R E S E A R C H
Open Access
Demiclosedness principle and approximation
theorems for certain classes of multivalued
mappings in Hilbert spaces
Felicia O Isiogugu
**Correspondence:
[email protected] Department of Mathematics, University of Nigeria, Nsukka, Nigeria
Abstract
We prove weak and strong convergence theorems and the demiclosedness property for classes of multivalued mappingsTsuch thatPTis not nonexpansive, where PTx={u∈Tx:x–u=d(x,Tx)}. Thus our results extend and improve the results on multivalued and single-valued mappings in the contemporary literature.
MSC: 47H10; 54H25
Keywords: proximinal sets; Hilbert spaces; nonexpansive-type mappings;k-strictly pseudocontractive-type mappings; pseudocontractive-type mappings
1 Introduction
LetEbe a normed space. A subsetKofEis called proximinal if for eachx∈Ethere exists k∈Ksuch that
x–k=infx–y:y∈K=d(x,K).
It is known that every closed convex subset of a uniformly convex Banach space is prox-iminal. In fact, ifKis a closed and convex subset of a uniformly convex Banach spaceX, then for anyx∈Xthere exists a unique pointux∈Ksuch that (see,e.g., [, ])
x–ux=inf
x–y:y∈K=d(x,K).
We will denote the family of all nonempty proximinal subsets ofXbyP(X), the family of all nonempty closed, convex and bounded subsets ofXbyCVB(X), the family of all nonempty closed and bounded subsets ofXbyCB(X) and the family of all nonempty subsets ofX by Xfor a nonempty setX. LetCB(E) be the family of all nonempty closed and bounded
subsets of a normed spaceE. LetHbe the Hausdorff metric induced by the metricdon E, that is, for everyA,B∈E,
H(A,B) =maxsup
a∈A
d(a,B),sup
b∈B
d(b,A).
IfA,B∈CB(E), then
H(A,B) =inf> :A⊆N(,B) andB⊆N(,A),
whereN(,C) =c∈C{x∈E:d(x,c)≤}. LetEbe a normed space. LetT:D(T)⊆E→E
be a multivalued mapping onE. A pointx∈D(T) is called afixed pointofTifx∈Tx. The setF(T) ={x∈D(T) :x∈Tx}is called a fixed point set ofT. A pointx∈D(T) is called astrict fixed pointofT ifTx={x}. The setFs(T) ={x∈D(T) :Tx={x}}is called a strict
fixed point set ofT. A multivalued mappingT:D(T)⊆E→Eis calledL-Lipschitzianif
there existsL> such that, for any pairx,y∈D(T),
H(Tx,Ty)≤Lx–y. ()
In () ifL∈[, ),Tis said to be acontraction, whileTisnonexpansiveifL= .Tis called quasi-nonexpansiveifF(T) ={x∈D(T) :x∈Tx} =∅and, for allp∈F(T),
H(Tx,Tp)≤ x–p. ()
Clearly, every nonexpansive mapping with nonempty fixed point set is quasi-nonexpan-sive.
In recent years, several works have been done on the approximation of fixed points of multivalued nonexpansive mappings by many authors (see, for example, [–] and ref-erences therein). Different iterative schemes have been introduced by several authors to approximate the fixed points of nonexpansive mappings (see, for example, [–]). Among the iterative schemes, Sastry and Babu [] introduced Mann and Ishikawa iteration as fol-lows.
LetT:X→P(X) andpbe a fixed point ofT. The sequence of Mann iterates is given for x∈Xby
xn+= ( –αn)xn+αnyn ∀n≥, ()
whereyn∈Txnis such thatyn–p=d(Txn,p) andαnis a real sequence in (, )
∞
n=αn=
∞.
The sequence of Ishikawa iterates is given by
yn= ( –βn)xn+βnzn,
xn+= ( –αn)xn+αnun,
()
wherezn∈Txn,un∈Tynare suchzn–p=d(p,Txn),un–p=d(Tyn,p) and{αn},{βn}
are real sequences satisfying: (i) ≤αn,βn< ; (ii)limn→∞βn= ; (iii)
∞
n=αnβ=∞.
Using the above iterative schemes, Panyanak [] generalized the result proved by []. Nadler [] made the following useful remark.
Lemma Let A,B∈CB(X)and a∈A.Ifγ > ,then there exists b∈B such that
d(a,b)≤H(A,B) +γ. ()
LetK be a nonempty convex subset ofX, letαn,βn∈[, ] andγn∈(,∞) such that
limn→∞γn= . Choosex∈K,z∈Tx. Let
y= ( –β)x+βz.
Chooseu∈Tysuch thatz–u ≤H(Tx,Ty) +γand
x= ( –α)x+αu.
Choosez∈Txsuch thatz–u ≤H(Tx,Ty) +γand
y= ( –β)x+βz.
Chooseu∈Tysuch thatz–u ≤H(Tx,Ty) +γand
x= ( –α)x+αu.
Inductively, we have
yn= ( –βn)xn+βnzn,
xn+= ( –αn)xn+αnun,
()
wherezn∈Txn,un∈Tynsatisfyzn–un ≤H(Txn,Tyn) +γn,zn+–un ≤H(Txn+,Tyn) +
γnand{αn},{βn}are real sequences in [, ) satisfyinglimn→∞βn= ,
∞
n=αnβn=∞.
Using the above iteration, they then proved the following theorem.
Theorem (Theorem , []) Let K be a nonempty compact convex subset of a uniformly convex Banach space X.Suppose that T:K→CB(K)is a multivalued nonexpansive map-ping such that F(T)=∅and T(p) ={p}for all p∈F(T).Then the Ishikawa sequence defined as above converges strongly to a fixed point of T.
Shahzad and Zegeye [] observed that ifXis a normed space andT:D(T)⊆E→P(X) is any multivalued mapping, then the mappingPT:D(T)→P(X) defined for eachxby
PT(x) =
y∈Tx:d(x,Tx) =x–y
has the property thatPT(q) ={q}for allq∈F(T). Using this idea, they removed the strong
condition ‘T(p) ={p}for allp∈F(T)’ introduced by Song and Wang [].
Recently, Khan and Yildirim [] introduced a new iteration scheme for multivalued nonexpansive mappings using the idea of the iteration scheme for a single-valued nearly asymptotically nonexpansive mapping introduced by Agarwalet al.[] as follows:
⎧ ⎪ ⎨ ⎪ ⎩
x∈K,
xn+= ( –λ)vn+λun,
yn= ( –η)xn+ηvn ∀n∈N,
()
wherevn∈PT(xn),un∈PT(yn) andλ∈[, ). Also, using a lemma in Schu [], the idea of
[] and the method of direct construction of a Cauchy sequence as indicated by Song and Cho [], they stated the following theorems.
Theorem (Theorem , []) Let X be a uniformly convex Banach space satisfying Opial’s condition and K be a nonempty closed convex subset of X.Let T:K→P(K)be a mul-tivalued mapping such that F(T)=∅and PTis a nonexpansive mapping.Let{xn}be the
sequence as defined in().Let(I–PT)be demiclosed with respect to zero.Then{xn}
con-verges weakly to a point of F(T).
However, we observe that there are many multivalued mappingsT for which neither T nor PT is nonexpansive. Based on the above observation, it is our purpose in this
paper to firstly introduce the new classes of multivalued nonexpansive-type,k-strictly pseudocontractive-type and pseudocontractive-type mappings which are more general than the class of multivalued nonexpansive mappings. Secondly, we prove that ifHis a real Hilbert space andKis a nonempty weakly closed subset ofH,T:K⊆H→P(H) is a multivalued mapping fromKinto the family of all nonempty proximinal subsets ofH. Sup-pose thatPTis ak-strictly pseudocontractive-type mapping. Then (I–PT) is demiclosed
atzero(i.e., the graph ofI–PT is closed atzeroinσ(E,E*)×(E, · ) or weakly
demi-closed atzero), whereI denotes the identity onE,σ(E,E*) the weak topology, (E, · ) the norm (or strong) topology andPTx={u∈Tx:x–u=d(x,Tx)}. Lastly, we prove
weak and strong convergence theorems for these classes of multivalued mappings with-out the compactness condition on the domain of the mappings using Mann and Ishikawa iteration schemes. Thus our results extend and improve the results on single-valued and multivalued mappings in the contemporary literature.
In the sequel, we will need the following definitions and lemmas.
Definition (See,e.g., []) LetEbe a Banach space. LetT:D(T)⊆E→Ebe a
mul-tivalued mapping. I–T is said to be strongly demiclosed at zero if for any sequence {xn}∞n=⊆D(T) such thatxnconverges strongly topand a sequence{yn} withyn∈Txn
for alln∈Nsuch that{xn–yn}converges strongly tozero, thenp∈Tp(i.e.,F(T)=∅).
Observe that ifT is a multivalued Lipschitzian mapping, thenI–T is strongly demi-closed.
Definition (See,e.g., [, ]) LetE be a Banach space. LetT :D(T)⊆E→E be a
multivalued mapping.I–T is said to beweakly demiclosedatzeroif for any sequence {xn}∞n=⊆D(T) such that{xn}converges weakly topand a sequence{yn}withyn∈Txnfor
alln∈Nsuch that{xn–yn}converges strongly tozero, thenp∈Tp(i.e., ∈(I–T)p).
Definition LetEbe a Banach space. LetT:D(T)⊆E→Ebe a multivalued mapping. A pointp∈Eis called anasymptotic fixed point of Tif there exists a sequence{xn}∞n=⊆
D(T) such that{xn}converges weakly topand a sequence{yn}withyn∈Txnfor alln∈N
such that{xn–yn}converges strongly tozero. We denote the set of asymptotic fixed points
ofTbyF(T).
Definition (See,e.g., [, ]) Let Ebe a Banach space. LetT :D(T)⊆E→E be a
converges weakly topand a sequence{yn}withyn∈Txnfor alln∈Nsuch that{xn–yn}
converges strongly toy, theny∈(I–T)p(i.e.,y=p–vfor somev∈Tp).
Definition ([]) LetHbe a real Hilbert space. LetT:H→CB(H) be a multivalued
mapping.T is said to be monotone if givenx,y∈Handu∈Tx, there existsv∈Tysuch that
u–v,x–y ≥,
d(x,Tx)≥f(dx,F(T) ∀x∈K.
Definition ([]) A multivalued mappingT:K→P(K) is said to satisfy condition () (see, for example, []) if there exists a nondecreasing functionf : [,∞)→[,∞) with f() = andf(r) > for allr∈(,∞) such that
d(x,Tx)≥f(dx,F(T) ∀x∈K.
Definition LetEbe a real Banach space. LetT:E→Ebe a multivalued mapping.I–T
is said to be monotone in the sense of [] if given any pairx,y∈Eandu∈Tx, there exists v∈Tysuch that
x–u– (y–v),x–y≥.
Lemma ([]) Let{an},{βn}and{γn}be sequences of nonnegative real numbers satisfying
the following relation:
an+≤( +βn)an+γn, n≥n,
where nis a nonnegative integer.If
βn<∞,
γn<∞,thenlimn→∞anexists.
Lemma ([]) Let K be a normed space.Let T:K→P(K)be a multivalued mapping
and PT(x) ={y∈Tx:x–y=d(x,Tx)}.Then the following are equivalent:
() x∈Tx; () PTx={x};
() x∈F(PT).
Moreover,F(T) =F(PT).
Lemma Let H be a real Hilbert space.Then the following well-known result holds:if {xn}∞n=is a sequence in H which converges weakly to z∈H,then
lim sup
n→∞xn–y
=lim sup
n→∞xn–z
+z–y ∀y∈H. ()
2 Main results
Definition LetXbe a normed space. A multivalued mappingT:D(T)⊆X→Xis said
existsk∈[, ) such that given anyx,y∈D(T) andu∈Tx, there existsv∈Tysatisfying u–v ≤H(Tx,Ty) and
H(Tx,Ty)≤ x–y+kx–u– (y–v). ()
Ifk= in (),Tis said to be apseudocontractive-typemapping.Tis called nonexpansive-type ifk= . Clearly, every multivalued nonexpansive mapping is a nonexpansive-type mapping.
From the definitions, it is clear that every multivalued nonexpansive-type mapping is k-strictly pseudocontractive-type and everyk-strictly pseudocontractive-type mapping is pseudocontractive-type. The following examples show that the class of nonexpansive-type mappings is properly contained in the class ofk-strictly pseudocontractive-type mappings and the class ofk-strictly pseudocontractive-type mappings is properly contained in the class of pseudocontractive-type mappings.
Example LetX=R(the reals with usual metric). DefineT: [,∞)→Rby
Tx=
–x , –x
. ()
ThenPTx={–x}for allx∈[,∞), hence it is not nonexpansive.
H(Tx,Ty) =max(x–y), (x–y)
= |x–y|
=|x–y|+
|x–y|
.
Also, for eachu∈Tx,u= –αx, ≤α≤, choosev= –αy. Then
|u–v|=–αx– (–αy)=α|x–y| ≤
|x–y|=H(Tx,Ty)
and
x–u– (y–v)
=x– (–αx) –y– (–αy)
=( +α)x– ( +α)y= ( +α)|x–y|.
Hence
H(Tx,Ty) =|x–y|+ |x–y|
=|x–y|+
( +α)( +α) |x–y|
=|x–y|+
( +α)x–u– (y–v)
≤ |x–y|+
( + )x–u– (y–v)
=|x–y|+
x–u– (y–v)
Consequently,T isk-strictly pseudocontractive-type withk=. It then follows thatT is pseudocontractive-type. Observe thatT is not nonexpansive-type so that the class of multivalued nonexpansive-type mappings is properly contained in the class of multival-uedk-strictly pseudocontractive-type mappings. Next, we show that the class of multi-valued pseudocontractive-type mappings properly contains the class of multimulti-valued k -strictly pseudocontractive-type mappings.
Example LetX=R(the reals with usual metric). DefineT: [,∞)→Rby
Tx= [–√x, –x]. ()
ThenPTx={–x}which is not nonexpansive. Also,
H(Tx,Ty) =max√(x–y),(x–y)= |x–y|=|x–y|+ |x–y|.
Furthermore, for eachu∈Tx,u= –αx, ≤α≤√, choosev= –αy. Then
|u–v|=–αx– (–αy)=α|x–y| ≤√|x–y|=H(Tx,Ty)
and
x–u– (y–v)=x– (–αx) –y– (–αy)=( +α)x– ( +α)y= ( +α)|x–y|.
It then follows that
H(Tx,Ty) =|x–y|+ |x–y|
=|x–y|+ ( + )|x–y|
≤ |x–y|+ ( +α)|x–y|
=|x–y|+x–u– (y–v).
Now, for any pairx, ∈[,∞),T ={}. It then follows that for anyu∈Txthe corre-spondingv= . In particular, foru= –x,
H(Tx,T) = |x– |
=|x– |+ |x– |
=|x– |+x– (–x)
=|x– |+x–u– (y–v)
>|x–y|+kx–u– (y–v) ∀k∈(, ).
It then follows thatTis a pseudocontractive-type mapping but not ak-strictly pseudocon-tractive-type mapping.
Proposition Let E be a real Banach space.Suppose T:D(T)⊂E→Eis a
pseudocon-tractive-type mapping.Then I–T is monotone.
Proof SinceT is pseudocontractive-type, for any pairx,y∈D(T) andu∈Tx, there exists v∈Tysuch that
u–v≤H(Tx,Ty)≤ x–y+x–u– (y–v).
Now,
u–v =x–y+x–u– (y–v)– x–u– (y–v),x–y ≤ x–y+x–u– (y–v).
Hence,x–u– (y–v),x–y ≥. Hence,I–Tis monotone.
Proposition Let E be a normed space. And let T :D(T)⊆E→E be a k-strictly
pseudocontractive-type mapping.Then T is a L-Lipschitzian mapping.
Proof LetT be ak-strictly pseudocontractive-type mapping. Then there existsk∈[, ) such that, for allx,y∈D(T) andu∈Tx, there existsv∈Tysatisfyingu–v ≤H(Tx,Ty) and
H(Tx,Ty)≤ x–y+kx–u– (y–v).
We then have that
H(Tx,Ty)≤ x–y+kx–u– (y–v)
≤x–y+√kx–u– (y–v).
It then follows that
H(Tx,Ty)≤ x–y+√kx–y+u–v
≤ x–y+√kx–y+H(Tx,Ty).
Hence
H(Tx,Ty)≤( + √
k)
( –√k)x–y.
Proposition Let H be a real Hilbert space.Let K be a nonempty weakly closed sub-set of H.Let T:K⊆H→P(H)be a multivalued mapping from K into the family of all nonempty proximinal subsets of H.Suppose that PTis a k-strictly pseudocontractive-type
mapping.Then(I–PT)is demiclosed at zero(i.e.,the graph of I–PT is closed at zero in
Proof Let{xn}∞n=⊆Kbe such that{xn}converges weakly topand a sequence{yn}with
yn∈PTxn for alln∈Nsuch that{xn–yn}converges strongly to . We prove that ∈
(I–PT)p(i.e.,p=vfor somev∈PTp). Since{xn}∞n=converges weakly, it is bounded. Let
q∈PTpbe arbitrary. From the definition ofk-strictly pseudocontractive-type, for each
n∈N, there existsvn∈PTxnsuch that
vn–q ≤H(PTxn,PTp), ()
and
H(PTxn,PTp)≤ xn–p+kxn–vn– (p–q)
()
for alln∈N. Also, sincevn,yn∈PTxn,xn–vn=xn–yn. Thus we have
lim
n→∞xn–vn=nlim→∞xn–yn= .
For eachx∈X, definef :E→[,∞) by
f(x) :=lim sup
n→∞ xn–x
.
Then, from Lemma , we obtain
f(x) =lim sup
n→∞ xn–p
+p–x ∀x∈E.
Thus
f(x) =f(p) +p–x ∀x∈E.
Therefore,
f(q) =f(p) +p–q. ()
Observe also that
f(q) =lim sup
n→∞ xn–q
=lim sup
n→∞
xn–vn+ (vn–q)
=lim sup
n→∞
vn–q
≤lim sup
n→∞
H(PTxn,PTp)
≤lim sup
n→∞
xn–p+kxn–vn– (p–q)
=lim sup
n→∞ xn–p
+k(p–q)
=f(p) +kp–q. ()
Theorem Let K be a nonempty closed and convex subset of a real Hilbert space H. Sup-pose that T:K→P(K)is a k-strictly pseudocontractive-type mapping from K into the family of all proximinal subsets of K with k∈(, )such that F(T)=∅and T(p) ={p}for all p∈F(T).Suppose(I-T)is weakly demiclosed at zero.Then the Mann type sequence defined by
xn+= ( –αn)xn+αnyn
converges weakly to q∈F(T),where yn∈Txnwithxn–yn=d(xn,Txn)andαnis a real
sequence in(, )satisfying: (i)αn→α< –k; (ii)α> ; (iii)
∞
n=αn( –αn) =∞.
Proof Using the well-known identity
tx+ ( –t)y=tx+ ( –t)y–t( –t)x–y,
which holds for allx,y∈Hand for allt∈[, ], we obtain
xn+–p=( –αn)xn+αnyn–p
=( –αn)(xn–p) +αn(yn–p)
= ( –αn)xn–p+αnyn–p–αn( –αn)xn–yn
≤( –αn)xn–p+αnH(Txn,Tp) –αn( –αn)xn–yn
≤( –αn)xn–p+αn
xn–p+kd(xn,Txn)
–αn( –αn)xn–yn
≤ xn–p+αnkxn–yn
–αn( –αn)xn–yn
=xn–p–αn
– (αn+k)
xn–yn.
It then follows thatlimn→∞xn–pexists, hence{xn}is bounded. Also,
∞
n=
αn
– (αn+k)
xn–yn≤ x–p≤ ∞.
Sinceα> , from (ii), we have thatlimn→∞xn–yn= . Also, sinceKis closed and{xn} ⊆
K with{xn}bounded, there exists a subsequence{xnt} ⊆ {xn} such that{xnt}converges weakly to some q∈K. Also,limn→∞xn–yn= implies thatlimn→∞xnt –ynt= .
Since (I–T) is weakly demiclosed atzero, we have thatq∈Tq. SinceH satisfies Opial’s condition [], we have that{xn}converges weakly toq∈F(T).
Corollary Let H be a real Hilbert space and K be a nonempty closed and convex subset of H.Let T:K→P(K)be a multivalued mapping from K into the family of all proximinal subsets of K with F(T)=∅.Suppose PTis a k-strictly pseudocontractive-type mapping with
k∈(, ).Then the Mann sequence{xn}defined in Theoremconverges weakly to a point
Proof The proof follows easily from Lemma , Proposition and Theorem .
Remark It is easy to see that Examples and satisfy the condition ‘given any pair x,y∈D(T) andu∈Txwithx–u=d(x,Tx), there existsv∈Tywithy–v=d(y,Ty) satisfying the conditions of Definition ’. Also, ifTis a multivalued mapping such thatPT
is a pseudocontractive-type mapping, then given any pairx,y∈D(T) andu∈PTxwith the
correspondingv∈PTysatisfying the conditions of Definition , it is the case thatx–u=
d(x,PTx) andy–v=d(y,PTy).
Based on Lemma and Remark above, we will first prove weak and strong conver-gence for the new class of pseudocontractive-type mappings with the following two con-ditions: (i) given any pair x,y∈D(T) and u∈Tx withx–u=d(x,Tx), there exists v∈Tywithy–v=d(y,Ty) satisfying the conditions of Definition ; (ii)T(p) ={p}for allp∈F(T), then obtain the case for an arbitrary multivalued mappingTsuch thatPTis
a pseudocontractive-type mapping without the two conditions onPTas corollary.
Theorem Let K be a nonempty closed and convex subset of a real Hilbert space X. Sup-pose that T:K→P(K)is an L-Lipschitzian pseudocontractive-type mapping from K into the family of all proximinal subsets of K such that F(T)=∅and T(p) ={p}for all p∈F(T). Suppose,for any pair x,y∈K and u∈Tx withx–u=d(x,Tx),there exists v∈Ty with y–v=d(y,Ty)satisfying the conditions of Definition.Suppose T satisfies condition(). Then the Ishikawa sequence defined by
yn= ( –βn)xn+βnun,
xn+= ( –αn)xn+αnwn
()
converges strongly to p∈F(T),where un∈Txnwithxn–un=d(xn,Txn),wn∈Tynwith
yn–wn=d(yn,Tyn)satisfying the conditions in Definitionand{αn}and{βn}are real
sequences satisfying: (i) ≤αn≤βn< ; (ii)lim infn→∞αn=α> ; (iii)supn≥βn≤β≤
√
+L+.
Proof
xn+–p=( –αn)xn+αnwn–p
=( –αn)(xn–p) +αn(wn–p)
= ( –αn)xn–p+αnwn–p
–αn( –αn)xn–wn
≤( –αn)xn–p+αnH(Tyn,Tp)
–αn( –αn)xn–wn
≤( –αn)xn–p+αn
yn–p
+d(yn,Tyn)
–αn( –αn)xn–wn
= ( –αn)xn–p+αnyn–p+αnd(yn,Tyn)
Also,
d(yn,Tyn)≤ yn–wn
=( –βn)xn+βnun–wn
=( –βn)(xn–wn) +βn(un–wn)
= ( –βn)xn–wn+βnun–wn
–βn( –βn)xn–un. ()
() and () imply that
xn+–p≤( –αn)xn–p+αnyn–p
+αn
( –βn)xn–wn+βnun–wn
–βn( –βn)xn–un
–αn( –αn)xn–wn, ()
yn–p=( –βn)xn+βnun–p
=( –βn)(xn–p) +βn(un–p)
= ( –βn)xn–p+βnun–p
–βn( –βn)xn–un
≤( –βn)xn–p+βnH(Txn,Tp)
–βn( –βn)xn–un,
≤( –βn)xn–p+βn
xn–p+d(xn,Txn)
–βn( –βn)xn–un
≤( –βn)xn–p+βnxn–p+βnxn–un
–βn( –βn)xn–un
=xn–p+βnxn–un. ()
() and () imply that
xn+–p≤( –αn)xn–p
+αn
xn–p+βnxn–un
+αn
( –βn)xn–wn+βnun–wn
–βn( –βn)xn–un
–αn( –αn)xn–wn
= ( –αn)xn–p+αnxn–p+αnβnxn–un
–αnβn( –βn)xn–un–αn( –αn)xn–wn
≤ xn–p+αnβnxn–un+αnβnH(Txn,Tyn)
–αn(βn–αn)xn–wn
–αnβn( –βn)xn–un
≤ xn–p+αnβnxn–un+αnβnLxn–un
–αnβn( –βn)xn–un
–αn(βn–αn)xn–wn
=xn–p–αnβn
– βn–Lβn
xn–un
–αn(βn–αn)xn–wn
=xn–p–αnβn
– βn–Lβn
xn–un. ()
It then follows from Lemma thatlimn→∞xn–pexists. Hence{xn}is bounded, so are
{un}and{wn}. We then have from (), (ii) and (iii) that
∞
n=
α – β–Lβxn–un≤
∞
n=
αnβn
– βn–Lβn
xn–un
≤ ∞
n=
xn–p–xn+–p
≤ x–p+D<∞.
It then follows thatlimn→∞xn–un= . Sinceun∈Txn, we have thatd(xn,Txn)≤ xn–
un → asn→ ∞. SinceT satisfies condition (),limn→∞d(xn,F(T)) = . Thus there
exists a subsequence{xnk}of{xn}such thatxnk –pk ≤
k for some{pk} ⊆F(T). From
()
xnk+–pk ≤ xnk–pk.
We now show that{pk}is a Cauchy sequence inF(T).
pk+–pk ≤ pk+–xnk++xnk+–pk
≤
k+ +
k=
k–.
Therefore{pk}is a Cauchy sequence and converges to someq∈KbecauseK is closed.
Now,
xnk–q ≤ xn–pk+pk–q.
Hencexnk→qask→ ∞,
d(q,Tq)≤ q–pk+pk–xnk+d(xnk,Txnk) +H(Txnk,Tq)
Hence,q∈Tqand{xnk}converges strongly toq. Sincelimxn–qexists, we have thatxn converges strongly toq∈F(T).
Corollary Let H be a real Hilbert space and K be a nonempty closed and convex subset of H.Let T:K→P(K)be a multivalued mapping from K into the family of all proximinal subsets of K such that F(T)=∅.Suppose PTis an L-Lipschitzian pseudocontractive-type
mapping.If T satisfies condition(),then the Ishikawa sequence{xn}defined in()
con-verges strongly to p∈F(T).
Proof The proof follows easily from Lemma , Remark and Theorem .
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This work was carried out at the University of Kwazulu Natal, South Africa when the author visited under the OWSDW [formally TWOWS], Abdus Salam International Centre for Theoretical Physics (ICTP) Trieste, Italy, Postgraduate Training Fellowship. She is grateful to OWSDW for the Fellowship and to University of Kwazulu Natal for making their facilities available and for hospitality.
Received: 19 July 2012 Accepted: 24 February 2013 Published: 18 March 2013
References
1. Landers, D, Rogge, L: Martingale representation in uniformly convex Banach spaces. Proc. Am. Math. Soc.75(1), 108-110 (1979)
2. Fitzpatrick, S: Metric projections and the differentiability of distance functions. Bull. Aust. Math. Soc.22, 291-312 (1980)
3. Sastry, KPR, Babu, GVR: Convergence of Ishikawa iterates for a multivalued mapping with a fixed point. Czechoslov. Math. J.55, 817-826 (2005)
4. Panyanak, B: Mann and Ishikawa iteration processes for multivalued mappings in Banach spaces. Comput. Math. Appl.54, 872-877 (2007)
5. Song, Y, Wang, H: Erratum to “Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces” [Comput. Math. Appl.54, 872-877 (2007)]. Comput. Math. Appl.55, 2999-3002 (2008)
6. Nadler, SB Jr.: Multivalued contraction mappings. Pac. J. Math.30, 475-488 (1969)
7. Shahzad, N, Zegeye, H: On Mann and Ishikawa iteration schemes for multi-valued mappings in Banach spaces. Nonlinear Anal.71(3-4), 838-844 (2009)
8. Khan, SH, Yildirim, I: Fixed points of multivalued nonexpansive mappings in a Banach spaces. Fixed Point Theory Appl.2012, 73 (2012). doi:10.1186/1687-1812-2012-73
9. Agarwal, RP, O’Regan, D, Sabu, DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal.8(1), 61-79 (2007)
10. Schu, J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc.43, 153-159 (1991)
11. Song, Y, Cho, YJ: Some notes on Ishikawa iteration for multivalued mappings. Bull. Korean Math. Soc.48(3), 575-584 (2011). doi:10.4134/BKMS.2011.48.3.575
12. Garcia-Falset, J, Lorens-Fuster, E, Suzuki, T: Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl.375, 185-195 (2011)
13. Dozo, EL: Multivalued nonexpansive mappings and Opial’s condition. Proc. Am. Math. Soc.38(2), 286-292 (1973) 14. Markin, JT: A fixed point theorem for set valued mappings. Bull. Am. Math. Soc.74, 639-640 (1968)
15. Tan, KK, Xu, HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl.178, 301-308 (1993)
16. Browder, FE, Petryshyn, WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl.
20, 197-228 (1967)
17. Osilike, MO, Isiogugu, FO: Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces. Nonlinear Anal.74, 1814-1822 (2011)
18. Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc.73, 591-597 (1967)
doi:10.1186/1687-1812-2013-61
