R E S E A R C H
Open Access
Strong convergence theorems for solving a
general system of finite variational
inequalities for finite accretive operators and
fixed points of nonexpansive semigroups
with weak contraction mappings
Nawitcha Onjai-uea
1, Phayap Katchang
2and Poom Kumam
1**Correspondence:
1Department of Mathematics,
Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok, 10140, Thailand
Full list of author information is available at the end of the article
Abstract
In this paper, we prove a strong convergence theorem for finding a common solution of a general system of finite variational inequalities for finite different inverse-strongly accretive operators and solutions of fixed point problems for a nonexpansive semigroup in a Banach space based on a viscosity approximation method by using weak contraction mappings. Moreover, we can apply the above results to find the solutions of the class ofk-strictly pseudocontractive mappings and apply a general system of finite variational inequalities into a Hilbert space. The results presented in this paper extend and improve the corresponding results of Cenget al.(2008), Katchang and Kumam (2011), Wangkeeree and Preechasilp (2012), Yaoet al.(2010) and many other authors.
MSC: Primary 47H05; 47H10; 47J25
Keywords: inverse-strongly accretive operator; fixed point; general system of finite variational inequalities; sunny nonexpansive retraction; weak contraction;
nonexpansive semigroups
1 Introduction
LetEbe a real Banach space with norm · andCbe a nonempty closed convex sub-set ofE. LetE*be the dual space ofEand·,·denote the pairing betweenEandE*. For q> , thegeneralized duality mapping Jq:E→E
*
is defined byJq(x) ={f ∈E*:x,f=
xq,f=xq–}for allx∈E. In particular, ifq= , the mappingJ is called the nor-malized duality mappingand, usually, writeJ=J. Further, we have the following prop-erties of the generalized duality mappingJq: (i)Jq(x) =xq–J(x) for allx∈Ewithx= ;
(ii)Jq(tx) =tq–Jq(x) for allx∈Eandt∈[,∞); and (iii)Jq(–x) = –Jq(x) for allx∈E. It is
known that ifEis smooth, thenJis single-valued, which is denoted byj. Recall that the duality mappingjis said to be weakly sequentially continuous if for eachxn→xweakly,
we havej(xn)→j(x) weakly-*. We know that ifEadmits a weakly sequentially continuous
duality mapping, thenEis smooth (for the details, see [, , ]).
Let f :C→C be ak-contraction mapping if there existsk∈[, ) such that f(x) –
f(y) ≤kx–y,∀x,y∈C. LetS:C→Ca nonlinear mapping. We useF(S) to denote the
set of fixed points ofS, that is,F(S) ={x∈C:Sx=x}. A mappingSis callednonexpansive
if Sx–Sy ≤ x–y,∀x,y∈C. A mapping f is called weakly contractiveon a closed convex setCin the Banach spaceEif there existsϕ: [,∞)→[,∞) is a continuous and strictly increasing function such thatϕ is positive on (,∞),ϕ() = ,limt→∞ϕ(t) =∞
andx,y∈C
f(x) –f(y)≤ x–y–ϕx–y. (.)
If ϕ(t) = ( –k)t, thenf is called to becontractivewith the contractive coefficientk. If
ϕ(t)≡, thenf is said to benonexpansive.
A familyS={T(t) :t≥}of mappings ofCinto itself is called anonexpansive semigroup
(see also []) onCif it satisfies the following conditions:
(i) T()x=xfor allx∈C;
(ii) T(s+t) =T(s)T(t)for alls,t≥;
(iii) T(s)x–T(s)y ≤ x–yfor allx,y∈Cands≥; (iv) for allx∈C,s→T(s)xis continuous.
We denote byF(S) the set of all commonfixed points ofS, that is,
F(S) =
∞
t=
FT(t)=x∈C:T(t)x=x, ≤t<∞.
It is known thatF(S) is closed and convex. Moreover, for the study of nonexpansive semi-group mapping, see [, –, ] for more details.
In , Suzuki [] was the first one to introduce the following implicit iteration process in Hilbert spaces:
xn=αnu+ ( –αn)T(tn)(xn), n≥ (.)
for the nonexpansive semigroup. In , Xu [] established a Banach space version of the sequence (.) of Suzuki []. In [], Chen and He considered the viscosity approx-imation process for a nonexpansive semigroup and proved another strong convergence theorems for a nonexpansive semigroup in Banach spaces, which is defined by
xn+=αnf(xn) + ( –αn)T(tn)xn, ∀n∈N, (.)
wheref :C→Cis a fixed contractive mapping. Recall that an operatorA:C→Eis said to beaccretiveif there existsj(x–y)∈J(x–y) such that
Ax–Ay,j(x–y) ≥
for allx,y∈C. A mappingA:C→E is said to beβ-strongly accretiveif there exists a constantβ> such that
Ax–Ay,j(x–y) ≥βx–y, ∀x,y∈C.
An operatorA:C→Eis said to beβ-inverse strongly accretiveif, for anyβ> ,
for allx,y∈C. Evidently, the definition of the inverse strongly accretive operator is based on that of the inverse strongly monotone operator. To convey an idea of thevariational inequality, letCbe a closed and convex set in a real Hilbert spaceH. For a given operator
A, we consider the problem of findingx*∈Csuch that
Ax*,x–x* ≥
for allx∈C, which is known as the variational inequality, introduced and studied by Stam-pacchia [] in in the field of potential theory. In , Aoyamaet al.[] first consid-ered the following generalized variational inequality problem in a smooth Banach space. LetAbe an accretive operator ofCintoE. Find a pointx∈Csuch that
Ax,j(y–x) ≥ (.) for ally∈C. This problem is connected with the fixed point problem for nonlinear map-pings, the problem of finding a zero point of an accretive operator and so on. For the prob-lem of finding a zero point of an accretive operator by the proximal point algorithm, see Kamimura and Takahashi [, ]. In order to find a solution of the variational inequal-ity (.), Aoyamaet al.[] proved the strong convergence theorem in the framework of Banach spaces which is generalized by Iidukaet al.[] from Hilbert spaces.
Motivated by Aoyamaet al.[] and also Cenget al.[], Qinet al.[] and Yaoet al.
[] first considered the followingnew general system of variational inequalitiesin Banach spaces:
LetA:C→Ebe aβ-inverse strongly accretive mapping. Find (x*,y*)∈C×Csuch that ⎧
⎨ ⎩
λAy*+x*–y*,j(x–x*) ≥, ∀x∈C,
μAx*+y*–x*,j(x–y*) ≥, ∀x∈C. (.)
LetCbe nonempty closed convex subset of a real Banach spaceE. For two given oper-atorsA,B:C→E, consider the problem of finding (x*,y*)∈C×Csuch that
⎧ ⎨ ⎩
λAy*+x*–y*,j(x–x*) ≥, ∀x∈C,
μBx*+y*–x*,j(x–y*) ≥, ∀x∈C, (.)
whereλandμare two positive real numbers. This system is called thegeneral system of variational inequalitiesin a real Banach spaces. If we add up the requirement thatA=B, then the problem (.) is reduced to the system (.).
By the following general system of variational inequalities, we extend into thegeneral system of finite variational inequalitieswhich is to find (x*
,x*, . . . ,x*M)∈C×C× · · · ×C
and is defined by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
λMAMx*M+x*–xM* ,j(x–x*) ≥, ∀x∈C,
λM–AM–x*M–+x*M–x*M–,j(x–x*M) ≥, ∀x∈C,
.. .
λAx*
+x*–x*,j(x–x*) ≥, ∀x∈C, λAx*+x*–x*,j(x–x*) ≥, ∀x∈C,
where{Al}Ml=:C→Eis a family of mappings,λl≥,l∈ {, , . . . ,M}. The set of solutions
of (.) is denoted byGSVI(C,Al). In particular, ifM= ,A=B,A=A,λ=μ,λ=λ, x*
=x*andx*=y*, then the problem (.) is reduced to the problem (.).
In this paper, motivated and inspired by the idea of Cenget al.[], Katchang and Ku-mam [] and Yaoet al.[], we introduce a new iterative scheme with weak contraction for finding solutions of a new general system of finite variational inequalities (.) for fi-nite different inverse-strongly accretive operators and solutions of fixed point problems for nonexpansive semigroups in a Banach space. Consequently, we obtain new strong con-vergence theorems for fixed point problems which solve the general system of variational inequalities (.). Moreover, we can apply the above theorem to finding solutions of zeros of accretive operators and the class ofk-strictly pseudocontractive mappings. The results presented in this paper extend and improve the corresponding results of Cenget al.[], Katchang and Kumam [], Wangkeeree and Preechasilp [], Yaoet al.[] and many other authors.
2 Preliminaries
We always assume thatEis a real Banach space andCis a nonempty closed convex subset ofE.
LetU={x∈E:x= }. A Banach spaceE is said to beuniformly convexif, for any
∈(, ], there existsδ> such that, for anyx,y∈U,x–y ≥impliesx+y ≤ –δ. It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space Eis said to besmoothif the limitlimt→x+tyt–x exists for allx,y∈U. It is also
said to beuniformly smoothif the limit is attained uniformly forx,y∈U. Themodulus of smoothnessofEis defined by
ρ(τ) =sup
x+y+x–y– :x,y∈E,x= ,y=τ
,
whereρ: [,∞)→[,∞) is a function. It is known thatEis uniformly smooth if and only iflimτ→ρτ(τ) = . Letqbe a fixed real number with <q≤. A Banach spaceEis said
to beq-uniformly smoothif there exists a constantc> such thatρ(τ)≤cτqfor allτ> :
see, for instance, [, ].
We note thatEis a uniformly smooth Banach space if and only ifJqis single-valued and
uniformly continuous on any bounded subset ofE. Typical examples of both uniformly convex and uniformly smooth Banach spaces areLp, where p> . More precisely,Lp is
min{p, }-uniformly smooth for everyp> . Note also that no Banach space isq-uniformly smooth forq> ; see [, ] for more details.
LetDbe a subset ofCandQ:C→D. ThenQis said to besunnyif
QQx+t(x–Qx)=Qx,
Proposition .([]) Let E be a smooth Banach space and let C be a nonempty subset of E. Let Q:E→C be a retraction and let J be the normalized duality mapping on E. Then the following are equivalent:
(i) Q is sunny and nonexpansive;
(ii) Qx–Qy≤ x–y,J(Qx–Qy),∀x,y∈E; (iii) x–Qx,J(y–Qx) ≤,∀x∈E,y∈C.
Proposition .([]) Let C be a nonempty closed convex subset of a uniformly convex
and uniformly smooth Banach space E, and let T be a nonexpansive mapping of C into itself with F(T)=∅. Then the set F(T)is a sunny nonexpansive retract of C.
A Banach spaceEis said to satisfyOpial’s conditionif for any sequence{xn}inE,xnx
(n→ ∞) implies
lim sup
n→∞ xn
–x<lim sup
n→∞ xn
–y, ∀y∈Ewithx=y.
By [, Theorem ], it is well known that, ifE admits a weakly sequentially continuous duality mapping, thenEsatisfies Opial’s condition andEis smooth.
We need the following lemmas for proving our main results.
Lemma .([]) Let E be a real -uniformly smooth Banach space with the best smooth
constant K . Then the following inequality holds:
x+y≤ x+ y,Jx+ Ky, ∀x,y∈E.
Lemma .([]) Let{xn}and{yn}be bounded sequences in a Banach space X and let
{βn} be a sequence in[, ] with <lim infn→∞βn≤lim supn→∞βn< . Suppose xn+=
( –βn)yn+βnxnfor all integers n≥andlim supn→∞(yn+–yn–xn+–xn)≤. Then,
limn→∞yn–xn= .
Lemma .(Lemma . in []) Let{an}and{bn}be two nonnegative real number
se-quences and{αn}a positive real number sequence satisfying the conditions:
∞
n=αn=∞
andlimn→∞bnαn= . Let the recursive inequality
an+≤an–αnϕ(an) +bn, n≥,
whereϕ(a)is a continuous and strict increasing function for all a≥withϕ() = . Then limn→∞an= .
Lemma .([]) Let E be a uniformly convex Banach space and Br() :={x∈E:x ≤r}
be a closed ball of E. Then there exists a continuous strictly increasing convex function g: [,∞)→[,∞)with g() = such that
λx+μy+γz≤λx+μy+γz–λμgx–y
Lemma .([]) Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E and let T be nonexpansive mapping of C into itself. If{xn}is a sequence of
C such that xn→x weakly and xn–Txn→strongly, then x is a fixed point of T .
Lemma .(Yaoet al.[, Lemma .]; see also [, Lemma .]) Let C be a nonempty
closed convex subset of a real -uniformly smooth Banach space E. Let the mapping A:
C→E beβ-inverse-strongly accretive. Then, we have
(I–λA)x– (I–λA)y≤ x–y+ λλK–βAx–Ay.
Ifβ≥λK, then I–λA is nonexpansive.
3 Main results
In this section, we prove a strong convergence theorem. In order to prove our main results, we need the following two lemmas.
Lemma . Let C be a nonempty closed convex subset of a real -uniformly smooth Banach
space E. Let QC be the sunny nonexpansive retraction from E onto C. Let the mapping
Al:C→E be aβl-inverse-strongly accretive such thatβl≥λlKwhere l∈ {, , . . . ,M}. If Q:C→C is a mapping defined by
Q(x) =QC(I–λMAM)QC(I–λM–AM–)· · ·QC(I–λA)QC(I–λA)x, ∀x∈C,
thenQis nonexpansive.
Proof TakingQl
C=QC(I–λlAl)QC(I–λl–Al–)· · ·QC(I–λA)QC(I–λA),l∈ {, , , . . . , M}andQC=I, whereIis the identity mapping onE, we haveQ=QM
C. For anyx,y∈C,
we have
Q(x) –Q(y)=QMCx–QMCy
=QC(I–λMAM)QMC–x–QC(I–λMAM)QMC–y
≤(I–λMAM)QMC–x– (I–λMAM)QMC–y
≤QM–
C x–QMC–y
.. .
≤Q
Cx–QCy
=x–y.
Therefore,Qis nonexpansive.
Lemma . Let C be a nonempty closed convex subset of a real smooth Banach space E.
Let QCbe the sunny nonexpansive retraction from E onto C. Let Al:C→E be nonlinear
problem (.) if and only if ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x*=QC(I–λMAM)x*M,
x*
=QC(I–λA)x*, x*=QC(I–λA)x*,
.. .
x*M=QC(I–λM–AM–)x*M–,
(.)
that is
x*=QC(I–λMAM)QC(I–λM–AM–)· · ·QC(I–λA)QC(I–λA)x*. Proof From (.), we rewrite as
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x*
– (x*M–λMAMx*M),j(x–x*) ≥, ∀x∈C, x*
M– (x*M––λM–AM–x*M–),j(x–x*M) ≥, ∀x∈C,
.. .
x*
– (x*–λAx*),j(x–x*) ≥, ∀x∈C, x*
– (x*–λAx*),j(x–x*) ≥, ∀x∈C.
(.)
Using Proposition .(iii), the system (.) is equivalent to (.). Throughout this paper, the set of fixed points of the mappingQis denoted byF(Q). The next result states the main result of this work.
Theorem . Let E be a uniformly convex and -uniformly smooth Banach space which
admits a weakly sequentially continuous duality mapping and C be a nonempty closed convex subset of E. LetS={T(t) :t≥} be a nonexpansive semigroup on C and QC be
a sunny nonexpansive retraction from E onto C. Let Al:C→E be aβl-inverse-strongly
accretive such thatβl≥λlK, where l∈ {, , . . . ,M}, and K be the best smooth constant.
Let f be a weakly contractive mapping on C into itself with functionϕ. SupposeF:=F(Q)∩
F(S)=∅, whereQis defined by Lemma .. For arbitrary given x=x∈C, the sequence {xn}is generated by
⎧ ⎨ ⎩
yn=QC(I–λMAM)QC(I–λM–AM–)· · ·QC(I–λA)QC(I–λA)xn,
xn+=αnf(xn) +βnxn+γnT(μn)yn,
(.)
where the sequences{αn},{βn}and{γn}are in(, )and satisfy{αn}+{βn}+{γn}= , n≥,
{μn} ⊂(,∞), andλl, l= , , . . . ,M are positive real numbers. The following conditions are
satisfied:
(C) limn→∞αn= and∞n=αn=∞;
(C) <lim infn→∞βn≤lim supn→∞βn< ;
(C) limn→∞μn= ;
Then{xn}converges strongly tox¯ =QFf(x¯ )and(x¯ ,x¯ , . . . ,x¯M)is a solution of the problem
(.) where QFis the sunny nonexpansive retraction of C ontoF.
Proof First, we prove that{xn}is bounded. Letp∈F, taking Ql
C=QC(I–λlAl)QC(I–λl–Al–)· · ·QC(I–λA)QC(I–λA), l∈ {, , , . . . ,M}, Q
C=I, whereIis the identity mapping onE. From the definition ofQCis nonexpansive
thenQl
C,l∈ {, , , . . . ,M}also. We note that
yn–p=QlCxn–QlCp≤ xn–p. (.)
From (.) and (.), we also have
xn+–p=αnf(xn) +βnxn+γnT(μn)yn–p
≤αnf(xn) –p+βnxn–p+γnT(μn)yn–T(μn)p
≤αn
xn–p–ϕ
xn–p
+αnf(p) –p+βnxn–p+γnyn–p
≤ xn–p–αnϕ
xn–p
+αnf(p) –p
≤maxx–p,ϕx–p,f(p) –p. (.)
This implies that{xn}is bounded, so are{f(xn)},{yn}, and{T(μn)yn}.
Next, we show thatlimn→∞xn+–xn= . Notice that
yn+–yn=QCMxn+–QMCxn
=QC(I–λMAM)QMC–xn+–QC(I–λMAM)QMC–xn
≤(I–λMAM)QMC–xn+– (I–λMAM)QMC–xn
≤QM–
C xn+–QMC–xn
.. .
≤Q
Cxn+–QCxn
=xn+–xn.
Settingxn+= ( –βn)zn+βnxnfor alln≥, we see thatzn=xn+––ββnnxn. Then we have
zn+–zn=
xn+–βn+xn+
–βn+
–xn+–βnxn –βn
=αn+f(xn+) +γn+T(μn+)yn+ –βn+
–αnf(xn) +γnT(μn)yn –βn
=αn+f(xn+) +γn+T(μn+)yn+ –βn+
–αn+f(xn) –βn+
+αn+f(xn) –βn+
–γn+T(μn)yn –βn+
+γn+T(μn)yn –βn+
–αnf(xn) +γnT(μn)yn –βn
= αn+ –βn+
f(xn+) –f(xn)
+ γn+ –βn+
T(μn+)yn+–T(μn)yn
+
αn+
–βn+
– αn –βn
f(xn) +
γn+
–βn+
– γn –βn
T(μn)yn
≤ αn+
–βn+
f(xn+) –f(xn)+ γn+
–βn+
T(μn+)yn+–T(μn)yn
+ αn+ –βn+
– αn –βn
f(xn)+
γn+
–βn+
– γn –βn
T(μn)yn
≤ αn+
–βn+
f(xn+) –f(xn)
+ –βn+–αn+ –βn+
yn+–yn+T(μn+)yn–T(μn)yn
+ αn+ –βn+
– αn –βn
f(xn)+
–βn+–αn+
–βn+
– –βn–αn –βn
T(μn)yn
≤ αn+
–βn+
xn+–xn–ϕ
xn+–xn
+
– αn+ –βn+
yn+–yn+T(μn+)yn–T(μn)yn
+ αn+ –βn+
– αn –βn
f(xn)+
αn+
–βn+
– αn –βn
T(μn)yn
≤ αn+
–βn+
xn+–xn+yn+–yn+T(μn+)yn–T(μn)yn
+ αn+ –βn+
– αn –βn
f(xn)+T(μn)yn
≤ αn+
–βn+xn+
–xn+xn+–xn+ sup y∈{yn}
T(μn+)y–T(μn)y
+ αn+ –βn+
– αn –βn
f(xn)+T(μn)yn.
Therefore,
zn+–zn–xn+–xn ≤ αn+
–βn+xn+
–xn+ sup y∈{yn}
T(μn+)y–T(μn)y
+ αn+ –βn+
– αn –βn
f(xn)+T(μn)yn.
It follows from the conditions (C), (C) and (C), which implies that
lim sup
n→∞
zn+–zn–xn+–xn
≤.
Applying Lemma ., we obtainlimn→∞zn–xn= and also
xn+–xn= ( –βn)zn–xn →
asn→ ∞. Therefore, we have
lim
Next, we show thatlimn→∞T(μn)yn–yn= . Sincep∈F, from Lemma ., we obtain
xn+–p =αnf(xn) +βnxn+γnT(μn)yn–p
≤αnf(xn) –p
+ ( –αn–γn)xn–p+γnyn–p
=αnf(xn) –p+ ( –αn)xn–p–γn
xn–p–yn–p
=αnf(xn) –p
+ ( –αn)xn–p
–γn
xn–p–yn–p
xn–p+yn–p
≤αnf(xn) –p+xn–p–γnxn–yn.
Therefore, we have
γnxn–yn≤αnf(xn) –p+xn–p–xn+–p ≤αnf(xn) –p
+xn–p+xn+–p
xn–xn+.
From the condition (C) and (.), this implies thatxn–yn → asn→ ∞. Now, we
note that
xn–T(μn)yn≤ xn–xn++xn+–T(μn)yn
=xn–xn++αnf(xn) +βnxn+γnT(μn)yn–T(μn)yn
=xn–xn++αn
f(xn) –T(μn)yn
+βn
xn–T(μn)yn
≤ xn–xn++αnf(xn) –T(μn)yn+βnxn–T(μn)yn.
Therefore, we get
xn–T(μn)yn≤
–βnxn
–xn++
αn
–βn
f(xn) –T(μn)yn.
From the conditions (C), (C) and (.), this implies thatxn–T(μn)yn → asn→ ∞.
Since
xn–T(μn)xn≤xn–T(μn)yn+T(μn)yn–T(μn)xn
≤xn–T(μn)yn+yn–xn,
and hence it follows thatlimn→∞T(μn)xn–xn= .
Next, we prove thatz∈F:=F(Q)∩F(S). By the reflexivity ofEand boundedness of the sequence{xn}, we may assume thatxnizfor somez∈C.
(a) First, we show thatz∈F(S). Putxi=xni,αi=αni,βi=βni,γi=γniandμi=μnifor
i∈N, letti≥ be such that
μi→ and T
(μi)xi–xi μi
Fixt> . Notice that
xi–T(t)p≤
[t/μi]–
k=
T(k+ )μi
xi–T(kμi)xi+T
[t/μi]μi
xi–T
[t/μi]μi
z
+T[t/μi]μi
z–T(t)z
≤[t/μi]T(μi)xi–xi+xi–p+T
t– [t/μi]μi
z–z ≤tT(μi)xi–xi
μi
+xi–p+T
t– [t/μi]μi
z–z ≤tT(μi)xi–xi
μi
+xi–p+maxT(s)z–z: ≤s≤μi
.
For alli∈N, we have
lim sup
i→∞
xi–T(t)z≤lim sup i→∞
xi–z.
Since the Banach spaceEwith a weakly sequentially continuous duality mapping satisfies Opial’s condition, this impliesT(t)z=z. Therefore,z∈F(S).
(b) Next, we show thatz∈F(Q). From Lemma ., we know thatQ=QMC is nonexpan-sive; it follows that
yn–Qyn=QMCxn–QMCyn≤ xn–yn.
Thuslimn→∞yn–Qyn= . SinceQis nonexpansive, we get
xn–Qxn ≤ xn–yn+yn–Qyn+Qyn–Qxn
≤xn–yn+yn–Qyn,
and so
lim
n→∞xn–Qxn= . (.)
By Lemma . and (.), we havez∈F(Q). Therefore,z∈F.
Next, we show thatlim supn→∞(f–I)x¯ ,J(xn–x¯ ) ≤, wherex¯ =QFf(x¯ ). Since{xn}
is bounded, we can choose a sequence{xni}of{xn}wherexnizsuch that
lim sup
n→∞
(f–I)x¯ ,J(xn–x¯ ) = lim i→∞
(f –I)x¯ ,J(xni–x¯ ). (.)
Now, from (.), Proposition .(iii) and the weakly sequential continuity of the duality mappingJ, we have
lim sup
n→∞
(f–I)x¯ ,J(xn–x¯ ) = lim i→∞
(f–I)x¯ ,J(xni–x¯ )
From (.), it follows that
lim sup
n→∞
(f–I)x¯,J(xn+–x¯) ≤. (.)
Finally, we show that{xn}converges strongly tox¯ =QFf(x¯ ). We compute that
xn+–x¯ =
xn+–x¯ ,J(xn+–x¯ )
=αnf(xn) +βnxn+γnT(μn)yn–x¯ ,J(xn+–x¯ )
=αn
f(xn) –x¯
+βn(xn–x¯ ) +γn
T(μn)yn–x¯
,J(xn+–x¯ )
=αn
f(xn) –f(x¯ ),J(xn+–x¯ ) +αn
f(x¯ ) –x¯ ,J(xn+–x¯ )
+βn
xn–x¯ ,J(xn+–x¯ ) +γn
T(μn)yn–x¯ ,J(xn+–x¯ ) ≤αn
xn–x¯ –ϕ
xn–x¯
xn+–x¯ +αn
f(x¯ ) –x¯ ,J(xn+–x¯ )
+βnxn–xx¯ n+–x¯ +γnyn–xx¯ n+–x¯ ≤αnxn–xx¯ n+–x¯ –αnϕ
xn–x¯
xn+–x¯
+αn
f(x¯ ) –x¯ ,J(xn+–x¯ )
+βnxn–xx¯ n+–x¯ +γnxn–xx¯ n+–x¯
=xn–x¯xn+–x¯–αnϕ
xn–x¯
xn+–x¯
+αn
f(x¯ ) –x¯ ,J(xn+–x¯ )
=
xn–x¯ +xn+–x¯
–αnϕ
xn–x¯
xn+–x¯
+αn
f(x¯ ) –x¯ ,J(xn+–x¯ ).
By (.) and since{xn+–x}¯ is bounded,i.e., there existsM> such thatxn+–x ≤¯ M,
which implies that
xn+–x¯ ≤ xn–x¯ – αnMϕ
xn–x¯
+ αn
f(x¯ ) –x¯ ,J(xn+–x¯ ). (.)
Now, from (C) and applying Lemma . to (.), we getxn–x →¯ asn→ ∞. This
completes the proof.
Corollary . Let E be a uniformly convex and -uniformly smooth Banach space which
admits a weakly sequentially continuous duality mapping and C be a nonempty closed convex subset of E. LetS={T(t) :t≥}be a nonexpansive semigroup on C and QCbe a
sunny nonexpansive retraction from E onto C. Let A:C→E be aβ-inverse-strongly accre-tive such thatβ≥λKwhere K is the best smooth constant. Let f be a weakly contractive mapping of C into itself with functionϕ. Let the sequences{αn},{βn}and{γn}be in(, )
with{αn}+{βn}+{γn}= , n≥,{μn} ⊂(,∞)and satisfy the conditions (C)-(C) in
Theorem .. SupposeF:=F(Q)∩F(S)=∅, whereQis defined by
andλbe a positive real number. For arbitrary given x=x∈C, the sequences{xn}are
generated by ⎧ ⎨ ⎩
yn=QC(I–λA)QC(I–λA)· · ·QC(I–λA)xn,
xn+=αnf(xn) +βnxn+γnT(μn)yn.
(.)
Then{xn}converges strongly tox¯ =QFf(x¯ ), where QFis the sunny nonexpansive
retrac-tion of C ontoF.
Proof PuttingA=AM=AM–=· · ·=A=A,β=βM=βM–=· · ·=β=βandλ=λM= λM–=· · ·=λ=λin Theorem ., we can conclude the desired conclusion easily. This
completes the proof.
Corollary . Let E be a uniformly convex and -uniformly smooth Banach space which
admits a weakly sequentially continuous duality mapping and C be a nonempty closed convex subset of E. LetS={T(t) :t≥} be a nonexpansive semigroup on C and QC be
a sunny nonexpansive retraction from E onto C. Let Al:C→E be aβl-inverse-strongly
accretive such thatβl≥λlK, where l∈ {, }and K be the best smooth constant. Let f be a
weakly contractive mapping of C into itself with functionϕ. Let the sequences{αn},{βn}and
{γn}be in(, )with{αn}+{βn}+{γn}= , n≥,{μn} ⊂(,∞)and satisfy the conditions
(C)-(C) in Theorem .. SupposeF:=F(Q)∩F(S)=∅, whereQis defined by
Q(x) =QC(I–λA)QC(I–λA)x, ∀x∈C,
andλ,λare positive real numbers. For arbitrary given x=x∈C, the sequences{xn}are
generated by ⎧ ⎨ ⎩
yn=QC(I–λA)QC(I–λA)xn,
xn+=αnf(xn) +βnxn+γnT(μn)yn.
(.)
Then{xn}converges strongly tox¯ =QFf(x¯ )and(x¯ ,x¯ )is a solution of the problem (.),
where QFis the sunny nonexpansive retraction of C ontoF.
Proof TakingM= in Theorem ., we can conclude the desired conclusion easily. This
completes the proof.
4 Some applications
4.1 (I) Application to strictly pseudocontractive mappings
LetEbe a Banach space and letCbe a subset ofE. Recall that a mappingT:C→Cis said to bek-strictly pseudocontractiveif there existk∈[, ) andj(x–y)∈J(x–y) such that
Tx–Ty,j(x–y) ≤ x–y– –k
(I–T)x– (I–T)y
(.) for allx,y∈C. Then (.) can be written in the following form:
(I–T)x– (I–T)y,j(x–y) ≥ –k
(I–T)x– (I–T)y
Moreover, we know thatAis –k-inverse strongly monotone andA– =F(T) (see also
[]).
Theorem . Let E be a uniformly convex and -uniformly smooth Banach space and C
be a nonempty closed convex subset of E. LetS={T(t) :t≥}be a nonexpansive semi-group on C and Tl:C→C be a kl-strictly pseudocontractive mapping withλl≤ (–Kkl), l∈ {, , . . . ,M}. Let f be a weakly contractive mapping of C into itself with functionϕand suppose the sequences{αn},{βn}and{γn}in(, )satisfy{αn}+{βn}+{γn}= , n≥and
{μn} ⊂(,∞). SupposeF:=F(S)∩(
M
l=F(Tl))=∅and letλl, l= , , . . . ,M be positive
real numbers. If the following conditions are satisfied: (i) limn→∞αn= and
∞
n=αn=∞;
(ii) <lim infn→∞βn≤lim supn→∞βn< ;
(iii) limn→∞μn= ;
(iv) limn→∞supy∈ ˜CT(μn+)y–T(μn)y= ,C˜ bounded subset ofC.
Then the sequence{xn}is generated by x=x∈C and
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
yn= (( –λM) +λMTM)(( –λM–) +λM–TM–)· · ·(( –λ) +λT) ×(( –λ) +λT)xn,
xn+=αnf(xn) +βnxn+γnT(μn)yn
(.)
converges strongly to QF, where QFis the sunny nonexpansive retraction of E ontoF.
Proof PuttingAl=I–Tl, l∈ {, , . . . ,M}. From (.), we get Al is –kl-inverse
strong-ly accretive operator. It follows that GSVI(C,Al) = GSVI(C,I – Tl) =F(Tl) =∅ and
(Ml=GSVI(C,I–Tl)) =F(Q)⇔is the solution of the problem (.) (see also Cenget al.
[, Theorem ., pp.-] and Aoyamaet al.[, Theorem ., p.]).
( –λ) +λT
xn=QC
( –λ) +λT
xn
.. .
( –λM) +λMTM
· · ·( –λ) +λT
xn
=QC
( –λM) +λMTM
· · ·QC
( –λ) +λT
xn.
Therefore, by Theorem .,{xn}converges strongly to some elementx¯ofF.
4.2 (II) Application to Hilbert spaces
In real Hilbert spacesH, by Lemma ., we obtain the following lemma:
Lemma . For given(x*
,x*, . . . ,x*M), a solution of the problem is as follows:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
λMAMx*M+x*–x*M,x–x* ≥, ∀x∈C,
λM–AM–x*M–+x*M–x*M–,x–x*M ≥, ∀x∈C,
.. .
λAx*
+x*–x*,x–x* ≥, ∀x∈C, λAx*+x*–x*,x–x* ≥, ∀x∈C,
if and only if
x*=PC(I–λMAM)PC(I–λM–AM–)· · ·PC(I–λA)PC(I–λA)x* is a fixed point of the mappingP:C→C defined by
P(x) =PC(I–λMAM)PC(I–λM–AM–)· · ·PC(I–λA)PC(I–λA)x, ∀x∈C,
where PCis a metric projection H onto C.
It is well known that the smooth constantK=
√
in Hilbert spaces. From Theorem .,
we can obtain the following result immediately.
Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H. Let
Al:C→H be aβl-inverse-strongly monotone mapping withλl∈(, βl), l∈ {, , . . . ,M}.
LetS={T(t) :t≥}be a nonexpansive semigroup on C and f be a weakly contractive mapping of C into itself with function ϕ. Assume thatF:=F(P)∩F(S)=∅, whereP is defined by Lemma . and letλl, l= , , . . . ,M be positive real numbers. Let the sequences
{αn},{βn}and{γn}in(, )with{αn}+{βn}+{γn}= , n≥and the following conditions
be satisfied:
(i) limn→∞αn= and
∞
n=αn=∞;
(ii) <lim infn→∞βn≤lim supn→∞βn< ;
(iii) limn→∞μn= ;
(iv) limn→∞supy∈ ˜CT(μn+)y–T(μn)y= ,C˜ bounded subset ofC.
For arbitrary given x=x∈C, the sequences{xn}are generated by
⎧ ⎨ ⎩
yn=PC(I–λMAM)PC(I–λM–AM–)· · ·PC(I–λA)PC(I–λA)xn,
xn+=αnf(xn) +βnxn+γnT(μn)yn.
(.)
Then{xn}converges strongly tox¯=PFf(x¯)and(x¯,x¯, . . . ,x¯M)is a solution of the problem
(.).
Remark . We can replace a contraction mappingf to a weak contractive mapping by
settingϕ(t) = ( –k)t. Hence, our results can be obtained immediately.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod,
Bangkok, 10140, Thailand.2Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology,
Rajamangala University of Technology Lanna Tak, Tak, 63000, Thailand.
Acknowledgements
(NRU-CSEC No. 55000613) for financial support. Finally, the authors are grateful to the reviewers for careful reading of the paper and for the suggestions which improved the quality of this work.
Received: 26 April 2012 Accepted: 27 June 2012 Published: 20 July 2012
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