# Some Results on Iterative Proximal Convergence and Chebyshev Center

Authors: Laishram Shanjit; Yumnam Rohen; Sumit Chandok; M. Bina Devi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we prove a sufficient condition that every nonempty closed convex bounded pairM,N in a reflexive Banach space B satisfying Opial’s condition has proximal normal structure. We analyze the relatively nonexpansive self-mapping T on M∪N satisfying TM⊆M and TN⊆N, to show that Ishikawa’s and Halpern’s iteration converges to the best proximity point. Also, we prove that under relatively isometry self-mapping T on M∪N satisfying TN⊆N and TM⊆M, Ishikawa’s iteration converges to the best proximity point in the collection of all Chebyshev centers of N relative to M. Some illustrative examples are provided to support our results.


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## Body

## 1. Introduction and Preliminaries

LetM and N be nonempty subsets of a Banach space B. A mapping T:M∪N→M∪N satisfying ∥Ts−Tt∥≤∥s−t∥ (respectively, ∥Ts−Tt∥=∥s−t∥) for all s∈M,t∈N is called relatively nonexpansive mapping (respectively, relatively isometry mapping). For more results on relatively nonexpansive (respectively, relatively isometry) mappings, readers can see the research papers in [1, 2] and references therein.For any two nonempty bounded subsetsM and N of a Banach space B, we denote some notations as follows:
(1)Rs,N≔sup∥s−t∥:t∈N,M0≔s∈M:∥s−t∥=distM,Nforsomet∈N,N0≔t∈N:∥s−t∥=distM,Nforsomes∈M,where distM,N≔inf∥s−t∥:s∈Mandt∈N. Here, it is to note that if M∩N≠∅, then M0=N0=M∩N.LetM be a nonempty convex subset of a normed linear space X, and let T:M→M be a mapping with FixT≠∅, where FixT=s∈M:Ts=s. A set M is said to have approximate fixed point property (AFPP) if the nonexpansive mapping T has an approximate fixed point sequence, that is, a sequence pn in M satisfies limn→+∞∥Tpn−pn∥=0.Definition 1 [3].
A normed spaceX is said to be uniformly convex (or uniformly rotund) if and only if for every ε∈0,2 there exists δ>0 such that ∥s+t∥/2≤1−δ whenever s,t∈X implies ∥s∥=1, ∥t∥=1, and ∥s−t∥≥ε.Definition 2 [4].
A nonempty convex subsetM of a Banach space B is said to have normal structure if for any nonempty convex closed bounded subset S of M with diamS>0 there exists s∈S such that Rs,S<diamS, where diamS=diamS,S=supRs,S:s∈S.Eldred et al. [1] introduced the notions of proximal pair and proximal normal structure.Definition 3 [1].
A nonempty pairM,Nof a normed linear spaceXis known as a proximal pair if, for everys,t∈M×N,there existss′,t′∈M×Nsuch that(2)∥s−t′∥=distM,N=∥s′−t∥.A nonempty convex pairM,N in a Banach space B is said to have proximal normal structure if M1,N1⊆M,N is a closed bounded convex pair for which distM1,N1=distM,N and diamM1,N1>distM1,N1, there exists s1,t1∈M1×N1 such that
(3)Rs1,N1<diamM1,N1=supRs1,N1:s1∈M1andRt1,M1<diamM1,N1.Here, it is to note that every nonempty convex weakly compact pair in a uniformly convex Banach space has proximal normal structure. IfM=N, then proximal normal structure becomes normal structure of Definition 2.Definition 4 [5].
A proximal pairM,Nin a Banach spaceBis known as a proximal parallel pair if(1)
for every elements,t in M×N, there exists a unique element s1,t1 in M×N such that ∥s−t1∥=∥t−s1∥=distM,N and(2)
N=M+h, where h is a unique element in BFurther, Espinola [5] proved the following lemma.Lemma 5 [5].
IfM,Nis a nonempty proximal pair in a strictly convex Banach spaceB, then proximal pairM,Nis a proximal parallel pair.Definition 6 [6].
The nonempty proximal parallel pairM,Nin a Banach spaceBis said to have rectangle property if for anys,t∈M,
(4)∥s+h−t∥=∥t+h−s∥,where h∈B and N=M+h.Eldred et al. [1] proved the following result.Theorem 7 [1].
LetM,Nbe a nonempty closed bounded convex pair in a uniformly convex Banach spaceB. LetTbe a relatively nonexpansive self-mapping onM∪NsatisfyingTM⊆MandTN⊆N. Lets0∈Mbe an initial point, and define a sequence (Krasnoselskii’s iteration formula) bysn+1=sn+Tsn/2, n≥0. Then,limn→+∞Tsn−sn=0. IfTMis a subset of some compact set inB, then the limit point ofsnunder the norm topology is the best proximity point ofT.It is ascertained that the geometric property, that is, proximal normal structure, was used in the following result of Eldred et al. [1].Theorem 8 [1].
LetBbe a strictly convex Banach space, and letM,Nbe a nonempty weakly compact convex pair having proximal normal structure. LetTbe a self-mapping onM∪Nsatisfying(5)TM⊆M,TN⊆Nand∥Ts−Tt∥≤∥s−t∥foralls∈M,t∈N,then T has fixed points s∈M, t∈N, and ∥s−t∥=distM,N.Definition 9 [7].
LetMbe a nonempty convex subset of a real Hilbert spaceH, and letTbe a self-mapping onM. Lets0∈Mbe an initial point, andsnis a sequence defined by(6)sn+1=1−ξnsn+ξnTtn,tn=1−ηnsn+ηnTsn,where 0≤ξn≤1, 0≤ηn≤1,n≥0.The iterative sequence defined in (6) is called Ishikawa’s iteration. If ηn=0, then Ishikawa’s iteration sequence reduces to Mann’s iteration sequence. Eldred and Praveen [8] generalized and extended Theorem 7 of Eldred et al. [1] by using Mann’s iteration method.Definition 10 [9].
LetMbe a nonempty convex subset of a real Hilbert spaceH, and letTbe a self-mapping onM. Fixu∈M. Letp0∈Mbe an initial point, and a sequencepnis defined by(7)pn+1=ξnu+1−ξnTpn,0≤ξn≤1,n≥0.The iterative sequence defined in (7) is called Halpern’s iteration.The following interesting result will be used extensively in the sequel.Proposition 11 [10].
LetXbe a uniformly convex normed linear space,0<α<1, andε>0. For anyr>0, ifs,t∈Xare such that∥s∥≤r,∥t∥≤r, ∥s−t∥≥ε, then there existsδ=δε/r>0such that(8)∥αs+1−αt∥≤1−2δεrminα,1−αr.Almezel et al. [11] modified the result of Xu [12] in the following way.Lemma 12 [11, 12].
Letxnbe a sequence of nonnegative real numbers satisfying(9)xn+1≤1−ηnxn+ηnνn,n≥0,where ηn and νn satisfy the following conditions.
(1)
ηn is a sequence in 0,1, where ∑n=1+∞ηn=+∞(2)
νn is a sequence in ℝ; either limsupn→+∞ηn≤0 or ∑n=0+∞∣ηnνn∣<+∞
Then,xn→0 as n→+∞.LetM and N be nonempty bounded subsets of a Banach space B. The number RM,N≔infRs,N:s∈M is the Chebyshev radius of N relative to M and CMN≔s∈M:Rs,N=RM,N is the set of all Chebyshev centers of N relative to M. Since the function R is convex and continuous on X, R is lower semicontinuous with respect to the weak topology. Consequently, if M is a nonempty weakly compact convex set, then CMN is a nonempty convex weakly compact subset of M. Rajesh and Veeramani [2] proved the following proposition.Proposition 13 [2].
LetM,Nbe a nonempty convex weakly compact proximal parallel pair in a Banach spaceB. Let the nonempty pairM,Nhave the rectangle property. Then,Rs,N=Rs+h,Mfors∈M, andRt,M=Rt−h,Nfort∈N. Moreover,CNM=CMN+h.Definition 14 [13].
LetBbe a Banach space. We say thatBsatisfies Opial’s condition if for any sequencepninBconverges weakly to somes, thenlimsupn→+∞pn−p>limsupn→+∞pn−sfor allp≠s∈B. If a reflexive Banach spaceBsatisfies Opial’s condition, thenBhas a normal structure.Proposition 15 (demiclosed principle [13]).
LetBbe a Banach space, and letMbe a nonempty weakly compact subset ofB. Also, letTbe a nonexpansive self-mapping onMwith FixT≠∅. If a sequencepninMconverges weakly tosand a sequenceI−Tpnconverges strongly top, thenI−Ts=p. Moreover, ifp=0, thenI−Tis demiclosed at zero.We need the following result of Dutta and Veeramani [14] to prove Proposition 17.Theorem 16 [14].
If a nonempty convex pairM,Nin a Banach spaceBdoes not have a proximal normal structure, then there exist sequencessn⊂M, tn⊂Nsuch that∥sn−tn∥=distM,Nfor alln, ∥sm−tn∥>distM,Nforsomem,nand(10)limn→+∞distsn+1,convt1,t2,⋯,tn=diamsn,tn,
or(11)limn→+∞disttn+1,convs1,s2,⋯,sn=diamsn,tn,where diamsn,tn=diams1,⋯,sn,⋯,t1,⋯,tn,⋯.


## 2. Opial’s Condition and Ishikawa’s Iteration for Relatively Nonexpansive Mappings

The geometrical property, that is, the proximal normal structure, is the sufficient condition for the existence of the best proximity [1]. For details about the best proximity point, one can see research papers in [1, 2, 5, 15–19]. We now prove the following result, which shows that the above condition can be dropped if a reflexive Banach space satisfies Opial’s condition.Proposition 17.
Every closed bounded convex pairM,Nin reflexive Banach spaceBsatisfying Opial’s condition has proximal normal structure.Proof.
Suppose the pairM,N does not have a proximal normal structure. Then, by Theorem 16, there exist sequences sn⊂M,tn⊂N such that ∥sn−tn∥=distM,N for all n, ∥sm−tn∥>distM,N for some m,n, and limn→+∞disttn+1,convs1,s2,⋯,sn=Rsn,tn. Let the sequence sn converges weakly to 0. Therefore, 0∈conv¯s1,s2,⋯,sn.
Supposes∈convs1,s2,⋯,sn, then limn→+∞∥s−tn∥=Rsn,tn, and the same holds as s∈conv¯s1,s2,⋯,sn. Therefore, when taking s=0, we get limn→+∞∥tn∥=Rsn,tn, and limn→+∞∥s1−tn∥=Rsn,tn, which is a contradiction, hence the result.After analyzing the theorems, definitions, lemma, and propositions mentioned above, we have some impressive new results herewith.Theorem 18.
LetM,Nbe a nonempty convex closed bounded proximal pair ofB, a uniformly convex Banach space. LetTbe a relatively nonexpansive self-mapping onM∪NsatisfyingTM⊆MandTN⊆N. Lets0∈Mbe an initial point, and a sequencesnis defined as(12)sn+1=1−ξnsn+ξnTtn,tn=1−ηnsn+ηnTsn,θ<ηn≤ξn<1−θ,0<θ≤12,limn→+∞ξnηn=0.
Then,limn→+∞∥Tsn−sn∥=0. If TM is a subset of a compact set, then the limit point of sn under the norm topology is the best proximity point of T.Proof.
IfdistM,N=0, then it is not necessary to discuss. Suppose distM,N>0, then by applying the result of Theorem 8, there exists t∈N such that Tt=t. Since
(13)∥sn+1−t∥≤1−ξn∥sn−t∥+ξn∥Ttn−t∥≤1−ξn∥sn−t∥+ξn1−ηn∥sn−t∥+ξnηn∥Tsn−t∥≤∥sn−t∥,
∥sn−t∥ is a nonincreasing sequence, there exists k>0 such that limn→+∞∥sn−t∥=k.
Supposelimn→+∞∥Tsn−sn∥≠0, then there exists a subsequence sni of sn such that
(14)∥sni−Tsni∥≥ε>0.
Letα∈0,1 and ε1 such that ε/α>k and 0<ε1<minε/α−k,kδα/1−δα. Since B is a uniformly convex Banach space, the modulus of convexity function δ. is strictly increasing and continuous. Hence, 0<δα<δε/k+ε1. So, we can choose a small positive number ε1>0 such that 1−aδε/k+ε1k+ε1<k, where a>0.
Let∥sni−t∥≤k+ε1 and ∥tni−t∥≤k+ε1 for some i. Now,
(15)∥t−Ttni∥≤∥t−tni∥=∥t−1−ηnisni+ηniTsni∥=∥1−ηnit−sni+ηnit−Tsni∥≤1−2δεk+ε1minηni,1−ηnik+ε1≤1−a1δεk+ε1k+ε1,
where0<a1≤2minηni,1−ηni. Further
(16)∥t−sni+1∥=∥t−1−ξnisni+ξniTtni∥=∥1−ξnit−sni+ξnit−Ttni∥≤1−2δεk+ε1minξni,1−ξnik+ε1≤1−a2δεk+ε1k+ε1,
where0<a2≤2minξni,1−ξni.
By choosingε1>0 as small as we wish, we get
(17)max1−a1δεk+ε1k+ε1,1−a2δεk+ε1k+ε1<k,which is a contradiction. Hence, limn→+∞∥Tsn−sn∥=0 and limn→+∞∥sn+1−sn∥=0.
IfTM is compact, then the sequence sn has a subsequence sni such that limi→+∞sni=s∈M. Since M,N is a proximal pair, there exists v∈N such that ∥s−v∥=distM,N.
Now, we havelimi→+∞∥sni−v∥=distM,N, and ∥sn−v∥ is a nonincreasing sequence; it implies that limn→+∞∥sn−v∥=distM,N. This shows that limn→+∞sn=s∈M. By strict convexity of the norm, limn→+∞∥Tsn−v∥=distM,N and ∥Tsn−Tv∥≤∥sn−v∥→distM,N as n→+∞ give Tv=v. Since limn→+∞∥Tsn−sn∥=0, it follows that Ts=s.We obtain the following result from Theorem18 by taking ηn=0 for n∈ℕ.Corollary 19 [8].
LetM,Nbe a nonempty convex closed bounded proximal pair ofB, a uniformly convex Banach space, and letTbe a relatively nonexpansive self-mapping onM∪NsatisfyingTN⊆NandTM⊆M. Lets0∈Mbe an initial point, and a sequencesnis defined as(18)sn+1=1−ξnsn+ξnTsn,ε<ξn<1−ε,0<ε≤12Mann’siteration.Then,limn→+∞∥Tsn−sn∥=0. Moreover, if TM is a subset of a compact set, then the limit point of sn under norm topology is the best proximity point of T.


## 3. Halpern’s Iteration and Relatively Nonexpansive Mapping

LetM be a nonempty subset of a real Hilbert space H, and let PM:H→2M be the nearest point projection mapping from H onto M that is, PMs≔s′∈M:∥s′−s∥=dists,M. If M is nonempty convex closed, then PM is nonexpansive giving unique image for all s in H, and hence by Kolmogorov’s criterion <PMt−s,PMt−t>≤0 for all t∈X,s∈M. Here, we use the following notation MFixT=s∈M:Ts=s.Theorem 20.
LetM,N be a nonempty closed bounded convex proximal pair of a real Hilbert space H, and let T be a relatively nonexpansive self-mapping on M∪N satisfying TM⊆M and TN⊆N. Let 0<ξn<1, and s0∈M be an initial point. A sequence sn is defined as
(19)sn+1=ξnu+1−ξnTsn,where u∈M such that ∥sn−t∥≥∥u−t∥forallt∈N.Iflimn→+∞ξn=0, ∑n=1+∞ξn=+∞, and either ∑n=1+∞∣ξn+1−ξn∣<+∞ or limn→+∞ξn/ξn+1=1, then the sequence sn under the norm topology converges to s∈MFixT, closest to point u such that ∥s−t∥=distM,N for some t∈NFixT.Proof.
By applying Theorem8, it is found that there exists t∈N such that Tt=t. Now, we have
(20)∥sn+1−t∥=∥ξnu+1−ξnTsn−t∥≤ξn∥u−t∥+1−ξn∥sn−t∥≤ξn∥u−t∥+1−ξn∥sn−t∥≤ξn∥sn−t∥+1−ξn∥sn−t∥since∥sn−t∥≥∥u−t∥.
Hence∥sn−t∥ is nonincreasing and limn→+∞∥sn−t∥=k>0.Supposelimn→+∞∥Tsn−sn∥≠0, then there exists a subsequence sni of sn such that ∥sni−Tsni∥≥ε>0. Since H is a Hilbert space (and hence uniformly convex space), it is possible to choose a small positive number ε1>0, such that 1−aδε/k+ε1k+ε1<k, where a>0.Let∥sni−t∥≤k+ε1 for some i. Now,
(21)∥t−sni+1∥=∥t−ξniu+1−ξniTsni∥=∥1−ξnit−Tsni+ξnit−u∥≤1−2δεk+ε1minξni,1−ξnik+ε1≤1−a1δεk+ε1k+ε1,where0<a1≤2minξni,1−ξni.By choosingε1>0 as small as we wish, we have
(22)1−a1δεk+ε1k+ε1<k,which is a contradiction. Hence, limn→+∞∥Tsn−sn∥=0 and limn→+∞∥sn+1−sn∥=0.Letsni be a subsequence of sn such that
(23)limsupn→+∞<sn−s,s−u>=limsupi→+∞<sni−s,s−u>.Without loss of generality, we assume that subsequence sni converges weakly to p∈M such that ∥p−t∥=distM,N for some t∈NFixT. Since limn→+∞∥Tsn−sn∥=0, by applying the demiclosed principle, we have p∈MFixT. Hence, by applying Kolmogorov’s criterion, we have
(24)limsupi→+∞<sni−s,s−u>=<p−s,s−u>≥0.Now, we have(25)∥sn+1−s∥2=<ξnu+1−ξnTsn−s,sn+1−s>≤ξns<u−s,sn+1−s>+1−ξn∥sn−s∥.∥sn+1−s∥≤1−ξn2∥sn−s∥2+∥sn+1−s∥2+ξn<u−s,sn+1−s>.Hence,(26)⇒∥sn+1−s∥2≤2ξn1+ξn<u−s,sn+1−s>+1−2ξn1+ξn∥sn−s∥2=1−ηn∥sn−s∥2+ξnνn,where ηn=2ξn/1+ξn and νn=2/1+ξn<u−s,sn+1−s>.Since∑n=1+∞ηn=+∞ and limsupn→+∞νn≤0, by Lemma 12, we have limn→+∞sn=s∈MFixT, closest to point u so that ∥s−t∥=distM,N for some t∈NFixT.We obtain the following corollary from Theorem20 when M=N.Corollary 21 [9].
LetM be nonempty closed bounded convex subsets of a real Hilbert space H and T be a nonexpansive self-mapping on M. Let s0∈M be an initial point, and sn is a sequence defined as
(27)sn+1=ξnu+1−ξnTsn,
whereu∈M and 0<ξn<1 (Halpern’s iteration).Iflimn→+∞ξn=0, ∑n=1+∞ξn=+∞, and either ∑n=1+∞∣ξn+1−ξn∣<+∞ or limn→+∞ξn/ξn+1=1, then the sequence sn under the norm topology converges to s∈MFixT, closest to point u.


## 4. Ishikawa’s Iteration and Chebyshev Centre

Lim et al. [20] proved the following interesting theorem in the year 2003, by using the geometrical property, viz., normal structure.Theorem 22 [20].
LetB be a Banach space, and let T be an isometry self-mapping on M, a nonempty weakly compact convex subset of B. It is assumed that M has a normal structure. Then, there exists s∈CM=CMM, the set of all Chebyshev centers of M such that Ts=s.LetM,N be a nonempty convex weakly compact proximal parallel pair in a Banach space B. Suppose the pair M,N has the rectangle property. Let T:M∪N→M∪N be a relatively isometry mapping satisfying TM⊆M and TN⊆N. It is ascertained that TCMN⊆CMN if and only if Rs,N=RTs,N=RM,N for all s∈CMN. Similarly, TCNM⊆CNM if and only if Rt,M=RTt,M=RM,N for all t∈CNM. It is affirmed that CNM=CMN+h for some h∈B (for details, see [2, 21, 22]). We establish the following result.Lemma 23.
LetM,N be a nonempty weakly compact convex proximal pair in a strictly convex Banach space B. Suppose T is a relatively isometry self-mapping on M∪N satisfying TM⊆M and TN⊆N. If s∈M and Tns has a Cauchy subsequence in M, then Rs,N=RTs,N. Similarly, if t∈N and Tnt has a Cauchy subsequence in N, then Rt,M=RTt,M.Proof.
Lets∈M. Then,
(28)Rs,N=RTs,TN≤RTs,N.Lets,t∈M×N such that ∥s−t∥=distM,N. Suppose ∥Tjns−Tint∥<distM,N+1/n, where in,jn∈ℤ+, with in<jn, for every n∈ℤ+. Since T is a relatively isometry mapping, we get limn→+∞Tjn−ins=s.Letan be a nondecreasing subsequence of jn−in. Since R is a nonnegative continuous real valued function, then the sequence RTans,N is nondecreasing, and limn→+∞Tans=s. Therefore, limn→+∞RTans,N=Rs,N. Thus,
(29)RTs,N≤RTa1s,N≤limn→+∞RTans,N=Rs,N.From, (28) and (29), we have Rs,N=RTs,N. Similarly, we can show that Rt,M=RTt,M.Lemma 24.
LetM,N be a nonempty weakly compact convex proximal parallel pair in a strictly convex Banach space B. It is assumed that the pair M,N has the rectangle property. Suppose T is a relatively isometry self-mapping on M∪N satisfying TM⊆M and TN⊆N. If CMN,CNM is nonempty and contained in a totally bounded proximal parallel pair M1,N1 of M,N such that TM1⊆M1 and TN1⊆N1, then TCMN⊆CMN and TCNM⊆CNM.Proof.
Lets∈CMN, where CMN⊆M1, TM1⊆M1, and CNM=CMN+h, for some h∈B. Then, Tns:n∈ℤ+⊆M1, and Tns+h:n∈ℤ+⊆N1.
AsM1,N1 is a totally bounded proximal pair, the sequences Tns and Tns+h, respectively, have Cauchy subsequences in M1 and N1. So, by Lemma 23, we have Rs,N=RM,N=RTs,N.
Hence,TCMN⊆CMN. Similarly, TCNM⊆CNM.Example 25.
LetX=ℝ2,∥.∥1. Let M=s,10−10s:s∈0,1, and N=M+h, where h=0,1∈X. Let s,10−10s∈M and 1+t,10−10t∈N, where s,t∈0,1. Now, we have
(30)∥1+t,10−10t−s,10−10s∥1=∥1+t−s,10s−10t∥1=1+t−s+∣10s−10t∣=1+9s−9t,ift≤s,1+11t−11s,ifs≤t.In particular, take1,0,0,10∈M, and 2,0,1,10∈N, we have ∥1,0+1,0−0,10∥1=12 and ∥0,10+1,0−1,0∥1=10.It shows that there exists a proximal parallel pairM,N with distM,N=∥h∥1=1 which does not satisfy the rectangle property.Theorem 26.
LetM,N be a nonempty totally bounded convex closed proximal pair in a uniformly convex (and hence reflexive) Banach space B. It is also assumed that the pair M,N has the rectangle property. Suppose T is a relatively isometry self-mapping on M∪N satisfying TN⊆N and TM⊆M. Let s0∈CMN be an initial point, and a sequence sn is defined as
(31)sn+1=1−ξnsn+ξnTtn,tn=1−ηnsn+ηnTsn,0≤ηn≤ξn<1,limn→+∞ξnηn=0,−θ,0<θ≤12Then,limn→+∞∥Tsn−sn∥=0. If TCMN is a subset of a compact set, then the limit point s∈CMN of the sequence sn under norm topology is the best proximity point of T.Proof.
It is easy to see thatCMN,CNM is a nonempty convex weakly compact proximal parallel pair having the rectangle property in a uniformly convex Banach space B.
SinceM,N is totally bounded and T is a relatively isometry self-mapping on M∪N satisfying TN⊆N and TM⊆M, by applying Lemma 24, we have TCMN⊆CMN and TCNM⊆CNM.
Now, by Theorem8, there exist s∈CMN and t∈CNM such that Ts=s, Tt=t, and ∥s−t∥=distM,N.
By applying Theorem18, it is found that the sequence sn under norm topology converges to Ts=s∈CMN, such that ∥s−t∥=distM,N for some t∈NFixT.
We obtain the following result from Theorem26 if ηn=0 for n∈ℕ.Theorem 27.
LetM,N be a nonempty totally bounded convex closed proximal pair in a uniformly convex (and hence reflexive) Banach space B. It is also assumed that M,N has the rectangle property. Suppose T is a relatively isometry self-mapping on M∪N satisfying TN⊆N and TM⊆M. Let s0∈CMN be an initial point, and a sequence sn is defined as
(32)sn+1=1−ξnsn+ξnTsn,ε<ξn<1−ε,0<ε≤12.Then,limn→+∞∥Tsn−sn∥=0. If TCMN is a subset of a compact set, then the limit point s∈CMN of sn under the norm topology is the best proximity point of T.Proof.
The result is similar to that of Theorem26.Example 28.
LetX=ℝ2,., a Euclidean space. Let
(33)M=s,t:s=−2,−1≤t≤1,N=s,t:s=2,−1≤t≤1.Here,M,N is a proximal parallel pair having the rectangle property, RM,N=17, CMN=−2,0, CNM=2,0, and CNM=CMN+h, where h=4,0.Define(34)T:M→MbyTs,t=T1s,T2t=−2,−t,whereT1:−2→−2andT2:−1,1→−1,1.Lets,t∈M, and s′,t′∈N. Then
(35)∥Ts,t−Ts′,t′∥=∥−2,−t−2,−t′∥=∥−4,t′−t∥=−42+t′−t2=∥s,t−s′,t′∥.Hence,T is a relatively isometry (and hence relatively nonexpansive) mapping on M∪N satisfying TN⊆N and TM⊆M.From Theorem18, we take the initial point s,t∈M and set s1=1−ξ0s+ξ0T1s′0 and s′0=1−η0s+η0T1s. We have T1s=−2. Since s=−2, we obtain s′0=−2 which implies s1=−2.Similarly, sets2=1−ξ1s1+ξ1T1s′1 and s′1=1−η1s1+η1T1s1. Since s1=−2, we obtain s′1=−2 which implies s2=−2. In general, we obtain sn+1=−2. Therefore, sn→−2 as n→+∞.Again, sett1=1−ξ0t+ξ0T2t′0 and t′0=1−η0t+η0T2t. Since T2t=−t, we obtain t′0=1−2η0t which implies t1=1−2ξ0+2ξ0η0t. Similarly, set t2=1−ξ1t1+ξ1T2t′1 and t′1=1−η1t1+η1T2t1. Since T2t1=−t1, we obtain
(36)t′1=1−η1t1+η1T2t1=1−η11−2ξ0+2ξ0η0t+η1T21−2ξ0+2ξ0η0t=1−η11−2ξ0+2ξ0η0t−η11−2ξ0+2ξ0η0t=1−2η11−2ξ0+2ξ0η0t, which implies
(37)t2=1−ξ1t1+ξ1T2t′1=1−ξ11−2ξ0+2ξ0η0t−ξ11−2η11−2ξ0+2ξ0η0t=1−2ξ0+2ξ0η01−2ξ1+2ξ1η1t.In general,tn+1=1−2ξ0+2ξ0η01−2ξ1+2ξ1η1⋯1−2ξn+2ξnηnt. Therefore, tn→0 as n→+∞. Hence, limn→+∞sn,tn=−2,0, a fixed point of T. In a similar way, if s′,t′∈N, then limn→+∞s′n,t′n=2,0, a fixed point of T and ∥−2,0−2,0∥=distM,N.From Theorem26, if we take the initial point x,y∈CMN, then it is trivial that limn→+∞sn,tn=−2,0, a fixed point of T. In a similar way, if s′,t′∈CNM, then limn→+∞s′n,t′n=2,0, a fixed point of T and ∥−2,0−2,0∥=distM,N.


## 5. Open Problem

LetM,N be a nonempty weak compact convex pair in a Banach space (or Hilbert space) B. Can Ishikawa’s iteration and Halpern’s iteration converge to the best proximity point of relatively nonexpansive (or relatively isometry) mapping T:M∪N→M∪N satisfying TM⊆N and TN⊆M?


## 6. Conclusion

If a reflexive Banach space satisfies Opial’s condition, then every bounded convex pairM,N has a proximal normal structure. Also, we show that Ishikawa’s and Halpern’s iterative sequences converge to the best proximity point. Finally, we show that Ishikawa’s iterative sequence converges to the best proximity point, which is a Chebyshev center.



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*Source: 8863325-2021-01-07.xml*