# Approximating Stationary Points of Multivalued Generalized Nonexpansive Mappings in Metric Spaces

Authors: Kifayat Ullah; Junaid Ahmad; Muhammad Arshad; Manuel de la Sen; Muhammad Safi Ullah Khan
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this research, under some appropriate conditions, we approximate stationary points of multivalued Suzuki mappings through the modified Agarwal-O’Regan-Sahu iteration process in the setting of 2-uniformly convex hyperbolic spaces. We also provide an illustrative numerical example. Our results improve and extend some recently announced results of the current literature.


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## Body

## 1. Introduction

LetM=M,ρ be a metric space and Y be a nonempty subset of M. For a∈M, set
(1)ρa,Y=infρa,b:b∈Y,Ra,Y=supρa,b:b∈Y.LetCY represent the set of all nonempty compact subsets of Y. The function ℍ defined by
(2)ℍS,W=maxsupu∈Sρu,W,supv∈Wρv,S,foreachS,W∈CY,satisfies all the properties of the metric and is often called the Hausdorff metric on CY. Recall that a multivalued mapping F:Y⟶CY is called the Suzuki mapping [1] if for each a,b∈Y,
(3)12ρa,Fa≤ρa,b⇒ℍFa,Fb≤ρa,b.The class of Suzuki mappings and its extensions in the setting of single-valued mappings are widely studied by many authors (see, e.g., [2–7] and references therein). Here, we will only focus on the multivalued version of Suzuki mappings. We can easily observe that if F is nonexpansive, that is, ℍFa,Fb≤ρa,b for all a,b∈Y, then F is also a Suzuki mapping. Nevertheless, the following example shows that the converse of this statement may not hold in general.Example 1.
LetY=1,4 and F:Y⟶CY be defined by
(4)Fa=1,fora≠4,1.9,2,fora=4.Whena∈3,4 and b=4, then we have nothing to prove, because in this case, we have the following:
(5)12ρa,Fa=a−12>1>ρa,b,12ρb,Fb=1>ρa,b.Whena<b and a,b∈Y×Y−3,4×4. In this case, F is nonexpansive, and hence, F is a Suzuki mapping.By settinga=3.5 and b=4, we have ℍFa,Fb=1>0.5=ρa,b. Hence, F is not nonexpansive.An elementq in Y is called a stationary point (or called an endpoint) of F whenever q=Fq and is called a fixed point of F whenever q∈Fq. Throughout the paper, the notations SF and FF will represent the set of all stationary points and the set of all fixed points of F, respectively. Recall that a multivalued mapping F:Y⟶CY is called quasinonexpansive provided that ℍFa,Fq≤ρa,q for each a∈Y and q∈FF. Existence of stationary points for different types of multivalued mappings is studied in [8–15]. The following statements hold:
(i)
SF⊆FF(ii)
q∈FF if and only if ρq,Fq=0(iii)
q∈SF if and only if Rq,Fq=0In 2005, Sastry and Babu [16] published a paper on the strong convergence of the fixed point for multivalued nonexpansive mappings using modified Mann and Ishikawa iterative processes in the setting of Hilbert spaces. In the year 2008, Panyanak [17] showed that the results of Sastry and Babu [16] can be extended to the slightly general context of uniformly convex Banach spaces. Song and Wang [18] improved the results of Panyanak [17]. For more details in this direction, see [19, 20] and others. In the year 2018, Panyanak [21] published a paper on the approximation of stationary points of multivalued nonexpansive mappings in the framework of Banach spaces using the modified Ishikawa iterative process. In 2019, Ullah et al. [22] quickly noted that the results of Panyanak [21] can be extended to the general context of CAT(0) spaces. In 2020, Laokul and Panyanak [23] used the Ishikawa iterative process for finding stationary points of multivalued Suzuki mappings in 2-uniformly convex hyperbolic spaces. Recently, Abdeljawad et al. [24] used the modified Agarwal-O’Regan-Sahu iterative process for finding stationary points of multivalued nonexpansive mappings in Banach spaces. Very recently, Ullah et al. [25] extended the results of Abdeljawad et al. [24] to the general context of 2-uniformly convex hyperbolic spaces. The modified Agarwal-O’Regan-Sahu iteration process reads as follows:
(6)w1∈Y,sn=1−βnwn⊕βnun,wn+1=1−αnun⊕αnvn,n≥1,where un∈Fwn such that ρwn,un=Rwn,Fwn and vn∈Fsn such that ρsn,vn=Rsn,Fsn. The purpose of this work is to prove, under some appropriate conditions, the strong and Δ convergence results of stationary points for a wider class of multivalued nonexpansive mappings so-called multivalued Suzuki mappings using iterative process (6) in the general setting of 2-uniformly convex hyperbolic spaces. In this way, we improve and extend the corresponding results proved in [21–25].Now we recall some basic definitions and results, which will be used in the sequel.Definition 2 [26].
A hyperbolic space is a metric spaceM,ρ endowed with a function H:M×M×0,1⟶M such that for all a,b,z,w∈M and i,s∈0,1, we have
H1ρz,Ha,b,i≤1−iρz,a+iρz,b
H2ρ(Ha,b,i,Ha,b,j=∣i−j∣ρa,b
H3Ha,b,i=Hb,a,1−i
H4ρHa,z,i,Hb,w,i≤1−iρa,b+iρz,wIfa,b∈M and i∈0,1, we use the notation 1−ia⊕ib for Ha,b,i. It follows from H1 that
(7)ρa,1−ia⊕ib=iρa,b,ρb,1−ia⊕ib=1−iρa,b.The set∅≠Y⊆M is called convex if for any a,b∈Y, one has a,b=1−γa⊕γb:γ∈0,1⊆Y.Definition 3.
A hyperbolic spaceM,ρ,H is said to be uniformly convex if for every real number r>0 and ϵ∈0,2, we can choose a σ∈0,1 such that for each a,b,z∈M with ρa,z≤r, ρb,z≤r, and ρa,b≥2ϵ, one has
(8)ρ12a⊕12b≤1−σr.A functionη:0,∞×0,2⟶0,1 providing such σ=ηr,ϵ for given r∈0,∞ and ϵ∈0,2 is called a modulus of uniform convexity. The function η is called monotone provided that it is nonincreasing in r for each fixed ϵ.Definition 4.
LetM,ρ be a uniformly convex hyperbolic space. For every r∈0,∞ and ϵ∈0,2, set
(9)φr,ϵ=inf12ρ2a,z+12ρ2b,z−ρ212a⊕12b,z,where the infimum is taken over each a,b,z∈M such that ρa,z≤r, ρb,z≤r and ρa,b≥rϵ. We say that M,ρ is 2-uniformly convex if
(10)cM=infφr,εr2ε2:r∈0,∞,ϵ∈0,2>0.Remark 5.
Notice that uniformly convex Banach space and CAT(0) spaces as well as CAT(κ) spaces (κ>0 and diamM≤π/2−ϵ/κ1/2 for some ϵ∈0,π/2 are 2 -uniformly convex hyperbolic spaces (see [23, 27, 28]).Definition 6.
Letwn be any bounded sequence in a complete 2-uniformly convex hyperbolic space M and ∅≠Y⊆M. The asymptotic radius of wn relative to Y is rY,wn=inflimsupn→∞ρwn,w:w∈Y. Moreover, the asymptotic center of wn relative to Y is the set AY,wn)=w∈Y:limsupn→∞ρwn,w=rY,wn.Definition 7.
LetY be a nonempty closed convex subset in a complete 2-uniformly convex hyperbolic space M and w∈Y. Let wn be any bounded sequence in M. We say that wnΔ -converges to w if AY,zn=w for each subsequence zn of wn. In this case, we write Δ‐limn→∞wn=w and call w the Δ‐lim of wn.Now, we collect some basic facts about multivalued Suzuki mappings, which can be found in [29–31].Proposition 8.
LetY be a nonempty subset of a complete 2-uniformly convex hyperbolic space and F:Y⟶CY.
(i)
IfF is a Suzuki mapping with a nonempty fixed point set, then F is quasinonexpansive(ii)
IfF is a Suzuki mapping, then the following holds:(11)ρa,Fb≤3ρa,Fa+ρa,bforalla,b∈Y.The following facts are also needed.Lemma 9 (see [17]).
Letαn,βn∈0,1 be such that limn→∞βn=0 and ∑αnβn=∞. Let γn be a sequence of nonnegative real numbers such that ∑αnβn1−βnγn<∞. Then, γn has a subsequence which converges to 0.The following lemma is a characterization of 2-uniformly convex hyperbolic spaces.Lemma 10 (see [23]).
LetM,ρ be a 2 -uniformly convex hyperbolic space. Then,
(12)ρ21−ξa⊕ξb,p≤1−ξρ2a,p+ξρ2b,p−4cMξ1−ξρ2a,b,for each ξ∈0,1 and a,b,p∈M.Lemma 11 (see [23]).
LetY be a nonempty closed convex subset of a2-uniformly convex hyperbolic space M and F:Y⟶CY be a Suzuki mapping. Suppose that wn is a bounded sequence in Y such that limn→∞Rwn,Fwn=0 and ρwn,e converges for every e∈SF, then ωωwn⊆SF. Here, ωωwn=∪AY,zn where the union is taken over all subsequences zn of wn. Furthermore, ωωwn is a singleton.


## 2. Convergence Theorems in 2-Uniformly Convex Hyperbolic Spaces

Throughout the section,M will stand for a complete 2-uniformly convex hyperbolic space with monotone modulus of uniform convexity.The following lemma is crucial.Lemma 12.
LetY be a nonempty closed convex subset of M and F:Y⟶CY be a Suzuki mapping with SF≠∅. Let wn be the sequence defined by (6). Then, limn→∞ρwn,q exists for all q∈SF.Proof.
Letq∈SF. By Proposition 8(i), we have
(13)ρwn+1,q≤1−αnρun,q+αnρvn,q=1−αnρun,Fq+αnρvn,Fq≤1−αnℍFwn,Fq+αnℍFsn,Fq≤1−αnρwn,q+αnρsn,q≤1−αnρwn,q+αn1−βnρwn,q+βnρun,q=1−αnρwn,q+αn1−βnρwn,Fq+βnρwn,Fq≤1−αnρwn,q+αn1−βnρwn,q+βnℍFwn,Fq≤1−αnρwn,q+αn1−βnρwn,q+βnρwn,q=1−αnρwn,q+αnρwn,q=ρwn,q.
Hence,ρwn,q is a nonincreasing sequence, which implies limn→∞ρwn,q exists for every q∈SF.First, we establish ourΔ-convergence theorem.Theorem 13.
LetY be a nonempty closed convex subset of M and F:Y⟶CY be a Suzuki mapping with SF)≠∅. Let αn,βn∈a,b⊂0,1 and wn be the sequence defined by (6). Then, wnΔ -converges to a stationary point of F.Proof.
Fixq∈SF. By Lemma 10, we have
(14)ρ2sn,q≤1−βnρ2wn,q+βnρ2un,q−4cMβn1−βnρ2wn,un≤1−βnρ2wn,q+βnℍ2Fwn,Fq−4cMβn1−βnρ2wn,un≤1−βnρ2wn,q+βnρ2wn,q−4cMβn1−βnρ2wn,un≤ρ2wn,q−4cMβn1−βnρ2wn,un.
Thus,(15)ρ2wn+1,q≤1−αnρ2un,q+αnρ2vn,q−4cMαn1−αnρ2un,vn≤1−αnℍ2Fwn,Fq+αnℍ2Fsn,Fq−4cMαn1−αnρ2un,vn≤1−αnρ2wn,q+αnρ2sn,q−4cMαn1−αnρ2un,vn,≤1−αnρ2wn,q+αnρ2sn,q≤ρ2wn,q−4cMαnβn1−βnρ2wn,un.
SincecM>0, it follows that
(16)∑n=1∞a21−bρ2wn,un≤∑n=1∞αnβn1−βnρ2wn,un<∞.
Thus,limn→∞ρ2wn,un=0 and hence
(17)limn→∞Rwn,Fwn=limn→∞ρwn,un=0.
By Lemma12, ρwn,e converges for all e∈SF. By Lemma 11, ωωwn is a singleton and is contained in SF. This shows that wnΔ-converges to a point of SF.Definition 14 (see [21]).
LetY be a nonempty subset of M and F:Y⟶CY. F is said to satisfy condition J if there is a nondecreasing function λ:0,∞⟶0,∞ with the properties λ0=0, λr>0 for r>0 and Ra,Fa≥λρa,SF for all a∈Y. F is called semicompact if for each sequence wn in Y satisfying limn→∞Rwn,Fwn=0, one can find a strongly convergent subsequence wnk of wn. Moreover, a sequence wn in M is called a Fejér monotone with respect to Y provided that ρwn+1,z≤ρwn,z, for each z∈Y and n≥1.The following facts are in [31].Proposition 15.
LetY be a nonempty closed subset of M and wn be a Fejér monotone sequence with respect to Y. Then, wn converges strongly to a point of Y if and only if limn→∞ρwn,Y=0.The following theorem is based on the semicompactness ofF.Theorem 16.
LetY be a nonempty closed convex subset of M and F:Y⟶CY be a Suzuki mapping with SF≠∅. Suppose αn,βn∈0,1 be such that ∑αnβn=∞ and βn⟶0. If F is semicompact, then wn generated by (6) converges strongly to a stationary point of F.Proof.
In view of (16),
(18)∑n=1∞αnβn1−βnρ2wn,un<∞.
By Lemma9, there exists subsequences, namely, wnt and unt of wn and un, respectively, such that limt→∞ρ2wnt,unt=0. Hence,
(19)limt→∞Rwnt,Fwnt=limt→∞ρwnt,unt=0.
By the semicompactness of the mappingF, one can find a strongly convergent subsequence wnt of wn with the strong limit, say z. We shall prove that z∈SF. By Proposition 8(ii), we have
(20)ρz,Fz≤ρz,wnt+ρwnt,Fz≤ρwnt,z+3ρwnt,Fwnt+ρwnt,z=2ρwnt,z+3ρwnt,Fwnt⟶0.
Hence,z∈Fz. By Proposition 8(i),
(21)ℍFwnt,Fz≤ρwnt,z⟶0.
Now, we letv∈Fz and choose ynt∈Fwnt such that ρv,ynt=ρv,Fwnt. From (19) and (21), we have
(22)ρz,v≤ρz,wnt+ρwnt,ynt+ρynt,v=ρz,wnt+ρwnt,ynt+ρv,Fwnt≤ρz,wnt+Rwnt,Fwnt+ℍFwnt,Fz⟶0.
Hence,v=z for all v∈Fz, that is, z=Fz. Therefore, z∈SF. By Lemma 12, limn→∞ρwn,z exists. Hence, z is the strong limit of wn.Example 17.
LetY and F be as in Example 1. Then, F is a Suzuki mapping with SF=1. Semicompactness of F follows from the compactness of Y. For each n≥1, we let αn=1/2 and βn=1/n. The, limn→∞βn=0 and ∑αnβn=∞. Hence, by Theorem 16, the sequence of the modified Agarwal-O’Regan-Sahu iteration defined by (6) converges strongly to 1. However, we cannot directly apply any result in [22, 24, 25] because, in this situation, F is not nonexpansive.The following theorem requires conditionJ.Theorem 18.
LetY be a nonempty closed convex subset of M and F:Y⟶CY be a Suzuki mapping with SF≠∅. Suppose αn,βn∈a,b⊂0,1. If F satisfies condition J, then wn generated by (6) converges strongly to a stationary point of F.Proof.
From (17), we have
(23)limn→∞Rwn,Fwn=0.
SinceF satisfies condition J, we have
(24)Rwn,Fwn≥λρwn,SF.
So, from (23), we get
(25)limn→∞λρwn,SF=0.
Since the functionλ is nondecreasing and λ0=0 and λr>0 for every r>0. Hence,
(26)limn→∞ρwn,SF=0.
Closeness ofSF follows from the quasinonexpansiveness of F. In the view of Lemma 12, we have wn which is a Fejér monotone with respect to SF. By Proposition 15, wn converges strongly to an element of SF.Example 19.
LetY and F be as in Example 1. Then, F is a Suzuki mapping with SF=1 and satisfies the condition J. For each n≥1, we let αn=βn=1/3. Then, by Theorem 18, the sequence of modified Agarwal-O’Regan-Sahu iteration defined by (6) converges strongly to 1. However, we cannot directly apply any result in [22, 24, 25] because, in this situation, F is not nonexpansive.


## 3. Conclusions

Under some appropriate conditions, we have proven that the sequence of the modified Agarwal-O’Regan-Sahu iterative process defined by (6) converges to a stationary point of a multivalued Suzuki mapping. We have used a 2-uniformly convex hyperbolic space, which is more general than the uniformly convex Banach spaces, CAT(0) spaces, and some CAT(κ) spaces. Moreover, the class of multivalued Suzuki mappings properly includes the class of multivalued nonexpansive mappings as shown by Example 1. Our iterative process is independent of but better than the Ishikawa iterative process. Hence, our presented results extend and improve the corresponding results in [21–25].



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