R E S E A R C H
Open Access
General iterative methods for monotone
mappings and pseudocontractive mappings
related to optimization problems
Jong Soo Jung
**Correspondence: [email protected]
Department of Mathematics, Dong-A University, Busan, 604-714, Korea
Abstract
In this paper, we introduce two general iterative methods for a certain optimization problem of which the constrained set is the common set of the solution set of the variational inequality problem for a continuous monotone mapping and the fixed point set of a continuous pseudocontractive mapping in a Hilbert space. Under some control conditions, we establish the strong convergence of the proposed methods to a common element of the solution set and the fixed point set, which is the unique solution of a certain optimization problem. As a direct consequence, we obtain the unique minimum-norm common point of the solution set and the fixed point set.
MSC: 47H06; 47H09; 47H10; 47J20; 49J40; 47J25; 47J05
Keywords: general iterative method; optimization problem; continuous monotone mapping; continuous pseudocontractive mapping; variational inequality; metric projection; strongly positive bounded linear operator; contractive mapping
1 Introduction
LetHbe a real Hilbert space with the inner product·,·and the induced norm · . LetC be a nonempty closed convex subset ofH, and letS:C→Cbe a self-mapping onC. We denote byFix(S) the set of fixed points ofSand byPCthe metric projection ofHontoC.
A mappingFofCintoHis calledmonotoneif
x–y,Fx–Fy ≥, ∀x,y∈C.
A mappingFofCintoHis calledα-inverse-strongly monotone(see [, ]) if there exists a positive real numberαsuch that
x–y,Fx–Fy ≥αFx–Fy, ∀x,y∈C.
IfFis anα-inverse-strongly monotone mapping ofCintoH, then it is obvious thatFis
α-Lipschitz continuous, that is,Fx–Fy ≤
αx–yfor allx,y∈C. Clearly, the class of
monotone mappings includes the class ofα-inverse-strongly monotone mappings.
An operatorAis said to bestrongly positiveonHif there exists a constantγ > such that
Ax,x ≥γx, ∀x∈H.
A mappingF ofCintoH is calledγ-strongly monotone if there exists a positive real
numberγ such that
x–y,Fx–Fy ≥γx–y, ∀x,y∈C.
Clearly, the class of monotone mappings includes the class of strongly positive mappings. LetFbe a nonlinear mapping ofCintoH. The variational inequality problem is to find u∈Csuch that
v–u,Fu ≥, ∀v∈C. (.)
We denote the set of solutions of the variational inequality problem (.) byVI(C,F). The variational inequality problem has been extensively studied in the literature; see [–] and the references therein.
The class of pseudocontractive mappings is one of the most important classes of map-pings among nonlinear mapmap-pings. We recall that a mappingT:C→His said to be pseu-docontractiveif
Tx–Ty≤ x–y+(I–T)x– (I–T)y, ∀x,y∈C,
andTis said to bek-strictly pseudocontractiveif there exists a constantk∈[, ) such that
Tx–Ty≤ x–y+k(I–T)x– (I–T)y, ∀x,y∈C,
whereIis the identity mapping. Note that the class ofk-strictly pseudocontractive map-pings includes the class of nonexpansive mapmap-pings as a subclass. That is,Tis nonexpansive (i.e.,Tx–Ty ≤ x–y,∀x,y∈C) if and only ifTis -strictly pseudocontractive. Clearly, the class of pseudocontractive mappings includes the class of strictly pseudocontractive mappings as a subclass, and the class ofk-strictly pseudocontractive mappings falls into the one between the class of nonexpansive mappings and the class of pseudocontractive mappings. Moreover, this inclusion is strict due to an example in [] (see also Example .. and Example .. in []). Recently, many authors have been devoting the studies to the problems of finding fixed points for pseudocontractive mappings; see, for example, [–] and the references therein.
The following optimization problem has been studied extensively by many authors:
min
x∈
μ
Ax,x+ x–u
–h(x),
where =∞i=Ci, C,C, . . . are infinitely many closed convex subsets ofH such that
∞
i=Ci=∅;u∈H;μ≥ is a real number;Ais a strongly positive bounded linear
functionf onH). For this kind of minimization problems, see, for example, Bauschke and Borwein [], Combettes [], Deutsch and Yamada [], Jung [], and Xu [, ] when =Ni=Ciandh(x) =x,bfor a given pointbinH.
Iterative methods for nonexpansive mappings and strictly pseudocontractive mappings have recently been applied to solve the optimization problem, where the constraint set is the fixed point set of the mapping; see, for instance, [, , , , –] and the ref-erences therein. Some iterative methods for equilibrium problems, variational inequality problems, and fixed point problems to solve optimization problem, where the constraint set is the common set of the solution set of the problems and the fixed point set of the mappings, were also investigated by many authors recently; see, for instance, [, ] and the references therein. We can refer to [] for certain iterative methods for the integral boundary value problems with causal operators, and we can refer to [] for iterative meth-ods for solving certain random operator equations.
In particular, in , combining Moudafi’s method [] with Xu’s method [], Marino and Xu [] introduced the following general iterative method for a nonexpansive map-pingS:
xn+=αnγfxn+ (I–αnA)Sxn, ∀n≥, (.)
whereγ > andf is a contractive mapping onH. Under well-known control conditions on the sequence{αn} ⊂[, ], they proved the strong convergence of the sequence {xn} generated by (.) to a pointx∈Fix(S), which is the unique solution of the variational inequality
(A–γf)x,x–p≤, ∀p∈Fix(S),
which is the optimality condition for the optimization problem
min
x∈Fix(S)
Ax,x–h(x),
where his a potential function forγf. Very recently, Jung [] proposed the following general iterative method for ak-strictly pseudocontractive mappingTfor some ≤k< :
xn+=αn(u+γfxn) +
I–αn(I+μA)
PCSxn, ∀n≥, (.)
where u∈C; μ≥ is a real number; and S: C→H is a mapping defined by Sx=
kx+ ( –k)Tx. Under different control conditions on the sequence{αn} ⊂[, ] and the sequence{xn}generated by (.), he showed the strong convergence of the sequence{xn} to a pointx∈Fix(T), which is the unique solution of the optimization problem
min
x∈Fix(T) μ
Ax,x+ x–u
–h(x),
wherehis a potential function forγf.
an element ofVI(C,F)∩Fix(S), whereFis anα-inverse-strongly monotone mapping and S is a nonexpansive mapping; see [, –] and the references therein. Some iterative methods for finding an element ofVI(C,F)∩Fix(T) were also presented by many authors,
where F is a continuous monotone mapping and T is a continuous pseudocontractive
mapping; see [–] and the references therein. In the case thatEis a Banach space with the dualE∗, we can refer to [] for iterative methods for finding an element ofVI(C,F)∩ Fix(T), whereF:C→E∗is anα-inverse-strongly monotone mapping andTis a relatively weak nonexpansive mapping, and we can refer to [] for iterative methods for finding an element ofNi=Fix(Ti)∩VI(C,F), whereF is anα-inverse-strongly accretive mapping
andTi,i= , . . . ,N, areki-strictly pseudocontractive mappings. And we can consult []
for iterative methods for finding a common element ofVI(C,F)∩VI(C,F), whereF,F: C→E∗are two continuous monotone mappings.
Recently, researchers have also invented some iterative methods for finding the mini-mum norm element in the solution set of certain problems (for instance, variational in-equality problem, minimization problem, split feasibility problem,etc.) and the fixed point set of nonlinear mappings (for instance, nonexpansive mapping, strictly pseudocontrac-tive mapping, Lipschitzian pseudocontracpseudocontrac-tive mapping,etc.); see, for instance, [–] and the references therein.
In this paper, as a continuation of study for the above-mentioned optimization problems, we consider the following optimization problem of which the constrained set isVI(C,F)∩ Fix(T):
min
x∈VI(C,F)∩Fix(T) μ
Ax,x+ x–u
–h(x), (.)
whereFis a continuous monotone mapping;T is a continuous pseudocontractive map-pingu∈C;μ≥ is a real number; andhis a potential function forγf whenf is a con-tractive mapping and γ > . We present two general iterative methods for solving the optimization problem (.). First, we introduce an implicit general iterative method. Con-sequently, by discretizing the continuous implicit method, we provide an explicit general iterative method. Under some control conditions, we show the strong convergence of the proposed methods to an element ofVI(C,F)∩Fix(T), which is the unique solution of the optimization problem (.). As special cases, we obtain two iterative methods which converge strongly to the minimum norm point of VI(C,F)∩Fix(T). Our results unify, complement, develop, and improve upon the corresponding results of Jung [, ], Yao et al.[], and some recent results in the literature.
2 Preliminaries and lemmas
LetH be a real Hilbert space, and letCbe a nonempty closed convex subset ofH. We writexnxto indicate that the sequence{xn}converges weakly tox.xn→ximplies that {xn}converges strongly tox.
For every pointx∈H, there exists a unique nearest point inC, denoted byPC(x), such
that
PCis called themetric projectionofH ontoC. It is well known thatPCis nonexpansive
and is characterized by the properties
u=PC(x) ⇐⇒ x–u,u–y ≥, ∀x∈H,y∈C. (.)
In a Hilbert spaceH, the following equality holds:
x–y=x+y– x,y, ∀x,y∈H. (.)
We need the following lemmas for the proof of our main results.
Lemma . In a real Hilbert space H,there holds the following inequality:
x+y≤ x+ y,x+y, ∀x,y∈H.
Lemma .([]) Let{sn}be a sequence of nonnegative real numbers satisfying
sn+≤( –wn)sn+wnδn+νn, ∀n≥,
where{wn},{δn},and{νn}satisfy the following conditions:
(i) {wn} ⊂[, ]and ∞n=wn=∞or,equivalently,
∞
n=( –wn) = ; (ii) lim supn→∞δn≤or ∞n=wn|δn|<∞;
(iii) νn≥(n≥), ∞n=νn<∞.
Thenlimn→∞sn= .
The following lemmas can be easily proven, and therefore, we omit the proofs.
Lemma . Let H be a real Hilbert space,and let A:H→H be a strongly positive bounded linear operator with a constant γ > .Let f :H→H be a contractive mapping with a constant k∈(, ).Letμ≥and <γ <+μγk .Then
x–y,(I+μA) –γfx–(I+μA) –γfy≥( +μγ–γk)x–y, ∀x,y∈H.
That is, (I+μA) –γf is strongly monotone with a constant +μγ –γk.
Lemma .([]) Letμ> ,and let A:H→H be a strongly positive bounded linear
self-adjoint operator on a Hilbert space H with a constantγ > .Let <ξ≤( +μA)–.Then
I–ξ(I+μA)< –ξ( +μγ).
The following lemmas are Lemma . and Lemma . of Zegeye [], respectively.
Lemma .([]) Let C be a closed convex subset of a real Hilbert space H.Let F:C→H be a continuous monotone mapping.Then,for r> and x∈H,there exists z∈C such that
y–z,Fz+
For r> and x∈H,define Fr:H→C by
Frx=
z∈C:y–z,Fz+
ry–z,z–x ≥,∀y∈C
.
Then the following hold:
(i) Fris single-valued;
(ii) Fris firmly nonexpansive,that is,
Frx–Fry≤ x–y,Frx–Fry, ∀x,y∈H;
(iii) Fix(Fr) =VI(C,F);
(iv) VI(C,F)is a closed convex subset ofC.
Lemma .([]) Let C be a closed convex subset of a real Hilbert space H.Let T:C→H be a continuous pseudocontractive mapping.Then,for r> and x∈H,there exists z∈C such that
y–z,Tz– r
y–z, ( +r)z–x≤, ∀y∈C.
For r> and x∈H,define Tr:H→C by
Trx=
z∈C:y–z,Tz– r
y–z, ( +r)z–x≤,∀y∈C
.
Then the following hold:
(i) Tris single-valued;
(ii) Tris firmly nonexpansive,that is,
Trx–Try≤ x–y,Trx–Try, ∀x,y∈H;
(iii) Fix(Tr) =Fix(T);
(iv) Fix(T)is a closed convex subset ofC.
The following lemma can be found in [, ].
Lemma . Let C be a nonempty closed convex subset of a real Hilbert space H,and let g:C→R∪ {∞}be a proper lower semicontinuous differentiable convex function.If x∗is a solution of the minimization problem
gx∗=inf
x∈Cg(x),
then
gx∗,p–x∗≥, p∈C.
In particular,if x∗solves the optimization problem
min
x∈C
μ
Ax,x+ x–u
where A is a bounded linear self-adjoint operator on H,then
u+γf – (I+μA)x∗,p–x∗≤, p∈C,
where h is a potential function ofγf.
3 Main results
Throughout the rest of this paper, we always assume the following:
• His a real Hilbert space;
• Cis a nonempty closed subspace subset ofH;
• A:C→Cis a strongly positive linear bounded self-adjoint operator with a constant
γ > ;
• f:C→Cis a contractive mapping with a constantk∈(, ); • Constantsμ≥and <γ <+kμγ;
• F:C→His a continuous monotone mapping;
• VI(C,F)is the set of the variational inequality problem (.) forF; • T:C→Cis a continuous pseudocontractive mapping such that
VI(C,F)∩Fix(T)=∅;
• Frt:H→Cis a mapping defined by
Frtx=
z∈C:y–z,Fz+ rt
y–z,z–x ≥,∀y∈C
forrt∈(,∞),t∈(, ), andlim inft→rt> ; • Trt:H→Cis a mapping defined by
Trtx=
z∈C:y–z,Tz– rt
y–z, ( +rt)z–x
≤,∀y∈C
forrt∈(,∞),t∈(, ), andlim inft→rt> ; • Frn:H→Cis a mapping defined by
Frnx=
z∈C:y–z,Fz+ rn
y–z,z–x ≥,∀y∈C
forrn∈(,∞)andlim infn→∞rn> ; • Trn:H→Cis a mapping defined by
Trnx=
z∈C:y–z,Tz– rn
y–z, ( +rn)z–x
≤,∀y∈C
forrn∈(,∞)andlim infn→∞rn> ; • u∈C.
By Lemma . and Lemma ., we note that Frt, Trt, Frn, and Trn are nonexpansive,
Fix(Frn) =VI(C,F) =Fix(Frt), andFix(Trn) =Fix(T) =Fix(Trt).
In this section, first, we introduce the following general iterative method that generates a net{xt}t∈(,min{,
+μA})in an implicit way:
xt=t(u+γfxt) +
Now, fort∈(,min{,+μA}), consider a mappingQt:C→Cdefined by
Qtx=t(u+γfx) +
I–t(I+μA)TrtFrtx, ∀x∈C.
It is easy to see thatQtis a contractive mapping with a constant –t( +μγ–γk). Indeed,
sinceTrtFrtis nonexpansive, by Lemma ., we have
Qtx–Qty ≤I–t(I+μA)
TrtFrtx–
I–t(I+μA)TrtFrty
+t(u+γfx) – (u+γfy)
≤ –t( +μγ)x–y+tγkx–y
= –t( +μγ –γk)x–y.
Since <t<min{,+μA}, it follows that < –t( +μγ–γk) < .
HenceQtis a contractive mapping. By the Banach contraction principle,Qthas a unique
fixed point, denoted byxt, which uniquely solves the fixed point equation (.).
We summarize the basic properties of{xt}.
Proposition . Let{xt}be defined via(.).Then
(i) {xt}is bounded fort∈(,min{,+μA}); (ii) limt→xt–TrtFrtxt= ;
(iii) xt: (,min{,+μA})→His locally Lipschitzian,provided
rt: (,min{,+μA})→(,∞)is locally Lipschitzian;
(iv) xtdefines a continuous path from(,min{,+μA})intoH,provided
rt: (,min{,+μA})→(,∞)is continuous.
Proof (i) Letzt =Frtxt, and let ut =Trtzt. Letp∈VI(C,F)∩Fix(T). Then p=Frtpby Lemma .(iii) andp=Trtp(=Tp) by Lemma .(iii), and from the nonexpansivity of TrtandFrt, it follows that
ut–p=Trtzt–Trtp ≤ zt–p, (.)
and
zt–p=Frtxt–Frtp ≤ xt–p. (.)
LetA=I+μA. By (.) and (.), we have
xt–p=t(u+γfxt) + (I–tA)Trtzt–p
=(I–tA)Ttzt– (I–tA)Trtp+tγfxt–tγfp+t(u+γf –A)p
≤(I–tA)Trtzt– (I–tA)Trtp+tγkxt–p
≤ –t( +μγ)zt–p+tγkxt–p+t
u+(γf–A)p
≤ –t( +μγ)xt–p+tγkxt–p+t
u+(γf–A)p
= –t( +μγ–γk)xt–p+t
u+(γf –A)p.
So, it follows that
xt–p ≤
u+(γf–A)p +μγ –γk .
Hence{xt}is bounded and so are{zt}={Frtxt},{ut}={Trtzt}, and{ATrtzt}={ATrtFrtxt},
and{fxt}.
(ii) Letzt=Frtxt. By the definition of{xt}and the boundedness of{fxt}and{ATrtzt}in (i), we have
xt–TrtFrtxt=tATrtFrtxt– (u+γfxt) ≤tATrtzt+u+γfxt
→ ast→.
(iii) Lett,t∈(,min{,+μA}), and letzt=Frtxt andzt=Frtxt. Letut=Trtzt and
ut=Trtzt. Then we get
y–ut,Tut–
rt
y–ut, ( +rt)ut–zt
≤, ∀y∈C, (.)
and
y–ut,Tut–
rt
y–ut, ( +rt)ut–zt
≤, ∀y∈C. (.)
Puttingy=utin (.) andy=utin (.), we obtain
ut–ut,Tut–
rt
ut–ut, ( +rt)ut–zt
≤, (.)
and
ut–ut,Tut–
rt
ut–ut, ( +rt)ut–zt
≤. (.)
Adding up (.) and (.), we have
ut–ut,Tut–Tut–
ut–ut,
( +rt)ut–zt
rt
–( +rt)ut–zt rt
≤,
which implies that
ut–ut, (ut–Tut) – (ut–Tut)
–
ut–ut,
ut–zt
rt
–ut–zt rt
≤.
Now, using the fact thatTis pseudocontractive, we get
ut–ut,
ut–zt
rt
–ut–zt rt
and hence
ut–ut,ut–ut+ut–zt–
rt
rt
(ut–zt)
≥. (.)
Without loss of generality, let us assume that there exists a real numberrt >b> for
t∈(,min{,+μA}). Then, by (.), we have
ut–ut≤
ut–ut,zt–zt+
–rt
rt
(ut–zt)
≤ ut–ut
zt–zt+
–rt
rt
ut–zt
. (.)
Hence, from (.) we obtain
ut–ut ≤ zt–zt+
rt
|rt–rt|ut–zt
≤ zt–zt+
b|rt–rt|L, (.)
whereL=sup{ut–zt:t∈(,min{,+μA})}.
Moreover, sincezt=Frtxtandzt=Frtxt, we get
y–zt,Fzt+
rt
y–zt,zt–xt ≥, ∀y∈C, (.)
and
y–zt,Fzt+
rt
y–zt,zt–xt ≥, ∀y∈C. (.)
Puttingy=ztin (.) andy=ztin (.), we obtain
zt–zt,Fzt+
rt
zt–zt,zt–xt ≥, (.)
and
zt–zt,Fzt+
rt
zt–zt,zt–xt ≥. (.)
Adding up (.) and (.), we have
–zt–zt,Fzt–Fzt+
zt–zt,
zt–xt
rt
–zt–xt rt
≥.
SinceFis monotone, we get
zt–zt,
zt–xt
rt
–zt–xt rt
and hence
zt–zt,zt–zt+zt–xt–
rt
rt
(zt–xt)
≥. (.)
Then, using the method in (.) and (.), from (.) we have
zt–zt≤
zt–zt,zt–xt+xt–xt–
rt
rt
(zt–xt)
=
zt–zt,xt–xt+
–rt
rt
(zt–xt)
≤ zt–zt
xt–xt+
b|rt–rt|zt–xt
.
This implies that
zt–zt ≤ xt–xt+
b|rt–rt|M, (.)
whereM=sup{zt–xt:t∈(,min{,+μA})}. Combining (.) with (.), we get
ut–ut=Trtzt–Trtzt ≤ xt–xt+
b|rt–rt|(L+M). (.)
Now, using (.), we calculate
xt–xt
=(I–tA)TrtFrtxt+t(u+γfxt) – (I–tA)TrtFrtxt–t(u+γfxt)
≤(I–tA)Trtzt– (I–tA)Trtzt+(I–tA)Trtzt– (I–tA)Trtzt
+γ|t–t|fxt+|t–t|u+γtfxt–fxt
≤ |t–t|ATrtzt+
–t( +μγ)Trtzt–Trtzt
+γ|t–t|fxt+|t–t|u+tγkxt–xt
≤ |t–t|ATrtzt+
–t( +μγ)xt–xt+
b|rt–rt|(L+M)
+γ|t–t|fxt+|t–t|u+tγkxt–xt.
This implies that
xt–xt ≤
ATrtzt+γfxt+u
t( +μγ –γk) |t–t|
+( –t( +μγ))
b(L+M)
t( +μγ –γk) |rt–rt|.
Sincert: (,min{,+μA})→(,∞) is locally Lipschitzian,xtis also locally Lipschitzian.
We prove the following theorem for strong convergence of the net{xt}ast→, which guarantees the existence of solutions of the optimization problem (.).
Theorem . Let the net{xt}be defined via(.).Then xt converges strongly to a point
x∈VI(C,F)∩Fix(T)as t→,which solves the variational inequality
u+γf – (I+μA)x,p–x≤, ∀p∈VI(C,F)∩Fix(T). (.)
Thisx is the unique solution of the optimization problem(.).
Proof We first show the uniqueness of a solution of the variational inequality (.), which is indeed a consequence of the strong monotonicity of (I+μA) –γf. In fact, sinceAis a strongly positive bounded linear operator with a constantγ > , we know from Lemma . thatI+μA–γf is strongly monotone with a constant +μγ –γk∈(, ). Suppose that x∈VI(C,F)∩Fix(T) andx∈VI(C,F)∩Fix(T) both are solutions to (.). Then we have
u+γf – (I+μA)x,x–x≤ (.)
and
u+γf – (I+μA)x,x–x≤. (.)
Adding up (.) and (.) yields
(I+μA) –γfx–(I+μA) –γfx,x–x≤.
The strong monotonicity of (I+μA) –γf (Lemma .) implies thatx=xand the unique-ness is proved.
Next, we prove thatxt →xast→. LetA= (I+μA), and letzt =Frtxt. Observing
Fix(T) =Fix(Trt) (by Lemma .(iii)) andFix(Frt) =VI(C,F) (by Lemma .(iii)), from
(.) we write, for givenp∈VI(C,F)∩Fix(T),
xt–p= (I–tA)Trtzt– (I–tA)Trtp+tγ(fxt–fp) +t
u+ (γf–A)p
= (I–tA)(Trtzt–Trtp) +tγ(fxt–fp) +t
u+ (γf –A)p
to derive that
xt–p=
(I–tA)(Trtzt–Trtp),xt–p
+tγfxt–fp,xt–p
+tu+ (γf–A)p,xt–p
≤ –t( +μγ)zt–pxt–p+tγkxt–p
+tu+ (γf–A)p,xt–p
≤ –t( +μγ)xt–p+tγkxt–p
+tu+ (γf–A)p,xt–p
Therefore we have
xt–p≤
+μγ –γk
u+ (γf –A)p,xt–p
. (.)
Since{xt}is bounded ast→ (by Proposition .(i)), there exists a subsequence{tn}in (,min{,+μA}) such thattn→ andxtnx∗. First of all, we prove thatx∗∈VI(C,F)∩
Fix(T). To this end, we divide its proof into four steps.
Step. We show thatlimn→∞xtn–ztn= , whereztn=Frtnxtn. To show this, letp∈
VI(C,F)∩Fix(T). Sincep=Frtnp, from (.) we deduce
ztn–p =F
rtnxtn–Frtnp
≤ ztn–p,xtn–p
=
xtn–p+ztn–p–xtn–ztn
,
and hence
ztn–p ≤ x
tn–p –x
tn–ztn .
Thus, from (.) we have
Trnztn–p ≤ z
tn–p ≤ x
tn–p –x
tn–ztn .
This implies
xtn–ztn ≤ x
tn–p –T
rnztn–p
≤xtn–p+Trnztn–p
xtn–p–Trnztn–p
≤xtn–p+Trnztn–p
xtn–Trnztn
=xtn–p+Trnztn–p
xtn–TrnFrnxtn.
Sincetn→ andxtn–TrnFrnxtn → by Proposition .(ii), we getxtn–ztn → by the boundedness of{xt}and{Trtzt}.
Step. We show thatlimn→∞utn–ztn= , whereutn=Trtnztn. Indeed, from Proposi-tion .(ii) and Step it follows that
utn–ztn ≤ utn–xtn+xtn–ztn → (asn→ ∞).
Step. We show thatx∗∈VI(C,F). In fact, from the definition ofztn=Frtnxtnwe have
y–ztn,Fztn+
y–ztn, ztn–xtn
rtn
Setwt=tv+ ( –t)x∗for allt∈(, ] andv∈C. Thenwt∈C. From (.) it follows that
wt–ztn,Fwt ≥ wt–ztn,Fwt–wt–ztn,Fztn–
wt–ztn, ztn–xtn
rtn
=wt–ztn,Fwt–Fztn–
wt–ztn, ztn–xtn
rtn
. (.)
By Step , we have ztn–xtn
rtn → asn→ ∞. Moreover, sincextnx∗, by Step , we have ztnx∗asn→ ∞. SinceFis monotone, we also have thatwt–ztn,Fwt–Fztn ≥. Thus, from (.) it follows that
≤ lim
n→∞wt–ztn,Fwt=
wt–x∗,Fwt
,
and hence
v–x∗,Fwt
≥, ∀v∈C.
Ift→, the continuity ofFyields that
v–x∗,Fx∗≥, ∀v∈C.
This implies thatx∗∈VI(C,F).
Step. We show thatx∗∈Fix(T). In fact, from the definition ofutn=Trtnztn, we have
y–utn,Tutn– rtn
y–utn, ( +rtn)utn–ztn
≤, ∀y∈C. (.)
Putwt=tv+ ( –t)x∗for allt∈(, ] andv∈C. Thenwt∈C, and from (.) and
pseu-docontractivity ofTit follows that
utn–wt,Twt ≥ utn–wt,Twt+wt–utn,Tutn
–
rtn
wt–utn, ( +rtn)utn–ztn
= –wt–utn,Twt–Tutn– rtn
wt–utn,utn–ztn
–wt–utn,utn
≥–wt–utn –
rtn
wt–utn,utn–ztn–wt–utn,utn
= –wt–utn,wt–
wt–utn,
utn–ztn rtn
. (.)
By Step , we get utn–ztn
rtn → asn→ ∞. Moreover, sincextnx
∗, by Step and Step ,
we haveutnx∗asn→ ∞. Therefore, from (.), asn→ ∞, it follows that
x∗–wt,Twt
≥x∗–wt,wt
and hence
–v–x∗,Twt
≥–v–x∗,wt
, ∀v∈C.
Lettingt→ and using the fact thatT is continuous, we get
–v–x∗,Tx∗≥–v–x∗,x∗, ∀v∈C.
Now, let v=Tx∗. Then we obtain x∗ =Tx∗ and hence x∗ ∈ Fix(T). Therefore, x∗ ∈ VI(C,F)∩Fix(T).
Now, we substitutex∗forpin (.) to obtain
xtn–x∗
≤
+μγ–γk
u+ (γf–A)x∗,xtn–x∗
. (.)
Note that xtn x∗ and limn→∞tn= . These facts and inequality (.) imply that xtn→x∗strongly.
Finally, we prove thatx∗is a solution of the variational inequality (.). In fact, putting xtnin place ofxtin (.) and taking the limit astn→, we obtain
x∗–p≤ +μγ–γk
u+ (γf–A)p,x∗–p, ∀p∈VI(C,F)∩Fix(T).
In particular,x∗solves the following variational inequality:
x∗∈VI(C,F)∩Fix(T), u+ (γf–A)p,p–x∗≤, ∀p∈VI(C,F)∩Fix(T),
or the equivalent dual variational inequality (see [])
x∗∈VI(C,F)∩Fix(T), u+ (γf–A)x∗,p–x∗≤, ∀p∈VI(C,F)∩Fix(T).
That is, x∗ ∈VI(C,F)∩Fix(T) is a solution of the variational inequality (.); hence x∗=x by uniqueness. In summary, we have shown that each cluster point of{xt} (at t→) equalsx. Thereforext→xast→. By (.) and Lemma ., we deduce
im-mediately the desired result. This completes the proof.
If we takeμ= ,u= andf≡ in Theorem ., then we have the following corollary.
Corollary . Let{xt}be defined by
xt= (I–t)TrtFrtxt.
Then{xt}converges strongly as t→to a pointx∈VI(C,F)∩Fix(T),which is the
mini-mum norm point ofVI(C,F)∩Fix(T).
Corollary . Let{xt}be defined by
xt=t(u+γfxt) +
I–t(I+μA)Frtxt.
Then{xt}converges strongly as t→to a pointx∈VI(C,F),where is the unique solution of the optimization problem
min
x∈VI(C,F) μ
Ax,x+ x–u
–h(x). (.)
Proof IfT≡I, thenTrin Lemma . is the identity mapping. Thus the result follows from
Theorem ..
TakingF≡ in Theorem ., we get the following corollary.
Corollary . Let{xt}be defined by
xt=t(u+γfxt) +
I–t(I+μA)Trtxt.
Then{xt}converges strongly as t→to a pointx∈Fix(T),where is the unique solution of the optimization problem
min
x∈Fix(T) μ
Ax,x+ x–u
–h(x). (.)
Proof IfF≡, thenFrin Lemma . is the identity mapping. Thus the result follows from
Theorem ..
If, in Theorem ., we takeC≡H, then we obtain the following corollary.
Corollary . Let T:H→H be a continuous pseudocontractive mapping,and let F : H→H be a continuous monotone mapping.Let{xt}be defined by(.).Then{xt}
con-verges strongly as t→to a pointx∈F–()∩Fix(T),which is the unique solution of the optimization problem
min
x∈F–()∩Fix(T)
μ
Ax,x+ x–u
–h(x). (.)
Proof SinceD(F) =H, we haveVI(H,F) =F–(). So, by Theorem ., we obtain the
de-sired result.
Now, we propose the following general iterative method which generates a sequence in an explicit way:
xn+=αn(u+γfxn) +
I–αn(I+μA)
TrnFrnxn, ∀n≥, (.)
Theorem . Let{xn} be the sequence generated by the explicit scheme(.).Let{αn} and{rn} ⊂(,∞)satisfy the following conditions:
(C) {αn} ⊂[, ]andαn→asn→ ∞; (C) ∞n=αn=∞;
(C) |αn+–αn| ≤o(αn+) +σn, ∞n=σn<∞(the perturbed control condition); (C) lim infn→∞rn> and ∞n=|rn+–rn|<∞.
Then{xn}converges strongly to a pointx∈VI(C,F)∩Fix(T),which is the unique solution of the variational inequality(.).Thisx is the unique solution of the optimization problem (.).
Proof First, note that from condition (C), without loss of generality, we assume that αn( +μγ –γk) < and αn–(+αμγnγ–kγk) < for alln≥. Letx∈VI(C,F)∩Fix(T) be the
unique solution of the variational inequality (.). (The existence ofxfollows from The-orem ..)
From now on, we putA=I+μA,zn=Frnxnandun=Trnzn. Letp∈VI(C,F)∩Fix(T).
Thenp=Trnpby Lemma .(iii) andp=Frnpby Lemma .(iii). Moreover, from the non-expansivity ofFrnit follows that
zn–p=Frnxn–Frnp ≤ xn–p. (.)
We divide the proof into several steps as follows.
Step. We show that{xn}is bounded. First of all, by (.), we deduce
xn+–p
=αn(u+γfxn) + (I–αnA)Trnzn–p
=(I–αnA)Trnzn– (I–αnA)Trnp+αnγ(fxn–fp) +αn
u+ (γf–A)p
≤(I–αnA)Trnzn– (I–αnA)Trnp+αnγkxn–p+αn
u+(γf–A)p
≤ –αn( +μγ)
zn–p+αnγkxn–p+αn
u+(γf–A)p
≤ –αn( +μγ)
xn–p+αnγkxn–p+αn
u+(γf –A)p
= –αn( +μγ –γk)
xn–p+αn
u+(γf –A)p
≤max
xn–p,
u+(γf–A)p +μγ –γk
.
By induction, we derive
xn–p ≤max
x–p,u+(γf –A)p +μγ –γk
, ∀n≥.
This implies that {xn} is bounded and so are {zn}={Frnxn}, {un}={Trnzn}, {fxn}, and
{ATrnzn}. As a consequence, with the control condition (C), we get
xn+–Trnzn ≤αn
u+γfxn–ATrnzn
→ (n→ ∞). (.)
un–=Trn–zn–instead ofzt=Frtxt,zt=Frtxt,ut=Trtzt, andut=Trtzt, respectively,
we have
Trnzn–Trn–zn– ≤ xn–xn–+
b|rn–rn–|(M+M), (.)
whereM=sup{un–zn:n≥},M=sup{zn–xn:n≥}, andrn>b> ,n≥ for
someb. Thus, by (.) and Lemma ., we derive
xn+–xn
=αn(u+γfxn) + (I–αnA)Trnzn
–αn–(u+γfxn–) – (I–αn–A)Trn–zn–
≤(I–αnA)(Trnzn–Trn–zn–)+|αn–αn–|ATrn–zn–
+αnγfxn–fxn–+|αn–αn–|u
≤ –αn( +μγ)
Trnzn–Trn–zn–
+αnγkxn–xn–+|αn–αn–|
ATrn–zn–+u
≤ –αn( +μγ)
xn–xn–+
b|rn–rn–|(M+M)
+αnγkxn–xn–+|αn–αn–|M
≤ –αn( +μγ–γk)
xn–xn–
+|αn–αn–|M+
b|rn–rn–|(M+M)
≤ –αn( +μγ–γk)
xn–xn–
+o(αn) +σn–
M+
b|rn–rn–|(M+M), (.)
whereM=sup{ATrnzn+u:n≥}. By takingsn+=xn+–xn,wn=αn( +μγ –
γk),wnδn=Mo(αn) andνn=σn–M+b|rn–rn–|(M+M), from (.) we deduce
sn+≤( –wn)sn+wnδn+νn.
Hence, by conditions (C), (C), (C) and Lemma ., we obtain
lim
n→∞xn+–xn= .
Step. We show thatlimn→∞xn–zn= . By takingxnandzninstead ofxtnandztnin Step of the proof of Theorem ., the result follows from Step in the proof of Theo-rem ., (.) and Step .
Step. We show thatlimn→∞xn–un= , whereun=Trnzn. In fact, from (.) and Step , we have
xn–un=xn–Trnzn
Step. We show thatlimn→∞un–zn= , whereun=Trnzn. In fact, from Step and Step , we have
un–zn ≤ un–xn+xn–zn → (asn→ ∞).
Step. We show thatlim supn→∞u+ (γf –A)x,xn–x ≤. To this end, take a
subse-quence{xnk}of{xn}such that
lim sup
n→∞
u+ (γf –A)x,xn–x
= lim
k→∞
u+ (γf–A)x,xnk–x .
Without loss of generality, we may assume thatxnkp. Takexnk andznk in place ofxtn andztnin Step and Step of the proof of Theorem .. Then, from Step and Step in the proof of Theorem . along with Step , we derivep∈VI(C,F)∩Fix(T). Hence, from (.) we conclude
lim sup
n→∞
u+ (γf –A)x,xn–x
= lim
k→∞
u+ (γf–A)x,xnk–x
=u+ (γf–A)x,p–x≤.
Step. We show thatlimn→∞xn–x= . Note thatx∈VI(C,F)∩Fix(T). Letzn=Frnxn.
By (.),x=Frnx, andx=Trnx, we deduce
xn+–x= (I–αnA)(Trnzn–Trnx) +αnγ(fxn–fx) +αn
u+ (γf –A)x.
Applying Lemma . and Lemma ., we obtain
xn+–x
=(I–αnA)(Trnzn–Trnx) +αnγ(fxn–fx) +αn
u+ (γf –A)x
≤(I–αnA)(Trnzn–Trnx)
+ αnγfxn–fx,xn+–x
+ αn
u+ (γf–A)x,xn+–x
≤ –αn( +μγ)
zn–x+ αnγkxn–xxn+–x
+ αn
u+ (γf–A)x,xn+–x
≤ –αn( +μγ)
xn–x+αnγk
xn–x+xn+–x
+ αn
u+ (γf–A)x,xn+–x
≤ – αn( +μγ) +αnγk
xn–x+αn( +μγ)xn–x+αnγkxn+–x
+ αn
u+ (γf–A)x,xn+–x
. (.)
It then follows from (.) that
xn+–x≤
– αn( +μγ) +αnγk
–αnγk
xn–x
+α
n( +μγ)
–αnγk
xn–x+
αn
–αnγk
=
–αn( +μγ –γk) –αnγk
xn–x
+α
n( +μγ)
–αnγk
xn–x+
αn
–αnγk
u+ (γf –A)x,xn+–x
≤( –wn)xn–x+wnδn,
where
wn=
αn( +μγ–γk)
–αnγk
and
δn=
( +μγ –γk)
αn( +μγ)M+
u+ (γf –A)x,xn+–x
,
whereM=sup{xn–x:n≥}. It can be easily seen from conditions (C) and (C) and
Step thatwn→, ∞n=wn=∞andlim supn→∞δn≤. From Lemma . withνn= ,
we conclude thatlimn→∞xn–x= . This completes the proof.
If we takeμ= ,u= andf≡ in Theorem ., then we have the following corollary.
Corollary . Let{xn}be defined by
xn+= ( –αn)TrnFrnxn.
Assume that the sequences{αn}and{rn}satisfy conditions(C)-(C)in Theorem..Then
{xn}converges strongly to a pointx∈VI(C,F)∩Fix(T),which is the minimum norm point ofVI(C,F)∩Fix(T).
TakingT≡Iin Theorem ., we have the following corollary.
Corollary . Let{xn}be generated by the following iterative scheme:
xn+=αn(u+γfxn) +
I–αn(I+μA)
Frnxn, ∀n≥.
Assume that the sequences{αn}and{rn}satisfy conditions(C)-(C)in Theorem..Then
{xn}converges strongly to a pointx∈VI(C,F),which is the unique solution of the optimiza-tion problem(.).
TakingF≡ in Theorem ., we get the following corollary.
Corollary . Let{xn}be generated by the following iterative scheme:
xn+=αn(u+γfxn) +
I–αn(I+μA)
Trnxn, ∀n≥.
Assume that the sequences{αn}and{rn}satisfy conditions(C)-(C)in Theorem..Then
{xn}converges strongly to a pointx∈Fix(T),which is the unique solution of the optimiza-tion problem(.).
Corollary . Let T:H→H be a continuous pseudocontractive mapping,and let F : H→H be a continuous monotone mapping.Let{xn}be generated by(.).Assume that the sequences{αn} and{rn}satisfy conditions(C)-(C)in Theorem..Then{xn} con-verges strongly to a pointx∈F–()∩Fix(T),which is the unique solution of the optimiza-tion problem(.).
Remark . () For finding an element ofVI(C,F)∩Fix(T),VI(C,F),Fix(T), andF–()∩ Fix(T), which is the unique solution of the optimization problems (.), (.), (.), and (.), respectively, whereT is a continuous pseudocontractive mapping andFis a con-tinuous monotone mapping, our results are new ones different from previous those in-troduced by several authors. Consequently, our results supplement, develop, and improve upon the corresponding results given recently by several authors in this direction (for ex-ample, see [–] forVI(C,F)∩Fix(T) in the case of a continuous monotone mapping Fand a continuous pseudocontractive mappingT; see [, –] forVI(C,F)∩Fix(S) in the case of anα-inverse-strongly monotone mappingFand a nonexpansive mappingS; see [, , ] forFix(T) of a strictly pseudocontractive mappingT; and see [, ] for Fix(S) of a nonexpansive mappingS).
() As in Corollary . and Corollary ., from Corollaries ., ., ., ., ., and ., we can obtain the minimum norm point ofVI(C,F),Fix(T), andF–()∩Fix(T) for the continuous monotone mappingFand the continuous pseudocontractive mappingT, respectively.
() We can replace the perturbed control condition|αn+–αn| ≤o(αn+) +σn, ∞n=σn< ∞on the control parameter{αn}in (C) of Theorem . by the following conditions [, ]:
(a) ∞n=|αn+–αn|<∞; or
(b) limn→∞ααnn+ = or, equivalently,limn→∞
αn–αn+
αn+ = .
Competing interests
The author declares to have no competing interests.
Acknowledgements
The author would like to thank the anonymous reviewers for their valuable suggestions and comments along with providing some recent related papers.
This study was supported by research funds from Dong-A University.
Received: 23 May 2015 Accepted: 30 October 2015 References
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