R E S E A R C H
Open Access
General composite implicit iteration process
for a finite family of asymptotically
pseudo-contractive mappings
Balwant Singh Thakur
1, Rajshree Dewangan
1and Mihai Postolache
2**Correspondence:
2Department of Mathematics and
Informatics, University Politehnica of Bucharest, Bucharest, 060042, Romania
Full list of author information is available at the end of the article
Abstract
In this paper, a modified general composite implicit iteration process is used to study the convergence of a finite family of asymptotically nonexpansive mappings. Weak and strong convergence theorems have been proved, in the framework of a Banach space.
MSC: 47H09; 47H10
Keywords: implicit iteration process; asymptotically pseudo-contractive mapping; common fixed points
1 Introduction
LetKbe a nonempty subset of a real Banach spaceEand letJ:E→E∗is the normalized duality mapping defined by
J(x) =f ∈E∗:x,f=xf;x=f, ∀x∈E,
whereE∗denotes the dual space ofEand·,·denotes the generalized duality pairing. It is well known that ifE∗is strictly convex, thenJis single valued.
In the sequel, we shall denote the single valued normalized duality mapping byj. LetK be a nonempty subset ofE. A mappingT:K→Kis said to beL-Lipschitzianif there exists a constantL> such that for allx,y∈K, we haveTx–Ty ≤Lx–y. It is
said to benonexpansiveifTx–Ty ≤ x–y, for allx,y∈K.T is calledasymptotically nonexpansive[] if there exists a sequence{hn} ⊆[,∞) withlimn→∞hn= such that Tnx–Tny ≤hnx–y, for all integersn≥ and allx,y∈K.
A mappingTis said to bepseudo-contractive[, ], if there existsj(x–y)∈J(x–y) such thatTx–Ty,j(x–y) ≤ x–y, for allx,y∈K.Tis calledstrongly pseudo-contractive, if there exists a constantβ∈(, ),j(x–y)∈J(x–y) such thatTx–Ty,j(x–y) ≤βx–y, for allx,y∈K. It is said to be asymptotically pseudo-contractive [] if there exists a se-quence{hn} ⊆[,∞) withlimn→∞hn= andj(x–y)∈J(x–y) such that
Tnx–Tny,j(x–y)≤hnx–y, ∀x,y∈K,∀n≥. (.)
It follows from Kato [] that
x–y ≤x–y+rhnI–Tn x–
hnI–Tn y, ∀x,y∈K,∀n≥,r> . (.)
We useF(T) to denote the set of fixed points ofT; that is,F(T) ={x∈K:x=Tx}.
It follows from the definition that ifTis asymptotically nonexpansive, then for allj(x–
y)∈J(x–y),
Tnx–Tny,j(x–y)=x–yTnx–Tny≤hnx–y.
Hence every asymptotically nonexpansive mapping is asymptotically pseudo-contrac-tive.
It can be observed from the definition that an asymptotically nonexpansive mapping is uniformlyL-Lipschitzian, whereL=supn≥{hn}.
Now consider an example of non-Lipschitzian mapping due to Rhoades []. Define a mappingT: [, ]→[, ] by the formulaTx={ –x}, forx∈[, ]. Schu [] used this
example to show that the class of asymptotically nonexpansive mappings is a subclass of the class of pseudo-contractive mappings. SinceTis not Lipschitzian, it cannot be asymp-totically nonexpansive. AlsoTis the identity mapping andTis monotonically decreasing, and it follows that
|x–y|Tnx–Tny=|x–y| for alln= m,m∈N
and
(x–y)Tnx–Tny = (x–y)(Tx–Ty)
≤
≤ |x–y| for alln= m– ,m∈N.
HenceTis asymptotically pseudo-contractive mapping with constant sequence{}. The iterative approximation problems for a nonexpansive mapping, an asymptotically nonexpansive mapping, and an asymptotically pseudo-contractive mapping were studied extensively by Browder [], Kirk [], Goebel and Kirk [], Schu [], Xu [, ], Liu [] in the setting of Hilbert space or uniformly convex Banach space.
In , Xu and Ori [] introduced the following implicit iteration process for a finite family of nonexpansive self-mappings in Hilbert space:
x∈K arbitrary,
xn=αnxn–+ ( –αn)Tnxn, n≥,
(.)
where{αn}be a sequence in (, ) andTn=TnmodN. They proved in [] that the sequence {xn}converges weakly to a common fixed point ofTn,n= , , . . . ,N.
and Thakur [] introduced a modified general composite implicit iteration process for a finite family of random asymptotically nonexpansive mapping and proved strong conver-gence theorems.
The purpose of this paper is to consider a finite family{Ti}N
i=of asymptotically pseudo-contractive mappings and to establish convergence results in Banach spaces based on the modified general composite implicit iteration:
Forx∈K, construct a sequence{xn}by
xn=αnxn–+ ( –αn)Tik((nn))yn,
yn=rnxn+snxn–+tnTik(n(n))xn+wnTik(n(n))xn–
(.)
for eachn≥, which can be written asn= (k(n) – )N+i(n), wherei(n) = , , . . . ,Nand
k(n)≥ is a positive integer, withk(n)→ ∞asn→ ∞. The sequences{αn},{rn},{sn},{tn}
and{wn}are in (, ) such thatrn+sn+tn+wn= for alln≥.
2 Preliminaries
In what follows we shall use the following results.
Lemma .[] Let E be a Banach space, K be a nonempty closed convex subset of E,
and T: K→K be a continuous and strong pseudo-contraction.Then T has a unique fixed point.
Lemma .[] Let{an},{bn},and{cn}be three nonnegative sequences satisfying the fol-lowing condition:
an+≤( +bn)an+cn for all n≥n,
where nis some nonnegative integer,
∞
n=bn<∞and
∞
n=cn<∞.
Then
(i) limn→∞anexists;
(ii) if,in addition,there exists a subsequence{ani} ⊂ {an}such thatani→,then an→asn→ ∞.
Lemma .[] Let E be a uniformly convex Banach space and let a,b be two constants with <a<b< .Suppose that{tn} ⊂[a,b]is a real sequence and{xn},{yn}are two se-quences in E.Then the conditions
lim sup
n→∞
xn ≤d, lim sup
n→∞
yn ≤d and lim
n→∞tnxn+ ( –tn)yn=d
imply thatlimn→∞xn–yn= ,where d≥is some constant.
Lemma .[] Let E be a reflexive smooth Banach space with a weakly sequential con-tinuous duality mapping J.Let K be a nonempty bounded and closed convex subset of E and T:K→K be a uniformly L-Lipschitzian and asymptotical pseudo-contraction.Then I–T is demiclosed at zero,where I is the identical mapping.
A Banach spaceEis said to satisfy Opial’s condition if for any sequence{xn} ∈E,xnx asn→ ∞implies
lim sup
n→∞
xn–x<lim sup n→∞
xn–y, ∀y∈Ewithx=y.
We know that a Banach space with a sequentially continuous duality mapping satisfies Opial’s condition (for details, see []).
3 The main results
Throughout this section, E is a uniformly convex Banach space,K a nonempty closed convex subset ofE.Ndenotes the set of natural numbers andI={, , . . . ,N}, the set of
the firstNnatural numbers.Ti(i∈I) areNuniformly Lipschitzian asymptotically pseudo-contractive self-mappings onK. LetF=i∈IF(Ti)=∅.
SinceTi(i∈I) are uniformly Lipschitzian, there exist constantsLi> such thatTinx–
Tiny ≤Lix–y, for allx,y∈K,n∈Nandi∈I. Also, sinceTi(i∈I) are asymptotically pseudo-contractive; therefore there exist sequences{h(ni)}such thatTinx–Tiny,j(x–y) ≤
h(ni)x–yfor allx,y∈Kandi∈I. TakeL=maxi∈I(Li) andhn=maxi∈I(h(ni)).
Before presenting the main results, we first show that the proposed iteration (.) is well defined.
LetTbe uniformly Lipschitzian asymptotically pseudo-contractive mapping. For every fixedu∈Kandα∈(LL++LL+, ), define a mappingSn:K→Kby the formula
Snx=αu+ ( –α)Tna,
a=rx+su+tTnx+wTnufor allx∈K, (.)
whereα,r,s,t,w∈(, ), with ( –α)(L+L) < . Then, for allx,y∈K,j(x–y)∈J(x–y), we have
Sny=αu+ ( –α)Tnb,
b=ry+su+tTny+wTnufor allx∈K. (.)
Now
Tna–Tnb,j(x–y)=Tna–Tnbx–y
≤La–bx–y
=Lr(x–y) +tTnx–Tny x–y
≤Lrx–y+tLx–y x–y
=Lr+tL x–y
≤L+L x–y,
so
Snx–Sny,j(x–y)
= ( –α)Tna–Tnb,j(x–y)
Since ( –α)(L+L)∈(, ),S
nis strongly pseudo-contractive, which is also continuous, by Lemma .,Snhas a unique fixed pointx∗∈K,i.e.
Snx∗=αu+ ( –α)Tna,
a=rx∗+su+tTnx∗+wTnufor allx∈K. (.) Thus the implicit iteration (.) is defined in K for a finite family{Ti} of uniformly Lipschitzian asymptotically pseudo-contractive self-mappings onK, providedαn∈(α, ), whereα= L+L
L+L+, for alln∈N,L=maxi∈I(Li).
Lemma . Let E,K,and Ti(i∈I)be as defined above and let{xn}be the sequence
de-fined by(.),where{αn}is a sequence of real numbers such that <α<αn≤β < for α = L+L
L+L+ andβ is some constant and satisfying the conditions
∞
n=( –αn) <∞and
limn→∞h–n–αn = .Let b> be a real number such that tn+wn≤b/L< .Then (i) limn→∞xn–pexists,for allp∈F,
(ii) limn→∞d(xn,F)exists,whered(xn,F) =infp∈Fxn–p,
(iii) limn→∞xn–Tlxn= ,∀l∈I.
Proof Letp∈F. Using (.), we have
xn–p=
xn–p,j(xn–p)
≤αn
xn––p,j(xn–p)
+ ( –αn)
Tik((nn))yn–Tik(n(n))xn,j(xn–p)
+ ( –αn)hk(n)xn–p
=αnxn––pxn–p+ ( –αn)Lyn–xnxn–p
+ ( –αn)hk(n)xn–p. (.)
Using (.), we obtain
yn–xn=sn(xn––xn) +tn
Tik(n(n))xn–xn +wn
Tik((nn))xn––xn
≤snxn––p+snxn–p+tnLxn–p+tnxn–p
+wnLxn––p+wnxn–p. (.)
Substituting (.) in (.), we get
xn–p≤
αn+ ( –αn)L(sn+wnL)xn––pxn–p
+ ( –αn)(sn+tn+wn+tnL)L+hk(n)
xn–p ≤αn+ ( –αn)( +L)L xn––pxn–p
+ ( –αn)
( +L)L+hk(n)
xn–p ≤αn+ ( –αn)( +L)L xn––pxn–p
+( –αn)( +L)L+ ( –αn+μk(n))
xn–p, (.)
whereμk(n)=hk(n)– for alln≥, by condition
∞
n=(hk(n)– ) <∞, we have
∞
Therefore, we have
xn–p ≤
(αn+ ( –αn)( +L)L) αn–μk(n)– ( –αn)( +L)L
xn––p
≤
+ μk(n)+ ( –αn)( +L)L αn–μk(n)– ( –αn)( +L)L
xn––p
≤
+ μk(n)+ ( –αn)( +L)L – ( –αn+μk(n)+ ( –αn)( +L)L)
xn––p. (.)
Sincelimn→∞hk(n)–
–αn =limn→∞
μk(n)
–αn = , there exists aMsuch that
μk(n)
–αn <M.
Now, we consider the second term on the right side of (.). We have
–αn+μk(n)+ ( –αn)( +L)L ≤( –αn)
+M+ ( +L)L.
By condition∞n=( –αn) <∞, we havelimn→∞( –αn) = , then there exists a natural
numberNsuch that ifn>N, then
– –αn+μk(n)+ ( –αn)( +L)L ≥ .
Therefore, it follows from (.) that
xn–p ≤
+ μk(n)+ ( –αn)( +L)L
xn––p
= ( +σn)xn––p, (.)
whereσn= {μk(n)+ ( –αn)( +L)L}. Taking the infimum overp∈F, we have
d(xn,F)≤( +σn)d(xn–,F). (.)
Since∞n=μk(n)<∞and
∞
n=( –αn) <∞, we have
∞
n=
σn<∞.
Thus, by Lemma .,limn→∞xn–pandlimn→∞d(xn,F) exist. Without loss of generality, we assume
lim
n→∞xn–p=d
. (.)
Setvk(n)= hk(n)–
hk(n) , and from (.), we have
xn–p ≤
xn–p+
–αn αnhk(n)
hk(n)I–Tik(n(n)) xn–
hk(n)I–Tik(n(n)) p
≤xn–p+ –αn
αn
αn
xn––Tik(n(n))xn + ( –αn)
+
–αn αn
hk(n)–
hk(n)
Tik((nn))xn–p
=xn–p+ –αn
xn––Tik((nn))xn +
( –αn) αn
Tik((nn))yn–Tik(n(n))xn
+
–αn αn
vk(n)Tik(n(n))xn–p
≤xn–p+
(xn––xn)
+
–αn αn
vk(n)Tik((nn))xn–p
+( –αn)
αn
Lyn–xn
≤
(xn–p) +
(xn––p)
+
–αn αn
vk(n)Tik((nn))xn–p
+( –αn)
αn
Lyn–xn.
Thus
lim inf
n→∞ xn–p ≤lim infn→∞
(xn–p) +
(xn––p)
+lim inf
n→∞
–αn αn
vk(n)Tik((nn))xn–p
+lim inf
n→∞
( –αn) αn
Lyn–xn.
Sincevk(n)= hk(n) –
hk(n) ∈(, ), we havelimn→∞vk(n)= and from
∞
n=( –αn) <∞, we havelimn→∞( –αn) = and using (.), we have
lim inf
n→∞
(xn–p) +
(xn––p)
≥d. (.)
On the other hand, we obtain
lim sup
n→∞
(xn–p) +
(xn––p)
≤lim sup
n→∞
xn–p+
xn––p
=d, (.)
from (.) and (.), we have
lim
n→∞
(xn–p) +
(xn––p)
=d.
It follows from Lemma . that
lim
n→∞xn–xn–= . (.)
Thus, for anyi∈I, we have
lim
Since <α<αn≤β< and from (.) and (.), we get
lim
n→∞xn–T
k(n)
i(n)yn= lim n→∞
αn –αn
xn–xn–
≤
–βnlim→∞xn–xn–= . (.)
On the other hand, from (.) and (.)
lim
n→∞xn––T k(n)
i(n)yn≤nlim→∞xn––xn+nlim→∞xn–T k(n)
i(n)yn= . (.)
Now,
Tik((nn))xn–xn≤ xn–xn–+Tik((nn))yn–xn–+Tik(n(n))yn–Tik(n(n))xn
≤( +L)xn–xn–+Tik(n(n))yn–xn–+Lyn–xn–. (.)
Again, by using (.), we obtain
yn–xn– ≤rnxn+snxn–+tnTik((nn))xn+wnTik((nn))xn––xn–
≤tnTik(n(n))xn–xn+wnTik((nn))xn––xn+ (rn+tn+wn)xn–xn–
≤(tn+wn)Tik((nn))xn–xn+ (rn+tn+wn+wnL)xn–xn–. (.)
Substituting (.) into (.), we get
Tik((nn))xn–xn≤( +L)xn–xn–+Tik((nn))yn–xn–+L(tn+wn)Tik((nn))xn–xn
+L(rn+tn+wn+wnL)xn–xn–.
Sincetn+wn≤b/L< , the above inequality gives
( –b)Tik((nn))xn–xn≤
+L( +rn+tn+wn+wnL)
xn–xn–+Tik((nn))yn–xn–.
Then from (.), (.), and the above inequality, we have
lim
n→∞T
k(n)
i(n)xn–xn= . (.)
From (.), (.), and (.), we get
lim
n→∞yn–xn–= . (.)
On the other hand, from (.) and (.) we have
lim
n→∞yn–xn ≤nlim→∞yn–xn–+nlim→∞xn––xn= . (.)
LetAn=Tik(n(n))yn–xn–, then from (.), we haveAn→. Also,
xn––Tnxn ≤xn––Tik((nn))yn+Tik((nn))yn–Tnxn
=An+Tik(n(n))yn–Ti(n)xn≤An+LTik(n(n))–yn–xn
≤An+LTik((nn))–yn–Tik((nn–)–N)xn–N
+Tik((nn–)–N)xn–N–Tik((nn–)–N)yn–N
+Tik(n(n–)–N)yn–N–x(n–N)–+x(n–N)––xn
. (.)
Since for eachn>N,n= (n–N) (modN) andn= (k(n) – )N+i(n),n–N= ((k(n) – ) – )N+i(n) = (k(n–N) – )N+i(n–N),i.e.
k(n–N) =k(n) – and i(n–N) =i(n).
Therefore from (.), we have
xn––Tnxn ≤An+LTik((nn))–yn–Tik((nn))–xn–N
+Tik((nn––NN))xn–N–Tik(n(n––NN))yn–N
+Tik((nn––NN))yn–N–x(n–N)–+x(n–N)––xn
≤An+LLyn–xn–N+Lxn–N–yn–N
+An–N+x(n–N)––xn
≤An+Lyn–xn+xn–xn–N+xn–N–yn–N
+LAn–N+x(n–N)––xn . (.)
From (.), (.), andAn→, we have
lim
n→∞xn––Tnxn= . (.)
It follows from (.) and (.) that
lim
n→∞xn–Tnxn ≤nlim→∞
xn–xn–+xn––Tnxn
= . (.)
Consequently, for anyi∈I, from (.), (.), we obtain
xn–Tn+ixn ≤ xn–xn+i+xn+i–Tn+ixn+i+Tn+ixn+i–Tn+ixn ≤( +L)xn–xn+i+xn+i–Tn+ixn+i →,
asn→ ∞. This implies that the sequence
N
i=
xn–Tn+ixn
∞
Since for eachl= , , . . . ,N,{xn–Tlxn}is a subsequence of
N
i={xn–Tn+ixn}, there-fore, we have
lim
n→∞xn–Tlxn= , ∀l∈I. (.)
This completes the proof.
3.1 Strong convergence theorems
First, we prove necessary and sufficient conditions for the strong convergence of the modi-fied general composite implicit iteration process to a common fixed point of a finite family of asymptotically pseudo-contractive mappings.
Theorem . Let E,K,and Ti(i∈I)be as defined above and{αn}be a sequence of real
numbers as in Lemma..Then the sequence{xn}generated by(.)converges strongly to a member ofFif and only iflim infn→∞d(xn,F) = .
Proof The necessity of the condition is obvious. Thus, we will only prove the sufficiency. Letlim infn→∞d(xn,F) = . Then from (ii) in Lemma ., we havelimn→∞d(xn,F) = .
Next, we show that {xn} is a Cauchy sequence in K. For any given ε> , since
limn→∞d(xn,F) = , there exists a natural numbernsuch thatd(xn,F) <ε/ whenn≥n. Sincelimn→∞xn–pexists for allp∈F, we havexn–p<M, for alln≥ and some positive numberM.
Furthermore ∞n=σn <∞ implies that there exists a positive integer n such that
∞
j=nσj<ε/Mfor alln≥n. LetN=max{n,n}. It follows from (.) that xn–p ≤ xn––p+Mσn.
Now, for alln,m≥Nand for allp∈F, we have xn–xm ≤ xn–p+xm–p
≤ xN–p+M n
j=N+
σj+xN–p+M m
j=N+ σj
≤xN–p+ M
∞
j=N σj.
Taking the infimum over allp∈F, we obtain
xn–xm ≤d
xN,F + M
∞
j=N σj<ε.
This implies that{xn}is a Cauchy sequence. SinceEis complete, therefore{xn}is conver-gent.
Supposelimn→∞xn=q.
Notice that
d(q,F) –d(xn,F)≤ q–xn, ∀n∈N,
sincelimn→∞xn=qandlimn→∞d(xn,F) = , we obtainq∈F.
This completes the proof.
Corollary . Suppose that the conditions are the same as in Theorem..Then the se-quence{xn}generated by(.)converges strongly to u∈F if and only if{xn}has a subse-quence{xnj}which converges strongly to u∈F.
A mappingT: K→KwithF(T)=∅is said to satisfycondition(A) [] onK if there exists a nondecreasing functionf: [,∞)→[,∞), withf() = andf(r) >r, for allr∈
(,∞), such that for allx∈K,
x–Tx ≥fdx,F(T) .
A family{Ti}N
i=ofNself-mappings ofKwithF=
i∈IF(Ti)=∅is said to satisfy
() condition(B) onK[] if there is a nondecreasing functionf: [,∞)→[,∞)with f() = andf(r) >rfor allr∈(,∞)such that for allx∈Ksuch that
max
≤l≤N
x–Tlx
≥fd(x,F);
() condition(C) onK[] if there is a nondecreasing functionf: [,∞)→[,∞)with f() = andf(r) >rfor allr∈(,∞)such that for allx∈Ksuch that
x–Tlx
≥fd(x,F)
for at least oneTl,l= , , . . . ,Nor, in other words, at least one of theTl’s satisfies
condition(A).
Condition (B) reduces to condition (A) when all but one of theTl’s are identities. Also condition (B) and condition (C) are equivalent (see []).
Senter and Dotson [] established a relation betweencondition(A) and demicompact-ness that thecondition(A) is weaker than demicompactness for a nonexpansive mapping
T defined on a bounded set. Every compact operator is demicompact. Since every com-pletely continuous mappingT: K→Kis continuous and demicompact, it satisfies con-dition(A).
Therefore in the next result, instead of complete continuity of mappingsT,T, . . . ,TN, we use condition (C).
Theorem . Let E and K be as defined above,Ti(i∈I)be N asymptotically
pseudo-contractive mappings as defined above and satisfyingcondition (C)and{αn}be a sequence of real numbers as in Lemma.. Then the sequence {xn} generated by(.) converges strongly to a member ofF.
Let one of theTi’s, sayTl,l∈I, satisfy condition (A).
By Lemma ., we have limn→∞xn–Tlxn= . Therefore we have limn→∞f(d(xn, F)) = . By the nature off and the fact thatlimn→∞d(xn,F) exists, we havelimn→∞d(xn, F) = . By Theorem ., we find that{xn}converges strongly to a common fixed point inF.
This completes the proof.
A mappingT: K→Kis said to be semicompact, if the sequence{xn}inK such that xn–Txn →, asn→ ∞, has a convergent subsequence.
Theorem . Let E and K be as defined above,and let Ti(i∈I)be N asymptotically
pseudo-contractive mappings as defined above such that one of the mappings in {Ti}N i=
is semicompact,and let{αn} be a sequence of real numbers as in Lemma.. Then the sequence{xn}generated by(.)converges strongly to a member ofF.
Proof Without loss of generality, we may assume thatTsis semicompact for some fixed
s∈ {, , . . . ,N}. Then by Lemma ., we havelimn→∞xn–Tsxn= . So by definition of semicompactness, there exists a subsequence{xnj} of{xn} such that{xnj}converges
strongly tox∗∈K. Now again by Lemma ., we have
lim
nj→∞
xnj–Tlxnj=
for alll∈I. By continuity ofTl, we haveTlxnj→Tlx
∗for alll∈I.
Thuslimj→∞xnj–Tlxnj=x∗–Tlx∗= for alll∈I. This implies thatx∗∈F. Also, lim infn→∞d(xn,F) = . By Theorem ., we find that{xn}converges strongly to a common
fixed point inF.
3.2 Weak convergence theorem
Theorem . Let E be a uniformly convex and smooth Banach space which admits a
weakly sequentially continuous duality mapping,and let K and Ti (i∈I)be as defined
above and{αn} be a sequence of real numbers as in Lemma..Then the sequence{xn} generated by(.)converges weakly to a member ofF.
Proof Since{xn}is a bounded sequence inK, there exists a subsequence{xnk} ⊂ {xn}such
that{xnk}converges weakly toq∈K. Hence from Lemma ., we have
lim
n→∞xnk–Tlxnk= , ∀l∈I.
By Lemma ., we find that (I–Tl) is demiclosed at zero,i.e.(I–Tl)q= , so thatq∈
F(Tl). By the arbitrariness ofl∈I, we know thatq∈F=
l∈IF(Tl). Next we prove that{xn}converges weakly toq.
If{xn}has another subsequence{xnj}which converges weakly toq=q, then we must
haveq∈F, and sincelimn→∞xn–qexists and since the Banach spaceEhas a weakly sequentially duality mapping, it satisfies Opial’s condition, and it follows from a standard argument thatq=q. Thus{xn}converges weakly toq∈F.
LetKbe a nonempty subset of a real Banach spaceE. LetDbe a nonempty bounded subset ofK. The set-measure of noncompactness ofD,γ(D), is defined as
γ(D) =inf{d> :Dcan be covered by a finite number of sets of diameter≤d}.
The ball-measure of compactness ofD,χ(D), is defined as
χ(D) =inf{r> :Dcan be covered by a finite family of balls with centers inE
and radiusr}.
A bounded continuous mappingT: K→Eis called
() k-set-contractive ifγ(T(D))≤kγ(D), for each bounded subsetDofKand some constantk≥;
() k-set-condensing ifγ(T(D)) <γ(D), for each bounded subsetDofKwithγ(D) > ; () k-ball-contractive ifχ(T(D))≤kχ(D), for each bounded subsetDofKand some
constantk≥;
() k-ball-condensing ifχ(T(D)) <χ(D), for each bounded subsetDofKwithχ(D) > .
A mappingT:K→Eis called
() compact ifcl(T(A))is compact wheneverA⊂Kis bounded;
() completely continuous if it maps weakly convergence sequences into strongly convergent sequences;
() a generalized contraction if for eachx∈Kthere existsk(x) < such that
Tx–Ty ≤k(x)x–yfor ally∈K;
() a mappingT:E→Eis called uniformly strictly contractive (relative toE) if for each x∈Ethere existsk(x) < such thatTx–Ty ≤k(x)x–yfor ally∈K. Every
k-set-contractive mapping withk< is set-condensing and also every compact
mapping is set-condensing.
LetKbe a nonempty closed bounded subset ofEandT:K→Ea continuous mapping. Then
(a) T is strictly semicontractive if there exists a continuous mappingV:E×E→E
withT(x) =V(x,x)forx∈Esuch that for eachx∈E,V(·,x)is ak-contraction with k< andV(x,·)is compact;
(b) T is of strictly semicontractive type if there exists a continuous mapping
V: K×K→EwithT(x) =V(x,x), forx∈Ksuch that for eachx∈K,V(·,x)is a
k-contraction with somek< independent ofxandx→V(·,x)is compact fromK
into the space of continuous mapping ofKintoEwith the uniform metric;
(c) T is of strongly semicontractive type relative toXif there exists a mapping
V: E×K→EwithT(x) =V(x,x), forx∈Ksuch thatx∈K,V(·,x)is uniformly strictly contractive onKrelative toEandV(x,·)is a completely continuous fromK toE, uniformly forx∈K.
For details refer to [–].
Let K be a nonempty closed convex bounded subset of a uniformly convex Banach spaceE. SupposeT:K→K. ThenTis semicompact ifTsatisfies any one of the following conditions [, Proposition .]:
(i) Tis either set-condensing or ball-condensing (or compact);
(iii) Tis uniformly strictly contractive; (iv) Tis strictly semicontractive;
(v) Tis of strictly semicontractive type;
(vi) Tis of strongly semicontractive type.
Remark . In view of the above, it is possible to replace the semicompactness assump-tion in Theorem . with any of the contractive assumpassump-tions (i)-(vi).
We now give an example of asymptotically pseudo-contractive mapping with nonempty fixed point set.
Example .[] LetE=R= (–∞,∞) with usual norm andK= [, ] and defineT:K→
Kby
Tx=
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
ifx= ,
ifx= ,
x–n+ ifn+ ≤x<(n+ +n),
n–x if(n++n)≤x<n
for alln≥. ThenF(T) ={}and for anyx∈K, there existsj(x– )∈J(x– ) satisfying
Tnx–Tn,j(x– )=Tnx·x≤
x <x
for alln≥. That is,T is an asymptotically pseudo-contractive mapping with sequence {kn}= .
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
1School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, CG 492010, India.2Department of
Mathematics and Informatics, University Politehnica of Bucharest, Bucharest, 060042, Romania.
Received: 3 January 2014 Accepted: 25 March 2014 Published:07 Apr 2014
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10.1186/1687-1812-2014-90
