# A Unified Iterative Treatment for Solutions of Problems of Split Feasibility and Equilibrium in Hilbert Spaces

Authors: Young-Ye Huang; Chung-Chien Hong
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We at first raise the so called split feasibility fixed point problem which covers the problems of split feasibility, convex feasibility, and equilibrium as special cases and then give two types of algorithms for finding solutions of this problem and establish the corresponding strong convergence theorems for the sequences generated by our algorithms. As a consequence, we apply them to study the split feasibility problem, the zero point problem of maximal monotone operators, and the equilibrium problem and to show that the unique minimum norm solutions of these problems can be obtained through our algorithms. Since the variational inequalities, convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, each type of our algorithms can be considered as a generalized methodology for solving the aforementioned problems.


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## Body

## 1. Introduction

Throughout this paper,ℋ denotes a real Hilbert space with inner product 〈·,·〉 and the norm ∥·∥, I the identity mapping on ℋ, ℕ the set of all natural numbers, and ℝ the set of all real numbers. For a self-mapping T on ℋ, Fix(T) denotes the set of all fixed points of T. If M:ℋ→2ℋ is a set-valued mapping; then 𝒟(M) denotes its domain, that is, 𝒟(M)={x∈ℋ:M(x)≠∅}.LetC and Q be nonempty closed convex subsets of two Hilbert spaces ℋ1 and ℋ2, respectively, and let A:ℋ1→ℋ2 be a bounded linear mapping. The split feasibility problem (SFP) is the problem of finding a point with the property
(1)x*∈C,Ax*∈Q.
The SFP was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and medical image reconstruction. Recently, it has been found that the SFP can also be used to model the intensity-modulated radiation therapy. For details, the readers are referred to Xu [2] and the references therein.Assume that the SFP has a solution. There are many iterative methods designed to approximate its solutions. The most popular algorithm is theCQ algorithm introduced by Byrne [3, 4].Start with any
x
1
∈
ℋ
1
and generate a sequence
{
x
n
}
through the iteration
(2)
x
n
+
1
=
P
C
[
I
-
γ
A
*
(
I
-
P
Q
)
A
]
x
n
,
where
γ
∈
(
0
,
2
/
∥
A
∥
2
)
,
A
*
the adjoint of
A
, and
P
C
and
P
Q
are the metric projections onto
C
and
Q
, respectively.The sequence{xn} generated by the CQ algorithm (2) converges weakly to a solution of SFP(1), cf. [2–4]. Under the assumption that SFP(1) has a solution, it is known that a point x*∈ℋ1 solves SFP(1) if and only if x* is a fixed point of the operator
(3)PC[I-γA*(I-PQ)A],
cf. [2], where Xu also proposed the regularized method,
(4)xn+1=PC[I-γn(A*(I-PQ)A+αnI)]xn,
and proved that the sequence {xn} converges strongly to the minimum norm solution of SFP(1) provided that the parameters {αn} and {γn} verify some suitable conditions. This regularized method was further investigated by Yao et al. [5] and Yao et al. [6].PuttingS=PC and T=PQ, SFP(1) is of the forms:
(5)findx*∈ℋ1sothatx*∈Fix(S),Ax*∈Fix(T).
As a metric projection is firmly nonexpansive, it is reasonable to require the S and T in (5) to be firmly nonexpansive only and call it a split feasibility fixed point problem (SFFP). Many interesting problems in the literature can be described as SFFP.(i) A set-valued mapM:ℋ→2ℋ is called monotone if
(6)〈x-y,u-v〉≥0
for all x,y∈𝒟(M) and for any u∈M(x), v∈M(y). M is said to be maximal monotone if its graph {(x,u):x∈ℋ,u∈M(x)} is not properly contained in the graph of any other monotone operator. A point v∈ℋ is called a zero point of a maximal monotone operator M if 0∈M(v). The set of all zero points of M is denoted by M-10, which is equal to Fix(JαM) for any α>0, where JαM denotes the resolvent of a monotone operator M; that is, JαM(x)=(I+αM)-1(x) for any x∈ℋ. It is known that for any α>0, JαM is firmly nonexpansive. Now, let M and N be two maximal monotone operators on ℋ1 and ℋ2, respectively. Replacing C and Q with M-10=Fix(JαM) and N-10=Fix(JβN), respectively, in (1), the SFP becomes a SFFP:
(7)findx*∈ℋ1sothatx*∈Fix(JαM),Ax*∈Fix(JβN).
Putting A=I, the previous SFFP is reduced to the common zero point problem of two maximal monotone operators:
(8)findx*∈ℋ1sothatx*∈M-10∩N-10.(ii) Letf:C×C→ℝ. An equilibrium problem is the problem of finding x^∈C such that
(9)f(x^,y)≥0,∀y∈C,
whose solution set is denoted by EP(f). For solving an equilibrium problem, we usually assume that the function f satisfies the following conditions:(A1)
f(x,x)=0,  for allx∈C;(A2)
fis monotone, that is,f(x,y)+f(y,x)≤0,  for allx∈C;(A3)
for allx,y,z∈C,limsupt↓0f((1-t)x+tz,y)≤f(x,y);(A4)
for allx∈C,f(x,·)  is convex and lower semicontinuous.Blum and Oettli [7] and Aoyama et al. [8] showed that there exists a uniquez∈Csuch that(10)f(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.
Moreover, For r>0, define Jrf:ℋ→C by
(11)Jrfx={z∈C:f(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}
for all x∈ℋ. Combettes and Hirstoaga [9] showed that there hold(a)
Jrf  is single-valued;(b)
Jrf  is firmly nonexpansive;(c)
Fix(Jrf)=EP(f);(d)
EP(f)  is closed and convex.Now, letf:C×C→ℝ and g:Q×Q→ℝ be two functions satisfying conditions (A1)–(A4). Replacing C and Q with EP(f)=Fix(Jαf) and EP(g)=Fix(Jβg), respectively, in (1), the SFP becomes a SFFP:
(12)findx*∈ℋ1sothatx*∈Fix(Jαf),Ax*∈Fix(Jβg).
Putting ℋ1=ℋ2, C=Q, and A=I, the previous SFFP is reduced to the common equilibrium problem:
(13)findx*∈Csothatf(x*,y)≥0,g(x*,y)≥0,∀y∈C.(iii) Whenℋ1=ℋ2=ℋ, and the bounded linear operator A is the identity mapping, SFP(1) is reduced to the convex feasibility problem (CFP):
(14)findx*∈ℋsothatx*∈C∩Q,
which in turn can be described as a SFFP:
(15)findx*∈ℋsothatx*∈Fix(S),Ax*∈Fix(T),
where S=PC and T=PQ. Although SFFP(5) contains CFP as a special case, it cannot cover the multiple-set split convex feasibility problem described in [10].In this paper, we are concerned with iterative methods for SFFP(5). We derive some weak convergence theorems for SFFP(5) in Section 3. In Section 4, we describe SFFP(5) in a more general form.Let
{
S
n
}
and
{
T
n
}
be two families of firmly nonexpansive self-mappings on
ℋ
1
and
ℋ
2
,  respectively, so that
∩
n
=
1
∞
Fix
(
S
n
)
≠
∅
and
∩
n
=
1
∞
Fix
(
T
n
)
≠
∅
(16)
f
i
n
d
x
*
∈
ℋ
1
s
o
t
h
a
t
x
*
∈
∩
n
=
1
∞
Fix
(
S
n
)
,
A
x
*
∈
∩
n
=
1
∞
Fix
(
T
n
)
and obtain the following main result.Let
{
a
n
}
,
{
b
n
}
,
{
c
n
}
,  and
{
d
n
}
be sequences in
[
0,1
]
with
a
n
+
b
n
+
c
n
+
d
n
=
1
and
a
n
∈
(
0,1
)
for all
n
∈
ℕ
. Let
{
γ
n
}
be a sequence in
(
0
,
2
/
∥
A
∥
2
)
and let
{
e
n
}
be a bounded sequence in
ℋ
1
. Suppose that the solution set
Ω
of  SFFP(16)  is nonempty. For any
u
∈
ℋ
1
, start with an arbitrary
x
1
∈
ℋ
1
and define a sequence
{
x
n
}
by
(17)
x
n
+
1
=
a
n
u
+
b
n
x
n
+
c
n
S
n
[
I
-
γ
n
A
*
(
I
-
T
n
)
A
]
x
n
+
d
n
e
n
.
Then, the sequence
{
x
n
}
converges strongly to
P
Ω
u
provided that the following conditions are satisfied:
(i)
lim
n
→
∞
a
n
=
lim
n
→
∞
(
d
n
/
a
n
)
=
0
,
∑
n
=
1
∞
a
n
=
∞
,
∑
n
=
1
∞
d
n
<
∞
;
(ii)
lim
n
→
∞
inf
c
n
(
2
γ
n
-
γ
n
2
∥
A
∥
2
)
>
0
,
lim
n
→
∞
inf
b
n
c
n
>
0
;
(iii)
there are two nonnegative real-valued functions  κ1andκ2onℕwith(18)∥Smx-x∥≤κ1(m)∥Snx-x∥,∀m∈ℕ,∀n≥m,∀x∈ℋ1,∥Tmy-y∥≤κ2(m)∥Tny-y∥,∀m∈ℕ,∀n≥m,∀y∈ℋ2.Based on the concept of using contractions to approximate nonexpansive mappings, another type of algorithms for SFFP(5) is also introduced, and the corresponding strong convergence theorem for the sequence generated by such algorithm is given too.In Section5, since resolvents of monotone operators are firmly nonexpansive, we replace the sequences {Sn} and {Tn} of firmly nonexpansive mappings in the previous condition (iii) by two sequences of resolvents of maximal monotone operators. Then, the proposed algorithm becomes a scheme to approach the minimum norm solution of zero point problem of maximal monotone operators and the equilibrium problem. It is worth noting that as Blum and Oettli [7] showed that the variational inequalities, convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, the proposed algorithm can be considered as a generalized methodology for solving all aforementioned problems.


## 2. Preliminaries

In order to facilitate our investigation in this paper, we recall some basic facts. A mappingS:ℋ→ℋ is said to be(i)
nonexpansive if(19)∥Sx-Sy∥≤∥x-y∥,∀x,y∈ℋ;(ii)
firmly nonexpansive if(20)∥Sx-Sy∥2≤〈x-y,Sx-Sy〉,∀x,y∈ℋ;(iii)
λ-averaged by G if
(21)S=(1-λ)I+λG,
for some λ∈(0,1) and some nonexpansive mapping G;(iv)
ν-inverse strongly monotone (ν-ism), with ν>0, if
(22)〈x-y,Sx-Sy〉≥ν∥Sx-Sy∥2,∀x,y∈ℋ.If S is nonexpansive, then the fixed point set Fix(S) of S is closed and convex, cf. [11]. If S=(1-λ)I+λG is averaged, then S is nonexpansive with Fix(S)=Fix(G). It is well known that S is firmly nonexpansive if and only if it is 1/2-averaged, cf. [11], and so is I-S. Here we would like to mention that the term “averaged mapping” originated in [12, 13]. In [12], Baillon el at. showed that if S is a λ-averaged mapping by G on a nonempty closed convex subset C of a uniformly convex Banach space, then Fix(G)=ϕ if and only if limn→∞∥Snx∥=∞ for all x in C. Moreover, in [13], Bruck and Reich showed that if the above C satisfies C=-C and S is odd, then {Snx} converges strongly to a fixed point of G.LetC be a nonempty closed convex subset of ℋ. The metric projection PC from ℋ onto C is the mapping that assigns each x∈ℋ the unique point PCx in C with the property
(23)∥x-PCx∥=miny∈C∥y-x∥.
It is known that PC is firmly nonexpansive and characterized by the inequality: for any x∈ℋ,
(24)〈x-PCx,y-PCx〉≤0,∀y∈C.We need some lemmas that will be quoted in the sequel.Lemma 1 (see [4]).
IfS is a self-mapping on ℋ, then the following assertions hold.(a)
S is nonexpansive if and only if the complement I-S is 1/2-averaged.(b)
IfS is ν-ism and γ>0, then γS is (ν/γ)-ism.(c)
S is α-averaged if and only if I-S is (1/2α)-ism.(d)
IfS is α1-averaged and T is α2-averaged, then the composite ST is α1+α2-α1α2-averaged.(e)
IfS and U are averaged on ℋ so that Fix(S)∩Fix(U)≠∅, then
(25)Fix(SU)=Fix(S)∩Fix(U).Forα>0, the resolvent JαA of a maximal monotone operator A on ℋ has the following properties.Lemma 2.
LetA be a maximal monotone operator on ℋ. Then,(a)
JαA is single-valued and firmly nonexpansive;(b)
𝒟(JαA)=ℋ and Fix(JαA)=A-10;(c)
(the resolvent identity) forμ,λ>0, the following identity holds:
(26)JμAx=JλA(λμx+(1-λμ)JμAx),∀x∈ℋ.We referred readers to [14–24] for maximal monotone operators and their related algorithms.Lemma 3.
Letx,y,z∈ℋ. Then,(a)
∥x+y∥2≤∥x∥2+2〈y,x+y〉;(b)
for anyλ∈ℝ,
(27)∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)∥x-y∥2;(c)
fora,b,c∈[0,1] with a+b+c=1,
(28)∥ax+by+cz∥2=a∥x∥2+b∥y∥2+c∥z∥2-ab∥x-y∥2-ac∥x-z∥2-bc∥y-z∥2.Lemma 4 (see [11] (demiclosedness principle)).
LetS be a nonexpansive self-mapping on ℋ and suppose that {xn} is a sequence in ℋ such that {xn} converges weakly to some z∈ℋ and limn→∞∥xn-Sxn∥=0. Then, Sz=z.Lemma 5 (see [23]).
Let{sn} be a sequence of nonnegative real numbers satisfying
(29)sn+1≤(1-αn)sn+αnμn+νn,n∈ℕ,
where {αn}, {μn}, and {νn} verify the following conditions:(i)
{αn}⊆[0,1], ∑n=1∞‍αn=∞;(ii)
limsupn→∞μn≤0;(iii)
{νn}⊆[0,∞) and ∑n=1∞‍νn<∞.Then, limn→∞sn=0.Lemma 6 (see [25]).
Let{sn} be a sequence in ℝ that does not decrease at infinity in the sense that there exists a subsequence {sni} such that
(30)sni<sni+1,∀i∈ℕ.
For any k∈ℕ, define mk=max{j≤k:sj<sj+1}. Then, mk→∞ as k→∞ and max{smk,sk}≤smk+1, for all k∈ℕ.


## 3. Weak Convergence Theorems

In this section, we at first transform SFFP(5) into a fixed point problem for the operator S(I-γA*(I-T)A), where γ is any positive real number. And then use fixed point algorithms to solve SFFP(5). From now on until the end of this paper, unless we state specifically, S and Sn, n∈ℕ (resp., T and Tn, n∈ℕ) are firmly nonexpansive self-mappings on ℋ1 (resp., ℋ2), and A denotes a bounded linear operator from ℋ1 into ℋ2 with adjoint A*.Lemma 7.
LetS be a firmly nonexpansive self-mapping on ℋ with Fix(S)≠∅. Then, for any x∈ℋ, one has
(31)〈x-Sx,v-Sx〉≤0,∀v∈Fix(S).Proof.
Sincev∈Fix(S) and S is firmly nonexpansive, we have
(32)∥Sx-v∥2=∥Sx-Sv∥2≤〈x-v,Sx-Sv〉=〈x-v,Sx-v〉,
and hence,
(33)〈x-Sx,v-Sx〉=〈x-v,v-Sx〉+〈v-Sx,v-Sx〉=∥v-Sx∥2-〈x-v,Sx-v〉≤∥v-Sx∥2-∥Sx-v∥2=0.Although〈x-Sx,v-Sx〉≤0, for all v∈Fix(S) is similar to the characterization inequality (24) for the metric projection PFix(S), as Sx needs not to be in Fix(S), it is in general different from PFix(S)(x). For example, let Sx=3x/4 for all x∈ℝ, which is obviously firmly nonexpansive with Fix(S)={0}. Thus, PFix(S)(x)=0 for all x∈ℝ, while Sx≠0 for all x≠0.Proposition 8.
For anyγ∈(0,2/∥A∥2), the operator I-γA*(I-T)A is γ∥A∥2/2-averaged and S[I-γA*(I-T)A] is (2+γ∥A∥2)/4-averaged.Proof.
Using the fact thatT is firmly nonexpansive, it is routine to show that A*(I-T)A is 1/∥A∥2-ism, and so γA*(I-T)A is 1/γ∥A∥2-ism by Lemma 1(b). Thus, Lemma 1(c) shows that I-γA*(I-T)A is γ∥A∥2/2-averaged. As S is firmly nonexpansive, it is 1/2-averaged. Therefore, S[I-γA*(I-T)A] is (1/2)+(γ∥A∥2/2)-(γ∥A∥2/4)=((2+γ∥A∥2)/4)-averaged by Lemma 1(d).Proposition 9.
LetΩ be the solution set of SFFP(5); that is, Ω=Fix(S)∩A-1(Fix(T)). For any γ∈(0,2/∥A∥2), let U=I-γA*(I-T)A. Suppose that Ω≠∅. Then, Fix(SU)=Fix(S)∩Fix(U)=Ω.Proof.
Ifx* solves SFFP(5), we have x*∈Fix(S) and Ax*∈Fix(T). Now, note that Ax*∈Fix(T) implies
(34)(I-T)Ax*=0,
and so
(35)(I-γA*(I-T)A)x*=x*,
which means that x*∈Fix(U). Consequently, x*∈Fix(S)∩Fix(U). This shows that Ω⊆Fix(S)∩Fix(U).
For the inverse inclusion, letx* be any member of Fix(S)∩Fix(U) and pick v∈Ω. It is readily seen from x*∈Fix(U) that
(36)A*(I-T)Ax*=0.
Since Av∈Fix(T), an application of Lemma 7 yields
(37)〈(I-T)Ax*,Av-TAx*〉≤0,
which together with (36) implies that
(38)∥(I-T)Ax*∥2=〈(I-T)Ax*,Ax*-Av〉+〈(I-T)Ax*,Av-TAx*〉≤〈(I-T)Ax*,Ax*-Av〉=〈A*(I-T)Ax,x*-v〉=0.
This comes to conclude that Ax*∈Fix(T), and hence x*∈Ω once we note that x*∈Fix(S). Finally, since Ω≠∅ by assumption, we have Fix(S)∩Fix(U)=Ω≠∅. Thus, Fix(SU)=Fix(S)∩Fix(U) follows from Lemma 1(e).Proposition 10 (see [2, 4]).
IfG is an averaged self-mapping on ℋ with Fix(G)≠∅, then for any x∈ℋ, the sequence {Gnx} converges weakly to a fixed point of G.An immediate consequence of Propositions8, 9, and 10 is the following convergence result.Theorem 11.
Assume that SFFP(5) has a solution. Then, for any γ∈(0,2/∥A∥2) and starting with any point x1∈ℋ, the sequence {xn} generated by
(39)xn+1=S[I-γA*(I-T)A]xn
converges weakly to a solution of SFFP(5).Proposition 12 (see [2]).
LetG be a α-averaged self-mapping on ℋ with Fix(G)≠∅ and assume that {αn} is a sequence in [0,1/α] such that
(40)∑n=1∞‍αn(1α-αn)=∞.
Then, for any x1∈ℋ, the sequence {xn} generated by the Mann's algorithm
(41)xn+1=(1-αn)xn+αnGxn
converges weakly to a fixed point of G.Applying Propositions8, 9, and 12, we have the following result.Theorem 13.
Assume that SFFP(5) has a solution and γ∈(0,2/∥A∥2). Let {αn} be a sequence in [0,4/(2+γ∥A∥2)] with
(42)∑n=1∞‍αn(42+∥A∥2-αn)=∞.
Then, for any x1∈ℋ, the sequence {xn} generated by the Mann's algorithm
(43)xn+1=(1-αn)xn+αnS[I-γA*(I-T)A]xn
converges weakly to a solution of SFFP(5).


## 4. Strong Convergence Theorems

In this section, we devise two algorithms, one for SFFP(16) and the other for SFFP(5). We deal with SFFP(16) firstly. To begin with, we need a lemma.Lemma 14.
For anyγ∈(0,2/∥A∥2) and all x,y∈ℋ1, one has
(44)∥[I-γA*(I-T)A]x-[I-γA*(I-T)A]y∥2≤∥x-y∥2-(2γ-γ2∥A∥2)×∥(I-T)Ax-(I-T)Ay∥2.Proof.
SinceT is firmly nonexpansive, so is I-T. Hence, for all x,y∈ℋ1, we have
(45)∥(I-T)Ax-(I-T)Ay∥2≤〈Ax-Ay,(I-T)Ax-(I-T)Ay〉.
Consequently,
(46)∥[I-γA*(I-T)A]x-[I-γA*(I-T)A]y∥2=∥x-y∥2-2γ×〈x-y,A*(I-T)Ax-A*(I-T)Ay〉+γ2∥A*(I-T)Ax-A*(I-T)Ay∥2=∥x-y∥2-2γ×〈Ax-Ay,(I-T)Ax-(I-T)Ay〉+γ2∥A*(I-T)Ax-A*(I-T)Ay∥2≤∥x-y∥2-2γ∥(I-T)Ax-(I-T)Ay∥2+γ2∥A∥2∥(I-T)Ax-(I-T)Ay∥2=∥x-y∥2-(2γ-γ2∥A∥2)×∥(I-T)Ax-(I-T)Ay∥2.Theorem 15.
Let{an}, {bn}, {cn}, and {dn} be sequences in [0,1] with an+bn+cn+dn=1 and an∈(0,1) for all n∈ℕ. Let {γn} be a sequence in (0,2/∥A∥2) and let {en} be a bounded sequence in ℋ1. Suppose that the solution set Ω of SFFP(16) is nonempty. For any u∈ℋ1, start with any x1∈ℋ1 and define a sequence {xn} by
(47)xn+1=anu+bnxn&+cnSn[I-γnA*(I-Tn)A]xn+dnen.
Then the sequence {xn} converges strongly to PΩu provided that the following conditions are satisfied: (i)
limn→∞an=limn→∞(dn/an)=0,∑n=1∞an=∞,∑n=1∞dn<∞;(ii)
limn→∞infcn(2γn-γn2∥A∥2)>0,limn→∞infbncn>0;(iii)
there are two nonnegative real-valued functionsκ1andκ2onℕwith(48)∥Smx-x∥≤κ1(m)∥Snx-x∥,∀m∈ℕ,∀n≥m,∀x∈ℋ1,∥Tmy-y∥≤κ2(m)∥Tny-y∥,∀m∈ℕ,∀n≥m,∀y∈ℋ2.Proof.
Putp=PΩu. For simplicity, put Gn=Sn(I-γnA*(I-Tn)A). In view of Proposition 9, Gn is nonexpansive, so
(49)∥xn+1-p∥=∥an(u-p)+bn(xn-p)+cn(Gnxn-p)+dn(en-p)∥≤an∥u-p∥+(bn+cn)×∥xn-p∥+dn∥en-p∥,
from which follows that {xn} is a bounded sequence. Taking into account Lemma 3, we get
(50)∥xn+1-p∥2=∥an(u-p)+bn(xn-p)+cn(Gnxn-p)+dn(en-p)∥2≤∥bn(xn-p)+cn(Gnxn-p)+dn(en-p)∥2+2an〈u-p,xn+1-p〉=(1-an)2∥bn1-an(xn-p)dddddddddd+cn1-an(Gnxn-p)dddddddddd+dn1-an(en-p)∥2+2an〈u-p,xn+1-p〉≤(1-an)2(bn1-an∥xn-p∥2+cn1-anddddddddd×∥Gnxn-p∥2+dn1-an∥en-p∥2)-bncn∥xn-Gnxn∥2+2an〈u-p,xn+1-p〉≤bn∥xn-p∥2+cn∥Gnxn-p∥2+dn∥en-p∥2-bncn∥xn-Gnxn∥2+2an〈u-p,xn+1-p〉.
Meanwhile, we have by Lemma 14 that
(51)∥Gnxn-p∥2=∥Sn(I-γnA*(I-Tn)A)xnddd-Sn(I-γnA*(I-Tn)A)p∥2≤∥(I-γnA*(I-Tn)A)xn-(I-γnA*(I-Tn)A)p∥2≤∥xn-p∥2-(2γn-γn2∥A∥2)×∥(I-Tn)Axn-(I-Tn)Ap∥2.
Therefore, we deduce that
(52)∥xn+1-p∥2≤bn∥xn-p∥2+cn[∥(I-Tn)Axn-(I-Tn)Ap∥2(2γn-γn2∥A∥2)∥xn-p∥2-(2γn-γn2∥A∥2)dddddd×∥(I-Tn)Axn-(I-Tn)Ap∥2(2γn-γn2∥A∥2)]+dn∥en-p∥2-bncn∥xn-Gnxn∥2+2an〈u-p,xn+1-p〉=(bn+cn)∥xn-p∥2+dn∥en-p∥2+2an〈u-p,xn+1-p〉-cn(2γn-γn2∥A∥2)∥(I-Tn)Axn∥2-bncn∥xn-Gnxn∥2.
We now carry on with the proof by considering the following two cases: (I) {∥xn-p∥} is eventually decreasing and (II) {∥xn-p∥} is not eventually decreasing.
Case I. Suppose that {∥xn-p∥} is eventually decreasing; that is, there is n0∈ℕ such that {∥xn-p∥}n≥n0 is decreasing. In this case, limn→∞∥xn-p∥ exists in ℝ. From inequality (52) we have
(53)cn(2γn-γn2∥A∥2)∥(I-Tn)Axn∥2+bncn∥xn-Gnxn∥2≤(bn+cn)∥xn-p∥2+dn∥en-p∥2+2an〈u-p,xn+1-p〉-∥xn+1-p∥2,
which together with the boundedness of {xn} and conditions (i) and (ii) implies
(54)limn→∞∥xn-Gnxn∥=0,limn→∞∥Axn-TnAxn∥=0.
Then, an application of condition (iii) follows that for all m∈ℕ,
(55)limn→∞∥xn-Gmxn∥=limn→∞∥Axn-TmAxn∥=0.
Since {xn} is bounded, it has a subsequence {xnk} such that {xnk} converges weakly to some z∈ℋ and
(56)limsupn→∞〈u-p,xn+1-p〉=limk→∞〈u-p,xnk-p〉=〈u-p,z-p〉≤0,
where the last inequality follows from (24) since z∈Ω by Lemma 4. Choose M>0 so that sup{∥en-p∥2+2∥u-p∥∥xn+1-p∥:n∈ℕ}≤M. From (52) we have
(57)∥xn+1-p∥2≤(1-(an+dn))∥xn-p∥2+(an+dn)·2〈u-p,xn+1-p〉+dn(∥en-p∥2+2∥u-p∥∥xn+1-p∥)≤(1-(an+dn))∥xn-p∥2+(an+dn)·2〈u-p,xn+1-p〉+dnM.
Accordingly, because of (56) and condition (i), we can apply Lemma 5 to inequality (57) with sn=∥xn-p∥2, αn=an+bn, μn=2〈u-p,xn+1-p〉, and νn=dnM to conclude that
(58)limn→∞xn=p.
Case II. Suppose that {∥xn-p∥} is not eventually decreasing. In this case, by Lemma 6, there exists a nondecreasing sequence {mk} in ℕ such that mk→∞ and
(59)max{∥xmk-p∥,∥xk-p∥}≤∥xmk+1-p∥,∀k∈ℕ.
Then, it follows from (52) and (59) that
(60)∥xmk-p∥2≤∥xmk+1-p∥2≤(bmk+cmk)∥xmk-p∥2+dmk∥emk-p∥2+2amk〈u-p,xmk+1-p〉-cmk(2γmk-γmk2∥A∥2)×∥(I-Tmk)Axmk∥2-bmkcmk∥xmk-Gmkxmk∥2.
Therefore,
(61)0≤cmk(2γmk+γmk2∥A∥2)∥(I-Tmk)Axmk∥2+bmkcmk∥xmk-Gmkxmk∥2≤-(1-(bmk+cmk))∥xmk-p∥2+dmk∥emk-p∥2+2amk〈u-p,xmk+1-p〉=-(amk+dmk)∥xmk-p∥2+dmk∥emk-p∥2+2amk〈u-p,xmk+1-p〉,
and then proceeding just as in the proof in Case I, we obtain
(62)limk→∞∥xmk-Gmkxmk∥=limk→∞∥Axmk-TmkAxmk∥=0,
which in conjunction with condition (iii) shows that for all mj(63)limk→∞∥xmk-Gmjxmk∥=limk→∞∥Axmk-TmjAxmk∥=0,
and then follows that
(64)limsupk→∞〈u-p,xmk+1-p〉≤0.
From (60) we have
(65)(1-(bmk+cmk))∥xmk-p∥2≤dmk∥emk-p∥2+2amk〈u-p,xmk+1-p〉,
and thus,
(66)∥xmk-p∥2≤dmkamk+dmk∥emk-p∥2+2amkamk+dmk〈u-p,xmk+1-p〉≤dmkamk∥emk-p∥2+2〈u-p,xmk+1-p〉.
Letting k→∞ and using (64) and condition (i), we obtain
(67)limk→∞∥xmk-p∥=0.
Also, since
(68)∥xmk+1-xmk∥≤amk∥u-xmk∥+cmk∥Gmkxmk-xmk∥+dmk∥emk-xmk∥,
which together with (62) and condition (i) implies that limk→∞∥xmk+1-xmk∥=0, and so
(69)limk→∞∥xmk+1-p∥=0
by virtue of (67). Consequently, we conclude that limk→∞∥xk-p∥=0 via (59) and (69). This completes the proof.This theorem says that the sequence{xn} converges strongly to a point of Ω which is the nearest to u. In particular, if u is taken to be 0, then the limit point v of the sequence {xn} is the unique minimum solution of SFFP(16).Corollary 16.
Let{an}, {bn}, and {cn} be sequences in [0,1] with an+bn+cn=1 and an∈(0,1) for all n∈ℕ. Let {γn} be a sequence in (0,2/∥A∥2) and let {en} be a bounded sequence in ℋ1. Suppose that the solution set Ω of SFFP(16) is nonempty. For any u∈ℋ1, start with any x1∈ℋ1 and define a sequence {xn} by
(70)xn+1=anu+bnxn+cnSn[I-γnA*(I-Tn)A]xn+en.
Then, the sequence {xn} converges strongly to PΩu provided that the following conditions are satisfied: (i)
limn→∞an=limn→∞(dn/an)=0,∑n=1∞an=∞,∑n=1∞dn<∞;(ii)
limn→∞infcn(2γn-γn2∥A∥2)>0,limn→∞infbncn>0;(iii)
there are two nonnegative real-valued functionsκ1andκ2onℕwith(71)∥Smx-x∥≤κ1(m)∥Snx-x∥,∀m∈ℕ,∀n≥m,∀x∈ℋ1,(72)∥Tmy-y∥≤κ2(m)∥Tny-y∥,∀m∈ℕ,∀n≥m,∀y∈ℋ2;(iv)
eitherlimn→∞(∥en∥/an)=0 or ∑n=1∞∥en∥<∞.Proof.
Putp=PΩu and Gn=Sn[I-γnA*(I-Tn)A]. Let z1=x1 and define a sequence {zn} iteratively by
(73)zn+1=anu+bnzn+cnGnzn.
We have limn→∞zn=p by Theorem 15. Since
(74)∥xn+1-zn+1∥≤bn∥xn-zn∥+cn∥Gnxn-Gnzn∥+∥en∥≤(bn+cn)∥xn-zn∥+∥en∥=(1-an)∥xn-zn∥+∥en∥,
the limit limn→∞∥xn-zn∥=0 follows by applying Lemma 5 to (74), and thus,
(75)limn→∞xn=p.If the sequence{Sn} (resp., {Tn}) of firmly nonexpansive mappings consists of a single mapping S (resp., T), then {Sn} and {Tn} obviously verify condition (iii), and hence, we have the following corollary.Corollary 17.
Let{an}, {bn}, {cn}, and {dn} be sequences in [0,1] with an+bn+cn+dn=1 and an∈(0,1) for all n∈ℕ. Let {γn} be a sequence in (0,2/∥A∥2) and let {en} be a bounded sequences in ℋ1. Assume that the solution set Ω of SFFP(5) is nonempty. For any u∈ℋ1, start with an arbitrary x1∈ℋ1 and define the sequence {xn} by
(76)xn+1=anu+bnxn+cnS[I-γnA*(I-T)A]xn+dnen.
Then, {xn} converges strongly to PΩu provided that the following conditions are satisfied: (i)
limn→∞an=limn→∞(dn/an)=0,∑n=1∞an=∞,∑n=1∞dn<∞;(ii)
limn→∞infcn(2γn-γn2∥A∥2)>0,limn→∞infbncn>0.When the sequence{γn} is taken to be a constant γ∈(0,2/∥A∥2), then because S[I-γnA*(I-T)A] is an averaged mapping, we can apply Corollary 3.4 of Huang and Hong [26] to obtain the following result.Theorem 18.
Let{an}, {bn}, {cn}, and {dn} be sequences in [0,1] with an+bn+cn+dn=1 and an∈(0,1) for all n∈ℕ. Suppose that γ∈(0,2/∥A∥2) and suppose that {en} is a bounded sequence in ℋ1. Assume that the solution set Ω of SFFP(5) is nonempty. For any u∈ℋ1, start with an arbitrary x1∈ℋ1 and define the sequence {xn} by
(77)xn+1=anu+bnxn+cnS[I-γA*(I-T)A]xn+dnen.
Then, {xn} converges strongly to PΩu provided that the following conditions are satisfied: (i)
limn→∞an=limn→∞(dn/an)=0,∑n=1∞an=∞,∑n=1∞dn<∞;(ii)
limn→∞infbn>0,limn→∞infcn>0.Since both condition (ii) of Corollary17 and Theorem 18 are equivalent provided that γn=γ for every n∈ℕ, Theorem 18 also follows from Corollary 17.We now turn to SFFP(5) for another algorithm, which essentially follows the argument of Wang and Xu [27]. For the sake of completeness, we still give a detailed proof.Theorem 19.
Let{an} be a sequence in (0,1). Suppose that γ∈(0,2/∥A∥2) and assume that the solution set Ω of SFFP(5) is nonempty. Start with any x1∈ℋ1 and define a sequence {xn} by
(78)xn+1=S[(1-an)(I-γA*(I-T)A)]xn.
Then, the sequence {xn} converges strongly to the minimum norm solution of SFFP(5) provided that the following conditions are satisfied: (i)
limn→∞an=0;(ii)
∑n=1∞an=∞;(iii)
either∑n=1∞|an+1-an|<∞orlimn→∞(|an+1-an|/an)=0.Proof.
PutU=I-γA*(I-T)A, G=SU, and Gn=S[(1-an)]U for all n∈ℕ. Then,
(79)xn+1=Gnxn=S[(1-an)U]xn,∥Gnx-Gny∥≤(1-an)∥x-y∥,∀x,y∈ℋ1.
Take p∈Ω. From Proposition 9, we have
(80)∥Gnp-p∥=∥S[(1-an)U]p-SUp∥≤∥(1-an)Up-Up∥=an∥Up∥=an∥p∥.
Hence,
(81)∥xn+1-p∥=∥Gnxn-p∥≤∥Gnxn-Gnp∥+∥Gnp-p∥≤(1-an)∥xn-p∥+an∥p∥,
from which follows that {xn} is bounded and so is {Uxn}. Choose M>0 so that ∥Uxn∥≤M for all n∈ℕ. We have
(82)∥xn+1-xn∥=∥Gnxn-Gn-1xn-1∥≤∥Gnxn-Gnxn-1∥+∥Gnxn-1-Gn-1xn-1∥=(1-an)∥xn-xn-1∥+∥S[(1-an)Uxn-1]-S[(1-an-1)Uxn-1]∥≤(1-an)∥xn-xn-1∥+∥(1-an)Uxn-1-(1-an-1)Uxn-1∥=(1-an)∥xn-xn-1∥+|an-an-1|∥Uxn-1∥≤(1-an)∥xn-xn-1∥+M|an-an-1|.
In view of conditions (i), (ii), and (iii), we can apply Lemma 5 to (82) to get
(83)limn→∞∥xn+1-xn∥=0,
and then from
(84)∥xn-Gxn∥≤∥xn-xn+1∥+∥Gnxn-Gxn∥≤∥xn-xn+1∥+∥(1-an)Uxn-Uxn∥≤∥xn-xn+1∥+M∥Uxn∥,
we see that
(85)limn→∞∥xn-Txn∥=0.
Consequently, the demiclosedness principle ensures that each weak limit point of {xn} is a fixed point of the averaged mapping SU. And then we conclude from Proposition 9 that each weak limit point of {xn} lies in Ω.
Letx^ be the minimum norm element of Ω; that is, x^=PΩ(0). We shall show that {xn} converges strongly to x^. To see this, we compute ∥xn+1-x^∥2 as follows:
(86)∥xn+1-x^∥2=∥S[(1-an)Uxn]-SUx^∥2≤∥(1-an)Uxn-Ux^∥2=∥(1-an)Uxn-x^∥2=∥(1-an)(Uxn-x^)-anx^∥2=(1-an)2∥Uxn-x^∥2+2an(1-an)〈Uxn-x^,-x^〉+an2∥x^∥2≤(1-an)∥xn-x^∥2+an[2(1-an)〈Uxn-x^,-x^〉+an∥x^∥2].
If limsupn→∞2(1-an)〈Uxn-x^,-x^〉+an∥x^∥2≤0, then an application of Lemma 5 to (86) yields that limn→∞xn=x^. Hence, to complete the proof, it suffices to show that limsupn→∞2(1-an)〈Uxn-x^,-x^〉+an∥x^∥2≤0. For this, taking into account Proposition 8, we can write U=(1-β)I+βV for some β∈(0,1) and some nonexpansive mapping V. Then, from
(87)∥xn+1-p∥2=∥S[(1-an)U]xn-p∥2≤∥(1-an)Uxn-p∥2=∥(1-an)(Uxn-p)-anp∥2≤(1-an)∥Uxn-p∥2+an∥p∥2≤∥(1-β)(xn-p)+β(Vxn-p)∥2+an∥p∥2=(1-β)∥xn-p∥2+β∥Vxn-p∥2-β(1-β)∥xn-Vxn∥2+an∥p∥2≤∥xn-p∥2-β(1-β)∥xn-Vxn∥2+an∥p∥2,
we obtain
(88)β(1-β)∥xn-Vxn∥2≤∥xn-p∥2-∥xn+1-p∥2+an∥p∥2,
which ensures that limn→∞∥xn-Vxn∥=0, and hence,
(89)limn→∞∥xn-Uxn∥=0
once we note that I-U=β(I-V). Since {xn} is bounded, we can take a subsequence {xni} so that it is weakly convergent to x*∈Ω and
(90)limsupn→∞〈xn-x^,-x^〉=limi→∞〈xni,-x^〉=〈x*-x^,-x^〉≤0,
where the last inequality comes from the characterization inequality of a metric projection. Now, applying (89) and (90) to the equality
(91)〈Uxn-x^,-x^〉=〈Uxn-xn,-x^〉+〈xn-x^,-x^〉,
we obtain
(92)limsupn→∞〈Uxn-x^,-x^〉≤0,
and thus,
(93)limsupn→∞2(1-an)〈Uxn-x^,-x^〉+an∥x^∥2≤0.
This completes the proof.


## 5. Applications

In this section, we shall apply the results in Section4 to approximate zeros of maximal monotone operators and solutions of equilibrium problems.Theorem 20.
Suppose thatM and N are two maximal monotone operators on ℋ1 and ℋ2, respectively, and suppose that A:ℋ1→ℋ2 is a bounded linear operator with adjoint A*. Suppose further that {an}, {bn}, {cn}, and {dn} are sequences in [0,1] with an+bn+cn+dn=1 and an∈(0,1) for all n∈ℕ. Let {αn} and {βn} be sequences in (0,∞), {γn} a sequence in (0,2/∥A∥2), and {en} a bounded sequence in ℋ1. Assume that the solution set Ω of the problem
(94)findx*∈ℋ1sothatx*∈M-10,Ax*∈N-10
is nonempty. For any u∈ℋ1, start with an arbitrary x1∈ℋ and define a sequence {xn} by
(95)xn+1=anu+bnxn+cnJαnM[I-γnA*(I-JβnN)]xn+dnen.
Then, the sequence {xn} converges strongly to PΩu provided that the following conditions are satisfied: (i)
limn→∞an=limn→∞(dn/an)=0, ∑n=1∞an=∞, ∑n=1∞dn<∞;(ii)
limn→∞infcn(2γn-γn2∥A∥2)>0, limn→∞infbncn>0;(iii)
limn→∞infαn>0, limn→∞infβn>0.Proof.
For anyn∈ℕ, letting Sn=JαnM and Tn=JβnN, then we have by Lemma 2(b) that M-10=Fix(Sn) and N-10=Fix(Tn) for all n∈ℕ, and hence, as shown in Section 1, the problem (94) becomes SFFP(16). Since all the requirements of Theorem 15 are satisfied except condition (iii), we have to check this condition. Because liminfn→∞αn=0, liminfn→∞βn=0 by assumption, we may assume that there is τ∈(0,1) so that τ<αn and τ<βn for all n∈ℕ. Let κ1(n)=2+(αn/τ) and κ2(n)=2+(βn/τ). Then, by virtue of the resolvent identity and the nonexpansiveness of JαmM, one has for all m∈ℕ that
(96)∥JαnMx-JαmMx∥=∥JαmM(αmαnx+(1-αmαn)JαnMx)-JαmMx∥≤|1-αmαn|∥JαnMx-x∥≤(1+αmαn)∥JαnMx-x∥,
and thus,
(97)∥JαmMx-x∥≤∥JαmMx-JαnMx∥+∥JαnMx-x∥≤(1+αmαn+1)∥JαnMx-x∥≤(2+αmτ)∥JαnMx-x∥=κ1(m)∥JαnMx-x∥,∀n≥m,∀x∈ℋ.
The same argument shows for all m∈ℕ that
(98)∥JβmNx-x∥≤κ2(m)∥JβnNx-x∥,∀n≥m,∀x∈ℋ.
Therefore, condition (iii) of Theorem 15 is true for {JαnM} and {JβnN}, and then the desired conclusion follows from Theorem 15.ReplacingS and T with JαM and JβN, respectively, in Theorem 19, we obtain the following result.Theorem 21.
Suppose thatM and N are two maximal monotone operators on ℋ1 and ℋ2, respectively, and suppose that A:ℋ1→ℋ2 is a bounded linear operator with adjoint A*. Let {an} be a sequence in (0,1), γ∈(0,2/∥A∥2), and α,β∈(0,∞) and assume that the solution set Ω of problem (94) is nonempty. Start with any x1∈ℋ1 and define a sequence {xn} by
(99)xn+1=JαM[(1-an)(I-γA*(I-JβN)A)]xn.
Then, the sequence {xn} converges strongly to the minimum norm solution of problem (94) provided that the following conditions are satisfied: (i)
limn→∞an=0;(ii)
∑n=1∞an=∞;(iii)
either∑n=1∞|an+1-an|<∞ or limn→∞(|an+1-an|/an)=0.Recall some facts in Section1. For a nonempty closed convex subset C of ℋ, let f:C×C→ℝ be a function satisfying conditions (A1)–(A4), which are described in Section 1. The solution set of the equilibrium problem
(100)findx*∈Csuchthatf(x*,y)≥0,∀y∈C
is denoted by EP(f), which is equal to Fix(Jαf) for any α>0, where Jαf is a function on ℋ into C defined by
(101)Jαfx={z∈C:f(z,y)+1α〈y-z,z-x〉≥0,∀y∈C+1α〈y-z,z-x〉}
for all x∈ℋ. Jαf is a single-valued firmly nonexpansive mapping.Theorem 22.
Suppose thatC and Q are two nonempty closed convex subsets of ℋ1 and ℋ2, respectively, and suppose that A:ℋ1→ℋ2 is a bounded linear operator with adjoint A*. Let f:C×C→ℝ and g:Q×Q→ℝ be functions satisfying conditions (A1)–(A4). Suppose further that {an}, {bn}, {cn}, and {dn} are sequences in [0,1] with an+bn+cn+dn=1 and an∈(0,1) for all n∈ℕ. Let {αn} and {βn} be sequences in (0,∞), {γn} a sequence in (0,2/∥A∥2), and {en} a bounded sequence in ℋ1. Assume that the solution set Ω of the problem
(102)findx*∈ℋ1sothatx*∈EP(f),Ax*∈EP(g)
is nonempty. For any u∈ℋ1, start with an arbitrary x1∈ℋ and define a sequence {xn} by
(103)xn+1=anu+bnxn+cnJαnf(I-γnA*(I-Jβng)A)xn+dnen.
Then, the sequence {xn} converges strongly to PΩu provided that the following conditions are satisfied: (i)
limn→∞an=limn→∞(dn/an)=0, ∑n=1∞an=∞, ∑n=1∞dn<∞;(ii)
limn→∞infcn(2γn-γn2∥A∥2)>0, limn→∞infbncn>0;(iii)
limn→∞infαn>0, limn→∞infβn>0.Proof.
Define two set-valued mappingsMf and Ng on ℋ1 and ℋ2, respectively, by
(104)Mf(x)={{z∈ℋ1:f(x,y)≥〈y-x,z〉,∀y∈C},∀x∈C,∅,∀x∉C;Ng(u)={{〈v-u,w〉w∈ℋ2:g(u,v)≥〈v-u,w〉,∀v∈Qℋ},∀u∈Q,∅,∀u∉Q.
As shown in Takahashi et al. [21], the set-valued mapping Mf (resp., Ng) is a maximal monotone operator with 𝒟(Mf)⊆C (resp., 𝒟(Ng)⊆Q), and JαMf=Jαf=EP(f) for any α>0 (resp., JβNg=Jβg=EP(g) for any β>0). With M=Mf and N=Ng in Theorem 20, the desired conclusion follows.Ifℋ1=ℋ2=ℋ, C=Q, f=g, and A is the identity transformation on ℋ, then (102) is reduced to the usual equilibrium problem, and we have the following corollary.Corollary 23.
Suppose thatC is a nonempty closed convex subsets of ℋ and f:C×C→ℝ is a function satisfying conditions (A1)–(A4). Let {an}, {bn}, {cn}, and {dn} be sequences in [0,1] with an+bn+cn+dn=1 and an∈(0,1) for all n∈ℕ. Let {αn} and {βn} be sequences in (0,∞), and let {γn} be a sequence in (0,2/∥A∥2) and suppose that {en} is a bounded sequence in ℋ1. Assume that the solution set Ω of the problem
(105)findx*∈ℋsothatx*∈EP(f)
is nonempty. For any u∈ℋ1, start with an arbitrary x1∈ℋ and define a sequence {xn} by
(106)xn+1=anu+bnxn+cnJαnf[I-γn(I-Jβnf)]xn+dnen.
Then, the sequence {xn} converges strongly to PΩu provided that the following conditions are satisfied: (i)
limn→∞an=limn→∞(dn/an)=0, ∑n=1∞an=∞, ∑n=1∞dn<∞;(ii)
limn→∞infcn(2γn-γn2∥A∥2)>0, limn→∞infbncn>0;(iii)
limn→∞infαn>0, limn→∞infβn>0.As shown in Blum and Oettli [7] that the variational inequalities, convex differentiable optimization, Nash equilibria in noncooperative games and so on can be formulated as equilibrium problems, we see that the pervious corollary may be applied to approximate solutions of all aforementioned problems. The readers may readily write down the details.Just as Theorem21 to Theorem 20, there are similar results corresponding to Theorem 22 and Corollary 23. We leave the work to readers.



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*Source: 613928-2013-10-28.xml*