R E S E A R C H
Open Access
Iterative common solutions for monotone
inclusion problems, fixed point problems and
equilibrium problems
Wataru Takahashi
1, Ngai-Ching Wong
2and Jen-Chih Yao
3**Correspondence:
3Center for General Education,
Kaohsiung Medical University, Kaohsiung, 80702, Taiwan Full list of author information is available at the end of the article
Abstract
LetHbe a real Hilbert space, and letCbe a nonempty closed convex subset ofH. Let
α
> 0, and letAbe anα
-inverse strongly-monotone mapping ofCintoH. LetTbe a generalized hybrid mapping ofCintoH. LetBandWbe maximal monotoneoperators onHsuch that the domains ofBandWare included inC. Let 0 <k< 1, and letgbe ak-contraction ofHinto itself. LetVbe a
γ
-strongly monotone andL-Lipschitzian continuous operator with
γ
> 0 andL> 0. Takeμ
,γ
∈Ras follows:0 <
μ
<2γ
L2 , 0 <
γ
<γ
–L22μk .
Suppose thatF(T)∩(A+B)–10∩W–10=∅, whereF(T) and (A+B)–10,W–10 are the set
of fixed points ofTand the sets of zero points ofA+BandW, respectively. In this paper, we prove a strong convergence theorem for finding a pointz0of
F(T)∩(A+B)–10∩W–10, wherez0is a unique fixed point of
PF(T)∩(A+B)–10∩W–10(I–V+
γ
g). This pointz0∈F(T)∩(A+B)–10∩W–10 is also a uniquesolution of the variational inequality
(V–
γ
g)z0,q–z0≥0, ∀q∈F(T)∩(A+B)–10∩W–10.Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space. In particular, we solve a problem posed by Kurokawa and Takahashi (Nonlinear Anal. 73:1562-1568, 2010).
MSC: 47H05; 47H10; 58E35
Keywords: maximal monotone operator; resolvent; inverse-strongly monotone mapping; generalized hybrid mapping; fixed point; iteration procedure; equilibrium problem
1 Introduction
LetHbe a real Hilbert space, and letCbe a nonempty closed convex subset ofH. LetNand Rbe the sets of positive integers and real numbers, respectively. A mappingT:C→His calledgeneralized hybrid[] if there existα,β∈Rsuch that
α Tx–Ty + ( –α) x–Ty ≤β Tx–y + ( –β) x–y
for allx,y∈C. We call such a mapping an (α,β)-generalized hybridmapping. Kocourek, Takahashi and Yao [] proved a fixed point theorem for such mappings in a Hilbert space. Furthermore, they proved a nonlinear mean convergence theorem of Baillon’s type [] in a Hilbert space. Notice that the mapping above covers several well-known mappings. For example, an (α,β)-generalized hybrid mappingTis nonexpansive forα= andβ= ,i.e.,
Tx–Ty ≤ x–y , ∀x,y∈C.
It is also nonspreading [, ] forα= andβ= ,i.e.,
Tx–Ty ≤ Tx–y + Ty–x , ∀x,y∈C.
Furthermore, it is hybrid [] forα= andβ=,i.e.,
Tx–Ty ≤ x–y + Tx–y + Ty–x , ∀x,y∈C.
We can also show that ifx=Tx, then for anyy∈C,
α x–Ty + ( –α) x–Ty ≤β x–y + ( –β) x–y ,
and hence x–Ty ≤ x–y . This means that an (α,β)-generalized hybrid mapping with a fixed point is quasi-nonexpansive. The following strong convergence theorem of Halpern’s type [] was proved by Wittmann []; see also [].
Theorem Let C be a nonempty closed convex subset of H,and let T be a nonexpansive mapping of C into itself with F(T)=∅.For any x=x∈C,define a sequence{xn}in C by
xn+=αnx+ ( –αn)Txn, ∀n∈N,
where{αn} ⊂(, )satisfies
lim
n→∞αn= ,
∞
n=
αn=∞ and
∞
n=
|αn–αn+|<∞.
Then{xn}converges strongly to a fixed point of T.
Kurokawa and Takahashi [] also proved the following strong convergence theorem for nonspreading mappings in a Hilbert space; see also Hojo and Takahashi [] for general-ized hybrid mappings.
Theorem Let C be a nonempty closed convex subset of a real Hilbert space H.Let T be a nonspreading mapping of C into itself.Let u∈C and define two sequences{xn}and{zn}
in C as follows:x=x∈C and ⎧
⎨ ⎩
xn+=αnu+ ( –αn)zn,
zn=n n–
for all n = , , . . . , where {αn} ⊂ (, ), limn→∞αn = and
∞
n=αn =∞. If F(T) is
nonempty,then{xn}and{zn}converge strongly to Pu,where P is the metric projection of H
onto F(T).
Remark We do not know whether Theorem for nonspreading mappings holds or not; see [] and [].
In this paper, we provide a strong convergence theorem for finding a pointzofF(T)∩ (A+B)–∩W– such that it is a unique fixed point of
PF(T)∩(A+B)–∩W–(I–V+γg)
and a unique solution of the variational inequality
(V–γg)z,q–z
≥, ∀q∈F(T)∩(A+B)–∩W–,
whereT,A,B,W,gandVdenote a generalized hybrid mapping ofCintoH, anα-inverse strongly-monotone mapping ofCintoHwithα> , maximal monotone operators onH
such that the domains ofBandWare included inC, ak-contraction ofHinto itself with <k< and aγ-strongly monotone andL-Lipschitzian continuous operator withγ > andL> , respectively. Using this result, we obtain new and well-known strong conver-gence theorems in a Hilbert space. In particular, we solve a problem posed by Kurokawa and Takahashi [].
2 Preliminaries
LetH be a real Hilbert space with inner product·,·and norm · , respectively. When
{xn}is a sequence inH, we denote the strong convergence of{xn}tox∈Hbyxn→xand the weak convergence byxnx. We have from [] that for anyx,y∈Handλ∈R,
x+y ≤ x + y,x+y (.)
and
λx+ ( –λ)y =λ x + ( –λ) y –λ( –λ) x–y . (.)
Furthermore, we have that forx,y,u,v∈H,
x–y,u–v= x–v + y–u – x–u – y–v . (.)
All Hilbert spaces satisfy Opial’s condition, that is,
lim inf
n→∞ xn–u <lim infn→∞ xn–v (.)
ifxnuandu=v; see []. LetCbe a nonempty closed convex subset of a Hilbert space
for allx∈Candy∈F(T). IfT :C→H is quasi-nonexpansive, thenF(T) is closed and convex; see []. For a nonempty closed convex subsetCofH, the nearest point projection ofHontoCis denoted byPC, that is, x–PCx ≤ x–y for allx∈Handy∈C. Such
PCis called the metric projection ofH ontoC. We know that the metric projectionPC is firmly nonexpansive; PCx–PCy ≤ PCx–PCy,x–yfor allx,y∈H. Furthermore,
x–PCx,y–PCx ≤ holds for allx∈Handy∈C; see []. The following result is in [].
Lemma Let H be a Hilbert space,and let C be a nonempty closed convex subset of H.Let
T:C→H be a generalized hybrid mapping.Suppose that there exists{xn} ⊂C such that
xnz and xn–Txn→.Then z∈F(T).
LetBbe a mapping ofHinto H. The effective domain ofBis denoted byD(B), that is,
D(B) ={x∈H:Bx=∅}. A multi-valued mappingBis said to be a monotone operator on
H ifx–y,u–v ≥ for allx,y∈D(B),u∈Bx, andv∈By. A monotone operatorBon
H is said to be maximal if its graph is not properly contained in the graph of any other monotone operator onH. For a maximal monotone operatorBonHandr> , we may define a single-valued operatorJr= (I+rB)–:H→D(B), which is called the resolvent of
Bforr. We denote byAr=r(I–Jr) the Yosida approximation ofBforr> . We know from [] that
Arx∈BJrx, ∀x∈H,r> . (.)
LetBbe a maximal monotone operator onH, and letB– ={x∈H: ∈Bx}. It is known thatB– =F(J
r) for allr> and the resolventJris firmly nonexpansive,i.e.,
Jrx–Jry ≤ x–y,Jrx–Jry, ∀x,y∈H. (.)
We also know the following lemma from [].
Lemma Let H be a real Hilbert space,and let B be a maximal monotone operator on H.
For r> and x∈H,define the resolvent Jrx.Then the following holds:
s–t
s Jsx–Jtx,Jsx–x ≥ Jsx–Jtx
for all s,t> and x∈H.
From Lemma , we have that
Jλx–Jμx ≤
|λ–μ|/λ x–Jλx
for allλ,μ> andx∈H; see also [, ]. To prove our main result, we need the following lemmas.
Lemma ([]; see also []) Let{sn}be a sequence of nonnegative real numbers,let{αn}be
with∞n=βn<∞,and let{γn}be a sequence of real numbers withlim supn→∞γn≤.
Suppose that
sn+≤( –αn)sn+αnγn+βn
for all n= , , . . . .Thenlimn→∞sn= .
Lemma ([]) Let{n}be a sequence of real numbers that does not decrease at infinity
in the sense that there exists a subsequence{ni}of{n}which satisfiesni<ni+for all
i∈N.Define the sequence{τ(n)}n≥nof integers as follows:
τ(n) =max{k≤n:k<k+},
where n∈Nsuch that{k≤n:k<k+} =∅.Then the following hold:
(i) τ(n)≤τ(n+ )≤ · · · andτ(n)→ ∞;
(ii) τ(n)≤τ(n)+andn≤τ(n)+,∀n∈N.
3 Strong convergence theorems
LetHbe a real Hilbert space. A mappingg:H→His a contraction if there existsk∈(, ) such that g(x) –g(y) ≤k x–y for allx,y∈H. We call such a mappinggak-contraction. A nonlinear operatorV:H→His called strongly monotone if there existsγ > such that
x–y,Vx–Vy ≥γ x–y for allx,y∈H. SuchV is also calledγ-strongly monotone. A nonlinear operatorV:H→His called Lipschitzian continuous if there existsL> such that Vx–Vy ≤L x–y for allx,y∈H. SuchVis also calledL-Lipschitzian continuous. We know the following three lemmas in a Hilbert space; see Lin and Takahashi [].
Lemma ([]) Let H be a Hilbert space, and let V be aγ-strongly monotone and L-Lipschitzian continuous operator on H withγ > and L> .Let t> satisfyγ >tLand > tγ.Then < –t(γ –tL) < and I–tV:H→H is a contraction,where I is the
identity operator on H.
Lemma ([]) Let H be a Hilbert space,and let C be a nonempty closed convex subset of H.Let PCbe the metric projection of H onto C,and let V be aγ-strongly monotone and
L-Lipschitzian continuous operator on H withγ > and L> .Let t> satisfyγ >tL
and > tγ,and let z∈C.Then the following are equivalent:
() z=PC(I–tV)z;
() Vz,y–z ≥,∀y∈C; () z=PC(I–V)z.
Such z∈C always exists and is unique.
Lemma ([]) Let H be a Hilbert space,and let g:H→H be a k-contraction with <
k< .Let V be aγ-strongly monotone and L-Lipschitzian continuous operator on H with
γ > and L> .Let a real numberγ satisfy <γ<γk.Then V–γg:H→H is a(γ–γk) -strongly monotone and(L+γk)-Lipschitzian continuous mapping.Furthermore,let C be a nonempty closed convex subset of H.Then PC(I–V+γg)has a unique fixed point zin C.
This point z∈C is also a unique solution of the variational inequality
(V–γg)z,q–z
Now, we prove the following strong convergence theorem of Halpern’s type [] for find-ing a common solution of a monotone inclusion problem for the sum of two monotone mappings, of a fixed point problem for generalized hybrid mappings and of an equilibrium problem for bifunctions in a Hilbert space.
Theorem Let H be a real Hilbert space,and let C be a nonempty closed convex
sub-set of H.Letα> ,and let A be anα-inverse strongly-monotone mapping of C into H.
Let B and W be maximal monotone operators on H such that the domains of B and W are included in C.Let Jλ= (I+λB)– and Tr= (I+rW)–be resolvents of B and W
for λ> and r > , respectively. Let S be a generalized hybrid mapping of C into H.
Let <k< ,and let g be a k-contraction of H into itself.Let V be aγ-strongly mono-tone and L-Lipschitzian continuous operator withγ > and L> .Takeμ,γ ∈Ras fol-lows:
<μ<γ
L, <γ < γ–Lμ
k .
Suppose F(S)∩(A+B)–∩W–=∅.Let x
=x∈H,and let{xn} ⊂H be a sequence
generated by
xn+=βnxn+ ( –βn)
αnγg(xn) + (I–αnV)SJλn(I–λnA)Trnxn
for all n∈ N, where {αn} ⊂(, ), {βn} ⊂(, ), {λn} ⊂(,∞) and {rn} ⊂(,∞)
sat-isfy
lim
n→∞αn= ,
∞
n=
αn=∞, <lim inf
n→∞ βn≤lim supn→∞ βn< ,
lim inf
n→∞ rn> and <a≤λn≤b< α.
Then{xn}converges strongly to z∈F(S)∩(A+B)–∩W–,where zis a unique fixed
point in F(S)∩(A+B)–∩W–of P
F(S)∩(A+B)–∩W–(I–V+γg). Proof Letz∈F(S)∩(A+B)–∩W–. We have thatz=Sz,z=J
λn(I–λnA)zandz=Trnz. Puttingwn=Jλn(I–λnA)Trnxnandun=Trnxn, we obtain that
Swn–z ≤ wn–z
= Jλn(un–λnAun) –Jλn(z–λnAz)
≤ un–λnAun– (z–λnAz)
= un–z–λn(Aun–Az)
= un–z – λnun–z,Aun–Az+λn Aun–Az
≤ un–z – λnα Aun–Az +λn Aun–Az
≤ xn–z +λn(λn– α) Aun–Az
Putτ=γ–Lμ. Usinglimn→∞αn= , we have that for anyx,y∈H, (I–αnV)x– (I–αnV)y
= x–y–αn(Vx–Vy)
= x–y – αnx–y,Vx–Vy+αn Vx–Vy
≤ x–y – αnγ x–y +αnL x–y
= – αnγ +αnL
x–y
= – αnτ–αnLμ+αnL
x–y
≤ – αnτ–αn
Lμ–αnL
+αnτ x–y
≤ – αnτ+αnτ
x–y
= ( –αnτ) x–y . (.)
Since –αnτ> , we obtain that for anyx,y∈H,
(I–αnV)x– (I–αnV)y ≤( –αnτ) x–y . (.)
Puttingyn=αnγg(xn) + (I–αnV)SJλn(I–λnA)Trnxn, fromz=αnVz+z–αnVz, (.) and (.) we have that
yn–z = αn
γg(xn) –Vz
+ (I–αnV)Swn– (I–αnV)z
≤αnγk xn–z +αn γg(z) –Vz + ( –αnτ) Swn–z
≤ –αn(τ–γk)
xn–z +αn γg(z) –Vz .
Using this, we get
xn+–z = βn(xn–z) + ( –βn)(yn–z)
≤βn xn–z + ( –βn) yn–z
≤βn xn–z
+ ( –βn)
–αn(τ–γk)
xn–z +αn γg(z) –Vz
= – ( –βn)αn(τ–γk)
xn–z
+ ( –βn)αn(τ–γk)
γg(z) –Vz
τ–γk .
PuttingK=max{ x–z , γgτ(–z)–γkVz }, we have that xn–z ≤Kfor alln∈N. Then{xn}is bounded. Furthermore,{un},{wn}and{yn}are bounded. Using Lemma , we can take a uniquez∈F(S)∩(A+B)–∩W– such that
z=PF(S)∩(A+B)–∩W–(I–V+γg)z.
From the definition of{xn}, we have that
xn+–xn=βnxn+ ( –βn)
αnγg(xn) + (I–αnV)Swn
and hence
xn+–xn– ( –βn)αnγg(xn) =βnxn+ ( –βn)(I–αnV)Swn–xn
= ( –βn)
(I–αnV)Swn–xn
= ( –βn)(Swn–xn–αnVSwn).
Thus, we have that
xn+–xn– ( –βn)αnγg(xn),xn–z
=( –βn)(Swn–xn–αnVSwn),xn–z
= –( –βn)xn–Swn,xn–z– ( –βn)αnVSwn,xn–z. (.)
From (.) and (.), we have that
xn–Swn,xn–z = xn–z + Swn–xn – Swn–z
≥ xn–z + Swn–xn – xn–z
= Swn–xn . (.)
From (.) and (.), we also have that
–xn–xn+,xn–z= ( –βn)αn
γg(xn),xn–z
– ( –βn)xn–Swn,xn–z– ( –βn)αnVSwn,xn–z
≤( –βn)αn
γg(xn),xn–z
– ( –βn) Swn–xn – ( –βn)αnVSwn,xn–z. (.)
Furthermore, using (.) and (.), we have that
xn+–z – xn–xn+ – xn–z
≤( –βn)αn
γg(xn),xn–z
– ( –βn) Swn–xn – ( –βn)αnVSwn,xn–z.
Settingn= xn–z , we have that
n+–n– xn–xn+
≤( –βn)αn
γg(xn),xn–z
– ( –βn) Swn–xn – ( –βn)αnVSwn,xn–z. (.)
Noting that
xn+–xn = ( –βn)αn
γg(xn) –VSwn
+ ( –βn)(Swn–xn)
≤( –βn)
Swn–xn +αn γg(xn) –VSwn
we have that
xn+–xn ≤( –βn)
Swn–xn +αn γg(xn) –VSwn
= ( –βn) Swn–xn + ( –βn)αn Swn–xn γg(xn) –VSwn
+ ( –βn)αn γg(xn) –VSwn
. (.)
Thus, we have from (.) and (.) that
n+–n≤ xn–xn+ + ( –βn)αn
γg(xn),xn–z
– ( –βn) Swn–xn – ( –βn)αnVSwn,xn–z
≤( –βn) Swn–xn + ( –βn)αn Swn–xn γg(xn) –VSwn
+ ( –βn)αn γg(xn) –VSwn + ( –βn)αn
γg(xn),xn–z
– ( –βn) Swn–xn – ( –βn)αnVSwn,xn–z
and hence
n+–n+βn( –βn) Swn–xn
≤( –βn)αn Swn–xn γg(xn) –VSwn
+ ( –βn)αn γg(xn) –VSwn
+ ( –βn)αn
γg(xn),xn–z
– ( –βn)αnVSwn,xn–z. (.)
We divide the proof into two cases.
Case : Suppose thatn+≤nfor alln∈N. In this case,limn→∞nexists and then
limn→∞(n+–n) = . Using <lim infn→∞βn≤lim supn→∞βn< andlimn→∞αn= , we have from (.) that
lim
n→∞ Swn–xn = . (.)
Using (.), we also have that
lim
n→∞ xn+–xn = . (.)
Sincexn+–xn= ( –βn)(yn–xn), we have from (.) that
lim
n→∞ yn–xn = . (.)
We also have from (.) that
un–z = Trnxn–Trnz
≤xn–z,un–z
and hence
un–z ≤ xn–z – un–xn . (.)
From (.) we have that
Swn–z ≤ un–z ≤ xn–z – un–xn
and hence
un–xn ≤ xn–z – Swn–z ≤M Swn–xn ,
whereM=sup{ xn–z + Swn–z :n∈N}. Thus, from (.) we have that
lim
n→∞ un–xn = . (.)
We showlimn→∞ Swn–wn = . Since · is a convex function, we have that
xn+–z ≤βn xn–z + ( –βn) yn–z . (.)
Fromz=αnVz+z–αnVzand (.), we also have that
yn–z = αn
γg(xn) –Vz
+ (I–αnV)Swn– (I–αnV)z
≤( –αnτ) Swn–z + αn
γg(xn) –Vz,yn–z
≤( –αnτ) wn–z + αn
γg(xn) –Vz,yn–z
≤ wn–z + αn
γg(xn) –Vz,yn–z
≤ xn–z +λn(λn– α) Aun–Az
+ αn
γg(xn) –Vz,yn–z
. (.)
Using (.) and (.), we have that
xn+–z ≤βn xn–z + ( –βn) xn–z
+ ( –βn)
λn(λn– α) Aun–Az + αn
γg(xn) –Vz,yn–z
= xn–z + ( –βn)
λn(λn– α) Aun–Az
+ αnγg(xn) –Vz,yn–z
. (.)
Thus, we have that
( –βn)λn(α–λn) Aun–Az
≤ xn–z – xn+–z + ( –βn)αn
γg(xn) –Vz,yn–z
. (.)
Then we have that
lim
SinceJλnis firmly nonexpansive, we have that
wn–z = Jλn(un–λnAun) –Jλn(z–λnAz)
≤un–λnAun– (z–λnAz),wn–z
= un–λnAun– (z–λnAz)
+ wn–z
– un–λnAun– (z–λnAz) – (wn–z)
≤ un–z + wn–z
– un–wn–λn(Aun–Az)
≤ xn–z + wn–z – un–wn
+ λnun–wn,Aun–Az–λn Aun–Az .
Thus, we get
wn–z ≤ xn–z – un–wn
+ λnun–wn,Aun–Az–λn Aun–Az . (.)
Using (.), we obtain
xn+–z ≤βn xn–z + ( –βn) yn–z
≤βn xn–z + ( –βn)
wn–z + αn
γg(xn) –Vz,yn–z
≤βn xn–z + ( –βn) xn–z
– ( –βn) un–wn + ( –βn)λnun–wn,Aun–Az
– ( –βn)λn Aun–Az + ( –βn)αn
γg(xn) –Vz,yn–z
= xn–z – ( –βn) un–wn
+ ( –βn)λnun–wn,Aun–Az– ( –βn)λn Aun–Az
+ ( –βn)αn
γg(xn) –Vz,yn–z
,
from which it follows that
( –βn) xn–wn ≤ xn–z
– xn+–z + λnun–wn,Aun–Az
–λn Aun–Az + αn
γg(xn) –Vz,yn–z
.
Then we have
lim
n→∞ un–wn = . (.)
From (.) and (.), we have that
lim
Since Swn–wn ≤ Swn–xn + xn–wn , we have that
lim
n→∞ Swn–wn = . (.)
Takeλ∈Rwith <a≤λ≤b< α arbitrarily. Putsn= (I–λnA)un. Usingun=Trnxn andwn=Jλn(I–λnA)un, we have from Lemma that
Jλ(I–λA)un–wn = Jλ(I–λA)un–Jλn(I–λnA)un
= Jλ(I–λA)un–Jλ(I–λnA)un
+Jλ(I–λnA)un–Jλn(I–λnA)un
≤ (I–λA)un– (I–λnA)un + Jλsn–Jλnsn
≤ |λ–λn| Aun +
|λ–λn| λ
Jλsn–sn . (.)
We also have from (.) that
un–Jλ(I–λA)un ≤ un–wn + wn–Jλ(I–λA)un . (.)
We will use (.) and (.) later.
Let us show thatlim supn→∞(V–γg)z,xn–z ≥. Put
A=lim sup
n→∞
(V–γg)z,xn–z
.
Without loss of generality, we may assume that there exists a subsequence {xni}of{xn} such thatA=limi→∞(V–γg)z,xni–zand{xni}converges weakly to some pointw∈H. From xn–wn → and xn–un →, we also have that{wni}and{uni}converge weakly tow∈C. On the other hand, from{λni} ⊂[a,b] there exists a subsequence{λnij}of{λni} such thatλnij→λfor someλ∈[a,b]. Without loss of generality, we assume thatwni→
w,uni→wandλni→λ. From (.) we knowlimn→∞ Swn–wn = . Thus, we have from Lemma thatw=Sw. SinceWis a monotone operator andxnir–uni
ni ∈Wuni, we have that for any (u,v)∈W,
u–uni,v–
xni–uni
rni
≥.
Sincelim infn→∞rn> ,uniwandxni–uni→, we have
u–w,v ≥.
SinceW is a maximal monotone operator, we have ∈Wwand hencew∈W–. Since λni→λ, we have from (.) that
Jλ(I–λA)uni–wni →.
Furthermore, we have from (.) that
SinceJλ(I–λA) is nonexpansive, we have thatw=Jλ(I–λA)w. This means that ∈ Aw+Bw. Thus, we have
w∈F(T)∩(A+B)–∩W–.
Then we have
A=lim
i→∞
(V–γg)z,xni–z
=(V–γg)z,w–z
≥. (.)
Sinceyn–z=αn(γg(xn) –Vz) + (I–αnV)Swn– (I–αnV)z, we have
yn–z ≤( –αnτ) Swn–z + αn
γg(xn) –Vz,yn–z
.
Thus, we have
yn–z ≤( –αnτ) xn–z + αn
γg(xn) –Vz,yn–z
.
Consequently, we have that
xn+–z ≤βn xn–z + ( –βn) yn–z
≤βn xn–z
+ ( –βn)
( –αnτ) xn–z + αn
γg(xn) –Vz,yn–z
=βn+ ( –βn)( –αnτ) xn–z
+ ( –βn)αn
γg(xn) –Vz,yn–z
≤ – ( –βn)
αnτ– (αnτ)
xn–z
+ ( –βn)αnγk xn–z + ( –βn)αn
γg(z) –Vz,yn–z
= – ( –βn)αn(τ–γk)
xn–z
+ ( –βn)(αnτ) xn–z + ( –βn)αn
γg(z) –Vz,yn–z
= – ( –βn)αn(τ–γk)
xn–z
+ ( –βn)αn(τ–γk)
αnτ xn–z (τ–γk) +
γg(z) –Vz,yn–z τ–γk
.
By (.) and Lemma , we obtain thatxn→z, where
z=PF(S)∩(A+B)–∩W–(I–V+γg)z.
Case : Suppose that there exists a subsequence{ni} ⊂ {n}such thatni<ni+for all
i∈N. In this case, we defineτ:N→Nby
Then we have from Lemma thatτ(n)<τ(n)+. Thus, we have from (.) that for all
n∈N,
βτ(n)( –βτ(n)) Swτ(n)–xτ(n)
≤( –βτ(n))ατ(n) Swτ(n)–xτ(n) γg(xτ(n)) –VSwτ(n)
+ ( –βτ(n))ατ(n) γg(xτ(n)) –VSwτ(n)
+ ( –βτ(n))ατ(n)
γg(xτ(n)),xτ(n)–z
– ( –βτ(n))ατ(n)VSwτ(n),xτ(n)–z. (.)
Usinglimn→∞αn= and <lim infn→∞βn≤lim supn→∞βn< , we have from (.) and Lemma that
lim
n→∞ Swτ(n) –xτ(n) = . (.)
As in the proof of Case , we also have that
lim
n→∞ xτ(n)+–xτ(n) = (.)
and
lim
n→∞ yτ(n)–xτ(n) = . (.)
Furthermore, we have that limn→∞ uτ(n) – xτ(n) = , limn→∞ Auτ(n) – Az = ,
limn→∞ uτ(n) –wτ(n) = and limn→∞ xτ(n) –wτ(n) = . From these we have that
limn→∞ Swτ(n)–wτ(n) = . As in the proof of Case , we can show that
lim sup
n→∞
(V–γg)z,xτ(n)–z
≥.
We also have that
yτ(n)–z ≤( –ατ(n)τ) xτ(n)–z + ατ(n)
γg(xτ(n)) –Vz,yτ(n)–z
and hence
xτ(n)+–z ≤
– ( –βτ(n))ατ(n)(τ–γk)
xτ(n)–z
+ ( –βτ(n))(ατ(n)τ) xτ(n)–z
+ ( –βτ(n))ατ(n)
γg(z) –Vz,yτ(n)–z
.
Fromτ(n)<τ(n)+, we have that
( –βτ(n))ατ(n)(τ–γk) xτ(n)–z
≤( –βτ(n))(ατ(n)τ) xτ(n)–z
+ ( –βτ(n))ατ(n)
γg(z) –Vz,yτ(n)–z
Since ( –βτ(n))ατ(n)> , we have that
(τ–γk) xτ(n)–z
≤ατ(n)τ xτ(n)–z +
γg(z) –Vz,yτ(n)–z
.
Thus, we have that
lim sup
n→∞
(τ–γk) xτ(n)–z ≤
and hence xτ(n)–z → asn→ ∞. Sincexτ(n)–xτ(n)+→, we have xτ(n)+–z → asn→ ∞. Using Lemma again, we obtain that
xn–z ≤ xτ(n)+–z →
asn→ ∞. This completes the proof.
4 Applications
In this section, using Theorem , we can obtain well-known and new strong convergence theorems in a Hilbert space. LetHbe a Hilbert space, and letf be a proper lower semi-continuous convex function ofHinto (–∞,∞]. Then the subdifferential∂f off is defined as follows:
∂f(x) =z∈H:f(x) +z,y–x ≤f(y),∀y∈H
for allx∈H. From Rockafellar [], we know that∂f is a maximal monotone operator. Let
Cbe a nonempty closed convex subset ofH, and letiCbe the indicator function ofC,i.e.,
iC(x) = ⎧ ⎨ ⎩
, x∈C,
∞, x∈/C.
Then,iCis a proper lower semicontinuous convex function onH. So, we can define the resolventJλof∂iCforλ> ,i.e.,
Jλx= (I+λ∂iC)–x
for allx∈H. We know thatJλx=PCxfor allx∈Handλ> ; see [].
Theorem Let H be a real Hilbert space,and let C be a nonempty closed convex subset of H.Let S be a generalized hybrid mapping of C into C.Suppose F(S)= ∅.Let u,x∈C,
and let{xn} ⊂C be a sequence generated by
xn+=βnxn+ ( –βn)
αnu+ ( –αn)Sxn
for all n∈N,where{βn} ⊂(, )and{αn} ⊂(, )satisfy
lim
n→∞αn= , ∞
and
<lim inf
n→∞ βn≤lim supn→∞ βn< .
Then the sequence{xn}converges strongly to z∈F(S),where z=PF(S)u.
Proof PutA= ,B=W=∂iCandλn=rn= for alln∈Nin Theorem . Then we have
Jλn=Trn=PCfor alln∈N. Furthermore, putg(x) =uandV(x) =xfor allx∈H. Then we can takeγ =L= . Thus, we can takeμ= . On the other hand, since g(x) –g(y) = ≤
x–y for allx,y∈H, we can takek=
. So, we can takeγ = . Then foru,x∈C, we get that
xn+=βnxn+ ( –βn)
αnu+ (I–αn)Sxn
for alln∈N. So, we have{xn} ⊂C. We also have
z=PF(S)∩C(I–V+γg)z=PF(S)(z–z+ ·u) =PF(S)u.
Thus, we obtain the desired result by Theorem .
Theorem solves the problem posed by Kurokawa and Takahashi []. The following result is a strong convergence theorem of Halpern’s type [] for finding a common solution of a monotone inclusion problem for the sum of two monotone mappings, of a fixed point problem for nonexpansive mappings and of an equilibrium problem for bifunctions in a Hilbert space.
Theorem Let H be a real Hilbert space,and let C be a nonempty closed convex subset of H.Letα> ,and let A be anα-inverse strongly-monotone mapping of C into H.Let B and W be maximal monotone operators on H such that the domains of B and W are included in C.Let Jλ= (I+λB)–and Tr= (I+rW)–be resolvents of B and W forλ>
and r> ,respectively.Let S be a nonexpansive mapping of C into H.Let <k< ,and let
g be a k-contraction of H into itself.Let V be aγ-strongly monotone and L-Lipschitzian
continuous operator withγ > and L> .Takeμ,γ ∈Ras follows:
<μ<γ
L, <γ < γ–Lμ
k .
Suppose F(S)∩(A+B)–∩W–=∅.Let x
=x∈H,and let{xn} ⊂H be a sequence
generated by
xn+=βnxn+ ( –βn)
αnγg(xn) + (I–αnV)SJλn(I–λnA)Trnxn
for all n∈N,where{αn} ⊂(, ),{βn} ⊂(, ),{λn} ⊂(,∞)and{rn} ⊂(,∞)satisfy
lim
n→∞αn= , ∞
n=
αn=∞, <lim inf
n→∞ βn≤lim supn→∞ βn< ,
lim inf
Then the sequence{xn}converges strongly to z∈F(S)∩(A+B)–∩W–,where z=
PF(S)∩(A+B)–∩W–(I–V+γg)z.
Proof We know that a nonexpansive mappingTofCintoHis a (, )-generalized hybrid
mapping. So, we obtain the desired result by Theorem .
Letf :C×C→Rbe a bifunction. The equilibrium problem (with respect toC) is to findxˆ∈Csuch that
f(xˆ,y)≥, ∀y∈C. (.)
The set of such solutionsxˆis denoted byEP(f),i.e.,
EP(f) =xˆ∈C:f(xˆ,y)≥,∀y∈C.
For solving the equilibrium problem, let us assume that the bifunction f :C×C→R
satisfies the following conditions:
(A) f(x,x) = for allx∈C;
(A) f is monotone,i.e.,f(x,y) +f(y,x)≤for allx,y∈C; (A) for allx,y,z∈C,
lim sup
t↓
ftz+ ( –t)x,y≤f(x,y);
(A) for allx∈C,f(x,·)is convex and lower semicontinuous.
The following lemmas were given in Combettes and Hirstoaga [] and Takahashi, Taka-hashi and Toyoda []; see also [, ].
Lemma ([]) Let H be a real Hilbert space,and let C be a nonempty closed convex subset of H.Assume that f :C×C→Rsatisfies(A)-(A).For r> and x∈H,define a
mapping Tr:H→C as follows:
Trx=
z∈C:f(z,y) +
ry–z,z–x ≥,∀y∈C
for all x∈H.Then the following hold:
() Tris single-valued;
() Tris a firmly nonexpansive mapping,i.e.,for allx,y∈H,
Trx–Try ≤ Trx–Try,x–y;
() F(Tr) =EP(f);
() EP(f)is closed and convex.
We call suchTrthe resolvent off forr> .
defined by
Afx= ⎧ ⎨ ⎩
{z∈H:f(x,y)≥ y–x,z,∀y∈C}, ∀x∈C,
∅, ∀x∈/C.
Then EP(f) =A–f and Af is a maximal monotone operator with D(Af)⊂C.Furthermore,
for any x∈H and r> ,the resolvent Trof f coincides with the resolvent of Af,i.e.,
Trx= (I+rAf)–x.
Using Lemmas , and Theorem , we also obtain the following result for generalized hybrid mappings ofCintoHwith equilibrium problem in a Hilbert space; see also [– ].
Theorem Let H be a real Hilbert space,and let C be a nonempty closed convex subset of H.Let S be a generalised hybrid mapping of C into H.Let f be a bifunction of C×C into
Rsatisfying(A)-(A).Let <k< ,and let g be a k-contraction of H into itself.Let V be aγ-strongly monotone and L-Lipschitzian continuous operator of H into itself withγ>
and L> .Takeμ,γ ∈Ras follows:
<μ<γ
L, <γ < γ–Lμ
k .
Suppose that F(S)∩EP(f)=∅.Let x=x∈H,and let{xn} ⊂H be a sequence generated by
f(un,y) +
rn
y–un,un–xn ≥, ∀y∈C,
xn+=βnxn+ ( –βn)
αnγg(xn) + (I–αnV)Sun
for all n∈N,where{βn} ⊂(, ),{αn} ⊂(, )and{rn} ⊂(,∞)satisfy
lim
n→∞αn= , ∞
n=
αn=∞, lim inf n→∞ rn> ,
and <lim inf
n→∞ βn≤lim supn→∞ βn< .
Then the sequence{xn}converges strongly to z∈F(S)∩EP(f),where z=PF(S)∩EP(f)(I–V+ γg)z.
Proof PutA= andB=∂iCin Theorem . Furthermore, for the bifunctionf:C×C→
R, defineAf as in Lemma . PutW=Af in Theorem , and letTrnbe the resolvent of
Af forrn> . Then we obtain that the domain ofAf is included inCandTrnxn=unfor all
n∈N. Thus, we obtain the desired result by Theorem .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Author details
1Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, 152-8552, Japan. 2Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan.3Center for General
Education, Kaohsiung Medical University, Kaohsiung, 80702, Taiwan.
Acknowledgements
The first author was partially supported by Grant-in-Aid for Scientific Research No. 23540188 from Japan Society for the Promotion of Science. The second and the third authors were partially supported by the grant Taiwan NSC
99-2115-M-110-007-MY3 and the grant Taiwan NSC 99-2115-M-037-002-MY3, respectively.
Received: 25 May 2012 Accepted: 30 September 2012 Published: 17 October 2012
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