# Viscosity Iterative Schemes for Finding Split Common Solutions of Variational Inequalities and Fixed Point Problems

Authors: Zhenhua He; Wei-Shih Du
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

We introduce some new iterative schemes based on viscosity approximation method for finding a split common element of the solution set of a pair of simultaneous variational inequalities for inverse strongly monotone mappings in  real Hilbert spaces with a family of infinitely nonexpansive mappings. Some strong convergence theorems are also given. Our results generalize and improve some well-known results in the literature and references therein.


---

## Body

## 1. Introduction

Throughout this paper, we denote byℕ and ℝ, the sets of positive integers and real numbers, respectively. Let H be a real Hilbert space, whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let I be the identity mapping on H and C be a nonempty closed convex subset of H. Let T:C→H be a nonlinear operators. Then the canonical variational inequality problem for the operator T ((VIP)T or (VIP), for short) is to find u∈C such that
(1.1)〈Tu,v-u〉≥0,∀v∈C.
We use the symbol VI(C,T) to denote the solution set of (VIP), that is
(1.2)VI(C,T)={u∈C:〈Tu,v-u〉≥0,∀v∈C}.(VIP) was extensively investigated and generalized to the vector variational inequality problems for single-valued or multivalued maps and contains optimization problems, quasi-variational inequality problems, equilibrium problems, fixed-point problems, complementary problems, bilevel problems, and semi-infinite problems as special cases and applications; see [1–6] and references therein.LetS,T:C→H be two nonlinear operators. In [7], some authors have considered the following pair of simultaneous variational inequality problems for operators S and T ((PSVIP)S,T, for short):
(1.3)(PSVIP)S,TFindu∈Csuchthat〈Su,v-u〉≥0and〈Tu,v-u〉≥0,∀v∈C.An elementu∈C is a solution of (PSVIP)S,T if and only if u∈VI(C,S)∩VI(C,T). Clearly, (PSVIP)S,T reduces to (VIP) if S=T.Example 1.1.
Letℝ with usual inner product and let a,b∈ℝ with a<b. Define two real-valued functions T1,T2 by T1x=x2,T2x=x4, for allx∈[a,b]. Then T1′x=2x, T2′x=4x3 and there exists x0∈[a,b] such that T1x0=minx∈[a,b]T1x and T2x0=minx∈[a,b]T2x. If x0∈(a,b), then T1x0=T2x0=0; if x0=a, then T1′x0≥0 and T2′x0≥0; if x0=b, then T1′x0≤0and T2′x0≤0. So we have
(1.4)〈T1′x0,x-x0〉=T1′x0(x-x0)≥0,〈T2′x0,x-x0〉=T2′x0(x-x0)≥0,∀x∈[a,b]
or x0∈VI(C,T1′)∩VI(C,T2′) which means that x0 is the solution of (PSVIP)T1′,T2′.
Obviously, the problem(PSVIP)S,T is considered in the same subset of the same space. But many cases, two variational inequality problems often lie in different subset of spaces. So, as a further development of the problem (PSVIP)S,T, Censor et al. [8] presented a split variational inequality problem. Let H1,H2 be two real Hilbert spaces and C⊂H1 and K⊂H2 two closed convex sets. Let A:H1→H2 be a bounded linear operator. T:C→H1 and S:K→H2 are two nonlinear operators. The split variational inequality problem for T and S ((SVIP)T,S, for short) is defined as follows:
(1.5)(PSVIP)S,TFindp∈Csuchthat〈Tp,v-p〉≥0,∀v∈C,andu:=Ap∈Ksolves〈Su,w-u〉≥0,∀w∈K.It is well known to find a solution of (VIP) or a common element of the solution set of (VIP) and a fixed point of nonlinear operators, which has been studied by many authors (see [9–16]) using all kinds of auxiliary techniques and formulations. In 2005, Iiduka and Takahashi [9] established the following iteration scheme: let x1∈H be arbitrary, define
(1.6)xn+1=αnu+(1-αn)S1PC(xn-λnTxn),
where S1 is a nonexpansive mapping. They proved that the sequence {xn} defined by (1.6) strongly converge to x*∈F(S1)⋂‍VI(C,T), if the coefficient αn,λn satisfy the following conditions:
(1.7)limn→∞αn=0,∑n=1∞‍αn=∞,∑n=1∞‍|αn+1-αn|<∞,∑n=1∞‍|λn+1-λn|<∞.In 2007, Chen et al. [10] studied the following iterative process:
(1.8)xn+1=αnf(xn)+(1-αn)S1PC(xn-λnTxn),
where S1 is a nonexpansive mapping. If limn→∞αn=0,∑n=1∞‍αn=∞,∑n=1∞‍|αn+1-αn|<∞and ∑n=1∞‍|λn+1-λn|<∞, then they proved that {xn} converges strongly to q∈F(S1)⋂‍VI(C,T), which solves the variational inequality:
(1.9)〈fq-q,p-q〉≤0,∀p∈F(S1)∩VI(C,T).For some split common solution problems, they have been studied by some authors; see [17, 18] and therein references. In this paper, we continue to study the (SVIP) and introduce some new iterative schemes based on viscosity approximation method for finding a common element of the fixed points set of nonexpansive mappings and the split solution set of a pair of variational inequalities for inverse strongly monotone mappings in real Hilbert spaces. Our results are new development of finding a common element of fixed point of nonlinear operators and variational inequality problems.


## 2. Preliminaries

In this paper, we use symbols→ and ⇀ to denote strong and weak convergence, respectively. A Banach space (X, ∥·∥) is said to satisfy Opial's condition, if for each sequence {xn} in X with xn⇀x∈X, we have
(2.1)liminfn→∞∥xn-x∥<liminfn→∞∥xn-y∥,∀y∈X,y≠x.
It is well known that each Hilbert space satisfies Opial's condition; see, for example, [19]. Let T:X→X be a mapping. In this paper, the set of fixed points of T is denoted by F(T).A set-valued mappingT1:H→2H is said to be monotone, if for all x,y∈H,f∈T1x, and g∈T1y imply that 〈f-g,x-y〉≥0. A monotone mapping T1:H→H is said to be maximal, if the graph G(T1) of T1 is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T1 is maximal, if and only if for (x,f)∈H×H, 〈f-g,x-y〉≥0 for every (y,g)∈G(T1) implies that f∈T1x. Let T:C→H be a monotone mapping and let NCv be the normal cone to C at v∈C, that is, NCv={w∈H:〈v-u,w〉≥0,forallu∈C}. Define
(2.2)T1v={Tv+NCv,v∈C,∅,v∉C.
Then T1 is maximal monotone and 0∈T1v if and only if v∈VI(C,T), where 0 is the zero vector of H; see, for example, [9, 20, 21] for more details.For anyx∈H, there exists a unique nearest point in C, denoted by PC(x), such that ∥x-PC(x)∥≤∥x-y∥ for all y∈C. The mapping PC is called the projection operator (or metric projection) from H onto C.LetH1 and H2 be two Hilbert spaces. Let A:H1→H2 and B:H2→H1 be two bounded linear operators. B is called the adjoint operator (or adjoint) of A, if for all z∈H1,w∈H2, B satisfies 〈Az,w〉=〈z,Bw〉. It is known that the adjoint operator of a bounded linear operator on a Hilbert space always exists and is bounded linear and unique. Moreover, it is not hard to show that if B is an adjoint operator of A, then ∥A∥=∥B∥.A mappingT:C→C is said to be(1)
v-expansive if there exists a constant v>0 such that ∥Tx-Ty∥≥v∥x-y∥ for all x,y∈C. In particular, if v=1, then T is called expansive.(2)
v-strongly monotone if there exists a constant v>0 such that
(2.3)〈Tx-Ty,x-y〉≥v∥x-y∥2,∀x,y∈C.
Clearly, any v-strongly monotone mapping is v-expansive.(3)
u-inverse strongly monotone if there exists a constant u>0 such that
(2.4)〈Tx-Ty,x-y〉≥u∥Tx-Ty∥2,∀x,y∈C.(4)
Relaxedu-cocoercive if there exists a constant u>0 such that
(2.5)〈Tx-Ty,x-y〉≥(-u)∥Tx-Ty∥2,∀x,y∈C.(5)
Relaxed(u,v)-cocoercive if there exists constants u,v>0 such that
(2.6)〈Tx-Ty,x-y〉≥(-u)∥Tx-Ty∥2+v∥x-y∥2,∀x,y∈C.Especially, if u=0, then T is v-strongly monotone. So this class of mapping is more general than the class of strongly monotone mapping.(6)
Anα-Lipschitz mapping if there exists a  constant α>0 such that ∥Tx-Ty∥≤α∥x-y∥ for all x,y∈C. In particular, if 0<α<1 (α=1, resp.), then T is called a contraction (a nonexpansive mapping, resp.)Remark ST (see [9]).
IfT is v-strongly monotone and μ-Lipschitz continuous, that is, ∥Tx-Ty∥≤μ∥x-y∥ for all x,y∈C, then T is (v/μ2)-inverse strongly monotone.Example 2.1.
LetTx=-2x, for allx∈ℝ. Then it is easy to see that for any x,y∈ℝ,
(2.7)〈Tx-Ty,x-y〉=-2|x-y|2≥(-1)|Tx-Ty|2+|x-y|2.
Hence T is a relaxed (1,1)-cocoercive mapping, but T is not a strongly monotone mapping.Now, let{Ti}i∈ℕ be a family of infinitely nonexpansive mappings. In [22], a mapping Wn is defined by the following:
(2.8)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,
where {λi}i∈ℕ⊂[0,1]. Such a mapping Wn is called the W-mapping generated by Tn,Tn-1,…,T1 and  λn,λn-1,…,λ1.The following properties for aW-mapping are well known.Theorem 2.2 (see [22, 23]).
LetC be a nonempty closed convex subset of a Hilbert space E, let T1,T2,… be a family of infinitely nonexpansive mappings from C into itself such that ⋂i=1∞‍F(Ti) is nonempty, and let λ1,λ2,… be real numbers such that 0<λi≤b<1 for any i∈ℕ. Then the following statements hold:(1)
Wn is a nonexpansive mapping and F(Wn)=⋂i=1n‍F(Ti).(2)
For eachx∈C and for each positive integer k, the limit limn→∞Un,kx exists.(3)
The mappingW:C→C defined by Wx:=limn→∞Wnx=limn→∞Un,1x,x∈C, is a nonexpansive mapping satisfying F(W)=⋂i=1∞‍F(Ti) and it is called the W-mapping  generated by T1,T2,… and λ1,λ2….Theorem 2.3 (see [23]).
LetC be a nonempty closed convex subset of a Hilbert space H, T1,T2,… be nonexpansive mappings with ⋂i=1∞‍F(Ti)=∅, {λi} be a real sequence such that 0<λi≤b<1 for any i∈ℕ. If K is any bounded subset of C, then
(2.9)limn→∞supx∈K∥Wx-Wnx∥=0.
In particular, if {xn}n∈ℕ is a bounded sequence in C, then limn→∞∥Wxn-Wnxn∥=0.The following results are crucial in this paper.Lemma 2.4 (see [19]).
For a givenz∈H, x∈C satisfies the inequality 〈x-z,y-x〉≥0,forally∈C if and only if x=PC(z), where PC is a projection operator from H onto C.
It is well known that the projection operatorPC is nonexpansive and satisfies
(2.10)∥PCx-PCy∥2≤〈PCx-PCy,x-y〉,∀x,y∈H.Lemma 2.5 (see [9]).
The elementu∈C is a solution of (VIP)T if and only if u∈C satisfies the relation u=PC(u-ρTu), where PC is the projection operator, ρ>0 is a constant.Lemma 2.6 (see [24]).
Let{an} be a nonnegative real sequence satisfying the following condition:
(2.11)an+1≤(1-λn)an+λnbn,∀n≥n0,
where n0 is some nonnegative integer, {λn} is a sequence in (0,1) and {bn} is a sequence in R such that (i)
∑n=0∞λn=∞;(ii)
limsupn→∞bn≤0 or ∑n=0∞λnbn is convergent.Thenlimn→∞an=0.Lemma 2.7 (see [25]).
Let{xn} and {yn} be bounded sequences in a Banach space E and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose xn+1=βnyn+(1-βn)xn  for all integers n≥0 and limsupn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0, then limn→∞∥yn-xn∥=0.Lemma 2.8 (see [26]).
LetE be a real Banach space and J:E→2E* be the normalized duality mapping, then for any x,y∈E the following inequality holds:
(2.12)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉,∀j(x+y)∈J(x+y).
Especially, whenE=H, then J=I. So, from Lemma 2.8, one has
(2.13)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.The following result is simple, but it is very useful in this paper.Lemma 2.9.
Let{an}, {bn} be two nonnegative real sequences. If limn→∞an=0, then liminfn→∞(an+bn)=liminfn→∞bn.


## 3. Main Results

In this section, we construct an iteration scheme including a pair of mappingsT:C→H1 and S:K→H2 which are u-inverse strongly monotone to solve the split variational inequality problem. For the purpose we first give the following Lemmas.Lemma 3.1 (see page 3 in [9]).
LetT:C→H be a u-inverse strongly monotone mapping. Then I-λT is nonexpansive for any λ∈[0,2u].Example 3.2.
LetTx=3x for all x∈ℝ and u=1/6. Since
(3.1)〈Tx-Ty,x-y〉=3|x-y|2≥u|Tx-Ty|2,T is u-inversely monotone. Let λ∈[0,1/3]=[0,2u]. It is easy to see that
(3.2)|(I-λT)x-(I-λT)y|=(1-3λ)|x-y|≤|x-y|.
So I-λT is nonexpansive for all λ∈[0,2u].Applying Lemma3.1, we have the following important result.Lemma 3.3.
LetT,S:C→H be two u-inverse strongly monotone mappings and S1:C→C be a nonexpansive mapping. Then for any given sequences {rn} and {sn} in [0,2u], PC(I-snT), PC(I-rnS), S1PC(I-snT) and S1PC(I-rnS) are all nonexpansive for all n∈ℕ.The following conclusion is immediate from Lemma2.5.Lemma 3.4.
The elementu∈C is a solution of (SVIP)T,S if and only if u∈C satisfies the relation
(3.3)u=PC(u-ρTu),Au=PK(u-ρSAu),
where PC and PK are the projection operators, ρ>0 is a constant.Theorem 3.5.
LetH1, H2 be two real Hilbert spaces and C⊂H1, K⊂H2 two nonempty closed convex sets. Let T:C→H1 and S:K→H2 be u-inversely monotone. Let A:H1→H2 be a bounded linear operator with adjoint operator A*. Let f:C→C be a contraction with contraction constant α. Let {Ti}i∈ℕ be a family of infinitely nonexpansive mappings of C into itself and U a nonexpansive mapping of K into itself such that Ω={p∈(⋂i=1∞‍F(Ti))⋂‍VI(C,T):Ap∈F(U)⋂‍VI(K,S)}≠∅. Let ξ be a real number and {λi}i∈ℕ be a sequence of real numbers such that 0<λi≤ξ<1 for every i∈ℕ. For each n∈ℕ, let Wn be the W-mapping of C into itself generated by Tn,Tn-1,…,T1 and λn,λn-1,…,λ1. Let {xn} be a sequence generated by the following algorithm:
(3.4)x0=x∈Cchosenarbitrarily,yn=PC(I-βnT)xn,ln=PK(I-βnS)Ayn,zn=(1-θ)xn+θWnPC(yn+rA*(Uln-Ayn)),xn+1=αnf(xn)+(1-αn)zn,∀n∈ℕ∪{0},
where r∈(0,1/∥A∥2) and θ∈(0,1) are two constants and {αn}n=0∞ and {βn}n=0∞ are two sequences in (0,1). If {αn}n=0∞ and {βn}n=0∞ further satisfy the following conditions:(
C
1
)
lim
n
→
∞
α
n
=
0 and ∑n=1∞‍αn=∞,(
C
2
)
{
β
n
}
⊂
[
a
,
b
] and limn→∞|βn+1-βn|=0, where 0<a,b<2u,then the following statements hold:(a)
there exists a uniqueq∈Ω such that PΩf(q)=q;(b)
{xn} converges strongly to q.Proof.
Letp∈Ω. By Lemma 3.3, PC(I-βnT), and PK(I-βnS) are nonexpansive for all n∈ℕ∪{0}. For each n∈ℕ∪{0}, by (3.4) and Lemma 3.4, we obtain the following inequalities:
(3.5)∥yn-p∥=∥PC(I-βnT)xn-PC(I-βnT)p∥≤∥xn-p∥,∥ln-Ap∥=∥PK(I-βnS)Ayn-PK(I-βnS)Ap∥≤∥Ayn-Ap∥,∥xn+1-p∥≤αn∥f(xn)-p∥+(1-αn)∥zn-p∥≤αn∥f(xn)-p∥+∥zn-p∥.
Let hn=PC(yn+rA*(Uln-Ayn)) for n∈ℕ∪{0}. Then
(3.6)∥Wnhn-p∥2≤∥hn-p∥2=∥PC(yn+rA*(Uln-Ayn))-p∥2≤∥yn+rA*(Uln-Ayn)-p∥2=∥yn-p∥2+∥rA*(Uln-Ayn)∥2+2r〈yn-p,A*(Uln-Ayn)〉≤∥yn-p∥2+∥rA*(Uln-Ayn)∥2+2r〈A(yn-p)+Uln-Ayn-(Uln-Ayn),Uln-Ayn〉=∥yn-p∥2+∥rA*(Uln-Ayn)∥2+2r{12∥Uln-Ap∥2+12∥Uln-Ayn∥2-∥Ayn-Ap∥2-∥Uln-Ayn∥2}≤∥yn-p∥2+r2∥A*∥2∥Uln-Ayn∥2-r∥Uln-Ayn∥2=∥yn-p∥2-r(1-r∥A*∥2)∥Uln-Ayn∥2,(3.7)∥zn-p∥2=∥(1-θ)(xn-p)+θ(Wnhn-p)∥2≤(1-θ)∥xn-p∥2+θ∥Wnhn-p∥2≤(1-θ)∥xn-p∥2+θ∥hn-p∥2≤(1-θ)∥xn-p∥2+θ∥yn-p∥2≤∥xn-p∥2,
for all n∈ℕ∪{0}. Next, we will show that the conclusion is true by several steps.
Step
1. We show that all {xn},{yn},{zn},{Txn},{SAyn},{ln}, and {Wnhn} are bounded.
To prove it, it suffices to show{xn} is bounded. Let p∈Ω. We claim that
(3.8)∥xn-p∥≤ℒ:=max{∥x0-p∥,∥f(p)-p∥1-α}∀n∈ℕ∪{0}.
Indeed, it is obvious that (3.8) is true for n=0. Assume that (3.7) is true for n=k, k∈ℕ. Since ∥yk-p∥≤∥xk-p∥ and ∥zk-p∥≤∥xk-p∥ by (3.5) and (3.7), it follows from (3.5) that
(3.9)∥xk+1-p∥≤αk∥f(xk)-p∥+(1-αk)∥zk-p∥=αk∥f(xk)-f(xk)+f(xk)-p∥+(1-αk)∥zk-p∥≤αkα∥xk-p∥+αk∥f(p)-p∥+(1-αk)∥xk-p∥≤(1-αk(1-α))∥xk-p∥+αk∥f(p)-p∥≤ℒ,
which prove that (3.8) is true for n=k+1. By induction, (3.8) holds for all n∈ℕ∪{0}. Hence, by (3.8), we know that {xn} is bounded and so are {xn},{yn},{zn},{Txn},{SAyn},{ln},{hn}, {Wnhn}, {ln}, and {Uln}. This also means that there exists a bounded subset C1⊂C such that
(3.10){xn},{yn},{zn},{Txn},{hn},{Wnhn}⊂C1.
Step
2. Prove limn→∞∥xn+1-xn∥=0.
For eachn∈ℕ∪{0}, by Lemma 3.1,
(3.11)∥yn+1-yn∥≤∥PC(I-βn+1T)xn+1-PC(I-βn+1T)xn∥+∥PC(I-βn+1T)xn-PC(I-βnT)xn∥≤∥xn+1-xn∥+|βn+1-βn|∥Txn∥.
Similarly,
(3.12)∥ln+1-ln∥≤∥PK(I-βn+1S)Ayn+1-PK(I-βn+1S)Ayn∥+∥PK(I-βn+1S)Ayn-PK(I-βnS)Ayn∥≤∥Ayn+1-Ayn∥+|βn+1-βn|∥SAyn∥,∀n∈ℕ∪{0}.
Since hn=PC(yn+rA*(Uln-Ayn)), n∈ℕ∪{0}, we have
(3.13)∥hn+1-hn∥2≤∥yn+1+rA*(Uln+1-Ayn+1)-(yn+rA*(Uln-Ayn))∥2≤∥yn+1-yn∥2+∥rA*(Uln+1-Ayn+1-(Uln-Ayn))∥2+2r〈yn+1-yn,A*(Uln+1-Ayn+1-(Uln-Ayn))〉≤∥yn+1-yn∥2+r2∥A*∥2∥Uln+1-Ayn+1-(Uln-Ayn)∥2+2r〈Ayn+1-Ayn+Uln+1-Ayn+1-(Uln-Ayn),Uln+1-Ayn+1-(Uln-Ayn)〉-2r〈Uln+1-Ayn+1-(Uln-Ayn),Uln+1-Ayn+1-(Uln-Ayn)〉=∥yn+1-yn∥2+r2∥A*∥2∥Uln+1-Ayn+1-(Uln-Ayn)∥2+2r{12∥Uln+1-Uln∥2+12∥Uln+1-Ayn+1-(Uln-Ayn)∥-12∥Ayn+1-Ayn∥2}-2r∥Uln+1-Ayn+1-(Uln-Ayn)∥2=∥yn+1-yn∥2-r(1-r∥A*∥2)∥Uln+1-Ayn+1-(Uln-Ayn)∥2+r{∥Uln+1-Uln∥2-∥Ayn+1-Ayn∥2}=∥yn+1-yn∥2-r(1-r∥A*∥2)∥Uln+1-Ayn+1-(Uln-Ayn)∥2+r|βn+1-βn|(∥Uln+1-Uln∥+∥Ayn+1-Ayn∥)∥SAyn∥≤∥xn+1-xn∥2+|βn+1-βn|(∥Txn∥∥xn+1-xn∥+|βn+1-βn|∥Txn∥2)-r(1-r∥A*∥2)∥Uln+1-Ayn+1-(Uln-Ayn)∥2+r|βn+1-βn|(∥Uln+1-Uln∥+∥Ayn+1-Ayn∥)∥SAyn∥≤∥xn+1-xn∥2+|βn+1-βn|M1,
where M1 is a constant such that
(3.14)(∥Txn∥∥xn+1-xn∥+|βn+1-βn|∥Txn∥2)+r(∥Uln+1-Uln∥+∥Ayn+1-Ayn∥)∥SAyn∥≤M1
for any n∈ℕ∪{0}. Since {hn}⊂C1, for each n∈ℕ∪{0}, we have
(3.15)∥Wn+1hn+1-Wnhn∥≤∥Wn+1hn+1-Whn+1∥+∥Whn+1-Whn∥+∥Whn-Wnhn∥≤supx∈C1∥Wn+1x-Wx∥+supx∈C1∥Wx-Wnx∥+∥hn+1-hn∥,(3.16)∥Wn+1hn+1-Wnhn∥2≤∥hn+1-hn∥2+ωn,
where
(3.17)ωn=(supx∈C1∥Wn+1x-Wx∥+supx∈C1∥Wx-Wnx∥)×(supx∈C1∥Wn+1x-Wx∥+supx∈C1∥Wx-Wnx∥+2∥hn+1-hn∥).
So, we have
(3.18)∥zn+1-zn∥2≤(1-θ)∥xn+1-xn∥2+θ∥Wn+1hn+1-Wnhn∥2≤(1-θ)∥xn+1-xn∥2+θ∥hn+1-hn∥2+ωn≤(1-θ)∥xn+1-xn∥2+θ∥xn+1-xn∥2+|βn+1-βn|M1+ωn=∥xn+1-xn∥2+|βn+1-βn|M1+ωn,
for any n∈ℕ∪{0}.
Choose a sequence{y¯n} such that xn+1=γny¯n+(1-γn)xn, where γn=1-(1-θ)(1-αn), then we have
(3.19)y¯n=αnf(xn)+(1-αn)θWnhnγn,∀n∈ℕ∪{0}.
It follows that
(3.20)∥y¯n+1-y¯n∥≤αnγn∥f(xn)∥+αn+1γn+1∥f(xn+1)∥+(1-αn+1)θγn+1∥Wn+1hn+1-Wnhn∥+|(1-αn+1)θγn+1-(1-αn)θγn|∥Wnhn∥≤(αn+αn+1)M3+(1-αn+1)θγn+1∥Wn+1hn+1-Wnhn∥+|αn-αn+1γn+1γn|θ∥Wnhn∥≤2(αn+αn+1)M3+(1-αn+1)θγn+1∥Wn+1hn+1-Wnhn∥,
where M3 is a constant such that supn∈ℕ∪{0}{∥f(xn)/γn∥,∥Wnhn∥}≤M3. From (3.20), (3.16), and (3.13) we have
(3.21)∥y¯n+1-y¯n∥2≤4(αn+1+αn)2M32+(1-αn+1)2θ2γn+12∥Wn+1hn+1-Wnhn∥2+4(αn+αn+1)M3(1-αn+1)θγn+1∥Wn+1hn+1-Wnhn∥≤4(αn+1+αn)2M32+(1-αn+1)2θ2γn+12∥hn+1-hn∥2+(1-αn+1)2θ2γn+12ωn+4(αn+αn+1)M3(1-αn+1)θγn+1∥Wn+1hn+1-Wnhn∥≤4(αn+1+αn)2M32+(1-αn+1)2θ2γn+12∥xn+1-xn∥2+(1-αn+1)2θ2γn+12|βn+1-βn|M1+(1-αn+1)2θ2γn+12ωn+4(αn+αn+1)M3(1-αn+1)θγn+1∥Wn+1hn+1-Wnhn∥.
Applying the condition (C2), it follows from (3.21) that
(3.22)limsupn→∞{∥y¯n+1-y¯n∥2-∥xn+1-xn∥2}=0,
which implies
(3.23)limsupn→∞{∥y¯n+1-y¯n∥-∥xn+1-xn∥}=0.
Applying Lemma2.7, we obtain limn→∞∥y¯n-xn∥→0 which implies that
(3.24)limn→∞∥xn+1-xn∥=limn→∞γn∥y¯n-xn∥=0.
Step
3. Prove limn→∞∥Txn-Tp∥=limn→∞∥SAyn-SAp∥=0.
For anyn∈ℕ∪{0}, we have
(3.25)∥yn-p∥2=∥PC(I-βnT)xn-PC(I-βnT)p∥2≤∥(I-βnT)xn-(I-βnT)p∥2=∥xn-p∥2-2βn〈xn-p,Txn-Tp〉+βn2∥Txn-Tp∥2≤∥xn-p∥2-βn(2u-βn)∥Txn-Tp∥2.
Similarly,
(3.26)∥Uln-Ap∥2≤∥ln-Ap∥2=∥PK(I-βnS)Ayn-Ap∥2≤∥Ayn-Ap∥2-βn(2u-βn)∥SAyn-SAp∥2.
From (3.5) again, we have
(3.27)∥xn+1-p∥2≤(αn∥f(xn)-p∥+∥zn-p∥)2≤αnM4+∥zn-p∥2,
where M4 is a constant such that supn∈ℕ∪{0}{αn∥f(xn)-p∥2+2∥f(xn)-p∥∥zn-p∥}≤M4. It follows that
(3.28)0<θβn(2u-βn)∥Txn-Tp∥2≤θ∥xn-p∥2-θ∥yn-p∥2(by(3.24))≤∥xn-p∥2-∥zn-p∥2(by(3.7))≤∥xn-p∥2+αnM4-∥xn+1-p∥2(by(3.26))≤(∥xn-p∥+∥xn+1-p∥)∥xn+1-xn∥+αnM4→0asn→∞,
which yields that limn→∞∥Txn-Tp∥=0 (by the condition 0<a≤βn≤b<2u).
For anyn∈ℕ∪{0}, by  (3.6), (3.7), and (3.27), we have
(3.29)θr(1-r∥A*∥2)∥Uln-Ayn∥2≤θ∥yn-p∥2-θ∥Wnhn-p∥2≤θ∥xn-p∥2-θ∥zn-p∥2≤θ∥xn-p∥2+αnM4-∥xn+1-p∥2=(∥xn-p∥+∥xn+1-p∥)∥xn+1-xn∥+αnM4.
So,
(3.30)limn→∞∥Uln-Ayn∥=0.
From (3.26) and (3.30) again, we have
(3.31)0<βn(2u-βn)∥SAyn-SAp∥2≤∥Ayn-Ap∥2-∥Uln-Ap∥2=(∥Ayn-Ap∥+∥Uln-Ap∥)(∥Ayn-Ap∥-∥Uln-Ap∥)≤(∥Ayn-Ap∥+∥Ap∥)∥Uln-Ayn∥→0asn→∞,
which implies that limn→∞∥SAyn-SAp∥=0 (by the condition 0<a≤βn≤b<2u).
On the other hand, since(3.32)∥ln-Ap∥2=∥PK(I-βnS)Ayn-PK(I-βnS)Ap∥2≤〈(I-βnS)Ayn-(I-βnS)Ap,In-Ap〉=12{∥(I-βnS)Ayn-(I-βnS)Ap∥2+∥ln-Ap∥2-∥ln-Ayn-βn(SAyn-SAp)∥2}≤12{∥Ayn-Ap∥2+∥ln-Ap∥2-∥ln-Ayn∥2+2βn〈ln-Ayn,SAyn-SAp〉-βn2∥SAyn-SAp∥2}≤12{∥Ayn-Ap∥2+∥ln-Ap∥2-∥ln-Ayn∥2+2βn∥ln-Ayn∥∥SAyn-SAp∥},
we get
(3.33)∥ln-Ap∥2≤∥Ayn-Ap∥2-∥ln-Ayn∥2+2βn∥ln-Ayn∥∥SAyn-SAp∥,∀n∈ℕ∪{0}.
By (3.6) and (3.33), we have
(3.34)∥Wnhn-p∥2≤∥yn-p∥2+∥rA*(Uln-Ayn)∥2+2r{12∥Uln-Ap∥2+12∥Uln-Ayn∥2-∥Ayn-Ap∥2-∥Uln-Ayn∥2}≤∥yn-p∥2+∥rA*(Uln-Ayn)∥2+2r{12∥ln-Ap∥2+12∥Uln-Ayn∥2-∥Ayn-Ap∥2-∥Uln-Ayn∥2}≤∥yn-p∥2+∥rA*(Uln-Ayn)∥2+2r{-12∥ln-Ayn∥2+βn∥ln-Ayn∥∥SAyn-SAp∥-12∥Uln-Ayn∥2}≤∥yn-p∥2+r2∥A*∥2∥Uln-Ayn∥2-r∥Uln-Ayn∥2-r∥ln-Ayn∥2+2rβn∥ln-Ayn∥∥SAyn-SAp∥=∥yn-p∥2+r(1-r∥A*∥2)∥Uln-Ayn∥2-r∥ln-Ayn∥2+2rβn∥ln-Ayn∥∥SAyn-SAp∥≤∥yn-p∥2-r∥ln-Ayn∥2+2rβn∥ln-Ayn∥∥SAyn-SAp∥,
Using (3.7), (3.27), and (3.34), we obtain
(3.35)θr∥ln-Ayn∥2≤θ∥yn-p∥2-θ∥Wnhn-p∥2+2rθβn∥ln-Ayn∥∥SAyn-SAp∥≤θ∥xn-p∥2-θ∥zn-p∥2+2rθβn∥ln-Ayn∥∥SAyn-SAp∥≤θ∥xn-p∥2+αnM4-∥xn+1-p∥2+2rθβn∥ln-Ayn∥∥SAyn-SAp∥=(∥xn-p∥+∥xn+1-p∥)∥xn+1-xn∥+αnM4+2rθβn∥ln-Ayn∥∥SAyn-SAp∥→0asn→∞,
which implies
(3.36)limn→∞∥ln-Ayn∥=0.
According to (3.30) and (3.36), we derive that
(3.37)limn→∞∥Uln-ln∥=0.
Step
4. Prove  limn→∞∥xn-zn∥=limn→∞∥yn-xn∥=limn→∞∥hn-xn∥=0.
Since(3.38)limn→∞∥xn+1-xn∥=0,limn→∞∥xn+1-zn∥=limn→∞αn∥f(xn)-zn∥=0,
we have limn→∞∥xn-zn∥=0. For any n∈ℕ∪{0}, since
(3.39)∥yn-p∥2=∥PC(I-βnT)xn-PC(I-βnT)p∥2≤〈(I-βnT)xn-(I-βnT)p,yn-p〉=12{∥(I-βnT)xn-(I-βnT)p∥2+∥yn-p∥2-∥xn-yn-βn(Txn-Tp)∥2}≤12{∥xn-p∥2+∥yn-p∥2-∥xn-yn∥2+2βn〈xn-yn,Txn-Tp〉-βn2∥Txn-Tp∥2}≤12{∥xn-p∥2+∥yn-p∥2-∥xn-yn∥2+2βn∥xn-yn∥∥Txn-Tp∥},
we get
(3.40)∥yn-p∥2≤∥xn-p∥2-∥xn-yn∥2+2βn∥xn-yn∥∥Txn-Tp∥.
It follows from (3.27), (3.7), and (3.40) that
(3.41)∥xn+1-p∥2≤αnM4+∥zn-p∥2≤αnM4+(1-θ)∥xn-p∥2+θ∥yn-p∥2≤αnM4+∥xn-p∥2-θ∥xn-yn∥2+2θβn∥xn-yn∥∥Txn-Tp∥,
which yields that
(3.42)θ∥xn-yn∥2≤αnM4+∥xn-p∥2-∥xn+1-p∥2+2βn∥xn-yn∥∥Txn-Tp∥≤αnM4+(∥xn-p∥+∥xn+1-p∥)∥xn+1-xn∥+2βn∥xn-yn∥∥Txn-Tp∥,
By Steps 1–3 and limn→∞αn=0, it follows from (3.42) that limn→∞∥xn-yn∥=0.
Since∥hn-yn∥≤∥γA*∥∥ln-Ayn∥ for all n∈ℕ∪{0}, we have limn→∞∥hn-yn∥=0. Using it with limn→∞∥xn-yn∥=0 and limn→∞∥xn-zn∥=0, we get limn→∞∥xn-hn∥=0.
Step
5. Prove  limn→∞∥Wnxn-xn∥=0 and  limn→∞∥Wxn-xn∥=0.
Indeed, since(3.43)∥Wnhn-xn∥=1θ∥zn-xn∥→0asn→∞,
we have
(3.44)∥Wnxn-xn∥≤∥Wnxn-Wnhn∥+∥Wnhn-xn∥≤∥xn-hn∥+∥Wnhn-xn∥→0asn→∞.
By Theorem 2.3,  limn→∞∥Wxn-Wnxn∥=0. Since
(3.45)∥Wxn-xn∥≤∥Wxn-Wnxn∥+∥Wnxn-xn∥∀n,
we obtain  limn→∞∥Wxn-xn∥=0.
Step
6. There exists a unique q∈Ω⊂H such that PΩf(q)=q.
Indeed, for anyx, y∈H,
(3.46)∥PΩf(x)-PΩf(y)∥≤∥f(x)-f(y)∥≤α∥x-y∥.
Since α∈[0,1), PΩf is a contraction on H. Applying Banach contraction principle, there exists a unique q∈H such that q=PΩf(q)∈Ω.
Step
7. Prove limsupn→∞〈fq-q,xn-q〉≤0.
For this purpose, we may choose subsequence{xni} of {xn} such that
(3.47)limsupn→∞〈fq-q,xn-q〉=limi→∞〈fq-q,xni-q〉.
Since {xni} is a bounded sequence, there exists a subsequence of {xni}, which is still denoted by {xni}, such that xni⇀x*∈H. Therefore, we have
(3.48)limsupi→∞〈fq-q,xni-q〉=〈fq-q,x*-q〉.
Next we prove x*∈Ω.
(a)Wx*=x*. In fact, if Wx*≠x*, then we have
(3.49)liminfi→∞∥xni-x*∥<liminfi→∞∥xni-Wx*∥≤liminfi→∞(∥xni-Wxni∥+∥Wxni-Wx*∥)≤liminfi→∞(∥xni-Wxni∥+∥xni-x*∥)=liminfi→∞∥xni-x*∥(byStep5andLemma2.9).
This is a contradiction. Hence, Wx*=x*, which implies that x*∈⋂i=1∞‍F(Ti) by Theorem 2.2.
(b) Provex*∈VI(C,T). Since xni⇀x* and limi→∞∥xni-yni∥=0, we have yni⇀x*. Let
(3.50)T1x={Tx+NCx,x∈C,∅,x∉C.
Since T is u-inversely monotone, T is monotone and hence T1 is a maximal monotone mapping. For any given (x,z)∈G(T1), since z-Tx∈NCx and yn∈C, by the definition of NC, we have
(3.51)〈x-yn,z-Tx〉≥0∀n.
On the other hand, since yn=PC(I-βnT)xn, we have
(3.52)〈x-yn,yn-(xn-βnTxn)〉≥0.
In particular,
(3.53)〈x-yn,yn-xnβn+Txn〉≥0∀n.
From (3.51) and (3.53) we have
(3.54)〈x-yni,z〉≥〈x-yni,Tx〉≥〈x-yni,Tx〉-〈x-yni,yni-xniβni+Txni〉=〈x-yni,Tx-Tyni〉+〈x-yni,Tyni-Txni〉-〈x-yni,yni-xniβni〉≥〈x-yni,Tyni-Txni〉-〈x-yni,yni-xniβni〉,
which implies
(3.55)〈x-x*,z〉=limi→∞〈x-yni,z〉≥0.
This shows 0∈T1x*, that is, x*∈VI(C,T).
(c) ProveAx*∈VI(K,S) and Ax*∈F(U). Let
(3.56)T2x={Sx+NKx,x∈K,∅,x∉K.
Then T2 is a maximal monotone mapping. For any given (x,z)∈G(T2), since z-Sx∈NKx and ln∈K, by the definition of NK we have
(3.57)〈x-ln,z-Sx〉≥0∀n.
On the other hand, since ln=PK(I-βnS)Ayn, we have
(3.58)〈x-ln,ln-(Ayn-βnSAyn)〉≥0,
and hence
(3.59)〈x-ln,ln-Aynβn+SAyn〉≥0∀n.
Since xni⇀x* and limn→∞∥xni-yni∥=0 and limn→∞∥lni-Ayni∥=0, we have lni⇀Ax*. From (3.57) and (3.59) we have
(3.60)〈x-lni,z〉≥〈x-lni,Sx〉≥〈x-lni,Sx-Slni〉+〈x-lni,Slni-SAyni〉-〈x-lni,lni-Ayniβni〉≥〈x-lni,lni-Ayni〉-〈x-lni,lni-Ayniβni〉.
So
(3.61)〈x-Ax*,z〉=limi→∞〈x-lni,z〉≥0.
This shows 0∈T2Ax*, that is, Ax*∈VI(K,S). In addition, by (3.36), (3.37), and Opial's condition, we can prove easily Ax*∈F(U).
By (a), (b), and (c),x*∈Ω is proved. Hence
(3.62)limsupn→∞〈fq-q,xn-q〉=〈fq-q,x*-q〉≤0.
Step
8. Prove {xn} converges strongly to q=PΩf(q)∈Ω.
In fact, by Step 7 we have(3.63)limsupn→∞〈fq-q,xn-q〉≤0.
It follows from (3.5) and Lemma 2.8 that
(3.64)∥xn+1-q∥2=∥αnf(xn)+(1-αn)zn-q∥2≤(1-αn)2∥zn-q∥2+2αn〈f(xn)-q,xn+1-q〉≤(1-αn)2∥xn-q∥2+2αn〈f(xn)-f(q)+f(q)-q,xn+1-q〉≤(1-αn)2∥xn-q∥2+2αnα∥xn-q∥∥xn+1-q∥+2αn〈f(q)-q,xn+1-q〉≤(1-αn(2-2α))∥xn-q∥2+αn2∥xn-q∥2+2αnα∥xn-q∥∥xn+1-xn∥+2αn〈f(q)-q,xn+1-q〉∀n∈ℕ∪{0}.
Let λn=αn(2-2α), n∈ℕ∪{0}. Then, for any n∈ℕ∪{0}, we obtain
(3.65)αn2∥xn-q∥2=λnαn2-2α∥xn-q∥2,2αnα∥xn-q∥∥xn+1-xn∥=λnα1-α∥xn-q∥∥xn+1-xn∥,2αn〈f(q)-q,xn+1-q〉=λn1-α〈f(q)-q,xn+1-q〉.
Set
(3.66)bn=αn2-2α∥xn-q∥2+α1-α∥xn-q∥∥xn+1-xn∥+11-α〈f(q)-q,xn+1-q〉.
The condition (C1) and the boundedness of {xn} ensure limn→∞(αn/(2-2α))∥xn-q∥2=0, Step 2 ensure limn→∞(1/(1-α))α∥xn-q∥∥xn+1-xn∥=0, and Step 7 ensure limsupn→∞(1/(1-α))〈f(q)-q,xn+1-q〉≤0, so limsupn→∞bn≤0. Applying Lemma 2.6 and the inequality
(3.67)∥xn+1-q∥2≤(1-λn)∥xn-q∥2+λnbn,
we obtain that limn→∞∥xn-q∥=0 which means that the sequence {xn} strongly converge to q. This completes the proof of the Theorem 3.5.The following convergence theorems can be established by applying Theorem3.5 with U=I.Corollary 3.6.
LetH1, H2 be two real Hilbert spaces and C⊂H1, K⊂H2 two nonempty closed convex sets. Let T:C→H1 and S:K→H2 be u-inversely monotone. A:H1→H2 is a bounded linear operator with adjoint operator A*. Let f:C→C be a contraction with contraction constant α. Let {Ti}i∈ℕ be a family of infinitely nonexpansive mappings of C into itself such that Ω={p∈⋂i=1∞‍F(Ti)⋂‍VI(C,T):Ap∈VI(K,S)}≠∅. Let ξ be a real number and {λi}i∈ℕ be a sequence of real numbers such that 0<λi≤ξ<1, for every i∈ℕ. For each n∈ℕ, let Wn be the W-mapping of C into itself generated by Tn,Tn-1,…,T1 and λn,λn-1,…,λ1. Let {xn} be a sequence generated by the following algorithm:
(3.68)x0=x∈Cchosenarbitrarily,yn=PC(I-βnT)xn,ln=PK(I-βnS)Ayn,zn=(1-θ)xn+θWnPC(yn+rA*(ln-Ayn)),xn+1=αnf(xn)+(1-αn)zn,∀n∈ℕ∪{0},
where r∈(0,1/∥A∥2) and θ∈(0,1) are two constants and {αn}n=0∞ and {βn}n=0∞ are two sequences in (0,1). If {αn}n=0∞ and {βn}n=0∞ further satisfy the following conditions:
(C1) limn→∞αn=0 and ∑n=1∞‍αn=∞;
(C2) {βn}⊂[a,b] and limn→∞|βn+1-βn|=0, where 0<a,b<2u;then the following statements hold:(a)
there exists a uniqueq∈Ω such that PΩf(q)=q;(b)
{xn} converges strongly to q.IfTi=I for all i∈ℕ in Theorem 3.5, then we have the following result.Corollary 3.7.
LetH1, H2 be two real Hilbert spaces and C⊂H1, K⊂H2 two nonempty closed convex sets. Let T:C→H1 and S:K→H2 be u-inversely monotone. A:H1→H2 is a bounded linear operator with adjoint operator A*. Let f:C→C be a contraction with contraction constant α. Let U be a nonexpansive mapping of K into itself such that Ω={p∈VI(C,T):Ap∈F(U)⋂‍VI(K,S)}≠∅. Let ξ be a real number and {λi}i∈ℕ be a sequence of real numbers such that 0<λi≤ξ<1, for every i∈ℕ. For each n∈ℕ, let Wn be the W-mapping of C into itself generated by Tn,Tn-1,…,T1 and λn,λn-1,…,λ1. Let {xn} be a sequence generated by the following algorithm:
(3.69)x0=x∈Cchosenarbitrarily,yn=PC(I-βnT)xn,ln=PK(I-βnS)Ayn,zn=(1-θ)xn+θPC(yn+rA*(Uln-Ayn)),xn+1=αnf(xn)+(1-αn)zn,∀n∈ℕ∪{0},
where r∈(0,1/∥A∥2) and θ∈(0,1) are two constants and {αn}n=0∞ and {βn}n=0∞ are two sequences in (0,1). If {αn}n=0∞ and {βn}n=0∞ further satisfy the following conditions:
(C1) limn→∞αn=0 and ∑n=1∞‍αn=∞;
(C2) {βn}⊂[a,b] and limn→∞|βn+1-βn|=0, where 0<a,b<2u;
then the following statements hold:(a)
there exists a uniqueq∈Ω such that PΩf(q)=q;(b)
{xn} converges strongly to q.In Theorem3.5, if Ti=I for all i∈ℕ and U=I, then we obtain Corollary 3.8.Corollary 3.8.
LetH1, H2 be two real Hilbert spaces and C⊂H1, K⊂H2 two nonempty closed convex sets. Let T:C→H1 and S:K→H2 be u-inversely monotone. A:H1→H2 is a bounded linear operator with adjoint operator A*. Let f:C→C be a contraction with contraction constant α. Let {Ti}i∈ℕ be a family of infinitely nonexpansive mappings of C into itself and U a nonexpansive mapping of K into itself such that Ω={p∈VI(C,T):Ap∈VI(K,S)}≠∅. Let ξ be a real number and {λi}i∈ℕ be a sequence of real numbers such that 0<λi≤ξ<1, for every i∈ℕ. For each n∈ℕ, let Wn be the W-mapping of C into itself generated by Tn,Tn-1,…,T1 and λn,λn-1,…,λ1. Let {xn} be a sequence generated by the following algorithm:
(3.70)x0=x∈Cchosenarbitrarily,yn=PC(I-βnT)xn,ln=PK(I-βnS)Ayn,zn=(1-θ)xn+θPC(yn+rA*(ln-Ayn)),xn+1=αnf(xn)+(1-αn)zn,∀n∈ℕ∪{0},
where r∈(0,1/∥A∥2) and θ∈(0,1) are two constants and {αn}n=0∞ and {βn}n=0∞ are two sequences in (0,1). If {αn}n=0∞ and {βn}n=0∞ further satisfy the following conditions:
(C1) limn→∞αn=0 and ∑n=1∞‍αn=∞;
(C2) {βn}⊂[a,b] and limn→∞|βn+1-βn|=0, where 0<a,b<2u;
then the following statements hold:(a)
there exists a uniqueq∈Ω such that PΩf(q)=q;(b)
{xn} converges strongly to q.In Theorem3.5, if H1=H2, C=K and A=A*=I, then we have Corollary 3.9.Corollary 3.9.
LetH be a real Hilbert space and C⊂H a nonempty closed convex set. Let T:C→H and S:C→H be u-inversely monotone. Let f:C→C be a contraction with contraction constant α. Let U:C→C be a nonexpansive mapping and {Ti}i∈ℕ:C→C a family of infinitely nonexpansive mappings such that Ω=⋂i=1∞‍F(Ti)⋂‍F(U)⋂‍VI(C,T)⋂‍VI(K,S)≠∅. Let ξ be a real number and {λi}i∈ℕ be a sequence of real numbers such that 0<λi≤ξ<1, for every i∈ℕ. For each n∈ℕ, let Wn be the W-mapping of C into itself generated by Tn,Tn-1,…,T1 and λn,λn-1,…,λ1. Let {xn} be a sequence generated by the following algorithm:
(3.71)x0=x∈Cchosenarbitrarily,yn=PC(I-βnT)xn,ln=PC(I-βnS)yn,zn=(1-θ)xn+θWn(yn+r(Uln-yn)),xn+1=αnf(xn)+(1-αn)zn,∀n∈ℕ∪{0},
where r∈(0,1) and θ∈(0,1) are two constants and {αn}n=0∞ and {βn}n=0∞ are two sequences in (0,1). If {αn}n=0∞ and {βn}n=0∞ further satisfy the following conditions:
(C1) limn→∞αn=0 and ∑n=1∞‍αn=∞;
(C2) {βn}⊂[a,b] and limn→∞|βn+1-βn|=0, where 0<a,b<2u;
then the following statements hold:(a)
there exists a uniqueq∈Ω such that PΩf(q)=q;(b)
{xn} converges strongly to q.In Theorem3.5, if H1=H2, C=KA=A*=I, and U=I, then we get the following result.Corollary 3.10.
LetH be a real Hilbert space and C⊂H a nonempty closed convex set. Let T:C→H and S:C→H be u-inversely monotone. Let f:C→C be a contraction with contraction constant α. Let {Ti}i∈ℕ:C→C be a family of infinitely nonexpansive mappings such that Ω=⋂i=1∞‍F(Ti)⋂‍VI(C,T)⋂‍VI(K,S)≠∅. Let ξ be a real number and {λi}i∈ℕ be a sequence of real numbers such that 0<λi≤ξ<1, for every i∈ℕ. For each n∈ℕ, let Wn be the W-mapping of C into itself generated by Tn,Tn-1,…,T1 and λn,λn-1,…,λ1. Let {xn} be a sequence generated by the following algorithm:
(3.72)x0=x∈Cchosenarbitrarily,yn=PC(I-βnT)xn,ln=PC(I-βnS)yn,zn=(1-θ)xn+θWn(yn+r(ln-yn)),xn+1=αnf(xn)+(1-αn)zn,∀n∈ℕ∪{0},
where r∈(0,1) and θ∈(0,1) are two constants and {αn}n=0∞ and {βn}n=0∞ are two sequences in (0,1). If {αn}n=0∞ and {βn}n=0∞ further satisfy the following conditions:
(C1) limn→∞αn=0 and ∑n=1∞‍αn=∞;
(C2) {βn}⊂[a,b] and limn→∞|βn+1-βn|=0, where 0<a,b<2u;
then the following statements hold:(a)
there exists a uniqueq∈Ω such that PΩf(q)=q;(b)
{xn} converges strongly to q.Finally, if we letH1=H2, C=KA=A*=I, Ti=I, for all i∈ℕ and U=I in Theorem 3.5, then the following result can be established.Corollary 3.11.
LetH be a real Hilbert space and C⊂H a nonempty closed convex set. Let T:C→H and S:C→H be u-inversely monotone. Let f:C→C be a contraction with contraction constant α. Suppose that Ω=VI(C,T)⋂‍VI(K,S)≠∅. Let {xn} be a sequence generated by the following algorithm:
(3.73)x0=x∈Cchosenarbitrarily,yn=PC(I-βnT)xn,ln=PC(I-βnS)yn,zn=(1-θ)xn+θ(yn+r(ln-yn)),xn+1=αnf(xn)+(1-αn)zn,∀n∈ℕ∪{0},
where r∈(0,1) and θ∈(0,1) are two constants and {αn}n=0∞ and {βn}n=0∞ are two sequences in (0,1). If {αn}n=0∞ and {βn}n=0∞ further satisfy the following conditions:
(C1) limn→∞αn=0 and ∑n=1∞‍αn=∞;
(C2) {βn}⊂[a,b] and limn→∞|βn+1-βn|=0, where 0<a,b<2u;
then the following statements hold:(a)
there exists a uniqueq∈Ω such that PΩf(q)=q;(b)
{xn} converges strongly to q.Remark 3.12.
(a) In [11, 16], the authors gave some algorithms for (u,v)-cocoercive and μ-Lipschitz continuous operator and obtain some strongly convergence theorems; see [11, Theorems 2.1 and 2.2] and [16, Corollary 3.3]. However, the (u,v)-cocoercive and μ-Lipschitz continuous operator considered by [11, 16] is actually a strongly monotone and μ-Lipschitz continuous operator. Then, by Remark ST, such operators studied in [11, 16] are u-inverse strongly monotone. Hence our results obtained in this paper conclude some results in [11, 16] as special cases.
(b) Our results are different from the main results in [9–11, 16] and references therein.



---

*Source: 470354-2012-11-21.xml*