# Iterative Algorithms for a Finite Family of Multivalued Quasi-Nonexpansive Mappings

Authors: C. Diop; M. Sene; N. Djitté
Journal: Advances in Numerical Analysis
 (2014)

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## Abstract

LetK be a nonempty closed and convex subset of a uniformly convex real Banach space E and let T1,…,Tm:K→2K be m multivalued quasi-nonexpansive mappings. A new iterative algorithm is constructed and the corresponding sequence xn is proved to be an approximating fixed point sequence of each Ti; that is, limdxn;Txn=0. Then, convergence theorems are proved under appropriate additional conditions. Our results extend and improve some important recent results (e.g., Abbas et al. (2011)).


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## Body

## 1. Introduction

Let(X,d) be a metric space, K a nonempty subset of X, and T:K→2K a multivalued mapping. An element x∈K is called a fixed point of T if x∈Tx. For single valued mapping, this reduces to Tx=x. The fixed point set of T is denoted by F(T)∶={x∈K:xTx}.For several years, the study of fixed point theory formultivalued nonlinear mappings has attracted, and continues to attract, the interest of several well known mathematicians (see, e.g., Brouwer [1], Kakutani [2], Nash [3, 4], Geanakoplos [5], Nadler Jr. [6], and Downing and Kirk [7]).Interest in the study of fixed point theory for multivalued nonlinear mappings stems, perhaps, mainly from its usefulness in real-world applications such asGame Theory andNonsmooth Differential Equations.Game Theory. Nash showed the existence of equilibria for noncooperativestatic games as a direct consequence ofmultivalued Brouwer or Kakutani fixed point theorem. More precisely, under some regularity conditions, given a game, there always exists amultivalued mapping whose fixed points coincide with the equilibrium points of the game. This, among other things, made Nash a recipient of Nobel Prize in Economic Sciences in 1994. However, it has been remarked that the applications of this theory to equilibrium are mostlystatic: they enhance understanding conditions under which equilibrium may be achieved but do not indicate how to construct a process starting from a nonequilibrium point and convergent to equilibrium solution. This is part of the problem that is being addressed byiterative methods for fixed point of multivalued mappings.Nonsmooth Differential Equations. A large number of problems from mechanics and electrical engineering lead to differential inclusions and differential equations with discontinuous right-hand sides, for example, a dry friction force of some electronic devices. Below are two models:
(1)dudt=f(t,u),a.e.t∈I∶=[-a,a],u(0)=u0,
where a and u0 are fixed in R. These types of differential equations do not have solutions in the classical sense. A generalized notion of solution is what is called a solution in the sense of Fillipov.Consider the followingmultivalued initial value problem:
(2)-d2udt2∈u-14-14sign⁡u-1on⁡Ω=(0,π);u(0)=0;u(π)=0.
Under some conditions, the solutions set of (1) and (2) coincides with the fixed point set of some multivalued mappings.LetD be a nonempty subset of a normed linear space E. The set D is calledproximinal (see, e.g., [8–10]) if for each x∈E, there exists u∈D such that
(3)dx,u=inf⁡x-y:y∈D=dx,D,
where d(x,y)=x-y for all x,y∈E. Every nonempty, closed, and convex subset of a real Hilbert space is proximinal. We denote by CB(D) the families of nonempty closed and bounded subsets of D, K(D) the families of nonempty compact subsets of D, and C(D) the families of nonempty compact convex subsets of D. TheHausdorff metric on CB(K) is defined by
(4)HA,B=max⁡sup⁡⁡a∈Ada,B,sup⁡⁡b∈Bdb,A
for all A,B∈CB(K). A multivalued mapping T:D(T)⊆E→CB(E) is calledLipschitzian if there exists L>0 such that
(5)H(Tx,Ty)≤Lx-y∀x,y∈D(T).
When L∈(0,1) in (5), we say that T is acontraction, and T is callednonexpansive if L=1. Finally, A multivalued mapping T:K→CB(K) is said to be quasi-nonexpansive if F(T)≠∅ and H(Tx,Tp)≤x-p for all x∈K and p∈F(T).Several papers deal with the problem ofapproximating fixed points ofmultivalued nonexpansive mappings (see, e.g., [8–12] and the references therein) and their generalizations (see, e.g., [13–15]).On the other hand, Abbas et al. [11] introduced a new one-step iterative process for approximating a common fixed point of two multivaluednonexpansive mappings in a real uniformly convex Banach space and established weak and strong convergence theorems for the proposed process under some basic boundary conditions. Let S,T:K→CB(K) be two multivalued nonexpansive mappings. They introduced the following iterative scheme:
(6)x1∈K,xn+1=anxn+bnyn+cnzn,
where yn∈Txn and zn∈Sxn are such that
(7)yn-p≤d(p,Sxn);zn-p≤d(p,Txn)
and {an}, {bn}, and {cn} are real sequences in (0,1) satisfying an+bn+cn=1.The following lemma is a consequence of the definition of Hausdorff metric, as remarked by Nadler Jr. [6].Lemma 1.
LetA,B∈CB(X) and a∈A. For every γ>0, there exists b∈B such that
(8)d(a,b)≤H(A,B)+γ.Following the work of Abbas et al. [11], Rashwan and Altwqi [16] introduced a new scheme for approximation of a common fixed point of three multivalued nonexpansive mappings in uniformly convex Banach space. Let T,S,R:K→CB(K) be three multivalued nonexpansive mappings. They employed the following iterative process:
(9)x1∈K,xn+1=anyn+bnzn+cnwn,n≥1,
where yn∈Txn, zn∈Sxn, and wn∈Rxn are such that
(10)yn-yn+1≤HTxn,Txn+1+ηn;zn-zn+1≤HSxn,Sxn+1+ηn;wn-wn+1≤HRxn,Rxn+1+ηn
and {an}, {bn}, and {cn} are real sequences in (0,1) satisfying an+bn+cn=1.Remark 2.
Note that ifyn, zn, and wn are known, then the existence of yn+1, zn+1, and wn+1 satisfying (10) is guaranteed by Lemma 1.Before we state the result of Rashwan and Altwqi [16], we need the following definition.The mappingsT,S,R:K→CB(K) are said to satisfy condition (C) if d(x,y)≤d(z,y), for z∈Tx and y∈Sx, or d(x,y)≤d(z,y), for z∈Tx and y∈Rx, or d(x,y)≤d(z,y), for z∈Rx and y∈Sx.LetF=F(T)∩F(S)∩F(R) be the set of all common fixed points of the mappings T, S, and R.Theorem RA (Rashwan and Altwqi [16]).LetEbe a uniformly convex Banach space andKa nonempty closed and convex subset ofE. LetT,S,R:K→CB(K)be multivalued nonexpansive mappings satisfying condition(C)and{xn}the sequence defined by (9) and (10). IfF≠∅andTp=Sp=Rp={p}for anyp∈F, then(11)lim⁡n→∞d(xn,Txn)=lim⁡n→∞d(xn,Sxn)=0=lim⁡n→∞d(xn,Rxn).Recently, Bunyawat and Suantai [17] introduced a one-step iterative scheme for finding a common fixed point of a finite family of multivalued quasi-nonexpansive mappings in a uniformly convex real Banach space. They proposed the following algorithm: let K be a nonempty closed and convex subset of a uniformly convex real Banach space E and T1,T2,…,Tm:K→CB(K) multivalued quasi-nonexpansive mappings. Let {xn} be the sequence defined iteratively from x1∈K by
(12)xn+1=an,0xn,0+an,1xn,1+⋯+an,mxn,m,n≥1,
where the sequence {an,i}⊂[0,1) satisfies ∑i=0man,i=1  ∀n≥1 and xn,i∈Tixn with d(p,xn,i)=d(p,Tixn).Then, they proved the following theorem.Theorem BS (Bunyawat and Suantai [17]).LetEbe a real Banach space andKa closed convex subset ofX. Let{Ti:i=1,2,…,m}be a finite family of multivalued quasi-nonexpansive mappings fromKintoC(K)withF∶=∩i=1mF(Ti)≠∅. Then the sequencexndefined by (12) converges strongly to a common fixed point of{Ti:i=1,2,…,m}if and only ifliminf⁡d(xn,F)=0.It is our purpose in this paper to construct a new iterative algorithm and prove strong convergence theorems for approximating a common fixed point of a finite family of multivalued quasi-nonexpansive mappings in uniformly convex real Banach spaces. The class of mappings used in our theorems is much more larger than that of multivalued nonexpansive mappings. Our theorems generalize and extend those of Abbas et al. [11], Rashwan and Altwqi [16], and Bunyawat and Suantai [17] and many other important results.


## 2. Preliminaries

Lemma 3 (Chang et al. [18]).
LetE be a uniformly convex real Banach space. For arbitrary r>0, let Br(0)={x∈E:x≤r}. Then, for any given subset {u1,u2,…,uN}⊂Br(0) and for any positive real numbers λ1,λ2,…,λN with λ1+λ2+⋯+λN=1, there exists a continuous, strictly increasing, and convex function g:[0,2r]→[0,+∞) with g(0)=0 such that, for any i, j∈{1,2,…,N} with i<j,
(13)∑k=1N‍λkuk2≤∑k=1N‍λkuk2-λiλjgui-uj.


## 3. Main Results

In this paper, we propose the following iterative algorithm.Letm≥1 and K a nonempty closed convex subset of a uniformly convex real Banach space E. Let T1,…,Tm:K→CB(K) be multivalued quasi-nonexpansive mappings. Let {xn} be a sequence defined iteratively as follows:
(14)x1∈K,xn+1=λ0xn+λ1un1+⋯+λmunm,
where uni∈Tixn, i=1,…,m, λi∈(0,1), i=0,1,…,m, with λ0+λ1+⋯λm=1. In the sequel, we will write F∶=∩i=1mF(Ti) for the set of common fixed points of the mappings Ti, i=1,…,m.Lemma 4.
LetK be a nonempty, closed, and convex subset of a real Banach space E. For m≥1, let {Ti:i=1,2,…,m} be a finite family of multivalued quasi-nonexpansive mappings from K into CB(K) with F∶=∩i=1mF(Ti)≠∅ and such that Ti(p)={p}∀p∈F and i=1,2,…,m. Let {xn} be the sequence defined by (14). Then, for all p∈F, lim⁡n→∞xn-p exits.Proof.
Letp∈F. By (14) and the quasi-nonexpansiveness of Ti we have
(15)xn+1-p=λ0xn-p+∑i=1m‍λiuni-p≤λ0xn-p+∑i=1m‍λiuni-p≤λ0xn-p+∑i=1m‍λiHTixn,Tip≤λ0xn-p+∑i=1m‍λixn-p=xn-p.
Equation (15) implies that the sequence {xn-p} is monotonically decreasing. Since it is bounded from below by 0, we conclude that lim⁡n→∞xn-p exists.Next, we prove the following theorem.Theorem 5.
LetK be a nonempty, closed, and convex subset of a real uniformly convex Banach space E. For i=1,…,m let Ti:K→CB(K) be a multivalued quasi-nonexpansive mapping. Suppose that F≠∅ and that Ti(p)={p} for all p∈F. Let {xn} be the sequence defined by (14). Then, for all i=1,…,m,
(16)lim⁡n→∞d(xn,Tixn)=0.Proof.
Using Lemma4 and the quasi-nonexpansiveness of Ti, there exists some positive real r such that
(17)uni-p≤xn-p≤r.
This implies that, for each i, 1≤i≤m,
(18)uni-xn≤2r.
Next, let 1≤i≤m; using inequality (17) and Lemma 3, we have
(19)xn+1-p2=λ0(xn-p)+∑k=1m‍λk(unk-p)2≤λ0xn-p2+∑k=1m‍λkunk-p2-λ0λig(uni-xn)≤λ0xn-p2+∑k=1m‍λkxn-p2-λ0λig(uni-xn)=xn-p2-λ0λiguni-xn.
Therefore
(20)λ0λiguni-xn≤xn-p2-xn+1-p2.
This implies that
(21)lim⁡n→∞guni-xn=0.
It follows that
(22)lim⁡n→∞uni-xn=0.
Since uni∈Tixn, we have
(23)lim⁡n→∞d(xn,Tixn)=0∀i=1,…,m.
This completes the proof.We now approximate common fixed points of{T1,T2,…,Tm} through strong convergence of the sequence {xn} defined by (14). We start with the following definition.Definition 6.
A family{T1,…,Tm:K→CB(K)} is said to satisfyCondition(I*), if there exists a strictly increasing function f:[0,∞)→[0,∞) with f(0)=0, f(r)>0 for all r∈(0,∞) and i0, 1≤i0≤m, such that
(24)d(x,Ti0x)≥f(d(x,F))∀x∈K.Theorem 7.
LetK be a nonempty, closed, and convex subset of a uniformly convex real Banach space E. For m≥1, let T1,T2,…,Tm:K→CBK be a multivalued quasi-nonexpansive mapping. Assume that T1,…,Tm satisfies condition I*. If F≠∅ and Tip=p for all p∈F, then the sequence xn converges strongly to a common fixed point of {Ti,i=1,…,m}.Proof.
From Theorem5, we have lim⁡n→∞d(xn,Tixn)=0 for all i=1,2,…,m. Using the fact that {T1,…,Tm} satisfies condition (I*), it follows that there exists some i0 such that lim⁡n→∞f(d(xn,F(Ti0)))=0. Thus there exist a subsequence {xnj} of {xn} and a sequence {pj}⊂F(Ti0) such that
(25)xnj-pj<12j∀j∈N.
By setting pj in place of p and following the same arguments as in the proof of Lemma 4, we obtain from inequality (15) that
(26)xnj+1-pj≤xnj-pj<12j.
We now show that {pj} is a Cauchy sequence in K. Notice that
(27)pj+1-pj≤pj+1-xnj+1+xnj+1-pj<12j+1+12j<12j-1.
This shows that {pj} is a Cauchy sequence in K and thus converges strongly to some p∈K. Using the fact that Ti is quasi-nonexpansive and pj→p, we have
(28)dpj,Tip≤HTipj,Tip≤pj-p,
so that d(p,Ti(p))=0 and thus p∈Ti(p). Therefore, p∈F(Ti) and {xnj} converges strongly to p. Setting pj in place of p in the proof of Lemma 4, it follows from inequality (15) that lim⁡n→∞xn-p exists. So, {xn} converges strongly to p. This completes the proof.Definition 8.
A mappingT:K→CB(K) is calledhemicompact if, for any sequence {xn} in K such that d(xn,Txn)→0 as n→∞, there exists a subsequence {xnj} of {xn} such that xnj→p∈K. One notes that if K is compact, then every multivalued mapping T:K→CB(K) is hemicompact.Theorem 9.
LetK be a nonempty, closed, and convex subset of a uniformly convex real Banach space E. For i=1,…,m let Ti:K→CB(K) be a multivalued continuous and quasi-nonexpansive mapping. Assume that Ti0 is hemicompact for some i0. If F≠∅ and Ti(p)={p} for all p∈F, then the sequence {xn} defined by (14) converges strongly to a common fixed point of {Ti,i=1,…,m}.Proof.
From Theorem5, we have that lim⁡n→∞d(xn,Tixn)=0. Since Ti0 is hemicompact, there exists a subsequence {xnj} of {xn} such that xnj→p for some p∈K. Since for each i=1,2,…,m, Ti is continuous, we have d(xnj,Tixnj)→d(p,Ti(p)). Therefore, d(p,Ti(p))=0 and so p∈F(Ti) for all i=1,2,…,m. Setting x*=p in the proof of Theorem 5, it follows from inequality (15) that the sequence {xn-p} is decreasing and bounded from bellow. Therefore, lim⁡n→∞xn-p exists. So, {xn} converges strongly to p. This completes the proof.The following result gives a necessary and sufficient condition for the strong convergence of the sequence in (14) to a common fixed point of a finite family of multivalued nonexpansive maps Ti, i=1,2,…,m.Theorem 10.
LetK be a nonempty, closed, and convex subset of a real Banach space E. For i=1,…,m let Ti:K→CB(K) be a multivalued nonexpansive mapping. Assume that F≠∅ and that Ti(p)={p} for all p∈F. Then the sequence {xn} defined by (14) converges strongly to a common fixed point of {Ti,i=1,…,m} if and only if liminf⁡d(xn,F)=0.Proof.
The necessity is obvious. Conversely, suppose thatliminf⁡d(xn,F)=0. Let p∈F. By (15), we have xn+1-p≤xn-p. This gives d(xn+1,F)≤d(xn,F). Hence, lim⁡d(xn,F) exists. By hypothesis, liminf⁡d(xn,F)=0 so we must have lim⁡d(xn,F)=0. Next, we show that {xn} is a Cauchy sequence in K. Let ϵ>0 be given and since lim⁡d(xn,F)=0, there exists N≥0 such that, for all n≥N, we have
(29)d(xn,F)<ϵ4.
In particular, inf⁡{xN-p:p∈F}<ϵ/4, so there exists p∈F such that
(30)xN-p<ϵ2.
Now for m,n≥N, we have
(31)xn+m-xn≤xn+m-p+xn-p≤2xN-p<2ϵ2=ϵ.
Hence, {xn} is a Cauchy sequence in a closed subset K of a Banach space E and therefore, it converges in K. Let lim⁡xn=x*. Now, for each i=1,2,3,…,m, we have
(32)dx*,Tix*≤d(x*,xn)+d(xn,Tixn)+H(Tixn,Tix*)≤d(p,xn)+d(xn,Tixn)+d(xn,p).
By letting n→+∞, it follows that d(x*,Tix*)=0, i=1,2,3,…,m, which implies that x*∈Tix*. Consequently, x*∈F.Corollary 11 (Abbas et al. [11]).
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(T)∩F(S)≠∅. Let {an}, {bn}, and {cn} be sequences in (0,1) satisfying an+bn+cn=1. Let {xn} be a sequence defined iteratively by
(33)x1∈D,xn+1=anxn+bnyn+cnzn,n≥1,
where yn∈Txn, zn∈Sxn are such that yn-p≤d(p,Sxn) and zn-p≤d(p,Txn) whenever p is a fixed point of any one of the mappings T and S. Then, {xn} converges strongly to a common fixed point of F(T)∩F(S) if and only if liminf⁡d(xn,F)=0.Corollary 12 (Bunyawat and Suantai [17]).
LetE be a real Banach space and K a closed convex subset of X. Let {Ti:i=1,2,…,m} be a finite family of multivalued quasi-nonexpansive mappings from K into C(K) with F∶=∩i=1mF(Ti)≠∅. Then the sequence xn defined by (12) converges strongly to a common fixed point of {Ti:i=1,2,…,m} if and only if liminf⁡d(xn,F)=0.Remark 13.
The recursion formula (14) used in our theorems is easier to use than the recursion formula (6) of Abbas et al., the one of Rashwan and Altwqi (9), and the one of Bunyawat and Suantai (12) in the following sense: in our algorithm, uni∈Tixn for i=1,…,m and do not have to satisfy the restrictive conditions: (7) in the recursion formula (6), (10) in the recursion formulas (9), and similar restrictive conditions in the recursion formula (12).



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