasibility
Problem via the Hierarchical Generalized
Variational Inequality Problems
Nopparat Wairojjana and Poom Kumam,
Member, IAENG
Abstract—Let C be a nonempty closed and convex subset of a real Hilbert space H. Let Am, Bm :C → H be relaxed cocoercive mappings for each1≤m≤r, wherer≥1is integer. Let f :C→C be a contraction with coefficient k∈(0,1). Let
G:C→C beξ-strongly monotone andL-Lipschitz continuous mappings. Under the assumption ∩r
m=1GV I(C, Bm, Am) 6= ∅, whereGV I(C, Bm, Am)is the solution set of a generalized vari-ational inequality. Consequently, we prove a strong convergence theorem for finding a point x˜∈ ∩r
m=1GV I(C, Bm, Am) which is a unique solution of the hierarchical generalized variational inequalityh(γf−µG)˜x, x−x˜i ≤0, ∀x∈ ∩r
m=1GV I(C, Bm, Am). Index Terms—Relaxed cocoercive mapping, convex feasibility problem, generalized variational inequality problem, hierarchi-cal generalized variational inequality problem.
I. INTRODUCTION
A
CONVEX feasibility problem, CFP, is the problem of finding a point in the intersection of finitely many closed convex sets in a real Hilbert spaces H. That is, finding an x ∈ ∩rm=1Cm, where r ≥ 1 is an integer and
each Cm is a nonempty closed and convex subset of H.
Many problems in mathematics, for example in physical sciences, in engineering and in real-world applications of various technological innovations can be modeled as CFP. There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [1] computer tomography [2] and radiation therapy treatment planning [3].
LetH be a real Hilbert space with inner product and norm are denoted by h., .i and k.k, respectively and let C be a nonempty closed convex subset ofH.
A mapping T:C→C is callednonexpansive if kT x−T yk ≤ kx−yk, ∀x, y∈C.
We useF(T) to denote the set off ixed points of T, that is, F(T) ={x∈C:T x=x}. It is well known that F(T) is a closed convex set, if T isnonexpansive.
Consider the set of solutions of the following generalized variational inequality:given nonlinear mappingsA, B:C→
H find ax∈C such that
hx−λBxˆ +λAx, x−yi ≥0,∀y∈C, (1)
where λˆ and λ are two positive constants. We use
GV I(C, B, A)to denotethe set of solutions of the generalized
This work was supported by Valaya Alongkorn Rajabhat University under the Royal Patronage.
N. Wairojjana and P. Kumam are with the Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thrung Kru, Bangkok 10140, Thailand e-mails: noparatw@windowslive.com (N. Wairojjana) and poom.kum@kmutt.ac.th (P. Kumam)
variational inequality (1). It is easy to see that an element
x∈ C is a solution to the variational inequality (1) if and only ifxis a fixed point of the mappingPC(ˆλB−λA), where
PC denotes the metric projection fromH ontoC. That is
x=F(PC(ˆλB−λA))x⇔x∈GV I(C, B, A). (2)
Therefore, fixed point algorithms can be applied to solve
GV I(C, B, A). Next, we consider a special case of (1). If
B=I, the identity mapping and ˆλ= 1, then the generalized
variational inequality (1.1) is reduced to the variational inequalityas follow: findx∈C such that
hAx, x−yi ≥0,∀y∈C. (3)
We use V I(C, A) to denote the set of solutions of the variational inequality (3). It is well known that the variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral, and equilibrium problems; which arise in several branches of pure and applied sciences in a unified and general framework. Several numerical methods have been developed for solving variational inequalities and related optimization problems, see [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] and the references therein.
A lot of times, we may need to find a pointx∈GV I(C, B, A) with the property that hF x, x−xi ≤0,∀x∈ GV I(C, B, A)
where GV I(C, B, A) is the solution set of the generalized variational inequality. We will describe this situation by the termhierachical generalized variational inequality problems (HGVIP). If the set GV I(C, B, A) is replaced by the set
V I(C, A), the solution set of the variational inequality, then the HGVIP is called a hierarchical variational inequality problems (HVIP). Many problems in mathematics, for example the signal recovery[16], the power control problem[17] and the beamforming problem[18] can be modeled as HGVIP.
In 2011, Yu and Liang [15] proved the following theorem for finding solutions to the HGVIP for a cocoercive mapping.
Theorem I.1. Let C be a nonempty closed and convex subset of a real Hilbert space H,Am:C →Hbe a relaxed (ηm, ρm)-cocoercive and νm-Lipschitz continuous mapping,
Bm : C → H be a relaxed (ˆηm,ρˆm)-cocoercive and ˆνm
-Lipschitz continuous mapping for each 1≤m ≤r. Assume that ∩r
m=1GV I(C, Bm, Am) 6= ∅. Given {xn} is a sequence
generated by
xn+1=αnu+βnxn+γnΣrm=1δ(m,n)PC(Tmxn), ∀n≥1,
where Tm = ˆλmBm−λmAm, u is fixed element in C and
{αn},{βn},{γn},{δ(1,n)},{δ(2,n)}, ...,{δ(r,n)} are sequences in (0,1), satisfying the following conditions:
(C3) 0<lim infn→∞βn≤lim supn→∞βn<1; (C4) limn→∞δ(m,n)=δm∈(0,1),∀1≤m≤r,
and {λm}rm=1,{λˆm}rm=1 are two positive sequences such
that for each 1≤m≤r
1 ≤
q
1−2λmρm+λ2mνm2 + 2λmηmν2m +
q
1−2ˆλmˆρm+ ˆλ2mˆνm2 + 2ˆλmˆηmˆν2m.
Then the sequence {xn} converges strongly to a common
element x˜ ∈ ∩r
m=1GV I(C, Bm, Am), which is the unique
solution of the following:
hu−x, x˜ −x˜i ≤0, ∀x∈ ∩rm=1GV I(C, Bm, Am). (4)
On the other hand, the hierarchical fixed point problems, i.e, find x∗ ∈ F(T) such that hAx∗, x−x∗i ≥ 0,∀x ∈ F(T),
have attracted many authors attention due to their link with some convex programming problems. See [19], [20], [21], [22], [23], [24], [25], [26]. In 2010, Tian [27] introduced a general iterative method for nonexpansive mappings and proved the following theorem.
Theorem I.2. LetCbe a nonempty closed and convex subset of a real Hilbert space H,f :C→C be a contraction with coefficient k ∈ (0,1), G : C → C be ξ-strongly monotone and L-Lipschitz continuous mapping, Let S : C → C be a nonexpansive mapping with F(S) 6= ∅, ξ > 0, L > 0,
0 < µ < 2ξ/L2 and 0 < γ < µ(ξ−µL2/2)/k = π/k. Given the initial guessx1∈C and{xn}is a sequence generated by
xn+1=αnγf(xn) + (I−αnµG)Sxn, ∀n≥1, (5)
where{αn} is a sequence in (0,1), satisfying the following
conditions:
(C1) limn→∞αn= 0; (C2) Σ∞
n=1αn=∞; (C3) Σ∞
n=1|αn+1−αn|<∞.
Then the sequence {xn} converges strongly to a common
element ˜x ∈ F(S), which is the unique solution of the hierarchical fixed point problem:
h(γf−µG)˜x, x−˜xi ≤0, ∀x∈F(S). (6)
Motivated and inspired by Yu and Liang’s results and Tian’s results, we consider and study the CFP in the case that eachCmis a solution set of generalized variational inequality
GV I(C, Bm, Am) and are devoted to solve the following the
HGVIP: find ˜x∈ ∩r
m=1GV I(C, Bm, Am) such that
h(γf−µG)˜x, x−x˜i ≤0,∀x∈ ∩r
m=1GV I(C, Bm, Am). (7)
Which is the problem (7) is general than the problem (4) and (6). Consequently, we prove a strong convergence theorem for finding a pointx˜which is a unique solution of the HGVIP (7).
II. PRELIMINARIES
This section collects some definitions and lemma which be use in the proofs for the main results in the next section. Some of them are known; others are not hard to derive. Let A: C → H and G : C → C be a nonlinear mappings. Recall the following definitions: for allx, y∈C
(a) Ais said to be monotone if
hAx−Ay, x−yi ≥0.
(b) A is said to be ρ-strongly monotone if there exists a positive real number ρ >0 such that
hAx−Ay, x−yi ≥ρkx−yk2.
(c) A is said to be η-cocoercive if there exists a positive real number η >0 such that
hAx−Ay, x−yi ≥ηkAx−Ayk2.
(d) A is said to be relaxed η-cocoercive if there exists a positive real numberη >0 such that
hAx−Ay, x−yi ≥(−η)kAx−Ayk2.
(e) Ais said to be relaxed(η, ρ)-cocoercive if there exists a positive real numberη, ρ >0 such that
hAx−Ay, x−yi ≥(−η)kAx−Ayk2+ρkx−yk2.
(f) G is said to be L-Lipschitzian on C if there exists a positive real numberL >0such that
kG(x)−G(y)k ≤Lkx−yk.
(g) G is said to be k-contraction if there exists a positive real numberk∈(0,1)such that
kG(x)−G(y)k ≤kkx−yk.
Lemma II.1. [30] Let H be a Hilbert space, C a closed convex subset of H and T : C → C be a nonexpansive mapping with F(T)6=∅. If {xn} is a sequence in C weakly
converging tox and if {(I−T)xn} converges strongly to y,
then(I−T)x=y; in particular, ify= 0thenx∈F(T).
Lemma II.2. [28]Let C be a nonempty closed and convex subset of a real Hilbert space H. Let S1 :C →C and S2 : C→Cbe nonexpansive mappings onC. Suppose thatF(S1)∩ F(S2) is nonempty. Define a mappingS:C→C by
Sx=aS1+ (1−a)S2, ∀x∈C,
whereais a constant in(0,1). Then Sis nonexpansive with F(S) =F(S1)∩F(S2).
Lemma II.3. [27])LetF:C→C be aη-strongly monotone and L-Lipschitzian operator withL >0, η >0. Assume that
0< µ <2η/L2, τ =µ(η−µL2/2) and 0< t <1. Then k(I− µtF)x−(I−µtF)yk ≤(1−tτ)kx−yk.
Lemma II.4. In a real Hilbert space H, we have the equations hold:
(1) kx+yk2≤ kxk2+ 2hy, x+yi, ∀x, y∈H; (2) kx+yk2≥ kxk2+ 2hy, xi, ∀x, y∈H.
Lemma II.5. [29] Assume that {an} is a sequence of
nonnegative numbers such that
an+1≤(1−γn)an+δn, ∀n≥0,
where{γn}is a sequence in(0,1)and{δn}is a sequence in
Rsuch that 1) P∞
n=1γn=∞,
2) lim supn→∞ δn γn≤0or
P∞
n=1|δn|<∞.
Thenlimn→∞an= 0.
Lemma II.6. [27]LetH be a real Hilbert space,f:H→H be a contraction with coefficient 0 < k < 1, and G : H →
H be a L-Lipschitzian continuous operator and ξ-strongly monotone operator withL >0,ξ >0. Then for0< γ < µξ/k and for allx, y∈H,
hx−y,(µG−γf)x−(µG−γf)yi ≥(µξ−γk)kx−yk2.
That is, µG−γf is(µξ−γk)- strongly monotone.
Lemma II.7. [31]LetCbe a closed convex subset ofH. Let
{xn}be a bounded sequence in H. Assume that
(1) The weakω-limit setωw(xn)⊂C,
(2) For eachz∈C,limn→∞kxn−zk exists.
Then{xn} is weakly convergent to a point inC.
III. MAIN RESULT
Theorem III.1. Let C be a nonempty closed and convex subset of a real Hilbert space H such that C ±C ⊂
C. Let f : C → C be a contraction with coefficient k ∈ (0,1). Let G : C → C be a ξ-strongly monotone and L-Lipschitz continuous mapping. let Am : C → H
be a relaxed (ηm, ρm)-cocoercive and νm-Lipschitz
contin-uous mapping and Bm : C → H be a relaxed (ˆηm,ρˆm)
-cocoercive and νˆm-Lipschitz continuous mapping for each 1 ≤ m ≤ r. Let pm =
p
1−2λmρm+λ2mν2m+ 2λmηmνm2
andqm=
q
1−2ˆλmρˆm+ ˆλ2mˆνm2 + 2ˆλmηˆmˆνm2, where{λm}and
{ˆλm}are two positive sequences for each1≤m≤r. Assume
that ∩r
m=1GV I(C, Bm, Am)6=∅, ξ >0, L >0, 0 < µ <2ξ/L2, 0 < γ < µ(ξ−µL2/2)/k=π/k and p
m, qm ∈ [0,12), for each 1 ≤ m ≤ r. Given the initial guess x1 ∈ C and {xn} is a
sequence generated by
xn+1=αnγf(xn) + (I−αnµG)Σrm=1β(m,n)PC(Tmxn), (8)
where Tm = PC(ˆλmBm − λmAm),∀1 ≤ m ≤ r and
{αn},{β(1,n)},{β(2,n)}, ...,{β(r,n)}are sequences in(0,1), sat-isfying the following conditions:
(C1) limn→∞αn= 0,Σ∞n=1αn=∞,Σ∞n=1|αn+1−αn|<∞; (C2) Σr
m=1β(m,n)= 1,∀n≥1,Σn∞=1|β(m,n+1)−β(m,n)|<∞, limn→∞β(m,n)=βm∈(0,1),∀1≤m≤r.
Then the sequence {xn} converges strongly to a common
element x˜ ∈ ∩r
m=1GV I(C, Bm, Am), which is the unique
solution of the HGVIP:
h(γf−µG)˜x, x−˜xi ≤0, ∀x∈ ∩r
m=1GV I(C, Bm, Am). (9)
Proof: For eachx, y∈C and for eachm≥1, we have
kTmx−Tmyk ≤ k(ˆλmBm−λmAm)x−(ˆλmBm−λmAm)yk
≤ k(x−y)−λm(Amx−Amy)k
+k(x−y)−λˆm(Bmx−Bmy)k. (10)
It follows from the assumption that each Am is relaxed (ηm, ρm)-cocoercive andνm-Lipschitz continuous that
k(x−y)−λm(Amx−Amy)k2 = kx−yk2+λ2
mkAmx−Amyk2
−2λmhAmx−Amy, x−yi
≤ kx−yk2−2λm(−ηm)kAmx−Amyk2 +ρmkx−yk2+λ2mν2mkx−yk2
≤ (1−2λmρm+λ2mνm)2 kx−yk2 +2λmηmνm2kx−yk2
= p2mkx−yk2.
This shows that
k(x−y)−λm(Amx−Amy)k ≤pmkx−yk. (11)
In a similar way, we can obtain that
k(x−y)−λˆm(Bmx−Bmy)k ≤qmkx−yk. (12)
Substituting (3.4) and (3.5) into (3.3), we have
kTmx−Tmyk ≤ (pm+qm)kx−yk
≤ kx−yk.
Hence Tm is a nonexpansive mapping and F(Tm) =
F(PC(ˆλmBm−λmAm)) =GV I(C, Bm, Am)for each1≤m≤r.
Put Sn = Σrm=1β(m,n)Tm. By Lemma II.2, we
con-clude that Sn is a nonexpansive mapping and F(Sn) =
∩r
m=1GV I(C, Bm, Am),∀n≥1. We can rewrite the algorithm
(8) as
xn+1=αnγf(xn) + (I−αnµG)Snxn. (13)
Step 1: We will show that{xn} is bounded.
Take v ∈ F(Sn) = ∩r
m=1GV I(C, Bm, Am), from (13) and
lemma II.3, we have
kxn+1−vk = kαnγf(xn) + (I−αnµG)Snxn−vk = kαn(γf(xn)−µGv) + (I−αnµG)Snxn
−(I−αnµG)vk
≤ αnkγ(f(xn)−f(v)) +γf(v)−µGvk +(1−αnπ)kxn−vk
≤ αnγkkxn−vk+αnkγf(v)−µGvk +(1−αnπ)kxn−vk
= (1−αn(π−γk))kxn−vk+αnkγf(v)−µGvk
≤ max
kxn−vk,kγf(v)−µGvk
π−γk
.
By induction, we obtain
kxn−vk ≤max
kx1−vk,kγf(v)−µGvk π−γk
.
Hence{xn}is bounded.
SinceSn is nonexpansive mappings forn≥1, we see that
kSnxn−vk = kSnxn−Snvk
≤ kxn−vk
≤ max
kx1−vk,
kγf(v)−µGvk
π−γk
.
Therefore, {Snxn} is bounded. SinceG is a L-Lipschitz
continuous mapping, we have
kGSnxn−Gvk = kGSnxn−GSnvk
≤ LkSnxn−Snvk
≤ Lkxn−vk
≤ max
Lkx1−vk, L
kγf(v)−µGvk
π−γk
.
Hence{GSnxn}is bounded. Sincef is contraction, sof(xn)
is bounded.
Step 2:We will show that limn→∞kxn+1−xnk= 0.
From (13), we consider
xn+1−xn
= [αnγf(xn) + (I−αnµG)Snxn]
−[αn−1γf(xn−1) + (I−αn−1µG)Sn−1xn−1] = αnγ(f(xn)−f(xn−1)) + [(I−αnµG)Snxn
−(I−αnµG)Sn−1xn−1] + (αn−αn−1)γf(xn−1) +(αn−1−αn)µGSn−1xn−1,
it follows that
kxn+1−xnk
≤ αnγkkxn−xn−1k+ (1−αnπ)kSnxn−Sn−1xn−1k +|αn−αn−1|(γkf(xn−1)k+µkGSn−1xn−1k)
≤ αnγkkxn−xn−1k+ (1−αnπ)kSnxn−Sn−1xn−1k
+|αn−αn−1|M1, (14)
where M1 = supn≥1{γkf(xn)k+µkGSnxnk}. On the other
hand, we note that
kSnxn−Sn−1xn−1k
≤ kSnxn−Snxn−1k+kSnxn−1−Sn−1xn−1k
≤ kxn−xn−1k+kΣrm=1β(m,n)Tmxn−1
−Σr
m=1β(m,n−1)Tmxn−1k
≤ kxn−xn−1k+M2Σmr=1|β(m,n)−β(m,n−1)|, (15)
where M2 is appropriate constant such that M2= max{supn≥1kTmxnk,∀1≤m≤r}.
kxn+1−xnk
≤ αnγkkxn−xn−1k+ (1−αnπ)kxn−xn−1k +M1|αn−αn−1|
+M2Σrm=1|β(m,n)−β(m,n−1)|
≤ αnγkkxn−xn−1k+ (1−αnπ)kxn−xn−1k +M3(|αn−αn−1|+ Σmr=1|β(m,n)−β(m,n−1)|),
where M3 is an appropriate constant such that M3 ≥ max{M1, M2}.
By conditions (C1) and (C2) and Lemma II.5, we obtain that
lim
n→∞kxn+1−xnk= 0. (16) Step 3: We will show thatlimn→∞kSxn−xnk= 0.
Define a mapping S:C→C by
Sx= Σr
m=1βmTmx,∀x∈C,
whereβm= limn→∞β(m,n). From Lemma II.2, we see that S is a nonexpansive mapping and
F(S) =∩r
m=1F(Tm) =∩rm=1GV I(C, Bm, Am),∀n≥1.
From (13), we observe that
kxn+1−Snxnk = αnkγf(xn) +µGSnxnk
≤ αn(γkf(xn)−f(v)k+kγf(v) +µGSnvk +µkGSnxn−GSnvk).
It follows from the condition (C1)and the boundedness of {f(xn)}and{GSnxn}, we obtain that
lim
n→∞kxn+1−Snxnk= 0. (17)
We observe that
kxn−Snxnk = kxn−xn+1+xn+1−Snxnk
≤ kxn−xn+1k+kxn+1−Snxnk.
From (16) and (17), we obtain
lim
n→∞kxn−Snxnk= 0. (18)
Now, we show that Sxn−xn→0 asn→ ∞. Note that
kSxn−xnk = kSxn−Snxn+Snxn−xnk
≤ kΣr
m=1βmTmxn−Σmr=1β(m,n)Tmxnk +kSnxn−xnk
≤ M2(Σrm=1|βm−β(m,n)|) +kSnxn−xnk.
By the condition(C2)and (18), we have
lim
n→∞kxn−Sxnk= 0. (19)
From the boundedness ofxn, we deduced thatxnconverges
weakly in F(S), say xn ⇀ p, by Lemma II.1 and (19), we
obtain p=Sp. So, we have
ωw(xn)⊂F(S). (20)
By Lemma II.6, µG − γf is strongly monotone, so the variational inequality (9) has a unique solution
˜
x∈F(S) =∩r
m=1GV I(C, Bm, Am).
Step 4: We show thatlim supn→∞h(γf−µG)˜x, xn−˜xi ≤0.
Indeed, since {xn} is bounded, then there exists a
subsequence{xni} ⊂ {xn} such that lim sup
n→∞
h(γf−µG)˜x, xn−x˜i= lim
i→∞h(γf−µG)˜x, xni−x˜i.
Without loss of generality, we may further assume thatxni⇀
p. It follows from (20) that p∈F(S). Since ˜xis the unique solution of (9), we obtain
lim sup n→∞
h(γf−µG)˜x, xn−x˜i = lim
i→∞h(γf−µG)˜x, xni−x˜i = h(γf−µG)˜x, p−x˜i ≤0.(21)
Step 5:Finally, we will show that xn→˜xasn→ ∞.
From Lemma II.4, we have
kxn+1−x˜k2 = kαn(γf(xn)−µGx˜) + (I−αnµG)Snxn
−µG(I−αnµG)˜xk2
≤ (1−αnπ)2kxn−˜xk2
+2αnhγf(xn)−µG˜x, xn+1−˜xi
≤ (1−αnπ)2kxn−˜xk2
+2αnγhf(xn)−f(˜x), xn+1−x˜i +2αnhγf(˜x)−µGx, x˜ n+1−x˜i
≤ (1−αnπ)2kxn−˜xk2 +2αnγkkxn−x˜kkxn+1−x˜k +2αnhγf(˜x)−µGx, x˜ n+1−x˜i
≤ (1−αnπ)2kxn−˜xk2
+αnγk(kxn−x˜k2+kxn+1−˜xk2) +2αnhγf(˜x)−µGx, x˜ n+1−x˜i
≤ 1−2αnπ+ (αnπ)
2+α nγk 1−αnγk
kxn−x˜k2
+ 2αn 1−αnγk
hγf(˜x)−µGx, x˜ n+1−x˜i
= [1−2αn(π−γk)
1−αnγk
]kxn−x˜k2
+ (αnπ) 2
1−αnγk
kxn−x˜k2
+ 2αn 1−αnγk
hγf(˜x)−µGx, x˜ n+1−x˜i = (1−θn)kxn−x˜k2+δn,
where θn := 2α1−n(απ−γk)
nγk and δn:= αn
1−αnγk[αnπ
2kx
n−x˜k2+ 2hγf(˜x)−µGx, x˜ n+1−x˜i].
Note that,
θn:=
2αn(π−γk) 1−αnγk
≤2(π−γk)
1−γk αn.
By the condition(C1), we obtain that
lim
n→∞θn= 0. (22)
On the other hand, we have
θn:= 2αn(π−γk) 1−αnγk
≥2αn(π−γk).
From the condition(C1), we have
∞
X
n=1
θn=∞. (23)
PutM= supn∈N{kxn−x˜k}, we have
δn
θn
= 1
2(π−γk)[αnπ
2M+ 2hγf(˜x)−µGx, x˜
n+1−x˜i].
From the condition(C1) and (21), we have
lim sup n→∞
δn
θn
≤0. (24)
Hence, by Lemma II.5, (22), (23) and (24), we conclude that
lim
n→∞kxn−x˜k= 0.
This completes the proof.
IfBm=I, the identity mapping andλˆm= 1, then Theorem
III.1 is reduced to the following result on the classical variational inequality (3).
Corollary III.2. Let C be a nonempty closed and convex subset of a real Hilbert space H such that C±C ⊂C. Let f:C→C be a contraction with coefficientk∈(0,1). LetG:
C→C be aξ-strongly monotone andL-Lipschitz continuous mapping. LetAm:C→H be a relaxed (ηm, ρm)-cocoercive
and νm-Lipschitz continuous mapping, for each 1 ≤ m ≤
r. Let pm =
p
is a positive sequence, for each 1 ≤ m ≤ r. Assume that
∩r
m=1V I(C, Am)6=∅,ξ >0, L >0,0< µ <2ξ/L2,0< γ < µ(ξ− µL2/2)/k=π/k and p
m∈[0,1)], for each1≤m≤r. Given
the initial guessx1∈C and{xn}is a sequence generated by
xn+1=αnγf(xn) + (I−αnµG)Σmr=1β(m,n)PC(xn−λmAmxn),
where {αn},{β(1,n)},{β(2,n)}, ...,{β(r,n)} are sequences in (0,1), satisfying the following conditions:
(C1) limn→∞αn= 0,Σ∞n=1αn=∞,Σ∞n=1|αn+1−αn|<∞; (C2) Σr
m=1β(m,n)= 1,∀n≥1,Σn∞=1|β(m,n+1)−β(m,n)|<∞, limn→∞β(m,n)=βm∈(0,1),∀1≤m≤r,
Then the sequence {xn} converges strongly to a common
element x˜∈ ∩r
m=1V I(C, Am), which is the unique solution of the HVIP:
h(γf−µG)˜x, x−x˜i ≤0, ∀x∈ ∩r
m=1V I(C, Am).
If r= 1, then Theorem III.1 is reduced to the following Corollary.
Corollary III.3. Let C be a nonempty closed and convex subset of a real Hilbert space H such that C ±C ⊂ C. Let f : C → C be a contraction with coefficient k ∈
(0,1). Let G : C → C be a ξ-strongly monotone and L -Lipschitz continuous mapping. Let A:C→H be a relaxed
(η, ρ)-cocoercive and ν-Lipschitz continuous mapping. Let B : C → H be a relaxed (ˆη,ρˆ)-cocoercive and νˆ-Lipschitz continuous mapping. Let p = p1−2λρ+λ2ν2+ 2λην2 and q=
q
1−2ˆλρˆ+ ˆλ2νˆ2+ 2ˆληˆˆν2, whereλand ˆλare two positive
real numbers. Assume that GV I(C, B, A)6= ∅, ξ >0, L > 0,
0< µ <2ξ/L2,0< γ < µ(ξ−µL2/2)/k=π/kand p, q∈[0,1 2). Given the initial guess x1 ∈ C and {xn} is a sequence
generated by
xn+1=αnγf(xn) + (I−αnµG)PC(ˆλBxn−λAxn),
where{αn} is a sequences in(0,1), satisfying the following
conditions:
lim
n→∞αn= 0, Σ ∞
n=1αn=∞ and Σ∞n=1|αn+1−αn|<∞.
Then the sequence {xn} converges strongly to a common
elementx˜∈GV I(C, B, A), which is the unique solution of the HGVIP:
h(γf−µG)˜x, x−x˜i ≤0, ∀x∈GV I(C, B, A).
For the variational inequality (3), we can obtain from Corollary III.3 the following immediately.
Corollary III.4. Let C be a nonempty closed and convex subset of a real Hilbert space H such that C ±C ⊂ C. Let f :C →C be a contraction with coefficient k ∈ (0,1). Let G : C → C be a ξ-strongly monotone and L-Lipschitz continuous mapping. Let A : C → H be a relaxed (η, ρ) -cocoercive and ν-Lipschitz continuous mapping. Let p =
p
1−2λρ+λ2ν2+ 2λην2, where λ is a positive real number. Assume that V I(C, A) 6= ∅, ξ > 0, L > 0, 0 < µ < 2ξ/L2, 0< γ < µ(ξ−µL2/2)/k=π/k and p∈[0,1). Given the initial guess x1∈C and{xn} is a sequence generated by
xn+1=αnγf(xn) + (I−αnµG)PC(xn−λAxn),
where{αn} is a sequences in(0,1), satisfying the following
conditions:
lim
n→∞αn= 0, Σ ∞
n=1αn=∞ and Σ∞n=1|αn+1−αn|<∞.
Then the sequence {xn} converges strongly to a common
element ˜x ∈ V I(C, A), which is the unique solution of the HVIP:
h(γf−µG)˜x, x−x˜i ≤0, ∀x∈V I(C, A).
Remark III.5. (1) If we take G = A and µ = 1, where
A is a strongly positive linear bounded operator on
C in Theorem III.1, then our iterative algorithm de-fine by (8) converges strongly to a common element
˜
x∈ ∩r
m=1GV I(C, Bm, Am), such thath(γf−A)˜x, x−x˜i ≤
0, ∀x ∈ ∩r
m=1GV I(C, Bm, Am), Equivalently, x˜ is the
unique solution to the minimization problem:
min x∈∩r
m=1GV I(C,Bm,Am) 1
2hAx, xi −h(x),
wherehis a potential function forγf(i.e.,h′(x) =γf(x)
forx∈H).
(2) If we takingG=I andγ=µ= 1, whereI is a identity mapping in Theorem III.1, then our iterative algorithm define by (8) converges strongly to a common element
˜
x∈ ∩r
m=1GV I(C, Bm, Am), such that h(f−I)˜x, x−x˜i ≤ 0, ∀x∈ ∩r
m=1GV I(C, Bm, Am).
In case, f = 0, our iterative algorithm define by (8) converges strongly tox˜which is the unique solution to the quadratic minimiztion problem:
z= arg min
x∈∩r
m=1GV I(C,Bm,Am)
kxk2. (25)
In case, f = u, where u is fixed element in C, our iterative algorithm define by (8) converges strongly to a common elementx˜∈ ∩r
m=1GV I(C, Bm, Am), such that
hu−˜x, x−x˜i ≤0, ∀x∈ ∩r
m=1GV I(C, Bm, Am).
(3) Note that, our iterative algorithm define by (8) are more flexible in solving the HGVIP than the one introduced by Yu and Liang.
IV. CONCLUSION
We studied the convex feasibility problem (CFP) in the case that each closed convex set is a solution set of gen-eralized variational inequality and exhibits an algorithm for finding solution of the hierarchical generalized variational inequality problem (HGVIP). The result of this paper extends and generalizes the corresponding results given by Yu and Liang [15] and some authors in the literature.
ACKNOWLEDGMENTS
This work was supported by the Higher Education Re-search Promotion and National ReRe-search University Project of Thailand, Office of the Higher Education Commission and Valaya Alongkorn Rajabhat University under the Royal Patronage.
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