# Convergence for a Fixed Point of Evolution Families in Banach Space via Iterative Process

Authors: Musharaf Shah; Gul Rahmat; Syed Inayat Ali Shah; Ebenezer Bonyah; Zahir Shah; Meshal Shutaywi
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we present the convergence of Krasnoselskii Mann iterative process for common fixed points of pointwise Lipschitzian evolution family of bounded nonlinear operators on a Banach spaceℜ.


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## Body

## 1. Introduction

Letℜ be a Banach space and B be a nonempty, bounded, closed, and convex subset of ℜ. Let us consider a pointwise Lipschitzian evaluation family of nonlinear mappings, that is, a family of mappings Φζ,ξ:B⟶B satisfying the conditions Φζ,ζμ=μ, Φζ,λΦλ,ξμ=Φζ,ξμ, ζ↦Φζ,ξx is strongly continuous for every x∈B, and each Φζ,ξ is pointwise Lipschitzian. The latter means that there exists a family of functions αt:B⟶0,∞ such that Φζ,ξμ−Φζ,ξν≤αtuμ−ν. The existence of common fixed points for function of contraction and nonexpansive mappings have been investigated since the early 1960s, see Bruck[1, 2] Lim [3], Browder [4], Demarr [5], Belluce and Kirk [6, 7]. The asymptotic approach for finding common points of semigroups of Lipschitzian (but not pointwise Lipschitzian) mappings has been also investigated for some time, see Xu and Tan [8] and the references there in. Kirk and Xu [4] proved the existence of fixed points for asymptotic pointwise contraction and asymptotic pointwise nonexpansive mappings in Banach space, and latter on, Khamsi and Hussain extend these results to metric spaces [9], and Kozlowski and Khamsi to modular function spaces [10, 11].The generalization of known iterative fixed point construction process of pointwise asymptotically nonexpansive mapping has been studied by several authors, such as the Mann process [12, 13] or the Ishikawa process for detail see [14]. The iterative fixed point construction process for asymptotically nonexpansive mapping exists in Hilbert space, Banach space, and Metric spaces, see for further detail [8, 9, 15–28] and the works referred there in. The modified Mann iterative process for a fixed point of asymptotically nonexpansive mappings in uniformly convex Banach space ℜ which possess the Opial property of the weak convergence and the strong convergence for asymptotically nonexpansive mapping has been proved by Schu [20]. Xu and Tan [19] proved for weak convergence of the modified Ishikawa and modified Mann iterative process for asymptotically nonexpansive mapping on Banach space ℜ that satisfy the Opial property. The generalized Ishikawa and Mann process weakly converges to a common fixed point of a pointwise nonexpansive mapping Φζ,ξ:B⟶B (B is a nonempty, bounded, closed, and convex subset of a Banach space ℜ) has been proved by Kozlowski in [29] which satisfying the Opial conditions.In [30], Kozlowski and Braily Sims examine the convergence of generalized Mann and Ishikawa iteration process for Tt:B⟶B. In this paper, we proved more general form of these results We extend this results to an evolution class of bounded nonlinear mappings U=Φζ,ξ;ζ≥ξ≥0 on the set B.


## 2. Definitions and Preliminary Results

Definition 1.
A familyU on B is pointwise Lipschitzian if there exists a family of mappings αt:0,∞⟶0,∞ such that(1)Φζ,ξμ−Φζ,ξν≤αtμ−νfor allζ,ξ∈B.
Ifαt<1 for each t, then the family is said to be a point-wise contraction family. If αt≤1 for all t, then the family is called nonexpansive point-wise evolution family.Definition 2.
A pointwise Lipschitzian evolution family denoted byU and defined as U=Φζ,ξ;ζ≥ξ≥0 is said to be pointwise nonexpansive if suplimn⟶∞αnt≤1 for each t∈B. By letting a0=1 and ant=maxαnt,1 for n>0, we say that Φζ,ξ is asymptotically nonexpansive if(2)Φζ,ξμ−Φζ,ξν≤αntμ−νfor allμ,ν∈B,n∈N.where(3)limt⟶∞αnt=1,
αnt≥1wheret∈B,n∈N.
By definingβnt=αnt−1, we have(4)limn⟶∞βnt=0.Definition 3.
BySB, we will represent the class of all asymptotic pointwise nonexpansive evolution families on B such that(5)Mn=supαns:s∈B<∞,∀n∈N,limn⟶∞supMn=1.Lemma 1.
(see [31]). Letℜbe a uniformly convex Banach space andUbe an asymptotically nonexpansive pointwise Lipschitzian evolution family onB. Then, the setUhas a common fixed point, and the setFUof common fixed points is closed and convex.Lemma 2.
(see [15]). Assumerkbe a bounded sequence of real numbers anddk,nbe double index sequence of real number satisfying(6)limk⟶∞suplimn⟶∞supdk,n≤0,and rk+n≤rk+dk,n for each k,n≥1. Then, the sequence rk converged to r, where r∈ℝ.Definition 4.
A subsetA⊂N is said to be a generating set for N if for each u∈N,u>0 there exists m∈ℕ, and ζ,r∈A such that u=mζ+r.
The next lemmas will be used in the proof of our main result.Lemma 3.
(see [20, 32]). Ifℜis uniformly convex Banach space, letcn⊂0,1be bounded away from 0 to 1, andun,vn⊂ℜbe such that(7)limn⟶∞supun≤a,limn⟶∞supvn≤a,and limn⟶∞cnun+1−cnvn=a, then limn⟶∞un−vn=0.


## 3. Main Result

A sequencecn⊂0,1 is called bounded away from 0 if exists a number a∈0,1 such that cn>a, for all natural n. Similarly, cn⊂0,1 is called bounded away from 1 if exists a number b∈0,1 such that cn<b for every natural number of n.Lemma 4.
Letℒ:=Φζ,0;ζ≥0be a subset of an evolution familyUon a Banach spaceℜ. If for eachm∈N, there existsαmsuch that(8)Φmζ,0μn−Φm−1ζ,0νn≤αmμμ−νfor allζ≥0.
LetΦζ,0μn−μn⟶0, then Φmζ,0μn−μn⟶0 for all m∈N.Proof 1.
It follows from the fact that everyαm is a bounded function that there exists a finite number M>0, such that(9)∑j=1m−1supαjsμ;μ∈B≤M,Φmζ,0μn−μn=Φmζ,0μn−Φm−1ζ,0+Φm−1ζ,0−Φm−2ζ,0+Φm−2ζ,0−⋯−Φ2ζ,0μn+Φ2ζ,0μn−Φζ,0μn+Φζ,0μn−μn≤Φmζ,0μn−Φm−1ζ,0+Φm−1ζ,0−Φm−2ζ,0+⋯+Φζ,0μn−μn.Then, by using the inequality, we have(10)Φmζ,0μn−μn≤αm−1Φζ,0μn−μn+αm−2Φζ,0μn−μn+⋯+α2Φζ,0μn−μn+α1Φζ,0μn−μn+1Φζ,0μn−μn,or(11)Φmζ,0μn−μn≤αm−1+αm−2+⋯+α1+1Φζ,0μn−μn=∑j=1m−1αjs+1Φζ,0μn−μn≤M+1Φζ,0μn−μn⟶0,limn⟶∞Φmζ,0μn−μn=0.
This completes the proof.Lemma 5.
IfBbe a nonempty, bounded, closed, and convex subset of a Banach spaceℜandμn⊂Bbe an approximate fixed point sequence ofℒfor eachn∈A, then the sequenceμnis an approximate fixed point sequence ofΦmζ+r,0.Proof 2.
Consider(12)Φmζ+r,0μn−μn=Φmζ+r,0μn−Φm−1ζ,0μn+Φm−1ζ,0μn−μn≤Φmζ+r,0μn−Φm−1ζ,0μn+Φm−1ζ,0μn−μn,
implies that(13)Φmζ+r,0μn−μn≤αμΦζ,0μn−μn+Φm−1ζ,0μn−μn⟶0.
The result follows by lemma4 and the boundedness of αt.
Let us recall the definition of the Opial property which will play an important role in our paper.Definition 5.
A Banach spaceℜ is said to have the Opial property if for every sequence μn⊂ℜ weakly converging to a point μ∈ℜ (denoted by μn⇀w) and for each ν∈ℜ such that μ≠ν there holds(14)limn⟶∞infμn−μ<limn⟶∞infμn−ν.
or equivalently(15)limn⟶∞supμn−μ<limn⟶∞supμn−ν.
In the next result, we will present the main work of this paper.Theorem 1.
Letℜbe a uniformly convex Banach space with the Opial property. LetBbe a nonempty, convex, and bounded subset ofℜ. Assume that there existsw∈ℜandμn⊂Bsuch thatμn⇀w. Assume there exists anm∈Jsuch thatΦmζ,0μn−μn⟶0asn⟶∞. Then,w∈FUkmζ,0for everyk∈ℕ.Proof 3.
By defining a functionϕμ=limn⟶∞supμn−μ for μ∈B. For fixed m∈ℕ, m>2, we have(16)Φmζ,0μn−μn≤Φmζ,0μn−Φm−1ζ,0+Φm−1ζ,0−Φm−2ζ,0+⋯+Φζ,0μn−μn+μn−μ≤αm−1+αm−2+⋯+α1+1Φζ,0μn−μn+μn−μ≤Σj=1m−1αjs+1Φζ,0μn−μn+μn−μ≤M+1Φζ,0μn−μn+μn−μ.
Since all functionsαi are bounded and hence Φζ,0μn−μn⟶0, it follows that(17)limn⟶∞supΦmζ,0μn−μn≤limn⟶∞supμn−μ=ϕμ.
By Lemma4, we have(18)ϕμ≤limn⟶∞supμn−Φmζ,0μn+limn⟶∞supΦmζ,0μn−μ=limn⟶∞supΦmζ,0μn−μ.
Hence,(19)ϕμ=limn⟶∞supΦmζ,0μn−μ.
SinceΦζ,0 is asymptotic pointwise nonexpansive, therefore,(20)ϕΦmζ,0μ≤amsμϕμ,for each μ∈B. By applying this to w and passing with m⟶∞, we have(21)limm⟶∞ϕΦmζ,0w≤ϕw,A.
Sinceμn⇀w, for μ,w∈ℜ with μ≠w, then by Opial property of ℜ, we have(22)ϕw≐limn⟶∞supμn−w<limn⟶∞supμn−μ=ϕμ.
⇒ϕw=infϕμ;μ∈B comparing this with A give us
limn⟶∞ϕΦmζ,0w=ϕw.
By Theorem2, [33] and Proposition 3.4, [4] for every d>0, there exists a continuous map(23)ψ:ℝ+⟶ℝ+.
such thatψn=0 if and only if n=0. This yields(24)αμ+1−αν2≤αμ2+1−αν2−1−αψμ−ν,for each α∈ℝ+, for every μ,ν∈ℜ such that μ≤d, ν≤d.
Putμ=μn−w, ν=μn−Φmζ,0w
andα=1/2, we get(25)μn−12w+Φmζ,0w2≤12μn−w2+12μn−Φmζ,0w2−14ψΦmζ,0w−w.By applying on both sides limn⟶∞sup and ϕw=infϕμ;μ∈B, we get(26)ϕw2≤12ϕw2+12ϕΦmζ,02−14ψΦmζ,0w−w,⇒ψΦmζ,0w−w≤2ϕΦmζ,0w2−2ϕw2.
Letn⟶∞, we get that(27)limn⟶∞ψΦmζ,0w−w=0.
Due to the property ofψ, Φmζ,0w⟶w for each fix k∈N using the same argument, we have(28)Φkζ,kξUmξ,0w⟶Ukζ,0w,⇒Φkt,0w=w,or(29)w∈FΦkt,0.Definition 6.
Letℒ=Φζ,0;ζ≥0⊆U=Φζ,ξ;ζ≥ξ≥0, and tk⊂J, and ck⊂0,1. In the generalized GKMℒ,ck,tk iterative process generated by the evolution family ℱ, the iterative formula is defined by the following:(30)μk+1=ckΦζk,0μk+1−ckμk,where μ1∈B is chosen arbitrarily, (i) ck is bounded in 0 and 1. (ii) limk⟶∞ζk=∞. (iii) ∑n=1∞bζnμ<∞ for all μ∈B.Definition 7.
Iflimk⟶∞supatkμk=1, then the above process(31)GKMℒ,ck,tkis well defined.
The following lemmas are important for the Mann convergence theorems.Lemma 6.
LetBis a closed, bounded, and convex subset of a Banach spaceℜ. Letw∈Fℒ, andμkbe a sequence generated byGKMℒ,ck,tk. Then,∃an elementr∈ℝsuch thatlimk⟶∞μk−w=r.Proof 4.
Letw∈FΦζ,0. Then,(32)μk+1−w=ckΦζk,0μk+1−ckμk−w=ckΦζk,0μk+1−ckμk−ckw+ckw−w.
This implies that(33)μk+1−w=ckΦζk,0μk−ckΦζk,0w+1−ckμk−ckw+ckw−w≤ckΦζk,0μk−ckΦζk,0w+1−ckμk−ckw+ckw−w≤ck1+bζkwμk−w+1−ckμk−w≤ck1+bζkwμk−w+1−ckμk−w≤ckbζkwμk−w+μk−w≤bζkwdiamB+μk−w.It follows for each natural number n∈ℕ,(34)μk+n−w≤ck+1Φζk+1,0μk+1−w+1−ck+1μk+1−w≤ck+11+bζk+1μk+1−w+1−ck+1μk+1−w≤ck+1bζk+1μk+1−w+μk+1−w≤ck+1bζk+1diamB+μk+1−w≤ck+1bζk+1diamB+bζkwdiamB+μk−w≤∑i=kk+2−1bζidiamB+μk−w.
Now, for anyn∈ℕ,(35)μk+n−w≤μk−w+diamB∑i=kk+n−1bζiw.
Letμp−w=rp for every natural number p, we have(36)dk,n=diamterB∑i=kk+n−1bζiw.
We observe thatlimk⟶∞suplimn⟶∞supdk,n=0. By lemma 2, there exists an element r∈ℝ, such that limk⟶∞μk−w=r.Lemma 7.
LetBis a closed, convex, and bounded subset of a uniform convex Banach spaceℜ. Letμkbe a sequence generated by the generalized Krasnoselskii-Mann processGKMℒ,ck,ζk. Then,(37)limk⟶∞Φζtk,0μk−μk=0,and(38)limk⟶∞μx+1−μk=0.Proof 5.
LetFΦζ,0≠ϕ. For a fix element, w∈FΦζ,0. By lemma 6, ∃ an element r∈ℝ such that limk⟶∞μk−w=r, where w∈FΦζ,0, then it is true(39)limk⟶∞supΦζk,0μk−w=limk⟶∞supΦζk,0μk−Φζk,0w≤limk⟶∞supaζkμk−w=r.
We observe that(40)limk⟶∞ckΦζk,o−w+1−ckμk−w=ckvk+1−ckuk.
By Lemma3 apply for Φζk,o−w=vk and μk−w=uk(41)⇒limk⟶∞vk−uk=0,limk⟶∞Φζk,0μk−w−μk+w=0,limk⟶∞Φζk,0μk−μk=0.
Now, the construction of sequenceμk is equivalent to(42)limk⟶∞μk+1−μk=limk⟶∞ckΦζk,0μk+1−ckμk−μklimk⟶∞ckΦζk,0μk−μk⟶0⇒limk⟶∞μk+1−μk=0.
This completes the proof.
Now, we are to show the consequential lemma which proves that under the appropriate assumption of sequence,μk generated by the following:(43)GKMℒ,ck,ζk.
The iterative process to become an approximate fixed point sequence will give the final determination of proving the process for convergence.Lemma 8.
LetBbe a convex and bounded subset of the Banach spaceℜ. Let theGKMℒ,ck,ζkis well defined, andAis a subset ofNsuch that for alls∈A∃a strictly increasing sequencejk∈ℕwith the following conditions:(i)
μk−μjk⟶0 as. k⟶∞(ii)
limk⟶∞Φdk,0μjk−μjk=0, where dk=ζjk+1−ζjk−s
Then,μk is an approximate fixed points sequence for each bounded linear operators Φmζ,0 for each s∈A and m∈ℕ, that is,(44)limk⟶∞Φmζ,o.μk−μk=0.
IfA⊂N, then(45)limk⟶∞Φζ,0μk−μk=0,ζ∈N.Proof 6.
By Lemma4, it is sufficient to show that limk⟶∞Φmζ,0μk−μk=0 for m=1. To this end, let s∈A, and noted that μjk−μjk+1⟶0 as k⟶∞. We have(46)μjk−μjk+1≤μjk−μk+μk−μk+1+μk+1−μjk+1⟶0.
In view of Lemma7, we observe that
μjk−Φζ,0μjk⟶0 as k⟶∞.
Indeed,(47)μjk−Φζ,0μjk≤μjk−μjk+1+μjk+1−Φζjk+1,0μjk+1+Φζjk+1,0μjk+1−Φζjk+1,0μjk+Φζjk+1,0μjk−Φζ+ξjk,0μjk+Φζ+ξjk,0μjk−Φζ,0μjk≤μjk−μjk+1+μjk+1−Φζjk+1,0μjk+1+aζjk+1μjk+1μjk+1−μjk+aζ+ξjkμjkΦdk,0−μjk−μjk+supμ∈BaζμΦζjk,0μjk−μjk.which approaches to zero as k tends to ∞ because of Lemma 7 and the fact that the process is well defined, and by boundedness of at.
Also,(48)μk−Φζ,0μk≤μk−μjk+μjk−Φζjk,0μjk+Φζjk,0μjk−Φζ+ξjk,0μjk+Φζ+ξjk,0μjk−Φζ,0μjk+Φζ,0μjk−Φζ,0μk≤μk−μjk+μjk−Φζjk,0μjk+aζjkμjkμjk−Φζ,0μjk+asμjsΦζjk,0μjk−μjk+asμkμjk−μk,which approaches to zero where k⟶∞ by Lemma 7 the facts that the process is well defined because the evolution family is asymptotic pointwise nonexpansive. Also, A⊂N, so by Lemma 5, μk is a fixed point approximate sequence of Φζ,0.
Now, we prove the weak convergence of the sequenceμk and generate the GKMℒ,ck,ζk iteration process.Theorem 2.
IfB⊂ℜwhich is closed, convex, and bounded, with Opial property. Let the sequenceμk,GKMℒ,ck,ζkiteration process is well defined and is an approximate fixed point sequence for eachs∈A⊂N, then a sequenceμkweakly converges to the common fixed pointw∈FΦζ,0.Proof 7.
Letν,ω∈B, be two weak limit points of the sequence μk. Then, ∃ two subsequences νk and ωk of μk, such that νk⇀ν and ωk⇀ω. For any fix s∈A, and since μk is approximate fixed point sequence for s, it satisfies(49)limk⟶∞Φζ,0μk−μk=0.
By the Demiclosedness principle theorem, we haveΦζ,0ν=ν and Φζ,0ω=ω. Now, by Lemma 6, the following limits limk⟶∞μk−ν=r1 and limk⟶∞μk−ω=r2 exist. We claim that ν=ω. On the contrary base let ν≠ω, then by Opial property(50)r1=limk⟶∞infνk−ν<limk⟶∞infνk−ω=r2=limk⟶∞infωk−ω<limk⟶∞infωk−ν=r1,which is an absurd hence ν=ω. This shows that the sequence μk has mostly one weak limit point w∈B. Since B is weakly sequentially compact, therefore, μk has only one weak limit point, that is, μ∈B. Again, using the Demiclosedness principle, we get that Φζ,0μ=μ. Since s∈A is chosen to be arbitrary and by the construction of μ does not depend on s, where A⊂N, we conclude that Φζ,0μ=μ for each ζ∈J.
Now, we can use the result for some more distinct situations. First, we start the discrete case.Theorem 3.
IfB⊂ℜwhich is nonempty, convex, closed, and bounded with Opial property. Letℒbe an evolution family with discrete generating setA=a1,a2,a3,…, andGKMℒ,ck,ζkiterative process withm≤cardA. If for eachn∈ℕ,∃a quasiperiodic strictly increasing sequencejkmof natural numbers with quasiperiodpm, and for each natural numberk∈ℕ,ζjk+1m:=am+ζjkm. Then, the sequenceμkgenerated byGKMℒ,ck,ζkis the approximate fixed point sequence for eachs∈A⊂N, andμkweakly converges to a common fixed pointw∈Fℒ.Proof 8.
To prove the result, we will verify both the conditions of Lemma8. The condition ii of Lemma 8 is clearly satisfied because ζjk+1m−ζjkm−am=0. To show i we noted by quasiperiodicity of jkm for each natural number k, ∃jkm such that k−jkm≤pm. Assuming that k−pm≤jkm≤k. Fix ϵ>0. Noted that by Lemma 7, μk+1−μk<ε/pm for k. Therefore, for natural number k sufficiently large, it is true that(51)μk−μjk≤μk−μk−1+⋯+μjkm+1−μjkm≤pmεpm=ε.
This shows that (i) is also satisfied; hence, the sequenceμk generated by GKMℒ,ck,ζk is the approximate fixed point sequence for each s∈A⊂N. By Theorem 2, the sequence μk is weakly converges to the common fixed point μ∈ℒ.Remark 1.
We noted that Theorem 1 in [29] is a special case of Theorem3withA=1.



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