R E S E A R C H
Open Access
Convergence theorems for new classes of
multivalued hemicontractive-type mappings
Felicia O Isiogugu
*and Micah O Osilike
*Correspondence:
[email protected] Department of Mathematics, University of Nigeria, Nsukka, Nigeria
Abstract
Weak and strong convergence theorems are proved in Hilbert spaces for new classes of multivalued demicontractive-type and hemicontractive-type mappings which are related to the class of multivalued pseudocontractive-type mappings studied by Isiogugu (Fixed Point Theory Appl. 2013:61, 2013). Thus our results extend and improve several corresponding results in the contemporary literature.
MSC: 47H10; 54H25
Keywords: Hilbert spaces; demicontractive-type mappings; hemicontractive-type mappings; demiclosedness principle; weak and strong convergence
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 well known that every closed convex subset of a uniformly convex Banach space is proximinal. For a nonempty setE, we shall denote the family of all nonempty proximinal subsets ofEbyP(E), the family of all nonempty closed and bounded subsets ofEbyCB(E), the family of all nonempty closed, convex, and bounded subsets ofEbyCVB(E), the family of all nonempty closed subsets ofEbyC(X), the family of all nonempty subsets ofE by E, the identity onEbyI, the weak topology ofEbyσ(E,E∗), and the norm (or strong)
topology ofEby (E, · ).
LetHdenote the Hausdorff metric induced by the metricdonE, that is, for everyA,B∈ E,
H(A,B) =maxsup
a∈Ad(a,B),
sup
b∈Bd(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 the 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 the strict
fixed point set ofT. A multivalued mappingT:D(T)⊆E→Eis calledL-Lipschitzianif there existsL≥ such that for allx,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-nonexpansive. Several authors have studied various classes of multivalued mappings. In [], Shahzad and Zegeye studied certain classes of multivalued nonself mappings in Banach spaces and constructed an appropriate net which converges strongly to a fixed point of the classes of the mappings. Recently, Isiogugu [] introduced new classes of multivalued mappings as follows.
Definition .([]) LetXbe a normed space. A multivalued mappingT:D(T)⊆X→X
is said to bek-strictly pseudocontractive-typein the sense of Browder and Petryshyn [] if there existsk∈[, ) such that given anyx,y∈D(T) andu∈Tx, there existsv∈Ty satisfyingu–v ≤H(Tx,Ty) and
H(Tx,Ty)≤ x–y+kx–u– (y–v). (.)
Ifk= in (.)T is said to be apseudocontractive-typemapping.T is called nonexpan-sive-typeifk= . Clearly, every multivalued nonexpansive mapping is 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. Examples to show that the class of nonexpansive-type mappings is properly contained in the class ofk-strictly pseudocontractive-type mappings and that the class ofk-strictly pseudocontractive-type mappings is properly contained in the class of pseudocontractive-type mappings were given in []. The following theorems were also proved in [].
Theorem . Let K be a nonempty closed and convex subset of a real Hilbert space H. Suppose 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)
∞
Theorem . Let K be a nonempty closed and convex subset of a real Hilbert space X. Suppose 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 withy–v=d(y,Ty)satisfying the conditions of Definition..Suppose T satisfies condition() (i.e.,if there exists a nondecreasing function f : [,∞)→[,∞)with f() = and f(r) > for all r∈(,∞)such that d(x,Tx)≥f(d(x,F(T))),∀x∈K.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 Definition.and{αn}and{βn}are real
sequences satisfying: (i) ≤αn≤βn< ; (ii)lim infn→∞αn=α> ; (iii)supn≥βn≤β≤
√ +L+.
In [], Chidumeet al.also considered a class of multivaluedk-strictly pseudocontractive mappings defined as follows.
LetHbe a real Hilbert space. A multivalued mappingT:D(T)⊆H→CB(H) is said to bek-strictly pseudocontractive if there existsk∈(, ) such that for allx,y∈D(T) one has
H(Tx,Ty)≤ x–y+kx–u– (y–v), ∀u∈Tx,v∈Ty.
If k= ,T is said to be pseudocontractive mapping. They constructed a Mann-type it-eration scheme which is an approximate fixed point sequence and obtain some strong convergence theorems for the class ofk-strictly pseudocontractive mappings.
The following example shows that the class of multivalued pseudocontractive-type map-pings considered by Isiogugu [] is not a subclass of the multivalued pseudocontractive mappings considered by Chidumeet al.[].
Example . LetX=R(the reals with usual metric). DefineT: [,∞)→CB(R) by
Tx=
–x , –x
. (.)
It was shown in [] thatTisk-strictly pseudocontractive-type mapping hence pseudo-contractive-type. However, forx= ,y= if we chooseu= –∈Txandv= –∈Tythen H(Tx,Ty) = andx–y+x–u– (y–v)= . Consequently,
H(Tx,Ty) >x–y+x–u– (y–v),
which implies thatTis not pseudocontractive and hence notk-strictly pseudocontractive mapping in the sense of Chidumeet al.[].
[], single-valued mappings of Browder and Petryshyn [], Hicks and Kubicek [] and Naimpally and Singh []. We also prove weak and strong convergence theorems for ap-proximation of fixed points of our classes of mappings.
2 Preliminaries
We shall 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., ∈(I–T)p).
Observe that ifT is a multivalued Lipschitzian mapping, thenI–T is strongly demi-closed.
Definition .(see,e.g., [, ]) Let Ebe a Banach space. LetT :D(T)⊆E→E be a
multivalued mapping.I–T is said to beweakly demiclosed at zeroif 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 .(see,e.g., [, ]) Let Ebe a Banach space. LetT :D(T)⊆E→E be a multivalued mapping. The graph ofI–T is said to beclosedinσ(E,E∗)×(E, · ) (i.e., I–Tisweakly demiclosed or demiclosed) if 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 toy. Theny∈(I–T)p(i.e.,y=p–vfor somev∈Tp).
Definition . A BanachXis said to satisfy Opial’s condition if whenever a sequence{xn} converges weakly tox∈Xthen it is the case that
lim infxn–x<lim infxn–y,
for ally∈X,y=x.
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)≥fdx,F(T), ∀x∈K.
Lemma .([]) Let{an},{βn},and{γn}be sequences of nonnegative real numbers satis-fying the following relation:
an+≤( +βn)an+γn, n≥n,
where nis a nonnegative integer.If
βn<∞,
γn<∞,thenlimn→∞anexists.
() x∈Tx; () PTx={x};
() x∈F(PT).
Moreover,F(T) =F(PT).
Lemma .([]) Let A,B∈CB(X)and a∈A.Ifγ > ,then there exists b∈B such that
d(a,b)≤H(A,B) +γ.
3 Main results
We now introduce the new classes of multivalued demicontractive-type and hemicon-tractive-type mappings and prove some convergence theorems for these classes of map-pings.
Definition . LetXbe a real normed space. A mappingT :D(T)⊆X→X is said to
bedemicontractivein the terminology of Hicks and Kubicek [] ifF(T)=∅and for all p∈F(T),x∈D(T) there existsk∈[, ) such that
H(Tx,Tp)≤ x–p+kd(x,Tx), (.)
whereH(Tx,Tp) = [H(Tx,Tp)]andd(x,p) = [d(x,p)].
Ifk= in (.) thenTis called a hemicontractive mapping.
The following are some examples of demicontractive mappings.
Example . Every multivalued quasi-nonexpansive mapping is demicontractive.
Example . LetXbe a normed space. Suppose thatTis a multivalued mapping such that F(T)=∅and thatPTis ak-strictly pseudocontractive-type mapping; thenPTis
demicon-tractive.
Example . LetX be a normed space. Let T : D(T)⊆X →P(X) be a multivalued k-strictly pseudocontractive-type with a nonempty fixed point set. SupposeTp={p}for allp∈F(T); then for anyx∈D(T),p∈F(T) andu∈Txwithu–x=d(x,Tx) we have
H(Tx,Tp)≤ x–p+kx–u=x–p+kd(x,Tx);
therefore,Tis demicontractive-type.
Example . LetX=R(the reals with usual metric). DefineT:R→Rby
Tx= [–
x
, –x], x∈(–∞, ],
[–x, –x], x∈(,∞). (.)
ThenF(T) ={}. For eachx∈(–∞, )∪(,∞),
H(Tx,T) =|–x– |= |x– |=|x– |+ |x|,
d(x,Tx) =x–
–x
=x
=
|x|
Therefore,
H(Tx,T) =|x– |+ |x|=|x– |+ d
(x,Tx)≤ |x– |+kd(x,Tx).
Consequently,T is demicontractive-type withk=. It then follows thatTis hemicon-tractive. Observe thatTis not nonexpansive so that the class of multivalued quasi-nonexpansive mappings is properly contained in the class of multivalued demicontractive-type mappings.
Next is an example of a multivalued mappingTwithF(T)=∅,Tp={p}for allp∈Tpfor whichPTis a demicontractive-type but not ak-strictly pseudocontractive-type mapping.
Example . LetX=R(the reals with usual metric). DefineT: [–, ]→[–,]by
Tx=
⎧ ⎪ ⎨ ⎪ ⎩
[–,xsinx], x∈(, ],
{}, x= ,
[xsinx, ], x∈[–, ).
(.)
ThenF(T) ={}. For eachx∈[–, ],
PTx= { xsin
x}, x= ,
{}, x= , (.)
which is demicontractive-type but notk-strictly pseudocontractive-type (see for exam-ple []).
The following example shows that the class of demicontractive mapping is properly con-tained in the class of hemicontractive mappings.
Example . LetX=R(the reals with the usual metric). DefineT:R→Rby
Tx= [–
√
x, ], x∈[,∞),
[, –√x], x∈(–∞, ). (.)
ThenF(T) ={}. For eachx∈(–∞, )∪(,∞),
H(Tx,T) =|√x– |= |x– |=|x– |+|x– |,
d(x,Tx) =|x– |=|x– |.
Therefore,
H(Tx,T)≤ |x– |+|x– |=|x– |+d(x,Tx) >|x– |+kd(x,Tx), (.)
∀x∈Band∀k∈[, ). Therefore,T is hemicontractive but not demicontractive.
Example . LetXbe a normed space. SupposeT is a multivalued mapping such that F(T)=∅andPTis pseudocontractive-type mapping; thenPTis hemicontractive.
Example . Let X be a normed space. Let T : D(T)⊆X →P(X) be a multivalued pseudocontractive-type with a nonempty fixed point set. SupposeTp={p}for allp∈F(T); then for anyx∈D(T),p∈F(T) andu∈Txwithu–x=d(x,Tx) we have
H(Tx,Tp)≤ x–p+x–u=x–p+d(x,Tx).
The following lemma shows that Lemma . is also valid for allA,B∈P(E) andγ = .
Lemma . Let E be a metric space.If A,B∈P(E)and a∈A,then it is a simple consequence of the Hausdorff metric H that there exists b∈B such that
d(a,b)≤H(A,B). (.)
Proof LetEbe a metric space andP(E) be the family of all nonempty proximinal subsets ofE. LetA,B∈P(E) anda∈A. SinceBis proximinal, there existsba∈Bsuch that
d(a,B) =d(a,ba).
Observe that
H(A,B) =max
sup
u∈Ad(u,B),
sup
v∈Bd(v,A)
≥sup
u∈Ad(u,B)≥d(a,B) =d(a,ba).
Hence the result follows.
Remark . Lemma . holds ifEis a reflexive real Banach space andP(E) is replaced withCB(K) withBweakly closed (see for example []).
We now prove the following theorems.
Theorem . Let K be a nonempty closed and convex subset of a real Hilbert space H. Suppose that T:K→P(K)is a demicontractive mapping from K into the family of all proximinal subsets of K with k∈(, )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∈Txnandαnis a real sequence in(, )satisfying:
(i)αn→α< –k; (ii)α> .
Proof Using the well-known identity:
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 limn→∞xn–yn= . Thuslimn→∞d(xn,Txn) = . Also
sinceKis closed and{xn} ⊆Kwith{xn}bounded, there exist a subsequence{xnt} ⊆ {xn}
such that{xnt}converges weakly to someq∈K. Alsolimn→∞xn–yn= implies that
limn→∞xnt–ynt= . Since (I–T) is weakly demiclosed atzerowe haveq∈Tq. SinceH
satisfies Opial’s condition [] we find that{xn}converges weakly toq∈F(T).
Corollary . Let K be a nonempty closed and convex subset of a real Hilbert space H. Suppose that T:K→P(K)is 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 sequence{xn} defined in Theorem.converges weakly to a point of F(T).
Proof The proof follows easily from Example . and Theorem ..
Corollary . Let H be a real Hilbert space and K 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.Suppose PTis a demicontractive mapping with k∈(, )and(I–PT)is weakly
demiclosed at zero.Then the Mann sequence{xn}defined in Theorem.converges weakly to a point of F(T).
Proof The proof follows easily from Lemma . and Theorem ..
Remark . Since the choice ofyn∈Txnin the Mann-type iteration scheme is
indepen-dent ofd(xn,Txn), we can also replaceP(K) withCB(K) in Theorem . and its corollaries.
Furthermore, sincelimn→∞d(xn,Txn) = , one can impose standard conditions onTorK
Theorem . Let K be a nonempty closed and convex subset of a real Hilbert space X. Suppose that T:K→P(K)is an L-Lipschitzian hemicontractive mapping from K into the family of all proximinal subsets of K and Tp={p} for all p∈F(T).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 ∈ Txn, wn ∈ Tyn satisfying the conditions
of Lemma . and {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)
–αn( –αn)xn–wn, (.)
d(yn,Tyn)≤ yn–wn
=( –βn)xn+βnun–wn
=( –βn)(xn–wn) +βn(un–wn)
= ( –βn)xn–wn+βnun–wn
–βn( –βn)xn–un. (.)
Equations (.) 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+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. (.)
Equations (.) 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
+αn( –βn)xn–wn+αnβnun–wn
–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+αnβn xn–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{un}
and{wn}also are. We then have from (.), (ii), and (iii)
∞
n=
α – β–Lβxn–un≤ ∞
n=
αnβn
– βn–Lβn
xn–un
≤ ∞
n=
xn–p–xn+–p
It then follows thatlimn→∞xn–un= . Sinceun∈Txnwe haved(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). We have
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→ ∞. We have
d(q,Tq)≤ q–pk+pk–xnk+d(xnk,Txnk) +H(Txnk,Tq)
≤ q–pk+pk–xnk+d(xnk,Txnk) +Lxnk–q.
Hence,q∈Tqand{xnk}converges strongly toq. Sincelimxn–qexists we see thatxn
converges strongly toq∈F(T).
Corollary . Let K be a nonempty closed and convex subset of a real Hilbert space X. Suppose 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 T satisfies condition().Then the Ishikawa sequence {xn} defined in (.)converges strongly to p∈F(T).
Proof The proof follows easily from Example ., Lemma ., and Theorem ..
Corollary . Let H be a real Hilbert space and K 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 hemicontractive mapping.
If T satisfies condition ().Then the Ishikawa sequence{xn} defined in(.) converges strongly to p∈F(T).
Proof The proof follows easily from Lemma . and Theorem ..
Remark . In Theorem . and its corollaries we can replaceP(K) withCB(K) with addi-tional condition thatTis weakly closed for allx∈D(T) =Kin order to ensure thatunand
wnsatisfy Lemma . as indicated in Remark .. Furthermore, sincelimn→∞d(xn,Txn) =
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.
Acknowledgements
The second author is grateful to the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy for their facilities and for hospitality. He contributed to the work during his visits to the Centre as a regular associate. The work was completed while the first author was visiting the University of Kwazulu Natal, South Africa under the OWSD (formally TWOWS) Postgraduate Training Fellowship. She is grateful to OWSD (formally TWOWS) for the Fellowship and to University of Kwazulu Natal for making facilities available and for hospitality.
Received: 27 September 2013 Accepted: 13 February 2014 Published:09 Apr 2014 References
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