# Iterative Schemes for Convex Minimization Problems with Constraints

Authors: Lu-Chuan Ceng; Cheng-Wen Liao; Chin-Tzong Pang; Ching-Feng Wen
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We first introduce and analyze one implicit iterative algorithm for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: the generalized mixed equilibrium problem, the system of generalized equilibrium problems, and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another implicit iterative algorithm for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.


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## Body

## 1. Introduction

LetC be a nonempty closed convex subset of a real Hilbert space H and let PC be the metric projection of H onto C. Let S:C→H be a nonlinear mapping on C. We denote by Fix⁡(S) the set of fixed points of S and by R the set of all real numbers. A mapping V is called strongly positive on H if there exists a constant γ¯>0 such that
(1)〈Vx,x〉≥γ¯∥x∥2,∀x∈H.
A mapping S:C→H is called L-Lipschitz continuous if there exists a constant L≥0 such that
(2)∥Sx-Sy∥≤L∥x-y∥,∀x,y∈C.
In particular, if L=1, then S is called a nonexpansive mapping; if L∈[0,1), then S is called a contraction.LetA:C→H be a nonlinear mapping on C. We consider the following variational inequality problem (VIP): find a point x∈C such that
(3)〈Ax,y-x〉≥0,∀y∈C.
The solution set of VIP (3) is denoted by VI⁡(C,A).Letφ:C→R be a real-valued function, A:H→H a nonlinear mapping, and Θ:C×C→R a bifunction. The generalized mixed equilibrium problem (GMEP) introduced in [1] is to find x∈C such that
(4)Θ(x,y)+φ(y)-φ(x)+〈Ax,y-x〉≥0,∀y∈C.
We denote the set of solutions of GMEP (4) by GMEP⁡(Θ,φ,A).We assume as in [1] that Θ:C×C→R is a bifunction satisfying conditions (H1)–(H4) and φ:C→R is a lower semicontinuous and convex function with restriction (H5), where(H1)
Θ(x,x)=0 for all x∈C;(H2)
Θ is monotone, that is, Θ(x,y)+Θ(y,x)≤0 for any x,y∈C;(H3)
Θ is upper-hemicontinuous, that is, for each x,y,z∈C,
(5)limsup⁡t→0+Θ(tz+(1-t)x,y)≤Θ(x,y);(H4)
Θ(x,·) is convex and lower semicontinuous for each x∈C;(H5)
for eachx∈H and r>0, there exist a bounded subset Dx⊂C and yx∈C such that, for any z∈C∖Dx,
(6)Θ(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0.Given a positive number,r>0. Let Sr(Θ,φ):H→C be the solution set of the auxiliary mixed equilibrium problem; that is, for each x∈H,
(7)Sr(Θ,φ)(x)∶={1ry∈C:Θ(y,z)+φ(z)-φ(y)+1r〈K′(y)-K′(x),z-y〉≥0,∀z∈C}.
In particular, whenever K(x)=(1/2)∥x∥2, for  all x∈H,Sr(Θ,φ)(x) is rewritten as Tr(Θ,φ)(x).LetΘ1,Θ2:C×C→R be two bifunctions and A1,A2:C→H two nonlinear mappings. Consider the following system of generalized equilibrium problems (SGEP) [2]: find (x*,y*)∈C×C such that
(8)Θ1(x*,x)+〈A1y*,x-x*〉+1ν1〈x*-y*,x-x*〉≥0,∀x∈C,Θ2(y*,y)+〈A2x*,y-y*〉+1ν2〈y*-x*,y-y*〉≥0,∀y∈C,
where ν1>0 and ν2>0 are two constants.In 2010, Ceng and Yao [2] transformed the SGEP into a fixed point problem in the following way.Proposition CY (see [2]).
Let
Θ
1
,
Θ
2
:
C
×
C
→
R
be two bifunctions satisfying conditions (H1)–(H4) and let
A
k
:
C
→
H
be
ζ
k
-inverse-strongly monotone for
k
=
1,2
. Let
ν
k
∈
(
0,2
ζ
k
)
for
k
=
1,2
. Then,
(
x
*
,
y
*
)
∈
C
×
C
is a solution of SGEP (8) if and only if
x
*
is a fixed point of the mapping
G
:
C
→
C
defined by
G
=
T
ν
1
Θ
1
(
I
-
ν
1
A
1
)
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
where
y
*
=
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
x
*
. Here, we denote the fixed point set of
G
by
S
G
E
P
(
G
)
.Let{Tn}n=1∞ be an infinite family of nonexpansive mappings on H and {λn}n=1∞ a sequence of nonnegative numbers in [0,1]. For any n≥1, define a mapping Wn on H as follows:
(9)Un,n+1=I,Un,n=λnTnUn,n+1+(1-λn)I,Un,n-1=λn-1Tn-1Un,n+(1-λn-1)I,⋮Un,k=λkTkUn,k+1+(1-λk)I,Un,k-1=λk-1Tk-1Un,k+(1-λk-1)I,⋮Un,2=λ2T2Un,3+(1-λ2)I,Wn=Un,1=λ1T1Un,2+(1-λ1)I.
Such a mapping Wn is called the W-mapping generated by Tn,Tn-1,…,T1 and λn,λn-1,…,λ1.LetQ:H→H be a contraction and V a strongly positive bounded linear operator on H. Assume that φ:H→R is a lower semicontinuous and convex functional, that Θ,Θ1,Θ2:H×H→R satisfy conditions (H1)–(H4), and that A,A1,A2:H→H are inverse-strongly monotone. Let the mapping G be defined as in Proposition CY. Very recently, Ceng et al. [3] introduced the following hybrid extragradient-like iterative scheme:
(10)zn=Srn(Θ,φ)(xn-rnAxn),xn+1=αn(u+γQxn)+βnxn+((1-βn)I-αn(I+μV))WnGzn,∀n≥0,
for finding a common solution of GMEP (4), SGEP (8), and the fixed point problem of an infinite family of nonexpansive mappings {Tn}n=1∞ on H, where {rn}⊂(0,∞),{αn}, {βn}⊂(0,1), νk∈(0,2ζk),  k=1,2, and x0,u∈H are given. The authors proved the strong convergence of the sequence {xn} to a point x*∈Ω:=∩n=1∞Fix⁡(Tn)∩GMEP⁡(Θ,φ,A)∩SGEP(G) under some suitable conditions. This point x* also solves the following optimization problem:
(OP1)min⁡x∈Ωμ2〈Vx,x〉+12∥x-u∥2-h(x),
where h:H→R is the potential function of γQ.LetB be a single-valued mapping of C into H and R a multivalued mapping with D(R)=C. Consider the following variational inclusion: find a point x∈C such that
(11)0∈Bx+Rx.
We denote by I(B,R) the solution set of the variational inclusion (11). In 1998, Huang [4] studied problem (11) in the case where R is maximal monotone and B is strongly monotone and Lipschitz continuous with D(R)=C=H.Letf:C→R be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing f over the constraint set C:
(12)minimize{f(x):x∈C}.
We denote by Γ the set of minimizers of CMP (12).Very recently, Ceng and Al-Homidan [5] introduced an implicit iterative algorithm for finding a common solution of the CMP (12), finitely many GMEPs and finitely many variational inclusions, and derived its strong convergence under appropriate conditions.Algorithm CA (see [5, Theorem  18]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let M,N be two integers. Let Θk be a bifunction from C×C to R satisfying (H1)–(H4) and φk:C→R∪{+∞} a proper lower semicontinuous and convex function, where k∈{1,2,…,M}. Let Ri:C→2H be a maximal monotone mapping and let Ak:H→H and Bi:C→H be μk-inverse strongly monotone and ηi-inverse strongly monotone, respectively, where k∈{1,2,…,M},  i∈{1,2,…,N}. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<(2η/κ2) and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Let {xn} be a sequence generated by
(13)un=TrM,n(ΘM,φM)(I-rM,nAM)TrM-1,n(ΘM-1,φM-1)×(I-rM-1,nAM-1)⋯Tr1,n(Θ1,φ1)(I-r1,nA1)xn,vn=JRN,λN,n(I-λN,nBN)JRN-1,λN-1,n×(I-λN-1,nBN-1)⋯JR1,λ1,n(I-λ1,nB1)un,xn=snγQxn+(I-snμF)Tnvn,∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn (here Tn is nonexpansive, sn=((2-λnL)/4)∈(0,(1/2)) for each λn∈(0,(2/L))), and the following conditions hold:(i)
sn∈(0,(1/2)) for each λn∈(0,(2/L)), lim⁡n→∞sn=0 (⇔lim⁡n→∞λn=(2/L));(ii)
{λi,n}⊂[ai,bi]⊂(0,2ηi),  for  all  i∈{1,2,…,N};(iii)
{rk,n}⊂[ek,fk]⊂(0,2μk),  for  all  k∈{1,2,…,M}.
Motivated and inspired by the above facts, we first introduce and analyze one implicit iterative algorithm for finding a solution of the CMP (12) with constraints of several problems: the GMEP (4), the SGEP (8), and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. The iterative algorithm is based on Korpelevich's extragradient method, hybrid steepest-descent method in [6], viscosity approximation method, averaged mapping approach to the GPA in [7], and strongly positive bounded linear operator technique. On the other hand, we also propose another implicit iterative algorithm for finding a fixed point of infinitely many nonexpansive mappings with the same constraints. We derive its strong convergence under mild assumptions.


## 2. Preliminaries

Throughout this paper, we assume thatH is a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let C be a nonempty closed convex subset of H. We write xn⇀x to indicate that the sequence {xn} converges weakly to x and xn→x to indicate that the sequence {xn} converges strongly to x. Moreover, we use ωw(xn) to denote the weak ω-limit set of the sequence {xn}; that is,
(14)ωw(xn)∶={x∈H:xni⇀xforsomesubsequence{xni}of{xn}}.Recall that a mappingA:C→H is called(i)
monotone if(15)〈Ax-Ay,x-y〉≥0,∀x,y∈C;(ii)
η-strongly monotone if there exists a constant η>0 such that
(16)〈Ax-Ay,x-y〉≥η∥x-y∥2,∀x,y∈C;(iii)
α-inverse-strongly monotone if there exists a constant α>0 such that
(17)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C.It is obvious that ifA is α-inverse-strongly monotone, then A is monotone and (1/α)-Lipschitz continuous.The metric (or nearest point) projection fromH onto C is the mapping PC:H→C which assigns to each point x∈H the unique point PCx∈C satisfying the property
(18)∥x-PCx∥=inf⁡y∈C∥x-y∥=:d(x,C).
Some important properties of projections are gathered in the following proposition.Proposition 1.
For given
x
∈
H
and
z
∈
C
,
(i)
z
=
P
C
x
⇔
〈
x
-
z
,
y
-
z
〉
≤
0
,  for  all
y
∈
C
;
(ii)
z
=
P
C
x
⇔
∥
x
-
z
∥
2
≤
∥
x
-
y
∥
2
-
∥
y
-
z
∥
2
,  for  all
y
∈
C
;
(iii)
〈
P
C
x
-
P
C
y
,
x
-
y
〉
≥
∥
P
C
x
-
P
C
y
∥
2
,  for  all
y
∈
H
.
Consequently,
P
C
is nonexpansive and monotone.IfA is an α-inverse-strongly monotone mapping of C into H, then it is obvious that A is (1/α)-Lipschitz continuous. If λ≤2α, then it is easy to see that I-λA is a nonexpansive mapping from C to H.Definition 2.
A mappingT:H→H is said to be(a)
nonexpansive if(19)∥Tx-Ty∥≤∥x-y∥,∀x,y∈H;(b)
firmly nonexpansive if2T-I is nonexpansive, or equivalently, if T is 1-inverse strongly monotone (1-ism),
(20)〈x-y,Tx-Ty〉≥∥Tx-Ty∥2,∀x,y∈H; alternatively, T is firmly nonexpansive if and only if T can be expressed as
(21)T=12(I+S),
where S:H→H is nonexpansive; projections are firmly nonexpansive.
It can be easily seen that ifT is nonexpansive, then I-T is monotone. It is also easy to see that a projection PC is 1-ism.Definition 3.
A mappingT:H→H is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping; that is,
(22)T≡(1-α)I+αS,
where α∈(0,1) and S:H→H is nonexpansive. More precisely, when the last equality holds, we say that T is α-averaged. Thus firmly nonexpansive mappings (in particular, projections) are (1/2)-averaged mappings.Proposition 4 (see [8]).
Let
T
:
H
→
H
be a given mapping.
(i)
T
is nonexpansive if and only if the complement
I
-
T
is
(
1
/
2
)
-ism.
(ii)
If
T
is
ν
-ism, then, for
γ
>
0
,
γ
T
is
(
ν
/
γ
)
-ism.
(iii)
T
is averaged if and only if the complement
I
-
T
is
ν
-ism for some
ν
>
1
/
2
. Indeed, for
α
∈
(
0,1
)
,
T
is
α
-averaged if and only if
I
-
T
is
(
1
/
2
α
)
-ism.Proposition 5 (see [8]).
Let
S
,
T
,
V
:
H
→
H
be given operators.
(i)
If
T
=
(
1
-
α
)
S
+
α
V
for some
α
∈
(
0,1
)
and if
S
is averaged and
V
is nonexpansive, then
T
is averaged.
(ii)
T
is firmly nonexpansive if and only if the complement
I
-
T
is firmly nonexpansive.
(iii)
If
T
=
(
1
-
α
)
S
+
α
V
for some
α
∈
(
0,1
)
and if
S
is firmly nonexpansive and
V
is nonexpansive, then
T
is averaged.
(iv)
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings
{
T
i
}
i
=
1
N
is averaged, then so is the composite
T
1
⋯
T
N
. In particular, if
T
1
is
α
1
-averaged and
T
2
is
α
2
-averaged, where
α
1
,
α
2
∈
(
0,1
)
, then the composite
T
1
T
2
is
α
-averaged, where
α
=
α
1
+
α
2
-
α
1
α
2
.
(v)
If the mappings
{
T
i
}
i
=
1
N
are averaged and have a common fixed point, then
(23)
⋂
i
=
1
N
‍
F
i
x
(
T
i
)
=
F
i
x
(
T
1
·
·
·
T
N
)
.
The notation
F
i
x
(
T
)
denotes the set of all fixed points of the mapping
T
; that is,
F
i
x
(
T
)
=
{
x
∈
H
:
T
x
=
x
}
.
By using the technique in [9], we can readily obtain the following elementary result.Proposition 6 (see [3, Lemma 1 and Proposition 1]).
Let
C
be a nonempty closed convex subset of a real Hilbert space
H
and let
φ
:
C
→
R
be a lower semicontinuous and convex function. Let
Θ
:
C
×
C
→
R
be a bifunction satisfying the conditions (H1)–(H4). Assume that
(i)
K
:
H
→
R
is strongly convex with constant
σ
>
0
and the function
x
↦
〈
y
-
x
,
K
′
(
x
)
〉
is weakly upper semicontinuous for each
y
∈
H
;
(ii)
for each
x
∈
H
and
r
>
0
, there exists a bounded subset
D
x
⊂
C
and
y
x
∈
C
such that, for any
z
∈
C
∖
D
x
,
(24)
Θ
(
z
,
y
x
)
+
φ
(
y
x
)
-
φ
(
z
)
+
1
r
〈
K
′
(
z
)
-
K
′
(
x
)
,
y
x
-
z
〉
<
0
.
Then the following hold:
(a)
for each
x
∈
H
,
S
r
(
Θ
,
φ
)
(
x
)
≠
∅
;
(b)
S
r
(
Θ
,
φ
)
is single-valued;
(c)
S
r
(
Θ
,
φ
)
is nonexpansive if
K
′
is Lipschitz continuous with constant
ν
>
0
and
(25)
〈
K
′
(
x
1
)
-
K
′
(
x
2
)
,
u
1
-
u
2
〉
≤
〈
K
′
(
u
1
)
-
K
′
(
u
2
)
,
u
1
-
u
2
〉
,
∀
(
x
1
,
x
2
)
∈
H
×
H
,
where
u
i
=
S
r
(
Θ
,
φ
)
(
x
i
)
for
i
=
1,2
;
(d)
for all
s
,
t
>
0
and
x
∈
H
(26)
〈
K
′
(
S
s
(
Θ
,
φ
)
x
)
-
K
′
(
S
t
(
Θ
,
φ
)
x
)
,
S
s
(
Θ
,
φ
)
x
-
S
t
(
Θ
,
φ
)
x
〉
≤
s
-
t
s
〈
K
′
(
S
s
(
Θ
,
φ
)
x
)
-
K
′
(
x
)
,
S
s
(
Θ
,
φ
)
x
-
S
t
(
Θ
,
φ
)
x
〉
;
(e)
F
i
x
(
S
r
(
Θ
,
φ
)
)
=
M
E
P
(
Θ
,
φ
)
;
(f)
M
E
P
(
Θ
,
φ
)
is closed and convex.Remark 7.
In Proposition4, whenever Θ:C×C→R is a bifunction satisfying the conditions (H1)–(H4) and K(x)=(1/2)∥x∥2,  for  all  x∈H, we have for any x,y∈H,
(27)∥Sr(Θ,φ)x-Sr(Θ,φ)y∥2≤〈Sr(Θ,φ)x-Sr(Θ,φ)y,x-y〉
(Sr(Θ,φ) is firmly nonexpansive) and
(28)∥Ss(Θ,φ)x-St(Θ,φ)x∥≤|s-t|s∥Ss(Θ,φ)x-x∥,hhhhhhhhhhhhhhhhhh∀s,t>0,x∈H.
In this case, Sr(Θ,φ) is rewritten as Tr(Θ,φ). If, in addition, φ≡0, then Tr(Θ,φ) is rewritten as TrΘ; see [2, Lemma 2.1] for more details.We need some facts and tools in a real Hilbert spaceH which are listed as lemmas below.Lemma 8.
LetX be a real inner product space. Then there holds the following inequality:
(29)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈X.Lemma 9.
LetA:C→H be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 1(i)) implies
(30)u∈VI(C,A)⟺u=PC(u-λAu),λ>0.Lemma 10 (see [10, demiclosedness principle]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C. Then I-T is demiclosed. That is, whenever {xn} is a sequence in C weakly converging to some x∈C and the sequence {(I-T)xn} strongly converges to some y, it follows that (I-T)x=y. Here I is the identity operator of H.Lemma 11 (see [11, Lemma 3.2]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Tn}n=1∞ be a sequence of nonexpansive self-mappings on C such that ∩n=1∞Fix(Tn)≠∅ and let {λn} be a sequence in (0,b] for some b∈(0,1). Then, for every x∈C and k≥1 the limit lim⁡n→∞Un,kx exists where Un,k is defined as in (9).Lemma 12 (see [11, Lemma 3.3]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Tn}n=1∞ be a sequence of nonexpansive self-mappings on C such that ∩n=1∞Fix(Tn)≠∅, and let {λn} be a sequence in (0,b] for some b∈(0,1). Then, Fix(W)=∩n=1∞Fix(Tn).The following lemma can be easily proved and, therefore, we omit the proof.Lemma 13.
LetV:H→H be a γ¯-strongly positive bounded linear operator with constant γ¯>1. Then for γ¯-1>0,
(31)〈(V-I)x-(V-I)y,x-y〉≥(γ¯-1)∥x-y∥2,∀x,y∈H.
That is, V-I is strongly monotone with constant γ¯-1.LetC be a nonempty closed convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in (0,1] and let μ>0. Associating with a nonexpansive mapping T:C→H, we define the mapping Tλ:C→H by
(32)Tλx:=Tx-λμF(Tx),∀x∈C,
where F:H→H is an operator such that, for some positive constants κ,η>0, F is κ-Lipschitzian and η-strongly monotone on H; that is, F satisfies the conditions
(33)∥Fx-Fy∥≤κ∥x-y∥,〈Fx-Fy,x-y〉≥η∥x-y∥2
for all x,y∈H.Lemma 14 (see [12, Lemma  3.1]).
T
λ is a contraction provided 0<μ<(2η/κ2); that is,
(34)∥Tλx-Tλy∥≤(1-λτ)∥x-y∥,∀x,y∈C,
where τ=1-1-μ(2η-μκ2)∈(0,1].We observe that sinceF is κ-Lipschitzian and η-strongly monotone on H, we get 0<η≤κ. Hence, whenever 0<μ<(2η/κ2), we have τ=1-1-μ(2η-μκ2)∈(0,1]. Also in Lemma 14, put F=(1/2)I and μ=2. Then we know that κ=η=(1/2),0<μ=2<(2η/κ2)=4 and τ=1.Recall that a set-valued mappingR:D(R)⊂H→2H is called monotone if for all x,y∈D(R), f∈R(x) and g∈R(y) imply
(35)〈f-g,x-y〉≥0.
A set-valued mapping R is called maximal monotone if R is monotone and (I+λR)D(R)=H for each λ>0, where I is the identity mapping of H. We denote by G(R) the graph of R. It is known that a monotone mapping R is maximal if and only if, for (x,f)∈H×H, 〈f-g,x-y〉≥0 for every (y,g)∈G(R) implies f∈R(x).LetA:C→H be a monotone, k-Lipschitz-continuous mapping and let NCv be the normal cone to C at v∈C; that is,
(36)NCv={w∈H:〈v-u,w〉≥0,∀u∈C}.
Define
(37)Tv={Av+NCv,ifv∈C,∅,ifv∉C.
It is well known that T is maximal monotone and 0∈Tv if and only if v∈VI(C,A).Assume thatR:D(R)⊂H→2H is a maximal monotone mapping. Then, for λ>0, associated with R, the resolvent operator JR,λ can be defined as
(38)JR,λx=(I+λR)-1x,∀x∈H.
From Huang [4] (see also [13]), there holds the following property for the resolvent operator JR,λ:H→D(R)¯.Lemma 15.
J
R
,
λ is single-valued and firmly nonexpansive; that is,
(39)〈JR,λx-JR,λy,x-y〉≥∥JR,λx-JR,λy∥2,∀x,y∈H.
Consequently, JR,λ is nonexpansive and monotone.Lemma 16 (see [14]).
LetR be a maximal monotone mapping with D(R)=C. Then for any given λ>0, u∈C is a solution of problem (11) if and only if u∈C satisfies
(40)u=JR,λ(u-λBu).Lemma 17 (see [13]).
LetR be a maximal monotone mapping with D(R)=C and let B:C→H be a strongly monotone, continuous, and single-valued mapping. Then for each z∈H, the equation z∈(B+λR)x has a unique solution xλ for λ>0.Lemma 18 (see [14]).
LetR be a maximal monotone mapping with D(R)=C and B:C→H a monotone, continuous, and single-valued mapping. Then (I+λ(R+B))C=H for each λ>0. In this case, R+B is maximal monotone.


## 3. Convex Minimization Problems with Constraints

In this section, we will introduce and analyze one implicit iterative algorithm for finding a solution of the CMP (12) with constraints of several problems: the GMEP (4), the SGEP (8), and finitely many variational inclusions in a real Hilbert space. We po prove strong convergence theorem for the iterative algorithm under suitable conditions.Theorem 19.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let A:H→H and let Ak,Bi:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone, and ηi-inverse-strongly monotone, respectively, for k=1,2 and i=1,2,…,N. Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<(2η/κ2) and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=GMEP(Θ,φ,A)∩SGEP(G)∩∩i=1NI(Bi,Ri)∩Γ≠∅ where G is defined as in Proposition CY. Let {rn} be a sequence in (0,2ζ] and {αn} a sequence in (0,1] such that lim⁡n→∞αn=0. Let {xn} be the sequence generated by
(41)un=Srn(Θ,φ)(I-rnA)xn,vn=JRN,λN,n(I-λN,nBN)JRN-1,λN-1,n×(I-λN-1,nBN-1)⋯JR1,λ1,n(I-λ1,nB1)un,xn=(I-snV)TnGvn+sn[Tnxn-αn(μF(Tnxn)-γQxn)],∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn (here Tn is nonexpansive, sn=((2-λnL)/4)∈(0,(1/2)) for each λn∈(0,(2/L))). Suppose that the following conditions hold.(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that, for any y∉Dx,
(42)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
sn∈(0,(1/2)) for each λn∈(0,(2/L)), lim⁡n→∞sn=0  (⇔lim⁡n→∞λn=(2/L));(iv)
νk∈(0,2ζk),  k=1,2 and {λi,n}⊂[ai,bi]⊂(0,2ηi),  i=1,2,…,N;(v)
0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ. Assume that Sr(Θ,φ) is firmly nonexpansive. Then {xn} converges strongly as λn→(2/L)(⇔sn→0) to a point q∈Ω, which is a unique solution in Ω to the VIP:
(43)〈(I-V)q,p-q〉≤0,∀p∈Ω.Proof.
First of all, let us show that the sequence{xn} is well defined. Indeed, since ∇f is L-Lipschitzian, it follows that ∇f is 1/L-ism. By Proposition 4(ii) we know that, for λ>0,λ∇f is (1/λL)-ism. So by Proposition 4(iii) we deduce that I-λ∇f is (λL/2)-averaged. Now since the projection PC is (1/2)-averaged, it is easy to see from Proposition 5(iv) that the composite PC(I-λ∇f) is ((2+λL)/4)-averaged for λ∈(0,(2/L)). Hence we obtain that, for each n≥1, PC(I-λn∇f) is ((2+λnL)/4)-averaged for each λn∈(0,(2/L)). Therefore, we can write
(44)PC(I-λn∇f)=2-λnL4I+2+λnL4Tn=snI+(1-sn)Tn,
where Tn is nonexpansive and sn:=sn(λn)=((2-λnL)/4)∈(0,(1/2)) for each λn∈(0,(2/L)). It is clear that
(45)λn⟶2L⟺sn⟶0.
As lim⁡n→∞sn=0 and 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ, we may assume, without loss of generality, that {rn}⊂[a,a^]⊂(0,2ζ) and sn∥V∥≤1 for all n≥1. Since V is a γ¯-strongly positive bounded linear operator on H, we know that
(46)∥V∥=sup⁡{〈Vu,u〉:u∈H,∥u∥=1}≥γ¯>1.
Taking into account that sn∥V∥≤1 for all n≥1, we have
(47)〈(I-snV)u,u〉=1-sn〈Vu,u〉≥1-sn∥V∥≥0,
that is, (1-βn)I-snV is positive. It follows that
(48)∥I-snV∥=sup⁡{〈(I-snV)u,u〉:u∈H,∥u∥=1}=sup⁡{1-sn〈Vu,u〉:u∈H,∥u∥=1}≤1-snγ¯.
Put
(49)Λni=JRi,λi,n(I-λi,nBi)JRi-1,λi-1,n×(I-λi-1,nBi-1)⋯JR1,λ1,n(I-λ1,nB1)
for all i∈{1,2,…,N}, and Λn0=I, where I is the identity mapping on H. Then we have vn=ΛnNun.
Consider the following mappingGn on H defined by
(50)Gnx=(I-snV)TnGΛnNSrn(Θ,φ)(I-rnA)x+sn[Tnx-αn(μF(Tnx)-γQx)],∀x∈H,n≥1,
where sn=((2-λnL)/4)∈(0,(1/2)) for each λn∈(0,(2/L)). Since Ak is ζk-inverse-strongly monotone with 0≤νk≤2ζk for k=1,2, we deduce that. for any x,y∈H,
(51)∥Gx-Gy∥2=∥Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)x-Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)y∥2≤∥(I-ν1A1)Tν2Θ2(I-ν2A2)x-(I-ν1A1)Tν2Θ2(I-ν2A2)y∥2=∥[Tν2Θ2(I-ν2A2)x-Tν2Θ2(I-ν2A2)y]-ν1[A1Tν2Θ2(I-ν2A2)x-A1Tν2Θ2(I-ν2A2)y]∥2≤∥Tν2Θ2(I-ν2A2)x-Tν2Θ2(I-ν2A2)y∥2+ν1(ν1-2ζ1)×∥A1Tν2Θ2(I-ν2A2)x-A1Tν2Θ2(I-ν2A2)y∥2≤∥Tν2Θ2(I-ν2A2)x-Tν2Θ2(I-ν2A2)y∥2≤∥(I-ν2A2)x-(I-ν2A2)y∥2=∥(x-y)-ν2(A2x-A2y)∥2≤∥x-y∥2+ν2(ν2-2ζ2)∥A2x-A2y∥2≤∥x-y∥2.
By the nonexpansivity of Sr(Θ,φ) and Lemma 14 we obtain from (41), (48), and (51) that for all x,y∈C(52)∥Gnx-Gny∥≤∥(I-snV)(TnGΛnNSrn(Θ,φ)(I-rnA))x-TnGΛnNSrn(Θ,φ)(I-rnA)y(I-snV)(TnGΛnNSrn(Θ,φ)(I-rnA)x∥+sn∥(I-αnμF)Tnx-(I-αnμF)Tny∥+snαnγ∥Qx-Qy∥≤(1-snγ¯)×∥TnGΛnNSrn(Θ,φ)(I-rnA)x-TnGΛnNSrn(Θ,φ)(I-rnA)y∥+sn(1-αnτ)∥x-y∥+snαnγl∥x-y∥≤(1-snγ¯)×∥GΛnNSrn(Θ,φ)(I-rnA)x-GΛnNSrn(Θ,φ)(I-rnA)y∥+sn(1-αnτ)∥x-y∥+snαnγl∥x-y∥≤(1-snγ¯)×∥ΛnNSrn(Θ,φ)(I-rnA)x-ΛnNSrn(Θ,φ)(I-rnA)y∥+sn(1-αnτ)∥x-y∥+snαnγl∥x-y∥≤(1-snγ¯)×∥(I-λN,nBN)ΛnN-1Srn(Θ,φ)(I-rnA)x-(I-λN,nBN)ΛnN-1Srn(Θ,φ)(I-rnA)y∥+sn(1-αnτ)∥x-y∥+snαnγl∥x-y∥≤(1-snγ¯)×∥ΛnN-1Srn(Θ,φ)(I-rnA)x-ΛnN-1Srn(Θ,φ)(I-rnA)y∥+sn(1-αnτ)∥x-y∥+snαnγl∥x-y∥⋮≤(1-snγ¯)∥Λn0Srn(Θ,φ)(I-rnA)x-Λn0Srn(Θ,φ)(I-rnA)y∥+sn(1-αnτ)∥x-y∥+snαnγl∥x-y∥=(1-snγ¯)∥Srn(Θ,φ)(I-rnA)x-Srn(Θ,φ)(I-rnA)y∥+sn(1-αnτ)∥x-y∥+snαnγl∥x-y∥≤(1-snγ¯)∥(I-rnA)x-(I-rnA)y∥+sn(1-αnτ)∥x-y∥+snαnγl∥x-y∥≤(1-snγ¯)∥x-y∥+sn(1-αnτ)∥x-y∥+snαnγl∥x-y∥≤(1-sn)∥x-y∥+sn(1-αnτ)∥x-y∥+snαnγl∥x-y∥=(1-snαn(τ-γl))∥x-y∥.
Since 0<1-snαn(τ-γl)<1, Gn:H→H is a contraction. Therefore, by the Banach contraction principle, Gn has a unique fixed point xn∈H, which uniquely solves the fixed point equation
(53)xn=(I-snV)TnGvn+sn[Tnxn-αn(μF(Tnxn)-γQxn)].
This shows that the sequence {xn} is defined well.
Note that(54)〈(V-I)x-(V-I)y,x-y〉≥(γ¯-1)∥x-y∥2,hhhhhhhhhhhhhhhhhhhhhhhhhhhh∀x,y∈H.
That is, V-I is (γ¯-1)-strongly monotone for γ¯-1>0. Moreover, it is clear that V-I is Lipschitz continuous. So the VIP (43) has only one solution in Ω. Below we use q∈Ω to denote the unique solution of the VIP (43).
Now, let us show that{xn} is bounded. In fact, take p∈Ω arbitrarily. Then from the nonexpansivity of Sr(Θ,φ) and {rn}⊂[a,a^]⊂(0,2ζ) we have
(55)∥un-p∥2=∥Srn(Θ,φ)(I-rnA)xn-Srn(Θ,φ)(I-rnA)p∥2≤∥(I-rnA)xn-(I-rnA)p∥2=∥(xn-p)-rn(Axn-Ap)∥2=∥xn-p∥2-2rn〈xn-p,Axn-Ap〉+rn2∥Axn-Ap∥2≤∥xn-p∥2-2rnζ∥Axn-Ap∥2+rn2∥Axn-Ap∥2=∥xn-p∥2+rn(rn-2ζ)∥Axn-Ap∥2≤∥xn-p∥2.
Since p=JRi,λi,n(I-λi,nBi)p, Λnip=p and Bi is ηi-inverse strongly monotone, where λi,n∈(0,2ηi),  i∈{1,2,…,N}, by Lemma 15 we deduce that for each n≥1(56)∥vn-p∥2=∥ΛnNun-p∥2≤∥Λniun-p∥2=∥JRi,λi,n(I-λi,nBi)Λni-1un-JRi,λi,n(I-λi,nBi)p∥2≤∥(I-λi,nBi)Λni-1un-(I-λi,nBi)p∥2≤∥Λni-1un-p∥2+λi,n(λi,n-2ηi)×∥BiΛni-1un-Bip∥2≤∥un-p∥2+λi,n(λi,n-2ηi)∥BiΛni-1un-Bip∥2≤∥un-p∥2.
Combining (55) and (56), we have
(57)∥vn-p∥≤∥xn-p∥.
Since p=Gp=Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)p,Ak is ζk-inverse-strongly monotone for k=1,2, and 0≤νk≤2ζk for k=1,2, we deduce that, for any n≥1,
(58)∥Gvn-p∥2=∥Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)vn-Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)p∥2≤∥(I-ν1A1)Tν2Θ2(I-ν2A2)vn-(I-ν1A1)Tν2Θ2(I-ν2A2)p∥2=∥[Tν2Θ2(I-ν2A2)vn-Tν2Θ2(I-ν2A2)p]-ν1[A1Tν2Θ2(I-ν2A2)vn-A1Tν2Θ2(I-ν2A2)p]∥2≤∥Tν2Θ2(I-ν2A2)vn-Tν2Θ2(I-ν2A2)p∥2+ν1(ν1-2ζ1)×∥A1Tν2Θ2(I-ν2A2)vn-A1Tν2Θ2(I-ν2A2)p∥2≤∥Tν2Θ2(I-ν2A2)vn-Tν2Θ2(I-ν2A2)p∥2≤∥(I-ν2A2)vn-(I-ν2A2)p∥2=∥(vn-p)-ν2(A2vn-A2p)∥2≤∥vn-p∥2+ν2(ν2-2ζ2)∥A2vn-A2p∥2≤∥vn-p∥2.
Thus, utilizing Lemmas 8 and 14, from (41), (43), (55), (56), and (58) we have
(59)∥xn-p∥2=∥(I-snV)(TnGvn-p)+sn[Tnxn-αn(μF(Tnxn)-γQxn)-p]+(I-snV)p-(1-sn)p∥2=∥(I-snV)(TnGvn-p)+sn[αn(γQxn-μFp)+(I-αnμF)Tnxn-(I-αnμF)Tnp]+sn(I-V)p∥2=∥(I-snV)(TnGvn-p)+sn[αnγ(Qxn-Qp)+(I-αnμF)Tnxn-(I-αnμF)Tnp]+sn[(I-V)p+αn(γQ-μF)p]∥2≤∥(I-snV)(TnGvn-p)+sn[αnγ(Qxn-Qp)+(I-αnμF)Tnxn-(I-αnμF)Tnp]∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤[∥(I-snV)(TnGvn-p)∥+sn∥αnγ(Qxn-Qp)+(I-αnμF)Tnxn-(I-αnμF)Tnp∥]2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤[(1-snγ¯)∥Gvn-p∥+sn(αnγl∥xn-p∥+(1-αnτ)∥xn-p∥)]2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤[(1-snγ¯)∥Gvn-p∥+sn(1-αn(τ-γl))∥xn-p∥]2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉=[(1-snγ¯)∥Gvn-p∥+snγ¯(1-αn(τ-γl))γ¯∥xn-p∥]2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-snγ¯)∥Gvn-p∥2+snγ¯(1-αn(τ-γl))2γ¯2∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-snγ¯)∥Gvn-p∥2+sn(1-αn(τ-γl))∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-snγ¯)∥Gvn-p∥2+sn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-snγ¯)∥vn-p∥2+sn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-snγ¯)∥un-p∥2+sn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉=(1-sn(γ¯-1))∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥,
which implies that
(60)∥xn-p∥≤2γ¯-1(∥(I-V)p∥+∥(γQ-μF)p∥).
Hence {xn} is bounded. So, according to (55) and (57) we know that {un},{vn},{Gvn}, {TnGvn},{Tnxn},{Qxn}, and {FTnxn} are bounded.
Next let us show that∥un-xn∥→0, ∥vn-un∥→0, ∥vn-Gvn∥→0 and ∥vn-PC(I-(2/L)∇f)vn∥→0 as n→∞. Indeed, combining (55) and (59), we obtain
(61)∥xn-p∥2≤(1-snγ¯)∥un-p∥2+sn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-snγ¯)[∥xn-p∥2+rn(rn-2ζ)∥Axn-Ap∥2]+sn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-sn(γ¯-1))∥xn-p∥2+(1-snγ¯)rn(rn-2ζ)∥Axn-Ap∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤∥xn-p∥2+(1-snγ¯)rn(rn-2ζ)∥Axn-Ap∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥,
which immediately yields
(62)(1-snγ¯)a(2ζ-a^)∥Axn-Ap∥2≤(1-snγ¯)rn(2ζ-rn)∥Axn-Ap∥2≤2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥.
From lim⁡n→∞sn=0 and the boundedness of {xn} we conclude that
(63)lim⁡n→∞∥Axn-Ap∥=0.
Furthermore, from the firm nonexpansivity of Srn(Θ,φ), we have
(64)∥un-p∥2=∥Srn(Θ,φ)(I-rnA)xn-Srn(Θ,φ)(I-rnA)p∥2≤〈(I-rnA)xn-(I-rnA)p,un-p〉=12[∥(I-rnA)xn-(I-rnA)p∥2+∥un-p∥2-∥(I-rnA)xn-(I-rnA)p-(un-p)∥2]≤12[∥xn-p∥2+∥un-p∥2-∥xn-un-rn(Axn-Ap)∥2]=12[∥xn-p∥2+∥un-p∥2-∥xn-un∥2+2rn〈Axn-Ap,xn-un〉-rn2∥Axn-Ap∥2],
which leads to
(65)∥un-p∥2≤∥xn-p∥2-∥xn-un∥2+2rn∥Axn-Ap∥∥xn-un∥.
From (59) and (65), we get
(66)∥xn-p∥2≤(1-snγ¯)∥un-p∥2+sn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-snγ¯)×[∥xn-p∥2-∥xn-un∥2+2rn∥Axn-Ap∥∥xn-un∥∥xn-p∥2-∥xn-un∥2]+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤(1-sn(γ¯-1))∥xn-p∥2-(1-snγ¯)∥xn-un∥2+2rn∥Axn-Ap∥∥xn-un∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤∥xn-p∥2-(1-snγ¯)∥xn-un∥2+2rn∥Axn-Ap∥∥xn-un∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥,
which hence implies that
(67)(1-snγ¯)∥xn-un∥2≤2rn∥Axn-Ap∥∥xn-un∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤2a^∥Axn-Ap∥∥xn-un∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥.
Since lim⁡n→∞sn=0 and {xn} and {un} are bounded sequences, it follows from (63) that
(68)lim⁡n→∞∥xn-un∥=0.
Next we show thatlim⁡n→∞∥BiΛniun-Bip∥=0,  i=1,2,…,N. As a matter of fact, combining (55), (56), and (59), we have
(69)∥xn-p∥2≤(1-snγ¯)∥vn-p∥2+sn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-snγ¯)[∥BiΛni-1un-Bip∥2∥un-p∥2+λi,n(λi,n-2ηi)×∥BiΛni-1un-Bip∥2]+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤(1-snγ¯)×[∥xn-p∥2+λi,n(λi,n-2ηi)∥BiΛni-1un-Bip∥2]+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥=(1-sn(γ¯-1))∥xn-p∥2+(1-snγ¯)λi,n(λi,n-2ηi)∥BiΛni-1un-Bip∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤∥xn-p∥2+(1-snγ¯)λi,n(λi,n-2ηi)×∥BiΛni-1un-Bip∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥,
which leads to
(70)(1-snγ¯)λi,n(2ηi-λi,n)∥BiΛni-1un-Bip∥2≤2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥.
Since lim⁡n→∞sn=0,{λi,n}⊂[ai,bi]⊂(0,2ηi),  i=1,2,…,N, and {xn} is bounded, it follows that
(71)lim⁡n→∞∥BiΛni-1un-Bip∥=0.
Also, by Lemma 15, we obtain that for each i∈{1,2,…,N}(72)∥Λniun-p∥2=∥JRi,λi,n(I-λi,nBi)Λni-1un-JRi,λi,n(I-λi,nBi)p∥2≤〈(I-λi,nBi)Λni-1un-(I-λi,nBi)p,Λniun-p〉=12(∥(I-λi,nBi)Λni-1un-(I-λi,nBi)p∥2+∥Λniun-p∥2-∥(I-λi,nBi)Λni-1un-(I-λi,nBi)p-(Λniun-p)∥2)≤12(∥Λni-1un-p∥2+∥Λniun-p∥2-∥Λni-1un-Λniun-λi,n(BiΛni-1un-Bip)∥2)≤12(∥un-p∥2+∥Λniun-p∥2-∥Λni-1un-Λniun-λi,n(BiΛni-1un-Bip)∥2),
which yields
(73)∥Λniun-p∥2≤∥un-p∥2-∥Λni-1un-Λniun-λi,n(BiΛni-1un-Bip)∥2=∥un-p∥2-∥Λni-1un-Λniun∥2-λi,n2∥BiΛni-1un-Bip∥2+2λi,n〈Λni-1un-Λniun,BiΛni-1un-Bip〉≤∥un-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥BiΛni-1un-Bip∥.
Thus, utilizing Lemma 8, from (41), (55), (56), (59), and (73) we have
(74)∥xn-p∥2≤(1-snγ¯)∥vn-p∥2+sn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-snγ¯)∥Λniun-p∥2+sn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-snγ¯)×[∥un-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥BiΛni-1un-Bip∥∥un-p∥2-∥Λni-1un-Λniun∥2]+sn∥xn-p∥2+2sn(∥(I-V)p∥+αn∥(γQ-μF)p∥)∥xn-p∥≤(1-snγ¯)[∥xn-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥BiΛni-1un-Bip∥∥xn-p∥2-∥Λni-1un-Λniun∥2]+sn∥xn-p∥2+2sn(∥(I-V)p∥+αn∥(γQ-μF)p∥)×∥xn-p∥≤(1-sn(γ¯-1))×∥xn-p∥2-(1-snγ¯)∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥BiΛni-1un-Bip∥+2sn(∥(I-V)p∥+αn∥(γQ-μF)p∥)∥xn-p∥≤∥xn-p∥2-(1-snγ¯)∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥BiΛni-1un-Bip∥+2sn(∥(I-V)p∥+αn∥(γQ-μF)p∥)∥xn-p∥,
which leads to
(75)(1-snγ¯)∥Λni-1un-Λniun∥2≤2λi,n∥Λni-1un-Λniun∥∥BiΛni-1un-Bip∥+2sn(∥(I-V)p∥+αn∥(γQ-μF)p∥)∥xn-p∥.
Since lim⁡n→∞sn=0,{λi,n}⊂[ai,bi]⊂(0,2ηi),  i=1,2,…,N and {xn} and {un} are bounded sequences, it follows from (71) that
(76)lim⁡n→∞∥Λni-1un-Λniun∥=0.
From (76) we get
(77)∥un-vn∥=∥Λn0un-ΛnNun∥≤∥Λn0un-Λn1un∥+∥Λn1un-Λn2un∥+⋯+∥ΛnN-1un-ΛnNun∥⟶0asn⟶∞.
Taking into account that ∥xn-vn∥≤∥xn-un∥+∥un-vn∥, we conclude from (68) and (77) that
(78)lim⁡n→∞∥xn-vn∥=0.
On the other hand, for simplicity, we writep~=Tν2Θ2(I-ν2A2)p, v~n=Tν2Θ2(I-ν2A2)vn and wn=Gvn=Tν1Θ1(I-ν1A1)v~n for all n≥1. Then
(79)p=Gp=Tν1Θ1(I-ν1A1)p~=Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)p.
We now show thatlim⁡n→∞∥Gvn-vn∥=0, that is, lim⁡n→∞∥wn-vn∥=0. As a matter of fact, for p∈Ω, it follows from (56), (57), and (59) that
(80)∥xn-p∥2≤(1-snγ¯)∥Gvn-p∥2+sn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉=(1-snγ¯)∥wn-p∥2+sn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-snγ¯)∥wn-p∥2+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤(1-snγ¯)×[∥v~n-p~∥2+ν1(ν1-2ζ1)∥A1v~n-A1p~∥2]+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)×∥xn-p∥≤(1-snγ¯)×[∥vn-p∥2+ν2(ν2-2ζ2)∥A2vn-A2p∥2+ν1(ν1-2ζ1)∥A1v~n-A1p~∥2]+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)×∥xn-p∥≤(1-snγ¯)×[∥xn-p∥2+ν2(ν2-2ζ2)∥A2vn-A2p∥2+ν1(ν1-2ζ1)∥A1v~n-A1p~∥2]+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥=(1-sn(γ¯-1))∥xn-p∥2+(1-snγ¯)×[ν2(ν2-2ζ2)∥A2vn-A2p∥2+ν1(ν1-2ζ1)∥A1v~n-A1p~∥2]+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤∥xn-p∥2+(1-snγ¯)×[ν2(ν2-2ζ2)∥A2vn-A2p∥2+ν1(ν1-2ζ1)∥A1v~n-A1p~∥2]+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥,
which immediately implies that
(81)(1-snγ¯)[ν2(2ζ2-ν2)∥A2vn-A2p∥2+ν1(2ζ1-ν1)∥A1v~n-A1p~∥2]≤2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥,
Since lim⁡n→∞sn=0, νk∈(0,2ζk),  k=1,2 and {xn} is bounded, it follows that
(82)lim⁡n→∞∥A2vn-A2p∥=0,lim⁡n→∞∥A1v~n-A1p~∥=0.
Also, in terms of the firm nonexpansivity of TνkΘk and the ζk-inverse strong monotonicity of Ak for k=1,2, we obtain from νk∈(0,2ζk),  k=1,2, and (58) that
(83)∥v~n-p~∥2=∥Tν2Θ2(I-ν2A2)vn-Tν2Θ2(I-ν2A2)p∥2≤〈(I-ν2A2)vn-(I-ν2A2)p,v~n-p~〉=12[∥(I-ν2A2)vn-(I-ν2A2)p∥2+∥v~n-p~∥2-∥(I-ν2A2)vn-(I-ν2A2)p-(v~n-p~)∥2]≤12[∥vn-p∥2+∥v~n-p~∥2-∥(vn-v~n)-ν2(A2vn-A2p)-(p-p~)∥2]=12[∥vn-p∥2+∥v~n-p~∥2-∥(vn-v~n)-(p-p~)∥2+2ν2〈(vn-v~n)-(p-p~),A2vn-A2p〉-ν22∥A2vn-A2p∥2],∥wn-p∥2=∥Tν1Θ1(I-ν1A1)v~n-Tν1Θ1(I-ν1A1)p~∥2≤〈(I-ν1A1)v~n-(I-ν1A1)p~,wn-p〉=12[∥(I-ν1A1)v~n-(I-ν1A1)p~∥2+∥wn-p∥2-∥(I-ν1A1)v~n-(I-ν1A1)p~-(wn-p)∥2]≤12[∥v~n-p~∥2+∥wn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1〈A1v~n-A1p~,(v~n-wn)+(p-p~)〉-ν12∥A1v~n-A1p~∥2]≤12[∥vn-p∥2+∥wn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1〈A1v~n-A1p~,(v~n-wn)+(p-p~)〉∥vn-p∥2+∥wn-p∥2].
Thus, we have
(84)∥v~n-p~∥2≤∥vn-p∥2-∥(vn-v~n)-(p-p~)∥2+2ν2〈(vn-v~n)-(p-p~),A2vn-A2p〉-ν22∥A2vn-A2p∥2,(85)∥wn-p∥2≤∥vn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1∥A1v~n-A1p~∥∥(v~n-wn)+(p-p~)∥.
Consequently, from (57), (80), and (84) it follows that
(86)∥xn-p∥2≤(1-snγ¯)×[∥v~n-p~∥2+ν1(ν1-2ζ1)∥A1v~n-A1p~∥2]+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤(1-snγ¯)×∥v~n-p~∥2+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤(1-snγ¯)×[∥vn-p∥2-∥(vn-v~n)-(p-p~)∥2+2ν2〈(vn-v~n)-(p-p~),A2vn-A2p〉-ν22∥A2vn-A2p∥2]+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤(1-snγ¯)×[∥xn-p∥2-∥(vn-v~n)-(p-p~)∥2+2ν2∥(vn-v~n)-(p-p~)∥∥A2vn-A2p∥∥xn-p∥2-∥(vn-v~n)-(p-p~)∥2]+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤(1-sn(γ¯-1))×∥xn-p∥2-(1-snγ¯)∥(vn-v~n)-(p-p~)∥2+2ν2∥(vn-v~n)-(p-p~)∥∥A2vn-A2p∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤∥xn-p∥2-(1-snγ¯)∥(vn-v~n)-(p-p~)∥2+2ν2∥(vn-v~n)-(p-p~)∥∥A2vn-A2p∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥,
which hence leads to
(87)(1-snγ¯)∥(vn-v~n)-(p-p~)∥2≤2ν2∥(vn-v~n)-(p-p~)∥∥A2vn-A2p∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥.
Since lim⁡n→∞sn=0 and {xn},{vn}, and {v~n} are bounded sequences, we conclude from (82) that
(88)lim⁡n→∞∥(vn-v~n)-(p-p~)∥=0.
Consequently, from (57), (80), and (85) it follows that
(89)∥xn-p∥2≤(1-snγ¯)∥wn-p∥2+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤(1-snγ¯)[∥vn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1∥A1v~n-A1p~∥∥(v~n-wn)+(p-p~)∥∥vn-p∥2-∥(v~n-wn)+(p-p~)∥2]+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)×∥xn-p∥≤(1-snγ¯)×[∥xn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1∥A1v~n-A1p~∥∥(v~n-wn)+(p-p~)∥∥xn-p∥2-∥(v~n-wn)+(p-p~)∥2]+sn∥xn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)×∥xn-p∥≤(1-sn(γ¯-1))×∥xn-p∥2-(1-snγ¯)∥(v~n-wn)+(p-p~)∥2+2ν1∥A1v~n-A1p~∥∥(v~n-wn)+(p-p~)∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥≤∥xn-p∥2-(1-snγ¯)∥(v~n-wn)+(p-p~)∥2+2ν1∥A1v~n-A1p~∥∥(v~n-wn)+(p-p~)∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥,
which hence yields
(90)(1-snγ¯)∥(v~n-wn)+(p-p~)∥2≤2ν1∥A1v~n-A1p~∥∥(v~n-wn)+(p-p~)∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥xn-p∥.
Since lim⁡n→∞sn=0 and {xn},{wn}, and {v~n} are bounded sequences, we conclude from (82) that
(91)lim⁡n→∞∥(v~n-wn)+(p-p~)∥=0.
Note that
(92)∥vn-wn∥≤∥(vn-v~n)-(p-p~)∥+∥(v~n-wn)+(p-p~)∥.
Hence from (88) and (91) we get
(93)lim⁡n→∞∥vn-Gvn∥=lim⁡n→∞∥vn-wn∥=0.
Also, it is clear from (41) that
(94)xn-TnGvn=sn[Tnxn-αn(μF(Tnxn)-γQxn)-VTnGvn].
So, it follows from lim⁡n→∞sn=0 that
(95)lim⁡n→∞∥xn-TnGvn∥=0.
Observe that
(96)∥vn-Tnvn∥≤∥vn-xn∥+∥xn-TnGvn∥+∥TnGvn-Tnvn∥≤∥vn-xn∥+∥xn-TnGvn∥+∥Gvn-vn∥.
Hence, from (78), (93), and (95) we have
(97)lim⁡n→∞∥vn-Tnvn∥=0.
It is easy to see that
(98)∥PC(I-λn∇f)vn-vn∥=∥snvn+(1-sn)Tnvn-vn∥=(1-sn)∥Tnvn-vn∥≤∥Tnvn-vn∥,
where sn=((2-λnL)/4)∈(0,(1/2)) for each λn∈(0,(2/L)). Hence we have
(99)∥PC(I-2L∇f)vn-vn∥≤∥PC(I-2L∇f)vn-PC(I-λn∇f)vn∥+∥PC(I-λn∇f)vn-vn∥≤∥(I-2L∇f)vn-(I-λn∇f)vn∥+∥PC(I-λn∇f)vn-vn∥≤(2L-λn)∥∇f(vn)∥+∥Tnvn-vn∥.
From the boundedness of {vn},sn→0  (⇔λn→(2/L)) and ∥Tnvn-vn∥→0 (due to (97)), it follows that
(100)lim⁡n→∞∥vn-PC(I-2L∇f)vn∥=0.
Furthermore, we show thatωw(xn)⊂Ω. Indeed, since {xn} is bounded, there exists a subsequence {xni}  of  {xn} which converges weakly to some w. Note that lim⁡n→∞∥xn-un∥=0 (due to (68)). Hence uni⇀w. Since C is closed and convex, C is weakly closed. So, we have w∈C. From (68), (76), and (78) we have that uni⇀w, vni⇀w and Λnimuni⇀w, where m∈{1,2,…,N}. First, we prove that w∈∩m=1NI(Bm,Rm). However, the argument to show that w lies in I(Bm,Rm) is quite standard by using maximality and hence is omitted. Next, we show that w∈GMEP⁡(Θ,φ,A). In fact, from un=Srn(Θ,φ)(I-rnA)xn, we know that
(101)Θ(un,y)+φ(y)-φ(un)+〈Axn,y-un〉+1rn〈K′(un)-K′(xn),y-un〉≥0,∀y∈C.
From (H2) it follows that
(102)φ(y)-φ(un)+〈Axn,y-un〉+1rn×〈K′(un)-K′(xn),y-un〉≥Θ(y,un),∀y∈C.
Replacing n by ni, we have
(103)φ(y)-φ(uni)+〈Axni,y-uni〉+〈K′(uni)-K′(xni)rni,y-uni〉≥Θ(y,uni),∀y∈C.
Put ut=ty+(1-t)w for all t∈(0,1] and y∈C. Then, from (103) we have
(104)〈ut-uni,Aut〉≥〈ut-uni,Aut〉-φ(ut)+φ(uni)-〈ut-uni,Axni〉-〈K′(uni)-K′(xni)rni,ut-uni〉+Θ(ut,uni)≥〈ut-uni,Aut-Auni〉+〈ut-uni,Auni-Axni〉-φ(ut)+φ(uni)-〈K′(uni)-K′(xni)rni,ut-uni〉+Θ(ut,uni).
Since ∥uni-xni∥→0 as i→∞, we deduce from the Lipschitz continuity of A and K′ that ∥Auni-Axni∥→0 and ∥K′(uni)-K′(xni)∥→0 as i→∞. Furthermore, from the monotonicity of A, we have 〈ut-uni,Aut-Auni〉≥0. So, from (H4), the weakly lower semicontinuity of φ,((K′(uni)-K′(xni))/rni)→0, and uni⇀w, we have
(105)〈ut-w,Aut〉≥-φ(ut)+φ(w)+Θ(ut,w),asi⟶∞.
From (H1), (H4), and (105) we also have
(106)0=Θ(ut,ut)+φ(ut)-φ(ut)≤tΘ(ut,y)+(1-t)Θ(ut,w)+tφ(y)+(1-t)φ(w)-φ(ut)=t[Θ(ut,y)+φ(y)-φ(ut)]+(1-t)[Θ(ut,w)+φ(w)-φ(w)-φ(ut)]≤t[Θ(ut,y)+φ(y)-φ(ut)]+(1-t)〈ut-w,Aut〉=t[Θ(ut,y)+φ(y)-φ(ut)]+(1-t)t〈y-w,Aut〉,
and hence
(107)0≤Θ(ut,y)+φ(y)-φ(ut)+(1-t)〈y-w,Aut〉.
Letting t→0, we have, for each y∈C,
(108)0≤Θ(w,y)+φ(y)-φ(w)+〈Aw,y-w〉.
This implies that w∈GMEP⁡(Θ,φ,A). In addition, let us show that w∈SGEP(G)∩Γ. As a matter of fact, from (93), (100), vni⇀w, and Lemma 10, we deduce that w∈SGEP(G) and w∈Fix⁡(PC(I-(2/L)∇f))=VI⁡(C,∇f)=Γ. Hence we get w∈SGEP(G)∩Γ. Therefore, w∈GMEP⁡(Θ,φ,A)∩SGEP(G)∩∩i=1NI(Bi,Ri)∩Γ=:Ω. This shows that ωw(xn)⊂Ω.
Finally, we prove that{xn} converges strongly as λn→(2/L) (⇔sn→0) to q∈Ω, which is the unique solution in Ω to the VIP (43). In fact, we note that, for w∈Ω with xni⇀w,
(109)xn-w=(I-snV)(TnGvn-w)+sn[αnγ(Qxn-Qw)+(I-αnμF)Tnxn-(I-αnμF)Tnw]+sn[(I-V)w+αn(γQ-μF)w].
By (48), (57), and Lemma 14, we obtain that
(110)∥xn-w∥2=〈(I-snV)(TnGvn-w),xn-w〉+sn[αnγ〈(Qxn-Qw),xn-w〉+〈(I-αnμF)Tnxn-(I-αnμF)Tnw,xn-w〉]+sn〈(I-V)w+αn(γQ-μF)w,xn-w〉≤(1-snγ¯)∥TnGvn-w∥∥xn-w∥+sn[+(1-αnτ)∥xn-w∥2αnγ∥Qxn-Qw∥∥xn-w∥+(1-αnτ)∥xn-w∥2]+sn〈(I-V)w+αn(γQ-μF)w,xn-w〉≤(1-snγ¯)∥vn-w∥∥xn-w∥+sn[αnγl∥xn-w∥2+(1-αnτ)∥xn-w∥2]+sn〈(I-V)w+αn(γQ-μF)w,xn-w〉≤(1-snγ¯)∥xn-w∥2+sn[αnγl∥xn-w∥2+(1-αnτ)∥xn-w∥2]+sn〈(I-V)w+αn(γQ-μF)w,xn-w〉=(1-snγ¯)∥xn-w∥2+sn(1-αn(τ-γl))∥xn-w∥2+sn〈(I-V)w+αn(γQ-μF)w,xn-w〉≤(1-snγ¯)∥xn-w∥2+sn∥xn-w∥2+sn〈(I-V)w+αn(γQ-μF)w,xn-w〉=(1-sn(γ¯-1))∥xn-w∥2+sn〈(I-V)w+αn(γQ-μF)w,xn-w〉,
which hence leads to
(111)∥xn-w∥2≤〈(I-V)w+αn(γQ-μF)w,xn-w〉γ¯-1.
In particular, we have
(112)∥xni-w∥2≤〈(I-V)w+αni(γQ-μF)w,xni-w〉γ¯-1.
Since xni⇀w and lim⁡n→∞αn=0, it follows from (112) that xni→w as i→∞.
Now we show thatw solves the VIP (43). As a matter of fact, from (55) and (59) we obtain that, for any p∈Ω,
(113)∥xn-p∥2≤(1-snγ¯)∥un-p∥2+sn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-snγ¯)∥xn-p∥2+sn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉=(1-sn(γ¯-1))∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉,
which immediately implies that
(114)〈(V-I)p+αn(μF-γQ)p,xn-p〉≤0.
Since αn→0 and xni→w, we get
(115)〈(V-I)p,w-p〉≤0,∀p∈Ω.
By Minty's lemma, w is a solution in Ω to the VIP (43). In terms of the uniqueness of solutions of VIP (43), we deduce that w=q and xni→q as n→∞. So, every weak convergence subsequence of {xn} converges strongly to the unique solution q of VIP (43). Therefore, {xn} converges strongly to the unique solution q of VIP (43). This completes the proof.Corollary 20.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and let φ:C→R be a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let A:H→H and Ak,Bi:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone, and ηi-inverse-strongly monotone, respectively, for k=1,2 and i=1,2. Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<(2η/κ2) and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=GMEP(Θ,φ,A)∩SGEP(G)∩I(B1,R1)∩I(B2,R2)∩Γ≠∅ where G is defined as in Proposition CY. Let {rn} be a sequence in (0,2ζ] and {αn} a sequence in (0,1] such that lim⁡n→∞αn=0. Let {xn} be the sequence generated by
(116)un=Srn(Θ,φ)(I-rnA)xn,vn=JR2,λ2,n(I-λ2,nB2)JR1,λ1,n(I-λ1,nB1)un,xn=(I-snV)TnGvn+sn[Tnxn-αn(μF(Tnxn)-γQxn)],∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn  (here Tn is nonexpansive, sn=((2-λnL)/4)∈(0,(1/2)) for each λn∈(0,(2/L))). Suppose that the following conditions hold.(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H.(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(117)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0.(iii)
sn∈(0,(1/2)) for each λn∈(0,(2/L)), lim⁡n→∞sn=0  (⇔lim⁡n→∞λn=(2/L)).(iv)
νk∈(0,2ζk),  k=1,2 and {λi,n}⊂[ai,bi]⊂(0,2ηi),  i=1,2.(v)
0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ. Assume that Sr(Θ,φ) is firmly nonexpansive. Then {xn} converges strongly as λn→(2/L)   (⇔sn→0) to a point q∈Ω, which is a unique solution in Ω to the VIP:
(118)〈(I-V)q,p-q〉≤0,∀p∈Ω.Corollary 21.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R a lower semicontinuous and convex functional. Let R:C→2H be a maximal monotone mapping and let A:H→H and Ak,B:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone, and ξ-inverse-strongly monotone, respectively, for k=1,2. Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<(2η/κ2) and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=GMEP(Θ,φ,A)∩SGEP(G)∩I(B,R)∩Γ≠∅ where G is defined as in Proposition CY. Let {rn} be a sequence in (0,2ζ] and {αn} be a sequence in (0,1] such that lim⁡n→∞αn=0. Let {xn} be the sequence generated by
(119)un=Srn(Θ,φ)(I-rnA)xn,vn=JR,ρn(I-ρnB)un,xn=(I-snV)TnGvn+sn[Tnxn-αn(μF(Tnxn)-γQxn)],hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn  (here Tn is nonexpansive, sn=((2-λnL)/4)∈(0,(1/2)) for each λn∈(0,(2/L))). Suppose that the following conditions hold(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(120)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
sn∈(0,(1/2)) for each λn∈(0,(2/L)), lim⁡n→∞sn=0  (⇔lim⁡n→∞λn=(2/L));(iv)
νk∈(0,2ζk),  k=1,2 and {ρn}⊂[a,b]⊂(0,2ξ);(v)
0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ. Assume that Sr(Θ,φ) is firmly nonexpansive. Then {xn} converges strongly as λn→(2/L) (⇔sn→0) to a point q∈Ω, which is a unique solution in Ω to the VIP:
(121)〈(I-V)q,p-q〉≤0,∀p∈Ω.
We next provide one example to illustrate Corollary21.Example 22.
LetH=R2 with inner product 〈·,·〉 and norm ∥·∥ which are defined by
(122)〈x,y〉=ac+bd,∥x∥=a2+b2,
for all x,y∈R2 with x=(a,b) and y=(c,d). Let C={(a,a):a∈R} and K(x)=(1/2)∥x∥2,  for  all  x∈R2. Clearly, C is a nonempty closed convex subset of a real Hilbert space H=R2. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f; for instance, putting S={2/31/31/32/3} and f(x)=(1/2)〈Sx,x〉, we obtain that ∥S∥=1 and ∇f=S with Lipschitz constant L=4. Put Θ=Θ1=Θ2=0 and φ=0. Then Θ,Θ1,Θ2 are three bifunctions from C×C to R satisfying (H1)–(H4) and let φ:C→R be a lower semicontinuous and convex functional. Let R:C→2H be a maximal monotone mapping, for instance, putting
(123)Rv={Sv+NCv,ifv∈C,∅,ifv∉C,
where NCv={w∈H:〈v-u,w〉≥0,  for  all  u∈C}. It is known that R is maximal monotone and 0∈Rv if and only if v∈VI(C,S). Put A={3/52/52/53/5}, B=0, A1=I-S={1/3-1/3-1/31/3}, A2=I-A={2/5-2/5-2/52/5}. Then A:H→H and Ak,B:C→H are ζ-inverse strongly monotone with ζ=(1/2), ζk-inverse strongly monotone with ζk=(1/2) and ξ-inverse-strongly monotone with ξ=(1/2), respectively, for k=1,2. Also, put μ=2, Q=0, V=(3/2)S, and F=(1/2)I, where I is the identity mapping of R2. Then V is a γ¯-strongly positive bounded linear operator with γ¯=(3/2). It is easy to see that Ω:=GMEP⁡(Θ,φ,A)∩SGEP(G)∩I(B,R)∩Γ={0} where G is defined as in Proposition CY. Let {αn}⊂(0,1], {rn}⊂(0,2ζ], and {ρn}⊂(0,2ξ] with ζ=ξ=(1/2); that is, {αn},{ρn},{rn}⊂(0,1]. In this case, for any given x1∈C, the iterative scheme (119) is equivalent to the following one:
(124)un=Srn(Θ,φ)(I-rnA)xn=PC(I-rnA)xn=(1-rn)xn,vn=JR,ρn(I-ρnB)un=JR,ρnun=PC(un-ρnSvn)=un-ρnvn,xn=(I-snV)TnGvn+sn[Tnxn-αn(μF(Tnxn)-γQxn)]=(I-32snS)TnPC(I-ν1A1)PC(I-ν2A2)vn+sn[Tnxn-αnμF(Tnxn)]=(I-32snS)Tnvn+sn[Tnxn-αnTnxn]=(I-32snS)Tnvn+sn(1-αn)Tnxn,∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn and sn=((2-λnL)/4)∈(0,(1/2)) for each λn∈(0,(2/L)) with L=4; that is, sn=((1-2λn)/2)∈(0,(1/2)) for each λn∈(0,(1/2)).
Next, taking into accountvn=JR,ρnun=(I+ρnR)-1un, we get un∈vn+ρnRvn, which leads to
(125)un-vnρn∈Rvn=Svn+NCvn.
Hence, we have 〈vn-u,((un-vn)/ρn)-Svn〉≥0,  for  all  u∈C; that is,
(126)〈u-vn,un-ρnSvn-vn〉≤0,∀u∈C,
which immediately implies that vn=PC(un-ρnSvn)=un-ρnvn. Thus,
(127)vn=11+ρnun=1-rn1+ρnxn.
Also, note that
(128)(1-λn)xn=PC(I-λn∇f)xn=snxn+(1-sn)Tnxn.
So, we obtain
(129)Tnxn=1-sn-λn1-snxn.
On the other hand, we have
(130)(1-λn)vn=PC(I-λn∇f)vn=snvn+(1-sn)Tnvn,
and hence
(131)Tnvn=1-sn-λn1-snvn=1-sn-λn1-sn·1-rn1+ρnxn,
which together with (124) implies that
(132)xn=(I-32snS)Tnvn+sn(1-αn)Tnxn=(I-32snS)1-sn-λn1-sn1-rn1+ρnxn+sn(1-αn)1-sn-λn1-snxn=(1-32sn)1-sn-λn1-sn1-rn1+ρnxn+sn(1-αn)1-sn-λn1-snxn=[(1-32sn)1-rn1+ρn+sn(1-αn)]1-sn-λn1-snxn.
It is clear that if sn=((1-2λn)/2)=((1-2λn)/2)∈(0,(1/2)) for each λn∈(0,(1/2)), then, for {rn},{αn},{ρn}⊂(0,1], we deduce from (132) that
(133)∥xn∥=[(1-32sn)1-rn1+ρn+sn(1-αn)]1-sn-λn1-sn∥xn∥≤(1-32sn)1-rn1+ρn+sn(1-αn)∥xn∥≤(1-32sn+sn)∥xn∥=(1-12sn)∥xn∥.
Therefore, xn=0 for all n≥1. There is no doubt that {xn} converges to the unique element 0 in Ω, which solves the VIP (121).


## 4. Fixed Point Problems with Constraints

In this section, we will introduce and analyze another implicit iterative algorithm for solving the fixed point problem of infinitely many nonexpansive mappings with constraints of several problems: the GMEP (4), the SGEP (8), and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under mild assumptions.Theorem 23.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let A:H→H and Ak,Bi:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone, and ηi-inverse-strongly monotone, respectively, for k=1,2 and i=1,2,…,N. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and {λn} a sequence in (0,b] for some b∈(0,1). Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<(2η/κ2) and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=∩n=1∞Fix(Tn)∩GMEP(Θ,φ,A)∩SGEP(G)∩∩i=1NI(Bi,Ri)≠∅ where G is defined as in Proposition CY. Let {rn} be a sequence in (0,2ζ] and let {αn} and {σn} be sequences in (0,1]. Let {xn} be the sequence generated by
(134)un=Srn(Θ,φ)(I-rnA)xn,vn=JRN,λN,n(I-λN,nBN)JRN-1,λN-1,n×(I-λN-1,nBN-1)⋯JR1,λ1,n(I-λ1,nB1)un,xn=(I-σnV)WnGvn+σn[Wnxn-αn(μF(Wnxn)-γQxn)],∀n≥1,
where Wn is the W-mapping defined by (9). Suppose that the following conditions hold:(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(135)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
lim⁡n→∞αn=0 and lim⁡n→∞σn=0;(iv)
νk∈(0,2ζk),k=1,2 and {λi,n}⊂[ai,bi]⊂(0,2ηi),  i=1,2,…,N;(v)
0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ. Assume that Sr(Θ,φ) is firmly nonexpansive. Then {xn} converges strongly to a point q∈Ω, which is a unique solution in Ω to the VIP:
(136)〈(I-V)q,p-q〉≤0,∀p∈Ω.Proof.
First of all, let us show that the sequence{xn} is well defined. As lim⁡n→∞σn=0 and 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ, we may assume, without loss of generality, that {rn}⊂[a,a^]⊂(0,2ζ) and σn∥V∥≤1 for all n≥1. Utilizing the arguments similar to those in the proof of Theorem 19, we get
(137)∥I-σnV∥≤1-σnγ¯.
Put
(138)Λni=JRi,λi,n(I-λi,nBi)JRi-1,λi-1,n×(I-λi-1,nBi-1)⋯JR1,λ1,n(I-λ1,nB1)
for all i∈{1,2,…,N}, and Λn0=I, where I is the identity mapping on H. Then we have vn=ΛnNun.
Consider the following mappingGn on H defined by
(139)Gnx=(I-σnV)TnGΛnNSrn(Θ,φ)(I-rnA)x+σn[Tnx-αn(μF(Tnx)-γQx)],∀x∈H,n≥1.
Since Ak is ζk-inverse-strongly monotone with 0≤νk≤2ζk for k=1,2, repeating the same arguments as in the proof of Theorem 19 we deduce that G is a nonexpansive mapping on H. Utilizing the arguments similar to those in the proof of Theorem 19, we deduce that, for any x,y∈H,
(140)∥Gnx-Gny∥2≤(1-σnαn(τ-γl))∥x-y∥.
Since 0<1-σnαn(τ-γl)<1, Gn:H→H is a contraction. Therefore, by the Banach contraction principle, Gn has a unique fixed point xn∈H, which uniquely solves the fixed point equation
(141)xn=(I-σnV)WnGvn+σn[Wnxn-αn(μF(Wnxn)-γQxn)].
This shows that the sequence {xn} is defined well.
It is easy to see the VIP (43) has only one solution in Ω. Below we use q∈Ω to denote the unique solution of the VIP (43).
Now, let us show that{xn} is bounded. In fact, take p∈Ω arbitrarily. Repeating the same arguments as in the proof of Theorem 19 we obtain
(142)∥un-p∥2≤∥xn-p∥2+rn(rn-2ζ)∥Axn-Ap∥2,(143)∥vn-p∥2≤∥Λniun-p∥2≤∥un-p∥2+λi,n(λi,n-2ηi)∥BiΛni-1un-Bip∥2,(144)∥Gvn-p∥2≤∥Tν2Θ2(I-ν2A2)vn-Tν2Θ2(I-ν2A2)p∥2+ν1(ν1-2ζ1)×∥A1Tν2Θ2(I-ν2A2)vn-A1Tν2Θ2(I-ν2A2)p∥2≤∥vn-p∥2+ν2(ν2-2ζ2)∥A2vn-A2p∥2,(145)∥xn-p∥2≤(1-σnγ¯)∥Gvn-p∥2+σn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-σnγ¯)∥vn-p∥2+σn∥xn-p∥2+2sn〈(I-V)p+αn(γQ-μF)p,xn-p〉=(1-σn(γ¯-1))∥xn-p∥2+2σn(∥(I-V)p∥+∥(γQ-μF)p∥)×∥xn-p∥.
From (145) we conclude that
(146)∥xn-p∥≤2γ¯-1(∥(I-V)p∥+∥(γQ-μF)p∥).
Hence {xn} is bounded. So, according to (142) and (143) we know that {un},{vn},{Gvn}, {WnGvn},{Wnxn},{Qxn}, and {FWnxn} are bounded.
Repeating the arguments similar to those of (68), (76), (78), (93), and (97) in the proof of Theorem 19 we obtain that ∥xn-un∥→0, ∥Λni-1un-Λniun∥→0, ∥xn-vn∥→0, ∥vn-Gvn∥→0 and ∥vn-Wnvn∥→0 as n→∞. In addition, note that
(147)∥vn-Wvn∥≤∥vn-Wnvn∥+∥Wnvn-Wvn∥.
So, from ∥vn-Wnvn∥→0 and [15, Remark 3.2] it follows that
(148)lim⁡n→∞∥vn-Wvn∥=0.
Further, we show thatωw(xn)⊂Ω. Indeed, since {xn} is bounded, there exists a subsequence {xni} of {xn} which converges weakly to some w. Note that lim⁡n→∞∥xn-un∥=0. Hence uni⇀w. Since C is closed and convex, C is weakly closed. So, we have w∈C. On the other hand, it is easy to see that uni⇀w, vni⇀w, and Λnimuni⇀w, where m∈{1,2,…,N}. Repeating the same arguments as in the proof of Theorem 19, we obtain that w∈∩i=1NI(Bi,Ri)∩GMEP⁡(Θ,φ,A). Next let us show that w∈SGEP(G)∩∩n=1∞Fix⁡(Tn). As a matter of fact, from ∥vn-Gvn∥→0, ∥vn-Wvn∥→0, vni⇀w and Lemma 10, we deduce that w∈SGEP(G) and w∈Fix⁡(W)=∩n=1∞Fix⁡(Tn) (due to Lemma 12). Hence we get w∈SGEP(G)∩∩n=1∞Fix⁡(Tn). Therefore, w∈∩n=1∞Fix⁡(Tn)∩GMEP⁡(Θ,φ,A)∩SGEP(G)∩∩i=1NI(Bi,Ri)=:Ω. This shows that ωw(xn)⊂Ω.
Finally, we prove that{xn} converges strongly to q∈Ω, which is the unique solution in Ω to the VIP (136). In fact, we note that, for w∈Ω with xni⇀w,
(149)xn-w=(I-σnV)(WnGvn-w)+σn[αnγ(Qxn-Qw)+(I-αnμF)Wnxn-(I-αnμF)Wnw]+σn[(I-V)w+αn(γQ-μF)w].
Utilizing the arguments similar to those in the proof of Theorem 19, we obtain that
(150)∥xn-w∥2≤(1-σn(γ¯-1))∥xn-w∥2+σn〈(I-V)w+αn(γQ-μF)w,xn-w〉,
which hence leads to
(151)∥xn-w∥2≤〈(I-V)w+αn(γQ-μF)w,xn-w〉γ¯-1.
In particular, we have
(152)∥xni-w∥2≤〈(I-V)w+αni(γQ-μF)w,xni-w〉γ¯-1.
Since xni⇀w and lim⁡n→∞αn=0, it follows from (152) that xni→w as i→∞.
Now we show thatw solves the VIP (136). As a matter of fact, from (142) and (145) we obtain that, for any p∈Ω,
(153)∥xn-p∥2≤(1-σnγ¯)∥un-p∥2+σn∥xn-p∥2+2σn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤(1-σnγ¯)∥xn-p∥2+σn∥xn-p∥2+2σn〈(I-V)p+αn(γQ-μF)p,xn-p〉=(1-σn(γ¯-1))∥xn-p∥2+2σn〈(I-V)p+αn(γQ-μF)p,xn-p〉≤∥xn-p∥2+2σn〈(I-V)p+αn(γQ-μF)p,xn-p〉,
which immediately implies that
(154)〈(V-I)p+αn(μF-γQ)p,xn-p〉≤0.
Since αn→0 and xni→w, we get
(155)〈(V-I)p,w-p〉≤0,∀p∈Ω.
By Minty's lemma, w is a solution in Ω to the VIP (136). In terms of the uniqueness of solutions of VIP (136), we deduce that w=q and xni→q as n→∞. So, every weak convergence subsequence of {xn} converges strongly to the unique solution q of VIP (136). Therefore, {xn} converges strongly to the unique solution q of VIP (136). This completes the proof.Corollary 24.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and let φ:C→R be a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let A,Ak:H→H and Bi:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone, and ηi-inverse strongly monotone, respectively, for k=1,2 and i=1,2. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and let {λn} be a sequence in (0,b] for some b∈(0,1). Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<(2η/κ2) and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=∩n=1∞Fix(Tn)∩GMEP(Θ,φ,A)∩SGEP(G)∩I(B1,R1)∩I(B2,R2)≠∅ where G is defined as in Proposition CY. Let {rn} be a sequence in (0,2ζ] and {αn} and {σn} sequences in (0,1]. Let {xn} be the sequence generated by
(156)un=Srn(Θ,φ)(I-rnA)xn,vn=JR2,λ2,n(I-λ2,nB2)JR1,λ1,n(I-λ1,nB1)un,xn=(I-σnV)WnGvn+σn[Wnxn-αn(μF(Wnxn)-γQxn)],∀n≥1,
where Wn is the W-mapping defined by (9). Suppose that the following conditions hold:(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that, for any y∉Dx,
(157)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
lim⁡n→∞αn=0 and lim⁡n→∞σn=0;(iv)
νk∈(0,2ζk),k=1,2 and {λi,n}⊂[ai,bi]⊂(0,2ηi),  i=1,2;(v)
0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ. Assume that Sr(Θ,φ) is firmly nonexpansive. Then {xn} converges strongly to a point q∈Ω, which is a unique solution in Ω to the VIP:
(158)〈(I-V)q,p-q〉≤0,∀p∈Ω.Corollary 25.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R a lower semicontinuous and convex functional. Let R:C→2H be a maximal monotone mapping and let A,Ak:H→H and B:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone, and ξ-inverse strongly monotone, respectively, for k=1,2. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and {λn} a sequence in (0,b] for some b∈(0,1). Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<(2η/κ2) and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=∩n=1∞Fix(Tn)∩GMEP(Θ,φ,A)∩SGEP(G)∩I(B,R)≠∅ where G is defined as in Proposition CY. Let {rn} be a sequence in (0,2ζ] and {αn} and {σn} sequences in (0,1]. Let {xn} be the sequence generated by
(159)un=Srn(Θ,φ)(I-rnA)xn,vn=JR,ρn(I-ρnB)un,xn=(I-σnV)WnGvn+σn[Wnxn-αn(μF(Wnxn)-γQxn)],∀n≥1,
where Wn is the W-mapping defined by (9). Suppose that the following conditions hold:(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that, for any y∉Dx,
(160)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
lim⁡n→∞αn=0 and lim⁡n→∞σn=0;(iv)
νk∈(0,2ζk),  k=1,2 and {ρn}⊂[a,b]⊂(0,2ξ);(v)
0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ. Assume that Sr(Θ,φ) is firmly nonexpansive. Then {xn} converges strongly to a point q∈Ω, which is a unique solution in Ω to the VIP:
(161)〈(I-V)q,p-q〉≤0,∀p∈Ω.



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*Source: 209372-2014-05-08.xml*