# Three-Step Fixed Point Iteration for Generalized Multivalued Mapping in Banach Spaces

Authors: Zhanfei Zuo
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The convergence of three-step fixed point iterative processes for generalized multivalued nonexpansive mapping was considered in this paper. Under some different conditions, the sequences of three-step fixed point iterates strongly or weakly converge to a fixed point of the generalized multivalued nonexpansive mapping. Our results extend and improve some recent results.


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## Body

## 1. Introduction

LetX be a Banach space and K a nonempty subset of X. The set K is called proximinal if for each x∈X, there exists an element y∈K such that ∥x-y∥=d(x,K), where d(x,K)=inf{∥x-z∥:z∈K}. Let CB(K),C(K),P(K),F(T) denote the family of nonempty closed bounded subsets, nonempty compact subsets, nonempty proximinal bounded subsets of K, and the set of fixed points, respectively. A multivalued mapping T:K→CB(K) is said to be nonexpansive (quasi-nonexpansive) if(1.1)H(Tx,Ty)≤‖x-y‖,x,y∈K,(H(Tx,Tp)≤‖x-p‖,x∈K,p∈F(T)),
where H(·,·) denotes the Hausdorff metric on CB(X) defined by(1.2)H(A,B):=max⁡{sup⁡x∈Ainf⁡y∈B‖x-y‖,sup⁡y∈Binf⁡x∈A‖x-y‖},A,B∈CB(X).
A point x is called a fixed point of T if x∈Tx. Since Banach's Contraction Mapping Principle was extended nicely to multivalued mappings by Nadler in 1969 (see [1]), many authors have studied the fixed point theory for multivalued mappings (e.g., see [2]). For single-valued nonexpansive mappings, Mann [3] and Ishikawa [4], respectively, introduced a new iteration procedure for approximating its fixed point in a Banach space as follows:(1.3)xn+1=(1-αn)xn+αnTxn,xn+1=(1-αn)xn+αnyn,yn=(1-bn)xn+bnTxn,
where {αn} and {bn} are sequences in [0,1]. Obviously, Mann iteration is a special case of Ishikawa iteration. Recently Song and Wang in [5, 6] introduce the following algorithms for multivalued nonexpansive mapping:(1.4)xn+1=(1-αn)xn+αnsn,
where sn∈Txn,γn∈(0,+∞) such that limn→∞γn=0 and ∥sn+1-sn∥≤H(Txn+1,Txn)+γn,(1.5)xn+1=(1-αn)xn+αnrn,yn=(1-bn)xn+bnsn,
where ∥sn-rn∥≤H(Txn,Tyn)+γn and ∥sn+1-rn∥≤H(Txn+1,Tyn)+γn for sn∈Txn and rn∈Tyn. They show some strong convergence results of the above iterates for multivalued nonexpansive mapping T under some appropriate conditions. However, the iteration scheme constructed by Song and Wang involves the following estimates,
(1.6)‖sn-rn‖≤H(Txn,Tyn)+γn,‖sn+1-rn‖≤H(Txn+1,Tyn)+γn,
which are not easy to be computed and the scheme is more time consuming. It is observed that Song and Wang [6] did not use the above estimates in their proofs and the assumption on T, namely, T(p)={p} for any p∈F(T) is quite strong. It is noted that the domain of T is compact, which is a strong condition. The aim of this paper is to construct an three iteration scheme for a generalized multivalued mappings, which removes the restriction of T, namely, T(p)={p} for any p∈F(T) and also relax compactness of the domain of T. The generalized multivalued mappings was introduced in [7], if(1.7)12d(x,Tx)≤‖x-y‖impliesH(Tx,Ty)≤‖x-y‖∀x,y∈K,
where d is induced by the norm. Obviously, the condition is weaker than nonexpansiveness and stronger than quasinonexpansiveness, furthermore, there are some examples of a generalized nonexpansive multivalued mapping which is not a nonexpansive multivalued mapping (see [7, 8]).LetT:K→P(K) be a generalized nonexpansive multivalued mapping and PT(x)={y∈T(x):∥x-y∥=d(x,T(x))}. The three-step mean multivalued iterative scheme is defined by x0∈K,(1.8)zn=(1-an)xn+ansn,yn=(1-bn-cn)xn+bntn+cnsn,xn+1=(1-αn-βn-γn)xn+αnrn+βntn+γnsn,
where {an},{bn},{cn},{bn+cn},{αn},{βn},{γn}, and {αn+βn+γn} are appropriate sequence in [0,1], furthermore sn∈PT(xn),tn∈PT(zn),rn∈PT(yn). If an=cn=βn=γn≡0 or an=bn=cn=βn=γn≡0, then iterative scheme (1.8) reduces to the Ishikawa and Mann multivalued iterative scheme. In fact let γn≡0 or cn=βn=γn≡0 or bn=cn=αn=γn≡0, we also have the other three algorithms.The mappingT:K→CB(K) is called hemicompact if, for any sequence xn in K such that d(xn,T(xn))→0 as n→∞, there exists a subsequence xnk of xn such that xnk→p∈K. We note that if K is compact, then every multivalued mapping T:K→CB(K) is hemicompact. The following definition was introduced in [9].Definition 1.1.
A multivalued mappingT:K→CB(K) is said to satisfy Condition (A) if there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0,f(x)>0 for x∈(0,∞) such that
(1.9)d(x,Tx)≥f(d(x,F(T)))∀x∈K.
where F(T)≠∅  is the fixed point set of the multivalued mapping T. From now on, F(T) stands for the fixed point set of the multivalued mapping  T.


## 2. Preliminaries

A Banach spaceX is said to be satisfy Opial's condition [10] if, for any sequence {xn} in X, xn⇀x(n→∞) implies the following inequality:(2.1)limsup⁡n→∞‖xn-x‖<limsup⁡n→∞‖xn-y‖,
for all y∈X with y≠x. It is known that Hilbert spaces and lp(1<p<∞) have the Opial's condition.Lemma 2.1 (see [7, 11]).
Let{xn},{yn}, and {zn} be sequence in uniformly convex Banach space X. Suppose that {αn},{βn}, and {γn} are sequence in [0,1] with αn+βn+γn=1,limsupn∥xn∥≤d,limsupn∥yn∥≤d,limsupn∥zn∥≤d, and limn∥αnxn+βnyn+γnzn∥=d. If liminfnαn>0 and  liminfnβn>0, then limn∥xn-yn∥=0.Lemma 2.2 (see [7, 11]).
LetX be a uniformly convex Banach space and Br:={x∈X:∥x∥≤r},r>0. Then there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(2.2)‖λx+μy+ξz+ϑω‖2≤λ‖x‖2+μ‖y‖2+ξ‖z‖2+ϑ‖ω‖2-13ϑ(λg(‖x-ω‖)+μg(‖y-ω‖)+ξg(‖z-ω‖)),
for all x,y,z,ω∈Br and λ,μ,ξ,ϑ∈[0,1] with λ+μ+ξ+ϑ=1.


## 3. Main Results

Lemma 3.1.
LetX be a real Banach space and K be a nonempty convex subset of X,T:K→P(K) be a generalized multivalued nonexpansive mapping with F(T)≠∅ such that PT is nonexpansive. Let {xn} be a sequence in K defined by (1.8), then one has the following conclusion:
(3.1)lim⁡n‖xn-p‖existsforanyp∈F(T).Proof.
Letp∈F(T), then p∈PT(p)={p}. Since T is quasi-nonexpansive, thus we obtain
(3.2)‖zn-p‖≤(1-an)‖xn-p‖+an‖sn-p‖≤(1-an)‖xn-p‖+and(sn,PT(p))≤(1-an)‖xn-p‖+anH(PT(xn),PT(p))≤(1-an)‖xn-p‖+an‖xn-p‖≤‖xn-p‖,
similarly ∥yn-p∥≤∥xn-p∥, then we have
(3.3)‖xn+1-p‖≤(1-αn-βn-γn)‖xn-p‖+αn‖rn-p‖+βn‖tn-p‖+γn‖sn-p‖≤(1-αn-βn-γn)‖xn-p‖+αnH(PT(yn),PT(p))+βnH(PT(zn),PT(p))+γnH(PT(xn),PT(p))≤‖xn-p‖.
Then {∥xn-p∥} is a decreasing sequence and hence limn∥xn-p∥ exists for any p∈F(T).Lemma 3.2.
LetX be a uniformly convex Banach space and K be a nonempty convex subset of X,T:K→P(K) be a generalized multivalued nonexpansive mapping with F(T)≠∅ such that PT is nonexpansive. Let {xn} be a sequence in K defined by (1.8), if the coefficient satisfy one of the following control conditions:(i)
liminfnαn>0 and one of the following holds:
(a)
limsupn(αn+βn+γn)<1 and limsupn(bn+cn)<1,(b)
0<liminfnβn≤limsupn(αn+βn+γn)<1 and limsupncn<1,(c)
0<liminfnbn≤limsupn(bn+cn)<1 and limsupnan<1,(d)
0<liminfncn≤limsupn(bn+cn)<1;(ii)
0<liminfnβn≤limsupn(αn+βn+γn)<1 and limsupnan<1;(iii)
0<liminfnγn≤limsupn(αn+βn+γn)<1;(iv)
0<liminfn(αnbn+βn) and 0<liminfnan≤limsupnan<1;then we have limnd(xn,Txn)=0.Proof.
By Lemma3.1, we know that limn∥xn-p∥ exists for any p∈F(T), then it follows that {sn-p},{tn-p}, and {rn-p} are all bounded. We may assume that these sequences belong to Br where r>0. Note that p∈PT(p)={p} for any fixed point p∈F(T) and T is quasi-nonexpansive. By Lemma 2.2, we get
(3.4)‖zn-p‖2≤(1-an)‖xn-p‖2+an‖sn-p‖2≤(1-an)‖xn-p‖2+anH(PT(xn),PT(p))2≤‖xn-p‖2,‖yn-p‖2≤(1-bn-cn)‖xn-p‖2+bn‖tn-p‖2+cn‖sn-p‖2-13(1-bn-cn)(bng(‖tn-xn‖)+cng(‖sn-xn‖))≤(1-bn-cn)‖xn-p‖2+bnH(PT(zn),PT(p))2+cnH(PT(xn),PT(p))2-13(1-bn-cn)bng(‖tn-xn‖)≤‖xn-p‖2-13(1-bn-cn)bng(‖tn-xn‖),
and therefore we have
(3.5)‖xn+1-p‖2≤(1-αn-βn-γn)‖xn-p‖2+αn‖rn-p‖2+βn‖tn-p‖2+γn‖sn-p‖2-13(1-αn-βn-γn)[αng(‖xn-rn‖)+βng(‖xn-tn‖)+γng(‖xn-sn‖)]≤(1-αn-βn-γn)‖xn-p‖2+αnH(PT(yn),PT(p))2+βnH(PT(zn),PT(p))2+γnH(PT(xn),PT(p))2-13(1-αn-βn-γn)[αng(‖xn-rn‖)+βng(‖xn-tn‖)+γng(‖xn-sn‖)]≤‖xn-p‖2-αn3(1-bn-cn)bng(‖tn-xn‖)-13(1-αn-βn-γn)×[αng(‖xn-rn‖)+βng(‖xn-tn‖)+γng(‖xn-sn‖)].
Then
(3.6)(1-αn-βn-γn)αng(‖xn-rn‖)≤3(‖xn-p‖2-‖xn+1-p‖2),(3.7)(1-αn-βn-γn)βng(‖xn-tn‖)≤3(‖xn-p‖2-‖xn+1-p‖2),(3.8)(1-αn-βn-γn)γng(‖xn-sn‖)≤3(‖xn-p‖2-‖xn+1-p‖2),(3.9)αn(1-bn-cn)bng(‖tn-xn‖)≤3(‖xn-p‖2-‖xn+1-p‖2).
Since limn∥xn-p∥ exists for any p∈F(T), it follows from (3.6) that limn(1-αn-βn-γn)αng(∥xn-rn∥)=0. From g is continuous strictly increasing with g(0)=0 and 0<liminfnαn≤limsupn(αn+βn+γn)<1, then
(3.10)lim⁡n‖xn-rn‖=0.
Using a similarly method together with inequalities (3.7) and 0<liminfnβn≤limsupn(αn+βn+γn)<1, then
(3.11)lim⁡n‖xn-tn‖=0.
Similarly, from (3.8) and 0<liminfnγn≤limsupn(αn+βn+γn)<1, we have limn∥xn-sn∥=0, since sn∈Txn, then 0≤limnd(xn,Txn)≤limn∥xn-sn∥=0, thus we get (iii). In the sequence we prove (i) (a). From iterative scheme (1.8), we have
(3.12)‖sn-xn‖≤‖sn-rn‖+‖rn-xn‖≤H(PT(xn),PT(yn))+‖rn-xn‖≤‖xn-yn‖+‖rn-xn‖≤bn‖xn-tn‖+cn‖xn-sn‖+‖rn-xn‖.
To show that limn∥xn-sn∥=0, it suffices to show that there exist a subsequence {nj} of {n} such that  limnj∥xnj-snj∥=0. If  liminfjbnj>0, it follows from (3.9) that
(3.13)αnj(1-bnj-cnj)bnjg(‖tnj-xnj‖)≤3(‖xnj-p‖2-‖xnj+1-p‖2).
Since limn∥xn-p∥ exists for any p∈F(T), we have
(3.14)lim⁡njαnj(1-bnj-cnj)bnjg(‖tnj-xnj‖)=0.
From g is continuous strictly increasing with g(0)=0,liminfjαnj>0 and 0<liminfnjbnj≤limsupnj(bnj+cnj)<1, we have
(3.15)lim⁡nj‖tnj-xnj‖=0.
This together with (3.10), (3.12), (3.15) gives
(3.16)lim⁡j(1-cnj)‖snj-xnj‖=0.
Since liminfnj(1-cnj)=1-limsupnjcnj>0, we have limj∥snj-xnj∥=0. On the other hand, if liminfjbnj=0, then we may extract a subsequence {bnk} of {bnj} so that limkbnk=0. This together with (i) (a) and (3.10), (3.12) gives
(3.17)lim⁡k(1-cnk)‖snk-xnk‖=0,andsolim⁡k‖snk-xnk‖=0.
By Double Extract Subsequence Principle, we obtain the result.
If0<liminfnβn≤limsupn(αn+βn+γn)<1 and limsupnan<1, we will prove (ii),
(3.18)‖sn-xn‖≤‖sn-tn‖+‖tn-xn‖≤H(PT(xn),PT(zn))+‖tn-xn‖≤‖xn-zn‖+‖tn-xn‖≤an‖xn-sn‖+‖tn-xn‖.
Since limsupnan<1, then
(3.19)liminf⁡n(1-an)=1-limsup⁡nan>0.
This together with (3.11), (3.18), we obtain the result.
We will prove (i) (b), letp∈F(T). By Lemma 3.1, we let limn∥xn-p∥=d for some d≥0. From iterative scheme (1.8), we know
(3.20)d=lim⁡n‖xn+1-p‖=lim⁡n‖(1-αn-βn-γn)(xn-p)+αn(rn-p)+βn(tn-p)+γn(sn-p)‖.
From Lemma 3.1, we have known that ∥zn-p∥≤∥xn-p∥ and ∥yn-p∥≤∥xn-p∥, then
(3.21)limsup⁡n‖rn-p‖≤limsup⁡nH(PT(yn),PT(p))≤limsup⁡n‖yn-p‖≤d,limsup⁡n‖tn-p‖≤limsup⁡nH(PT(zn),PT(p))≤limsup⁡n‖zn-p‖≤d,limsup⁡n‖sn-p‖≤limsup⁡nH(PT(xn),PT(p))≤limsup⁡n‖xn-p‖≤d.
From (3.20) and Lemma 2.1, we have
(3.22)lim⁡n‖xn-tn‖=lim⁡n‖rn-xn‖=0.
Notice that
(3.23)‖xn-sn‖≤‖xn-rn‖+‖rn-sn‖≤‖xn-rn‖+H(PT(yn),PT(xn))≤‖xn-yn‖+‖xn-rn‖≤bn‖xn-tn‖+cn‖xn-sn‖+‖xn-rn‖.
Since limsupncn<1, we have limn∥sn-xn∥=0, therefore 0≤limnd(xn,Txn)≤limn∥xn-sn∥=0.
We will prove (i) (c). From iterative scheme (1.8) and Lemma 3.1, we have
(3.24)‖xn+1-p‖≤(1-αn-βn-γn)‖xn-p‖+αn‖yn-p‖+βn‖zn-p‖+γn‖xn-p‖≤(1-αn)‖xn-p‖+αn‖yn-p‖.
which implies
(3.25)‖xn+1-p‖-‖xn-p‖+αn‖xn-p‖≤αn‖yn-p‖.
Notice that liminfnαn>0 and limn∥xn-p∥ exists. Hence from (3.25) we have
(3.26)d=lim⁡n‖xn-p‖≤liminf⁡n‖yn-p‖≤limsup⁡‖yn-p‖≤d.
Therefore, from iterative scheme (1.8) we have
(3.27)d=lim⁡n‖yn-p‖=lim⁡n‖(1-bn-cn)(xn-p)+bn(tn-p)+cn(sn-p)‖.
From Lemma 2.1, we have
(3.28)lim⁡n‖xn-tn‖=0.
Notice that
(3.29)‖sn-xn‖≤‖sn-tn‖+‖tn-xn‖≤H(PT(xn),PT(zn))+‖tn-xn‖≤‖xn-zn‖+‖tn-xn‖≤an‖xn-sn‖+‖tn-xn‖.
Since limsupnan<1, then 0≤limnd(xn,Txn)≤limn∥xn-sn∥=0.
By (3.27) and Lemma 2.1, we can similarly prove (i) (d).
Finally, we will prove (iv). From iterative scheme (1.8) and Lemma 3.1, we have
(3.30)‖xn+1-p‖≤(1-αn-βn-γn)‖xn-p‖+αn‖rn-p‖+βn‖tn-p‖+γn‖sn-p‖≤(1-αn-βn-γn)‖xn-p‖+αn‖yn-p‖+βn‖zn-p‖+γn‖xn-p‖≤(1-αn-βn)‖xn-p‖+αn[(1-bn)‖xn-p‖+bn‖zn-p‖]+βn‖zn-p‖,
which implies
(3.31)‖xn+1-p‖-‖xn-p‖+(αnbn+βn)‖xn-p‖≤(αnbn+βn)‖zn-p‖.
Notice that
(3.32)0<liminf⁡n(αnbn+βn),lim⁡n‖xn-p‖exists.
Hence we have
(3.33)d=lim⁡n‖xn-p‖≤liminf⁡n‖zn-p‖≤limsup⁡‖zn-p‖≤d.
Thus, we have
(3.34)d=lim⁡n‖zn-p‖=lim⁡n(1-an)‖xn-p‖+an‖sn-p‖.
By Lemma 2.1  and 0<liminfnan≤limsupnan<1, we have  0≤limnd(xn,Txn)≤limn∥xn-sn∥=0.Theorem 3.3.
LetX be a uniformly convex Banach space and K be a nonempty convex subset of X, T:K→P(K) be a generalized multivalued nonexpansive mapping with F(T)≠∅ such that PT is nonexpansive. Let {xn} be a sequence in K defined by (1.8), the coefficient satisfy the control conditions in Lemma 3.2 and T satisfies Condition (A) with respect to the sequence {xn}, then {xn} converges strongly to a fixed point of T.Proof.
By Lemma3.2, we have limnd(xn,Txn)=0. Since T satisfies Condition (A) with respect to {xn}. Then
(3.35)f(d(xn,F(T)))≤d(xn,Txn)⟶0.
Thus, we get limnd(xn,F(T))=0. The remainder of the proof is the same as in [6, Theorem 2.4], we omit it.Theorem 3.4.
LetX be a uniformly convex Banach space and K be a nonempty convex subset of X, T:K→P(K) be a generalized multivalued nonexpansive mapping with F(T)≠∅ such that PT is nonexpansive. Let {xn} be a sequence in K defined by (1.8), the coefficient satisfy the control conditions in Lemma 3.2 and T is hemicompact, then {xn} converges strongly to a fixed point of T.Proof.
By Lemma3.2, we have limnd(xn,Txn)=0. Since T is hemicompact, then there exist a subsequence {xnk} of {xn} such that limk→∞∥xnk-q∥=0 for some q∈K. Thus,
(3.36)d(q,Tq)≤‖q-xnk‖+d(xnk,Txnk)+H(Txnk,Tq)≤2‖q-xnk‖+d(xnk,Txnk)⟶0.
Hence, q is a fixed point of T. Now on take on q in place of p, we get that limn→∞∥xn-q∥ exists. It follows that xn→q as n→∞. This completes the proof.Theorem 3.5.
LetX,T and {xn} be the same as in Lemma 3.2. If K be a nonempty weakly compact convex subset of a Banach space X and X satisfies Opial's condition, then {xn} converges weakly to a fixed point of T.Proof.
The proof of the Theorem is the same as in [6, Theorem 2.5], we omit it.Remark 3.6.
From the definition of iterative scheme (1.8), Theorems 3.3, 3.4, and 3.5 extend some results in [6, 12], and also give some new results are different from the [5]. In fact, we can present an example of a multivalued map T for which PT is nonexpansive. A multivalued map T:D→CB(X) is *-nonexpansive [13] if for all x,y∈D and ux∈T(x) with d(x,ux)=inf{d(x,z):z∈T(x)}, there exists uy∈T(y) with d(y,uy)=inf{d(y,w):ω∈T(y)} such that
(3.37)d(ux,uy)≤d(x,y).
It is clear that if T is *-nonexpansive, then PT is nonexpansive. It is known that *-nonexpansiveness is different from nonexpansiveness for multivalued maps. Let D=[0,∞) and T be defined by Tx=[x,2x] for x∈D [14]. Then PT(x)=x for x∈D and thus it is nonexpansive. Note that T is *-nonexpansive but not nonexpansive (see [14]).



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