# Convergence Theorems for Finite Family of Multivalued Maps in Uniformly Convex Banach Spaces

Authors: Yekini Shehu
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

We introduce a new iterative process to approximate a common fixed point of a finite
family of multivalued maps in a uniformly convex real Banach space and establish strong convergence theorems
for the proposed process. Furthermore, strong convergence theorems for finite family of quasi-nonexpansive
multivalued maps are obtained. Our results extend important recent results.


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## Body

## 1. Introduction

LetD be a nonempty, closed, and convex subset of a real Hilbert space H. The set D is called proximinal if for each x∈H, there exists y∈D such that ∥x-y∥=d(x,D), where d(x,D)=inf{∥x-z∥:z∈D}. Let CB(D),K(D), and P(D) denote the families of nonempty, closed and bounded subsets, nonempty compact subsets, and nonempty proximinal bounded subsets of D, respectively. The Hausdorff metric on CB(D) is defined by(1.1)H(A,B)=max⁡{sup⁡x∈A⁡d(x,B),sup⁡y∈Bd(y,A)},
for A,B∈CB(D). A single-valued map T:D→D is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈D. A multivalued map T:D→CB(D) is said to be nonexpansive if H(Tx,Ty)≤∥x-y∥ for all x,y∈D. An element p∈D is called a fixed point of T:D→D (resp., T:D→CB(D)) if p=Tp (resp., p∈Tp). The set of fixed points of T is denoted by F(T). A multivalued map T:D→CB(D) is said to be quasi-nonexpansive if H(Tx,Tp)≤∥x-p∥ for all x∈D and for all p∈F(T).AT:D→CB(D) is said to satisfy Condition (I) if there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0,f(r)>0 for r∈(0,∞) such that (1.2)d(x,Tx)≥f(d(x,F(T))),
for all x∈D.The fixed point theory of multivalued nonexpansive mappings is much more complicated and difficult than the corresponding theory of single-valued nonexpansive mappings. However, some classical fixed point theorems for single-valued nonexpansive mappings have already been extended to multivalued mappings. The first results in this direction were established by Markin [1] in Hilbert spaces and by Browder [2] for spaces having weakly continuous duality mapping. Dozo [3] generalized these results to a Banach space satisfying Opial's condition.In 1974, by using Edelstein's method of asymptotic centers, Lim [4] obtained a fixed point theorem for a multivalued nonexpansive self-mapping in a uniformly convex Banach space.Theorem 1.1 (Lim [4]).
LetD be a nonempty, closed convex, and bounded subset of a uniformly convex Banach space E and T:D→C(E) a multivalued nonexpansive mapping. Then, T has a fixed point.In 1990, Kirk and Massa [5] gave an extension of Lim's theorem proving the existence of a fixed point in a Banach space for which the asymptotic center of a bounded sequence in a closed bounded convex subset is nonempty and compact.Theorem 1.2 (Kirk and Massa [5]).
LetD be a nonempty, closed convex, and bounded subset of a Banach space E and T:D→CB(E) a multivalued nonexpansive mapping. Suppose that the asymptotic center in E of each bounded sequence of E is nonempty and compact. Then, T has a fixed point.Banach contraction mapping principle was extended nicely multivalued mappings by Nadler [6] in 1969. (Below is stated in a Banach space setting).Theorem 1.3 (Nadler [6]).
LetD be a nonempty closed subset of a Banach space E and T:D→CB(D) a multivalued contraction. Then, T has a fixed point.In 1953, Mann [7] introduced the following iterative scheme to approximate a fixed point of a nonexpansive mapping T in a Hilbert space H:
(1.3)xn+1=αnxn+(1-αn)Txn,∀n≥1,
where the initial point x0 is taken arbitrarily in D and {αn}n=1∞ is a sequence in [0,1]. However, we note that Mann's iteration has only weak convergence; see, for example, [8].In 2005, Sastry and Babu [9] proved that the Mann and Ishikawa iteration schemes for a multivalued map T with a fixed point p converge to a fixed point q of T under certain conditions. They also claimed that the fixed point p may be different from q.In 2007, Panyanak [10] extended the results of Sastry and Babu to uniformly convex Banach spaces and proved the following theorems.Theorem 1.4 (Panyanak [10]).
LetE be a uniformly convex Banach space, D a nonempty closed bounded convex subset of E, and T:D→P(D) a multivalued nonexpansive mapping that satisfies condition (I). Assume that (i)0≤αn<1 and (ii)Σn=1∞αn=∞. Suppose that F(T) a nonempty proximinal subset of D. Then, the Mann iterates {xn} defined by x0∈D,
(1.4)xn+1=αnyn+(1-αn)xn,αn∈[a,b],0<a<b<1,n≥0,
where yn∈Txn such that ∥yn-un∥=d(un,Txn) and un∈F(T) such that ∥xn-un∥=d(xn,F(T)), converges strongly to a fixed point of T.Theorem 1.5 (Panyanak [10]).
LetE be a uniformly convex Banach space, D a nonempty compact convex subset of E, and T:D→P(D) a multivalued nonexpansive mapping with a fixed point p. Assume that (i) 0≤αn,βn<1; (ii) βn→0 and (iii) Σn=1∞αnβn=∞. Then, the Ishikawa iterates {xn} defined by x0∈D,
(1.5)yn=βnzn+(1-βn)xn,βn∈[0,1],n≥0,zn∈Txn such that ∥zn-p∥=d(p,Txn), and
(1.6)xn+1=αnzn′+(1-αn)xn,αn∈[0,1],n≠0,zn′∈Tyn such that ∥zn′-p∥=d(p,Tyn) converges strongly to a fixed point of T.Later, Song and Wang [11] noted there was a gap in the proofs of Theorem 1.5 above and of [9, Theorem 5]. They further solved/revised the gap and also gave the affirmative answer Panyanak [10] question using the Ishikawa iterative scheme.  In the main results, the domain of T is still compact, which is a strong condition (see [11, Theorem 1]) and T satisfies condition (I) (see [11, Theorem 1]).Recently, Shahzad and Zegeye [12] proved the following theorems for quasi-nonexpansive multivalued map and multivalued map in uniformly convex Banach space.Theorem 1.6 (Shahzad and Zegeye [12]).
LetE be a uniformly convex real Banach space and D a nonempty, closed and convex subset of E. Let T:D→CB(D) be a quasi-nonexpansive multivalued map with F(T)≠∅ for which Tp={p}, for allp∈F(T). Let {xn}n=1∞ be a sequence defined iteratively by x0∈D,
(1.7)yn=βnzn+(1-βn)xn,βn∈[0,1],n≥0,zn∈Txn, and
(1.8)xn+1=αnzn′+(1-αn)xn,αn∈[0,1],n≠0,zn′∈Tyn. Assume that T satisfies condition (I) and αn,βn∈[a,b]⊂(0,1). Then, {xn}n=1∞ converges strongly to a fixed point of T.Theorem 1.7 (Shahzad and Zegeye [12]).
LetE be a uniformly convex real Banach space and D a nonempty, closed, and convex subset of E. Let T:D→P(D) be a multivalued map with F(T)≠∅ such that PT is nonexpansive. Let {xn}n=1∞ be a sequence defined iteratively by x0∈D,
(1.9)yn=βnzn+(1-βn)xn,βn∈[0,1],n≥0,zn∈PT(xn), and
(1.10)xn+1=αnzn′+(1-αn)xn,αn∈[0,1],n≠0,zn′∈PT(yn). Assume that T satisfies condition (I) and αn,βn∈[a,b]⊂(0,1). Then, {xn}n=1∞ converges strongly to a fixed point of T.More recently, Abbas  et al. [13] introduced the following one-step iterative process to compute common fixed points of two multivalued nonexpansive mappings.(1.11)x1∈D,xn+1=anxn+bnyn+cnzn,∀n≥1.
Using (1.11), Abbas et al. [13] proved weak and strong convergence theorems for approximation of common fixed point of two multivalued nonexpansive mappings in Banach spaces.Motivated by the ongoing research and the above mentioned results, we introduce  a new iterative scheme for approximation of common fixed points of finite family of multivalued maps in a real Banach space. Furthermore, we  prove strong convergence theorems for approximation of common fixed points of finite family of multivalued maps in a uniformly convex real Banach space. Next, we  prove a necessary and sufficient condition for strong convergence of our new iterative process to a common fixed point of finite family of multivalued maps. Finally, we  introduce a new iterative scheme and prove strong convergence theorems for finite family of quasi-nonexpansive multivalued maps in a uniformly convex real Banach space. Our results extend the results of Sastry and Babu [9], Panyanak [10], Shahzad and Zegeye [12], and Song and Wang [11].


## 2. Preliminaries

LetE be Banach space and dimE≥2. The modulus of convexity of E is the function δE:(0,2]→[0,1] defined by (2.1)δE(ϵ):=inf⁡{1-||x+y2||:‖x‖=‖y‖=1;ϵ=‖x-y‖}.E is uniformly convex if for any ϵ∈(0,2], there exists a δ=δ(ϵ)>0 such that if x,y∈E with ∥x∥≤1,∥y∥≤1 and ∥x-y∥≥ϵ, then ∥(1/2)(x+y)∥≤1-δ. Equivalently, E is uniformly convex if and only if δE(ϵ)>0 for all ϵ∈(0,2].A family{Ti:D→CB(D),i=1,2,…,m} is said to satisfy Condition (II) if there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0,f(r)>0 for r∈(0,∞) such that (2.2)d(x,Tix)≥f(d(x,∩i=1mF(Ti))),
for all i=1,2,…,m and x∈D.The mappingT:D→CB(D) is called hemicompact if for any sequence {xn} in D such that d(xn,Txn)→0 as n→∞, there exists a subsequence {xnk} of {xn} such that xnk→p∈D. We note that if D is compact, then every multivalued mapping T:D→CB(D) is hemicompact.LetD be a nonempty, closed, and convex subset of a real Banach space E. Let T:D→P(D) be a multimap and (2.3)PT(x):={ux∈Tx:‖x-ux‖=d(x,Tx)}.
Then, PT(x):D→P(D) is nonempty and compact for every x∈D. Furthermore, we observe that PT(y)={y} if y is a fixed point of T.A mappingT:D→P(D) is  *-nonexpansive([14]) if for all x,y∈D and ux∈Tx with d(x,ux)=inf{d(x,z):z∈Tx}, there exists uy∈Ty with d(y,uy)=inf{d(y,w):w∈Ty} such that (2.4)d(ux,uy)≤d(x,y).It is known that*-nonexpansiveness is different from nonexpansiveness for multimaps. There are some *-nonexpansive multimaps which are not nonexpansive and some nonexpansive multimaps which are not *-nonexpansive ([15, 16]).By the definition of Hausdorff metric, we obtain that if a multimapT:D→P(D) is *-nonexpansive, then PT is nonexpansive.Throughout this paper, we writexn→x to indicate that the sequence {xn} converges strongly to x.Also, this following lemma will be used in the sequel.Lemma 2.1 (Schu [17]).
Suppose thatE is a uniformly convex Banach space and 0<p≤tn≤q<1 for all positive integers n. Also, suppose that {xn} and {yn} are two sequences of E such that limsupn→∞∥xn∥≤r,limsupn→∞∥yn∥≤r and limn→∞∥tnxn+(1-tn)yn∥=r hold for some r>0. Then, limn→∞∥xn-yn∥=0.


## 3. Main Results

We now introduce the  following iteration scheme. LetE be a real normed space and D a nonempty subset of E. Let T1,T2,…,Tm be multivalued maps of D into P(D) with F:=∩i=1mF(Ti)≠∅ such that PT1,PT2,…,PTm are nonexpansive and {αni}n=1∞,i=0,1,…,m a sequence in [ϵ,1-ϵ],ϵ∈(0,1) such that ∑i=0mαni=1 for all n≥1. Let {xn}n=1∞ be a sequence defined iteratively by(3.1)x1∈D,xn+1=αn0xn+αn1yn(1)+⋯+αnmyn(m),
where yn(i)∈PTixn,i=1,2,…,m.Lemma 3.1.
LetE be a real normed space and D a nonempty subset of E. Let T1,T2,…,Tm be multivalued maps of D into P(D) with F:=∩i=1mF(Ti)≠∅ such that PT1,PT2,…,PTm are nonexpansive. Let {αni}n=1∞,i=0,1,…,m a sequence in [ϵ,1-ϵ],ϵ∈(0,1) such that ∑i=0mαni=1 for all n≥1. Let {xn}n=1∞ be a sequence defined iteratively by (3.1). Then,
(3.2)lim⁡n→∞d(xn,Tixn)=0,∀i=1,2,…,m.Proof.
Letx*∈∩i=1mF(Ti). Then, from (3.1), we have the following estimates:
(3.3)‖xn+1-x*‖≤αn0‖xn-x*‖+αn1‖yn(1)-x*‖+⋯+αnm‖yn(m)-x*‖≤αn0‖xn-x*‖+αn1d(yn(1),PT1x*)+⋯+αnmd(yn(m),PTmx*)≤αn0‖xn-x*‖+αn1H(PT1xn,PT1x*)+⋯+αnmH(PTmxn,PTmx*)≤αn0‖xn-x*‖+αn1‖xn-x*‖+⋯+αnm‖xn-x*‖=‖xn-x*‖.
Thus, limn→∞∥xn-x*∥ exists. Let
(3.4)lim⁡n→∞‖xn-x*‖=c,
for some c≥0. Then,
(3.5)c=lim⁡n→∞‖xn+1-x*‖=lim⁡n→∞‖αn0(xn-x*)+αn1(yn(1)-x*)+⋯+αnm(yn(m)-x*)‖=lim⁡n→∞‖(1-αn0)[αn11-αn0(yn(1)-x*)+⋯+αnm1-αn0(yn(m)-x*)]+αn0(xn-x*)‖.
Since PTi,i=1,2,…,m is nonexpansive mapping and ∩i=1mF(Ti)≠∅, we have
(3.6)‖yn(i)-x*‖≤d(yn(i),PTix*)≤H(PTixn,PTix*)≤‖xn-x*‖,
for each x*∈∩i=1mF(Ti). Taking limsup on both sides, we obtain
(3.7)limsup⁡n→∞‖yn(i)-x*‖≤c,i=1,2,…,m.
Next,
(3.8)limsup⁡n→∞‖αn11-αn0(yn(1)-x*)+αn21-αn0(yn(2)-x*)+⋯+αnm1-αn0(yn(m)-x*)‖≤limsup⁡n→∞[αn11-αn0‖yn(1)-x*‖+αn21-αn0‖yn(2)-x*‖+⋯+αnm1-αn0‖yn(m)-x*‖]≤limsup⁡n→∞αn1+αn2+⋯+αnm1-αn0‖xn-x*‖=c.
Using (3.5), (3.8), and Lemma 2.1, we obtain
(3.9)lim⁡n→∞‖αn11-αn0(yn(1)-x*)+αn21-αn0(yn(2)-x*)+⋯+αnm1-αn0(yn(m)-x*)-(xn-x*)‖=0.
This yields
(3.10)0=lim⁡n→∞‖αn11-αn0yn(1)+αn21-αn0yn(2)+⋯+αnm1-αn0yn(m)-xn‖=lim⁡n→∞(11-αn0)‖αn1yn(1)+αn2yn(2)+⋯+αnmyn(m)-(1-αn0)xn‖=lim⁡n→∞(11-αn0)‖xn+1-xn‖.
Thus, limn→∞∥xn+1-xn∥=0. Furthermore,
(3.11)c=lim⁡n→∞‖xn+1-x*‖=lim⁡n→∞‖αn0(xn-x*)+αn1(yn(1)-x*)+⋯+αnm(yn(m)-x*)‖=lim⁡n→∞‖(1-αn1)[αn01-αn1(xn-x*)+αn21-αn1(yn(2)-x*)+⋯+αnm1-αn1(yn(m)-x*)]+αn1(yn(1)-x*)αnm1-αn1‖,lim⁡⁡sup⁡n→∞‖αn01-αn1(xn-x*)+αn21-αn1(yn(2)-x*)+⋯+αnm1-αn1(yn(m)-x*)‖≤lim⁡⁡sup⁡n→∞[αn01-αn1‖xn-x*‖+αn21-αn1‖yn(2)-x*‖+⋯+αnm1-αn1‖yn(m)-x*‖]≤lim⁡⁡sup⁡n→∞αn0+αn2+⋯+αnm1-αn1‖xn-x*‖=c.
Using (3.11) and Lemma 2.1, we obtain
(3.12)lim⁡n→∞‖αn01-αn1(xn-x*)+αn21-αn1(yn(2)-x*)+⋯+αnm1-αn1(yn(m)-x*)-(yn(1)-x*)‖=0.
This yields
(3.13)0=lim⁡n→∞‖αn01-αn1xn+αn21-αn1yn(2)+⋯+αnm1-αn1yn(m)-yn(1)‖=lim⁡n→∞(11-αn1)‖αn0xn+αn2yn(2)+⋯+αnmyn(m)-(1-αn1)yn(1)‖=lim⁡n→∞(11-αn1)‖xn+1-yn(1)‖.
Thus, limn→∞∥xn+1-yn(1)∥=0. So,
(3.14)‖xn-yn(1)‖≤‖xn+1-xn‖+‖xn+1-yn(1)‖⟶0,n⟶∞.
Then,
(3.15)d(xn,T1xn)≤d(xn,PT1xn)≤‖xn-yn(1)‖⟶0,n⟶∞.
In a similar way, we can show that
(3.16)lim⁡n→∞d(xn,Tixn)=0,i=2,3,…,m.
This completes the proof.Theorem 3.2.
LetE be a uniformly convex real Banach space and D a nonempty, closed, and convex subset of E. Let T1,T2,…,Tm be multivalued maps of D into P(D) with F:=∩i=1mF(Ti)≠∅ such that PT1,PT2,…,PTm are nonexpansive and {Ti}i=1m satisfying condition (II). Let {αni}n=1∞,i=0,1,…,m a sequence in [ϵ,1-ϵ],ϵ∈(0,1) such that ∑i=0mαni=1 for all n≥1. Let {xn}n=1∞ be a sequence defined iteratively by (3.1). Then, {xn}n=1∞ converges strongly to a common fixed point of {Ti}i=1m.Proof.
Since{Ti}i=1m satisfies condition (II), we have that d(xn,∩i=1mF(Ti))→0 as n→∞. Thus, there is a subsequence {xnk} of {xn} and a sequence {pk}⊂∩i=1mF(Ti) such that
(3.17)‖xnk-pk‖<12k,
for all k. By Lemma 3.1, we obtain
(3.18)‖xnk+1-pk‖≤‖xnk-pk‖<12k.
We now show that {pk} is a Cauchy sequence in D. Observe that
(3.19)‖pk+1-pk‖≤‖pk+1-xnk+1‖+‖xnk+1-pk‖<12k+1+12k<12k-1.
This shows that {pk} is a Cauchy sequence in D, and thus converges to p∈D. Since
(3.20)d(pk,Tip)‖≤d(pk,PTip)≤H(PTip,PTipk)≤‖p-pk‖,
and pk→p as k→∞, it follows that d(p,Tip)=0, and thus p∈∩i=1mF(Ti), and {xnk} converges strongly to p. Since limn→∞∥xn-p∥ exists, it follows that {xn} converges strongly to p. This completes the proof.Corollary 3.3.
LetE be a uniformly convex real Banach space and D a nonempty, closed, and convex subset of E. Let T1,T2,…,Tm be *-nonexpansive multimaps of D into P(D) with F:=∩i=1mF(Ti)≠∅ and {Ti}i=1m satisfying condition (II). Let {αni}n=1∞,i=0,1,…,m a sequence in [ϵ,1-ϵ],ϵ∈(0,1) such that ∑i=0mαni=1 for all n≥1. Let {xn}n=1∞ be a sequence defined iteratively by (3.1). Then, {xn}n=1∞ converges strongly to a common fixed point of {Ti}i=1m.Theorem 3.4.
LetE be a uniformly convex real Banach space and D a nonempty, closed and convex subset of E. Let T1,T2,…,Tm be multivalued maps of D into P(D) with F:=∩i=1mF(Ti)≠∅ such that PT1,PT2,…,PTm are nonexpansive and Ti is hemicompact and continuous for each i=1,2,…,m. Let {αni}n=1∞,i=0,1,…,m a sequence in [ϵ,1-ϵ],ϵ∈(0,1) such that ∑i=0mαni=1 for all n≥1. Let {xn}n=1∞ be a sequence defined iteratively by (3.1). Then, {xn}n=1∞ converges strongly to a common fixed point of {Ti}i=1m.Proof.
Sincelimn→∞d(xn,Tixn)=0,for alli=1,2,…,m and Ti is hemicompact for each i=1,2,…,m, there is a subsequence {xnk} of {xn} such that xnk→p as k→∞ for some p∈D. Since Ti is continuous for each i=1,2,…,m, we have d(xnk,Tixnk)→d(p,Tip). As a result, we have that d(p,Tip)=0,for alli=1,2,…,m, and so, p∈∩i=1mF(Ti). Since limn→∞∥xn-p∥ exists, it follows that {xn} converges strongly to p. This completes the proof.Corollary 3.5.
LetE be a uniformly convex real Banach space and D a nonempty, closed, and convex subset of E. Let T1,T2,…,Tm be *-nonexpansive multimaps of D into P(D) with F:=∩i=1mF(Ti)≠∅ and Ti is hemicompact and continuous for each i=1,2,…,m. Let {αni}n=1∞,i=0,1,…,m a sequence in [ϵ,1-ϵ],ϵ∈(0,1) such that ∑i=0mαni=1 for all n≥1. Let {xn}n=1∞ be a sequence defined iteratively by (3.1). Then, {xn}n=1∞ converges strongly to a common fixed point of {Ti}i=1m.Theorem 3.6.
LetE be a uniformly convex real Banach space and D a nonempty compact convex subset of E. Let T1,T2,…,Tm be multivalued maps of D into P(D) with F:=∩i=1mF(Ti)≠∅ such that PT1,PT2,…,PTm are nonexpansive. Let {αni}n=1∞,i=0,1,…,m a sequence in [ϵ,1-ϵ],ϵ∈(0,1) such that ∑i=0mαni=1 for all n≥1. Let {xn}n=1∞ be a sequence defined iteratively by (3.1). Then, {xn}n=1∞ converges strongly to a common fixed point of {Ti}i=1m.Proof.
From the compactness ofD, there exists a subsequence {xnk}n=k∞ of {xn}n=1∞ such that limk→∞∥xnk-q∥=0 for some q∈D. Thus,
(3.21)d(q,Tiq)≤d(q,PTiq)≤‖xnk-q‖+d(xnk,PTixnk)+H(PTixnk,PTiq)≤2‖xnk-q‖+d(xnk,PTixnk)⟶0ask⟶∞.
Hence, q∈∩i=1mF(Ti). Now, on taking q in place of x*, we get that limn→∞∥xn-q∥ exists. This completes the proof.The following result gives a necessary and sufficient condition for strong convergence of the sequence in (3.1) to a common fixed point of {Ti}i=1m.Theorem 3.7.
LetD be a nonempty, closed, and convex subset of a real Banach space E. Let T1,T2,…,Tm be multivalued maps of D into P(D) with F:=∩i=1mF(Ti)≠∅ such that PT1,PT2,…,PTm are nonexpansive. Let {αni}n=1∞,i=0,1,…,m a sequence in [ϵ,1-ϵ],ϵ∈(0,1) such that ∑i=0mαni=1 for all n≥1. Let {xn}n=1∞ be a sequence defined iteratively by (3.1). Then, {xn}n=1∞ converges strongly to a common fixed point of {Ti}i=1m if and only if liminfn→∞d(xn,F)=0.Proof.
The necessity is obvious. Conversely, suppose thatliminfn→∞d(xn,F)=0. By (3.3), we have
(3.22)‖xn+1-x*‖≤‖xn-x*‖.
This gives
(3.23)d(xn+1,F)≤d(xn,F).
Hence, limn→∞d(xn,F) exists. By hypothesis, liminfn→∞d(xn,F)=0, so we must have limn→∞d(xn,F)=0.
Next, we show that{xn}n=1∞ is a Cauchy sequence in D. Let ϵ>0 be given, and since liminfn→∞d(xn,F)=0, there exists n0 such that for all n≥n0, we have
(3.24)d(xn,F)<ϵ4.
In particular, inf{∥xn0-p∥:p∈F}<ϵ/4 so that there must exist a p*∈F such that
(3.25)‖xn0-p*‖<ϵ2.
Now, for m,n≥n0, we have
(3.26)‖xn+m-xn‖≤|‖xn+m-p*‖+‖xn-p*‖≤2‖xn0-p*‖<2(ϵ2)=ϵ.
Hence, {xn} is a Cauchy sequence in a closed subset D of a Banach space E, and therefore, it must converge in D. Let limn→∞xn=p. Now, for each i=1,2,…,m, we obtain
(3.27)d(p,Tip)≤d(p,PTip)≤d(p,xn)+d(xn,PTixn)+H(PTixn,PTip)≤d(p,xn)+d(xn,PTixn)+d(xn,p)⟶0asn⟶∞
gives that d(p,Tip)=0,i=1,2,…,m which implies that p∈Tip. Consequently, p∈F=∩i=1mF(Ti)≠∅.All the results we have obtained so far can be established for finite family of quasi-nonexpansive multivalued maps. LetT1,T2,…,Tm be quasi-nonexpansive multivalued maps of D into CB(D) such that F:=∩i=1mF(Ti)≠∅ for which Tip={p}, for all p∈∩i=1mF(Ti). Let {αni}n=1∞,i=0,1,…,m a sequence in [ϵ,1-ϵ],ϵ∈(0,1) such that ∑i=0mαni=1 for all n≥1. Let {xn}n=1∞ be a sequence defined iteratively by(3.28)x1∈D,xn+1=αn0xn+αn1yn(1)+⋯+αnmyn(m),
where yn(i)∈Tixn, i=1,2,…,m. Thus, we obtain the following theorems using iterative process (3.28).Theorem 3.8.
LetE be a uniformly convex real Banach space and D a nonempty, closed, and convex subset of E. Let T1,T2,…,Tm be quasi-nonexpansive multivalued maps of D into CB(D) such that F:=∩i=1mF(Ti)≠∅ for which Tip={p}, for all p∈∩i=1mF(Ti) and {Ti}i=1m satisfying condition (II). Let {αni}n=1∞,i=0,1,…,m a sequence in [ϵ,1-ϵ],ϵ∈(0,1) such that ∑i=0mαni=1 for all n≥1. Let {xn}n=1∞ be a sequence defined iteratively by (3.28). Then, {xn}n=1∞ converges strongly to a common fixed point of {Ti}i=1m.Theorem 3.9.
LetE be a uniformly convex real Banach space and D a nonempty, closed, and convex subset of E. Let T1,T2,…,Tm be quasi-nonexpansive multivalued maps of D into CB(D) such that F:=∩i=1mF(Ti)≠∅ for which Tip={p}, for all p∈∩i=1mF(Ti) and Ti is hemicompact and continuous for each i=1,2,…,m. Let {αni}n=1∞,i=0,1,…,m a sequence in [ϵ,1-ϵ],ϵ∈(0,1) such that ∑i=0mαni=1 for all n≥1. Let {xn}n=1∞ be a sequence defined iteratively by (3.28). Then, {xn}n=1∞ converges strongly to a common fixed point of {Ti}i=1m.Theorem 3.10.
LetE be a uniformly convex real Banach space and D a nonempty compact convex subset of E. Let T1,T2,…,Tm be quasi-nonexpansive multivalued maps of D into CB(D) such that F:=∩i=1mF(Ti)≠∅ for which Tip={p}, for all p∈∩i=1mF(Ti). {αni}n=1∞,i=0,1,…,m a sequence in [ϵ,1-ϵ],ϵ∈(0,1) such that ∑i=0mαni=1 for all n≥1. Let {xn}n=1∞ be a sequence defined iteratively by (3.28). Then, {xn}n=1∞ converges strongly to a common fixed point of {Ti}i=1m.Corollary 3.11 (Abbas et al. [13]).
LetE be a uniformly convex real Banach space satisfying Opial's condition. Let D be a nonempty, closed, and convex of E. Let T,S be multivalued nonexpansive mappings of D into K(D) such that F:=F(T)∩F(S)≠∅. Let {an}n=1∞,{bn}n=1∞, and {cn}n=1∞ be sequence in (0,1) satisfying an+bn+cn≤1. Let {xn}n=1∞ be a sequence defined iteratively by
(3.29)x1∈D,xn+1=anxn+bnyn+cnzn,∀n≥1,
where yn∈Txn,zn∈Sxn such that ∥yn-p∥≤d(p,Txn) and ∥zn-p∥≤d(p,Sxn) whenever p is a fixed point of any one of mappings T and S. Then, {xn}n=1∞ converges weakly to a common fixed point of F(T)∩F(S).Corollary 3.12 (Abbas et al. [13]).
LetE be a real Banach space and D a nonempty, closed, and convex subset of E. Let T,S be multivalued nonexpansive mappings of D into K(D) such that F:=F(T)∩F(S)≠∅. Let {an}n=1∞,{bn}n=1∞ and {cn}n=1∞ be sequence in (0,1) satisfying an+bn+cn≤1. Let {xn}n=1∞ be a sequence defined iteratively by
(3.30)x1∈D,xn+1=anxn+bnyn+cnzn,∀n≥1,
where yn∈Txn, zn∈Sxn such that ∥yn-p∥≤d(p,Txn) and ∥zn-p∥≤d(p,Sxn) whenever p is a fixed point of any one of mappings T and S. Then, {xn}n=1∞ converges strongly to a common fixed point of F(T)∩F(S) if and only if liminfn→∞d(xn,F)=0.Remark 3.13.
Our results extend the results of Sastry and Babu [9], Panyanak [10], and Song and Wang [11] from approximation of a fixed point of a single multivaued nonexpansive mapping to approximation of common fixed point of a finite family of quasi-nonexpansive multivaued mappings.Remark 3.14.
Our results extend the results of Shahzad and Zegeye [12] from approximation of a fixed point of a single quasi-nonexpansive multivaued mapping and single multivalued map to approximation of common fixed point of a finite family of quasi-nonexpansive multivaued mappings and a finite family of multivalued maps.



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*Source: 576108-2011-08-28.xml*