# Approximating Fixed Points of Enriched Nonexpansive Mappings in Geodesic Spaces

Authors: Rahul Shukla; Rekha Panicker
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we consider the class of enriched nonexpansive mappings in the setting of geodesic spaces. We obtain a number of fixed point theorems for these mappings in geodesic spaces. Further, we employ the SP iterative method and present some new convergence theorems for the class of enriched nonexpansive mappings under different assumptions. We present some results concerningΔ and strong convergence.


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## Body

## 1. Introduction

Nonexpansive mappings are those class of nonlinear mappings which have Lipschitz constant equal to one. A nonexpansive mapping needs not to admit a fixed point in a complete space. However, Browder [1], Göhde [2], and Kirk [3] independently ensured the existence of fixed points of nonexpansive mappings in Banach spaces under certain geometric assumptions. Many mathematicians have generalized and extended these results and considered a number of nonlinear mappings, see [4–9] (see also the references therein).In 2019, Berinde [10] considered a new class of nonlinear mappings by enriching nonexpansive mappings, known as enriched nonexpansive mappings. He obtained some fixed point theorems for these classes of mappings in Hilbert spaces. It was observed in [10, 11] that class of enriched nonexpansive mappings has strong relations with averaged and nonexpansive mappings.On the other hand, in 1970, Takahashi [12] considered the structure of convexity outside linear spaces. These spaces are fruitful in the context of fixed point theory. Goebel and Kirk [13] employed Krasnosel’skiĭ-Mann iterative method to find fixed points of nonexpansive mappings in hyperbolic type spaces. In the recent years, a number of papers have appeared in the literature dealing with the fixed point theorems in nonlinear spaces, see [14–24].The class of enriched nonexpansive mappings has been studied only in linear spaces. Now it is natural to extend this class of mappings outside of linear spaces (or in nonlinear spaces) and ensure the existence of fixed points. The aim of this paper is to study the class of enriched nonexpansive mappings in geodesic spaces. We observe that for everyb-enriched nonexpansive mapping, one can define a nonexpansive mapping, and the set of fixed points of both the mappings remains the same. Therefore, the existence of fixed points for b-enriched nonexpansive mappings is equivalent to existence of fixed points for nonexpansive mappings. However, the convergence of fixed points for b-enriched nonexpansive mappings is slightly different than the convergence of fixed points for nonexpansive mappings. We prove that Krasnosel’skiĭ method converges to fixed point of mapping. Further, we use SP iterative method to reckon fixed points of b-enriched nonexpansive mappings under certain assumptions. These results are new even in Hilbert spaces. Our results extend, complement, and generalize some results from [10, 11, 16, 19, 25–27].


## 2. Preliminaries

LetΓ,ρ be a metric space and 0,c⊂ℝ. A mapping g:0,c⟶Γ is called as geodesic path from ζ to ξ if
(1)g0=ζ,gc=ξ,ρgs,gs′=s−s′,for all s,s′∈0,c. The image g0,c of g forms a geodesic joining ζ and ξ. It is noted that the geodesic segment joining ζ and ξ is not unique, in general. For more details of geodesic spaces, see [14, 21].Definition 1 (see [28]).
A tripletΓ,ρ,Ω is called as a hyperbolic metric space if Γ,ρ is a metric space, and function Ω:Γ×Γ×0,1⟶Γ satisfies the following assumptions for all ζ,ξ,υ,w∈Γ and μ,θ∈0,1
(W1)ρυ,Ωζ,ξ,μ≤1−μρυ,ζ+μρυ,ξ
(W2)ρΩζ,ξ,μ,Ωζ,ξ,θ=μ−θρζ,ξ
(W3)Ωζ,ξ,μ=Ωξ,ζ,1−μ
(W4)ρΩζ,υ,μ,Ωξ,w,μ≤1−μρζ,ξ+μρυ,wRemark 2.
IfΩζ,ξ,μ=1−μζ+μξ for all ζ,ξ∈Γ,μ∈0,1, then it can be seen that all normed linear spaces are hyperbolic metric space.Remark 3.
If conditions (W1)–(W3) are satisfied, thenΓ,ρ,Ω is hyperbolic type space considered by Goebel and Kirk [13]. Reich and Shafrir [22] also obtained some important results in hyperbolic metric spaces.
We shall write(2)Ωζ,ξ,μ≔1−μζ⊕μξ,to denote a point Ωζ,ξ,μ of Γ,ρ,Ω space. For ζ,ξ∈Γ,(3)ζ,ξ=1−μζ⊕μξ:μ∈0,1,indicates geodesic segments. A subset Z of hyperbolic metric space (or hyperbolic space) Γ,ρ,Ω is called convex if ζ,ξ⊂Z whenever ζ,ξ∈Z.Remark 4.
Leustean [20] proved that the class of CAT0 spaces is the class of complete uniformly convex hyperbolic spaces (in short, complete UCΩ-hyperbolic space), see the definition of UCΩ-hyperbolic space in [19].
IfΓ,ρ,Ω is a Busemann space, then there is a unique convexity mapping Ω in such a way that Γ,ρ,Ω is Ω-hyperbolic space with unique geodesics. In other words, for all ζ≠ξ∈Γ and any μ∈0,1, there is an element υ∈Γ which is unique (say υ=Ωζ,ξ,μ) in such a way
(4)ρζ,υ=μρζ,ξandρξ,υ=1−μρζ,ξ.
Letζ,ξ,υ be three points in metric space Γ,ρ; the point ξ is said to lie between ζ and υ if
(5)ρζ,υ=ρζ,ξ+ρξ,υ,and these points are distinct pairwise. Thus, if ξ lies between ζ and υ, then ξ lies between υ and ζ.Lemma 5 (see [14]).
LetΓ be a uniquely geodesic space. Let ζ,ξ,υ∈Γ be pairwise distinct points. A point ξ lies between ζ and υ if and only if ξ∈ζ,υ.Proposition 6 (see [14]).
LetΓ be a metric space and ζ,ξ,υ,w∈Γ be pairwise distinct points. The following are equivalent:
(a)
ξ lies between ζ and υ, and υ lies between ζ and w(b)
ξ lies between ζ and w, and υ lies between ξ and wLetζn be a bounded sequence in a hyperbolic space Γ,ρ,Ω and Z⊆Γ with Z≠∅. A functional r.,ζn:Γ⟶0,∞ can be defined as follows:
(6)rξ,ζn=limsupn⟶∞ρξ,ζn.The asymptotic radius ofζn with respect to (in short, wrt) Z is defined as
(7)rZ,ζn=infrξ,ζnξ∈Z.A pointζ in Z is called an asymptotic center of ζn wrt Z if
(8)rζ,ζn=rZ,ζn.AZ,ζn is denoted as set of all asymptotic centers of ζn wrt Z. A bounded sequence ζn in a hyperbolic space Γ,ρ,Ω is said to Δ-converge to ζ if ζ is the unique asymptotic center for every subsequence un of ζn. A sequence ζn⊆Γ is called Fejér monotone wrt Z if for all ζ†∈Z(9)ρζ†,ζn+1≤ρζ†,ζn,for all n≥0.Definition 7 (see [29]).
A mappingϜ:Z⟶Z is called quasi-nonexpansive if
(10)ρϜζ,ζ†≤ρζ,ζ†∀ζ∈Z,ζ†∈FixϜ≠∅,where FixϜ=ζ†∈ZϜζ†=ζ†.Definition 8 (see [30]).
The mappingϜ:Z⟶Z with FixϜ≠∅ is said to have Condition (I) if the following assumptions are satisfied:
(a)
∃ a function f:0,∞⟶0,∞ which is nondecreasing(b)
Forr∈0,∞, fr>0 and f0=0(c)
For allζ∈Z, ρζ,Ϝζ≥fρζ,FixϜwhere ρζ,FixϜ=infρζ,ξ:ξ∈FixϜ.Definition 9.
LetΓ,ρ be a metric space and Z⊆Γ with Z≠∅. A mapping Ϝ:Z⟶Z is called as compact if ϜZ has a compact closure.Proposition 10 (see [20]).
LetΓ,ρ,Ω be a complete UCΩ-hyperbolic space, Z⊆Γ with Z≠∅. Suppose that Z is convex and closed, and ζn is bounded sequence in Γ. Then, ζn has a unique asymptotic center with respect to Z.Lemma 11 (see [17]).
LetΓ,ρ,Ω be same as in Proposition 10. Let w∈Γ and ωn be a sequence with ωn⊆a,b⊆0,1. For some r≥0, if ζn and ξn are sequences in Γ with limsupn⟶∞ρζn,w≤r, limsupn⟶∞ρξn,w≤r, and limn⟶∞ρωnξn⊕1−ωnζn,w=r. Then, limn⟶∞ρξn,ζn=0.Lemma 12.
LetΓ,ρ,Ω and Z be same as in Proposition 10. Let Ϝ:Z⟶Z be a mapping. For λ∈0,1, consider Ψ:Z⟶Z as follows:
(11)Ψζ=1−λζ⊕λϜζ,for all ζ∈Z. Then, FixΨ=FixϜ.Lemma 13 (see [20]).
Letζn be a bounded sequence in Γ and AZ,ζn=υ. Let κn and νn be two sequences in ℝ with for all n∈ℕ,κn∈0,∞,limsupκn≤1 and limsupνn≤0. Suppose that ξ∈Z and there exists m,q∈ℕ such that
(12)ρξ,ζn+m≤κnρυ,ζn+νn∀n≥q.
Then,ξ=υ.Lemma 14 (see [14]).
LetΓ,ρ,Ω be a metric space, Z⊆Γ such that Z≠∅. If ζn is Fejér monotone wrt Z, AZ,ζn=ζ and AΓ,un⊆Z for every subsequence un of ζn. Then, the sequence ζnΔ-converges to ζ∈Z.Lemma 15 (see [16]).
LetΓ,ρ,Ω be a complete UCΩ-hyperbolic space and Z⊆Γ such that Z≠∅ and Z is closed convex. Let ζn be a bounded sequence in Γ and τ:Z⟶0,∞ a function defined as follows:
(13)τζ=limsupn⟶∞ρζn,ζ,for any ζ∈Z. τ is called as type function, and it is unique.
Then, there is a minimum point (unique)w∈Z and τw=infτζ:ζ∈Z.Proposition 16 (see [13]).
LetΓ,ρ,Ω and Z be same as in Lemma 15 with Z is bounded. Let Ϝ:Z⟶Z be a nonexpansive mapping. Let ζ0∈Z and ϑ∈0,1. Define a sequence ζn in Z by Krasnosel’skiĭ iterative method [31].
(14)ζn+1=1−ϑζn⊕ϑϜζn,n∈ℕ∪0.
Then,limn⟶∞ρζn,Ϝζn=0.The proof of the following theorem is motivated from [16].Theorem 17.
LetΓ,ρ,Ω, Z, and Ϝ be same as in Proposition 16. Then, FixϜ≠∅.Proof.
For a givenζ0∈Z and for any ω∈0,1, a sequence can be defined:
(15)ζn+1=1−ωζn⊕ωΨζn.
From Proposition16, it implies that
(16)limn⟶∞ρζn,Ψζn=0.
From Lemma15, there is a minimum point (unique) υ†∈Z in such a way that
(17)τυ†=infτω:ω∈Z.
From the definition of mappingΨ,
(18)τΨυ†=limsupn⟶∞ρζn,Ψυ†≤limsupn⟶∞ρζn,Ψζn+limsupn⟶∞ρΨζn,Ψυ†≤limsupn⟶∞ρζn,υ†.
Then,Ψυ†=υ†.


## 3. Main Results

In 2019, Berinde [10] considered a new class of mappings which is defined below.Definition 18.
LetΓ,. be a Banach space and Ϝ:Γ⟶Γ a mapping. The mapping Ϝ is called b-enriched nonexpansive if ∃b∈0,∞ in such a way that
(19)bζ−ξ+Ϝζ−Ϝξ≤b+1ζ−ξ,for all ζ,ξ∈Γ.It can be noted that 0-enriched mapping is nonexpansive mapping. Even both the class of mappings, that is, quasi-nonexpansive andb-enriched nonexpansive, are independent in nature, cf. [27].Remark 19.
Takeb≠0, and it is straight forward from (19) that
(20)bb+1ζ−ξ+1b+1Ϝζ−Ϝξ≤ζ−ξ⇔1−1b+1ζ−ξ+1b+1Ϝζ−Ϝξ≤ζ−ξ⇔1−1b+1ζ+1b+1Ϝζ−1−1b+1ξ+1b+1Ϝξ≤ζ−ξ.
Take,λb=1/b+1∈0,1 then
(21)1−λbζ+λbϜζ−1−λbξ+λbϜξ≤ζ−ξ.
From the above inequality, we can take convex combination ofϜ and the identity mappings.In view of Remark19, we consider Definition 18 in Ω-hyperbolic spaces.Definition 20.
LetΓ,ρ,Ω be a Ω-hyperbolic space, Z a subset of Γ such that Z≠∅, and Ϝ:Z⟶Z a mapping. The mapping Ϝ is called b-enriched nonexpansive if ∃b∈0,∞ in such a way that
(22)ρ1−λbζ⊕λbϜζ,1−λbξ⊕λbϜξ≤ρζ,ξ,for all ζ,ξ∈Z, where λb=1/b+1.We prove the following important lemma which will be utilized throughout this paper.Lemma 21.
LetΓ,ρ,Ω be a uniquely geodesic space. For some λ,ω∈0,1, let ζ,ξ,υ∈Γ be pairwise distinct points with ξ=1−λζ⊕λυ and υ=1−ωζ⊕ωw. Then
(23)ξ=1−ϑζ⊕ϑw,where ϑ=ωλ.Proof.
From Lemma5, ξ lies between ζ and υ. And υ lies between ζ and w. From Proposition 6, ξ lies between ζ and w. Thus, ξ∈ζ,w and
(24)ξ=1−ϑζ⊕ϑw,for some ϑ∈0,1. Since Γ is uniquely geodesic space, we have
(25)ρζ,ξ=ϑρζ,w,(26)ρξ,w=1−ϑρζ,w.
Sinceξ=1−λζ⊕λυ, we have
(27)ρζ,ξ=λρζ,υ.
Again, sinceυ=1−ωζ⊕ωw, we have
(28)ρζ,υ=ωρζ,w.
From (25), (27), and (28), one can conclude
(29)ρζ,ξ=λρζ,υ=ωλρζ,w=ωλϑρζ,ξ.
Therefore,ωλ/ϑ=1, and ϑ=ωλ.Theorem 22.
LetΓ,ρ,Ω be a complete UCΩ-hyperbolic space and Z⊆Γ such that Z≠∅. Assume that Z is closed, bounded, and convex. Let Ϝ:Z⟶Z be a b-enriched nonexpansive mapping. Then, FixϜ≠∅. Moreover, for given ζ0∈Z, any ω∈0,1, there exists ωb=ω/b+1 such that the sequence ζn generated by (Krasnosel’skiĭ method)
(30)ζn+1=1−ωbζn+ωbϜζnforalln∈ℕ∪0.
Δ-converges to an element of FixϜ.Proof.
By the definition of mappingϜ, we get
(31)ρ1−λbζ⊕λbϜζ,1−λbξ⊕λbϜξ≤ρζ,ξ,for all ζ,ξ∈Z and λb=1/b+1. Set the mapping Ψ as follows:
(32)Ψζ=1−1b+1ζ⊕1b+1Ϝζforallζ∈Z.
Thus, from (31), we get, for all ζ,ξ∈Z,
(33)ρΨζ,Ψξ≤ρζ,ξ,and Ψ is a nonexpansive mapping. For any ω∈0,1 and a given ζ0∈Z, we can define a sequence
(34)ζn+1=1−ωζn⊕ωΨζn.
From Proposition16, it follows that
(35)limn⟶∞ρζn,Ψζn=0.
From Theorem17, FixΨ≠∅; thus, from Lemma 12, FixΨ=FixϜ≠∅. Further, for any υ†∈FixΨ,(36)ρΨζn,υ†≤ρζn,υ†foralln≥0.
Thus, from (W1)(37)ρζn+1,υ†=ρ1−ωζn⊕ωΨζn,υ†≤1−ωρζn,υ†+ωρΨζn,υ†≤ρζn,υ†.
Hence, the sequenceρζn,υ† is monotone nonincreasing. It implies that ζn is Fejér monotone sequence wrt FixϜ. In view of Proposition 10, the sequence ζn has unique asymptotic center w† wrt FixϜ. Suppose un is a subsequence of ζn and u† is unique asymptotic center of un wrt FixϜ. Now,
(38)ρun,Ψu†≤ρΨun,Ψu†+ρΨun,un≤ρun,u†+ρΨun,un.
From (35) and Lemma 13, it follows that Ψu†=u†. From Lemma 14, the sequence ζnΔ-converges to an element of FixϜ. From Lemma 21 with υ=Ψζ and w=Ϝζ, we have
(39)1−ωζ⊕ωΨζ=1−ωλbζ⊕ωλbϜζ,for all ζ∈Z since ω∈0,1 and λb=1/b+1. It follows that ωλb∈0,1/b+1. Thus, for any ωb=ωλb∈0,1/b+1, the sequence ζn defined by (30) Δ-converges to a point in FixϜ.Remark 23.
It can be seen that Theorem22 generalizes the results in [10] (Theorem 3.3) from Hilbert spaces to hyperbolic spaces.Theorem 24.
LetΓ,ρ,Ω and Ϝ be same as in Theorem 22. Suppose Z⊆Γ such that Z≠∅, and Z is closed and convex. Assume Ϝ satisfies Condition (I) with FixϜ≠∅. For fixed ζ0∈Z and any ω∈0,1, there exists ωb=ω/b+1 such that the sequence ζn generated by (Krasnosel’skiĭ method)
(40)ζn+1=1−ωbζn+ωbϜζn∀n∈ℕ∪0,strongly converges to an element of FixϜ.Proof.
By the similar technique in proof of Theorem22, one can set a mapping Ψ as in (32), and Ψ is nonexpansive. Let ω∈0,1 and define
(41)ζn+1=1−ωζn⊕ωΨζn.
For allυ†∈FixΨ(42)ρΨζn,υ†≤ρζn,υ†foralln≥1.
From (41), we have
(43)ρζn+1,υ†≤ρζn,υ†.
Thus,ρζn,υ† and ρζn,FixΨ are monotone nonincreasing sequences and limn⟶∞ρζn,υ† and limn⟶∞ρζn,FixΨ exist. Let
(44)limn⟶∞ρζn,υ†=r.
From (42)
(45)limn⟶∞ρΨζn,υ†≤r.
By (44), we have
(46)r=limn⟶∞ρζn+1,υ†=limn⟶∞ρ1−ωζn⊕ωΨζn,υ†.
In view of (44), (45), (46), and Lemma 11, it implies:
(47)limn⟶∞ρζn,Ψζn=0.
Since(48)Ψζ=1−1b+1ζ⊕1b+1Ϝζ,we have
(49)ρζ,Ψζ=1b+1ρζ,Ϝζforallζ∈Z.
SinceϜ satisfies Condition (I) and (49), we obtain
(50)b+1ρζn,Ψζn=ρζn,Ϝζn≥fρζn,FixϜ=fρζn,FixΨ.
By (47), limn⟶∞fρζn,FixΨ=0 and
(51)limn⟶∞ρζn,FixΨ=0.
One can easily show thatζn is a Cauchy sequence. For the sake of completeness, we prove this claim. From (51), for given ε>0, ∃ a n0∈ℕ in such a way that
(52)ρζn,FixΨ<ε4,for all n≥n0. Hence,
(53)infρζn0,υ†:υ†∈FixΨ<ε4,so there is υ†∈FixΨ in such a way
(54)ρζn0,υ†<ε2.
Therefore, for allm,n≥n0,
(55)ρζn+m,ζn≤ρζn+m,υ†+ρυ†,ζn≤2ρζn0,υ†<2ε2=ε,and ζn is a Cauchy sequence. By the closedness of Z in Γ, ζn converges to a point ζ†∈Z. Now
(56)ρζ†,Ψζ†≤ρζ†,ζn+ρζn,Ψζn+ρΨζn,Ψζ†≤2ρζ†,ζn+ρζn,Ψζn.
From (47), ζ†=Ψζ†. Therefore, the sequence ζn strongly converges to a point in FixϜ. Further,
(57)1−ωζ⊕ωΨζ=1−ωbζ⊕ωbϜζ,for all ζ∈Z with ωb=ω/b+1.Remark 25.
Theorem24 generalizes the results in [11] (Theorem 3.2) from Hilbert spaces to hyperbolic spaces.Theorem 26.
LetΓ,ρ,Ω and Ϝ be same as in Theorem 22. Let Z⊆Γ such that Z≠∅ and Z be a closed and convex. Suppose that Ϝ is compact mapping with FixϜ≠∅. For fixed λ∈0,1/b+1, ζn is a sequence generated as follows:
(58)ζn+1=1−λζn⊕λϜζn,strongly converges to an element of FixϜ.Proof.
We set the nonexpansive mappingΨ as in the proof of Theorem 22. For given ζ0∈Z and for any ω∈0,1, define a sequence
(59)ζn+1=1−ωζn⊕ωΨζn.
Following largely as in Theorem24 and from Lemma 11(60)limn⟶∞ρζn,Ψζn=0.
Since(61)Ψζ=1−1b+1ζ⊕1b+1Ϝζ,we get
(62)ρζ,Ψζ=1b+1ρζ,Ϝζforallζ∈Z.
From the above equation and (60)
(63)limn⟶∞ρζn,Ϝζn=0.
Since the range ofZ under Ϝ is subset of a compact set, there is a subsequence Ϝζnj of Ϝζn strongly converges to ζ†∈Z. By (63), the subsequence ζnj strongly converges to ζ†. Since Ψ is nonexpansive mapping and by the triangle inequality, we obtain
(64)ρζnj,Ψζ†≤ρζnj,Ψζnj+ρΨζnj,Ψζ†≤ρζnj,Ψζnj+ρζnj,ζ†.
Thus, subsequenceζnj strongly converges to Ψζ† and Ψζ†=ζ†. Since limn⟶∞ρζn,ζ† exists, it follows that the sequence ζn strongly converges to an element of FixϜ.


## 4. SP Iterative Method

In this section, we present some convergence results for SP iterative process. For a fixζ0∈Z and the mapping Ϝ:Z⟶Z, the SP iterative method in the setting of hyperbolic metric spaces can be defined as follows [26]:
(65)wn=1−σnζn⊕σnϜζn,ξn=1−ϑnwn⊕ϑnϜwn,ζn+1=1−ωnξn⊕ωnϜξn,where σn, ϑn, and ωn are sequences in 0,1.Similar to [25] (Lemma 4), we model the following lemma.Lemma 27.
LetΓ,ρ,Ω and Z be same as in Theorem 24. Let Ψ:Z⟶Z be a nonexpansive mapping with FixϜ≠∅. For fixed ζ0∈Z and for all n∈ℕ∪0, ωn,ϑn,σn∈ω,ϑ with ω,ϑ∈0,1, the sequence ζn is defined by
(66)wn=1−σnζn⊕σnΨζn,ξn=1−ϑnwn⊕ϑnΨwn,ζn+1=1−ωnξn⊕ωnΨξn.
Then, the following holds:(1)
limn⟶∞ρζn,υ exists ∀υ∈FixΨ(2)
limn⟶∞ρζn,Ψζn=0Proof.
From (W1), we get(67)ρwn,υ≤1−σnρζn,υ+σnρΨζn,υ≤1−σnρζn,υ+σnρζn,υ=ρζn,υ,(68)ρξn,υ≤1−ϑnρwn,υ+ϑnρΨwn,υ≤1−ϑnρwn,υ+ϑnρwn,υ=ρwn,υ.
Further, from (67) and (68), we get
(69)ρζn+1,υ≤1−ωnρξn,υ+ωnρΨξn,υ≤1−ωnρξn,υ+ωnρξn,υ=ρξn,υ≤ρwn,υ≤ρζn,υ.
Thus,ρζn,υ is a monotone nonincreasing sequence. Hence, limn⟶∞ρζn,υ exists. Let
(70)limn⟶∞ρζn,υ=r>0.
From (69) and (70), we have
(71)r≤liminfn⟶∞ρwn,υ,limsupn⟶∞ρwn,υ≤r.
Thus(72)limn⟶∞ρwn,υ=r.
Since the mappingΨ is nonexpansive
(73)limsupn⟶∞ρΨζn,υ≤limn⟶∞ρζn,υ=r,and from (72)
(74)limn⟶∞ρ1−σnζn⊕σnΨζn,υ=limn⟶∞ρwn,υ=r.
From (70), (73), (74), and Lemma 11, it follows:
(75)limn⟶∞ρζn,Ψζn=0.Theorem 28.
LetΓ,ρ,Ω and Z be same as in Theorem 24. Let Ϝ:Z⟶Z be a b-enriched nonexpansive mapping with FixϜ≠∅. For fixed ζ0∈Z, for all n∈ℕ∪0, ωn,ϑn,σn∈ω/b+1,ϑ/b+1 with ω,ϑ∈0,1, the sequence ζn generated by (65) Δ-converges to an element of FixϜ.Proof.
For givenζ0∈Z and for all n∈ℕ∪0, ωnb,ϑnb,σnb∈ω,ϑ with ω,ϑ∈0,1, we can consider a sequence ζn:
(76)wn=1−σnbζn⊕σnbΨζn,ξn=1−ϑnbwn⊕ϑnbΨwn,ζn+1=1−ωnbξn⊕ωnbΨξn,where Ψ is a mapping defined as in (32). Using Lemma 21, we have
(77)wn=1−σnζn⊕σnϜζn,ξn=1−ϑnwn⊕ϑnϜwn,ζn+1=1−ωnξn⊕ωnϜξn,where σn=σnb/b+1, ϑn=ϑnb/b+1, and ωn=ωnb/b+1. By Lemma 27 and repeating the technique of proof of Theorem 22, one can complete the proof.Theorem 29.
LetΓ,ρ,Ω, Z, and Ϝ be same as in Theorem 24. For fixed ζ0∈Z, for all n∈ℕ∪0, ωn,ϑn,σn∈ω/b+1,ϑ/b+1 with ω,ϑ∈0,1, the sequence ζn generated by (65) strongly converges to an element of FixϜ.Proof.
Using proof of Theorem28, Lemma 27, and Theorem 24, one can complete the proof.Theorem 30.
LetΓ,ρ,Ω and Z be same as in Theorem 24. Let Ϝ:Z⟶Z be a compact b-enriched nonexpansive mapping with FixϜ≠∅. For fixed ζ0∈Z, for all n∈ℕ∪0, ωn,ϑn,σn∈ω/b+1,ϑ/b+1 with ω,ϑ∈0,1, the sequence ζn generated by (65) strongly converges to a point in FixϜ.Proof.
Using proof of Theorem28, Lemma 27, and Theorem 26, one can complete the proof.Remark 31.
Theorems28–30 are new even in Hilbert spaces.



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*Source: 6161839-2022-05-04.xml*