# Iterative Approximation of Endpoints for Multivalued Mappings in Banach Spaces

Authors: Thabet Abdeljawad; Kifayat Ullah; Junaid Ahmad; Nabil Mlaiki
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The purpose of this paper is to introduce the modified Agarwal-O’Regan-Sahu iteration process (S-iteration) for finding endpoints of multivalued nonexpansive mappings in the setting of Banach spaces. Under suitable conditions, some weak and strong convergence results of the iterative sequence generated by the proposed process are proved. Our results especially improve and unify some recent results of Panyanak (J. Fixed Point Theory Appl. (2018)). At the end of the paper, we offer an example to illustrate the main results.


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## Body

## 1. Introduction and Preliminaries

Throughout this paper,ℕ stands for the set of natural numbers, and ℝ stands for the set of real numbers. Let X=X,⋅ be a Banach space and C be a nonempty subset of X. For x∈X, set(1)distx,C=infx−y:y∈C,Dx,C=supx−y:y∈C.We shall denote the set of all nonempty and compact subsets ofC by KC. Set HA,B=maxsupa∈Adista,B,supb∈Bdistb,A,for  eachA,B∈KC.H.,. is known as the Hausdorff metric on the set KC. A multivalued mapping T:C⟶KC is said to be nonexpansive if(2)HTx,Ty≤x−y,for eachx,y∈C.A pointq∈C is said to be a fixed point of T:C⟶KC if q∈Tq and is said to be an endpoint (or a stationary point) of T:C⟶KC if Tq=q. From now on, we will denote the set of all endpoints and the set of all fixed points of T by ET and FT, respectively. Note that, a multivalued mapping T:C⟶KC is said to satisfy the endpoint condition [1] if ET=FT. Different iteration processes have been developed to approximate the fixed points of multivalued mappings. Keep in mind, Sastry and Babu [2] proved Mann and Ishikawa-type convergence results for multivalued nonexpansive mappings in the framework of Hilbert spaces. Panyanak [3] extended the results of Sastry and Babu [2] to the framework of uniformly convex Banach spaces. Actually, Panyanak [3] showed some results using Ishikawa-type iteration process without the endpoint condition. Song and Wang [4] showed that without the endpoint condition, their process was not well-defined. They reconstructed the process using the endpoint condition which made it well-defined. After this, Shahzad and Zegeye [5] introduced two types of Ishikawa iterations. Note that, their first type iteration also requires the endpoint condition.For a multivalued mappingT:C⟶KC, if q∈C is an endpoint of T, then q is also a fixed point of T; but the converse is not always true (see the following example).Example 1.
LetC=0,1. Define T:C⟶KC by(3)Tx=0,x2for eachx∈C.
ClearyET=0 and FT=0,1.
For existence results of endpoints of multivalued mappings in the framework of Banach spaces, see [6–12]. Very recently, Panyanak [13] used an Ishikawa-type iteration process to approximate endpoints of multivalued nonexpansive mappings in the setting of Banach spaces.
Agarwal et al. [14] introduced an iteration process known as S-iteration process, which is independent of both Mann [15] and Ishikawa [16] iterations, for a single-valued mappings in Banach spaces:(4)x1∈C,yn=βnTxn+1−βnxn,xn+1=αnTyn+1−αnTxn,n∈ℕ,where αn,βn∈0,1. They proved that the rate of convergence of iteration process (4) is the same as Picard iteration process xn+1=Txn and faster than Mann [15] iteration process for the class of contraction mappings. Later, it was observed that this scheme also converges faster than Ishikawa [16] iteration process (see e.g., [17]; for more details and some recent literature of S-iteration process, see [18–24]).
Keeping above in mind, we introduce our iteration process as follows:(5)x1∈C,yn=βnun+1−βnxn,xn+1=αnvn+1−αnun,n∈ℕ,where un∈Txn such that xn−un=Dxn,Txn and vn∈Tyn such that yn−vn=Dyn,Tyn.
In this way, we approximate endpoints of multivalued nonexpansive mappings by an iteration process which is independent of but faster than Ishikawa iteration process. Thus, our results improve and unify corresponding results of Panyanak [13] and references therein.Definition 1.
A Banach spaceX is said to be uniformly convex if for each ε∈0,2, there is a λ>0 such that for every x,y∈X,(6)x≤1y≤1x−y>ε⟹12x+y≤1−λ.Definition 2.
(see [25]). A Banach space X is said to have Opial’s property if for each sequence xn in X which weakly converges to x∈X and for every y∈X−x, it follows that(7)limsupn⟶∞xn−x<limsupn⟶∞xn−y.
Examples of Banach spaces satisfying this condition are Hilbert spaces and alllp spaces (1<p<∞).Definition 3.
(see [13]). Let C be a nonempty subset of a Banach space X. A mapping T:C⟶KC is said to satisfy condition J if there is a nondecreasing function g:0,∞⟶0,∞ with g0=0, gt>0 for t∈0,∞ such that(8)Dx,Tx≥gdistx,ET,for each x∈C.Definition 4.
(see [13]). Let C be a nonempty subset of a Banach space X. A mapping T:C⟶KC is said to be semicompact if for every sequence xn in C such that(9)limn⟶∞Dxn,Txn=0,there is a subsequence xnj of xn such that limj⟶∞xnj=p for some p∈C. We see that, if C is compact then every multivalued mapping T:C⟶KC is semicompact.Definition 5.
LetC be a nonempty subset of a Banach space XA sequence xn in X is called Fejer-monotone with respect to C if(10)xn+1−c≤xn−c,for each c∈C and n∈ℕ.
The following important lemma is due to Xu [26].Lemma 1.
A Banach spaceX is uniformly convex if and only if for any number k>0, and there is a strictly increasing and continuous function ψ:0,∞⟶0,∞ with ψ0=0 such that(11)αx+1−αy2≤αx2+1−αy2−α1−αψx−y,for each x,y∈X with x≤k, y≤k, and α∈0,1.
The following lemma can be found in [13].Lemma 2.
For a multivalued mappingT:C⟶KC, the following statements hold.(a)
distx,Tx=0 ⟺ x is a fixed point of T(b)
Dx,Tx=0 ⟺ x is an endpoint of T(c)
IfT is nonexpansive, then the mapping h:C⟶ℝ defined by hx=Dx,Tx is continuousLemma 3.
(see [3]) Let αn, βn be two real sequences such that(a)
0≤αn,βn<1(b)
limn⟶∞βn=0(c)
∑αnβn=∞
Letγn be a sequence of nonnegative real numbers such that ∑αnβn1−βnγn<∞, then γn has a subsequence which converges to 0.Lemma 4.
(see [27]). Let C be a nonempty closed and convex subset of a uniformly convex Banach space and T:C⟶KC be a multivalued nonexpansive mapping. Then, the following is true:(12)xn⊆C,xn⇀x,Dxn,Txn⟶0⟹x∈ET.
The following fact is needed which can be found in [28].Proposition 1.
LetC be a nonempty closed subset of a Banach space. Let xn be a Fejer-monotone sequence with respect to C. Then, xn converges (strongly) to the point of C if and only if limn⟶∞distxn,C=0.


## 2. Main Results

The following lemma is crucial.Lemma 5.
LetC be a nonempty closed convex subset of a uniformly convex Banach space X and T:C⟶KC be a multivalued nonexpansive mapping with ET≠∅. Let xn be a sequence defined by (5). Then, limn⟶∞xn−q exists for each q∈ET.Proof.
Letq∈ET. For each n∈ℕ, we have(13)yn−q≤βnun−q+1−βnxn−q≤βndistun,Tq+1−βnxn−q≤βnHTxn,Tq+1−βnxn−q≤βnxn−q+1−βnxn−q≤xn−q,which implies that(14)xn+1−q≤αnvn−q+1−αnun−q≤αndistvn,Tq+1−αndistun,Tq≤αnHTyn,Tq+1−αnHTxn,Tq≤αnyn−q+1−αnxn−q≤xn−q.
Hence,xn−q is a nonincreasing sequence, which implies limn⟶∞xn−q exists for all q∈ET.
First, we prove our weak convergence result.Theorem 1.
LetX be a uniformly convex Banach space with the Opial property, C be a nonempty closed convex subset of X, and T:C⟶KC be a multivalued nonexpansive mapping with ET≠∅. Let αn,βn∈a,b⊂0,1 and xn be a sequence defined by (5). Then, xn converges weakly to an element of ET.
Proof. Fixq∈ET. By Lemma 1., there exists a strictly increasing continuous function ψ:0,∞⟶0,∞ with ψ0=0 such that(15)yn−q2=βnun+1−βnxn−q2≤βnun−q2+1−βnxn−q2−βn1−βnψun−xn≤βnH2Txn,Tq+1−βnxn−q2−βn1−βnψun−xn≤βnxn−q2+1−βnxn−q2−βn1−βnψun−xn≤xn−q2−βn1−βnψun−xn.Thus,(16)xn+1−q2=αnvn+1−αnun−q2≤αnvn−q2+1−αnun−q2−αn1−αnψvn−un≤αnH2Tyn,Tq+1−αnH2Txn,Tq−αn1−αnψvn−un≤αnyn−q2+1−αnxn−q2−αn1−αnψvn−un≤αnyn−q2+1−αnxn−q2≤αnxn−q2+1−αnxn−q2−αnβn1−βnψun−xn.It follows that(17)∑n=1∞a21−bψun−xn≤∑n=1∞αnβn1−βnψun−xn<∞.Thus,limn⟶∞ψun−xn=0. But ψ is strictly increasing and continuous, we have limn⟶∞un−xn=0. Hence,(18)limn⟶∞Dxn,Txn=limn⟶∞un−xn=0.To show thatxn converges weakly to an element of ET, it suffices to show that xn has a unique weak subsequential limit in ET. For this purpose, we assume that there are subsequences xnm and xnk of xn such that xnm⇀u and xnk⇀v. By (18), limm⟶∞Dxnm,Txnm=0. By Lemma 4, u∈ET. Similarly, it can be shown that v∈ET. Next, we prove u=v. On the contrary suppose u≠v, then by Lemma 5 together withOpial’s property, we have(19)limn⟶∞xn−u=limm⟶∞xnm−u<limm⟶∞xnm−v=limn⟶∞xn−v=limk⟶∞xnk−v<limk⟶∞xnk−u=limn⟶∞xn−u,which is a contradiction. So, u=v. Hence proved.Now, we approximate endpoints of the mappingT through strong convergence of the sequence xn defined by (5).Theorem 2.
LetC be a nonempty closed convex subset of a uniformly convex Banach space X and T:C⟶KC be a multivalued nonexpansive mapping with ET≠∅. Let αn,βn∈0,1 be such that limn⟶∞βn=0 and ∑αnβn=∞ and let xn be a sequence defined by (5). If T is semicompact, then xn converges strongly to an element of ET.
Proof. In view of (17),(20)∑n=1∞αnβn1−βnψun−xn<∞.By Lemma3, subsequences unj and xnj of un and xn exists, respectively, such that limj⟶∞ψunj−xnj=0. Since ψ is strictly increasing and continuous, we have limj⟶∞unj−xnj=0. Hence,(21)limj⟶∞Dxnj,Txnj=limj⟶∞unj−xnj=0.On the contrary,T is semicompact; we may choose by passing through a subsequence that limj⟶∞xnj=p for some p∈C. We need to show p∈ET and xn⟶p. By Lemma 2 part (c), together with (21), we have(22)Dp,Tp=limj⟶∞Dxnj,Txnj=0.It follows from Lemma2 part (b) that p∈ET. By Lemma 5, limn⟶∞xn−p exists, and hence p is the strong limit of xn.In the next strong convergence result, we relax semicompactness ofT with the help of condition (J).Theorem 3.
LetC be a nonempty closed convex subset of a uniformly convex Banach space X and T:C⟶KC be a multivalued nonexpansive mapping with ET≠∅. Let αn,βn∈a,b⊂0,1 and xn be a sequence defined by (5). If T satisfies condition (J), then xn converges strongly to an element of ET.Proof.
SinceT satisfies condition J, by (18), we get that limn⟶∞distxn,ET=0. Closeness of ET follows from the nonexpansiveness of T. In the view of Lemma 5, we have xn is Fejer-monotone with respect to ET. By Proposition 1, xn converges strongly to an element of ET.
Here is an example in support of the main theorems.Example 2.
LetX=ℝ and C=2,5 with absolute valued norm. Define a multivalued mapping T:C⟶KC by Tx=2,x for each x∈C. Clearly T is semicompact and nonexpansive with ET=2. We prove that T satisfies condition J with gt=t. When x∈C=2,5, we have(23)gdistx,ET=gdistx,2=gx−2=x−2=supx−y:y∈2,x=Dx,2,x=Dx,Tx.
Next, we prove thatxn defined by (5) strongly converges to 2.
Chooseαn=βn=1/n+1, for all n∈ℕ. Let x1=3, then Tx1=2,3. Take u1∈2,3 such that;(24)3−u1=D3,T3=D3,2,3=sup3−y:y∈2,3=3−2.
That is,u1=2. So,(25)y1=β1u1+1−β1x1=122+1−123=3−12.
Now,Ty1=2,3−1/2. Choose v1∈Ty1 such that y1−v1=Dy1,Ty1. Similar calculation to the above gives v1=2. Hence,(26)x2=α1v1+1−α1u1=122+1−122=2.
Continuing in this manner,xn=2, for all n≥2, and hence xn converges strongly to 2∈ET.Remark 1.
We see in Example 2. that,ET=2 and FT=C, i.e., T does not satisfy the endpoint condition. Therefore, we cannot directly apply any result in [19, 22, 23].



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