R E S E A R C H
Open Access
Convergence theorems of a new iteration for
asymptotically nonexpansive mappings in
Banach spaces
Jong Kyu Kim
*and Won Hee Lim
*Correspondence:
[email protected] Department of Mathematics Education, Kyungnam University, Changwon, Gyeongnam 631-701, Korea
Abstract
In this paper, the problem of modified iterative approximation of common fixed points of asymptotically nonexpansive is investigated in the framework of Banach spaces. Weak convergence theorems are established.
MSC: 47H09; 47J05; 47J25; 47H25
Keywords: asymptotically nonexpansive mapping; common fixed point; implicit Ishikawa-type iterative process
1 Introduction
Fixed-point theory as an important branch of nonlinear analysis has been applied in the study of nonlinear phenomena. The theory itself is a beautiful mixture of analysis, topol-ogy, and geometry. Recently, iterative algorithms for finding common fixed points of non-linear mappings have been considered by many authors. The well-known convex feasi-bility problem capture application in various disciplines such as image restorations, and radiation therapy treatment planning is to find a point in the intersection of common fixed-point sets of nonlinear mappings (see, [–]).
From the method of generating iterative sequence, we can divide iterative algorithms into explicit algorithms. Recently, both explicit Mann iterative algorithms and implicit Mann-iterative algorithms have been extensively studied for approximating common fixed points of nonlinear mappings (see [–]).
In this paper, we consider the problem of approximating a common fixed point of asymp-totically nonexpansive mappings based on a general implicit iterative algorithm, which includes an explicit process as a special case. The organization of this paper is as follows. In Section , we provide some necessary preliminaries. In Section , weak convergence theorems are established in a uniformly convex Banach space.
2 Preliminaries
LetEbe a real Banach space.Eis said to beuniformly convexif for any two sequences{xn}
and{yn}inEsuch thatxn=yn= andlimn→∞xn+yn= , thenlimn→∞xn–yn=
holds. It is known that a uniformly convex Banach space is reflexive.
In this paper, we use the symbolsand→denote weak convergence and strong con-vergence, respectively.Eis said to haveOpial’s condition(see []) if, for each sequence
{xn}inE,xnximplies that
lim inf
n→∞ xn–x<lim infn→∞ xn–y, ∀y∈E(y=x).
LetCbe a nonempty subset ofE, andT:C→Ca mapping. In this paper, the symbol
F(T) stands for the fixed point set ofT.T is said to benonexpansiveif
Tx–Ty ≤ x–y, ∀x,y∈C.
Tis said to beasymptotically nonexpansiveif there exists a sequence{kn} ⊂[,∞) with
kn→ asn→ ∞such that
Tnx–Tny≤knx–y, ∀x,y∈C,∀n≥.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [] as a generalization of the class of nonexpansive mappings. They proved that ifCis a nonempty, closed, convex, and bounded subset of a real uniformly convex Banach space, then every asymptotically nonexpansive self mapping has a fixed point (see []).
In order to prove our main results, we still need the following lemmas.
Lemma .[] Let C be a nonempty,closed,and convex subset of a uniformly convex Banach space E.Let T:C→C be an asymptotically nonexpansive mapping.Then I–T is demiclosed at zero,that is,xnx and xn–Txn→imply that x=Tx.
Lemma .[] Let{an},{bn},and{cn}be three nonnegative sequences satisfying the
fol-lowing condition:
an+≤( +bn)an+cn, ∀n≥n, where nis some nonnegative integer,
∞
n=bn<∞and
∞
n=cn<∞.Then thelimn→∞an
exists.
Lemma .[] Let E be a uniformly convex Banach space,r> a positive number and Br()a closed ball of E with the center at zero. Then there exists a continuous,strictly
increasing,and convex function g: [,∞)→[,∞)with g() = such that
m
s=
(αsxs)
≤
m
s=
αsxs
–αiαjg
xi–xj
, ∀i,j∈ {, , . . . ,r},
where x,x, . . . ,xm∈Br(),andα,α, . . . ,αm∈(, )with
m i=αi= .
3 Main Results
Before starting the main results in this paper, we give the implicit iterative process first. LetCbe a nonempty, closed, and convex subset of a Banach spaceE. LetT:C→Cbe an asymptotically nonexpansive mapping with the sequence{kn}. For every u∈Cand
tn∈(, ), Define a mappingTn:C→Cbelow
If ( –tn)kn< , for everyn≥, thenTnis a contraction. In the light of the Banach
con-traction principle, we see that there exists a unique fixed point ofTn, for everyn≥.
Letxbe chosen arbitrarily andr≥ a positive integer. Let{αn},{βn,},{βn,}, . . . ,{βn,r},
{γn,},{γn,}, . . . , and {γn,r},{an},{bn,},{bn,}, . . . ,{bn,r},{cn,},{cn,}, . . . , and {cn,r} be real
number sequences in (, ) such that
αn+ r
m= βn,m+
r
m=
γn,m=an+ r
m= bn,m+
r
m=
cn,m= .
LetSm,Tm:C→Cbe asymptotically nonexpansive mappings, for everym∈ {, , . . . ,r}.
Findx,yby solving the following equations:
x=αx+
r
m=
β,mSmx+
r
m=
γ,mTmy,
y=ax+
r
m=
b,mSmx+
r
m=
c,mTmx.
Findx,yby solving the following equations:
x=αx+
r
m=
β,mSmx+
r
m=
γ,mTmy,
y=ax+
r
m=
b,mSmx+
r
m=
c,mTmx,
.. .
Findxn,ynby solving the following equations:
xn=αnxn–+
r
m=
βn,mSnmxn–+
r
m=
γn,mTmnyn,
yn=anxn+ r
m=
bn,mSmnxn+ r
m=
cn,mTmnxn.
In view of the above, we have the following implicit iterative algorithm:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x∈C,
xn=αnxn–+ r
m=βn,mSnmxn–+ r
m=γn,mTmnyn,
yn=anxn+
r
m=bn,mSnmxn+
r
m=cn,mTmnxn, ∀n≥.
(ϒ)
Now we show that (ϒ) can be employed to approximate fixed points of asymptotically nonexpansive mapplings, which are assumed to be Lipschitz continuous. LetSm:C→C
be an asymptotically nonexpansive mapping with the sequence{sn,m}, andTm:C→Can
wherer≥ is some positive integer. Define a mappingCn:C→Cby
Cn(x) =αnxn–+
r
m=
βn,mSmnxn–
+
m
n
γn,mTmn
anx+
r
m=
bn,mSmnx+ n
m
cn,mTmnx
, ∀n≥.
It follows that
Cn(x) –Cn(y)
≤
r
m= γn,mtn
an(x–y) + r
m= bn,m
Snmx–Snmy+
r
m= cn,m
Tmnx–Tmny ≤ r m= γn,mtn
an+
r
m=
bn,msn+ r
m= cn,mtn
x–y, ∀x,y∈C,
wheretn=max{tn,m: ≤m≤r}andsn=max{sn,m: ≤m≤r}.
Ifrm=γn,mtn(an+
r
m=bn,msn+
r
m=cn,mtn) < for all ≤m≤r,n≥, thenCnis a
contraction. Hence, by the Banach contraction principle, there exists a unique fixed point
xn∈Csuch that
xn=Cn(x) =αnxn–+
r
m=
βn,mSmnxn–
+
m
n
γn,mTmn
anx+
r
m=
bn,mSnmx+ n
m
cn,mTmnx
, ∀n≥.
That is, the implicit iterative algorithm (ϒ) is well defined. Now, we are in a position to give our main results.
Theorem . Let C be a nonempty,closed,and convex subset of a uniformly convex Banach space E.Let Sm:C→C be an asymptotically nonexpansive mapping with the sequence
{sn,m},and Tm:C→C an asymptotically nonexpansive mapping with the sequence{tn,m},
for every m∈ {, , . . . ,r},where r≥is some positive integer.Assume that
F =
r
m= F(Sm)∩
r
m=
F(Tm)=∅.
Let tn=max{tn,m: ≤m≤r}and sn=max{sn,m: ≤m≤r}.Assume that
∞
n=(kn– ) <
∞, where kn =max{sn,tn : ≤ m≤ r}. Let {xn}∞n= be a sequence generated by (ϒ), where {αn},{βn,},{βn,}, . . . ,{βn,r},{γn,},{γn,}, . . . ,{γn,r},{an},{bn,},{bn,}, . . . ,{bn,r},{cn,},
{cn,}, . . . ,{cn,r}be real number sequences in(, )such thatαn+
r
m=βn,m+
r
m=γn,m=
an+
r
m=bn,m +
r
m=cn,m= . Assume that the following restrictions imposed on the
control sequence {αn},{βn,},{βn,}, . . . ,{βn,r},{γn,},{γn,}, . . . ,{γn,r},{an},{bn,},{bn,}, . . . ,
{bn,r},{cn,},{cn,}, . . . ,{cn,r}are satisfied
(a) lim infn→∞αnβn,m> ,lim infn→∞αnγn,m> andlim infn→∞anbn,m> ,
(b) rm=γn,mtn(an+
r
m=bn,msn+
r
m=cn,mtn) < .
Then
lim
n→∞xn–Smxn=nlim→∞xn–Tmxn= , ∀m∈ {, , . . . ,r}. Proof Step . Takingp∈F, we see that
xn–p ≤αnxn––p+
r
m=
βn,mSnmxn––p+
r
m=
γn,mTmnyn–p
≤
αn+
r
m= βn,mkn
xn––p+
r
m=
γn,mknyn–p (.)
and
yn–p ≤anxn–p+ r
m=
bn,mSmnxn–p+ r
m=
cn,mTmnxn–p
≤anknxn–p+ r
m=
bn,mknxn–p+ r
m=
cn,mknxn–p
≤knxn–p. (.)
Substituting (.) into (.), we have
xn–p ≤
αn+
r
m= βn,mkn
xn––p+
r
m=
γn,mknxn–p. (.)
In view oflim infn→∞αnβn,m> , andαn+
r
m=βn,m+
r
m=γn,m= , we see that there
exists some positive integern, and a real numberh, whereh∈(, ), such that
r
m=
γn,m≤h, n≥n.
Since∞n=(kn– ) <∞, we find that there exists some positive integernsuch thatkn≤
+–hh,∀n≥n. It follows that
r
m=
γn,mkn≤u< , ∀n≥n,
whereu=h( +–h
h), andn≥max{n,n}. It follows (.) that
xn–p ≤
αn+
r
m=βn,mkn
–rm=γn,mkn
xn––p
≤
+αn+
r
m=βn,mkn+
r
m=γn,mkn–
–rm=γn,mkn
xn––p
≤
+k
n–
–u
It follows from Lemma . thatlimn→∞xn–pexists. This implies that the sequence{xn}
is bounded.
On the other hand, we find from Lemma . that
xn–p
≤αnxn––p+
r
m=
βn,mSnmxn––p
+
r
m=
γn,mTmnyn–p
–αnβn,mPxn––Snmxn–
≤
αn+
r
m= βn,mkn
xn––p+
r
m=
γn,mknxn–p
–αnβn,mPxn––Smnxn–, ∀m∈ {, , . . . ,r}.
It implies that
αnβn,mPxn––Smnxn–
≤
αnkn+
r
m= βn,mkn
xn––p+
r
m=
γn,mknxn–p
–k
nxn–p+
k
n–
xn–p
≤
αnkn+
r
m= βn,mkn
xn––p–xn–p
+kn– xn–p, ∀m∈ {, , . . . ,r}.
Sincelimn→∞xn–pexists, we find from restriction (a) that
lim
n→∞Pxn––S n
mxn–= ,
for everym∈ {, , . . . ,r}. It follows that
lim
n→∞xn––S
n
mxn–= , ∀m∈ {, , . . . ,r}. (.)
In view of Lemma ., we have
xn–p≤αnxn––p+
r
m=
βn,mSnmxn––p
+
r
m=
γn,mTmnyn–p
–αnγn,mPxn––Tmnyn
≤
αn+
r
m= βn,mkn
xn––p+
r
m=
γn,mknxn–p
It implies that
αnγn,mPxn––Tmnyn
≤
αnkn+
r
m= βn,mkn
xn––p+
r
m=
γn,mknxn–p
–knxn–p+
kn– xn–p
≤
αnkn+
r
m= βn,mkn
xn––p–xn–p
+kn– xn–p, ∀m∈ {, , . . . ,r}.
Sincelimn→∞xn–pexists, from the condition (a), we have that
lim
n→∞Pxn––T
n
myn= ,
for everym∈ {, , . . . ,r}. It follows that
lim
n→∞xn––T n
myn= , ∀m∈ {, , . . . ,r}. (.)
Notice that
xn–xn– ≤
r
m=
βn,mSnmxn––xn–+
r
m=
γn,mTmnyn–xn–
In the light of (.), and (.), we find that
lim
n→∞xn––xn= . (.)
Step . Notice that
xn–Tmnyn≤ xn–xn–+xn––Tmnyn, ∀m∈ {, , . . . ,r}.
It implies from (.), and (.) that
lim
n→∞xn–T
n
myn= , ∀m∈ {, , . . . ,r}. (.)
On the other hand, we have
xn–Smnxn≤ xn–xn–+xn––Snmxn–
+Smnxn––Smnxn, ∀m∈ {, , . . . ,r}.
SinceSmis Lipschitz for everym∈ {, , . . . ,r}, we see from (.) and (.) that
lim
n→∞xn–S n
Step . In view oflim infn→∞anbn,m> , and
an+ r
m= bn,m+
r
m= cn,m= ,
we see that there exists some positive integern, and a real numberh, whereh∈(, ),
such that
r
m=
cn,m≤h, n≥n.
Since∞n=(kn– ) <∞, we find that there exists a positive integernsuch thatkn≤ +–h
h,
∀n≥n. It follows that
r
m=
cn,mkn≤u< , ∀n≥n,
whereu=h( +–hh), andn≥max{n,n}. It follows that
xn–yn ≤ r
m=
bn,mSmnxn–xn+ r
m=
cn,mTmnxn–xn
≤
r
m=
bn,mSmnxn–xn+ r
m=
cn,mTmnxn–Tmnyn+ r
m=
cn,mTmnyn–xn
≤
r
m=
bn,mSmnxn–xn+ r
m=
cn,mknxn–yn+ r
m=
cn,mTmnyn–xn.
This implies that
–
r
m= cn,mkn
xn–yn
≤
r
m=
bn,mSmnxn–xn+ r
m=
cn,mTmnyn–xn.
It follows from (.) and (.) that
lim
n→∞xn–yn= . (.)
Step . Notice that
yn–yn– ≤anxn–xn–+
r
m=
bn,mSmnxn–xn–
+
r
m=
cn,mTmnxn–xn–+xn––yn–
≤anxn–xn–+
r
m=
+
r
m=
bn,mxn–xn–+
r
m=
cn,mTmnxn–Tmnyn
+
r
m=
cn,mTmnyn–xn–+xn––yn–.
SinceTmis Lipschitz for everym∈ {, , . . . ,r}, we see from (.), (.), (.), and (.)
that
lim
n→∞yn–yn–= . (.)
Step . Notice that
xn–Smxn ≤ xn–xn++xn+–Snm+xn+
+Smn+xn+–Snm+xn+Snm+xn–Smxn
≤( +M)xn–xn++xn+–Snm+xn+
+MSmnxn–xn,
whereM=supn≥{kn}. It follows from (.) and (.) that
lim
n→∞xn–Smxn= , ∀m∈ {, , . . . ,r}. (.)
On the other hand, we have
xn–Tmxn ≤ xn–xn++xn+–Tmn+yn+
+Tmn+yn+–Tmn+yn+Tmn+yn–Tmxn
≤ xn–xn++xn+–Tmn+yn++Myn+–yn
+MTmnyn–xn.
It follows from (.), (.), and (.) that
lim
n→∞xn–Tmxn= , ∀m∈ {, , . . . ,r}. (.)
This completes the proof.
Next, we give the following weak convergence theorems with the help of Opial’s condi-tion.
Theorem . Let C be a nonempty,closed,and convex subset of a uniformly convex Ba-nach space E with Opial’s condition.Let Sm:C→C be an asymptotically nonexpansive
mapping with the sequence{sn,m},and Tm:C→C an asymptotically nonexpansive
map-ping with the sequence{tn,m},for every m∈ {, , . . . ,r},where r≥is some positive integer.
Assume that
F =
r
m= F(Sm)∩
r
m=
Let tn=max{tn,m: ≤m≤r},and sn=max{sn,m: ≤m≤r}.Assume that
∞
n=(kn– ) <
∞, where kn =max{sn,tn : ≤ m≤ r}. Let {xn}∞n= be a sequence generated by (ϒ), where {αn},{βn,},{βn,}, . . . ,{βn,r},{γn,},{γn,}, . . . , and {γn,r},{an},{bn,},{bn,}, . . . ,{bn,r},
{cn,},{cn,}, . . . ,and{cn,r}be real number sequences in(, )such that
αn+ r
m= βn,m+
r
m=
γn,m=an+ r
m= bn,m+
r
m=
cn,m= .
Assume that restrictions(a)and(b)as in Theorem.are satisfied.Then{xn}converges
weakly to some point inF.
Proof Since{xn}is bounded, there exists a subsequence{xni} ⊂ {xn}such that{xni} con-verges weakly to a pointx¯∈C. It follows from Lemma . thatx¯∈F. Assume that there exists another subsequence{xnj} ⊂ {xn}such that{xnj}converges weakly to a pointxˆ∈C. It follows from Lemma . thatxˆ∈F. Ifx¯=xˆ, then
lim
n→∞xn–x¯=lim infi→∞ xni–x¯<lim infi→∞ xni–xˆ
=lim inf
j→∞ xnj–xˆ<lim infj→∞ xnj–x¯
= lim
n→∞xn–x¯.
This is a contradiction. Hence,x¯=xˆ. This completes the proof.
Ifr= , then Theorem . is reduced to the following.
Corollary . Let C be a nonempty,closed,and convex subset of a uniformly convex Ba-nach space E with Opial’s condition.Let S:C→C be an asymptotically nonexpansive mapping with the sequence{sn},and T:C→C an asymptotically nonexpansive mapping
with the sequence{tn}.Assume that
F =F(S)∩F(T)=∅.
Assume that∞n=(kn– ) <∞,where kn=max{sn,tn: ≤m≤r}.Let{xn}∞n=be a sequence generated by the following:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x∈C,
xn=αnxn–+βnSnxn–+γnTnyn,
yn=anxn+bnSnxn+cnTnxn, ∀n≥.
where{αn}, {βn},{γn},{an}, {bn},and{cn}are real number sequences in(, )such that
αn+βn+γn=an+bn+cn= .Assume that the following restrictions imposed on the control
sequences{αn},{βn},{γn},{an},{bn},and{cn}are satisfied:
(a) lim infn→∞αnβn> ,lim infn→∞αnγn> andlim infn→∞anbn> ;
(b) γntn(an+bnsn+cntn) < .
Then{xn}converges weakly to some point inF.
Corollary . Let C be a nonempty,closed,and convex subset of a uniformly convex Ba-nach space E with Opial’s condition.Let Sm:C→C be an asymptotically nonexpansive
mapping with the sequence{sn,m},and Tm:C→C an asymptotically nonexpansive
map-ping with the sequence{tn,m},for every m∈ {, , . . . ,r}.where r≥is some positive integer.
Assume that
F =
r
m= F(Sm)∩
r
m=
F(Tm)=∅.
Let tn=max{tn,m: ≤m≤r},and sn=max{sn,m: ≤m≤r}.Assume that∞n=(kn– ) <
∞,where kn=max{sn,tn: ≤m≤r}.Let{xn}∞n=be a sequence generated by the following:
x∈C, xn=αnxn–+
r
m=
βn,mSnmxn–+
r
m=
γn,mTmnxn, ∀n≥
where{αn},{βn,},{βn,}, . . . ,{βn,r},{γn,},{γn,}, . . . ,and{γn,r},are real number sequences in
(, )such thatαn+
r
m=βn,m+
r
m=γn,m= .Assume that the following restrictions
im-posed on the control sequences{αn},{βn,},{βn,}, . . . ,{βn,r},{γn,},{γn,}, . . . ,and{γn,r}are
satisfied
(a) lim infn→∞αnβn,m> ,andlim infn→∞αnγn,m> ,∀m∈ {, , . . .r};
(b) rm=γn,mtn< .
Then{xn}converges weakly to some point inF.
Ifβn,m=bn,m= , than Theorem . reduced the following.
Corollary . Let C be a nonempty,closed,and convex subset of a uniformly convex Ba-nach space E with Opial’s condition.Let Tm:C→C be an asymptotically nonexpansive
mapping with the sequence{tn,m},for every m∈ {, , . . . ,r},where r≥is some positive
integer.Assume that
F =
r
m=
F(Tm)=∅.
Assume that∞n=(tn– ) <∞,where tn=max{tn,m: ≤m≤r}.Let{xn}∞n=be a sequence generated by the following:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x∈C,
xn=αnxn–+ r
m=γn,mTmnyn,
yn=anxn+
r
m=cn,mTmnxn, ∀n≥,
where{αn},{γn,},{γn,}, . . . ,{γn,r},{an},{cn,},{cn,}, . . . ,{cn,r} be real number sequences in
(, )such that
αn+ r
m=
γn,m=an+ r
m=
cn,m= .
Assume that the following restrictions imposed on the control sequence{αn},{γn,},{γn,},
(a) lim infn→∞αn> ,lim infn→∞αnγn,m> andlim infn→∞an> ,∀m∈ {, , . . . ,r};
(b) rm=γn,mtn(an+
r
m=cn,mtn) < .
Then{xn}converges weakly to some point inF.
IfSm=I, whereIstands for the identity mappings, then Theorem . reduced the
fol-lowing.
Corollary . Let C be a nonempty,closed,and convex subset of a uniformly convex
Ba-nach space E with Opial’s condition.Let Tm:C→C be an asymptotically nonexpansive
mapping with the sequence{tn,m},for every m∈ {, , . . . ,r},where r≥is some positive
integer.Assume that
F =
r
m=
F(Tm)=∅.
Asume that∞n=(tn– ) <∞,where tn=max{tn,m: ≤m≤r}.Let{xn}∞n=be a sequence generated by the following:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x∈C,
xn= (αn+
r
m=βn,m)xn–+ r
m=γn,mTmnyn,
yn= (an+
r
m=bn,m)xn+
r
m=cn,mTmnxn, ∀n≥,
where {αn},{βn,},{βn,}, . . . ,{βn,r},{γn,},{γn,}, . . . ,{γn,r},{an},{bn,},{bn,}, . . . ,{bn,r},{cn,},
{cn,}, . . . ,{cn,r}be real number sequences in(, )such thatαn+
r
m=βn,m+
r
m=γn,m=
an+
r
m=bn,m +
r
m=cn,m= . Assume that the following restrictions imposed on the
control sequences {αn},{βn,},{βn,}, . . . ,{βn,r},{γn,},{γn,}, . . . ,{γn,r},{an},{bn,},{bn,}, . . . ,
{bn,r},{cn,},{cn,}, . . . ,{cn,r}are satisfied
(a) lim infn→∞αnβn,m> ,lim infn→∞αnγn,m> andlim infn→∞anbn,m> ,
∀m∈ {, , . . . ,r}; (b) rm=γn,mtn(an+
r
m=bn,m+
r
m=cn,mtn) < .
Then{xn}converges weakly to some point inF.
IfTm=I,bn,m= , whereIstands for the identity mappings, then Theorem . reduced
the following.
Corollary . Let C be a nonempty,closed,and convex subset of a uniformly convex
Ba-nach spaces E with Opial’s condition.Let Sm:C→C be an asymptotically nonexpansive
mapping with the sequence{sn,m},for every m∈ {, , . . . ,r},where r≥is some positive
integer.Assume that
F =
r
m=
Assume that∞n=(sn– ) <∞,where sn=max{sn,m: ≤m≤r}.Let{xn}∞n=be a sequence generated by the following:
⎧ ⎨ ⎩
x∈C,
xn=–r αn
m=(an+rm=cn,m)xn–+
r m=βn,mSnm
–rm=(an+rm=cn,m)xn–, ∀n≥,
where{αn},{βn,},{βn,}, . . . ,{βn,r},{γn,},{γn,}, . . . ,{γn,r},{an},{cn,},{cn,}, . . . ,{cn,r} be real
number sequences in(, )such that
αn+ r
m= βn,m+
r
m=
γn,m=an+ r
m= cn,m= .
Assume that the following restrictions imposed on the control sequences{αn},{βn,},{βn,},
. . . ,{βn,r},{γn,},{γn,}, . . . ,{γn,r},{an},{cn,},{cn,}, . . . ,{cn,r}are satisfiedlim infn→∞αnβn,m>
andlim infn→∞αnγn,m> for all m∈ {, , . . . ,r}.Then{xn}converges weakly to some
point inF.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main idea of this paper is proposed by JKK. JKK and WHL prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the final manuscript.
Acknowledgements
The research was supported by Kyungnam University Research Fund, 2012.
Received: 28 November 2012 Accepted: 3 April 2013 Published: 16 April 2013
References
1. Bauschke, HH, Borwein, JM: On projection algorithms for solving convex feasibility problems. SIAM Rev.38, 367-426 (1996)
2. Kotzer, T, Cohen, N, Shamir, J: Image restoration by a novel method of parallel projection onto constraint sets. Optim. Lett.20, 1772-1774 (1995)
3. Byrne, C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl.20, 103-120 (2008)
4. Censor, Y, Elfving, T, Kopf, N, Bortfeld, T: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl.21, 2071-2084 (2005)
5. Censor, Y, Bortfeld, T, Martin, B, Trofimov, A: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol.51, 2353-2365 (2006)
6. Lopez, G, Martin, V, Xu, HK: Perturbation techniques for nonexpansive mappings with applications. Nonlinear Anal.
10, 2369-2383 (2009)
7. Reich, S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl.67, 274-276 (1979)
8. Schu, J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc.43, 153-159 (1991)
9. Khan, SH, Yildirim, I, Ozdemir, M: Convergence of an implicit algorithm for two families of nonexpansive mappings. Comput. Math. Appl.59, 3084-3091 (2010)
10. Acedo, GL, Xu, HK: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal.67, 2258-2271 (2007)
11. Qin, X, Kang, SM, Agarwal, RP: On the convergence of an implicit iterative process for generalized asymptotically quasi-nonexpansive mappings. Fixed Point Theory Appl.2010, Article ID 714860 (2010)
12. Qin, X, Kim, JK, Wang, TZ: On the convergence of implicit iterative processes for asymptotically pseudocontractive mappings in the intermediate sense. Abstr. Appl. Anal.2011, Article ID 468716 (2011)
13. Qin, X, Cho, SY: Implicit iterative algorithms for treating strongly continuous semigroups of Lipschitz pseudocontractions. Appl. Math. Lett.23, 1252-1255 (2010)
14. Kim, JK, Nam, YM, Sim, JY: Convergence theorems of implicit iterative sequences for a finite family of asymptotically quasi-nonexpansive type mappings. Nonlinear Anal.71, e2839-e2848 (2009)
16. Chang, SS, Tan, KK, Lee, HWJ, Chan, CK: On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings. J. Math. Anal. Appl.313, 273-283 (2006)
17. Tian, YK, Chang, SS, Huang, JL, Wang, X, Kim, JK: Implicit iteration process for common fixed point of strictlt asymptotically pseudocontractive mappings in Banach spaces. Fixed Point Theory Appl.2008, Article ID 324575 (2008)
18. Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc.73, 591-597 (1967)
19. Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc.35, 171-174 (1972)
20. Chang, SS, Cho, YJ, Zhou, HY: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings. J. Korean Math. Soc.38, 1245-1260 (2001)
21. Tan, KK, Xu, HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl.178, 301-308 (1993)
doi:10.1186/1029-242X-2013-179
