A new iterative process for a hybrid pair of generalized asymptotically nonexpansive single-valued and generalized nonexpansive multi-valued mappings in Banach spaces

(1)

R E S E A R C H

Open Access

A new iterative process for a hybrid pair of

generalized asymptotically nonexpansive

single-valued and generalized nonexpansive

multi-valued mappings in Banach spaces

Suthep Suantai

1

_{and Withun Phuengrattana}

2,3*

*_{Correspondence:} [email protected] 2_{Department of Mathematics,} Faculty of Science and Technology, Nakhon Pathom Rajabhat University, Nakhon Pathom, 73000, Thailand

3_{Research Center for Pure and} Applied Mathematics, Research and Development Institute, Nakhon Pathom Rajabhat University, Nakhon Pathom, 73000, Thailand Full list of author information is available at the end of the article

Abstract

In this paper, we construct an iterative process involving a hybrid pair of a ﬁnite family of generalized asymptotically nonexpansive single-valued mappings and a ﬁnite family of generalized nonexpansive multi-valued mappings and prove weak and strong convergence theorems of the proposed iterative process in Banach spaces. Our main results extend and generalize many results in the literature.

MSC: 47H09; 47H10; 54H25; 54E40

Keywords: ﬁxed point; generalized asymptotically nonexpansive mapping; nonexpansive mapping; Banach space

1 Introduction

Throughout this paper we denote byNthe set of all positive integers. LetXbe a Banach space and letDbe a nonempty subset ofX. LetCB(D) andKC(D) denote the families of nonempty, closed, and bounded subsets and nonempty, compact, and convex subsets ofD, respectively. TheHausdorﬀ metriconCB(D) is deﬁned by

H(A,B) =max

sup

x∈A

dist(x,B),sup

y∈B

dist(y,A) forA,B∈CB(D),

wheredist(x,D) =inf{x–y:y∈D} is the distance from a pointxto a subsetD. Let tbe a single-valued mapping ofDinto DandT be a multi-valued mapping ofDinto CB(D). The set of ﬁxed points oftandT will be denoted byF(t) ={x∈D:x=tx}and F(T) ={x∈D:x∈Tx}, respectively. A pointxis called acommon ﬁxed pointoftandT if x=tx∈Tx.

Deﬁnition . A single-valued mappingt:D→Dis said to begeneralized asymptotically nonexpansiveif there exist sequences{kn} ⊂[,∞) and{sn} ⊂[,∞) withlimn→∞kn= ,

limn→∞sn=  such that

tnx–tny≤knx–y+sn,

for allx,y∈Dandn∈N.

(2)

In the case ofsn= , for alln∈N, a single-valued mappingtis calledan asymptotically

nonexpansive mapping. In particular, ifkn=  andsn= , for alln∈N, a single-valued

mappingtreduce to anonexpansive mapping. The ﬁxed point property for generalized asymptotically nonexpansive single-valued mappings can be found in []. The following example shows that the ﬁxed point set of a generalized asymptotically nonexpansive map-ping is not necessarily closed; see also [].

Example .([]) Deﬁne a single-valued mappingt: [– ,

 ]→[–

 ,

 ] by

tx=

⎧ ⎪ ⎨ ⎪ ⎩

x, ifx∈[–_, ), 

, ifx= , x, ifx∈(,_].

Thentis generalized asymptotically nonexpansive andF(t) = [–_, ) which is not closed.

Deﬁnition . A multi-valued mappingT:D→CB(D) is said to be

(i) nonexpansiveifH(Tx,Ty)≤ x–y, for allx,y∈D;

(ii) quasi-nonexpansiveifF(T)=∅andH(Tx,Tp)≤ x–p, for allx∈Dandp∈F(T).

The study of fixed points for nonexpansive multi-valued mappings using the Hausdorff metric was initiated by Markin []. Different iterative processes have been used to approx-imate fixed points of nonexpansive and quasi-nonexpansive multi-valued mappings; in particular, Sastry and Babu [] considered Mann and Ishikawa iterates for a multi-valued mappingT with a fixed pointpand proved that these iterates converge to a fixed point q ofT under certain conditions. Moreover, they illustrated that the fixed point qmay be different fromp. Later in , Panyanak [] generalized results of Sastry and Babu [] to uniformly convex Banach spaces and proved a convergence theorem of Mann iter-ates for a mapping defined on a noncompact domain. In , Shahzad and Zegeye [] proved strong convergence theorems for the Ishikawa iteration scheme involving quasi-nonexpansive multi-valued mappings. They constructed an iterative process which re-moves the restriction ofT, namelyend-point condition,i.e.,Tp={p}for anyp∈F(T); see also [, ].

In , Garcia-Falsetet al.[] introduced a new condition on single-valued mappings, calledcondition(E), which is weaker than nonexpansiveness. Later, Abkar and Eslamian [] used a modiﬁed condition for multi-valued mappings as follows.

Deﬁnition . A multi-valued mappingT:D→CB(D) is said to satisfycondition(Eμ)

whereμ≥ if for eachx,y∈D,

dist(x,Ty)≤μdist(x,Tx) +x–y.

We say thatT satisﬁescondition(E) wheneverTsatisﬁes (Eμ) for someμ≥.

Remark . From the above deﬁnitions, it is clear that ifTis nonexpansive, thenT

(3)

In , Sokhuma and Kaewkhao [] introduced the following iterative process for ap-proximating a common ﬁxed point of a pair of a nonexpansive single-valued mappingt and a nonexpansive multi-valued mappingT:

yn= ( –αn)xn+αnzn,

xn+= ( –βn)xn+βntyn, n∈N,

(.)

wherex∈D,zn∈Txn, and  <a≤αn,βn≤b< . They also proved a strong convergence

theorem for the iterative process (.) in uniformly convex Banach spaces.

In , Eslamian [] extended the results of [, ] in uniformly convex Banach spaces. He used the following iterative process for a pair of a ﬁnite family of asymptotically non-expansive single-valued mappings{ti}Ni=and a ﬁnite family of quasi-nonexpansive multi-valued mapping{Ti}Ni=:

yn=βn()xn+

N

i=β (i)

nzn(i),

xn+=αn()xn+

N

i=α (i)

ntinyn, n∈N,

(.)

wherex∈D,zn(i)∈Tixn, and{αn(i)},{βn(i)}are sequences in [, ] for alli= , , . . . ,Nsuch

thatN_i_=αn(i)=

N

i=β (i)

n = .

In this paper, motivated by the above results, we propose an iterative process for approx-imating a common fixed point of a pair of a finite family of generalized asymptotically non-expansive single-valued mappings and a finite family of quasi-nonnon-expansive multi-valued mappings and prove weak and strong convergence theorems of the proposed iterative pro-cess in Banach spaces.

2 Preliminaries

A Banach spaceXis calleduniformly convexif for eachε>  there is aδ>  such that forx,y∈Xwithx ≤,y ≤, andx–y ≥ε,x+y ≤( –δ) holds. The following result was proved by Xu [].

Proposition . Let X be a uniformly convex Banach space and let r> .Then there exists a strictly increasing,continuous,and convex function g: [,∞)→[,∞)with g() = such that

λx+ ( –λ)y≤λx+ ( –λ)y–λ( –λ)gx–y

for all x,y∈Br={z∈X:z ≤r}andλ∈[, ].

A Banach spaceXis said to satisfy theOpial property(see []) if it is given that whenever {xn}converges weakly tox∈X,

lim sup

n→∞ xn–x<lim supn→∞ xn–y

for eachy∈Xwithy=x. The examples of Banach spaces which satisfy the Opial property are Hilbert spaces and allLp[, π] with  <p=  fail to satisfy the Opial property.

(4)

Deﬁnition . (see []) LetF be a nonempty subset of a Banach spaceX and let{xn}

be a sequence inX. We say that{xn}is ofmonotone type (I) with respect to Fif there exist

sequences{δn}and{εn}of nonnegative real numbers such that

_∞

n=δn<∞,

_∞

n=εn<∞,

andxn+–p ≤( +δn)xn–p+εnfor alln∈Nandp∈F.

Proposition .(see []) Let F be a nonempty subset of a Banach space X and let{xn}be a

sequence in X.If{xn}is of monotone type(I)with respect to F andlim infn→∞dist(xn,F) = ,

thenlimn→∞xn=p for some p∈X satisfyingdist(p,F) = .In particular,if F is closed,then

p∈F.

Lemma .(see []) Let{an},{bn},and{cn}be sequences of nonnegative real numbers

satisfy

an+≤( +cn)an+bn, for all n∈N,

where∞_n_=bn<∞and

_∞

n=cn<∞.Then:

(i) limn→∞anexists.

(ii) Iflim infn→∞an= ,thenlimn→∞an= .

Lemma .(see []) Let X be a uniformly convex Banach space,let{λn}be a sequence of

real numbers such that <a≤λn≤b< ,for all n∈N,and let{xn}and{yn}be sequences

of X satisfying,for some r≥, (i) lim sup_n_→∞xn ≤r,

(ii) lim sup_n_→∞yn ≤rand

(iii) limn→∞λnxn+ ( –λn)yn=r.

Thenlimn→∞xn–yn= .

Lemma .(see []) Let X be a Banach space which satisﬁes the Opial property and{xn}

be a sequence in X.Let u,v∈X be such thatlimn→∞xn–uandlimn→∞xn–vexist.If

{xni}and{xnj}are subsequences of{xn}which converge weakly to u and v,respectively,then u=v.

3 Main results

In this section, we prove weak and strong convergence theorems of the proposed iterative process in Banach spaces. We ﬁrst note that if{ti}Ni=is a ﬁnite family of generalized asymp-totically nonexpansive single-valued mappings ofDinto itself, whereDis a nonempty con-vex subset of a Banach spaceX. Then we havetn

ix–tniy ≤k

(i)

nx–y+sn(i), for allx,y∈D

and all i= , , . . . ,N, where{kn(i)} ⊂[,∞) and {sn(i)} ⊂[,∞) with limn→∞kn(i)=  and

lim_n_→∞s(ni)= . Putkn=max≤i≤N{kn(i)}andsn=max≤i≤N{sn(i)}. It is clear thatlimn→∞kn=

 andlimn→∞sn=  and

tn_ix–t_iny≤knx–y+sn

for allx,y∈D,i= , , . . . ,N, and alln∈N.

(5)

Lemma . Let D be a nonempty,closed,and convex subset of a Banach space X.Let{ti}Ni= be a ﬁnite family of generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences{kn} ⊂[,∞)and{sn} ⊂[,∞)such that

∞

n=(kn– ) <∞and

∞

n=sn<∞.Let{Ti}Ni= be a ﬁnite family of quasi-nonexpansive multi-valued mappings of D into CB(D).Assume thatF=N_i_=F(ti)∩

N

i=F(Ti)is nonempty closed and Tip={p}

for all p∈Fand i= , , . . . ,N.Let x∈D and the sequence{xn}be generated by

yn=βn()xn+

N

i=β (i)

nzn(i), zn(i)∈Tixn,

xn+=αn()xn+

N

i=α (i)

ntinyn, n∈N,

where{α(ni)}and{βn(i)}are sequences in[, ]for all i= , , . . . ,N such that

N

i=α (i)

n = and

N

i=β (i)

n = .Thenlimn→∞xn–pexists for all p∈F.

Proof Letp∈F, fori= , , . . . ,N, we have

xn+–p ≤αn()xn–p+ N

i=

α(_ni)t_inyn–p

≤α_n()xn–p+ N

i=

α(_ni)knyn–p+sn

=α_n()xn–p+kn N

i=

α(_ni)yn–p+sn N

i= α(_ni)

≤α_n()xn–p+kn N

i=

α_n(i)yn–p+sn

≤α_n()xn–p+kn N

i= α_n(i)

β_n()xn–p+ N

i=

β_n(i)z(_ni)–p

+sn

=

α_n()+knβn() N

i= α_n(i)

xn–p+kn N

i= α_n(i)

N

i=

β_n(i)z_n(i)–p+sn

=

α_n()+knβn() N

i= α_n(i)

xn–p+kn N

i= α_n(i)

N

i=

β_n(i)distz(_ni),Tip

+sn

≤

α()_n +knβn() N

i= α_n(i)

xn–p+kn N

i= α_n(i)

N

i=

β_n(i)H(Tixn,Tip) +sn

≤

α()_n +knβn() N

i= α_n(i)

xn–p+kn N

i= α_n(i)

N

i=

β_n(i)xn–p+sn

=

α_n()+kn N

i= α(_ni)

xn–p+sn

≤knxn–p+sn

= + (kn– )

xn–p+sn.

By Lemma .,∞_n_=(kn– ) <∞and

∞

n=sn<∞, we conclude thatlimn→∞xn–p

(6)

Theorem . Let D be a nonempty,closed,and convex subset of a Banach space X.Let {ti}Ni=be a ﬁnite family of generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences{kn} ⊂[,∞)and{sn} ⊂[,∞)such that

∞

n=(kn– ) <∞

and∞_n_=sn<∞.Let{Ti}Ni=be a ﬁnite family of quasi-nonexpansive multi-valued map-pings of D into CB(D).Assume thatF=N_i_=F(ti)∩

N

i=F(Ti)is nonempty closed and

Tip={p}for all p∈Fand i= , , . . . ,N.Let x∈D and the sequence{xn}be generated by

yn=βn()xn+

N

i=β (i)

xn+=αn()xn+

N

i=α (i)

ntinyn, n∈N,

where{αn(i)} and{βn(i)} are sequences in[, ]for all i= , , . . . ,N such thatNi=α (i)

n = 

andN_i_=βn(i)= .Then the sequence{xn}converges strongly to a point inFif and only if

lim infn→∞dist(xn,F) = .

Proof The necessity is obvious and thus we prove only the suﬃciency. Suppose that

lim infn→∞dist(xn,F) = . In the proof of Lemma ., we see that the sequence {xn}is

of monotone type (I) with respect toF. It follows by Proposition . that{xn}converges

to a point inF.

The closedness ofF=N_i_=F(ti)∩

N

i=F(Ti) can be dropped iftiis asymptotically

non-expansive for alli= , , . . . ,N. Then the following corollary is obtained directly from The-orem ..

Corollary . Let D be a nonempty, closed, and convex subset of a Banach space X. Let{ti}Ni= be a ﬁnite family of asymptotically nonexpansive single-valued mappings of D into itself with a sequence{kn} ⊂[,∞)such that

∞

n=(kn– ) <∞. Let{Ti}Ni= be a ﬁ-nite family of quasi-nonexpansive multi-valued mappings of D into CB(D).Assume that

F=N_i_=F(ti)∩

N

i=F(Ti)is nonempty and Tip={p}for all p∈Fand i= , , . . . ,N.Let

x∈D and the sequence{xn}be generated by

yn=βn()xn+

N

i=β (i)

xn+=αn()xn+

N

i=α (i)

ntinyn, n∈N,

where{αn(i)} and{βn(i)} are sequences in[, ]for all i= , , . . . ,N such that

N

i=α (i)

n = 

andN_i_=βn(i)= .Then the sequence{xn}converges strongly to a point inFif and only if

lim infn→∞dist(xn,F) = .

Recall that a mappingt:D→Dis calleduniformly L-Lipschitzianif there exists a con-stantL>  such thattnx–tny ≤Lx–yfor allx,y∈Dandn∈N. Next, we prove a strong convergence theorem in a uniformly convex Banach space.

Lemma . Let D be a nonempty,closed,and convex subset of a uniformly convex Banach

space X.Let{ti}Ni=be a ﬁnite family of uniformly L-Lipschitzian and generalized asymptot-ically nonexpansive single-valued mappings of D into itself with sequences{kn} ⊂[,∞)

and {sn} ⊂[,∞) such that

∞

n=(kn– ) <∞ and

∞

(7)

F=N_i_=F(ti)∩

N

i=F(Ti)is nonempty and Tip={p}for all p∈Fand i= , , . . . ,N.Let

x∈D and the sequence{xn}be generated by

yn=βn()xn+

N

i=β (i)

xn+=αn()xn+Ni=α (i)

ntinyn, n∈N,

where{αn(i)}and{βn(i)}are sequences in[, ]for all i= , , . . . ,N such that <a≤αn(i),βn(i)≤

b< ,N_i_=αn(i)= ,and

N

i=β (i)

n = .Then we have the following:

(i) limn→∞xn–zn(i)= for alli= , , . . . ,N;

(ii) limn→∞xn–tixn= for alli= , , . . . ,N.

Proof (i) By Lemma .,lim_n_→∞xn–pexists. Putlimn→∞xn–p=c. By the deﬁnition

of{xn}, we have

tn_iyn–p≤knyn–p+sn

≤kn

β_n()xn–p+ N

i=

β_n(i)z(_ni)–p

+sn

=knβn()xn–p+kn N

i=

β_n(i)z_n(i)–p+sn

=knβn()xn–p+kn N

i=

β_n(i)distz(_ni),Tip

+sn

≤knβn()xn–p+kn N

i=

β_n(i)H(Tixn,Tip) +sn

≤knβn()xn–p+kn N

i=

β_n(i)xn–p+sn

=kn

β_n()+

N

i= β_n(i)

xn–p+sn

=knxn–p+sn.

Then we have

lim sup

n→∞

tn_iyn–p≤lim sup n→∞

knyn–p+sn

≤lim sup

n→∞

knxn–p+sn

.

Bylimn→∞kn=  andlimn→∞sn= , we have

lim sup

n→∞

tn_iyn–p≤lim sup

n→∞ yn–p ≤lim supn→∞ xn–p=c. (.) Since c=limn→∞xn+–p=limn→∞α()n (xn–p) +

N

i=α (i)

n(tniyn–p), it follows by

Lemma . that

lim

n→∞xn–t n

(8)

Consider

xn+–p ≤αn()xn–p+ N

i=

=

 –

N

i= α(_ni)

xn–p+ N

i=

≤

 –

N

i= α_n(i)

xn–p+ N

i=

α_n(i)knyn–p+sn

.

This implies that

xn+–p–xn–p ≤ N

i=

α_n(i)knyn–p–xn–p+sn

.

Therefore,

xn+–p–xn–p

bN +xn–p ≤

xn+–p–xn–p

N

i=α (i)

n

+xn–p

≤knyn–p+sn.

By (.), we obtain

c=lim inf

n→∞

_x

n+–p–xn–p

bN +xn–p

≤lim inf

n→∞

knyn–p+sn

=lim inf

n→∞ yn–p ≤lim sup

n→∞ yn–p ≤c.

Thus,

c= lim

n→∞yn–p=nlim→∞

βn()(xn–p) +

N

i=

β_n(i)z(_ni)–p

.

Since

z(_ni)–p=distz(_ni),Tip

≤H(Tixn,Tip)≤ xn–p,

it implies that

lim sup

n→∞

z(_ni)–p≤lim sup

n→∞ xn–p=c.

Hence, by Lemma ., we have

lim

n→∞xn–z

(i)

(9)

(ii) Sincetiis generalized asymptotically nonexpansive, for alli= , , . . . ,N, we get

tn_ixn–xn≤tinxn–tinyn+tniyn–xn≤knxn–yn+sn+tniyn–xn.

By the deﬁnition of{xn}, we haveyn–xn=

N

i=β (i)

n (z(ni)–xn). This implies that

tn_ixn–xn≤kn

N

i=

β_n(i)z(_ni)–xn+tinyn–xn+sn

≤knz(ni)–xn+tniyn–xn+sn.

Then, by (i) and (.), we get

lim

n→∞xn–t n

ixn=  for alli= , , . . . ,N. (.)

Fori= , , . . . ,N, we have

xn–tixn ≤ xn–xn++xn+–tni+xn++tin+xn+–tni+xn+tin+xn–tixn

≤( +L)xn–xn++xn+–tin+xn++Ltnixn–xn

≤( +L)

N

i=

α_n(i)xn–tniyn+xn+–tni+xn++Ltnixn–xn.

By (.) and (.), we conclude thatlimn→∞xn–tixn=  for alli= , , . . . ,N. Theorem . Let D be a nonempty,compact,and convex subset of a uniformly convex Banach space X.Let{ti}Ni= be a ﬁnite family of uniformly L-Lipschitzian and generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences{kn} ⊂

[,∞)and{sn} ⊂[,∞)such that

∞

n=(kn– ) <∞and

∞

n=sn<∞.Let{Ti}Ni=be a ﬁnite family of quasi-nonexpansive multi-valued mappings of D into CB(D)satisfying condition (E).Assume thatF=N_i_=F(ti)∩

N

i=F(Ti)is nonempty and Tip={p}for all p∈Fand

i= , , . . . ,N.Let x∈D and the sequence{xn}be generated by

yn=βn()xn+Ni=β (i)

xn+=αn()xn+

N

i=α (i)

ntinyn, n∈N,

b< ,N_i_=αn(i)= ,and

N

i=β (i)

n = .Then the sequence{xn}converges strongly to a point

inF.

Proof By Lemma ., we have{xn}is bounded. SinceDis compact, there exists a

subse-quence{xnj}of{xn}converging strongly top∈D. By condition (E), there existsμ≥ such that fori= , , . . . ,N,

dist(p,Tip)≤ p–xnj+dist(xnj,Tip)

(10)

= xnj–p+μdist(xnj,Tixnj) ≤xnj–p+μxnj–z

(i)

nj.

Then, by Lemma .(i), we havep∈Tipfor alli= , , . . . ,N. Sop∈

N

i=F(Ti).

Sincetiis uniformlyL-Lipschitzian, for alli= , , . . . ,N, we have

tip–p ≤ tip–tixnj+tixnj–xnj+xnj–p ≤(L+ )xnj–p+tixnj–xnj.

By Lemma .(ii), it implies thattip=pfor alli= , , . . . ,N. Thus,p∈

N

i=F(ti).

There-fore,p∈F. Sincelimn→∞xn–pexists, we getlimn→∞xn–p=limj→∞xnj–p= . This shows that{xn}converges strongly to a point inF.

Next, we give a numerical example to support Theorem ..

Example . LetRbe the real line with the usual norm| · |and letD= [, ]. Deﬁne two single-valued mappingstandtonDas follows:

tx=sinx, tx=x.

Also we deﬁne two multi-valued mappingsTandTonDas follows:

Tx=

[,x_], x= ;

{}, x= ; Tx=

x ,

x 

.

Let{xn}and{yn}be generated by

yn=βn()xn+i=β (i)

xn+=αn()xn+



i=α (i)

ntinyn, n∈N,

(.)

where αn() = _n+_n, αn()= n_–_n , α()n = _n–_n, βn() = _n+_n , βn()= _n–_n, βn()= _n–_n , for all

n∈N. Then the sequences{xn}and{yn}converge strongly to , where{}=



i=F(ti)∩



i=F(Ti).

Solution It is shown in[]that both tand tare generalized asymptotically nonexpan-sive single-valued mappings.Moreover,they are uniformly L-Lipschitzian mappings and



i=F(ti) ={}.It is easy to see that both Tand Tare quasi-nonexpansive multi-valued mappings satisfying condition(E)and_i_=F(Ti) ={}.Thus,



i=F(ti)∩



i=F(Ti) ={}.

For every n∈N,αn()=_n+_n,αn()=n__n–,α()n =_n–_n,βn()=_n+_n ,βn()= _n–_n,βn()=_n–_n .

Then the sequences{αn()},{αn()},{α()n },{βn()},{βn()},and{βn()}satisfy all the conditions of

Theorem..Put zn()=x_n and zn()=x_n for all n∈N.Then the algorithm(.)becomes

yn= (_+__n)xn,

xn+= (_+__n–__n)xn+ (n__n–)tnyn, n∈N.

(.)

(11)

Table 1 The values of the sequences{xn}and{yn}in Example 3.6

n xn yn

1 2.5000000 1.5277778 2 2.1025927 1.2119111 3 1.5352877 0.8671533 4 1.0799923 0.6037457 5 0.7544605 0.4191447 .

. .

. . .

. . . 21 0.0023377 0.0012740 .

. .

. . .

. . . 38 0.0000040 0.0000022 39 0.0000027 0.0000015 40 0.0000019 0.0000010 41 0.0000013 0.0000007 42 0.0000009 0.0000005

Finally, we prove a weak convergence theorem in uniformly convex Banach spaces.

Theorem . Let D be a nonempty, closed, and convex subset of a uniformly convex Banach space X with the Opial property. Let {ti}Ni= be a ﬁnite family of uniformly L-Lipschitzian and generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences{kn} ⊂[,∞)and{sn} ⊂[,∞)such that

∞

n=(kn– ) <∞and

∞

n=sn<∞.Let{Ti}Ni= be a ﬁnite family of quasi-nonexpansive multi-valued mappings of D into KC(D)satisfying the condition(E).Assume thatF=N_i_=F(ti)∩

N

i=F(Ti)is

nonempty and Tip={p}for all p∈Fand i= , , . . . ,N.Let x∈D and the sequence{xn}

be generated by

yn=βn()xn+

N

i=β (i)

xn+=αn()xn+

N

i=α (i)

ntinyn, n∈N,

b< ,N_i_=αn(i)= ,and

N

i=β (i)

n = .Then the sequence{xn}converges weakly to a point

inF.

Proof By Lemma .,{xn}is bounded. SinceXis uniformly convex, there exists a

subse-quence{xnj}of{xn}converging weakly top∈D. By Lemma ., we havelimj→∞xnj– z(nij)=  andlimj→∞xnj–tixnj=  for alli= , , . . . ,N. We will show thatp∈F. Since Tpis compact, for allj∈N, we can choosewnj∈Tpsuch thatxnj–wnj=dist(xnj,Tp) and the sequence{wnj}has a convergent subsequence{wnk}withlimk→∞wnk=w∈Tp. By condition (E), we have

dist(xnk,Tp)≤μdist(xnk,Txnk) +xnk–p. Then we have

xnk–w ≤ xnk–wnk+wnk–w

=dist(xnk,Tp) +wnk–w

≤μdist(xnk,Txnk) +xnk–p+wnk–w ≤μxnk–z

(i)

(12)

This implies that

lim sup

k→∞ xnk–w ≤

lim sup

k→∞ xnk–p.

From the Opial property, we havep=w∈Tp. Similarly, it can be shown thatp∈Tipfor

alli= , . . . ,N. Thus,p∈N_i_=F(Ti).

Next, by mathematical induction, we can prove that, fori= , , . . . ,N,

lim

j→∞

xnj–t

m

i xnj=  for eachm∈N. (.)

Indeed, it is obvious that the conclusion it true form= . Suppose the conclusion holds form≥. Sincetiis uniformlyL-Lipschitzian, we have

xnj–t

m+

i xnj≤xnj–t

m

i xnj+t

m i xnj–t

m+

i xnj ≤xnj–t

m

i xnj+Lxnj–tixnj.

This shows thatlimj→∞xnj–t

m+

i xnj=  for alli= , , . . . ,N. Hence, (.) holds. From (.), we have for eachx∈D,m∈Nandi= , , . . . ,N,

lim sup

j→∞ xnj–x=lim supj→∞

t_imxnj–x. (.)

Sincetiis generalized asymptotically nonexpansive, we get

lim sup

j→∞

tm_i xn–timp≤lim sup j→∞

kmxnj–p+sm

.

Then we have

lim sup

m→∞

lim sup

j→∞

tm_i xnj–t

m

i p≤lim sup j→∞

xnj–p. (.)

By Proposition ., we have

xnj–

p+tm_i p 

=

(xnj–p) +  

xnj–t

m i p



≤

xnj–p ₊

xnj–t

m i p

 –

gp–t

m i p.

It implies that

lim sup

j→∞

xnj–

p+tm i p



≤

lim supj→∞ xnj–p ₊

lim supj→∞

_x_n

j–t

m i p



–

gp–t

m

i p. (.)

By the Opial property and{xnj}converging weakly top, we obtain

lim sup

j→∞ xnj–p

_≤_{lim sup}

j→∞

xnj–

p+tm i p



(13)

Then, by (.), we have

gp–tmp≤lim sup

j→∞

xnj–t

m i p



– lim sup

j→∞ xnj–p

_. _(.)

It implies by (.), (.), and (.) that

lim sup

m→∞

gp–t_imp≤lim sup

m→∞

lim sup

j→∞

xnj–t

m i p



– lim sup

j→∞ xnj–p 

≤.

This shows thatlimm→∞g(p–tmi p) =  for alli= , , . . . ,N. Then the properties ofg

yieldlim_m_→∞p–tm

i p=  for alli= , , . . . ,N. So we have

tip–p ≤tip–tim+p+tim+p–p

≤Lp–t_imp+t_im+p–p→ asm→ ∞.

This implies thattip=pfor alli= , , . . . ,N. Thus,p∈

N

i=F(ti).

Hence, we obtainp∈F.

Finally, we show that{xn}converges weakly top. To show this, suppose not. Then there

exists a subsequence{xnl}of{xn}such that{xnl}converges weakly toq∈Dandq=p. By the same method as given above, we can prove thatq∈F. By Lemma .,limn→∞xn–p

andlimn→∞xn–qexist. It follows by Lemma . thatq=p. Thus,{xn}converges weakly

to a point inF.

Remark . Theorem . extends and generalizes the results of Sokhuma and Kaewkhao [] to a pair of a finite family of generalized asymptotically nonexpansive single-valued mappings and a finite family of quasi-nonexpansive multi-valued mappings satisfying con-dition (E). Theorems . and . extend and generalize the results of Eslamian [] and Eslamian and Abkar [] to a pair of a finite family of generalized asymptotically nonex-pansive single-valued mappings and a finite family of quasi-nonexnonex-pansive multi-valued mappings satisfying condition (E).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand.}2_{Department of} Mathematics, Faculty of Science and Technology, Nakhon Pathom Rajabhat University, Nakhon Pathom, 73000, Thailand. 3_{Research Center for Pure and Applied Mathematics, Research and Development Institute, Nakhon Pathom Rajabhat} University, Nakhon Pathom, 73000, Thailand.

Acknowledgements

This paper was supported by the Thailand Research Fund under the project RTA5780007.

Received: 21 January 2015 Accepted: 1 April 2015 References

(14)

2. Saejung, S, Suantai, S, Yotkaew, P: A note on “Common Fixed Point of Multistep Noor Iteration with Errors for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mapping”. Abstr. Appl. Anal.2009, Article ID 283461 (2009) 3. Markin, JT: Continuous dependence of ﬁxed point sets. Proc. Am. Math. Soc.38, 545-547 (1973)

4. Sastry, KPR, Babu, GVR: Convergence of Ishikawa iterates for a multivalued mapping with a ﬁxed point. Czechoslov. Math. J.55, 817-826 (2005)

5. Panyanak, B: Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Comput. Math. Appl. 54, 872-877 (2007)

6. Shahzad, N, Zegeye, H: On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces. Nonlinear Anal.71, 838-844 (2009)

7. Khan, SH, Yildirim, I: Fixed points of multivalued nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2012, Article ID 73 (2012)

8. Khan, SH, Abbas, M, Ali, S: Fixed points of multivalued quasi-nonexpansive mappings using a faster iterative process. Acta Math. Sin. Engl. Ser.30(7), 1231-1241 (2014)

9. Garcia-Falset, J, Llorens-Fuster, E, Suzuki, T: Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl.375, 185-195 (2011)

10. Abkar, A, Eslamian, M: Common ﬁxed point results in CAT(0) spaces. Nonlinear Anal.74, 1835-1840 (2011) 11. Sokhuma, K, Kaewkhao, A: Ishikawa iterative process for a pair of single-valued and multivalued nonexpansive

mappings in Banach spaces. Fixed Point Theory Appl.2011, Article ID 618767 (2011)

12. Eslamian, M: Weak and strong convergence theorems of iterative process for two ﬁnite families of mappings. Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys.75(4), 81-90 (2013)

13. Eslamian, M, Abkar, A: One-step iterative process for a ﬁnite family of multivalued mappings. Math. Comput. Model. 54, 105-111 (2011)

14. Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal.16, 1127-1138 (1991)

15. Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc.73, 595-597 (1967)

16. Xu, HK: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc.65, 109-113 (2002)

17. Schu, J: Weak and strong convergence of ﬁxed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc.43, 153-159 (1991)

18. Suantai, S: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings. J. Math. Anal. Appl.311, 506-517 (2005)