R E S E A R C H
Open Access
On approximation of fixed points of
multivalued pseudocontractive mappings in
Hilbert spaces
Felicia Obiageli Isiogugu
**Correspondence:
[email protected] School of Mathematics, Statistics and Computer Sciences, University of KwaZulu-Natal, Westville Campus, Durban, 4000, South Africa Department of Mathematics, University of Nigeria, Nsukka, Nigeria
Abstract
It is proved that if a multivaluedk-strictly pseudocontractive mappingSof Chidume
et al.(Abstr. Appl. Anal. 2013:629468, 2013) is of type one, thenI–Sis demiclosed at zero. Also, under this condition, the Mann (respectively, Ishikawa) sequence weakly (respectively, strongly) converges to a fixed point of a multivaluedk-strictly pseudocontractive (respectively, pseudocontractive) mappingSwithout the condition that the fixed point set ofSis strict, whereSis of type one if for any pair
r,g∈D(S),
u–v ≤
(
Sr,Sg) for allu∈PSr,v∈PSg,and
denotes the Hausdorff metric. The results obtained give a partial answer to the problem of the removal of the strict fixed point set condition, which is usually imposed on multivalued mappings. Thus, the results extend, complement, and improve the results on multivalued and single-valued mappings in the contemporary literature.
MSC: 47H10; 54H25
Keywords: Hilbert spaces;k-strictly pseudocontractive mappings; pseudocontractive mappings; demiclosedness; type one
1 Introduction
The approximation of fixed points of multivalued mappings with respect to the Haus-dorff metric, using Mann [] or Ishikawa [] iteration scheme, has never been successful without imposing the condition that either the fixed point set ofSis strict or thatSis a multivalued mapping for whichPSsatisfies some contractive conditions (see,e.g., [, –
], and references therein). Among these recent studies, Chidumeet al.[] introduced the class of multivalued pseudocontractive mappings as follows.
Definition .([]) LetHbe a real Hilbert space. A multivalued mappingS:D(S)⊆H→ CB(H) from the domain ofSinto the family of all closed and bounded subsets ofHis said to bek-strictly pseudocontractive in the sense of Chidumeet al.[] if there existsk∈(, )
such that
(Sr,Sg)≤ r–g+kr–u– (g–v) for allu∈Sr,v∈Sg.
Ifk= , thenSis said to be pseudocontractive. Also, they proved some weak and strong convergence theorems for this class of mappings using the condition that either the fixed point set ofSis strict orS is a multivalued mapping such thatPS is pseudocontractive.
They proved the following theorems.
Theorem .([]) Let C be a nonempty,closed,and convex subset of a real Hilbert space H,
and let S:C→P(C)be a multivalued mapping such that F(S)=∅.Assume that PSis k-strictly pseudocontractive.Let{rn}be a sequence defined iteratively from arbitrary point r∈C by
rn+= ( –λ)rn+λgn,
where gn∈PSrnandλ∈(, –k).Then,limn→∞d(rn,Srn) = .
Theorem .([]) Let C be a nonempty,closed,and convex subset of a real Hilbert space H,
and let S:C→P(C)be a multivalued k-strictly pseudocontractive mapping with F(S)=∅
such that for every r∈C,Sr is weakly closed and Sz={z}for all z∈F(S).Suppose that S is hemicompact.Let{rn}be a sequence defined iteratively from arbitrary point r∈C by
rn+= ( –λ)rn+λgn,
where gn∈PSrnandλ∈(, –k).Then,the sequence{rn}strongly converges to a fixed point of S.
Recently, Isioguguet al.[] observed that there are multivalued Lipschitzian maps with nonempty fixed point set for which neither the fixed point set ofSis strict norPS
satis-fies any of the contractive conditions studied so far by authors. They also noted that these classes of maps have another interesting property of some existing maps considered re-cently by authors (see,e.g., [, –, , ], and references therein). They therefore sug-gested to approximate the fixed points of multivalued mappingsSdirectly instead ofPS
and without imposing the strict fixed point set condition onS. This suggestion was also due to the fact that it has not been established that if a multivalued mapSbelongs to a class of maps, then PS necessarily belongs to the same class of maps and that the fixed
point set ofSneed not be strict. Consequently, they introduced the ‘type-one’ condition, which guarantees the weak convergence of the Mann sequence{rn}without imposing the condition that the fixed point set of S is strict to a fixed point of a multivalued quasi-nonexpansive mappingSin a real Hilbert space. They also proved that under this condi-tion, ifSis nonexpansive, thenI–Sis demiclosed at zero. They obtained the following results.
that{rn}∞n=⊆C is such that{rn}weakly converges to p and a sequence{gn}with gn∈Srn andrn–gn=d(rn,Srn)for all n∈Nis such that{rn–gn}strongly converges to.Then
∈(I–S)p(i.e.,p=v for some v∈Sp).
Theorem .([]) Let C be a nonempty closed and convex subset of a real Hilbert space H.
Suppose that S:C→P(C)is a type-one nonexpansive mapping from C into the family of all proximinal subsets of C such that F(S)=∅.Then the Mann-type sequence defined by
rn+= ( –μn)rn+μngn
weakly converges to q∈F(S), where gn∈Srn withrn–gn=d(rn,Srn)andμn⊆(, ) satisfyingμn→μ∈(, ).
Theorem .([]) Let C be a nonempty closed and convex subset of a real Hilbert space H.
Suppose that S:C→P(C)is a type-one quasi-nonexpansive mapping from C into the family of all proximinal subsets of C.If(I–S)satisfies Proposition.,then the Mann-type sequence defined by
rn+= ( –μn)rn+μngn
weakly converges to q∈F(S), where gn∈Srn withrn–gn=d(rn,Srn)andμn⊆(, ) satisfyingμn→μ∈(, ).
Therefore, our purpose in this work is to establish that the type-one condition intro-duced by Isioguguet al.[] guarantees the weak (respectively, strong) convergence of the Mann (respectively, Ishikawa) sequence{rn}to a fixed point ofk-strictly pseudocontrac-tive (respecpseudocontrac-tively, pseudocontracpseudocontrac-tive) mappingSof Chidumeet al.[] without the condi-tion that the fixed point set ofSis strict in a real Hilbert space. It is also proved that, under this condition,I–Sis demiclosed at zero ifSisk-strictly pseudocontractive. The results obtained extend, complement, and improve the results on multivalued and single-valued mappings and also give a partial solution to the problem of removing the strict fixed point set condition usually imposed onS.
2 Preliminaries
In the sequel, we shall need the following definitions and lemmas.
Definition . LetXbe a nonempty set, and letS:X→Xbe a map. A pointr∈Xis called a fixed point ofSifr=Sr. IfS:X→Xis a multivalued map, thenris a fixed point
ofSifr∈Sr. IfSr={r}, thenris called a strict fixed point ofS. The setF(S) ={r∈D(S) :
r∈Sr}(respectively,F(S) ={r∈D(S) :r=Sr}) is called the fixed point set of a multivalued (respectively, single-valued) mapS, whereas the setFs(S) ={r∈D(S) :Sr={r}}is called
the strict fixed point set ofS.
Definition . LetXbe a normed space. A subsetCofXis said to be proximinal if given anyr∈X, there existsk∈Csuch that
It is well known that every closed convex subset of a uniformly convex Banach space is proximinal. For a nonempty setX, we shall denote the family of all nonempty closed and bounded subsets ofXbyCB(X), the family of all nonempty closed, convex, and bounded subsets ofXbyCVB(X), the family of all nonempty closed subsets ofXbyC(X), the family of all proximinal subsets ofXbyP(X), and the family of all nonempty subsets ofXby X.
Definition . Letdenote the Hausdorff metric induced by the metricdonX, that is, for allA,B∈CB(X),
(A,B) =max
sup a∈Ad(a,B),
sup b∈Bd(b,A)
.
Definition . LetXbe a normed space. LetS:D(S)⊆X→Xbe a multivalued mapping
onX.Sis calledL-Lipschitzian if there existsL≥ such that, for allr,g∈D(S),
(Sr,Sg)≤Lr–g. (.)
IfL∈[, ) in (.), thenSis said to be acontraction, whereasSisnonexpansiveifL= .
Sis calledquasi-nonexpansiveifF(S) ={r∈D(S) :r∈Sr} =∅and, for allz∈F(S),
(Sr,Sz)≤ r–z. (.)
Clearly, every nonexpansive mapping with nonempty fixed point set is quasi-non-expansive.Sis said to bek-strictly pseudocontractive in the sense of Chidumeet al.[] if there existsC∈(, ) such that
(Sr,Sg)≤ r–g+kr–u– (g–v) for allu∈Sr,v∈Sg. (.)
Ifk= in (.), thenSis said to be pseudocontractive.Sis said to be ofk-strictly pseu-docontractive type in the sense of Isiogugu [] if there existsk∈(, ) such that given any pairr,g∈D(S) andu∈Sr, there existsv∈Sgsatisfyingu–v ≤(Sr,Sg) and
(Sr,Sg)≤ r–g+kr–u– (g–v). (.)
Ifk= in (.), thenSis said to be of pseudocontractive type. It is easy to see that any proximinal, pseudocontractive mapSof Chidumeet al.[] is of pseudoconytractive type in the sense of Isiogugu [].
Definition .([, ]) LetXbe a Banach space. LetS:D(S)⊆X→Xbe a multivalued mapping. ThenI–Sis said to beweakly demiclosedatzeroif for any sequence{rn}∞n=⊆
D(S) such that{rn}converges weakly tozand a sequence{gn}withgn∈Srnfor alln∈N
such that{rn–gn}strongly converges tozero, we havez∈Sz(i.e., ∈(I–S)z).
Definition .([]) A Banach Xis said to satisfy Opial’s condition if whenever a se-quence{rn}converges weakly tor∈X, we have
lim infrn–r<lim infrn–g
Definition .([, ]) LetXbe a Banach space, andS:D(S)⊆X→Xa multivalued
mapping. The graph ofI–Sis said to beclosedinσ(X,X∗)×(X, · ) (i.e.,I–Sisweakly demiclosedordemiclosed) if for any sequence{rn}∞n=⊆D(S) such that{rn}weakly con-verges toz and a sequence{gn} withgn∈Srnfor alln∈Nsuch that{rn–gn}strongly
converges tog, we haveg∈(I–S)z(i.e.,g=z–vfor somev∈Sz). HereIdenotes the identity onX,σ(X,X∗) is the weak topology ofX, and (X, · ) is the norm (or strong) topology ofX.
Definition .([]) A multivalued mappingS:C→P(C) is said to satisfy condition () (see,e.g., []) if there exists a nondecreasing functionf : [,∞)→[,∞) withf() = and
f > on (,∞) such that
d(r,Sr)≥f(dr,F(S), ∀r∈C.
Definition .([]) LetXbe a normed space, andS:D(S)⊆X→Xbe a multivalued map.Sis said to be of type one if given any pairr,g∈D(S), we have
u–v ≤(Sr,Sg) for allu∈PSr,v∈PSg.
Lemma . Let H be a real Hilbert space,and{rn}∞n=a sequence in H weakly converging to z∈H.Then
lim sup
n→∞rn–g
=lim sup
n→∞rn–z
+z–g, ∀g∈H.
Lemma .([]) Let{an},{δn},and{γn}be sequences of nonnegative real numbers satis-fying the following relation:
an+≤( +δn)an+γn, n≥n,
where nis a nonnegative integer.If δn<∞and γn<∞,thenlimn→∞anexists.
Lemma .([]) Let X be a metric space.If A,B∈P(X),a∈A,andγ ≥,then,as a simple consequence of the Hausdorff metric,there exists b∈B such that
d(a,b)≤(A,B) +γ.
3 Main results
We now obtain a demiclosedness property in the sense that if{rn}∞n=⊆Cis such that{rn}
weakly converges tozand a sequence{gn}withgn∈Srnandrn–gn=d(rn,Srn) for all n∈Nis such that{rn–gn}strongly converges to , then ∈(I–S)z(i.e.,z=vfor some v∈Sz).
Proof We use a method similar to that of the proof of Proposition in []. Let{rn}∞n=⊆C
be such that{rn}weakly converges toz, and{gn}be a sequence withgn∈Srnandrn– gn=d(rn,Srn) for all n∈Nsuch that {rn–gn}strongly converges to . We prove that
∈(I–S)z(i.e.,z=vfor somev∈Sz). Since{rn}∞n=converges weakly, it is bounded. Let
q∈Szwithz–q=d(z,Sz), From the definition ofk-strictly pseudocontractive and type-one condition onS, for eachn∈N, we have
gn–q ≤(Srn,Sz) (.)
and
(Srn,Sz)≤ rn–z+krn–gn– (z–q)
. (.)
Therefore, for eachr∈H, definef:H→[,∞) by
f(r) :=lim sup n→∞ rn–r
.
Then from Lemma . we obtain
f(r) =lim sup n→∞ rn–z
+z–r, ∀r∈X.
Thus,
f(r) =f(z) +z–r, ∀r∈X.
Therefore,
f(q) =f(z) +z–q. (.)
Observe also that
f(q) =lim sup n→∞ rn–q
=lim sup n→∞
rn–gn+ (gn–q)
=lim sup
n→∞ gn–q
≤lim sup
n→∞
(Sr
n,Sz)
≤lim sup n→∞
rn–z+krn–gn– (z–q)
=lim sup n→∞ rn–z
+k(z–q)
=f(z) +kz–q. (.)
Hence, it follows from (.) and (.) that ( –k)z–q= .
We now obtain some strong and weak convergence results for the class of pseudocon-tractive mappings andk-strictly pseudocontractive mappings of Chidumeet al.[], re-spectively, in Hilbert spaces.
Theorem . Let C be a nonempty closed and convex subset of a real Hilbert space X.If S:
C→P(C)is a type-one L-Lipschitzian pseudocontractive mapping from C into the family of all proximinal subsets of C such that F(S)=∅.Suppose that S satisfies condition().
Then,the Ishikawa sequence defined by
gn= ( –δn)rn+δnun, rn+= ( –μn)rn+μnwn,
strongly converges to p∈F(S),where un∈Srnwithrn–un=d(rn,Srn),wn∈Sgnwith gn–wn=d(gn,Sgn),and{μn}and{δn}are real sequences satisfying(i)lim infn→∞μn=
μ> , (ii) ≤μn≤δn< ,and(iii)supn≥δn≤δ≤√+L+.
Proof Using a method similar to that of the proof of Theorem in [], we have that
rn+–z =( –μn)rn+μnwn–z
=( –μn)(rn–z) +μn(wn–z)
= ( –μn)rn–z+μnwn–z–μn( –μn)rn–wn
≤( –μn)rn–z+μn(Sgn,Sz)
–μn( –μn)rn–wn
≤( –μn)rn–z+μn
gn–z+gn–wn
–μn( –μn)rn–wn
= ( –μn)rn–z+μngn–z+μnd(gn,Sgn)
–μn( –μn)rn–wn. (.)
Also,
gn–wn=( –δn)rn+δnun–wn
=( –δn)(rn–wn) +δn(un–wn)
= ( –δn)rn–wn+δnun–wn
–δn( –δn)rn–un. (.)
From (.) and (.) it follows that
rn+–z≤( –μn)rn–z+μngn–z
+μn
( –δn)rn–wn+δnun–wn
–δn( –δn)rn–un
gn–z=( –δn)rn+δnun–z
=( –δn)(rn–z) +δn(un–z)
= ( –δn)rn–z+δnun–z–δn( –δn)rn–un
≤( –δn)rn–z+δn(Srn,Sz) –δn( –δn)rn–un ≤( –δn)rn–z+δn
rn–z+rn–un
–δn( –δn)rn–un
=rn–z+δnrn–un. (.)
From (.) and (.) it follows that
rn+–z≤( –μn)rn–z
+μn
rn–z+δnrn–un
+μn
( –δn)rn–wn+δnun–wn–δn( –δn)rn–un
–μn( –μn)rn–wn
= ( –μn)rn–z+μnrn–z+μnδnrn–un
+μn( –δn)rn–wn+μnδnun–wn
–μnδn( –δn)rn–un–μn( –μn)rn–wn
≤ rn–z+μnδnrn–un+μnδn(Srn,Sgn)
–μn(δn–μn)rn–wn
–μnδn( –δn)rn–un
≤ rn–z+μnδnrn–un+μnδnLrn–un
–μnδn( –δn)rn–un
–μn(δn–μn)rn–wn
=rn–z–μnδn
– δn–Lδn
rn–un
–μn(δn–μn)rn–wn
=rn–z–μnδn
– δn–Lδn
rn–un. (.)
It then follows from Lemma . thatlimn→∞rn–zexists. Hence,{rn}is bounded, so
also are{un}and{wn}. We then have from (.)(i), (iii) that
∞
n=
μ – δ–Lδrn–un≤ ∞
n=
μnδn
– δn–Lδn
rn–un
≤ ∞
n=
rn–z–rn+–z
≤ r–z+D<∞.
exists a subsequence{rnk}of{rn} such thatrnk –zk ≤
k for some{zk} ⊆F(S). From
(.) we have
rnk+–zk ≤ rnk–zk.
We now show that{zk}is a Cauchy sequence inF(S):
zk+–zk ≤ zk+–rnk++rnk+–zk
≤
k+ +
k
< k–.
Therefore,{zk}is a Cauchy sequence and converges to someq∈CbecauseCis closed. Now, we have
rnk–q ≤ rn–zk+zk–q.
Hence,rnk→qask→ ∞. We have
d(q,Sq)≤ q–zk+zk–rnk+d(rnk,Srnk) +(Srnk,Sq)
≤ q–zk+zk–rnk+d(rnk,Srnk) +Lrnk–q.
Hence,q∈Sq, and{rnk}strongly converges toq. Sincelimrn–qexists, we have thatrn
strongly converges toq∈F(S).
Theorem . Let C be a nonempty closed and convex subset of a real Hilbert space H.
Suppose that S:C→P(C)is k-strictly pseudocontractive mapping from C into the family of all proximinal subsets of C with k∈(, )such that F(S)=∅.If S is of type one,then the Mann-type sequence defined by
rn+= ( –μn)rn+μngn
weakly converges to q∈F(S), where gn∈Srn withrn–gn=d(rn,Srn)andμn⊆(, ) satisfying(i)μn→μ< –k, (ii)μ> ,and(iii) ∞n=μn( –μn) =∞.
Proof Using the well-known identity and a method similar to that of the proof of Theo-rem in [], we have that
tr+ ( –t)g=tr+ ( –t)g–t( –t)r–g,
which holds for allr,g∈Hand allt∈[, ], from which we obtain
rn+–z =( –μn)rn+μngn–z
=( –μn)(rn–z) +μn(gn–z)
≤( –μn)rn–z+μn(Srn,Sz) –μn( –μn)rn–gn ≤( –μn)rn–z+μn
rn–z+krn–gn
–μn( –μn)rn–gn
=rn–z–μn
– (μn+k)rn–gn.
It then follows Lemma . thatlimn→∞rn–zexists, and hence{rn}is bounded. Also,
∞
n=
μn
– (μn+k)
rn–gn≤ r–z≤ ∞.
Sinceμ> from (ii), we have thatlimn→∞rn–gn= . Also, sinceCis closed and{rn} ⊆C
with{rn}bounded, there exists a subsequence{rnt} ⊆ {rn}that weakly converges to some q∈C. Also,limn→∞rn–gn= implies thatlimn→∞rnt–gnt= . Since (I–S) is weakly
demiclosed atzero, we have thatq∈Sq. SinceHsatisfies Opial’s condition [], we have
that{rn}weakly converges toq∈F(S).
We now have the following corollaries with proofs easily following from the definition and Theorems . and ., respectively.
Corollary . ([]) Let C be a nonempty closed and convex subset of a real Hilbert space H.Suppose that S:C→P(C)is a type-one quasi-nonexpansive mapping from C into the family of all proximinal subsets of C.If(I–S)satisfies Proposition.,then the Mann-type sequence defined by
rn+= ( –μn)rn+μngn
weakly converges to q∈F(S),where gn∈Srnwithrn–gn=d(rn,Srn),andμnis a real sequence in(,)satisfyingμn→μ∈(, ).
Corollary . ([]) Let C be a nonempty closed and convex subset of a real Hilbert space H.Suppose that S:C→P(C)is a type-one nonexpansive mapping from C into the family of all proximinal subsets of C such that F(S)=∅. Then the Mann-type sequence defined by
rn+= ( –μn)rn+μngn
weakly converges to q∈F(S), where gn∈Srn withrn–gn=d(rn,Srn)andμn⊆(, ) satisfyingμn→μ∈(, ).
Concluding Remark It is not clear if Theorem . and Theorem . will still hold if the classes of pseudocontractive andk-strictly pseudocontractive mappings considered are replaced with the classesk-strictly pseudocontractive-type and pseudocontractive-type mappings, respectively, considered by Isiogugu [].
Competing interests
Acknowledgements
The author is a post doctoral fellow at the School of Mathematics, Statistics and Computer Sciences, University of Kwazulu Natal, South Africa. She is grateful to the University for the scholarship, making their facilities available, and for hospitality. She is also grateful to the University of Nigeria Nsukka for granting her study leave.
Received: 24 August 2015 Accepted: 22 April 2016 References
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