R E S E A R C H
Open Access
Iterative algorithms with regularization for
hierarchical variational inequality problems
and convex minimization problems
Lu-Chuan Ceng
1,2, Suliman Al-Homidan
3*and Qamrul Hasan Ansari
4 *Correspondence:[email protected] 3Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
Full list of author information is available at the end of the article
Abstract
In this paper, we consider a variational inequality problem which is defined over the set of intersections of the set of fixed points of a
ζ
-strictly pseudocontractive mapping, the set of fixed points of a nonexpansive mapping and the set of solutions of a minimization problem. We propose an iterative algorithm with regularization to solve such a variational inequality problem and study the strong convergence of the sequence generated by the proposed algorithm. The results of this paper improve and extend several known results in the literature.1 Introduction
LetH be a real Hilbert space with the inner product·,· and the norm · , letCbe a nonempty closed convex subset ofH, and letf :C→Rbe a convex and continuously Fréchet differentiable functional. We consider the following minimization problem (MP):
min
x∈Cf(x). (.)
We denote byΞ the set of minimizers of problem (.), and we assume thatΞ =∅. The gradient-projection algorithm (GPA) is one of the most elegant methods to solve the min-imization problem (.). The convergence of the sequence generated by the GPA depends on the behavior of the gradient∇f. If∇f is strongly monotone and Lipschitz continuous, then we get the strong convergence of the sequence generated by the GPA to a unique so-lution of MP (.). However, if the gradient∇f is assumed to be only Lipschitz continuous, then the sequence generated by the GPA converges weakly ifHis infinite-dimensional (a counterexample is given in []). Since the Lipschitz continuity of the gradient∇f implies that it is actually inverse strongly monotone (ism) [], its complement can be an averaged mapping (that is, it can be expressed as a proper convex combination of the identity map-ping and a nonexpansive mapmap-ping) []. Consequently, the GPA can be rewritten as the composite of a projection and an averaged mapping, which is again an averaged mapping. This shows that averaged mappings play an important role in the GPA. Very recently, Xu [] used averaged mappings to study the convergence analysis of the GPA, which is an operator-oriented approach. He showed that the sequence generated by the GPA con-verges in norm to a minimizer of MP (.), which is also a unique solution of a particular type of variational inequality problem (VIP). It is worth to emphasize that the
ization, in particular the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems. The advantage of a regularization method is its possible strong convergence to the minimum-norm solution of the optimization problem. In [], Xu introduced a hybrid gradient-projection algorithm with regularization and proved the strong convergence of the sequence to the minimum-norm solution of MP (.). Some iter-ative algorithms with or without regularization for MP (.) are proposed and analyzed in [–] for finding a common solution of MP (.) and the set of solutions of a nonexpansive mapping.
On the other hand, the theory of variational inequalities [, ] has emerged as an impor-tant tool to study a wide class of problems from science, engineering, social sciences. If the underlying set in the formulation of a variational inequality problem is a set of fixed points of a mapping or, more precisely, of a nonexpansive mapping, then the variational inequal-ity problem is called hierarchical variational problem. For further details on hierarchical variational inequalities, we refer to [–] and the references therein.
In this paper, we consider a variational inequality problem which is defined over the set of intersections of the set of fixed points of aζ-strictly pseudocontractive mapping, the set of fixed points of a nonexpansive mapping and the set of solutions of MP (.). We propose an iterative algorithm with regularization to solve such a variational inequality problem and study the strong convergence of the sequence generated by the proposed algorithm. The results of this paper improve and extend several known results in the literature.
2 Preliminaries and formulations
Throughout the paper, unless otherwise specified, we use the following assumptions and notations. LetHbe a real Hilbert space whose inner product and norm are denoted by·,· and · , respectively. LetCbe a nonempty closed convex subset ofH. We writexn→x
(respectively,xnx) to indicate that the sequence{xn}converges strongly (respectively,
weakly) tox. Moreover, we useωw(xn) to denote the weakω-limit set of the sequence{xn},
that is,
ωw(xn) :=
x∈H:xnixfor some subsequence{xni}of{xn}
.
The metric (or nearest point) projection fromH ontoCis the mapping PC:H→C
which assigns to each pointx∈Hthe unique pointPCx∈Csatisfying
x–PCx=inf
y∈Cx–y=:d(x,C).
Some important properties of projections are gathered in the following proposition.
Proposition . For given x∈H and z∈C,we have (a) z=PCx⇔ x–z,y–z ≤,∀y∈C;
(b) z=PCx⇔ x–z≤ x–y–y–z,∀y∈C;
(c) PCx–PCy,x–y ≥ PCx–PCy,∀y∈H,which concludes thatPCis nonexpansive and monotone.
(a) ζ-strictly pseudocontractive if there exists a constantζ∈[, )such that Tx–Ty≤ x–y+ζ(I–T)x– (I–T)y, ∀x,y∈H.
Ifζ= , then it is called nonexpansive;
(b) firmly nonexpansive ifT–Iis nonexpansive, or equivalently,
x–y,Tx–Ty ≥ Tx–Ty, ∀x,y∈H;
alternatively,Tis firmly nonexpansive if and only ifTcan be expressed as
T=
(I+S),
whereS:H→His a nonexpansive mapping.
It can be easily seen that the projection mappings are firmly nonexpansive. It is clear thatT:C⊆H→Cisζ-strictly pseudocontractive if and only if
Tx–Ty,x–y ≤ x–y– –ζ
(I–T)x– (I–T)y
, ∀x,y∈C.
Definition . LetTbe a nonlinear operator with domainD(T)⊆Hand rangeR(T)⊆H.
(a) T is said to be monotone if
x–y,Tx–Ty ≥, ∀x,y∈D(T).
(b) Given a numberβ> ,T is said to beβ-strongly monotone if x–y,Tx–Ty ≥βx–y, ∀x,y∈D(T).
(c) Given a numberν> ,Tis said to beν-inverse strongly monotone (ν-ism) if x–y,Tx–Ty ≥νTx–Ty, ∀x,y∈D(T).
Clearly,
• ifTis nonexpansive, thenI–Tis monotone; • a projectionPKis-ism;
• ifTis aζ-strictly pseudocontractive mapping, thenI–Tis –ζ-inverse strongly monotone.
Definition .[] A mappingT:H→His said to be an averaged mapping if it can be written as the average of the identityIand a nonexpansive mapping, that is,
T≡( –α)I+αS,
Proposition .[] Let T:H→H be a given mapping.
(a) Tis nonexpansive if and only if the complementI–Tis -ism. (b) IfTisν-ism,then forγ> ,γT is ν
γ-ism.
(c) Tis averaged if and only if the complementI–Tisν-ism for someν> /.Indeed, forα∈(, ),Tisα-averaged if and only ifI–T isα-ism.
Proposition .[, ] Let S,T,V:H→H be given operators.
(a) IfT= ( –α)S+αV for someα∈(, )and ifSis averaged andVis nonexpansive, thenTis averaged.
(b) Tis firmly nonexpansive if and only if the complementI–Tis firmly nonexpansive. (c) IfT= ( –α)S+αV for someα∈(, )and ifSis firmly nonexpansive andV is
nonexpansive,thenTis averaged.
(d) The composite of finitely many averaged mappings is averaged,that is,if each of the mappings{Ti}Ni=is averaged,then so is the compositeT· · ·TN.In particular,ifTis
α-averaged andTisα-averaged,whereα,α∈(, ),then the compositeTTis
α-averaged,whereα=α+α–αα.
Lemma .[, Proposition .] Let C be a nonempty closed convex subset of a real Hilbert space H,and let T:C→C be a mapping.
(a) IfTis aζ-strictly pseudocontractive mapping,thenTsatisfies the Lipschitz condition
Tx–Ty ≤ +ζ
–ζx–y, ∀x,y∈C.
(b) IfTis aζ-strictly pseudocontractive mapping,then the mappingI–Tis semiclosed at,that is,if{xn}is a sequence inCsuch thatxn→ ˜xweakly and(I–T)xn→
strongly,then(I–T)x˜= .
(c) IfTis aζ-(quasi-)strict pseudocontraction,then the fixed point setFix(T)ofTis closed and convex so that the projectionPFix(T)is well defined.
The following lemma is an immediate consequence of an inner product.
Lemma . In a real Hilbert space H,we have
x+y≤ x+ y,x+y, ∀x,y∈H.
The following elementary result on real sequences is quite well known.
Lemma .[] Let{an}be a sequence of nonnegative real numbers such that
an+≤( –sn)an+sntn+n, ∀n≥,
where{sn} ⊂(, ]and{tn}satisfy the following conditions:
(i) ∞n=sn=∞;
(ii) eitherlim supn→∞tn≤or∞n=sn|tn|<∞;
Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H,and let T:C→C be aζ-strictly pseudocontractive mapping.Letγ andδbe two nonnegative real numbers such that(γ+δ)ζ ≤γ.Then
γ(x–y) +δ(Tx–Ty)≤(γ +δ)x–y, ∀x,y∈C.
The following lemma appeared implicitly in the paper of Reineermann [].
Lemma .[] Let H be a real Hilbert space.Then,for all x,y∈H andλ∈[, ], λx+ ( –λ)y=λx+ ( –λ)y–λ( –λ)x–y.
LetCbe a nonempty closed convex subset of a real Hilbert spaceH, and letA:C→H
be a monotone mapping. The variational inequality problem (VIP) is to findx∈Csuch that
Ax,y–x ≥, ∀y∈C.
The solution set of the VIP is denoted byVI(C,A). It is well known that
x∈VI(C,A) ⇔ x=PC(x–λAx), ∀λ> .
A set-valued mappingV:H→His called monotone if for allx,y∈H,f∈Vxandg∈Vy
imply thatx–y,f–g ≥. A monotone set-valued mappingV:H→His called maximal
if its graphGph(V) is not properly contained in the graph of any other monotone set-valued mapping. It is known that a monotone set-set-valued mappingV:H→His maximal if and only if for (x,f)∈H×H,x–y,f –g ≥ for every (y,g)∈Gph(V) implies that
f ∈Vx. LetA:C→Hbe a monotone and Lipschitz continuous mapping andNCvbe the normal cone toCatv∈C, that is,
NCv=w∈H:v–u,w ≥,∀u∈C.
Define
Vv= ⎧ ⎨ ⎩
Av+NCv ifv∈C, ∅ ifv∈/C.
Lemma .[] Let A:C→H be a monotone mapping.Then (i) V is maximal monotone;
(ii) v∈V–⇔v∈VI(C,A).
Throughout the paper, we denote byFix(T) andFix(Γ) the set of fixed points ofTand
Γ, respectively. We also assume that the setFix(T)∩Fix(Γ)∩Ξis nonempty closed and convex.
hierarchical variational inequality problem which is defined onFix(T)∩Fix(Γ)∩Ξ. Findx˜∈Fix(T)∩Fix(Γ)∩Ξ such that
˜x–Sx˜,x˜–x ≤, ∀x∈Fix(T)∩Fix(Γ)∩Ξ. (.)
We denote byΩthe solution set of problem (.). It is not difficult to verify that solving (.) is equivalent to the fixed point problem of findingx˜∈Csuch that
˜
x=PFix(T)∩Fix(Γ)∩ΞSx˜,
wherePFix(T)∩Fix(Γ)∩Ξstands for the metric projection onto the closed convex setFix(T)∩
Fix(Γ)∩Ξ.
Problem (.) contains the hierarchical variational inequality problems considered and studied in [, , ] and the references therein.
By using the definition of the normal cone toFix(T)∩Fix(Γ)∩Ξ, we have the mapping
NFix(T)∩Fix(Γ)∩Ξ:H→H:
x→
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
{u∈H: (∀y∈Fix(T)∩Fix(Γ)∩Ξ)y–x,u ≤}, ifx∈Fix(T)∩Fix(Γ)∩Ξ;
∅, otherwise,
and we readily prove that (.) is equivalent to the variational inequality
∈(I–S)x˜+NFix(T)∩Fix(Γ)∩Ξx˜.
By combining the hybrid gradient-projection method of Xu [] and a two-step method of Yaoet al.[], we introduce the following three-step iterative algorithm:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
yn=θnSxn+ ( –θn)xn,
zn=βnQyn+ ( –βn)TPC(yn–λ∇fαn(yn)),
xn+=σnzn+γnPC(zn–λ∇fαn(zn)) +δnΓPC(zn–λ∇fαn(zn)), ∀n≥,
(.)
whereQ:C→Cis aρ-contraction mapping,{αn} ⊂(,∞),{βn},{θn},{σn} ⊂(, ) and
{γn},{δn} ⊂[, ] withσn+γn+δn= ,∀n≥. It is proven that under appropriate
assump-tions, the above iterative sequence {xn}converges strongly to an element x˜∈Fix(T)∩
Fix(Γ)∩Ξ. 3 Main results
Let us consider the following assumptions:
• the mappingQ:C→Cis aρ-contraction;
• the mappingΓ :C→Cis aζ-strict pseudocontraction; • S,T:C→Care two nonexpansive mappings;
• ∇f :C→His Lipschitz continuous with <λ< L; • {αn}is a sequence in(,∞)with∞n=αn<∞;
• {γn},{δn}are sequences in[, ]withσn+γn+δn= ,∀n≥; • lim infn→∞δn> and(γn+δn)ζ≤γn,∀n≥.
Theorem . Let{xn}be a bounded sequence generated from any given x∈C by(.). Assume that the following conditions hold:
(H) ∞n=βn=∞,limn→∞βn| – θn–
θn |= ;
(H) limn→∞β
n|
θn–
θn–|= ,limn→∞
θn| – βn–
βn |= ;
(H) limn→∞θn= andlimn→∞αnθ+nβn = ;
(H) limn→∞|αnβ–nαθnn–|= ,limn→∞
|σn–σn–| βnθn = ;
(H) limn→∞β
nθn| γn
–σn– γn– –σn–|= . Then the following assertions hold:
(i) limn→∞xn+θn–xn= ;
(ii) ωw(xn)⊂Ω.
Proof First of all, we show thatPC(I–λ∇fα) isξ-averaged for eachλ∈(,α+L), where
ξ= +λ(α+L) ∈(, ).
Indeed, the Lipschitz continuity of∇f implies that∇f isL-ism [], that is,
∇f(x) –∇f(y),x–y≥
L∇f(x) –∇f(y)
.
Observe that
(α+L)∇fα(x) –∇fα(y),x–y
= (α+L)αx–y+∇f(x) –∇f(y),x–y
=αx–y+α∇f(x) –∇f(y),x–y+αLx–y+L∇f(x) –∇f(y),x–y ≥αx–y+ α∇f(x) –∇f(y),x–y+∇f(x) –∇f(y)
=α(x–y) +∇f(x) –∇f(y) =∇fα(x) –∇fα(y).
Therefore, it follows that∇fα=αI+∇f is α+L-ism. Thus, by Proposition .(b),λ∇fαis
λ(α+L)-ism. From Proposition .(c), the complementI–λ∇fαis
λ(α+L)
-averaged.
There-fore, noting thatPCis-averaged and utilizing Proposition .(d), we obtain that for each
λ∈(,α+L),PC(I–λ∇fα) isξ-averaged with
ξ= +
λ(α+L)
–
·
λ(α+L)
=
+λ(α+L) ∈(, ).
This shows that PC(I –λ∇fα) is nonexpansive. For λ ∈(,L), utilizing the fact that
limn→∞α
n+L=
L, we may assume that
<λ<
αn+L
Consequently, it follows that for each integern≥,PC(I–λ∇fαn) isξn-averaged with
ξn=
+
λ(αn+L)
–
·
λ(αn+L)
=
+λ(αn+L)
∈(, ).
This implies thatPC(I–λ∇fαn) is nonexpansive for alln≥. The rest of the proof is divided into several steps.
Step .limn→∞xn+–xn θn = .
For simplicity, we put˜yn=PC(yn–λ∇fαn(yn)) andzn˜ =PC(zn–λ∇fαn(zn)) for everyn≥. Thenzn=βnQyn+ ( –βn)Ty˜nandxn+=σnzn+γn˜zn+δnΓz˜nfor everyn≥.
Taking into account <lim infn→∞σn≤lim supn→∞σn< , without loss of generality, we
may assume that{σn} ⊂[c,d] for somec,d∈(, ). We writexn=σn–zn–+ ( –σn–)vn–, ∀n≥, wherevn–=xn––σnσ–zn–n–. It follows that for alln≥,
vn–vn–=
xn+–σnzn
–σn
–xn–σn–zn– –σn–
=γn˜zn+δnΓzn˜ –σn
–γn–zn˜ –+δn–Γ˜zn– –σn–
=γn(zn˜ –zn˜ –) +δn(Γzn˜ –Γzn˜ –) –σn
+
γn
–σn
– γn– –σn–
˜ zn–+
δn
–σn
– δn– –σn–
Γ˜zn–. (.)
Since (γn+δn)ζ≤γnfor alln≥, by Lemma ., we have
γn(˜zn–˜zn–) +δn(Γzn˜ –Γzn˜ –)≤(γn+δn)˜zn–zn˜ –. (.)
Now, we estimatezn–zn–. Observe that for everyn≥,
˜yn–yn˜ – ≤PC(I–λ∇fαn)yn–PC(I–λ∇fαn)yn– +PC(I–λ∇fαn)yn––PC(I–λ∇fαn–)yn–
≤ yn–yn–+PC(I–λ∇fαn)yn––PC(I–λ∇fαn–)yn– ≤ yn–yn–+(I–λ∇fαn)yn–– (I–λ∇fαn–)yn–
=yn–yn–+λ∇fαn(yn–) –λ∇fαn–(yn–)
=yn–yn–+λ|αn–αn–|yn–. (.)
Similarly, for alln≥, we have
˜zn–˜zn– ≤ zn–zn–+λ|αn–αn–|zn–.
From (.), we have ⎧
⎨ ⎩
yn=θnSxn+ ( –θn)xn,
and therefore
yn–yn–=θn(Sxn–Sxn–) + (θn–θn–)(Sxn––xn–) + ( –θn)(xn–xn–),
which implies that
yn–yn– ≤θnSxn–Sxn–+|θn–θn–|Sxn––xn–+ ( –θn)xn–xn– ≤ xn–xn–+|θn–θn–|Sxn––xn–. (.)
Also, from (.) we have ⎧
⎨ ⎩
zn=βnQyn+ ( –βn)Ty˜n,
zn–=βn–Qyn–+ ( –βn–)Tyn˜ –, ∀n≥,
then simple calculations show that
zn–zn–= ( –βn)(Ty˜n–Ty˜n–) + (βn–βn–)(Qyn––Ty˜n–) +βn(Qyn–Qyn–),
and thus, from (.)-(.), we have
zn–zn–
≤( –βn)Ty˜n–Ty˜n–+|βn–βn–|Qyn––Ty˜n–+βnQyn–Qyn– ≤( –βn)˜yn–y˜n–+|βn–βn–|Qyn––T˜yn–+βnQyn–Qyn– ≤( –βn)
yn–yn–+λ|αn–αn–|yn–
+|βn–βn–|Qyn––T˜yn–
+βnρyn–yn– ≤ – ( –ρ)βn
yn–yn–+λ|αn–αn–|yn–+|βn–βn–|Qyn––Ty˜n– ≤ – ( –ρ)βn
xn–xn–+|θn–θn–|Sxn––xn–
+λ|αn–αn–|yn–+|βn–βn–|Qyn––Tyn˜ – ≤ – ( –ρ)βn
xn–xn–+|θn–θn–|Sxn––xn–
+λ|αn–αn–|yn–+|βn–βn–|Qyn––Tyn˜ – ≤ – ( –ρ)βn
xn–xn–+M
|θn–θn–|+|αn–αn–|+|βn–βn–|
, (.)
whereSxn–xn+λyn+Qyn–Tyn ≤˜ M,∀n≥ for someM> . This together with
(.)-(.) implies that
vn–vn–
≤γn(˜zn–˜zn–) +δn(Γzn˜ –Γ˜zn–)
–σn
+ γn –σn
– γn– –σn–
˜zn–
+ δn –σn
– δn– –σn–
Γzn˜ –
≤(γn+δn)˜zn–zn˜ –
–σn
+ γn –σn
– γn– –σn–
˜zn–+
γn
–σn
– γn– –σn–
=˜zn–zn˜ –+
γn
–σn
– γn– –σn–
˜zn–+Γyn˜ –
≤ zn–zn–+λ|αn–αn–|zn–+
γn
–σn
– γn– –σn–
˜zn–+Γzn˜ –
≤ – ( –ρ)βn
xn–xn–+M
|θn–θn–|+|αn–αn–|+|βn–βn–|
+λ|αn–αn–|zn–+
γn
–σn
– γn– –σn–
˜zn–+Γzn˜ –
≤ – ( –ρ)βn
xn–xn–+M
|θn–θn–|+ |αn–αn–|+|βn–βn–|
+ γn –σn
– γn– –σn–
, (.)
whereM+λzn+˜zn+Γ˜zn ≤M,∀n≥ for someM> .
Further, we observe that ⎧
⎨ ⎩
xn+=σnzn+ ( –σn)vn,
xn=σn–zn–+ ( –βn–)vn–, ∀n≥,
and then by simple calculations, we have
xn+–xn= ( –σn)(vn–vn–) + (σn–σn–)(zn––vn–) +σn(zn–zn–).
By taking norm and using (.)-(.), we get
xn+–xn
≤( –σn)vn–vn–+|σn–σn–|zn––vn–+σnzn–zn– ≤( –σn)
– ( –ρ)βn
xn–xn–+M
|θn–θn–|+ |αn–αn–|+|βn–βn–|
+ γn –σn
– γn– –σn–
+|σn–σn–|zn––vn–
+σn
– ( –ρ)βn
xn–xn–+M
|θn–θn–|+|αn–αn–|+|βn–βn–|
≤ – ( –ρ)βn
xn–xn–+M
|θn–θn–|+ |αn–αn–|+|βn–βn–|
+ γn –σn
– γn– –σn–
+|σn–σn–|zn––vn–
≤ – ( –ρ)βn
xn–xn–+M
|θn–θn–|+ |αn–αn–|+|βn–βn–|
+ γn –σn
– γn– –σn–
+|σn–σn–|
,
whereM+zn–vn ≤M,∀n≥ for someM≥. Therefore,
xn+–xn
θn
≤ – ( –ρ)βn
xn–xn–
θn
+M
|
θn–θn–|
θn
+ |αn–αn–|
θn
+|βn–βn–|
+
θn
γn
–σn
– γn– –σn–
+|σn–σn–|
θn
= – ( –ρ)βn
xn–xn–
θn–
+ – ( –ρ)βn
xn–xn–
θn
–xn–xn–
θn–
+M
|θn–θn–|
θn
+ |αn–αn–|
θn
+|βn–βn–|
θn
+
θn
γn
–σn
– γn– –σn–
+|σn–σn–|
θn
≤ – ( –ρ)βn
xn–xn–
θn–
+M
θn
–
θn–
+|θn–θn–|
θn
+ |αn–αn–|
θn
+|βn–βn–|
θn
+
θn
γn
–σn
– γn– –σn–
+|σn–σn–|
θn
= – ( –ρ)βn
xn–xn–
θn–
+ ( –ρ)βn· M
–ρ
βn
θn–
θn–
+
βn
–θn–
θn
+ |αn–αn–|
βnθn
+
θn
–βn–
βn
+
βnθn
γn
–σn
– γn– –σn–
+|σn–σn–|
βnθn
, (.)
where M+xn–xn– ≤M, ∀n≥ for some M≥. From (H)-(H), it follows that
∞
n=( –ρ)βn=∞and
lim n→∞
M
–ρ
βn
θn–
θn–
+
βn
–θn–
θn
+ |αn–αn–|
βnθn
+
θn
–βn–
βn
+
βnθn
γn
–σn
– γn– –σn–
+|σn–σn–|
βnθn
= .
Thus, by applying Lemma . to (.), we conclude that
lim n→∞
xn+–xn
θn
= ,
which implies that
lim
n→∞xn+–xn= . (.)
Step .limn→∞xn–zn= .
Indeed, letp∈Fix(T)∩Fix(Γ)∩Ξ. Then we have
˜yn–p=PC(I–λ∇fαn)yn–PC(I–λ∇f)p
≤PC(I–λ∇fαn)yn–PC(I–λ∇fαn)p +PC(I–λ∇fαn)p–PC(I–λ∇f)p ≤ yn–p+PC(I–λ∇fαn)p–PC(I–λ∇f)p
Similarly, we get
˜zn–p ≤ zn–p+λαnp.
By Lemma . and (.), we have
zn–p
=βn(Qyn–p) + ( –βn)(T˜yn–p) ≤βnQyn–p+ ( –βn)˜yn–p ≤βnQyn–p+˜yn–p ≤βnQyn–p+
yn–p+λαnp
=βnQyn–p+yn–p+λαnpyn–p+λαnp
≤βnQyn–p+θnSxn–p+ ( –θn)xn–p+λαnp
yn–p+λαnp
≤βnQyn–p+θnSxn–p+xn–p+λαnpyn–p+λαnp. Since (γn+δn)ζ≤γnfor alln≥, utilizing Lemma ., we obtain
xn+–p
=σn(zn–p) +γn(zn˜ –p) +δn(Γzn˜ –p)
=σn(zn–p) + (γn+δn)
γn+δn
γn(zn˜ –p) +δn(Γ˜zn–p)
=σnzn–p+ (γn+δn)
γn+δnγn(zn˜ –p) +δn(Γ˜zn–p)
–σn(γn+δn)
(zn–p) –
γn+δn
γn(zn˜ –p) +δn(Γzn˜ –p)
=σnzn–p+ (γn+δn)
γn+δnγn(z˜n–p) +δn(Γ˜zn–p)
–σn(γn+δn)
γn+δnγn(zn˜ –zn) +δn(Γzn˜ –zn)
=σnzn–p+ (γn+δn)
γn+δnγn(zn˜ –p) +δn(Γ˜zn–p)
– σn
γn+δnxn+
–zn
≤σnzn–p+ (γn+δn)˜zn–p–
σn
γn+δn
xn+–zn
=σnzn–p+ ( –σn)˜zn–p–
σn
–σn
xn+–zn
≤σnzn–p+ ( –σn)
zn–p+λαnpzn–p+λαnp
– σn –σnxn+
–zn
≤ zn–p+λαnp
zn–p+λαnp
– σn
–σn
≤βnQyn–p+θnSxn–p+xn–p+λαnpyn–p+λαnp
+λαnpzn–p+λαnp
– σn
–σn
xn+–zn
=xn–p+βnQyn–p+θnSxn–p
+ λαnpyn–p+zn–p+λαnp
– σn –σn
xn+–zn.
Since <lim infn→∞σn≤lim supn→∞σn< , we may assume that{σn} ⊂[c,d] for some c,d∈(, ). Therefore, we deduce
c
–cxn+–zn
≤ σn
–σnxn+
–zn
≤ xn–p–xn+–p+βnQyn–p+θnSxn–p
+ λαnpyn–p+zn–p+λαnp
≤xn–p+xn+–p
xn–xn++βnQyn–p+θnSxn–p
+ λαnpyn–p+zn–p+λαnp
.
Sinceαn→,βn→,θn→ andxn–xn+ → asn→ ∞, we conclude from the
boundedness of{xn},{yn}and{zn}thatxn+–zn → as n→ ∞. This together with xn–xn+ → implies that
lim
n→∞xn–zn= . (.)
Step .limn→∞yn–y˜n= andlimn→∞zn–z˜n= .
Letp∈Fix(T)∩Fix(Γ)∩Ξ. Then, by Lemmas . and ., we have
zn–p
=βn(Qyn–p) + ( –βn)(T˜yn–p)
≤βnQyn–p+ ( –βn)˜yn–p
=βnQyn–p+ ( –βn)PC(I–λ∇fαn)yn–PC(I–λ∇f)p
≤βnQyn–p+ ( –βn)(I–λ∇f)yn– (I–λ∇f)p–λαnyn ≤βnQyn–p+ ( –βn)(I–λ∇f)yn– (I–λ∇f)p
– λαn
yn, (I–λ∇fαn)yn– (I–λ∇f)p
≤βnQyn–p+ ( –βn)
yn–p+λ
λ–
L
∇f(yn) –∇f(p)
+ λαnyn(I–λ∇fαn)yn– (I–λ∇f)p
≤βnQyn–p+ ( –βn)
+λ
λ–
L
∇f(yn) –∇f(p)+ λαnyn(I–λ∇fαn)yn– (I–λ∇f)p
≤βnQyn–p+θnSxn–p+xn–p+ ( –βn)λ
λ–
L
∇f(yn) –∇f(p)
+ λαnyn(I–λ∇fαn)yn– (I–λ∇f)p.
Therefore, we obtain
( –βn)λ
L–λ
∇f(yn) –∇f(p)
≤βnQyn–p+θnSxn–p+xn–p–zn–p
+ λαnyn(I–λ∇fαn)yn– (I–λ∇f)p
≤βnQyn–p+θnSxn–p+xn–p+zn–pxn–p–zn–p + λαnyn(I–λ∇fαn)yn– (I–λ∇f)p
≤βnQyn–p+θnSxn–p+xn–p+zn–pxn–zn + λαnyn(I–λ∇fαn)yn– (I–λ∇f)p.
Sinceαn→,βn→,θn→,xn–zn → and <λ< L, from the boundedness of {xn},{yn}and{zn}, we obtainlimn→∞∇f(yn) –∇f(p)= , and hence
lim
n→∞∇fαn(yn) –∇f(p)= .
Also, since
yn–zn ≤ yn–xn+xn–zn=θnSxn–xn+xn–zn,
fromθn→ andxn–zn →, it follows that
lim
n→∞yn–zn= and n→∞lim∇fαn(zn) –∇f(p)= . (.)
Furthermore, from the firm nonexpansiveness ofPC, we obtain
˜yn–p=PC(I–λ∇fαn)yn–PC(I–λ∇f)p
≤(I–λ∇fαn)yn– (I–λ∇f)p,y˜n–p
=
(I–λ∇fαn)yn– (I–λ∇f)p
+˜yn–p
–(I–λ∇fαn)yn– (I–λ∇f)p– (˜yn–p)
≤
yn–p+ λ∇fαn(yn) –∇f(p)(I–λ∇fαn)yn– (I–λ∇f)p
+˜yn–p–yn–yn˜ + λyn–yn˜ ,∇fαn(yn) –∇f(p)
–λ∇fαn(yn) –∇f(p)
and so,
˜yn–p
≤ yn–p–yn–y˜n
+ λ∇fαn(yn) –∇f(p)(I–λ∇fαn)yn– (I–λ∇f)p + λyn–˜yn,∇fαn(yn) –∇f(p)
–λ∇fαn(yn) –∇f(p)
.
Similarly, we have
˜zn–p
≤ zn–p–zn–z˜n+ λ∇fαn(zn) –∇f(p)(I–λ∇fαn)zn– (I–λ∇f)p
+ λzn–zn˜ ,∇fαn(zn) –∇f(p)
–λ∇fαn(zn) –∇f(p)
. (.)
Thus, we have
zn–p≤βnQyn–p+ ( –βn)˜yn–p ≤βnQyn–p+˜yn–p
≤βnQyn–p+yn–p–yn–yn˜
+ λ∇fαn(yn) –∇f(p)(I–λ∇fαn)yn– (I–λ∇f)p + λyn–yn˜ ,∇fαn(yn) –∇f(p)
–λ∇fαn(yn) –∇f(p)
≤βnQyn–p+yn–p–yn–yn˜
+ λ∇fαn(yn) –∇f(p)(I–λ∇fαn)yn– (I–λ∇f)p + λyn–y˜n,∇fαn(yn) –∇f(p)
≤βnQyn–p+yn–p–yn–yn˜
+ λ∇fαn(yn) –∇f(p)(I–λ∇fαn)yn– (I–λ∇f)p+yn–˜yn
,
which implies that
yn–yn˜
≤βnQyn–p+yn–p–zn–p
+ λ∇fαn(yn) –∇f(p)(I–λ∇fαn)yn– (I–λ∇f)p+yn–yn˜
≤βnQyn–p+yn–p+zn–pyn–zn
+ λ∇fαn(yn) –∇f(p)(I–λ∇fαn)yn– (I–λ∇f)p+yn–yn˜
.
Sinceβn→,yn–zn → and∇fαn(yn) –∇f(p) →, from the boundedness of{xn}, {yn},{zn}and{˜yn}, it follows that
lim
In addition, since (γn+δn)ζ ≤γnfor alln≥, utilizing Lemma ., we get from (.) xn+–p≤σnzn–p+ (γn+δn)˜zn–p
=σnzn–p+ ( –σn)˜zn–p ≤σnzn–p+ ( –σn)
zn–p–zn–zn˜
+ λ∇fαn(zn) –∇f(p)(I–λ∇fαn)zn– (I–λ∇f)p + λzn–zn˜ ,∇fαn(zn) –∇f(p)
–λ∇fαn(zn) –∇f(p)
≤σnzn–p+ ( –σn)
zn–p–zn–zn˜
+ λ∇fαn(zn) –∇f(p)(I–λ∇fαn)zn– (I–λ∇f)p + λzn–zn˜ ∇fαn(zn) –∇f(p)
≤ zn–p– ( –σn)zn–z˜n
+ λ∇fαn(zn) –∇f(p)(I–λ∇fαn)zn– (I–λ∇f)p+zn–zn˜
,
which implies that
( –σn)zn–z˜n
≤ zn–p–xn+–p
+ λ∇fαn(zn) –∇f(p)(I–λ∇fαn)zn– (I–λ∇f)p+zn–˜zn
≤zn–p+xn+–p
zn–xn+
+ λ∇fαn(zn) –∇f(p)(I–λ∇fαn)zn– (I–λ∇f)p+zn–˜zn
.
Since{σn} ⊂[c,d],zn–xn+ → and∇fαn(zn) –∇f(p) →, from the boundedness of
{xn},{zn}and{˜zn}, it follows that
lim
n→∞zn–zn˜ = .
Step .ωw(xn)⊂Ω.
Letp∗∈ωw(xn). Then there exists a subsequence{xni}of{xn}such thatxnip∗. Since
zn–yn=βn(Qyn–yn) + ( –βn)(Tyn˜ –yn)
=βn(Qyn–yn) + ( –βn)(Tyn˜ –yn˜ ) + ( –βn)(yn˜ –yn),
we have
( –βn)Tyn˜ –˜yn=zn–yn–βn(Qyn–yn) – ( –βn)(yn˜ –yn) ≤ zn–yn+βnQyn–yn+ ( –βn)˜yn–yn ≤ zn–yn+βnQyn–yn+˜yn–yn.
Meanwhile, observe that
xn+–zn=γn(z˜n–zn) +δn(Γz˜n–z˜n) +δn(z˜n–zn)
= (γn+δn)(zn˜ –zn) +δn(Γzn˜ –zn˜ )
= ( –σn)(zn˜ –zn) +δn(Γ˜zn–˜zn).
Thus,
δnΓzn˜ –zn˜ =xn+–zn– ( –σn)(˜zn–zn) ≤ xn+–zn+ ( –σn)˜zn–zn
≤ xn+–zn+˜zn–zn → asn→ ∞.
This together withlim infn→∞δn> yieldslimn→∞Γz˜n–˜zn= . Sincexn–zn →
andzn–zn →˜ , we havezn˜ ip∗. By Lemma .(b) (demiclosedness principle), we havep∗∈Fix(Γ).
Further, let us showp∗∈Ξ. Indeed, fromxn–yn → and˜yn–yn →, we have
ynip∗and˜ynip∗. Define
Vv= ⎧ ⎨ ⎩
∇f(v) +NCv ifv∈C,
∅ ifv∈/C,
whereNCv={w∈H:v–u,w ≥,∀u∈C}. ThenV is maximal monotone and ∈Vvif and only ifv∈VI(C,∇f) (see []). Let (v,w)∈graph(V). Then we have
w∈Vv=∇f(v) +NCv,
and hence
w–∇f(v)∈NCv.
Therefore, we have
v–u,w–∇f(v)≥, ∀u∈C.
On the other hand, from
˜ yn=PC
yn–λ∇fαn(yn)
and v∈C,
we have
yn–λ∇fαn(yn) –y˜n,y˜n–v
≥,
and hence
v–˜yn, ˜ yn–yn
λ +∇fαn(yn)
Therefore, from
w–∇f(v)∈NC(v) and yn˜ i∈C,
we have
v–yn˜ i,w ≥
v–yn˜ i,∇f(v)
≥v–yn˜ i,∇f(v)–
v–yn˜ i,yn˜ i–yni
λ +∇fαni(yni)
=v–yn˜ i,∇f(v)
–
v–yn˜ i,
˜ yni–yni
λ +∇f(yni)
–αniv–˜yni,yni =v–yn˜ i,∇f(v) –∇f(˜yni)
+v–˜yni,∇f(yn˜ i) –∇f(yni)
–
v–yn˜ i,
˜ yni–yni
λ
–αniv–yn˜ i,yni
≥v–yn˜ i,∇f(yn˜ i) –∇f(yni)
–
v–yn˜ i,
˜ yni–yni
λ
–αniv–yn˜ i,yni.
Hence, we obtain
v–p∗,w≥ asi→ ∞.
SinceVis maximal monotone, we havep∗∈V–, and hencep∗∈VI(C,∇f), which leads
to p∈Ξ. Consequently,p∗∈Fix(T)∩Fix(Γ)∩Ξ. This shows thatωw(xn)⊂Fix(T)∩ Fix(Γ)∩Ξ.
Finally, let us show p∗ ∈Ω. Indeed, it follows from (.) that for every p∈Fix(T)∩
Fix(Γ)∩Ξ,
yn–p=( –θn)(xn–p) +θn(Sxn–Sp) +θn(Sp–p)
≤( –θn)(xn–p) +θn(Sxn–Sp)
+ θnSp–p,yn–p
≤( –θn)xn–p+θnSxn–Sp+ θnSp–p,yn–p ≤ xn–p+ θnSp–p,yn–p,
and hence
zn–p
≤βnQyn–p+ ( –βn)˜yn–p ≤βnQyn–p+˜yn–p ≤βnQyn–p+
yn–p+λαnp
≤βnQyn–p+yn–p+λαnpyn–p+λαnp ≤βnQyn–p+xn–p+ θnSp–p,yn–p
Since (γn+δn)ζ≤γnfor alln≥, by Lemma ., we have xn+–p
≤σnzn–p+ (γn+δn)˜zn–p ≤σnzn–p+ ( –σn)
zn–p+λαnp
≤ zn–p+λαnpzn–p+λαnp
≤βnQyn–p+xn–p+ θnSp–p,yn–p
+λαnpyn–p+λαnp+λαnpzn–p+λαnp
=xn–p+βnQyn–p+ θnSp–p,yn–p
+ λαnpyn–p+zn–p+λαnp,
which implies that
p–Sp,yn–p ≤
θn
xn–p–xn+–p
+βn
θnQyn
–p
+αn
θn
λpyn–p+zn–p+λαnp
≤xn–xn+
θn
xn–p+xn+–p
+βn
θn
Qyn–p
+αn
θn
λpyn–p+zn–p+λαnp.
Sinceαn+βn
θn → and
xn–xn+
θn → asn→ ∞, from the boundedness of{xn},{yn}and{zn}, we deduce that
lim sup
n→∞ p–Sp,yn–p ≤, ∀p∈Fix(T)∩Fix(Γ)∩Ξ.
So, fromynip∗, we get
p–Sp,p∗–p≤, ∀p∈Fix(T)∩Fix(Γ)∩Ξ.
Taking into consideration thatI–Sis monotone and continuous, utilizing Minty’s lemma [], we have
p∗–Sp∗,p∗–p≤, ∀p∈Fix(T)∩Fix(Γ)∩Ξ.
Remark . The following sequences satisfy the hypotheses on the parameter in Theo-rem ..
(a) αn=n+s+t,βn=ns andθn=nt, wheret∈(,)ands∈(t, –t);
(b) σn=+nandγn=δn=–nfor alln> .
Theorem . Let{xn}be the bounded sequence generated from any given x∈C by(.). Assume that hypotheses(H)-(H)of Theorem.hold and
(H) limn→∞θn βn = ;
(H) There is a constantk> such that
x–TPC(I–λ∇f)x ≥kdist(x,Fix(T)∩Fix(Γ)∩Ξ)for eachx∈C,where dist(x,Fix(T)∩Fix(Γ)∩Ξ) =infy∈Fix(T)∩Fix(Γ)∩Ξx–y.
Then the sequences{xn},{yn}and{zn}converge strongly to x∗=PΩQx∗providedxn–zn=
o(θn),where x∗solves the following variational inequality:
x∗–Sx∗,x∗–x≤, ∀x∈Fix(T)∩Fix(Γ)∩Ξ.
Proof Letp∈Fix(T)∩Fix(Γ)∩Ξ. From (.), we have
zn–p=βn(Qyn–Qp) +βn(Qp–p) + ( –βn)(Ty˜n–p),
and therefore,
zn–p
≤βn(Qyn–Qp) + ( –βn)(Tyn˜ –p)+ βnQp–p,zn–p ≤( –βn)Tyn˜ –p+βnQyn–Qp+ βnQp–p,zn–p ≤( –βn)˜yn–p+βnρyn–p+ βnQp–p,zn–p ≤( –βn)
yn–p+λαnp
+βnρyn–p+ βnQp–p,zn–p ≤ – ( –ρ)βn
yn–p+λαnp
yn–p+λαnp
+ βnQp–p,zn–p. (.)
Again from (.), we obtain
yn–p=( –θn)(xn–p) +θn(Sxn–Sp) +θn(Sp–p)
≤( –θn)(xn–p) +θn(Sxn–Sp)+ θnSp–p,yn–p ≤( –θn)xn–p+θnSxn–Sp+ θnSp–p,yn–p
≤ xn–p+ θnSp–p,yn–p. (.)
Substituting (.) into (.), we get
zn–p≤
– ( –ρ)βn
xn–p+ θnSp–p,yn–p
= – ( –ρ)βn
xn–p+ – ( –ρ)βn
θnSp–p,yn–p
+ βnQp–p,zn–p+λαnpyn–p+λαnp. (.)
Since (γn+δn)ζ≤γnfor alln≥, utilizing Lemma ., we get from (.) and (.) xn+–p≤σnzn–p+ (γn+δn)˜zn–p
≤σnzn–p+ ( –σ n)
zn–p+λαnp
≤ zn–p+λαnpzn–p+λαnp
≤ – ( –ρ)βn
xn–p+ – ( –ρ)βn
θnSp–p,yn–p + βnQp–p,zn–p+λαnpyn–p+λαnp
+λαnpzn–p+λαnp
= – ( –ρ)βn
xn–p+
– ( –ρ)βn
θnSp–p,yn–p
+ βnQp–p,zn–p+ λαnpyn–p+zn–p+λαnp
≤ – ( –ρ)βn
xn–p+ – ( –ρ)βn
θnSp–p,yn–p
+ βnQp–p,zn–p+Mαn, (.)
whereM=supn≥{λp(yn–p+zn–p+λαnp)}<∞.
Taking into consideration thatPΩ◦Qis a contractive mapping, we know thatPΩ◦Q has a unique fixed pointx∗∈Ω. That is, there is a unique solutionx∗∈Ωof the following variational inequality problem (VIP):
Qx∗–x∗,q–x∗≤, ∀q∈Ω. (.)
Sincex∗∈Ω, it is clear thatx∗=PFix(T)∩Fix(Γ)∩ΞSx∗, and hencex∗∈Fix(T)∩Fix(Γ)∩Ξ. Thus, from (.), we conclude that
xn+–x∗
≤ – ( –ρ)βnxn–x∗+
– ( –ρ)βn
θn
Sx∗–x∗,yn–x∗
+ βn
Qx∗–x∗,zn–x∗
+Mαn
= – ( –ρ)βnxn–x∗
+ ( –ρ)βn
( – ( –ρ)βn)
–ρ θn
βn
Sx∗–x∗,yn–x∗
+ –ρ
Qx∗–x∗,zn–x∗+Mαn. (.)
Consider a subsequence{xni}of{xn}such that
lim sup n→∞
Qx∗–x∗,xn–x∗
= lim
i→∞
Qx∗–x∗,xni–x
∗.
have
lim sup n→∞
Qx∗–x∗,zn–x∗=lim sup n→∞
Qx∗–x∗,zn–xn+Qx∗–x∗,xn–x∗
=lim sup n→∞
Qx∗–x∗,xn–x∗= lim i→∞
Qx∗–x∗,xni–x
∗
=Qx∗–x∗,x˜–x∗≤,
which implies that
lim sup n→∞
–ρ
Qx∗–x∗,zn–x∗≤. (.)
Meanwhile, fromx∗∈Ωand (H), we infer that
Sx∗–x∗,yn–x∗
=Sx∗–x∗,yn–PFix(T)∩Fix(Γ)∩Ξyn
+Sx∗–x∗,PFix(T)∩Fix(Γ)∩Ξyn–x∗
≤Sx∗–x∗,yn–PFix(T)∩Fix(Γ)∩Ξyn
≤Sx∗–x∗yn–PFix(T)∩Fix(Γ)∩Ξyn
=distyn,Fix(T)∩Fix(Γ)∩ΞSx∗–x∗ ≤
kSx
∗–x∗y
n–TPC(I–λ∇f)yn.
From (.), we have
zn–xn
θn
=βn
θn
(Qyn–xn) + –βn
θn
(T˜yn–xn).
This together withlimn→∞znθ–nxn= and βn
θn = implies that
lim n→∞
Tyn˜ –xn
θn
= .
Hence,
lim n→∞
θnTyn˜ –xn
βn
= lim n→∞
Tyn˜ –xn
θn
θ n
βn
= .
Observe that
yn–xn=θn(Sxn–xn).
Therefore, we get
lim n→∞
θn
βn
yn–xn= lim n→∞
θn βn
and hence
θn
βn
yn–TPC(I–λ∇f)yn
≤ θn
βn
yn–xn+xn–TPC(I–λ∇f)yn
≤ θn
βn
yn–xn+xn–TPC(I–λ∇fαn)yn
+TPC(I–λ∇fαn)yn–TPC(I–λ∇f)yn
≤ θn
βn
yn–xn+xn–Tyn˜ +(I–λ∇fαn)yn– (I–λ∇f)yn
=θn
βn
yn–xn+
θn
βn
xn–Ty˜n+
θn
βn
λαnyn
=θn
βn
yn–xn+
θn
βn
xn–Ty˜n+
θ n
βn
αn
θn
λyn → asn→ ∞.
Thus, it follows that
lim sup n→∞
θn
βn
Sx∗–x∗,yn–x∗
≤,
and hence
lim sup n→∞
( – ( –ρ)βn)
–ρ θn
βn
Sx∗–x∗,yn–x∗≤. (.)
Utilizing Lemma ., from∞n=Mαn<∞and (.)-(.), we conclude that the
se-quence{xn}converges strongly tox∗. Taking into consideration thatxn–yn → and
xn–zn →, we obtain thatyn–x∗ → andzn–x∗ → asn→ ∞. This completes
the proof.
Remark . The following parametric sequences satisfy the hypotheses of Theorem ..
(a) αn=n+s+t,βn=ns andθn=nt, wheret∈(,]ands∈(t, t)ort∈(,),
s∈(t, –t);
(b) σn=+n,γn=δn=–n,∀n> .
Remark . Theorems . and . improve, extend, supplement and develop [, Theo-rems . and .] and [, TheoTheo-rems . and .] in the following aspects:
(a) Three-step iterative algorithm (.) with regularization for MP (.), two
nonexpansive mappings and a strictly pseudocontractive mapping are more flexible and more subtle than the algorithms in [, ].
(b) The argument techniques in Theorems . and . are different from the ones in [, Theorems . and .] and the ones in [, Theorems . and .] because we use the properties of strict pseudocontractive mappings and maximal monotone mappings (see, for example, Lemmas ., . and .).
