# Relaxed Viscosity Approximation Methods with Regularization for Constrained Minimization Problems

Authors: Lu-Chuan Ceng; Hong-Kun Xu; Ching-Feng Wen
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

We introduce a new relaxed viscosity approximation method with regularization and prove the strong convergence of the method to a common fixed point of finitely many nonexpansive mappings and a strict pseudocontraction that also solves a convex
minimization problem and a suitable equilibrium problem.


---

## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, C a nonempty closed convex subset of H, and PC the metric projection of H, onto C. Let T:C→C be self-mapping on C. We denote by Fix(T) the set of fixed points of T and by R the set of all real numbers. A mapping T:C→C is called ζ-strictly pseudocontractive if there exists a constant ζ∈[0,1) such that
(1)∥Tx-Ty∥2≤∥x-y∥2+ζ∥(I-T)x-(I-T)y∥2,∀x,y∈C.
In particular, if ζ=0, then T is called a nonexpansive mapping. A mapping A:C→H is called α-inverse strongly monotone, if there exists a constant α>0 such that
(2)〈Ax-Ay,x-y〉≥α∥x-y∥2,∀x,y∈C.Letf:C→R be a convex and a continuous  Fréchet differentiable functional. Consider the minimization problem (MP) of minimizing f over the constraint set C(3)minx∈Cf(x),
where we assume the existence of minimizers. We denote by Γ the set of minimizers of  (3). The gradient-projection algorithm (GPA) generates a sequence {xn} determined by the gradient ∇f and the metric projection PC as follows:
(4)xn+1:=PC(xn-λ∇f(xn)),∀n≥0,
or more generally,
(5)xn+1:=PC(xn-λn∇f(xn)),∀n≥0,
where, in both (4) and (5), the initial guess x0 is taken from C arbitrarily, the parameters λ or λn are positive real numbers. The convergence of algorithms (4) and (5) depends on the behavior of the gradient ∇f. As a matter of fact, it is known that if ∇f is strongly monotone and Lipschitz continuous, then, for 0<λ<2α/L2, the operator
(6)S:=PC(I-λ∇f)
is a contraction. Hence, the sequence {xn} defined by the GPA (4) converges in norm to the unique solution of (3). More generally, if the sequence {λn} is chosen to satisfy the property
(7)0<liminfn→∞λn≤limsupn→∞λn<2αL2,
then the sequence {xn} defined by the GPA (5) converges in norm to the unique minimizer of (3). If the gradient ∇f is only assumed to be a Lipschitz continuous, then {xn} can only be weakly convergent if H is infinite dimensional. A counterexample is given by Xu in [1].Since the Lipschitz continuity of the gradient∇f implies that it is inverse strongly monotone (ism), it can be expressed as a proper convex combination of the identity mapping and a nonexpansive mapping. Consequently, the GPA can be rewritten as the composite of a projectionand an averaged mapping which is again an averaged mapping. This shows that averaged mappings play an important role in the GPA. Very recently, Xu [1] used averaged mappings to study the convergence analysis of the GPA which is an operator-oriented approach.We observe that the regularization, in particular, the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems. Consider the following regularized minimization problem:(8)minx∈Cfα(x):=f(x)+α2∥x∥2,
where α>0 is the regularization parameter and again f is convex with an L-Lipschitz continuous gradient ∇f.The advantage of a regularization method is that it is possible to get strong convergence to the minimum-norm solution of the optimization problem under investigation. The disadvantage is however its implicity, and hence explicit iterative methods seem more attractive. See, for example, [1].Given a mappingA:C→H, the classical variational inequality problem (VIP) is to find x*∈C such that
(9)〈Ax*,x-x*〉≥0,∀x∈C.
The solution set of VIP (9) is denoted by VI(C,A). It is well known that x*∈VI(C,A) if and only if x*=PC(x*-λAx*) for some λ>0. The variational inequality was first discussed by Lions [2] and now is well known. The variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, and equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. See, for example, [3–10] and the references therein.In this paper, we study the following equilibrium problem (EP) which is to findx*∈C such that
(10)F(x*,y)+h(x*,y)≥0,∀y∈C.
The solution set of EP (10) is denoted by EP(F,h). We will introduce and consider a relaxed viscosity iterative scheme with regularization for finding a common element of the solution set Γ of the minimization problem (3), the solution set EP(F,h) of the equilibrium problem (10), and the common fixed point set Fix(T)∩(⋂iFix(Si)) of finitely many nonexpansive mappings Si:C→C,i=1,…,N, and a strictly pseudocontractive mapping T in the setting of the infinite-dimensional Hilbert space. We will prove that this iterative scheme converges strongly to a common fixed point of the mappings T,Si:C→C,i=1,…,N, which is both a minimizer of MP (3) and an equilibrium point of EP (10).


## 2. Preliminaries

LetH be a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let K 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} and ωs(xn) to denote the strong ω-limit set of the sequence {xn}; that is,
(11)ωw(xn)≔{{xniforsomesubsequence{xni}of{xn}}x∈H:xni⇀xforsomesubsequence{xni}of{xn}},ωs(xn)≔{forsomesubsequence{xni}of{xn}x∈H:xni⟶xforsomesubsequence{xni}of{xn}}.The metric (or nearest point) projection fromH onto K is the mapping PK:H→K which assigns to each point x∈H the unique point PKx∈K satisfying the property
(12)∥x-PKx∥=infy∈K∥x-y∥=:d(x,K).Some important properties of projections are gathered in the following.Proposition 1.
For givenx∈H and z∈K(i)
z=PKx⇔〈x-z,y-z〉≤0, for all  y∈K;(ii)
z=PKx⇔∥x-z∥2≤∥x-y∥2-∥y-z∥2,  for all  y∈K;(iii)
〈PKx-PKy,x-y〉≥∥PKx-PKy∥2,  for all  y∈H, which hence implies that  PK  is nonexpansive and monotone.Definition 2.
A mappingT:H→H is said to be(a)
nonexpansive if(13)∥Tx-Ty∥≤∥x-y∥,∀x,y∈H;(b)
firmly nonexpansive if2T-I is nonexpansive, or equivalently,
(14)〈x-y,Tx-Ty〉≥∥Tx-Ty∥2,∀x,y∈H;
alternatively, T is firmly nonexpansive if and only if T can be expressed as
(15)T=12(I+S),
where S:H→H is nonexpansive; projections are firmly nonexpansive.Definition 3.
LetT be a nonlinear operator with domain D(T)⊆H and range R(T)⊆H.(a)
T is said to be monotone if
(16)〈x-y,Tx-Ty〉≥0,∀x,y∈D(T).(b)
Given a numberβ>0, T is said to be β strongly monotone if
(17)〈x-y,Tx-Ty〉≥β∥x-y∥2,∀x,y∈D(T).(c)
Given a numberν>0, T is said to be ν-inverse strongly monotone (ν-ism) if
(18)〈x-y,Tx-Ty〉≥ν∥Tx-Ty∥2,∀x,y∈D(T).It can be easily seen that ifT is nonexpansive, then I-T is monotone. It is also easy to see that a projection PK is 1-ism. Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.Definition 4.
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,
(19)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 maps.Proposition 5 (see [11]).
LetT: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  isν-ism, then for  γ>0,  γT  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  isα-averaged if and only if  I-T  is  (1/2α)-ism.Proposition 6 (see [11]).
LetS,T,V:H→H be given operators.(i)
If  T=(1-α)S+αV  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+αV  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  {Ti}i=1N  is averaged, then so is the composite  T1⋯TN. In particular, if  T1  is  α1  -averaged and  T2  is  α2  -averaged, where  α1,α2∈(0,1), then the composite  T1T2  is  α-averaged, where  α=α1+α2-α1α2.(v)
If the mappings  {Ti}i=1N  are averaged and have a common fixed point, then(20)⋂i=1N‍Fix(Ti)=Fix(T1⋯TN).
The notationFix(T) denotes the set of all fixed points of the mapping T, that is, Fix(T)={x∈H:Tx=x}.It is clear that, in a real Hilbert spaceH, T:C→C is ζ-strictly pseudocontractive if and only if there holds the following inequality:
(21)〈Tx-Ty,x-y〉≤∥x-y∥2-1-ζ2×∥(I-T)x-(I-T)y∥2,∀x,y∈C.
This immediately implies that if T is a ζ-strictly pseudocontractive mapping, then I-T is ((1-ζ)/2)-inverse strongly monotone; for further detail, we refer to [12] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings.Lemma 7 (see [12, Proposition 2.1]).
LetC be a nonempty closed convex subset of a real Hilbert space H and T:C→C be a mapping.(i)
IfT is a ζ-strictly pseudocontractive mapping, then T satisfies the Lipschitz condition where
(22)∥Tx-Ty∥≤1+ζ1-ζ∥x-y∥,∀x,y∈C.(ii)
IfT is a ζ-strictly pseudocontractive mapping, then the mapping I-T is semiclosed at 0; that is, if {xn} is a sequence in C such that xn→x~ weakly and (I-T)xn→0 strongly, then (I-T)x~=0.(iii)
IfT is a ζ-(quasi-)strict pseudocontraction, then the fixed point set Fix(T) of T is closed and convex so that the projection PFix(T) is well defined.The following lemma is an immediate consequence of an inner product.Lemma 8.
In a real Hilbert spaceH, there holds the following inequality:
(23)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.The following elementary result on real sequences is quite well known.Lemma 9 (see [13]).
Let{an} be a sequence of nonnegative real numbers satisfying the property
(24)an+1≤(1-sn)an+sntn+ϵn,∀n≥0,
where {sn}⊂(0,1] and {tn} are the real sequences such that(i)
∑n=0∞‍sn=∞;(ii)
eitherlimsupn→∞tn≤0 or ∑n=0∞‍sn|tn|<∞;(iii)
∑n=0∞‍ϵn<∞ where ϵn≥0, for all n≥0. Then, limn→∞an=0.Lemma 10 (see [14]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a ζ-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers such that (γ+δ)ζ≤γ. Then,
(25)∥γ(x-y)+δ(Tx-Ty)∥≤(γ+δ)∥x-y∥,∀x,y∈C.The following lemma appears implicitly in the paper of Reinermann [15].Lemma 11 (see [15]).
LetH be a real Hilbert space. Then, for all x,y∈H and λ∈[0,1],
(26)∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)∥x-y∥2.Lemma 12 (see [16]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F:C×C→R be a bifunction such that(f1)
F(x,x)=0 for all x∈C;(f2)
F is monotone and upper hemicontinuous in the first variable;(f3)
F is lower semicontinuous and convex in the second variable.
Leth:C×C→R be a bifunction such that(h1)
h(x,x)=0 for all x∈C;(h2)
h is monotone and weakly upper semicontinuous in the first variable;(h3)
h is convex in the second variable.
Moreover, let one suppose that(H)
for fixedr>0 and x∈C, there exists a bounded K⊂C and x^∈K such that for all z∈C∖K,-F(x^,z)+h(z,x^)+(1/r)〈x^-z,z-x〉<0.
Forr>0 and x∈H, let Tr:H→2C be a mapping defined by
(27)Trx={1rz∈C:F(z,y)+h(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}
called the resolvent of F and h. Then,(1)
Trx≠∅;(2)
Trx is a singleton;(3)
Tr is firmly nonexpansive;(4)
EP(F,h)=Fix(Tr) and it is closed and convex.Lemma 13 (see [16]).
Let one suppose that (f1)–(f3), (h1)–(h3) and (H) hold. Letx,y∈H, r1,r2>0. Then,
(28)∥Tr2y-Tr1x∥≤∥y-x∥+|r2-r1r2|∥Tr2y-y∥.Lemma 14 (see [17]).
Suppose that the hypotheses of Lemma12 are satisfied. Let {rn} be a sequence in (0,∞) with liminfn→∞rn>0. Suppose that {xn} is a bounded sequence. Then, the following statements are equivalent and true.(a)
if∥xn-Trnxn∥→0 as n→∞, each weak cluster point of {xn} satisfies the problem:
(29)F(x,y)+h(x,y)≥0,∀y∈C,that is,ωw(xn)⊆EP(F,h).(b)
The demiclosedness principle holds in the sense that, ifxn⇀x* and ∥xn-Trnxn∥→0 as n→∞, then (I-Trk)x*=0 for all k≥1.


## 3. Main Results

We now propose the following relaxed viscosity iterative scheme with regularization:(30)F(un,y)+h(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn,1=βn,1S1un+(1-βn,1)un,yn,i=βn,iSiun+(1-βn,i)yn,i-1,i=2,…,N,yn=βnQyn,N+(1-βn)PC(yn,N-λ∇fαn(yn,N)),xn+1=σnyn+γnPC(yn-λ∇fαn(yn))+δnTPC(yn-λ∇fαn(yn)),
for all n≥0, where the mapping Q:C→C is a ρ-contraction; the mapping T:C→C is a ζ-strict pseudocontraction;  Si:C→C is a nonexpansive mapping for each i=1,…,N;  ∇f:C→H satisfies the Lipschitz condition (10) with 0<λ<(2/L);   F,h:C×C→R are two bifunctions satisfying the hypotheses of Lemma 12;  {αn} is a sequence in (0,∞) with ∑n=0∞‍αn<∞;   {βn},{σn} are sequences in (0,1) with 0<liminfn→∞σn≤limsupn→∞σn<1;  {γn},{δn} are sequences in [0,1] with σn+γn+δn=1, for all n≥0;   {βn,i}i=1N are sequences in (0,1) and (γn+δn)ζ≤γn, for all n≥0; {rn} is a sequence in (0,∞) with liminfn→∞rn>0 and liminfn→∞δn>0.Before stating and proving the main convergence results, we first establish the following lemmas.Lemma 15.
Let one suppose thatΩ=Fix(T)∩(⋂iFix(Si))∩EP(F,h)∩Γ≠∅. Then, the sequences {xn}, {yn}, {yn,i} for all i, and {un} are bounded.Proof.
First of all, we can show as in [18] that PC(I-λ∇fα) is nonexpansive for λ∈(0,2/(α+L)), and PC(I-λ∇fαn) is nonexpansive for all n≥0 and λ∈(0,2/L). We observe that if  p∈Ω, then
(31)∥yn,1-p∥≤∥un-p∥≤∥xn-p∥.
For all, from i=2 to i=N, by induction, one proves that
(32)∥yn,i-p∥≤βn,i∥un-p∥+(1-βn,i)∥yn,i-1-p∥≤∥un-p∥≤∥xn-p∥.
Thus, we obtain that for every i=1,…,N,
(33)∥yn,i-p∥≤∥un-p∥≤∥xn-p∥.
For simplicity, put y~n,N=PC(yn,N-λ∇fαn(yn,N)) and y~n=PC(yn-λ∇fαn(yn)) for every n≥0. Then, yn=αnQyn,N+(1-αn)y~n,N and xn+1=σnyn+γny~n+δnTy~n for every n≥0. Taking into consideration that Tp=p and PC(I-λ∇f)p=p for λ∈(0,2/L), we have
(34)∥y~n,N-p∥=∥PC(I-λ∇fαn)yn,N-PC(I-λ∇f)p∥≤∥PC(I-λ∇fαn)yn,N-PC(I-λ∇fαn)p∥+∥PC(I-λ∇fαn)p-PC(I-λ∇f)p∥≤∥yn,N-p∥+∥PC(I-λ∇fαn)p-PC(I-λ∇f)p∥≤∥yn,N-p∥+λαn∥p∥.
Similarly, we get ∥y~n-p∥≤∥yn-p∥+λαn∥p∥. Thus, from (34) we have
(35)∥yn-p∥=∥βn(Qyn,N-p)+(1-βn)(y~n,N-p)∥≤βn∥Qyn,N-p∥+(1-βn)∥y~n,N-p∥≤βn∥Qyn,N-Qp∥+βn∥Qp-p∥+(1-βn)(∥yn,N-p∥+λαn∥p∥)≤βnρ∥yn,N-p∥+βn∥Qp-p∥+(1-βn)(∥yn,N-p∥+λαn∥p∥)=(1-βn(1-ρ))∥yn,N-p∥+βn∥Qp-p∥+(1-βn)λαn∥p∥=(1-βn(1-ρ))∥yn,N-p∥+βn(1-ρ)∥Qp-p∥1-ρ+(1-βn)λαn∥p∥≤max{∥yn,N-p∥,∥Qp-p∥1-ρ}+λαn∥p∥≤max{∥un-p∥,∥Qp-p∥1-ρ}+λαn∥p∥≤max{∥xn-p∥,∥Qp-p∥1-ρ}+λαn∥p∥.
Since (γn+δn)ζ≤γn for all n≥0, utilizing Lemma 10, we derive from (35)
(36)∥xn+1-p∥=∥σ(yn-p)+γn(y~n-p)+δn(Ty~n-p)∥≤σn∥yn-p∥+∥γn(y~n-p)+δn(Ty~n-p)∥≤σn∥yn-p∥+(γn+δn)∥y~n-p∥≤σn∥yn-p∥+(γn+δn)(∥yn-p∥+λαn∥p∥)≤∥yn-p∥+λαn∥p∥≤max{∥xn-p∥,∥Qp-p∥1-ρ}+λαn∥p∥+λαn∥p∥=max{∥xn-p∥,∥Qp-p∥1-ρ}+2λαn∥p∥.
By induction, we get
(37)∥xn+1-p∥≤max{∥x0-p∥,∥Qp-p∥1-ρ}+2λ∥p∥·∑i=0n‍αi,∀n≥0.
This implies that {xn} is bounded and so are {yn}, {un}, and {yn,i} for each i=1,…,N. It is clear that both {y~n,N} and {y~n} are also bounded. Since ∥Ty~n-p∥≤((1+ζ)/(1-ζ))∥y~n-p∥, {Ty~n} is also bounded.Lemma 16.
Let one suppose thatΩ≠∅. Moreover, let one suppose that the following hold:(H1)
limn→∞βn=0 and ∑n=0∞‍βn=∞;(H2)
∑n=1∞‍|βn-βn-1|<∞ or limn→∞(|βn-βn-1|/βn)=0;(H3)
∑n=1∞‍|βn,i-βn-1,i|<∞ or limn→∞(|βn,i-βn-1,i|/βn)=0 for each i=1,…,N;(H4)
∑n=1∞‍|rn-rn-1|<∞ or limn→∞(|rn-rn-1|/βn)=0;(H5)
∑n=1∞‍|σn-σn-1|<∞ or limn→∞(|σn-σn-1|/βn)=0;(H6)
∑n=1∞‍|γn/(1-σn)-γn-1/(1-σn-1)|<∞ or limn→∞(1/βn)|γn/(1-σn)-γn-1/(1-σn-1)|=0. Then, limn→∞∥xn+1-xn∥=0, that is, {xn} is asymptotically regular.Proof.
Taking into account0<liminfn→∞σn≤limsupn→∞σn<1, we may assume, without loss of generality, that {σn}⊂[c,d] for some c,d∈(0,1). First, we write xn=σn-1yn-1+(1-σn-1)vn-1, for all n≥1, where vn-1=(xn-σn-1yn-1)/(1-σn-1). It follows that for all n≥1(38)vn-vn-1=xn+1-σnyn1-σn-xn-σn-1yn-11-σn-1=γny~n+δnTy~n1-σn-γn-1y~n-1+δn-1Ty~n-11-σn-1=γn(y~n-y~n-1)+δn(Ty~n-Ty~n-1)1-σn+(γn1-σn-γn-11-σn-1)y~n-1+(δn1-σn-δn-11-σn-1)Ty~n-1.
Since (γn+δn)ζ≤γn for all n≥0, utilizing Lemma 10, we have
(39)∥γn(y~n-y~n-1)+δn(Ty~n-Ty~n-1)∥≤(γn+δn)∥y~n-y~n-1∥.
Next, we estimate∥yn-yn-1∥. Observe that for every n≥1(40)∥y~n,N-y~n-1,N∥≤∥PC(I-λ∇fαn)yn,N-PC(I-λ∇fαn)yn-1,N∥+∥PC(I-λ∇fαn)yn-1,N-PC(I-λ∇fαn-1)yn-1,N∥≤∥yn,N-yn-1,N∥+∥PC(I-λ∇fαn)yn-1,N-PC(I-λ∇fαn-1)yn-1,N∥≤∥yn,N-yn-1,N∥+∥(I-λ∇fαn)yn-1,N-(I-λ∇fαn-1)yn-1,N∥=∥yn,N-yn-1,N∥+∥λ∇fαn(yn-1,N)-λ∇fαn-1(yn-1,N)∥=∥yn,N-yn-1,N∥+λ|αn-αn-1|∥yn-1,N∥,
and similarly,
(41)∥y~n-y~n-1∥≤∥yn-yn-1∥+λ|αn-αn-1|∥yn-1∥.
Also, from (30), we have
(42)yn=βnQyn,N+(1-βn)y~n,N,yn-1=βn-1Qyn-1,N+(1-βn-1)y~n-1,N,∀n≥1.
Simple calculations show that
(43)yn-yn-1=(1-βn)(y~n,N-y~n-1,N)+(βn-βn-1)(Qyn-1,N-y~n-1,N)+βn(Qyn,N-Qyn-1,N).
Then, passing to the norm we get from (40) that
(44)∥yn-yn-1∥≤(1-βn)∥y~n,N-y~n-1,N∥+|βn-βn-1|∥Qyn-1,N-y~n-1,N∥+βn∥Qyn,N-Qyn-1,N∥≤(1-βn)(∥yn,N-yn-1,N∥+λ|αn-αn-1|∥yn-1,N∥)+|βn-βn-1|∥Qyn-1,N-y~n-1,N∥+βnρ∥yn,N-yn-1,N∥≤(1-(1-ρ)βn)∥yn,N-yn-1,N∥+λ|αn-αn-1|∥yn-1,N∥+|βn-βn-1|∥Qyn-1,N-y~n-1,N∥≤(1-(1-ρ)βn)∥yn,N-yn-1,N∥+M1(|αn-αn-1|+|βn-βn-1|),
where λ∥yn,N∥+∥Qyn,N-y~n,N∥≤M1, for all n≥0 for some M1≥0. Furthermore, by the definition of yn,i one obtains that, for all i=N,…,2(45)∥yn,i-yn-1,i∥≤βn,i∥un-un-1∥+∥Siun-1-yn-1,i-1∥|βn,i-βn-1,i|+(1-βn,i)∥yn,i-1-yn-1,i-1∥.
In the case of i=1, we have
(46)∥yn,1-yn-1,1∥≤βn,1∥un-un-1∥+∥S1un-1-un-1∥|βn,1-βn-1,1|+(1-βn,1)∥un-un-1∥=∥un-un-1∥+∥S1un-1-un-1∥|βn,1-βn-1,1|.
Substituting (46) in all (45) type one obtains for i=2,…,N(47)∥yn,i-yn-1,i∥≤∥un-un-1∥+∑k=2i‍∥Skun-1-yn-1,k-1∥|βn,k-βn-1,k|+∥S1un-1-un-1∥|βn,1-βn-1,1|.
This together with (44) implies that
(48)∥yn-yn-1∥≤(1-(1-ρ)βn)×[∑k=2N‍∥un-un-1∥+∑k=2N‍∥Skun-1-yn-1,k-1∥|βn,k-βn-1,k|+∥S1un-1-un-1∥|βn,1-βn-1,1|∑k=2N‍]+M1(|αn-αn-1|+|βn-βn-1|)≤(1-(1-ρ)βn)∥un-un-1∥+∑k=2N‍∥Skun-1-yn-1,k-1∥|βn,k-βn-1,k|+∥S1un-1-un-1∥|βn,1-βn-1,1|+M1(|αn-αn-1|+|βn-βn-1|).
By Lemma 13, we know that
(49)∥un-un-1∥≤∥xn-xn-1∥+κ|1-rn-1rn|,
where κ=supn≥0∥un-xn∥. So, substituting (49) in (48) we obtain
(50)∥yn-yn-1∥≤(1-(1-ρ)βn)(∥xn-xn-1∥+κ|1-rn-1rn|)+∑k=2N‍∥Skun-1-yn-1,k-1∥|βn,k-βn-1,k|+∥S1un-1-un-1∥|βn,1-βn-1,1|+M1(|αn-αn-1|+|βn-βn-1|)≤(1-(1-ρ)βn)∥xn-xn-1∥+κ|rn-rn-1|rn+∑k=2N‍∥Skun-1-yn-1,k-1∥|βn,k-βn-1,k|+∥S1un-1-un-1∥|βn,1-βn-1,1|+M1(|αn-αn-1|+|βn-βn-1|)≤(1-(1-ρ)βn)∥xn-xn-1∥+M2[|rn-rn-1|rn+∑k=2N‍|βn,k-βn-1,k|+|βn,1-βn-1,1|+M2+|αn-αn-1|+|βn-βn-1|∑k=2N‍]≤(1-(1-ρ)βn)∥xn-xn-1∥+M2[|rn-rn-1|b+∑k=1N‍|βn,k-βn-1,k|+M2+|αn-αn-1|+|βn-βn-1|∑k=2N‍],
where b>0 is a minorant for {rn} and κ+M1+∑k=2N‍∥Skun-yn,k-1∥+∥S1un-un∥≤M2, for all n≥0 for some M2≥0. This together with (38)-(39), implies that
(51)∥vn-vn-1∥≤∥γn(y~n-y~n-1)+δn(Ty~n-Ty~n-1)∥1-σn+|γn1-σn-γn-11-σn-1|∥y~n-1∥+|δn1-σn-δn-11-σn-1|∥Ty~n-1∥≤(γn+δn)∥y~n-y~n-1∥1-σn+|γn1-σn-γn-11-σn-1|∥y~n-1∥+|γn1-σn-γn-11-σn-1|∥Ty~n-1∥=∥y~n-y~n-1∥+|γn1-σn-γn-11-σn-1|(∥y~n-1∥+∥Ty~n-1∥)≤∥yn-yn-1∥+λ|αn-αn-1|∥yn-1∥+|γn1-σn-γn-11-σn-1|(∥y~n-1∥+∥Ty~n-1∥)≤(1-(1-ρ)βn)∥xn-xn-1∥+M2[|rn-rn-1|b+∑k=1N‍|βn,k-βn-1,k|+M2+|αn-αn-1|+|βn-βn-1|∑k=2N‍]+λ|αn-αn-1|∥yn-1∥+|γn1-σn-γn-11-σn-1|(∥y~n-1∥+∥Ty~n-1∥)≤(1-(1-ρ)βn)∥xn-xn-1∥+M3[|rn-rn-1|b+∑k=1N‍|βn,k-βn-1,k|+M2+|αn-αn-1|+|βn-βn-1|∑k=2N‍]+M3|αn-αn-1|+M3|γn1-σn-γn-11-σn-1|=(1-(1-ρ)βn)∥xn-xn-1∥+M3[|rn-rn-1|b+∑k=1N‍|βn,k-βn-1,k|+M3+2|αn-αn-1|+|βn-βn-1|+M3+|γn1-σn-γn-11-σn-1|∑k=1N‍],
where M2+λ∥yn∥+∥y~n∥+∥Ty~n∥≤M3, for all n≥0 for some M3≥0.
Further, we observe that(52)xn+1=σnyn+(1-σn)vn,xn=σn-1yn-1+(1-βn-1)vn-1,∀n≥1.
Simple calculations show that
(53)xn+1-xn=(1-σn)(vn-vn-1)+(σn-σn-1)(yn-1-vn-1)+σn(yn-yn-1).
Then, passing to the norm, we get from (51)
(54)∥xn+1-xn∥≤(1-σn)∥vn-vn-1∥+|σn-σn-1|∥yn-1-vn-1∥+σn∥yn-yn-1∥≤(1-σn)×{∑k=1N‍(1-(1-ρ)βn)∥xn-xn-1∥+M3[|rn-rn-1|b+∑k=1N‍|βn,k-βn-1,k|+2|αn-αn-1|+|βn-βn-1|++|γn1-σn-γn-11-σn-1|∑k=1N‍]}+|σn-σn-1|∥yn-1-vn-1∥+σn{∑k=1N‍(1-(1-ρ)βn)∥xn-xn-1∥+σn+M2[|rn-rn-1|b+∑k=1N‍|βn,k-βn-1,k|+σn+M2+|αn-αn-1|+|βn-βn-1|∑k=1N‍]}≤(1-(1-ρ)βn)∥xn-xn-1∥+M3[|rn-rn-1|b+∑k=1N‍|βn,k-βn-1,k|+M3+2|αn-αn-1|+|βn-βn-1|+M3+|γn1-σn-γn-11-σn-1|∑k=1N‍]+|σn-σn-1|∥yn-1-vn-1∥≤(1-(1-ρ)βn)∥xn-xn-1∥+M[|rn-rn-1|b+∑k=1N‍|βn,k-βn-1,k|+M+|βn-βn-1|+|σn-σn-1|+M+|γn1-σn-γn-11-σn-1|∑k=1N‍]+2M|αn-αn-1|,
where M3+∥yn-vn∥≤M, for all n≥0 for some M≥0. By hypotheses (H1)–(H6) and Lemma 9, from ∑n=0∞‍αn<∞, we obtain the claim.Lemma 17.
Let one suppose thatΩ≠∅. Let one suppose that {xn} is asymptotically regular. Then, ∥xn-yn∥→0 and ∥xn-un∥=∥xn-Trnxn∥→0 as n→∞.Proof.
We recall that, by the firm nonexpansivity ofTrn, a standard calculation (see [17]) shows that if v∈EP(F,h), then
(55)∥un-v∥2≤∥xn-v∥2-∥xn-un∥2.
Let p∈Ω. Then by Lemma 11, we have from (33)–(34) the following
(56)∥y~n-p∥2=∥βn(Qyn,N-p)+(1-βn)(y~n,N-p)∥2≤βn∥Qyn,N-p∥2+(1-βn)∥y~n,N-p∥2≤βn∥Qyn,N-p∥2+∥y~n,N-p∥2≤βn∥Qyn,N-p∥2+[∥yn,N-p∥+λαn∥p∥]2=βn∥Qyn,N-p∥2+∥yn,N-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)≤βn∥Qyn,N-p∥2+∥xn-p∥2+λαn∥p∥(2∥xn-p∥+λαn∥p∥),(57)∥y~n-p∥2=∥PC(I-λ∇fαn)yn-PC(I-λ∇f)p∥2≤(∥yn-p∥+λαn∥p∥)2=∥yn-p∥2+λαn∥p∥(2∥yn-p∥+λαn∥p∥).
Since (γn+δn)ζ≤γn for all n≥0, utilizing Lemma 10, we have
(58)∥xn+1-p∥2=∥σn(yn-p)+γn(y~n-p)+δn(Ty~n-p)∥2=∥σn(yn-p)+(γn+δn)1γn+δn×[γn(y~n-p)+δn(Ty~n-p)]1γn+δn∥2=σn∥yn-p∥2+(γn+δn)×∥1γn+δn[γn(y~n-p)+δn(Ty~n-p)]∥2-σn(γn+δn)∥(yn-p)-1γn+δn×[γn(y~n-p)+δn(Ty~n-p)]1γn+δn∥2=σn∥yn-p∥2+(γn+δn)×∥1γn+δn[γn(y~n-p)+δn(Ty~n-p)]∥2-σn(γn+δn)∥1γn+δn[γn(y~n-yn)+δn(Ty~n-yn)]∥2=σn∥yn-p∥2+(γn+δn)×∥1γn+δn[γn(y~n-p)+δn(Ty~n-p)]∥2-σnγn+δn∥xn+1-yn∥2≤σn∥yn-p∥2+(γn+δn)∥y~n-p∥2-σnγn+δn∥xn+1-yn∥2=σn∥yn-p∥2+(1-σn)∥y~n-p∥2-σn1-σn∥xn+1-yn∥2≤σn∥yn-p∥2+(1-σn)×[∥yn-p∥2+λαn∥p∥(2∥yn-p∥+λαn∥p∥)]-σn1-σn∥xn+1-yn∥2≤∥yn-p∥2+λαn∥p∥(2∥yn-p∥+λαn∥p∥)-σn1-σn∥xn+1-yn∥2≤βn∥Qyn,N-p∥2+∥xn-p∥2+λαn∥p∥(2∥xn-p∥+λαn∥p∥)+λαn∥p∥(2∥yn-p∥+λαn∥p∥)-σn1-σn∥xn+1-yn∥2=∥xn-p∥2+βn∥Qyn,N-p∥2+2λαn∥p∥(∥xn-p∥+∥yn-p∥+λαn∥p∥)-σn1-σn∥xn+1-yn∥2.
Taking into account 0<liminfn→∞σn≤limsupn→∞σn<1, we may assume that {σn}⊂[c,d] for some c,d∈(0,1). So, we deduce that
(59)c1-c∥xn+1-yn∥2≤σn1-σn∥xn+1-yn∥2≤∥xn-p∥2-∥xn+1-p∥2+βn∥Qyn,N-p∥2+2λαn∥p∥(∥xn-p∥+∥yn-p∥+λαn∥p∥)≤(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+βn∥Qyn,N-p∥2+2λαn∥p∥(∥xn-p∥+∥yn-p∥+λαn∥p∥).
Since αn→0, βn→0 and ∥xn-xn+1∥→0 as n→∞, we conclude from the boundedness of {xn},{yn}, and {yn,N} that ∥xn+1-yn∥→0 as n→∞. This together with ∥xn-xn+1∥→0, implies that
(60)limn→∞∥xn-yn∥=0.
Furthermore, from (33), (55), and (56), we have
(61)∥yn-p∥2≤βn∥Qyn,N-p∥2+∥yn,N-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)≤βn∥Qyn,N-p∥2+∥un-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)≤βn∥Qyn,N-p∥2+∥xn-p∥2-∥xn-un∥2+λαn∥p∥(2∥xn-p∥+λαn∥p∥),
which hence implies that
(62)∥xn-un∥2≤βn∥Qyn,N-p∥2+∥xn-p∥2-∥yn-p∥2+λαn∥p∥(2∥xn-p∥+λαn∥p∥)≤βn∥Qyn,N-p∥2+(∥xn-p∥+∥yn-p∥)∥xn-yn∥+λαn∥p∥(2∥xn-p∥+λαn∥p∥).
Since αn→0,βn→0 and ∥xn-yn∥→0 as n→∞, we deduce from the boundedness of {xn}, {yn}, and {yn,N} that
(63)limn→∞∥xn-un∥=0.Remark 18.
By the last lemma we haveωw(xn)=ωw(un) and ωs(xn)=ωs(un); that is, the sets of strong/weak cluster points of {xn} and {un} coincide.Of course, ifβn,i→βi≠0, as n→∞, for all index i, the assumptions of Lemma 16 are enough to assure that
(64)limn→∞∥xn+1-xn∥βn,i=0,∀i∈{1,…,N}.
In the next lemma, we examine the case in which at least one sequence {βn,k0} is a null sequence.Lemma 19.
Let one suppose thatΩ≠∅. Let one suppose that (H1) holds. Moreover, for an index k0∈{1,…,N}, limn→∞βn,k0=0, and the following hold:(H7)
for alli,
(65)limn→∞|βn,i-βn-1,i|βnβn,k0=limn→∞|αn-αn-1|βnβn,k0=limn→∞|βn-βn-1|βnβn,k0=limn→∞|σn-σn-1|βnβn,k0=limn→∞|rn-rn-1|βnβn,k0=limn→∞1βnβn,k0|γn1-σn-γn-11-σn-1|=0;(H8)
there exists a constantτ>0 such that (1/βn)|1/βn,k0-1/βn-1,k0|<τ for all n≥1. Then,
(66)limn→∞∥xn+1-xn∥βn,k0=0.Proof.
We start by (54). Dividing both the terms by βn,k0 we have
(67)∥xn+1-xn∥βn,k0≤[1-(1-ρ)βn]∥xn-xn-1∥βn,k0+M[|rn-rn-1|bβn,k0+∑k=1N|βn,k-βn-1,k|βn,k0+M+2|αn-αn-1|βn,k0+|βn-βn-1|βn,k0+|σn-σn-1|βn,k0+M+|γn/(1-σn)-γn-1/(1-σn-1)|βn,k0∑k=1N|βn,k-βn-1,k|βn,k0].
So, by (H8) we have
(68)∥xn+1-xn∥βn,k0≤[1-(1-ρ)βn]∥xn-xn-1∥βn-1,k0+[1-(1-ρ)βn]∥xn-xn-1∥|1βn,k0-1βn-1,k0|+M[|rn-rn-1|bβn,k0+∑k=1N|βn,k-βn-1,k|βn,k0+M+|γn/(1-σn)-γn-1/(1-σn-1)|βn,k0+M+2|αn-αn-1|βn,k0+|βn-βn-1|βn,k0+|σn-σn-1|βn,k0∑k=1N|βn,k-βn-1,k|βn,k0]≤[1-(1-ρ)βn]∥xn-xn-1∥βn-1,k0+∥xn-xn-1∥|1βn,k0-1βn-1,k0|+M[|rn-rn-1|bβn,k0+∑k=1N|βn,k-βn-1,k|βn,k0+M+|γn/(1-σn)-γn-1/(1-σn-1)|βn,k0+M+2|αn-αn-1|βn,k0+M+|βn-βn-1|βn,k0+|σn-σn-1|βn,k0∑k=1N|βn,k-βn-1,k|βn,k0]≤[1-(1-ρ)βn]∥xn-xn-1∥βn-1,k0+βnτ∥xn-xn-1∥+M[|rn-rn-1|bβn,k0+∑k=1N|βn,k-βn-1,k|βn,k0+M+|γn/(1-σn)-γn-1/(1-σn-1)|βn,k0+M+2|αn-αn-1|βn,k0+|βn-βn-1|βn,k0+|σn-σn-1|βn,k0∑k=1N|βn,k-βn-1,k|βn,k0]=[1-(1-ρ)βn]∥xn-xn-1∥βn-1,k0+(1-ρ)βn·11-ρ×{∑k=1N|βn,k-βn-1,k|βnβn,k0τ∥xn-xn-1∥+M[|rn-rn-1|bβnβn,k0+∑k=1N|βn,k-βn-1,k|βnβn,k0+|γn/(1-σn)-γn-1/(1-σn-1)|βnβn,k0+2|αn-αn-1|βnβn,k0+|βn-βn-1|βnβn,k0+|σn-σn-1|βnβn,k0∑k=1N|βn,k-βn-1,k|βnβn,k0]}.
Therefore, utilizing Lemma 9, from (H1), (H7), and the asymptotical regularity of {xn} (due to Lemma 16), we deduce that
(69)limn→∞∥xn+1-xn∥βn,k0=0.Lemma 20.
Let one suppose thatΩ≠∅. Let one suppose that (H1)–(H6) hold. Then,
(70)limn→∞∥y~n,N-yn,N∥=limn→∞∥y~n-yn∥=0.Proof.
Letp∈Ω. Then, by Lemma 11 we have
(71)∥yn-p∥2=∥βn(Qyn,N-p)+(1-βn)(y~n,N-p)∥2≤βn∥Qyn,N-p∥2+(1-βn)∥y~n,N-p∥2=βn∥Qyn,N-p∥2+(1-βn)×∥PC(I-λ∇fαn)yn,N-PC(I-λ∇f)p∥2≤βn∥Qyn,N-p∥2+(1-βn)×∥(I-λ∇f)yn,N-(I-λ∇f)p-λαnyn,N∥2≤βn∥Qyn,N-p∥2+(1-βn)×[∥(I-λ∇f)yn,N-(I-λ∇f)p∥2-2λαn〈yn,N,(I-λ∇fαn)yn,N-(I-λ∇f)p〉∥(I-λ∇f)yn,N-(I-λ∇f)p∥2]≤βn∥Qyn,N-p∥2+(1-βn)×[∥yn,N-p∥2+2λ(λ-2L)∥∇f(yn,N)-∇f(p)∥2+2λαn∥yn,N∥∥(I-λ∇fαn)yn,N-(I-λ∇f)p∥λ-2L]≤βn∥Qyn,N-p∥2+(1-βn)×[∥xn-p∥2+2λ(λ-2L)∥∇f(yn,N)-∇f(p)∥2+2λαn∥yn,N∥∥(I-λ∇fαn)yn,N-(I-λ∇f)p∥(λ-2L)].
So, we obtain
(72)(1-βn)2λ(2L-λ)∥∇f(yn,N)-∇f(p)∥2≤βn∥Qyn,N-p∥2+(1-βn)∥xn-p∥2-∥yn-p∥2+(1-βn)2λαn∥yn,N∥×∥(I-λ∇fαn)yn,N-(I-λ∇f)p∥≤βn∥Qyn,N-p∥2+(∥xn-p∥+∥yn-p∥)×(∥xn-p∥-∥yn-p∥)+2λαn∥yn,N∥∥(I-λ∇fαn)yn,N-(I-λ∇f)p∥≤βn∥Qyn,N-p∥2+(∥xn-p∥+∥yn-p∥)(∥xn-yn∥)+2λαn∥yn,N∥∥(I-λ∇fαn)yn,N-(I-λ∇f)p∥.
Since αn→0, βn→0, ∥xn-yn∥→0, and 0<λ<2/L, from the boundedness of {xn}, {yn}, and {yn,N} it follows that limn→∞∥∇f(yn,N)-∇f(p)∥=0, and hence
(73)limn→∞∥∇fαn(yn,N)-∇f(p)∥=0.
Moreover, from the firm nonexpansiveness of PC we obtain
(74)∥y~n,N-p∥2=∥PC(I-λ∇fαn)yn,N-PC(I-λ∇f)p∥2≤〈(I-λ∇fαn)yn,N-(I-λ∇f)p,y~n,N-p〉=12{∥(I-λ∇fαn)yn,N-(I-λ∇f)p∥2+∥y~n,N-p∥2-∥(I-λ∇fαn)yn,N-(I-λ∇f)p-(y~n,N-p)∥2}≤12{∥∇fαn(yn,N)-∇f(p)∥2∥yn,N-p∥2+2λ∥∇fαn(yn,N)-∇f(p)∥×∥(I-λ∇fαn)yn,N-(I-λ∇f)p∥+∥y~n,N-p∥2-∥yn,N-y~n,N∥2+2λ〈yn,N-y~n,N,∇fαn(yn,N)-∇f(p)〉-λ2∥∇fαn(yn,N)-∇f(p)∥2∥yn,N-p∥2},
and so
(75)∥y~n,N-p∥2≤∥yn,N-p∥2-∥yn,N-y~n,N∥2+2λ∥∇fαn(yn,N)-∇f(p)∥×∥(I-λ∇fαn)yn,N-(I-λ∇f)p∥+2λ〈yn,N-y~n,N,∇fαn(yn,N)-∇f(p)〉-λ2∥∇fαn(yn,N)-∇f(p)∥2.
Thus, we have
(76)∥yn-p∥2≤βn∥Qyn,N-p∥2+(1-βn)∥y~n,N-p∥2≤βn∥Qyn,N-p∥2+∥yn,N-p∥2-(1-βn)∥yn,N-y~n,N∥2+2λ∥∇fαn(yn,N)-∇f(p)∥×∥(I-λ∇fαn)yn,N-(I-λ∇f)p∥+2(1-βn)λ〈yn,N-y~n,N,∇fαn(yn,N)-∇f(p)〉-(1-βn)λ2∥∇fαn(yn,N)-∇f(p)∥2,
which implies that
(77)(1-βn)∥yn,N-y~n,N∥2≤βn∥Qyn,N-p∥2+∥yn,N-p∥2-∥yn-p∥2+2λ∥∇fαn(yn,N)-∇f(p)∥×∥(I-λ∇fαn)yn,N-(I-λ∇f)p∥+2(1-βn)λ〈yn,N-y~n,N,∇fαn(yn,N)-∇f(p)〉-(1-βn)λ2∥∇fαn(yn,N)-∇f(p)∥2≤βn∥Qyn,N-p∥2+∥xn-p∥2-∥yn-p∥2+2λ∥∇fαn(yn,N)-∇f(p)∥×∥(I-λ∇fαn)yn,N-(I-λ∇f)p∥+2(1-βn)λ〈yn,N-y~n,N,∇fαn(yn,N)-∇f(p)〉-(1-βn)λ2∥∇fαn(yn,N)-∇f(p)∥2≤βn∥Qyn,N-p∥2+(∥xn-p∥+∥yn-p∥)∥xn-yn∥+2λ∥∇fαn(yn,N)-∇f(p)∥×∥(I-λ∇fαn)yn,N-(I-λ∇f)p∥+2λ∥yn,N-y~n,N∥∥∇fαn(yn,N)-∇f(p)∥.
Since βn→0, ∥xn-yn∥→0, and ∥∇fαn(yn,N)-∇f(p)∥→0, from the boundedness of {xn}, {yn}, {yn,N}, and {y~n,N}, it follows that limn→∞∥yn,N-y~n,N∥=0. Observe that
(78)∥yn-y~n,N∥=βn∥Qyn,N-y~n,N∥⟶0asn⟶∞,
and hence
(79)∥y~n-yn∥≤∥y~n-y~n,N∥+∥y~n,N-yn∥=∥PC(I-λ∇fαn)yn-PC(I-λ∇fαn)yn,N∥+∥y~n,N-yn∥≤∥yn-yn,N∥+∥y~n,N-yn∥≤∥yn-y~n,N∥+∥y~n,N-yn,N∥+∥y~n,N-yn∥=2∥yn-y~n,N∥+∥y~n,N-yn,N∥⟶0asn⟶∞.
Thus, limn→∞∥y~n-yn∥=0.Lemma 21.
Let one suppose thatΩ≠∅. Let one suppose that 0<liminfn→∞βn,i≤limsupn→∞βn,i<1 for each i=1,…,N. Moreover, suppose that (H1)–(H6) are satisfied. Then, limn→∞∥Siun-un∥=0 for each i=1,…,N.Proof.
First of all, observe that(80)xn+1-yn=γn(y~n-yn)+δn(Ty~n-yn)=γn(y~n-yn)+δn(Ty~n-Tyn)+δn(Tyn-yn).
By Lemmas 16 and 20, we know that ∥xn+1-xn∥→0 and ∥y~n-yn∥→0 as n→∞. Hence, utilizing Lemma 7(i), we have
(81)∥δn(Tyn-yn)∥=∥xn+1-yn-γn(y~n-yn)-δn(Ty~n-Tyn)∥≤∥xn+1-yn∥+γn∥y~n-yn∥+δn∥Ty~n-Tyn∥≤∥xn+1-yn∥+∥y~n-yn∥+∥Ty~n-Tyn∥≤∥xn+1-xn∥+∥xn-yn∥+∥y~n-yn∥+1+ζ1-ζ∥y~n-yn∥=∥xn+1-xn∥+∥xn-yn∥+21-ζ∥y~n-yn∥,
which together with ∥xn-yn∥→0 implies that limn→∞∥δn(Tyn-yn)∥=0. Taking into account liminfn→∞δn>0, we have
(82)limn→∞∥Tyn-yn∥=0.
Let us show that for eachi∈{1,…,N}, one has ∥Siun-yn,i-1∥→0 as n→∞. Let p∈Ω. When i=N, by Lemma 11, we have from (33)-(34) the following:
(83)∥yn-p∥2≤βn∥Qyn,N-p∥2+(1-βn)∥y~n,N-p∥2≤βn∥Qyn,N-p∥2+∥y~n,N-p∥2≤βn∥Qyn,N-p∥2+(∥yn,N-p∥+λαn∥p∥)2=βn∥Qyn,N-p∥2+∥yn,N-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)=βn∥Qyn,N-p∥2+βn,N∥SNun-p∥2+(1-βn,N)∥yn,N-1-p∥2-βn,N(1-βn,N)∥SNun-yn,N-1∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)≤βn∥Qyn,N-p∥2+∥un-p∥2-βn,N(1-βn,N)∥SNun-yn,N-1∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)≤βn∥Qyn,N-p∥2+∥xn-p∥2-βn,N(1-βn,N)∥SNun-yn,N-1∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥).
So, we have
(84)βn,N(1-βn,N)∥SNun-yn,N-1∥2≤βn∥Qyn,N-p∥2+∥xn-p∥2-∥yn-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)≤βn∥Qyn,N-p∥2+(∥xn-p∥+∥yn-p∥)∥xn-yn∥+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥).
Since αn→0, βn→0, 0<liminfn→∞βn,N≤limsupn→∞βn,N<1, and limn→∞∥xn-yn∥=0, it is known that {∥SNun-yn,N-1∥} is a null sequence.
Leti∈{1,…,N-1}. Then, one has
(85)∥yn-p∥2≤βn∥Qyn,N-p∥2+∥y~n,N-p∥2≤βn∥Qyn,N-p∥2+(∥yn,N-p∥+λαn∥p∥)2=βn∥Qyn,N-p∥2+∥yn,N-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)≤βn∥Qyn,N-p∥2+βn,N∥SNun-p∥2+(1-βn,N)∥yn,N-1-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)≤βn∥Qyn,N-p∥2+βn,N∥xn-p∥2+(1-βn,N)∥yn,N-1-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)≤βn∥Qyn,N-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)+βn,N∥xn-p∥2+(1-βn,N)[∥yn,N-2-p∥2βn,N-1∥SN-1un-p∥2+(1-βn,N)+(1-βn,N-1)∥yn,N-2-p∥2]≤βn∥Qyn,N-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)+(βn,N+(1-βn,N)βn,N-1)∥xn-p∥2+∏k=N-1N‍(1-βn,k)∥yn,N-2-p∥2,
and so, after (N-i+1) iterations,
(86)∥yn-p∥2≤βn∥Qyn,N-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)+(βn,N+∑j=i+2N‍(∏l=jN‍(1-βn,l))βn,j-1)×∥xn-p∥2+∏k=i+1N‍(1-βn,k)∥yn,i-p∥2≤βn∥Qyn,N-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)+(βn,N+∑j=i+2N‍(∏l=jN‍(1-βn,l))βn,j-1)×∥xn-p∥2+∏k=i+1N‍(1-βn,k)×[βn,i∥Siun-p∥2+(1-βn,i)∥yn,i-1-p∥2-βn,i(1-βn,i)∥Siun-yn,i-1∥2]≤βn∥Qyn,N-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)+∥xn-p∥2-βn,i∏k=iN‍(1-βn,k)∥Siun-yn,i-1∥2.
Again, we obtain that
(87)βn,i∏k=iN‍(1-βn,k)∥Siun-yn,i-1∥2≤βn∥Qyn,N-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)+∥xn-p∥2-∥yn-p∥2≤βn∥Qyn,N-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)+(∥xn-p∥+∥yn-p∥)∥xn-yn∥.
Since αn→0, βn→0, 0<liminfn→∞βn,i≤limsupn→∞βn,i<1 for each i=1,…,N-1, and limn→∞∥xn-yn∥=0, it is known that
(88)limn→∞∥Siun-yn,i-1∥=0.
Obviously, fori=1, we have ∥S1un-un∥→0.
To conclude, we have that(89)∥S2un-un∥≤∥S2un-yn,1∥+∥yn,1-un∥=∥S2un-yn,1∥+βn,1∥S1un-un∥,
from which ∥S2un-un∥→0. Thus, by induction ∥Siun-un∥→0 for all i=2,…,N since it is enough to observe that
(90)∥Siun-un∥≤∥Siun-yn,i-1∥+∥yn,i-1-Si-1un∥+∥Si-1un-un∥≤∥Siun-yn,i-1∥+(1-βn,i-1)∥Si-1un-yn,i-2∥+∥Si-1un-un∥.Remark 22.
As an example, we considerN=2 and the following sequences:(a)
σn=1/2+2/n, γn=δn=1/4-1/n for all n>4;(b)
βn=1/n, rn=2-1/n, for all n>1;(c)
βn,1=1/2-1/n,βn,2=1/2-1/n2, for all n>2.
Then, they satisfy the hypotheses on the parameter sequences in Lemma21.Lemma 23.
Let one suppose thatΩ≠∅ and βn,i→βi for all i as n→∞. Suppose there exists k∈{1,…,N} such that βn,k→0 as n→∞. Let k0∈{1,…,N} be the largest index such that βn,k0→0 as n→∞. Suppose that(i)
(αn+βn)/βn,k0→0 as n→∞;(ii)
ifi≤k0 and βn,i→0, then βn,k0/βn,i→0 as n→∞;(iii)
ifβn,i→βi≠0, then βi lies in (0,1).
Moreover, suppose that (H1), (H7), and (H8) hold. Then,limn→∞∥Siun-un∥=0 for each i=1,…,N.Proof.
First of all, we note that if (H7) holds, then also (H2)–(H6) are satisfied. So{xn} is asymptotically regular. Let k0 be as in the hypotheses. As in Lemma 21, for every index i∈{1,…,N} such that βn,i→βi≠0 (which leads to 0<liminfn→∞βn,i≤limsupn→∞βn,i<1), one has ∥Siun-yn,i-1∥→0 as n→∞.
For all the other indexesi≤k0, we can prove that ∥Siun-yn,i-1∥→0 as n→∞ in a similar manner. By the following relation (due to (86)):
(91)∥xn+1-p∥2≤σn∥yn-p∥2+(γn+δn)∥y~n-p∥2=σn∥yn-p∥2+(1-σn)∥y~n-p∥2≤σn∥yn-p∥2+(1-σn)(∥yn-p∥+λαn∥p∥)2≤(∥yn-p∥+λαn∥p∥)2=∥yn-p∥2+λαn∥p∥(2∥yn-p∥+λαn∥p∥)≤βn∥Qyn,N-p∥2+λαn∥p∥(2∥yn,N-p∥+λαn∥p∥)+∥xn-p∥2-βn,i∏k=iN‍(1-βn,k)∥Siun-yn,i-1∥2+λαn∥p∥(2∥yn-p∥+λαn∥p∥)≤βn∥Qyn,N-p∥2+2λαn∥p∥×(∥xn-p∥+∥yn-p∥+λαn∥p∥)+∥xn-p∥2-βn,i∏k=iN‍(1-βn,k)∥Siun-yn,i-1∥2,
we immediately obtain that
(92)∏k=iN‍(1-βn,k)∥Siun-yn,i-1∥2≤βnβn,i∥Qyn,N-p∥2+αnβn,i2λ∥p∥×(∥xn-p∥+∥yn-p∥+λαn∥p∥)+(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥βn,i.
By Lemma 19 or by hypothesis (ii) of the sequences, we have
(93)∥xn-xn+1∥βn,i=∥xn-xn+1∥βn,k0·βn,k0βn,i⟶0.
So, the thesis follows.Remark 24.
Let us considerN=3 and the following sequences:(a)
αn=1/n5/4, βn=1/n1/2, rn=2-1/n2, for all n>1;(b)
σn=1/2+2/n2, γn=δn=1/4-1/n2, for all n>2;(c)
βn,1=1/n1/4, βn,2=1/2-1/n2, βn,3=1/n1/3, for all n>1.
It is easy to see that all hypotheses (i)–(iii), (H1), (H7), and (H8) of Lemma23 are satisfied.Remark 25.
Under the hypotheses of Lemma23, analogously to Lemma 21, one can see that
(94)limn→∞∥Siun-yn,i-1∥=0,∀i∈{2,…,N}.Corollary 26.
Let one suppose that the hypotheses of either Lemma21 or Lemma 23 are satisfied. Then, ωw(xn)=ωw(un)=ωw(yn,1),ωs(xn)=ωs(un)=ωs(yn,1), and ωw(xn)⊂Ω.Proof.
By Remark18, we have ωw(xn)=ωw(un) and ωs(xn)=ωs(un). Observe that
(95)∥xn-yn,1∥≤∥xn-un∥+∥yn,1-un∥=∥xn-un∥+βn,1∥S1un-un∥.
By Lemmas 17 and 21, ∥xn-un∥→0 and ∥S1un-un∥→0 as n→∞, and hence
(96)limn→∞∥xn-yn,1∥=0.
So, we get ωw(xn)=ωw(yn,1) and ωs(xn)=ωs(yn,1).
Letp∈ωw(xn). Since p∈ωw(un), by Lemma 21 and Lemma 7(ii) (demiclosedness principle), we have p∈Fix(Si) for all index i, that is, p∈⋂iFix(Si). Taking into consideration that T is ζ-strictly pseudocontractive, by Lemma 7(i), we get
(97)∥Txn-xn∥≤∥Txn-Tyn∥+∥Tyn-yn∥+∥yn-xn∥≤1+ζ1-ζ∥xn-yn∥+∥yn-Tyn∥+∥xn-yn∥=21-ζ∥xn-yn∥+∥yn-Tyn∥,
which together with ∥xn-yn∥→0 (by Lemma 17) and ∥yn-Tyn∥→0 (by (82)) implies that
(98)limn→∞∥xn-Txn∥=0.
Utilizing Lemma7(ii) (demiclosedness principle), we have p∈Fix(T). Furthermore, by Lemmas 14 and 17, we know that p∈EP(F,h). Finally, by similar argument as in [18], we can show that p∈Γ, and as a result p∈Ω.Theorem 27.
Let one suppose thatΩ≠∅. Let {βn}, {βn,i}, i=1,…,N, be sequences in (0,1) such that 0<liminfn→∞βn,i≤limsupn→∞βn,i<1 for all index i. Moreover, Let one suppose that (H1)–(H6) hold. Then, the sequences {xn}, {yn}, and {un}, explicitly defined by scheme (30), all converge strongly to the unique solution x*∈Ω of the following variational inequality:
(99)〈Qx*-x*,z-x*〉≤0,∀z∈Ω.Proof.
Since the mappingPΩQ is a ρ-contraction, it has a unique fixed point x*; it is the unique solution of (99). Since (H1)–(H6) hold, the sequence {xn} is asymptotically regular (by Lemma 16). In terms of Lemma 17, ∥xn-yn∥→0 and ∥xn-un∥→0 as n→∞. Moreover, utilizing Lemmas 8 and 10, we have from (33)-(34) the following:
(100)∥xn+1-x*∥2≤σn∥yn-x*∥2+(1-σn)∥y~n-x*∥2≤σn∥yn-x*∥2+(1-σn)(∥yn-x*∥+λαn∥p∥)2≤(∥yn-x*∥+λαn∥p∥)2=∥yn-x*∥2+λαn∥p∥(2∥yn-x*∥+λαn∥p∥)≤∥βn(Qyn,N-Qx*)+(1-βn)(y~n,N-x*)∥2+2βn〈Qx*-x*,yn-x*〉+λαn∥p∥(2∥yn-x*∥+λαn∥p∥)≤(βnρ∥yn,N-x*∥+(1-βn)∥yn,N-x*∥)2+2βn〈Qx*-x*,yn-x*〉+λαn∥p∥(2∥yn-x*∥+λαn∥p∥)=(1-(1-ρ)βn)2∥yn,N-x*∥2+2βn〈Qx*-x*,yn-x*〉+λαn∥p∥(2∥yn-x*∥+λαn∥p∥)≤(1-(1-ρ)βn)∥un-x*∥2+2βn〈Qx*-x*,yn-x*〉+λαn∥p∥(2∥yn-x*∥+λαn∥p∥)≤(1-(1-ρ)βn)∥xn-x*∥2+2βn〈Qx*-x*,yn-x*〉+λαn∥p∥(2∥yn-x*∥+λαn∥p∥)=(1-(1-ρ)βn)∥xn-x*∥2+(1-ρ)βn·21-ρ〈Qx*-x*,yn-x*〉+λαn∥p∥(2∥yn-x*∥+λαn∥p∥).
Now, let {xnk} be a subsequence of {xn} such that
(101)limsupn→∞〈Qx*-x*,xn-x*〉=limk→∞〈Qx*-x*,xnk-x*〉.
By the boundedness of {xn}, we may assume, without loss of generality, that xnk⇀z∈ωw(xn). According to Corollary 26, we know that ωw(xn)⊂Ω, and hence z∈Ω. Taking into consideration that x*=PΩQx* we obtain from (101) that
(102)limsupn→∞〈Qx*-x*,yn-x*〉=limsupn→∞[〈Qx*-x*,xn-x*〉+〈Qx*-x*,yn-xn〉]=limsupn→∞〈Qx*-x*,xn-x*〉=limk→∞〈Qx*-x*,xnk-x*〉=〈Qx*-x*,z-x*〉≤0.
Since ∑n=0∞‍αn<∞ and ∑n=0∞‍βn=∞, we deduce that ∑n=0∞‍λαn∥p∥(2∥yn-x*∥+λαn∥p∥)<∞ and ∑n=0∞‍(1-ρ)βn=∞. In terms of Lemma 9 we derive xn→x* as n→∞.In a similar way, we can derive the following result.Theorem 28.
Let one suppose thatΩ≠∅. Let {βn}, {βn,i}, i=1,…,N, be sequences in (0,1) such that βn,i→βi for all i as n→∞. Suppose that there exists k∈{1,…,N} for which βn,k→0 as n→∞. Let k0∈{1,…,N} the largest index for which βn,k0→0. Moreover, let one suppose that (H1), (H7), and (H8) hold, and(i)
(αn+βn)/βn,k0→0 as n→∞;(ii)
ifi≤k0 and βn,i→0, then βn,k0/βn,i→0 as n→∞;(iii)
ifβn,i→βi≠0, then βi lies in (0,1).
Then, the sequences{xn}, {yn}, and {un} explicitly defined by scheme (30) all converge strongly to the unique solution x*∈Ω of the following variational inequality:
(103)〈Qx*-x*,z-x*〉≤0,∀z∈Ω.Remark 29.
According to the above argument processes for Theorems27 and 28, we can readily see that if in scheme (30), the iterative step yn=βnQyn,N+(1-βn)PC(yn,N-λ∇fαn(yn,N)) is replaced by the iterative one yn=βnQxn+(1-βn)PC(yn,N-λ∇fαn(yn,N)), then Theorems 27 and 28 remain valid.Remark 30.
Theorems27 and 28 improve, extend, supplement, and develop [17, Theorems 3.12 and 3.13] and [1, Theorems 5.2 and 6.1] in the following aspects:(a)
the multistep iterative scheme (30) of [17] is extended to develop our relaxed viscosity iterative scheme (30) with regularization for MP (3), EP (10), and strict pseudocontraction T by virtue of Xu iterative schemes in [1];(b)
the argument techniques in Theorems27 and 28 are very different from the ones in [17, Theorems 3.12 and 3.13] and the ones in [1, Theorems 5.2 and 6.1] because we use the properties of strict pseudocontractive mappings and maximal monotone mappings (see, e.g., Lemmas 7 and 10);(c)
compared with the proof of Theorems 5.2 and 6.1 in [1], the proof of Theorems 27 and 28 shows that limn→∞∥yn,N-PC(I-λ∇fαn)yn,N∥=limn→∞∥yn-PC(I-λ∇fαn)yn∥=0 via the argument of limn→∞∥∇fαn(yn,N)-∇f(p)∥=0, for all p∈Ω (see Lemma 20 and its proof);(d)
the problem of finding an element ofFix(T)∩(⋂iFix(Si))∩EP(F,h)∩Γ in Theorems 27 and 28 is more general than the one of finding an element of Fix(T)∩(⋂iFix(Si))∩EP(F,h) in [17, Theorems 3.12 and 3.13] and the one of finding an element of Γ in [1, Theorems 5.2 and 6.1].


## 4. Applications

For a given nonlinear mappingA:C→H, we consider again the variational inequality problem (VIP) of finding x-∈C such that
(104)〈Ax-,y-x-〉≥0,∀y∈C.Recall that ifu is a point in C, then the following relation holds:
(105)u∈VI(C,A)⟺u=PC(I-λA)u,forsomeλ>0,
from which we have the following relation:
(106)u∈Γ⟺u∈VI(C,∇f)⟺u=PC(I-λ∇f)u,forsomeλ>0.An operatorA:C→H is said to be an α-inverse strongly monotone operator if there exists a constant α>0 such that
(107)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C.As an example, we recall that theα-inverse strongly monotone operators are firmly nonexpansive mappings if α≥1 and that every α-inverse strongly monotone operator is also a (1/α) Lipschitz continuous (see [19]). We observe that, if A is α-inverse strongly monotone, the mapping PC(I-μA) is nonexpansive for all μ∈(0,2α] since they are compositions of nonexpansive mappings (see [19, page 419]).Let us considerS~1,…,S~M a finite number of nonexpansive self-mappings on C and A1,…,AN be a finite number of α-inverse strongly monotone operators. Let T:C→C be a ζ-strict pseudocontraction on C with fixed points. Let us consider the following mixed problem of finding x*∈Fix(T)∩EP(F,h)∩Γ such that
(108)〈(I-S~1)x*,y-x*〉≥0,∀y∈Fix(T)∩EP(F,h)∩Γ,〈(I-S~2)x*,y-x*〉≥0,∀y∈Fix(T)∩EP(F,h)∩Γ,⋮〈(I-S~M)x*,y-x*〉≥0,∀y∈Fix(T)∩EP(F,h)∩Γ,〈A1x*,y-x*〉≥0,∀y∈C,〈A2x*,y-x*〉≥0,∀y∈C,⋮〈ANx*,y-x*〉≥0,∀y∈C.
We denote by (SVI) the set of solutions of the above (N+M) system. This problem is equivalent to finding a common fixed point of T, {PFix(T)∩EP(F,h)∩ΓS~i}i=1N, {PC(I-μAi)}i=1M. The following results are then consequences of Theorems 27 and 28.Theorem 31.
Let one suppose thatΩ=Fix(T)∩(SVI)∩EP(F,h)∩Γ≠∅. Fix μ∈(0,2α], and λ∈(0,2/L). Let {αn}, {βn,i}, i=1,…,(M+N), be sequences in (0,1) such that 0<liminfn→∞βn,i≤limsupn→∞βn,i<1 for all index i. Moreover, Let one suppose that (H1)–(H6) hold. Then the sequences {xn}, {yn}, and {un} explicitly defined by the following scheme:
(109)F(un,y)+h(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn,1=βn,1PFix(T)∩EP(F,h)∩ΓS~1un+(1-βn,1)un,yn,i=βn,iPFix(T)∩EP(F,h)∩ΓS~iun+(1-βn,i)yn,i-1,i=2,…,M,yn,M+j=βn,M+jPC(I-μAj)un+(1-βn,M+j)yn,M+j-1,j=1,…,N,yn=βnQyn,M+N+(1-βn)×PC(yn,M+N-λ∇fαn(yn,M+N)),xn+1=σnyn+γnPC(yn-λ∇fαn(yn))+δnTPC(yn-λ∇fαn(yn)),∀n≥0,
all converge strongly to the unique solution x*∈Ω of the following variational inequality:
(110)〈Qx*-x*,z-x*〉≤0,∀z∈Ω.Theorem 32.
Let one suppose thatΩ≠∅. Fixμ∈(0,2α] and λ∈(0,2/L). Let {βn}, {βn,i}, i=1,…,(M+N), be sequences in (0,1) and βn,i→βi for all i as n→∞. Suppose that there exists k∈{1,…,M+N} such that βn,k→0 as n→∞. Let k0∈{1,…,M+N} be the largest index for which βn,k0→0. Moreover, let one suppose that (H1), (H7), and (H8) hold, and(i)
(αn+βn)/βn,k0→0 as n→∞;(ii)
ifi≤k0 and βn,i→0, then βn,k0/βn,i→0 as n→∞;(iii)
ifβn,i→βi≠0, then βi lies in (0,1).
Then, the sequences{xn}, {yn}, and {un} explicitly defined by scheme (109) all converge strongly to the unique solution x*∈Ω of the following variational inequality:
(111)〈Qx*-x*,z-x*〉≤0,∀z∈Ω.Remark 33.
If we choose∇f=A1=⋯=AN=0 in system (108), we obtain a system of hierarchical fixed point problems introduced by Moudafi  and Maingé [20, 21].Further, utilizing Theorems27 and 28, we again give the following strong convergence theorems for finding a common element of the solution set Γ of MP (3), the solution set EP(F,h) of EP (10), and the common fixed point set (⋂iFix(Si)) of a finite family of nonexpansive mappings Si:C→C,i=1,…,N.Theorem 34.
Let one suppose thatΩ=(⋂iFix(Si))∩EP(F,h)∩Γ≠∅. Let {βn}, {βn,i}, i=1,…,N, be sequences in (0,1) such that 0<liminfn→∞βn,i≤limsupn→∞βn,i<1 for all index i. Moreover, Let one suppose that there hold (H1)–(H6) with γn=0, for all n≥0. Then, the sequences {xn}, {yn}, and {un} generated explicitly by
(112)F(un,y)+h(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn,1=βn,1S1un+(1-βn,1)un,yn,i=βn,iSiun+(1-βn,i)yn,i-1,i=2,…,N,yn=βnQyn,N+(1-βn)PC(yn,N-λ∇fαn(yn,N)),xn+1=σnyn+(1-σn)PC(yn-λ∇fαn(yn)),∀n≥0,
all converge strongly to the unique solution x*∈Ω of the following variational inequality:
(113)〈Qx*-x*,z-x*〉≤0,∀z∈Ω.Proof.
In Theorems27, put T=I the identity mapping and γn=0, for all n≥0. Then, T is a ζ-strictly pseudocontractive mapping with ζ=0. Hence, we deduce that Ω=Fix(T)∩(⋂iFix(Si))∩EP(F,h)∩Γ=(⋂iFix(Si))∩EP(F,h)∩Γ≠∅, (γn+δn)ζ≤γn, for all n≥0, and
(114)xn+1=σnyn+γnPC(yn-λ∇fαn(yn))+δnTPC(yn-λ∇fαn(yn))=σnyn+δnPC(yn-λ∇fαn(yn))=σnyn+(1-σn)PC(yn-λ∇fαn(yn)).
Thus, the conditions in Theorem 27 are all satisfied. and from which we obtain the desired result.Theorem 35.
Let one suppose thatΩ=(⋂iFix(Si))∩EP(F,h)∩Γ≠∅. Let {βn}, {βn,i}, i=1,…,N, be sequences in (0,1) such that βn,i→βi for all i as n→∞. Suppose that there exists k∈{1,…,N} for which βn,k→0 as n→∞. Let k0∈{1,…,N} be the largest index for which βn,k0→0. Moreover, let one suppose that there hold (H1), (H7), and (H8) with γn=0, for all n≥0, and(i)
(αn+βn)/βn,k0→0 as n→∞;(ii)
ifi≤k0 and βn,i→0, then βn,k0/βn,i→0 as n→∞;(iii)
ifβn,i→βi≠0, then βi lies in (0,1).
Then the sequences{xn}, {yn}, and {un} generated explicitly by (112)  all converge strongly to the unique solution x*∈Ω of the following variational inequality:
(115)〈Qx*-x*,z-x*〉≤0,∀z∈Ω.



---

*Source: 531859-2013-05-13.xml*