R E S E A R C H
Open Access
Explicit averaging cyclic algorithm for
common fixed points of a finite family of
asymptotically strictly pseudocontractive
mappings in
q
-uniformly smooth Banach
spaces
Ying Zhang
1,2*and Zhiwei Xie
3*Correspondence:
[email protected] 1School of Mathematics and Physics, North China Electric Power University, Baoding, Hebei 071003, P.R. China
2School of Economics, Renmin
University of China, Beijing, 100872, P.R. China
Full list of author information is available at the end of the article
Abstract
LetEbe a realq-uniformly smooth Banach space which is also uniformly convex and
Kbe a nonempty, closed and convex subset ofE. We obtain a weak convergence theorem of the explicit averaging cyclic algorithm for a finite family of asymptotically strictly pseudocontractive mappings ofKunder suitable control conditions, and elicit a necessary and sufficient condition that guarantees strong convergence of an explicit averaging cyclic process to a common fixed point of a finite family of asymptotically strictly pseudocontractive mappings inq-uniformly smooth Banach spaces. The results of this paper are interesting extensions of those known results. MSC: 47H09; 47H10
Keywords: asymptotically strictly pseudocontractive mappings; weak and strong convergence; explicit averaging cyclic algorithm; fixed points;q-uniformly smooth Banach spaces
1 Introduction
LetEandE* be a real Banach space and the dual space ofE, respectively. LetJ
q (q> )
denote the generalized duality mapping fromEinto E* given byJq(x) ={f ∈E*:x,f=
xqandf=xq–}for allx∈E, where·,· denotes the generalized duality pairing betweenEandE*. In particular,Jis called the normalized duality mapping and it is usually denoted byJ. IfEis smooth orE*is strictly convex, thenJ
qis single-valued. In the sequel,
we will denote the single-valued generalized duality mapping byjq.
Let K be a nonempty subset of E. A mappingT :K →K is calledasymptotically
κ-strictly pseudocontractivewith sequence{κn}∞n=⊆[,∞) such thatlimn→∞κn= (see, e.g., [–]) if for allx,y∈K, there exist a constantκ∈[, ) andjq(x–y)∈Jq(x–y) such
that
Tnx–Tny,jq(x–y)≤κnx–yq–κx–y–
Tnx–Tnyq, ∀n≥. ()
IfIdenotes the identity operator, then () can be written in the form
I–Tnx–I–Tny,jq(x–y)
≥κI–Tnx–I–Tnyq– (κn– )x–yq. ()
The class of asymptoticallyκ-strictly pseudocontractive mappings was first introduced in Hilbert spaces by Qihou []. In Hilbert spaces,jqis the identity, and it is shown by Osilike et al.[] that () (and hence ()) is equivalent to the inequality
Tnx–Tny≤λnx–y+λx–y–
Tnx–Tny,
wherelimn→∞λn=limn→∞[ + (κn– )] = ,λ= ( – κ)∈[, ).
A mappingTwith domainD(T) and rangeR(T) inEis calledstrictly pseudocontractive
of Browder-Petryshyn type [] if for allx,y∈D(T), there existκ∈[, ) andjq(x–y)∈
Jq(x–y) such that
Tx–Ty,jq(x–y)≤ x–yq–κx–y– (Tx–Ty)q. ()
IfIdenotes the identity operator, then () can be written in the form
(I–T)x– (I–T)y,jq(x–y)≥κ(I–T)x– (I–T)yq. ()
In Hilbert spaces, () (and hence ()) is equivalent to the inequality
Tx–Ty≤ x–y+kx–y– (Tx–Ty), k= ( – κ) < .
It is shown in [] that the class of asymptoticallyκ-strictly pseudocontractive mappings and the class ofκ-strictly pseudocontractive mappings are independent.
A mappingTis said to beuniformly L-Lipschitzianif there exists a constantL> such that, for allx,y∈K,
Tnx–Tny≤Lx–y, n≥.
Let{Tj}Nj=–beNasymptotically strictly pseudocontractive self-mappings ofK, and de-note the common fixed points set of{Tj}Nj=–byF:=jN=–F(Tj), whereF(Tj) :={x∈K:
Tjx=x}. We consider the following explicit averaging cyclic algorithm.
For a givenx∈K, and a real sequence{αn}n∞=⊆(, ), the sequence{xn}∞n=is generated as follows:
x=αx+ ( –α)Tx,
x=αx+ ( –α)Tx,
.. .
xN =αN–xN–+ ( –αN–)TN–xN–,
xN+=αN+xN++ ( –αN+)TxN+,
.. .
xN=αN–xN–+ ( –αN–)TN–xN–,
xN+=αNxN+ ( –αN)TxN,
xN+=αN+xN++ ( –αN+)TxN+,
.. .
The algorithm can be expressed in a compact form as
xn+=αnxn+ ( –αn)Tik((nn))xn, n≥, ()
wheren= (k– )N+iwithi=i(n)∈I={, , , . . . ,N– },k=k(n)≥ a positive integer andlimn→∞k(n) =∞. The cyclic algorithm was first studied by Acedo and Xu [] for the
iterative approximation of common fixed points of a finite family of strictly pseudocon-tractive mappings in Hilbert spaces, and it is better than implicit iteration methods.
In [] Xiaolong Qinet al.proved the following theorem in a Hilbert space.
Theorem QCKS Let K be a closed and convex subset of a Hilbert space H and N ≥
be an integer.Let,for each≤i≤N,Ti:K→K be an asymptoticallyκi-strictly pseudo-contractive mapping for some≤κi< and a sequence{kn,i}such that
∞
n=(kn,i– ) < ∞. Letκ =max{κi: ≤i≤N} andκn=max{κn,i: ≤i≤N}.Assume that F=∅. For any x∈K,let{xn}be the sequence generated by the cyclic algorithm().Assume that the
control sequence{αn}is chosen such thatκ+≤αn≤ –for all n≥and a small enough constant∈(, ).Then{xn}converges weakly to a common fixed point of the family{Ti}Ni=.
Osilike and Shehu [] extended the result of Theorem QCKS from a Hilbert space to -uniformly smooth Banach spaces which are also uniformly convex. They proved the following theorem.
Theorem OS Let E be a real-uniformly smooth Banach space which is also uniformly
convex,and K be a nonempty,closed and convex subset of E.Let{Tj}N–
j= be N
asymptoti-callyλj-strictly pseudocontractive self-mappings of K for some≤λj< with a sequence {κn(j)}∞n=⊂[,∞)such that
∞
n=(κ (j)
n – ) <∞,∀j∈J={, , , . . . ,N– },and F=∅.Let {αn}satisfy the conditions
(i*) ≤α
n< , n≥,
(ii*) <a≤ –αn≤b<
λ
C ,
whereλ=minj∈J{λj}and Cis the constant appearing in the inequality()with q= .Let
{xn}be the sequence generated by the cyclic algorithm().Then{xn}converges weakly to a common fixed point of the family{Tj}N–
j=.
result of Theorem OS from -uniformly smooth Banach spaces toq-uniformly smooth Banach spaces which are also uniformly convex. We prove that if{αn}satisfies the
condi-tions
(i) μ≤αn< , n≥,
(ii)
∞
n= ( –αn)
qλ–Cq( –αn)q–
=∞, ()
where μ=max{, – (Cqqλ)q– },λ=min
j∈J{λj}, then the iterative sequence () converges
weakly to a common fixed point of the family{Tj}Nj=–.
Furthermore, we elicit a necessary and sufficient condition that guarantees strong con-vergence of the iterative sequence () to a common fixed point of the family {Tj}Nj=–in
q-uniformly smooth Banach spaces. We will use the notation:
. for weak convergence.
. ωW(xn) ={x:∃xnjx}denotes the weakω-limit set of{xn}.
2 Preliminaries
LetEbe a real Banach space. Themodulus of smoothnessofEis the functionρE: [,∞)→
[,∞) defined by
ρE(τ) =sup
x+y+x–y– :x ≤,y ≤τ
.
Eisuniformly smoothif and only iflimτ→[ρE(τ)/τ] = .
Letq> .Eis said to beq-uniformly smooth(or to have a modulus of smoothness of power typeq> ) if there exists a constantc> such thatρE(τ)≤cτq. Hilbert spaces,Lp
(orlp) spaces ( <p<∞) and the Sobolev spacesWmp ( <p<∞) areq-uniformly smooth.
Hilbert spaces are -uniformly smooth while
Lp(orlp) or Wmp is
⎧ ⎨ ⎩
p-uniformly smooth if <p≤,
-uniformly smooth ifp≥.
Theorem HKX([, p.]) Let q> and let E be a real q-uniformly smooth Banach
space.Then there exists a constant Cq> such that,for all x,y∈E,
x+yq≤ xq+qy,jq(x)+Cqyq. ()
Eis said to have aFréchet differentiable normif, for allx∈U={x∈E:x= },
lim t→
x+ty–x t
exists and is attained uniformly iny∈U. In this case, there exists an increasing function
b: [,∞)→[,∞) withlimt→+[b(t)/t] = such that, for allx,h∈E,
x
+h,j(x)≤ x+h
≤ x
It is well known (see, for example, [, p.]) that aq-uniformly smooth Banach space has a Fréchet differentiable norm.
Lemma .([, p.]) Let E be a real q-uniformly smooth Banach space which is also
uniformly convex.Let K be a nonempty,closed and convex subset of E and T:K→K be an asymptoticallyκ-strictly pseudocontractive mapping with a nonempty fixed point set.
Then(I–T)is demiclosed at zero,that is,if whenever{xn} ⊂D(T)such that{xn}converges weakly to x∈D(T)and{(I–T)xn}converges strongly to,then Tx=x.
Lemma .([, p.]) Let{an}∞n=,{bn}∞n=,{δn}∞n=be sequences of nonnegative real
num-bers satisfying the following inequality:
an+≤( +δn)an+bn, ∀n≥.
If∞n=δn<∞and ∞
n=bn<∞,thenlimn→∞anexists.If,in addition,{an}∞n=has a
sub-sequence which converges strongly to zero,thenlimn→∞an= .
Lemma .([, p.]) Let E be a real Banach space,K be a nonempty subset of E and
T:K→K be an asymptoticallyκ-strictly pseudocontractive mapping.Then T is uniformly L-Lipschitzian.
Lemma . Let E be a real q-uniformly smooth Banach space which is also uniformly
convex,and let K be a nonempty,closed and convex subset of E.Let,for each≤j≤N– ,
Tj:K→K be an asymptoticallyλj-strictly pseudocontractive mapping with F=∅. Let {xn}∞n=be the sequence satisfying the following conditions:
(a) limn→∞xn–pexists for everyp∈F;
(b) limn→∞xn–Tjxn= ,for each≤j≤N– ;
(c) limn→∞txn+ ( –t)p–pexists for allt∈[, ]and for allp,p∈F.
Then the sequence{xn}converges weakly to a common fixed point of the family{Tj}Nj=–.
Proof Sincelimn→∞xn–pexists, then{xn}is bounded. By (b) and Lemma ., we have
ωW(xn)⊂F. Assume thatp,p∈ωW(xn) and that{xni} and{xmj}are subsequences of {xn}such thatxnip andxmjp, respectively. SinceEis a realq-uniformly smooth Banach space, which is also uniformly convex, thenEhas a Fréchet differentiable norm. Setx=p–p,h=t(xn–p) in (), we obtain
p–p
+txn–p
,j(p–p)
≤
txn+ ( –t)p–p
≤
p–p
+btx
n–p
+txn–p,j(p–p)
,
wherebis an increasing function. Sincexn–p ≤M,∀n≥, for someM> , then
p–p +tx
n–p,j(p–p)
≤
txn+ ( –t)p–p
≤
p–p
+b(tM) +txn–p
,j(p–p)
Therefore,
p–p
+tlim sup
n→∞
xn–p,j(p–p)
≤
nlim→∞txn+ ( –t)p–p
≤
p–p
+b(tM)
+tlim inf n→∞
xn–p,j(p–p)
.
Hence, lim supn→∞xn–p,j(p–p) ≤ lim infn→∞xn–p,j(p–p)+b(tM)/t. Since
limt→+[b(tM)/t] = , thenlimn→∞xn–p,j(p–p)exists. Sincelimn→∞xn–p,j(p– p)=p–p,j(p–p), for allp∈ωW(xn). Setp=p. We havep–p,j(p–p)= , that is,p=p. Hence,ωW(xn) is a singleton, so that{xn}converges weakly to a common
fixed point of the family{Tj}N–
j=.
3 Main results
Theorem . Let E be a real q-uniformly smooth Banach space which is also uniformly
convex and K be a nonempty,closed and convex subset of E.Let N ≥ be an integer and J={, , , . . . ,N– }.Let,for each j∈J,Tj:K→K be an asymptoticallyλj-strictly pseudocontractive mapping for some≤λj< with sequences{κn,j}∞n=⊂[,∞)such that
∞
n=(κn– ) <∞,whereκn=maxj∈J{κn,j},and F:= N–
j= F(Tj)=∅.Letλ=minj∈J{λj}.Let {αn}satisfy the conditions()and{xn}be the sequence generated by the cyclic algorithm
().Then{xn}converges weakly to a common fixed point of the family{Tj}Nj=–.
Proof Pick ap∈F. We firstly show thatlimn→∞xn–pexists. To see this, using () and
(), we obtain
xn+–pq =xn–p– ( –αn)
xn–p–Tik((nn))xn–pq
≤ xn–pq+Cq( –αn)qxn–p–
Tik((nn))xn–pq
–q( –αn)
xn–p–
Tik((nn))xn–p
,jq(xn–p)
≤ xn–pq+Cq( –αn)qxn–p–
Tik((nn))xn–pq
–q( –αn)
λi(n)xn–p–
Tik((nn))xn–pq
– (κk(n),i(n)– )xn–pq
= +q( –αn)(κk(n),i(n)– )
xn–pq
– ( –αn)
qλi(n)–Cq( –αn)q–xn–Tik((nn))xn q
≤ +q( –αn)(κk(n)– )
xn–pq
– ( –αn)
qλ–Cq( –αn)q–xn–Tik((nn))xn
q
, ()
whereκk(n)=maxi∈J{κk(n),i(n)}. Sinceμ≤αn< for alln, whereμ=max{, – (Cqqλ) q–}, we
get ( –αn)[qλ–Cq( –αn)q–]≥. Therefore, () implies
xn+–pq≤
+q( –αn)(κk(n)– )
Letδn= +q( –αn)(κk(n)– ). Since
∞
n=(κn– ) <∞, we have
∞
n=
(δn– ) =q
∞
n=
( –αn)(κk(n)– )≤qN
∞
n=
(κn– ) <∞,
then () implieslimn→∞xn–pexists by Lemma . (and hence the sequence{xn–p}
is bounded, that is, there exists a constantM> such thatxn–p<M). Then we provelimn→∞xn–Tjxn= ,∀j∈J. In fact, it follows from () that
( –αn)
qλ–Cq( –αn)q–xn–Tik((nn))xn q
≤ xn–pq–xn+–pq
+q( –αn)(κk(n)– )xn–pq.
Then
∞
n= ( –αn)
qλ–Cq( –αn)q–xn–Tik(n(n))xn
q
<x–pq+Mq
∞
n=
(δk(n)– )<∞. ()
Since ∞n=( –αn)[qλ–Cq( –αn)q–] =∞, then () implies that lim infn→∞xn–
Tik((nn))xn= . Thuslimn→∞xn–Tik(n(n))xn= .
For alln>N, we havek(n) – =k(n–N) andi(n) =i(n–N). By Lemma ., we know thatTjis uniformlyLj-Lipschitzian, then there exists a constantL=maxj∈J{Lj}, such that
Tjnx–Tjny≤Lx–y, ∀n≥,∀x,y∈Kand∀j∈J.
Thus
xn–Ti(n)xn ≤xn–Tik(n(n))xn+Tik(n(n))xn–Ti(n)xn
≤xn–Tik(n(n))xn+LTik((nn))–xn–xn
≤xn–Tik(n(n))xn+LTik((nn))–xn–Tik(n(n–)–N)xn–N
+LTik((nn–)–N)xn–N–xn–N–+Lxn–N––xn
≤xn–Tik(n(n))xn+Lxn–xn–N
+LTik((nn––NN))xn–N–xn–N–+Lxn–N––xn.
Observe that
xn–xn+= ( –αn)xn–Tik(n(n))xn→ asn→ ∞.
Consequently,
xn–xn+l → asn→ ∞, for all integerl.
Observe also that
Hence,
lim
n→∞xn–Ti(n)xn= .
Consequently, for allj∈J, we have
xn–Tn+jxn ≤ xn–xn+j+xn+j–Tn+jxn+j+Lxn–xn+j → asn→ ∞.
Thus,
lim
n→∞xn–Tjxn= , ∀j∈J.
Now we prove that for all p,p ∈F, limn→∞txn+ ( –t)p –p exists for all t∈ [, ]. Let an(t) =txn+ ( –t)p–p. It is obvious that limn→∞an() =p–pand
limn→∞an() =limn→∞xn–pexist. So, we only need to consider the case oft∈(, ). DefineAn:K→Kby
Anx=αnx+ ( –αn)Tik(n(n))x, x∈K.
Then for allx,y∈K
Anx–Anyq≤ x–yq–q( –αn)
I–Tik(n(n))x–I–Tik((nn))y,jq(x–y)
+Cq( –αn)qx–y–
Tik((nn))x–Tik((nn))yq
≤ +q( –αn)(κk(n)– )
x–yq
– ( –αn)
qλ–Cq( –αn)q–x–y–
Tik(n(n))x–Tik((nn))yq.
By the choice ofαn, we have ( –αn)[qλ–Cq( –αn)q–]≥, so it follows thatAnx– Anyq≤[ +q( –α
n)(κk(n)– )]x–yq=δnx–yq. For the convenience of the following
discussion, setηn= (δn)
q, thenAnx–Any ≤η
nx–y.
SetSn,m=An+m–An+m–· · ·An,m≥. We have
Sn,mx–Sn,my ≤ n+m–
j=n
ηj
x–y for allx,y∈K,
and
Sn,mxn=xn+m, Sn,mp=p for allp∈F.
Setbn,m=Sn,m(txn+ ( –t)p) –tSn,mxn– ( –t)Sn,mp. Ifxn–p= for somen, then
xn=p for anyn≥nso that limn→∞xn–p= , in fact, {xn} converges strongly to
p∈F. Thus we may assumexn–p> for anyn≥. Letδ denote the modulus of convexity ofE. It is well known (see, for example, [, p.]) that
tx+ ( –t)y≤ – mint, ( –t)δx–y
for allt∈[, ] and for allx,y∈Esuch thatx ≤,y ≤. Set
wn,m=
Sn,mp–Sn,m(txn+ ( –t)p)
t(jn=+nm–ηj)xn–p
, zn,m=
Sn,m(txn+ ( –t)p) –Sn,mxn
( –t)(jn=+nm–ηj)xn–p .
Thenwn,m ≤ andzn,m ≤ so that it follows from () that
t( –t)δwn,m–zn,m
≤ –twn,m+ ( –t)zn,m. ()
Observe that
wn,m–zn,m=
bn,m
t( –t)(jn=+nm–ηj)xn–p
and
twn,m+ ( –t)zn,m=
Sn,mxn–Sn,mp (nj=+nm–ηj)xn–p
,
it follows from () that
t( –t)
n+m–
j=n
ηj
xn–pδ
bn,m
t( –t)(jn=+nm–ηj)xn–p
≤ n+m–
j=n
ηj
xn–p–Sn,mxn–Sn,mp
=
n+m–
j=n
ηj
xn–p–xn+m–p. ()
SinceEis uniformly convex, thenδ(s)/sis nondecreasing, and since (nj=+nm–ηj)xn– p ≤(
n+m–
j=n ηj)ηn–xn––p ≤ · · · ≤(
n+m–
j=n ηj)( n–
j=ηj)x–p= (
n+m–
j= ηj)x–
p, hence it follows from () that
(nj=+m–ηj)x–p
δ
(nj=+m–ηj)x–p
bn,m
≤ n+m–
j=n
ηj
xn–p–xn+m–p
sincet( –t)≤
for allt∈[, ]
.
Sincelimn→∞nj=+m–ηjexits andlimn→∞ n+m–
j= ηj= . Also sincelimn→∞ n+m–
j=n ηj=
andlimn→∞xn–pexists, then the continuity ofδandδ() = yieldlimn→∞bn,m=
uniformly for allm≥. Observe that
an+m(t)≤txn+m+ ( –t)p–p+
Sn,m
txn+ ( –t)p
–tSn,mxn– ( –t)Sn,mp
+Sn,m
txn+ ( –t)p
=Sn,m
txn+ ( –t)p
–Sn,mp+bn,m
≤ n+m–
j=n
ηj
txn+ ( –t)p–p+bn,m= n+m–
j=n
ηj
an(t) +bn,m.
Hence lim supn→∞an(t)≤lim infn→∞an(t), this ensures thatlimn→∞an(t) exists for all
t∈(, ).
Now apply Lemma . to conclude that{xn}converges weakly to a common fixed point of the family{Tj}N–
j=.
Theorem . Let E be a real q-uniformly smooth Banach space,and let K be a nonempty,
closed and convex subset of E.Let N≥be an integer and J={, , , . . . ,N– }.Let,for each j∈J,Tj:K→K be an asymptoticallyλj-strictly pseudocontractive mapping for some≤
λj< with sequences{κn,j}∞n=⊂[,∞)such that
∞
n=(κn– ) <∞,whereκn=maxj∈J{κn,j}, and F:=Nj=–F(Tj)=∅. Letλ=minj∈J{λj}.Let{αn} satisfy the conditions()and{xn} be the sequence generated by the cyclic algorithm ().Then{xn} converges strongly to a common fixed point of the family{Tj}Nj=–if and only if
lim inf
n→∞ d(xn,F) = ,
where d(xn,F) =infp∈Fxn–p.
Proof It follows from () that
xn+–pq≤δnxn–pq.
Thus [d(xn+–p)]q≤δn[d(xn–p)]q, and it follows from Lemma . thatlimn→∞d(xn,F)
exists.
Now if{xn}converges strongly to a common fixed pointpof the family{Tj}Nj=–, then
limn→∞xn–p= . Since
≤d(xn,F)≤ xn–p,
we havelim infn→∞d(xn,F) = .
Conversely, supposelim infn→∞d(xn,F) = , then the existence oflimn→∞d(xn,F)
im-plies thatlimn→∞d(xn,F) = . Thus, for arbitrary> , there exists a positive integern such thatd(xn,F) <
for anyn≥n. From (), we have
xn+–pq≤ xn–pq+Mq(δn– ), n≥,
and for someM> ,xn–p<M. Now, an induction yields
xn–pq≤ xn––pq+Mq(δn–– )
≤ xn––pq+Mq(δn–– ) +Mq(δn–– )
≤ · · · ≤ xl–pq+Mq n–
j=l
Since∞n=(δn–) <∞, then there exists a positive integernsuch that
∞
j=n(δj–) < (M)q, ∀n≥n. ChooseN=max{n,n}, then for alln,m≥N+ and for allp∈F, we have
xn–xm ≤ xn–p+xm–p
≤
xN–pq+Mq n–
j=N
(δj– )
q
+
xN–pq+Mq m–
j=N
(δj– )
q
≤
xN–pq+Mq
∞
j=N
(δj– )
q
+
xN–pq+Mq
∞
j=N
(δj– )
q
=
xN–pq+Mq
∞
j=N
(δj– )
q
.
Taking infimum over allp∈F, we obtain
xn–xm ≤
d(xN,F)q+Mq
∞
j=N
(δj– ) q < q
+Mq M q q < .
Thus{xn}∞n=is Cauchy. Supposelimn→xn=u. Then for allj∈Jwe have
≤ u–Tju ≤ u–xn+xn–Tjxn+Lxn–u → asn→ ∞.
Thusu∈F(Tj),∀j∈J, and henceu∈F.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors have read and approved the final manuscript.
Author details
1School of Mathematics and Physics, North China Electric Power University, Baoding, Hebei 071003, P.R. China.2School of
Economics, Renmin University of China, Beijing, 100872, P.R. China.3Easyway Company Limited, Beijing, 100872, P.R. China.
Received: 9 June 2012 Accepted: 13 September 2012 Published: 2 October 2012
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