R E S E A R C H
Open Access
The modified Mann type iterative algorithm
for a countable family of totally
quasi-φ-asymptotically nonexpansive
mappings by the hybrid generalized
f
-projection method
Siwaporn Saewan
1*, Preedaporn Kanjanasamranwong
1*, Poom Kumam
2*and Yeol Je Cho
3**Correspondence:
[email protected]; [email protected]; [email protected]; [email protected]
1Department of Mathematics and
Statistics, Faculty of Science, Thaksin University (TSU), Pa Phayom, Phatthalung, 93110, Thailand 2Department 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
The purpose of this article is to introduce the modified Mann type iterative sequence, using a new technique, by the hybrid generalizedf-projection operator for a
countable family of totally quasi-φ-asymptotically nonexpansive mappings in a uniform smooth and strictly convex Banach space with the Kadec-Klee property. Then we prove that the modified Mann type iterative scheme converges strongly to a common element of the sets of fixed points of the given mappings. Our result extends and improves the results of Liet al.(Comput. Math. Appl. 60:1322-1331, 2010), Takahashiet al.(J. Math. Anal. Appl. 341:276-286, 2008) and many other authors. MSC: 47H05; 47H09; 47H10
Keywords: generalizedf-projection operator; modified Mann type iterative sequence; totally quasi-φ-asymptotically nonexpansive mapping
1 Introduction
LetEbe a real Banach space andCbe a nonempty closed and convex subset ofE. A map-pingT :C→Cis said to betotally asymptotically nonexpansive[] if there exist non-negative real sequencesνn,μnwithνn→,μn→ asn→ ∞and a strictly increasing
continuous functionϕ:R+→R+withϕ() = such that
Tnx–Tny≤ x–y+μnψ
x–y+νn, ∀x,y∈C,n≥.
A pointx∈Cis afixed pointofTprovidedTx=x. Denote byF(T) the fixed point set of
T, that is,F(T) ={x∈C:Tx=x}. A pointp∈Cis called anasymptotic fixed pointofT[] ifCcontains a sequence{xn}which converges weakly topsuch thatlimn→∞xn–Txn= .
The asymptotic fixed point set ofTis denoted byF(T).
LetE*be a dual space of the Banach spaceE. We recall that for allx∈Eandx∈E*, we denote the value ofx*atxby x,x*. Then thenormalized duality mapping J:E→E*is defined by
J(x) =x*∈E*:x,x*=x,x*=x , ∀x∈E.
IfE is a Hilbert space, thenJ=I, whereI is the identity mapping. Next, consider the functionalφ:E×E→R+∪ {}defined by
φ(x,y) =x– x,Jy+y, ∀x,y∈E, (.)
whereJis the normalized duality mapping and ·,·denotes the duality pairing ofEandE*.
IfEis a Hilbert space, thenφ(x,y) =x–y. It is obvious from the definition ofφthat
x–y≤φ(x,y)≤x+y, ∀x,y∈E. (.)
T is said to berelatively nonexpansive[, ] ifF(T) =F(T) and
φ(p,Tx)≤φ(p,x), ∀x∈C,p∈F(T).
Tis said to berelatively asymptotically nonexpansive[, ] ifF(T) =F(T)=∅and there exists a sequence{kn} ⊂[,∞) withkn→ asn→ ∞such that
φp,Tnx≤knφ(p,x), ∀x∈C,p∈F(T),n≥.
T is said to beφ-nonexpansive[, ] if
φ(Tx,Ty)≤φ(x,y), ∀x,y∈C.
T is said to bequasi-φ-nonexpansive[, ] ifF(T)=∅and
φ(p,Tx)≤φ(p,x), ∀x∈C,p∈F(T).
Tis said to beasymptoticallyφ-nonexpansive[] if there exists a sequence{kn} ⊂[,∞)
withkn→ asn→ ∞such that
φTnx,Tny≤knφ(x,y), ∀x,y∈C.
T is said to bequasi-φ-asymptotically nonexpansive[] ifF(T)=∅and there exists a sequence{kn} ⊂[,∞) withkn→ asn→ ∞such that
φp,Tnx≤knφ(p,x), ∀x∈C,p∈F(T),n≥.
T is said to betotally quasi-φ-asymptotically nonexpansiveifF(T)=∅and there exist nonnegative real sequencesνn,μnwithνn→,μn→ asn→ ∞and a strictly increasing
continuous functionϕ:R+→R+withϕ() = such that
φp,Tnx≤φ(p,x) +νnϕ
φ(p,x)+μn, ∀x∈C,p∈F(T),n≥.
non-expansive mapping and a quasi-φ-asymptotically nonexpansive mapping implies a totally quasi-φ-asymptotically nonexpansive mapping, but the converses are not true.
() A relatively quasi-nonexpansive mapping is sometimes called hemi-relatively non-expansive mapping. The class of relatively quasi-nonnon-expansive mappings is more general than the class of relatively nonexpansive mappings (see [, –]), which requires the strong restrictionF(T) =F(T).
() For other examples of relatively quasi-nonexpansive mappings such as the general-ized projections and others, see [, Examples . and .].
On the other hand, Alber [] introduced that thegeneralized projectionC:E→Cis
a mapping that assigns to an arbitrary pointx∈Ethe minimum point of the functional
φ(x,y), that is,Cx=x¯, wherex¯is a solution of the minimization problem
φ(x¯,x) =inf
y∈Cφ(y,x). (.)
In , Wu and Huang [] introduced a new generalized f-projection operator in Banach spaces. They extended the definition of generalized projection operators intro-duced by Abler [] and proved the properties of the generalizedf-projection operator.
Now, we recall the concept of the generalizedf-projection operator. LetG:C×E*→
R∪ {+∞}be a functional defined by
G(y,) =y– y,++ ρf(y), (.) wherey∈C,∈E*,ρis a positive number andf :C→R∪ {+∞}is proper, convex and
lower semicontinuous. From the definition ofG, Wu and Huang [] studied the following properties:
() G(y,)is convex and continuous with respect to whenyis fixed;
() G(y,)is convex and lower semicontinuous with respect toywhenis fixed.
Definition . LetEbe a real Banach space with the dual spaceE*andCbe a nonempty
closed and convex subset ofE. We say thatπCf :E*→Cis ageneralized f -projection op-eratorif
πCf=
u∈C:G(u,) =inf
y∈CG(y,),∀∈E
*.
In , Mann [] introduced the following iteration process, which is now well known as Mann’s iteration:
xn+=αnxn+ ( –αn)Txn, ∀n≥, (.)
where the initial guess elementx∈Cis arbitrary and{αn}is a sequence in [, ]. Mann’s
Recently, Takahashiet al.[] studied the strong convergence theorem by the new hybrid method{xn}for a family of nonexpansive mappings in Hilbert spaces:x∈H,C=C,
x=PCxand ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
yn=αnxn+ ( –αn)Tnxn,
Cn+={z∈C:yn–z ≤ xn–z}, xn+=PCn+x, ∀n≥,
(.)
where ≤αn≤a< for alln≥ and{Tn}is a sequence of nonexpansive mappings of Cinto itself such that∞n=F(Tn)=∅. They proved that if{Tn}satisfies some appropriate
conditions, then{xn}converges strongly toPn=∞ F(Tn)x.
The ideas to generalize the process (.) from Hilbert spaces to Banach spaces have re-cently been made. Especially, Matsushita and Takahashi [] proposed the following hy-brid iteration method with the generalized projection for a relatively nonexpansive map-pingTin a Banach spaceE:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,
yn=J–(αnJxn+ ( –αn)JTxn), Cn={z∈C:φ(z,yn)≤φ(z,xn)}, Qn={z∈C: xn–z,Jx–Jxn ≥}, xn+=Cn∩Qnx, ∀n≥.
(.)
They proved that{xn}converges strongly to a pointF(T)x. Many authors studied
meth-ods for approximating fixed points of a countable family of (relatively quasi-) nonexpansive mappings (see [–]).
In , Alberet al. [] proved a new strong convergence result of the regularized successive approximation method for a total asymptotically nonexpansive mapping in a Hilbert spaces. In , Liet al.[] introduced the following hybrid iterative scheme{xn}
for approximation fixed points of a relatively nonexpansive mapping using the generalized
f-projection operator in a uniformly smooth real Banach space which is also uniformly convex:x∈Cand
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
yn=J–(αnJxn+ ( –αn)JTxn),
Cn+={w∈Cn:G(w,Jyn)≤G(w,Jxn)}, xn+=fCn+x, ∀n≥.
(.)
They proved strong convergence theorems for finding an element in the fixed point set ofT.
One question is raised naturally as follows:
Are the results of Alber et al.[], Li et al.[]and Takahashi et al.[]true in the framework of strictly convex Banach spaces for totally quasi-φ-asymptotically nonexpan-sive mappings?
countable family of totally quasi-φ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Our result ex-tends and improves the results of Liet al.[], Takahashiet al.[] and many other au-thors.
2 Preliminaries
A Banach spaceEwith the norm · is calledstrictly convexifx+y< for allx,y∈E
withx=y= andx=y. LetU={x∈E:x= }be a unit sphere ofE. A Banach space
Eis calledsmoothif the limit
lim
t→
x+ty–x t
exists for eachx,y∈U. It is also calleduniformly smoothif the limit exists uniformly for allx,y∈U. Themodulus of smoothnessofEis the functionρE: [,∞)→[,∞) defined
by
ρE(t) =sup
x+y+x–y
– :x= ,y ≤t
.
Themodulus of convexityofE(see []) is the functionδE: [, ]→[, ] defined by
δE(ε) =inf
–x+y
:x,y∈E,x=y= ,x–y ≥ε
.
In this paper, we denote the strong convergence and weak convergence of a sequence
{xn}byxn→xandxnx, respectively.
Remark . The basic properties ofE,E*,JandJ–are as follows (see []):
() IfEis an arbitrary Banach space, thenJis monotone and bounded; () IfEis strictly convex, thenJis strictly monotone;
() IfEis smooth, thenJis single-valued and semi-continuous;
() IfEis uniformly smooth, thenJis uniformly norm-to-norm continuous on each bounded subset ofE;
() IfEis reflexive smooth and strictly convex, then the normalized duality mappingJis single-valued, one-to-one and onto;
() IfEis a reflexive strictly convex and smooth Banach space andJis the duality mapping fromEintoE*, thenJ–is also single-valued, bijective and is also the duality mapping fromE*intoEand thusJJ–=I
E*andJ–J=IE; () IfEis uniformly smooth, thenEis smooth and reflexive; () Eis uniformly smooth if and only ifE*is uniformly convex;
() IfEis a reflexive and strictly convex Banach space, thenJ–is
norm-weak*-continuous.
Remark . IfEis a reflexive, strictly convex and smooth Banach space, thenφ(x,y) = if and only ifx=y. It is sufficient to show that ifφ(x,y) = thenx=y. From (.) we have
x=y. This implies that x,Jy=x=Jy. From the definition ofJ, one hasJx=Jy.
Recall that a Banach spaceEhas theKadec-Klee property[, , ] if, for any sequence
{xn} ⊂Eandx∈Ewithxnxandxn → x,xn–x → asn→ ∞. It is well known
that ifEis a uniformly convex Banach space, thenEhas the Kadec-Klee property. Thegeneralized projection[] fromEintoCis defined by
C(x) =arg min y∈C
φ(y,x).
The existence and uniqueness of the operator C follows from the properties of the
functionalφ(y,x) and the strict monotonicity of the mappingJ(see, for example, [, , , , ]). IfE is a Hilbert space, thenφ(x,y) =x–y and
C becomes the metric
projectionPC:H→C. IfCis a nonempty closed and convex subset of a Hilbert spaceH,
thenPCis nonexpansive. This fact actually characterizes Hilbert spaces and, consequently,
it is not available in more general Banach spaces.
We also need the following lemmas for the proof of our main results.
LetTbe a nonlinear mapping,Tis said to beuniformly asymptotically regularonCif
lim
n→∞
sup
x∈C
Tn+x–Tnx= .
A mappingTfromCinto itself is said to beclosedif, for any sequence{xn} ⊂Csuch that
limn→∞xn=xandlimn→∞Txn=y, we haveTx=y.
Lemma .(Changet al.[]) Let C be a nonempty closed and convex subset of a uni-formly smooth and strictly convex Banach space E with the Kadec-Klee property. Let T :C→C be a closed and total quasi-φ-asymptotically nonexpansive mapping with the sequencesνnandμnof nonnegative real numbers withνn→,μn→as n→ ∞and a strictly increasing continuous functionψ:R+→R+withψ() = .Ifμ
= ,then the fixed
point set F(T)is a closed convex subset of C.
Lemma .(Wu and Hung []) Let E be a real reflexive Banach space with the dual space E*and C be a nonempty closed and convex subset of E.The following statements hold:
() πCfis a nonempty,closed and convex subset ofCfor all∈E*; () IfEis smooth,then for all∈E*,x∈πf
C if and only if
x–y,–Jx+ρf(y) –ρf(x)≥, ∀y∈C;
() IfEis strictly convex andf :C→R∪ {+∞}is positive homogeneous(i.e.,f(tx) =tf(x)
for allt> such thattx∈C,wherex∈C),thenπCf is a single-valued mapping.
In the following lemma, Fanet al.[] showed that Lemma .(iii) in [] can be removed.
Lemma .(Fanet al.[]) Let E be a real reflexive Banach space with its dual space E* and C be a nonempty closed and convex subset of E.If E is strictly convex,thenπCf is single-valued.
following:
G(y,Jx) =y– y,Jx+x+ ρf(y). (.) Now, we consider the second generalizedf-projection operator in Banach spaces (see []).
Definition . LetEbe a real smooth Banach space andCbe a nonempty, closed and convex subset ofE. We say thatfC:E→Cis thegeneralized f -projection operatorif
fCx=
u∈C:G(u,Jx) =inf
y∈CG(y,Jx),∀x∈E
.
Lemma .(Deimling []) Let E be a Banach space and f :E→R∪ {+∞}be a lower semicontinuous convex function.Then there exist x*∈E*andα∈Rsuch that
f(x)≥x,x*+α, ∀x∈E.
Lemma .(Liet al.[]) Let E be a reflexive smooth Banach space and C be a nonempty,
closed and convex subset of E.The following statements hold:
() fCxis nonempty,closed and convex subset ofCfor allx∈E; () For allx∈E,xˆ∈fCxif and only if
ˆx–y,Jx–Jxˆ+ρf(y) –ρf(xˆ)≥, ∀y∈C;
() IfEis strictly convex,thenfCis a single-valued mapping.
Lemma .(Liet al. []) Let E be a real reflexive smooth Banach space and C be a nonempty closed and convex subset of E.Ifxˆ∈fCx for all x∈E,then
φ(y,xˆ) +G(xˆ,Jx)≤G(y,Jx), ∀y∈C.
Remark . LetEbe a uniformly convex and uniformly smooth Banach space andf(x) = for allx∈E. Then Lemma . reduces to the property of the generalized projection operator considered by Alber [].
Iff(y)≥ for ally∈Candf() = , then the definition of totally quasi-φ-asymptotically nonexpansiveT is equivalent to the following:
IfF(T)=∅and there exist nonnegative real sequencesνn,μnwithνn→,μn→ as n→ ∞and a strictly increasing continuous functionζ :R+→R+withζ() = such that
Gp,JTnx≤G(p,Jx) +νnζG(p,Jx) +μn, ∀x∈C,p∈F(T),n≥.
3 Main results
Theorem . Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let{Ti}∞i= be a countable
convex and lower semicontinuous function with C⊂int(D(f))such that f(x)≥for all x∈C and f() = .Assume that Tiis uniformly asymptotically regular for all i≥and F=∞i=F(Ti)=∅.For an initial point x∈E,let C,i=C for each i≥and C=
∞
i=C,i and define the sequence{xn}by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
yn,i=J–(αnJxn+ ( –αn)JTinxn),
Cn+,i={z∈Cn:G(z,Jyn,i)≤G(z,Jxn) +βn}, Cn+=
∞
i=Cn+,i, xn+=fCn+x, ∀n≥,
(.)
where{αn}is a sequence in(, )andβn=νnsupq∈FψG(q,Jxn) +μn.Iflim infn→∞( –αn) >
,then{xn}converges strongly to a pointfFx.
Proof We split the proof into four steps.
Step . We first show thatCn+is closed and convex for alln≥. From the definition,
C=∞i=C,i=Cfor alli≥ is closed and convex. Suppose thatCn,iis closed and convex
for somen≥. For anyz∈Cn,i, we know thatG(z,Jyn,i)≤G(z,Jxn) +βnis equivalent to
the following:
z,Jxn–Jyn,i ≤ xn–yn,i+βn, ∀i≥.
Therefore,Cn+,iis closed and convex. HenceCn+=
∞
i=Cn+,iis closed and convex for all n≥.
Step . We show, by induction, thatF=∞i=F(Ti)=∅ ⊂Cnfor alln≥. It is obvious
thatF⊂C=C. Suppose thatF⊂Cnfor somen≥. Letq∈F. Since{Ti}is a totally
quasi-φasymptotically nonexpansive mapping, for eachi≥, we have
G(q,Jyn,i)
=Gq,αnJxn+ ( –αn)JTinxn
=q– q,αnJxn+ ( –αn)JTinxn
+αnJxn+ ( –αn)JTinxn
+ ρf(q)
≤ q– αn q,Jxn– ( –αn)
q,JTinxn
+αnJxn+ ( –αn)JTinxn
+ ρf(q) =αnG(q,Jxn) + ( –αn)G
q,JTinxn
≤αnG(q,Jxn) + ( –αn)
G(q,Jxn) +νnψ
G(q,Jxn)
+μn
≤G(q,Jxn) +νnsupψ
G(q,Jxn)
+μn
=G(q,Jxn) +βn. (.)
This shows thatq∈Cn+, which implies thatF⊂Cn+. HenceF⊂Cnfor alln≥.
Step . We show thatxn→pasn→ ∞. Sincef:E→Ris a convex and lower
semicon-tinuous function, from Lemma ., it follows that there existx*∈E*andα∈Rsuch that
f(x)≥ x,x*+αfor allx∈E. Sincex
n∈E, it follows that G(xn,Jx) =xn– xn,Jx+x+ ρf(xn)
≥ xn– xn,Jx+x+ ρ
xn,x*
=xn–
xn,Jx–ρx*
+x+ ρα ≥ xn– xnJx–ρx*+x+ ρα
=xn–Jx–ρx*
+x–Jx–ρx*
+ ρα. (.) For anyq∈F, sincexn=fCnx, we have
G(q,Jx)≥G(xn,Jx)≥
xn–Jx–ρx*
+x–Jx–ρx*
+ ρα.
This implies that{xn}is bounded and so are{G(xn,Jx)}and{yn,i}. From the fact thatxn+= fCn+x∈Cn+⊂Cnandxn=fCnx, it follows from Lemma . that
≤xn+–xn
≤φ(xn+,xn)≤G(xn+,Jx) –G(xn,Jx). (.)
This implies that{G(xn,Jx)}is nondecreasing. Hence we know thatlimn→∞G(xn,Jx)
ex-ists. Takingn→ ∞, we obtain
lim
n→∞φ(xn+,xn) = . (.)
Since{xn} is bounded,E is reflexive andCn is closed and convex for alln≥, we can
assume thatxnp∈Cn. From the fact thatxn=fCnxandp∈Cn, we get
G(xn,Jx)≤G(p,Jx), ∀n≥. (.)
Sincef is convex and lower semicontinuous, we have
lim inf
n→∞ G(xn,Jx) =lim infn→∞
xn– xn,Jx+x+ ρf(xn)
≥ p– p,Jx+x+ ρf(p)
=G(xn,Jx). (.)
By (.) and (.), we get
G(p,Jx)≤lim inf
n→∞ G(xn,Jx)≤lim supn→∞ G(xn,Jx)≤G(p,Jx).
That is,limn→∞G(xn,Jx) =G(p,x), which implies thatxn → pand so, by virtue of
the Kadec-Klee property ofE, it follows that
lim
n→∞xn=p. (.)
We also have
lim
n→∞xn+=p. (.)
Since{xn}is bounded (we denoteM=supn≥{xn}<∞), it follows that
lim
n→∞βn=nlim→∞
νnsup q∈Fψ
G(q,xn)
+μn
From (.) and (.), we havelimn→∞xn–xn+= . SinceJis uniformly norm-to-norm
continuous, it follows that
lim
n→∞Jxn–Jxn+= . (.)
Sincexn+=fCn+x∈Cn+⊂Cnand by the definition ofCn+, it follows that G(xn+,Jyn,i)≤G(xn+,Jxn) +βn, ∀i≥
⇐⇒ xn+– xn+,Jyn,i+yn,i+ ρf(xn+)
≤ xn+– xn+,Jxn+xn+ ρf(xn+) +βn, ∀i≥
⇐⇒ φ(xn+,yn,i) =xn+– xn+,Jyn,i+yn,i
≤ xn+– xn+,Jxn+xn+βn
=φ(xn+,xn) +βn, ∀i≥,
that is, we get
φ(xn+,yn,i)≤φ(xn+,xn) +βn, ∀i≥.
From (.) and (.), it follows that for eachi≥,
lim
n→∞φ(xn+,yn,i) = . (.)
Also, from (.), it follows that for eachi≥,
yn,i → p (n→ ∞). (.)
SinceJis uniformly norm-to-norm continuous, it follows that for eachi≥,
Jyn,i → Jp (n→ ∞). (.)
That is,{Jyn,i}bounded inE*for alli≥. SinceEis reflexive andE*is also reflexive,
we can assume thatJyn,iy*∈E*for alli≥. SinceEis reflexive, we see thatJ(E) =E*.
Hence there existsy∈Esuch thatJy=y*. It follows that for eachi≥,
φ(xn+,yn,i) =xn+– xn+,Jyn,i+yn,i=xn+– xn+,Jyn,i+Jyn,i.
Takinglim infn→∞on the both sides of the equality above and the property of weak lower
semicontinuity of the norm · , it follows that ≥ p– p,y*+y*
=p– p,Jy+Jy
=p– p,Jy+y
That is,p=y, which implies thaty*=Jp. It follows that for eachi≥,Jy
n,iJp∈E*. From
(.) and the Kadec-Klee property ofE*, we haveJy
n,i→Jpasn→ ∞for alli≥. Since J–:E*→E is norm-weak*-continuous, that is,yn,ip, it follows from (.) and the
Kadec-Klee property ofEthat
lim
n→∞yn,i=p, ∀i≥. (.)
From (.), (.) and the triangle inequality, we have
lim
n→∞xn+–yn,i= , ∀i≥. (.)
SinceJis uniformly norm-to-norm continuous, we obtain
lim
n→∞Jxn+–Jyn,i= , ∀i≥. (.)
From the definition ofyn,i, it follows that
Jxn+–Jyn,i=Jxn+–αnJxn– ( –αn)JTinxn
=( –αn)Jxn+– ( –αn)JTinxn+αnJxn+–αnJxn
≥( –αn)Jxn+–JTinxn–αnJxn–Jxn+, (.)
and so
Jxn+–JTinxn≤
( –αn)
Jxn+–Jyn,i+αnJxn–Jxn+
. (.)
Sincelim infn→∞( –αn) > , it follows from (.) and (.) that, for eachi≥,
lim
n→∞Jxn+–JT
n
ixn= . (.)
SinceJ–is uniformly norm-to-norm continuous, for eachi≥, we obtain
lim
n→∞xn+–T
n
ixn= . (.)
By using the triangle inequality, for eachi≥, we have
Tinxn–p≤Tinxn–xn++xn+–p.
From (.) andxn+→pasn→ ∞, it follows that for eachi≥,
lim
n→∞T n
ixn–p= . (.)
For eachi≥, we have
SinceTiis uniformly asymptotically regular for alli≥, it follows from (.) that Tin+xn–p= . (.)
That is,Tn+
i xn=TiTinxn→pasn→ ∞. FromTinxn→pasn→ ∞and the closedness
ofTi, we haveTip=pfor alli≥. We see thatp∈F(Ti) for alli≥, which implies that p∈∞i=F(Ti).
Step . We show thatp=fFx. SinceF is a closed and convex set, it follows from
Lemma . thatfFxis single-valued, which is denoted byv. By the definitionxn=fCnx
andv∈F⊂Cn, we also have G(xn,Jx)≤G(v,Jx), ∀n≥.
By the definition ofGandf, we know that, for anyx∈E, G(ξ,Jx) is convex and lower semicontinuous with respect toξand so
G(p,Jx)≤lim inf
n→∞ G(xn,Jx)≤lim supn→∞ G(xn,Jx)≤G(v,Jx).
From the definition offFx, sincep∈F, we conclude thatv=p=fFx andxn→pas n→ ∞. This completes the proof.
Settingνn≡ andμn≡ in Theorem ., we have the following.
Corollary . Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let{Ti}∞i= be a countable
family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings.Let f :E→ Rbe a convex and lower semicontinuous function with C⊂int(D(f))such that f(x)≥for all x∈C and f() = .Assume that Tiis uniformly asymptotically regular for all i≥and F=∞i=F(Ti)=∅.For an initial point x∈E,let C,i=C,C=
∞
i=C,iand define the sequence{xn}by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
yn,i=J–(αnJxn+ ( –αn)JTinxn), Cn+,i={z∈Cn:G(z,Jyn,i)≤G(z,Jxn)}, Cn+=
∞
i=Cn+,i, xn+=fCn+x, ∀n≥,
(.)
where{αn}is a sequence in(, ).Iflim infn→∞( –αn) > ,then{xn}converges strongly to a pointfFx.
Settingi= andTi=Tin Theorem ., we have the following.
function with C⊂int(D(f))such that f(x)≥for all x∈C and f() = .Assume that T is a uniformly asymptotically regular andF=F(T)=∅.For an initial point x∈E,let C=C
and define the sequence{xn}by ⎧
⎪ ⎪ ⎨ ⎪ ⎪ ⎩
yn=J–(αnJxn+ ( –αn)JTnxn),
Cn+={z∈Cn:G(z,Jyn,i)≤G(z,Jxn) +βn}, xn+=fCn+x, ∀n≥,
(.)
where{αn}is a sequence in(, )andβn=νnsupψG(p,xn) +μn.Iflim infn→∞( –αn) > , then{xn}converges strongly to a pointfFx.
Takingf(x) = for allx∈C, we haveG(ξ,Jx) =φ(ξ,x) andfCx=Cx. Thus, from
Theorem ., we obtain the following.
Corollary . Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let{Ti}∞i= be a countable
family of closed and uniformly totally quasi-φ-asymptotically nonexpansive mappings with the sequencesνn,μnof nonnegative real numbers withνn→,μn→as n→ ∞and a strictly increasing continuous functionψ :R+→R+withψ() = .Assume that Tiis uniformly asymptotically regular for all i≥andF=∞i=F(Ti)=∅.For an initial point x∈E,let C,i=C,C=
∞
i=C,iand define the sequence{xn}by ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
yn,i=J–(αnJxn+ ( –αn)JTinxn),
Cn+,i={z∈Cn:φ(z,yn,i)≤φ(z,xn) +βn}, Cn+=
∞
i=Cn+,i, xn+=Cn+x, ∀n≥,
(.)
where{αn}is a sequence in(, )andβn=νnsupψ(φ(p,xn)) +μn.Iflim infn→∞( –αn) > , then{xn}converges strongly to a pointFx.
Corollary . Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let{Ti}∞i= be a countable
family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings.Assume that Ti is uniformly asymptotically regular for all i≥andF=∞i=F(Ti)=∅.For an initial point x∈E,let C,i=C,C=
∞
i=C,iand define the sequence{xn}by ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
yn,i=J–(αnJxn+ ( –αn)JTinxn), Cn+,i={z∈Cn:φ(z,yn,i)≤φ(z,xn)}, Cn+=
∞
i=Cn+,i, xn+=Cn+x, ∀n≥,
(.)
where{αn}is a sequence in(, ).Iflim infn→∞( –αn) > ,then{xn}converges strongly to a pointFx.
quasi-φ-asymptotically nonexpansive mapping with the sequencesνn,μnof nonnegative real numbers withνn→,μn→as n→ ∞and a strictly increasing continuous func-tionψ :R+→R+withψ() = .Assume that T is uniformly asymptotically regular and
F=F(T)=∅.For an initial point x∈E,let C=C and define the sequence{xn}by ⎧
⎪ ⎪ ⎨ ⎪ ⎪ ⎩
yn=J–(αnJxn+ ( –αn)JTnxn),
Cn+={z∈Cn:φ(z,yn)≤φ(z,xn) +βn}, xn+=Cn+x, ∀n≥,
(.)
where{αn}is a sequence in(, )andβn=νnsupψ(φ(p,xn)) +μn.Iflim infn→∞( –αn) > , then{xn}converges strongly to a pointFx.
Remark . () Corollary . extends and improves the results of Liet al.[] from a rel-atively nonexpansive mapping to a totally quasi-φ-asymptotically nonexpansive mapping. () Corollary . extends and generalizes the result of Takahashiet al.[] from a Hilbert space to a Banach space and from a nonexpansive mapping to a totally quasi-φ asymptot-ically nonexpansive mapping.
() In the case of spaces, we extend Banach spaces from a uniformly smooth and uniformly convex Banach to a uniformly smooth and strictly convex Banach with the Kadec-Klee property, which can be found in the literature works by many authors (see [, , , ]).
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 and Statistics, Faculty of Science, Thaksin University (TSU), Pa Phayom, Phatthalung, 93110,
Thailand.2Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, KMUTT, Bangmod, Bangkok, 10140, Thailand.3Department of Mathematics Education and RINS, Gyeongsang National University, Chinju, 660-701, Korea.
Acknowledgements
This research was supported by Thaksin University. Moreover, the forth author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012-0008170).
Received: 27 July 2012 Accepted: 21 February 2013 Published: 18 March 2013 References
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doi:10.1186/1687-1812-2013-63
