R E S E A R C H
Open Access
New iterative methods for a common
solution of fixed points for
pseudo-contractive mappings and variational
inequalities
Rabian Wangkeeree
1and Kamonrat Nammanee
2**Correspondence:
2Department of Mathematics,
School of Science, University of Phayao, Phayao, 56000, Thailand Full list of author information is available at the end of the article
Abstract
In this paper, we introduce three iterative methods for finding a common element of the set of fixed points for a continuous pseudo-contractive mapping and the solution set of a variational inequality problem governed by continuous monotone mappings. Strong convergence theorems for the proposed iterative methods are proved. Our results improve and extend some recent results in the literature.
MSC: 47H05; 47H09; 47J25; 65J15
Keywords: pseudo-contractive mapping; monotone mapping; strong convergence theorem
1 Introduction
The theory of variational inequalities represents, in fact, a very natural generalization of the theory of boundary value problems and allows us to consider new problems arising from many fields of applied mathematics, such as mechanics, physics, engineering, the theory of convex programming, and the theory of control. While the variational theory of boundary value problems has its starting point in the method of orthogonal projection, the theory of variational inequalities has its starting point in the projection on a convex set.
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. The classical variational inequality problem is to findu∈Csuch thatv–u,Au ≥ for allv∈C, where
Ais a nonlinear mapping. The set of solutions of the variational inequality is denoted by
VI(C,A). The variational inequality problem has been extensively studied in the literature; see [–] and the references therein. In the context of the variational inequality problem, this implies thatu∈VI(C,A)⇔u=PC(u–λAu),∀λ> , wherePCis a metric projection ofHintoC.
LetAbe a mapping fromCtoH, thenAis calledmonotoneif and only if for eachx,y∈C,
x–y,Ax–Ax ≥. (.)
An operatorAis said to be strongly positive onHif there exists a constantγ > such that
Ax,x ≥γx, ∀x∈H.
A mappingAofCinto itself is calledL-Lipschitz continuous if there exits a positive num-berLsuch that
Ax–Ay ≤Lx–y, ∀x,y∈C.
A mappingAofCintoH is calledα-inverse-strongly monotoneif there exists a positive real numberαsuch that
x–y,Ax–Ay ≥αAx–Ay
for allx,y∈C; see [–]. IfAis anα-inverse strongly monotone mapping ofCintoH, then it is obvious thatAis α-Lipschitz continuous, that is,Ax–Ay ≤ αx–yfor all
x,y∈C. Clearly, the class of monotone mappings includes the class ofα-inverse strongly monotone mappings.
A mappingAofCintoH is calledγ¯-strongly monotoneif there exists a positive real numberγ¯ such that
x–y,Ax–Ay ≥ ¯γx–y
for allx,y∈C; see []. Clearly, the class ofγ¯-strongly monotone mappings includes the class of strongly positive mappings.
Recall that a mappingTofCintoHis calledpseudo-contractiveif for eachx,y∈C, we have
Tx–Ty,x–y ≤ x–y. (.)
Tis said to be ak-strict pseudo-contractive mapping if there exists a constant ≤k≤ such that
x–y,Tx–Ty ≤ x–y–k(I–T)x– (I–T)y for allx,y∈D(T).
A mappingTofCinto itself is callednonexpansiveifTx–Ty ≤ x–yfor allx,y∈C. We denote byF(T) the set of fixed points ofT. Clearly, the class of pseudo-contractive mappings includes the class of nonexpansive and strict pseudo-contractive mappings.
For a sequence{αn}of real numbers in (, ) and arbitraryu∈C, let the sequence{xn} inCbe iteratively defined byx∈Cand
xn+:=αn+u+ ( –αn+)Txn, n≥, (.)
In , Moudafi [] introduced a viscosity approximation method and proved that if
His a real Hilbert space, for givenx∈C, the sequence{xn}generated by the algorithm
xn+:=αnf(xn) + ( –αn)Txn, n≥, (.) wheref :C→Cis a contraction mapping and{αn} ⊂(, ) satisfies certain conditions, converges strongly to the unique solutionx∗inCof the variational inequality
(I–f)x∗,x–x∗≥, x∈C. (.)
Moudafi [] generalized Halpern’s theorems in the direction of viscosity approxima-tions. In [], Zegeyeet al.extended Moudafi’s result to the case of Lipschitz pseudo-contractive mappings in Banach spaces more general that Hilbert spaces.
In , Marino and Xu [] introduced the following general iterative method:
xn+:=αnγf(xn) + ( –αnA)Txn, n≥. (.) They proved that if the sequence{αn} of parameters satisfies appropriate conditions, then the sequence{xn}generated by (.) converges strongly to the unique solution of the variational inequality
(A–γf)x∗,x–x∗≥, x∈C, (.) which is the optimality condition for the minimization problem
min
x∈C
Ax,x–h(x),
wherehis a potential function forγf (i.e.,h(x) =γf(x) forx∈H).
Recently, Zegeye and Shahzad [] introduced an iterative method and proved that ifC
is a nonempty subset of a real Hilbert spaceH,T:C→Cis a pseudo-contractive mapping
andT:C→His a continuous monotone mapping such thatF:=F(T)∩VI(C,T)=∅.
For{rn} ⊂(,∞) definedTrn,Frn:H→Cby the following: forx∈Hand{rn} ⊂(,∞), define
Trnx:=
z∈C:y–z,Tz–
rn
y–z, ( +rn)z–x≤,y∈C
, (.)
Frnx:=
z∈C:y–z,Tz+
rn
y–z,z–x ≥,y∈C
. (.)
Then the sequence{xn}generated byx∈Cand
xn+:=αnf(xn) + ( –αn)TrnFrnxn, n≥, (.)
wheref :C→Cis a contraction mapping and{αn} ⊂[, ] and{rn}satisfy certain condi-tions, converges strongly toz∈F, wherez=PFf(z).
common fixed point of a set of fixed points of continuous pseudo-contractive mappings more general than nonexpansive mappings and a solution set of the variational inequality problem for continuous monotone mappings more general thanα-inverse strongly mono-tone mappings in a real Hilbert space. Our result extend and unify most of the results that have been proved for important classes of nonlinear operators.
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. LetT,T:C→ H be a continuous pseudo-contractive mapping and a continuous monotone mapping, respectively. Forx∈Hand{rn} ⊂(,∞), letTrn,Frn:H→Cbe defined by (.) and (.).
We consider the three iterative methods given as follows:
x∈H,
xn+:=αnγf(xn) + (I–αnA)TrnFrnxn, n≥, (.) y∈H,
yn+:=αnγf(TrnFrnyn) + (I–αnA)TrnFrnyn, n≥, (.) z∈H,
zn+:=TrnFrn
αnγf(zn) + (I–αnA)zn
, n≥, (.)
whereAis aγ¯-strongly monotone andL-Lipschitzian continuous operator andf:H→H
is a contraction mapping. We prove in Section that if{αn}and{rn}of parameters satisfy appropriate conditions, then the sequences {xn},{yn}and{zn} converge strongly toz=
PF(I–A+γf)(z).
2 Preliminaries
LetCbe a closed and convex subset of a real Hilbert spaceH. For everyx∈H, there exists a unique nearest point inC, denoted byPCx, such that
x–PCx=x–y, ∀y∈C. (.)
PCis called the metric projection ofHontoC. We know thatPCis a nonexpansive mapping ofHontoC. In connection with metric projection, we have the following lemma.
Lemma . Let H be a real Hilbert space.The following identity holds:
x+y≤ x+ y,x+y, ∀x,y∈H.
Lemma . Let C be a nonempty closed convex subset of a Hilbert space H.Let x∈H and y∈C.Then y=PCx if and only if
x–y,y–z ≥, ∀z∈C. (.)
Lemma .[] Let{an}be a sequence of nonnegative real numbers such that
an+≤( –γn)an+σn, ∀n≥,
(i) {γn} ⊂[, ], ∞n=γn=∞and (ii) lim supn→∞σn
γn≤or
∞
n=|σn|<∞.
Then an→as n→ ∞.
Lemma .[] Let C be a nonempty closed and convex subset of a real Hilbert space H.
Let A:C→H be a continuous monotone mapping.Then,for r> and x∈H,there exists z∈C such that
y–z,Az+
ry–z,z–x ≥, ∀y∈C. (.)
Moreover,by a similar argument as in the proof of Lemmas.and.in[],Zegeye[]
obtained the following lemmas.
Lemma .[] Let C be a nonempty closed and convex subset of a real Hilbert space H.
Let A:C→H be a continuous monotone mapping.For r> and x∈H,define a mapping Fr:H→C as follows:
Frx:=
z∈C:y–z,Az+
ry–z,z–x ≥,∀y∈C
for all x∈H.Then the following hold: () Fris single-valued;
() Fris a firmly nonexpansive type mapping,i.e.,for allx,y∈H,
Frx–Fry≤ Frx–Fry,x–y;
() F(Fr) =VI(C,A);
() VI(C,A)is closed and convex.
In the sequel,we shall make use of the following lemmas.
Lemma .[] Let C be a nonempty closed and convex subset of a real Hilbert space H.
Let T:C→H be a continuous pseudo-contractive mapping.Then,for r> and x∈H,
there exists z∈C such that
y–z,Tz–
r
y–z, ( +r)z–x≤, ∀y∈C. (.)
Lemma .[] Let C be a nonempty closed and convex subset of a real Hilbert space H.
Let T:C→C be a continuous pseudo-contractive mapping.For r> and x∈H,define a mapping Tr:H→C as follows:
Trx:=
z∈C:y–z,Tz+
r
y–z, ( +r)z–x≤,∀y∈C
for all x∈H.Then the following hold: () Tris single-valued;
() Tris a firmly nonexpansive type mapping,i.e.,for allx,y∈H,
() F(Tr) =F(T);
() F(T)is closed and convex.
Lemma .[] Let <α< and let f be anα-contraction of a real Hilbert space H into itself,and let A be aγ¯-strongly monotone and L-Lipschitzian continuous operator of H into itself withγ¯> and L> .Takeμ,γ to be real numbers as follows:
<μ<γ¯
L, <γ <
¯
γ–Lμ
α .
If{αn} ⊆(, ),limn→∞αn= andτ=γ¯–L μ
,then
(I–αnA)x– (I–αnA)y≤( –αnτ)x–y, ∀x,y∈H.
3 Main results
Now, we prove our main theorems.
Theorem . Let H be a real Hilbert space,T:C→C be a continuous pseudo-contractive mapping and T :C→H be a continuous monotone mapping such that F:=F(T)∩ VI(C,T)=∅.Let <α< and let f be anα-contraction of H into itself,and let A be a
¯
γ-strongly monotone and L-Lipschitzian continuous operator of C into H withγ¯> and L> .Takeμ,γ to be real numbers as follows:
<μ<γ¯
L, <γ <
¯
γ–Lμ
α .
For x∈H,let{xn}be a sequence generated by(.),where{αn} ⊂[, ]and{rn} ⊂(,∞)
are such thatlimn→∞αn= , ∞n=αn=∞, ∞n=|αn+–αn|<∞,lim infn→∞rn> and ∞
n=|rn+–rn|<∞.Then the sequence{xn}converges strongly to z∈F,where z=PF(I–
A+γf)(z).
Proof Sinceαn→ asn→ ∞, we may assume, without loss of generality,αn< for alln. ForQ=PF, it implies thatQ(I–A+γf) is a contraction ofHinto itself. SinceHis a real
Hilbert space, there exists a unique elementz∈Hsuch thatz=PF(I–A+γf)(z).
Letv∈F, and letun:=Trnwn, wherewn:=Frnxn. Then we have from Lemma (.) and
(.) that
un–v=Trnwn–Trnv ≤ wn–v=Frnxn–Frnv ≤ xn–v. (.)
Moreover, from (.) and (.), we get that
xn+–v=αnγf(xn) + (I–αnA)un–v =αn
γf(xn) –A(v)+ (I–αnA)un– (I–αnA)v
≤( –αnτ)xn–v+αnγ αxn–v+αnγf(v) –A(v)
≤ –αn(τ–γ α)xn–v+αnγf(v) –A(v) = –αn(τ–γ α)xn–v+αn(τ–γ α)
γf(v) –A(v)
It follows from induction that
xn–v ≤max
x–v,
γf(v) –A(v)
τ–γ α
, n≥. (.)
Thus{xn}is bounded, and hence so are{un},{wn}and{f(xn)}. Next, to show thatxn+– xn →, we have
xn+–xn=αnγf(xn) + (I–αnA)un–
αn–γf(xn–) + (I–αn–A)un–
=αnγf(xn) –αnγf(xn–) +αnγf(xn–) –αn–γf(xn–)
+ (I–αnA)un– (I–αnA)un–+ (I–αnA)un–– (I–αn–A)un–
≤αnγ αxn–xn–+|αn–αn–|γf(xn–)
+ ( –αnτ)un–un–+|αn–αn–|Aun–
≤αnγ αxn–xn–+ ( +γ)|αn–αn–|K+ ( –αnτ)un–un–
≤αnγ αxn–xn–+ ( +γ)|αn–αn–|K+ ( –αnτ)wn–wn–, (.)
whereK=sup{f(xn)+Aun:n∈N}<∞.
Moreover, sincewn=Frnxn,wn+=Frn+xn+, we get that
y–wn,Twn+
rn
y–wn,wn–xn ≥ for ally∈C, (.)
y–wn+,Twn++
rn+
y–wn+,wn+–xn+ ≥ for ally∈C. (.)
Puttingy=wn+in (.) andy=wnin (.), we get that
wn+–wn,Twn+
rn
wn+–wn,wn–xn ≥, (.)
wn–wn+,Twn++
rn+
wn–wn+,wn+–xn+ ≥. (.)
Adding (.) and (.), we have
wn+–wn,Twn–Twn++
wn+–wn, wn–xn
rn
–wn+–xn+
rn+
≥, (.)
which implies that
–wn+–wn,Twn+–Twn+
wn+–wn, wn–xn
rn
–wn+–xn+
rn+
≥. (.)
Now, using the fact thatTis monotone, we get that
wn+–wn, wn–xn
rn
–wn+–xn+
rn+
≥, (.)
and hence
wn+–wn,wn–wn++wn+–xn–
rn
rn+
(wn+–xn+)
Without loss of generality, let us assume that there exists a real number bsuch that
rn>b> for alln∈N. Then we have
wn+–wn≤
wn+–wn,xn+–xn+
– rn
rn+
(wn+–xn+)
≤ wn+–wn
xn+–xn+ – rn
rn+
wn+–xn+
, (.)
and hence from (.) we obtain that
wn+–wn ≤ xn+–xn+
rn+|
rn+–rn|wn+–xn+
≤ xn+–xn+
b|rn+–rn|L, (.)
whereL=sup{wn–xn:n∈N}<∞.
Furthermore, from (.) and (.) we have that
xn+–xn ≤αnγ αxn–xn–+ ( +γ)|αn–αn–|K
+ ( –αnτ)
xn–xn–+
b|rn–rn–|L
. (.)
Hence by Lemma ., we have
xn+–xn → asn→ ∞. (.)
Consequently, from (.) and (.), we have that
wn+–wn → asn→ ∞. (.)
Moreover, sinceun=Trnwn,un+=Trn+wn+, we get that
y–un,Tun–
rn
y–un, ( +rn)un–wn
≤ for ally∈C. (.)
and
y–un+,Tun+–
rn+
y–un+, ( +rn+)un+–wn+
≤ for ally∈C. (.)
Puttingy=un+in (.) andy=unin (.), we get that
un+–un,Tun–
rn
un+–un, ( +rn)un–wn
≤ (.)
and
un–un+,Tun+–
rn+
un–un+, ( +rn+)un+–wn+
Adding (.) and (.), we have
un+–un,Tun–Tun+
–
un+–un,
( +rn)un–wn
rn
–( +rn+)un+–wn+
rn+
≤, (.)
which implies that
un+–un, (un+–Tun+) – (un–Tun)
–
un+–un,
un–wn
rn
–un+–wn+
rn+
≤. (.)
Now, using the fact thatTis pseudo-contractive, we get that
un+–un, un–wn
rn
–un+–wn+
rn+
≥, (.)
and hence
un+–un,un–un++un+–wn–
rn
rn+
(un+–wn+)
≥. (.)
Thus, using the method in (.) and (.), we have that
un+–un ≤ xn+–xn+
rn+
|rn+–rn|un+–wn+
≤ xn+–xn+
b|rn+–rn|L, (.)
whereL=sup{un–wn:n∈N}<∞.
Therefore, from (.) and the property of{rn}, we have that
un+–un → asn→ ∞. (.)
Furthermore, sincexn=αn–γf(xn–) + (I–αn–A)un–, we have that
xn–un ≤ xn–un–+un––un
=αn–γf(xn–) + (I–αn–A)un––un–+un––un
=αn–γf(xn–) –Aun–+un––un. (.) Fromαn→, we havexn–un →.
Now, forv∈F, using Lemma ., we get that
wn–v=Frnxn–Frnv
≤ Frnxn–Frnv,xn–v
=wn–v,xn–v
=
wn–v+xn–v–xn–wn
and hence
wn–v≤ xn–v–xn–wn. (.)
Therefore, we have
xn+–v =αnγf(xn) + (I–αnA)un–v
=αn
γf(xn) –Av+ (I–αnA)(un–v)
≤( –αnτ)un–v+ αn
γf(xn) –Av,xn+–v
≤( –αnτ)wn–v+ αn
γf(xn) –γf(v) +γf(v) –Av,xn+–v
≤( –αnτ)wn–v+ αnγ
f(xn) –f(v),xn+–v
+ αn
γf(v) –Av,xn+–v
≤( –αnτ)wn–v+ αnγ αxn–vxn+–v
+ αnγf(v) –Avxn+–v
≤( –αnτ)
xn–v–xn–wn
+ αnγ αxn–vxn+–v
+ αnγf(v) –Avxn+–v
= – αnτ+ (αnτ)
xn–v– ( –αnτ)xn–wn
+αnγ αxn–vxn+–v+ αnγf(v) –Avxn+–v. (.)
Hence
( –αnτ)xn–wn
≤ xn–v–xn+–v+αnγ¯xn–v
+ αnγ αxn–vxn+–v+ αnγf(v) –Avxn+–v
≤ xn–xn+
xn–v+xn+–v
+αnτxn–v
+ αnγ αxn–vxn+–v+ αnγf(v) –Avxn+–v. (.)
So, we have
xn–wn → asn→ ∞. (.)
Sinceun–wn ≤ un–xn+xn–wn, it follows that
un–wn → asn→ ∞.
Next, we show that
lim sup
n→∞
(A–γf)z,z–xn
≤,
To show this equality, we choose a subsequence{xni}of{xn}such that
lim sup
n→∞
(A–γf)z,z–xni
=lim sup
n→∞
(A–γf)z,z–xn
.
Since {xni} is bounded, there exists a subsequence {xnij} of{xni} and w∈H such that xnijw. Without loss of generality, we may assume thatxniw. Since{xni} ⊂H and H is closed and convex, we get thatw∈H. Moreover, sincexn–wn→ asn→ ∞, we have thatwniw.
Now, we show thatw∈F. Note that from the definition ofwn, we have
y–wni,Twni+
y–wni,
wni–xni rni
≥ for ally∈C. (.)
Putzt=tv+ ( –t)wfor allt∈(, ] andv∈H. Consequently, we get thatzt∈H. From (.) it follows that
zt–wni,Tzt ≥ zt–wni,Tzt–zt–wni,Twni–
zt–wni,
wni–xni rni
=zt–wni,Tzt–Twni–
zt–wni,
wni–xni rni
.
From the fact thatwni–xni→ asn→ ∞, we obtain that
wni–xni
rni → asn→ ∞. SinceTis monotone, we have thatzt–wni,Tzt–Twni ≥. Thus, it follows that
≤ lim
i→∞zt–wni,Tzt=zt–w,Tzt,
and hence
z–w,Tzt ≥ for allz∈C. (.)
Lettingt→ and using the fact thatTis continuous, we obtain that
z–w,Tw ≥ for allv∈C. (.)
This implies thatw∈VI(C,T).
Furthermore, from the definition ofuniwe have that
y–uni,Tuni–
rni
y–uni, (rni+ )uni–xni
≤ for ally∈C. (.)
Putzt=tv+ ( –t)wfor allt∈(, ] andv∈H. Consequently, we get thatzt∈H. From (.) and pseudo-contractivity ofT, it follows that
uni–zt,Tzt ≥ uni–zt,Tzt+zt–uni,Tuni–
rni
zt–uni, (rni+ )uni–wni
= –zt–uni,Tzt–Tuni–
rni
≥–zt–uni –
rni
zt–uni,uni–wni–zt–uni,uni
= –zt–uni,zt–
zt–uni,
uni–wni rni
. (.)
Then, sincewn–xn→ asn→ ∞, we obtain that wni–xni
rni → asi→ ∞. Thus, asi→ ∞, it follows that
w–zt,Tzt ≥ w–zt,zt, (.)
and hence
–v–w,Tzt ≥–v–w,zt for allv∈C. (.)
Lettingt→ and using the fact thatTis continuous, we obtain that
–v–w,Tw ≥–v–w,w for allv∈C. (.)
Now, letv=Tw. Then we obtain thatw=Twand hencew∈F(T).
Therefore,w∈F(T)∩VI(C,T) and sincez=PF(I–A+γf)(z), by Lemma ., implies
that
lim sup
n→∞
(γf–A)(z),xn–z
=lim sup
i→∞
(I–A+γf)(z),xni–z
=(γf–A)(z),w–z≤. (.)
Now, we show thatxn→zasn→ ∞. Fromxn+–z=αn(γf(xn) –Az) + (I–αnA)(un–z), we have that
xn+–z=αn
γf(xn) –Az+ (I–αnA)(un–z)
≤( –αnτ)un–z+ αn
γf(xn) –Az,xn+–z
≤( –αnτ)xn–z + αn
γf(xn) –γf(z) +γf(z) –Az,xn+–z
≤( –αnτ)xn–z+ αnγ
f(xn) –f(z),xn+–z
+ αn
γf(z) –Az,xn+–z
≤( –αnτ)xn–z+ αnγ αxn–zxn+–z
+ αn
γf(z) –Az,xn+–z
≤( –αnτ)xn–z+αnγ α
xn–z+xn+–z
+ αn
γf(z) –Az,xn+–z
=( –αnτ)+αnγ α
xn–z+αnγ αxn+–z
+ αn
γf(z) –Az,xn+–z
This implies that
xn+–z≤
– αnτ+ (αnτ)+αnγ α –αnγ α
xn–z
+ αn –αnγ α
γf(z) –Az,xn+–z
=
–(τ–γ α)αn –αnγ α
xn–z+ (αnτ) –αnγ α
xn–z
+ αn –αnγ α
γf(z) –Az,xn+–z
≤
–(τ–γ α)αn –αnγ α
xn–z
+(τ–γ α)αn –αnγ α
(αnτ)M (γ¯–γ α)+
τ–γ α
γf(z) –Az,xn+–z
= ( –δn)xn–z+δnβn, (.) whereM=sup{xn–z:n∈N},δn=(–τ–αγ αnγ α)αn andβn=
(αnτ)M (τ–γ α)+
τ–γ αγf(z) –Az,xn+–
z. We putξn=δnβn. It is easy to see thatδn→, n∞=δn=∞andlim supn→∞ξδnn≤ by
(.). Hence, by Lemma ., the sequence{xn}converges strongly toz. This completes
the proof.
Theorem . Let H be a real Hilbert space, T : C → H be a continuous pseudo-contractive mapping and T:C→H be a continuous monotone mapping such thatF:= F(T)∩VI(C,T)=∅.Let <α< and let f be anα-contraction of H into itself,and let A be aγ¯-strongly monotone and L-Lipschitzian continuous operator of H into itself H with
¯
γ > and L> .Takeμ,γ to be real numbers as follows:
<μ<γ¯
L, <γ <
¯
γ–Lμ
α .
For y∈H,let{yn}be a sequence generated by(.),where{αn} ⊂[, ]and{rn} ⊂(,∞)
are such thatlimn→∞αn= , ∞n=αn=∞, ∞n=|αn+–αn|<∞,lim infn→∞rn> and ∞
n=|rn+–rn|<∞.The sequence{yn}converges strongly to z∈F,where z=PF(I–A+
γf)(z).
Proof Let{xn}be the sequence given byx=yand
xn+=αnγf(xn) + (I–αnA)TrnFrnxn, ∀n≥.
From Theorem .,xn→z. We claim thatyn→z. Indeed, we estimate
xn+–yn+ ≤αnγf(xn) –αnγf(TrnFrnyn)
+(I–αnA)TrnFrnxn– (I–αnA)TrnFrnyn
≤αnγ αTrnFrnyn–xn+ ( –αnτ)xn–yn
≤αnγ αyn–z+αnγ αz–xn+ ( –αnτ)xn–yn
≤αnγ αyn–xn+αnγ αxn–z+αnγ αz–xn+ ( –αnτ)xn–yn = –αn(τ–γ α)xn–yn+αn(τ–γ α)
γ α
τ–γ αxn–z. (.)
It follows from ∞n=αn=∞,limn→∞xn–z= and Lemma . thatxn–yn →.
Consequently,yn→zas required.
Theorem . Let H be a real Hilbert space, T : C → H be a continuous pseudo-contractive mapping and T:C→H be a continuous monotone mapping such thatF:= F(T)∩VI(C,T)=∅.Let <α< and let f be anα-contraction of H into itself,and let A be aγ¯-strongly monotone and L-Lipschitzian continuous operator of H into itself H with
¯
γ > and L> .Takeμ,γ to be real numbers as follows:
<μ<γ¯
L, <γ <
¯
γ–Lμ
α .
For z∈H,let{zn}be a sequence generated by(.),where{αn} ⊂[, ]and{rn} ⊂(,∞)
are such thatlimn→∞αn= , ∞n=αn=∞, ∞n=|αn+–αn|<∞,lim infn→∞rn> and ∞
n=|rn+–rn|<∞.The sequence{zn}converges strongly to z∈F,where z=PF(I–A+
γf)(z).
Proof Define the sequences{yn}and{βn}by
yn=αnγf(zn) + (I–αnA)zn and βn=αn+, ∀n≥.
Takingp∈F, we have
zn+–p=TrnFrnyn–TrnFrnp ≤ yn–p
=αnγf(zn) + (I–αnA)zn– (I–αnA)p–αnA(p)
≤( –αnτ)zn–p+αnγf(zn) –A(p) = ( –αnτ)zn–p+αnτ
γf(zn) –A(p)
τ . (.)
It follows from induction that
zn+–p ≤max
z–p,
γf(z) –A(p)
τ
, n≥. (.)
Thus both{zn}and{yn}are bounded. We observe that
yn+=αn+γf(zn+) + (I–αn+A)zn+=βnγf(TrnFrnyn) + (I–βnA)TrnFrnyn.
Thus Theorem . implies that{yn}converges to some pointz. In this case, we also have
Hence the sequence{zn}converges to some pointz. This completes the proof.
Settingγ = ,A≡I, whereIis the identity mapping in Theorem ., we have the fol-lowing result.
Corollary . Let H be a real Hilbert space, T : C →C be a continuous pseudo-contractive mapping and T:C→H be a continuous monotone mapping such thatF:= F(T)∩VI(C,T)=∅.Let f be a contraction of H into itself and let{xn}be a sequence
generated by x∈H and
xn+=αnf(xn) + ( –αn)FrnTrnxn, (.) where {αn} ⊂[, ] and {rn} ⊂ (,∞) are such that limn→∞αn = , ∞n=αn =∞,
∞
n=|αn+–αn|<∞,lim infn→∞rn> and ∞n=|rn+–rn|<∞.The sequence{xn}
con-verges strongly to z∈F,where z=PFf(z).
In Theorem .,γ = ,A≡I,f :=u∈His a constant mapping, then we getz=PF(u). In
fact, we have the following corollary.
Corollary . Let H be a real Hilbert space, T : C →C be a continuous pseudo-contractive mapping and T:C→H be a continuous monotone mapping such thatF:= F(T)∩VI(C,T)= ∅.Let{xn}be a sequence generated by x,u∈H and
xn+=αnu+ ( –αn)FrnTrnxn, (.) where {αn} ⊂[, ] and {rn} ⊂ (,∞) are such that limn→∞αn = , ∞n=αn =∞,
∞
n=|αn+–αn|<∞,lim infn→∞rn> and ∞n=|rn+–rn|<∞.The sequence{xn}
con-verges strongly to z∈F,where z=PF(u).
In Theorem .,γ = andA,T≡I, whereIis the identity mapping, then we have the
following corollary.
Corollary . Let H be a real Hilbert space and T:C→H be a continuous monotone mapping such that VI(C,T)=∅.Let f be a contraction of H into itself,and let{xn}be a
sequence generated by x∈H and
xn+=αnf(xn) + ( –αn)Trnxn, (.)
where {αn} ⊂[, ] and {rn} ⊂ (,∞) are such that limn→∞αn = , ∞n=αn =∞, ∞
n=|αn+–αn|<∞,lim infn→∞rn> and ∞n=|rn+–rn|<∞.The sequence{xn}
con-verges strongly to z∈F(T),where z=PF(T)(z).
Remark . Our results extend and unify most of the results that have been proved for these important classes of nonlinear operators. In particular, Theorem . extends The-orem . of Iiduka and Takahashi [] and Zegeyeet al.[], Corollary . of Suet al.
of Iiduka and Takahashi [] in the sense that our convergence is for the more general class of continuous pseudo-contractive and continuous monotone mappings.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand.2Department of
Mathematics, School of Science, University of Phayao, Phayao, 56000, Thailand.
Acknowledgements
The authors would like to thank Naresuan University and the university of Phayao. Moreover, the authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.
Received: 29 May 2013 Accepted: 18 August 2013 Published: 5 September 2013
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doi:10.1186/1687-1812-2013-233
