# Strong Convergence of Krasnoselskii–Mann Process for Nonexpansive Evolution Families

Authors: Musharaf Shah; Gul Rahmat; Syed Inayat Ali Shah; Zahir Shah; Wejdan Deebani
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This article is about investigating Krasnoselskii and Mann’s convergence theorems for nonexpansive evolution family with Bochner integral and without convexity of Banach space. We introduced a general form of these results that extends these results to general form of semigroup called evolution families.


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## Body

## 1. Introduction

In the 19th century, the fixed point theory was started by Poincaré, followed by many other mathematicians like Brouwer [1], Schauder [2], and Tarski [3] in the 20th century. The fixed point theory has a wide range of applications. It is one of the most important tools of modern mathematical analysis and is useful in various field like mathematics, engineering, physics, statistics, economics, and many more. Fixed point theory can be used as a tool to discuss the uniqueness and existence of solutions of many problems like integral equations, differential equations, numerical equations, and algebraic systems. We refer to [4] for more detailed study of fixed point theory and its applications in metric spaces.Throughout the paper, we will denote byH a Banach space and let B be a nonempty, closed, and convex subset of a Banach space H. We recall that an operator Tμ:μ≥0 on B is called a nonexpansive mapping if Tμx−Tμy≤x−y for all x,y∈B. The collection of all fixed points of T is denoted by FT. Let B be a compact subset of H; then, Schauder’s fixed point theorem [5] says that FT is nonempty. For details, we refer to [6–10] and others.Now we will again recall an important family which is known to a semigroup of linear operators onH. A family of mappings S:=Sμ:μ≥0 is called a nonexpansive strongly continuous semigroup if it satisfies the following conditions:(i)
S0=I, where I is the identity operator on H.(ii)
Sμ+ν=SμSν, for all μ,ν≥0.(iii)
For eachx∈H, the map μ⟶Tμx is strongly continuous. Such families arise from the solution of autonomous differential system: x˙t=Axt.The study of fixed point of a semigroup got attention in the last four decades. For nonexpansive semigroups, some theorems [11] and corollary [12] are given below in details.Theorem 1 (see [12]).
LetS=Sμ:μ≥0 be a one-parameter strongly continuous semigroup of nonexpansive mapping on B. Let an∈0,∞ and αn∈0,1 be two sequences such that limn⟶∞αn=0 and limn⟶∞an=∞. For u∈B, define a sequence un in B by(1)un=1−αn1an∫0tnTsunds+αnu,for n∈ℕ. Then, the sequence un converges strongly to a common fixed point of S.
Krasnoselskii and Mann’s convergence theorems for nonexpansive mappings have been studied by many researchers (see, e.g., [13–21] and the references therein). Recently, the following Krasnoselskii and Mann’s convergence theorem for nonexpansive semigroups has also been proved by Takahashi in [22].Theorem 2 (see [22]).
LetS=Sμ:μ≥0 be a one-parameter strongly continuous semigroup of nonexpansive mapping on B. Define a sequence xn in B by z1∈B.(2)zn+1=αnan∫oanSsznds+1−αnzn.
For every natural numbern, where αn∈0,1 and an∈0,∞ are two sequences that satisfy 0<liminfn⟶∞αn≤limsupn⟶∞αn<1 and limn⟶∞an=∞, limn⟶∞an+1/an=1. Then, the sequence zn strongly converges to a common fixed point of S.
Now we proved the following Browder’s type convergence theorem for nonexpansive evolution family without using the Bochner integral.Theorem 3 (see [23]).
LetB be a nonempty and convex subset of complete inner product space H. Let Ea,0a≥0 be a one-parameter strongly continuous evolution family of nonexpansive mapping on B. Let σ∈ℝ+andαn,an be sequences of real numbers satisfying the conditions 0<αn<1,0<σ+an, and an≠0 for natural number n∈ℕ,andlimn⟶∞an=limn⟶∞αn/an=0. For ordering u∈B, prescribe a sequence un∈B nearby(3)un=1−αnEτ+an,0un+αnu,for every natural number n. Then, the sequence un strongly converges to the member of Ea,0:a≥0.
In [13], Suzuki discussed the convergence of Krasnoselskii and Mann type convergence theorems for nonexpansive semigroup except the Bochner integral and without considering the strictly convexity of Banach space. In the current article, we introduced a general form of these results that extends these results to general form of semigroup called evolution family. For other such type of recent results, we refer to [24–26].
A familyU=Eμ,ν:μ≥ν≥0 is called a nonexpansive strongly continuous evolution family on B if it satisfies the following conditions:(i)
For everya≥0:Ea,0 is a nonexpansive mapping on B.(ii)
Eμ,μx=Ix.(iii)
Eμ,λEλ,ν=Eμ,ν∀μ≥ν≥0.(iv)
For everyx∈B, the mapping a⟶Ea,0x from 0,∞ into B is strongly continuous. Such families arise from the solution of nonautonomous differential system: x˙t=Atxt. It is obvious that evolution family is a generalized form of a semigroup.
The setFEa,0 represents all common fixed points of Ea,0:a≥0 and is denoted by FEa,0=∩a≥0FEa,0. As B is compact, then FEa,0 is nonempty (for details, see [8, 27–30]).


## 2. Preliminaries

Throughout this paper,ℕ, ℤ, ℚ, and ℝ are used for natural numbers, integers, rational numbers, and real numbers, respectively. For any real numbers r, we represent r as the greatest integer not greater than r. Obviously, for any positive real numbers n and m,(4)0≤m−mnn<n.It is true foru,v∈ℤ, and we substitute(5)ℤu,v=j∈ℤ:u≤j≤v.The following lemma is very necessary for our main work.Lemma 1.
Letan be a real sequence and τ∈ℝ satisfy liminfn⟶∞an≤σ≤limsupn⟶∞an. Furthermore, let the following be true:(1)
limsupn⟶∞an+1−an≤0.(2)
liminfn⟶∞an+1−an≥0.
Then, letσ be limit point of sequence an. Furthermore, considering that ϵ>0 and m,k are any positive integers, there exist m0≥msuchthataj−σ<ϵ for each j in ℕ as well m0≤j≤m0+k.
Before the proof of this lemma, we have the following remark.Remark 1.
Statement (1) is comparable tolimsupn⟶∞an+1−an=0 since there is no element σ in ℝ with liminfn⟶∞an≤σ≤limsupn⟶∞an in the case of limsupn⟶∞an+1−an<0. It is the same for statement (2).
Now we will prove our lemma.Proof.
Let the sequencean be convergent; then, the conclusion is clearly true. So, on the contrary basis, let the sequence an be not convergent, i.e., liminfn⟶∞an<limsupn⟶∞an, and we fix ϵ>0 and r,q∈ℕ. Also, we fix a∈ℝ with a−τ<ϵ/2 and liminfn⟶∞an<a<limsupn⟶∞an. Then, there exists δ in ℝ with 0<δϵ/2 and(6)liminfn⟶∞an<a−δ<a+δ<limsupn⟶∞an.
By the hypothesis, for any natural numbersp1,q such that p1>m and an+1−an<δ/r for each p≥p1, we also consider p2,p3∈ℕ such that p3>p2>p1.
Put(7)q0=maxp:p<p3,an≤a−δ+1>p2>p1.
Sinceaq0−1≤a−δ, we get for j∈ℤq0,q0+r,(8)aj=aq0−1+∑l=q0jal−al−1<aq0−1+∑l=q0jδr≤aq0−1+δr+1r≤a−δ+2δ=a+δ.
Asj<p3, then using q0, we get a−δ<aj.
Therefore,aj∈a−δ,a+δ⊂a−ϵ/2,a+ϵ/2 for j∈ℤq0,q0+r.
So, we have(9)aj−σ≤aj−a+a−σ<ϵ2+ϵ2=ϵfor j∈ℤq0,mq+r.
Now we will discuss Krasnoselskii and Mann’s sequences. Letxn and wn be two sequences in a Banach space H which satisfies(10)xn+1=αnwn+1−αnxn.
For every natural numbern, a sequenceαn in 0≤αn≤1. We say such a sequencesxn and wnKrasnoselskii and Mann’s sequence, for more detail, see [31, 32].Lemma 2 (see [21]).
Letxn and wn be two bounded sequences in a Banach space H and let αn be a sequence in 0,1 with the condition that 0<liminfn⟶∞αn≤limsupn⟶∞αn<1. Let(11)xn+1=αnwn+1−αnxn,for every n∈ℕ and(12)limsupn⟶∞wn+1−wn−xn+1−xn≤0.
Then,(13)limn⟶∞wn−xn=0.
Using Lemma2, we have the following lemma for evolution equations.Lemma 3.
LetH be a Banach space and Enμ,0 be a sequence of nonexpansive mappings on B. Defining aznbe a sequence in Bbyz1∈B(14)zn+1=αnEnμ,0zn+1−αnzn.
For every natural numbernsuch that αnn∈0,1along.
0<liminfn⟶∞αn≤limsupn⟶∞αn<1.Letznis bounded or equivalent to Enμ,0znis bounded, and that either,(15)limn⟶∞En+1μ,0zn−Enμ,0zn=0,0r,limn⟶∞En+1μ,0zn+1−Enμ,0zn+1
Holds. Then.limn⟶∞Enμ,0zn−zn=0.Proof.
We put forM=supzn<∞, anda=liminfn⟶∞αn/2>0. Then for a natural numbern1where0<a<αn. For n≥n1, we have(16)Enμ,0zn=1αnzn+1−1−αnzn,≤1αnzn+1+1−αnαnzn≤2aM.
Hence,Enμ,0zn is bounded. In the case limn⟶∞En+1μ,0zn−Enμ,0zn=0, we have(17)limsupn⟶∞En+1μ,0zn+1−Enμ,0zn−zn+1−zn≤limsupn⟶∞=‖En+1μ,0zn+1−En+1μ,0zn‖+En+1μ,0zn−Enμ,0zn−zn+1−zn≤limsupn⟶∞‖zn+1−zn‖+En+1μ,0zn−Enμ,0zn−zn+1−zn=limsupn⟶∞En+1μ,0zn−Enμ,0zn=0.
Similarly, we can prove thatlimn⟶∞En+1μ,0zn+1−Enμ,0zn+1=0, and we have(18)limsupn⟶∞En+1μ,0zn+1−Enμ,0zn−zn+1−zn≤limsupn⟶∞En+1μ,0zn+1−Enμ,0zn+1+Enμ,0zn+1−Enμ,0zn−zn+1−zn≤limsupn⟶∞En+1μ,0zn+1−Enμ,0zn+1+‖zn+1−zn‖−zn+1−zn=limsupn⟶∞En+1μ,0zn+1−Enμ,0zn+1=0.
Using Lemma2, we have(19)limn⟶∞Enμ,0xn−xn=0.
In both cases, it is proved.Remark 2.
(see [21]). We will prove the boundedness of the sequencezn and Enμ,0znare equivalent. Also, we have shown that the sequence Enμ,0zn is bounded when the sequence zn is bounded.
Conversely, ifEnμ,0zn is bounded, put M=SupEnμ,0zn+z1<∞. We have to prove zn<M for every natural number n. We suppose that zn≤M; then, we get(20)zn+1≤αnEnμ,0zn+1−αnzn≤αnM+1−αnM=M.
So, by induction, we havezn≤M and hence zn is a bounded sequence.Lemma 4.
LetEnμ,0 be a sequence of nonexpansive mappings on B. Defining a sequence zn in B by z1∈B,(21)zn+1=αnEnμ,0zn+1−αnzn.
For every natural numbern and αn∈0,1along, 0<liminfn⟶∞αn≤limsupn⟶∞αn<1.
Letzn be bounded or Enμ,0zn be bounded and limn⟶∞En+1μ,0z−Enμ,0z=0. Then, limn⟶∞Enμ,0zn−zn=0.


## 3. Main Result

In this section, we will present our main results.Theorem 4.
LetH be a Banach space and B be a compact convex subset H. Let Ea,0a≥0 be a one-parameter strongly continuous evolution family of nonexpansive mapping on B. Let an∈0,∞ be a sequence satisfying the following conditions: liminfn⟶∞an<limsupn⟶∞an and limn⟶∞an+1−an=0. Introduce zn a sequence in B by z1∈B with λ∈0,1.(22)zn+1=λEan,0zn+1−λzn,for every natural number n. Then, zn converges strongly to a common fixed point of Ea,0:a≥0.Proof.
First of all, we have to prove that(23)limsupn⟶∞Ean+1,0z−Ean,0z=0.
On the contrary, let (23) be not true; then, there exist a subsequenceank of an and a sequence bk in B and ζ>0 such that(24)Eank+1,0bk−Eank,0bk≥ζ,for every natural number k. Since B is compact, there exists a subsequence of bk which is strongly converged. Without loss of generality, we can assume that bn is self-strongly converged to some of the points of w∈B.
We get(25)0<ζ≤limsupn⟶∞Eank+1,0bk−Eank,0bk≤limsupn⟶∞Eank+1−ank,0bk−E0,0bk≤limsupn⟶∞Eank+1−ank,0bk−Eank+1−ank,0w+Eank+1−ank,0w−E0,0w+E0,0w−E0,0bk≤limsupn⟶∞2bk−w+Eank+1−ank,0w−E0,0w=0,which is a contradiction. Therefore, (23) is true. So, by using Lemma 4, we have(26)limn⟶∞Ean,0zn−zn=0.
We have(27)limsupn⟶∞E0,0zn−zn≤limsupn⟶∞E0,0zn−Ean,0zn+Ean,0zn−zn=limsupn⟶∞E0,0zn−E0,0Ean,0zn+Ean,0zn−zn≤limsupn⟶∞2Ean,0zn−zn=0.
We fix forδ,τ∈ℝ with(28)liminfn⟶∞an<τ<τ+δ<limsupn⟶∞an.
We considerni which is a subsequence of n with the condition(29)limi⟶∞Eani,0zni−zniani−τ=limi⟶∞Eani,0zni−zni=0,limi⟶∞ani=τ.
τ<ani for every natural i∈ℕ. From equation (24), there exists m1∈ℕ such that Ean,0zn−zn<1/32 for every n≥m1.
Using the result of Lemma1, we have that τ+δ/2 is a limit point of sequence an. Hence, there exists n1 greater than m1 such that τ+δ/3<an1<τ+δ. Again from (24), there exists m2>n1 such that Ean,0zn−zn<1/42 for each n≥m2. Again using Lemma 1, we note that τ+δ/3 is a limit point of an. Therefore, there exists n2≥m2 such that τ+δ/4<an2<τ+δ/2.
To continue this process, we defined a subsequenceni of n which satisfies Eani,0zni−zni<1/i+22 and τ+δ/i+2<ani<τ+δ/i for all i∈ℕ.
Then,τ<ani for each i∈ℕ,limi⟶∞ani=τ and limi⟶∞Eani,0zni−zni=0. Also, we have(30)limsupi⟶∞Eani,0zni−zniani−τ≤limsupi⟶∞1/i+22δ/i+2=limi⟶∞1δi+2=0.
SinceB is compact, there exists a subsequence znij ofzni such that znij strongly converges to z∈B.
Put forbj=znij∈B and sj=anij∈0,∞ for every natural number j. We have(31)limsupj⟶∞bj−Eτ,0z≤limsupj⟶∞bj−Esj,0bj+Esj,0bj−Esj,0z+Esj,0z−Eτ,0z≤limj⟶∞bj−Esj,0bj+bj−z+Esj−τ,0z−E0,0z=0.
Hence, the sequencebj converges to Eτ,0z. This implies that Eτ,0z=z.
Also, we get(32)E0,0z=Eτ,0E0,0z=Eτ,0z=z.
Fixa>0; then, for every natural number j with sj−τ<a, we get(33)bj−Ea,0x=bj−Ea,0Eτ,0asj−τx=bj−Easj−ττ+a,0x≤bj−E0,0bj+∑k=0a/sj−τ−1Ek+1sj,0bj−Eksj,0bj+Easj−τsj,0bj−Easj−τsj,0x+Easj−τsj,0x−Easj−ττ+a,0x≤bj−E0,0bj+asj−τEsj,obj−bj+bj−x+Ea−asj−τsj−τ,0x−E0,0x‖≤bj−E0,0bj+Esj,0bj−bjsj−τ+bj−x+maxEs,0x−E0,0x:for0≤s≤sj−τ.
Hence,limj⟶∞bj−Ea,0x=0. This implies Ea,0x=x. This shows that x is a common fixed point of Ea,0:a≥0. Also, x is a limit point of zn, and we get liminfn⟶∞zn−x=0. But since(34)zn+1−x≤λEan,0zn−x+1−λzn−x=λEan,0zn−Ean,0x+1−λzn−x≤λzn−x+1−λzn−x=zn−x,for every natural number n, we have to obtain limn⟶∞zn−x=0.
The following is another convergence theorem which has been proved in [33]; proof can also be seen in [34].Theorem 5 (see [33]).
LetEt,0:t≥0 be a strongly continuous evolution family of nonexpansive mappings on B. α,β∈ℝ+ satisfying α/β which does not belong to ℚ.. Then,(35)∩t≥0FEt,0=FEα,0∩Eβ,0holds.
In [21], the convergence theorem of infinite families for nonexpansive mappings without considering the strictly convexity has been proved by the author. Now we give alternate proof by using arguments to Theorem 4, firstly presenting the corollary related to the main theorem [21].Theorem 6 (see [33]).
LetEt,0 and Es,0 be nonexpansive mappings on B with Et,0Es,0=Es,0Et,0. Let λ∈0,1 be also a sequence αn∈0,1 satisfying liminfn⟶∞αn=0, limsupn⟶∞αn>0, and limn⟶∞αn+1−αn=0. Introduce a sequence zn in B by z1∈B and(36)zn+1=1−αnλEs,0zn+αnEt,0zn+1−λzn,for all n∈ℕ. Then, the sequence zn strongly converges to a common fixed point of Es,0 and Et,0.Proof.
LetM=supz∈Bz. We have(37)limsupn⟶∞supz∈B1−αn+1Es,0z+αn+1Et,0z−1−αnEs,0z+αnEt,0z≤limsupn⟶∞supz∈B‖1−αn+1Es,0z−1−αnEs,0z‖+1−αn+1Et,0z−αnEt,0z=limsupn⟶∞supz∈Bαn+1−αnEs,0z+αn+1−αnEt,0z=limn⟶∞2Mαn+1−αn=0.
Sincez↦1−αEs,0z+αEt,0z is a nonexpansive mapping on B for all α∈0,1, we get(38)limn⟶∞1−αnEs,0zn+αnEt,0zn−zn=0.
By Lemma4, for a real number a, with liminfn⟶∞αn=0<a<limsupn⟶∞αn, we consider that ni is a subsequence of n which satisfies(39)limi⟶∞1−αniEs,0zni+αniEt,0zni−zniαni=limi⟶∞1−αniEs,0zni+αniEt,0zni−zni=0.
limi⟶∞αni=0, and αni>0 for each i∈ℕ. From equation (25), there exists m1∈ℕ such that 1−αnEs,0zn+αnEt,0zn−zn<1/32 for each n≥m1. Using Lemma 1, we noted that α/2 is limit point of αn. Therefore, there exists n1>m1 such that α/3<αn1<a. Again from (25), ∃m2>n1 such that 1−αnEs,0zn+αnEt,0zn−zn<1/42 for each n≥m2.
Again by Lemma1, we noted that α/3 is a limit point of αn. Therefore, ∃n2 is greater than m2 such that a/4<αn2<a/2. By pursuing this method, we describe a subsequence ni of n satisfying(40)1−αniEs,0zni+αniEt,0zni−zni<1i+22,and a/i+2<αni<a/i for each i∈ℕ. Obviously, αni>0 for every i∈ℕ,limi⟶∞αni=0 and(41)limi⟶∞1−αniEs,0zni+αniEt,0zni−zni=0.
We have(42)limsupi⟶∞1−αniEs,0zni+αniEt,0zni−zniαni≤limsupi⟶∞1/i+22a/i+2,=limi⟶∞1ai+2=0.
But sinceB is compact, there exists a subsequence znij of zni such that znij strongly converges to some z∈B.
Letbj=znij∈B and βj=αnij∈0,1 for each j∈ℕ. We get limsupj⟶∞bj−Es,0x.(43)≤limsupj⟶∞bj−1−βjEs,0bj−βjEt,0bj+1−βjEs,0bj+βjEt,0bj−1−βjEs,0x−βjEt,0x+1−βjEs,0x+βjEt,0x−Es,0x≤limsupj⟶∞bj−1−βjEs,0bj−βjEt,0bj+bj−x+βjEt,0x−Es,0x≤limj⟶∞bj−1−βjEs,0bj−βjEt,0bj+bj−x+2βjM=0.
Hence, the sequencebj converges to Es,0x. This implies that Es,0x=x. So, we get Et,0x=Et,0Es,0x=Es,0Et,0x=x. We have Et,0x−bj.(44)≤Et,0x−1−βjEs,0bj−βjEt,0bj+1−βjEs,0bj+βjEt,0bj−bj≤1−βjEt,0x−Es,0bj+βjEt,0x−Et,0bj+1−βjEs,0bj+βjEt,0bj−bj=1−βjEs,0Et,0x−Es,bj+βjEt,0x−Et,0bj+1−βjEs,0bj+βjEt,0bj−bj≤1−βjEt,0x−bj+βjx−bj+1−βjEs,0bj+βjEt,0bj−bj,for every j∈ℕ, and therefore(45)limsupj⟶∞Et,0x−bj≤limsupj⟶∞x−bj+1−βjEs,0bj+βjEt,0bj−bjβj=0.
This implies that the sequencebj converges to Et,0x. So, we get Et,0x=x. Hence, x is a common fixed point of Es,0 and Et,0. Since x is a limit point of sequence zn, we have liminfn⟶∞zn−x=0.
Since(46)zn+1−x≤λ1−αnEs,0zn−x+λαnEt,0zn−x+1−λzn−x=λ1−αnEs,0zn−Es,0x+λαnEt,0zn−Et,0x+1−λzn−x≤λ1−αnzn−x+λαnzn−x+1−λzn−x=zn−x,for every natural number n, we get limn⟶∞zn−x=0.
By combining Theorems5 and 6, we get the following theorem.Theorem 7.
LetB be a compact convex subset of a Banach space H, and let Et,0:t≥0 be a one-parameter strongly continuous evolution family of nonexpansive mappings on B. Let τ, σ∈ℝ+ satisfying σ/τ which does not belong to ℚ. Let λ∈0,1, and let a sequence αn∈0,1 that satisfies liminfn⟶∞αn=0, limsupn⟶∞αn>0, and limn⟶∞αn+1−αn=0. Introduce a sequence zn in B by z1∈B and(47)zn+1=1−αnλEσ,0zn+αnλEτ,0zn+1−λzn,for all n∈ℕ. Then, the sequence zn converges strongly to a common fixed point of Et,0:t≥0.Remark 3.
If an evolution family is periodic of every positive real number, then it becomes a semigroup [25]. In such a case, the results given in [13] become special case of our results.


## 4. An Application

In this section, we are going to present an example which will illustrate the importance of evolution equations as well as our theoretical results.Example 1.
Letℋ:=L20,π,ℂ be the Hilbert space and let S=Sμ:μ≥0 be a semigroup defined by Sμvt=2/π∑n=1∞e−μn2cnvsinnt,t∈0,π,μ≥0, where cnv:=∫0πxssinnsds. Clearly, it is a strongly continuous and nonexpansive semigroup on ℋ, and it is generated by the linear operator A given by Av=v¨ and the maximal domain of A is the set DA of all x∈ℋ such that v and v˙ are absolutely continuous, v¨∈ℋ, and v0=vπ=0.
Now, consider the nonautonomous Cauchy problem(48)∂ut,ξ∂t=ht∂2ut,ξ∂2ξ,t>0,ξ∈0,πut,0=ut,π=0,t≥0u0,ξ=zξ,where z·∈ℋ and the function h:ℝ+⟶1,∞ is nonexpansive on ℝ+ and obeys the periodicity condition, i.e., ht+q=ht for all t∈ℝ+ for some q≥1. Let Ht=∫0thtdt. It is obvious that the solution x· of the above Cauchy problem will satisfy the evolution property:(49)xt=Et,sxs,where Et,s=SHt−Hs (see [13], example [2.9b]).
We can findν≥0 such that the function t↦eνtut is bounded on ℝ+. In fact, we have(50)∫0∞Et,0v2dt=2π∫0∞∑ν=1∞cν2ve−2ν2Htdt=2π∑ν=1∞cν2v∫0∞e−2ν2Htdt=v22∫0∞e−2ν2Htdt≤v22∫0∞e−2Htdt.
On the other hand,(51)∫0∞e−2Htdt=∑j=0∞∫jqj+1qe−2Htdt=∑j=0∞∫0qe−2Hjq+ada=∑j=0∞e−2jHq∫0qe−2Hada≤q∑j=0∞e−2jHq=qe2Hqe2Hq−1=C.
Hence,(52)∫0∞Et,0v2dt≤Cv22.
Using Theorem 3.2 in [35], we have ω0L≤−1/2M, where M≥1 and ω0E is the growth bound of the subfamily E (see [35] for further details). This shows that the evolution family is nonexpansive on ℋ, so Theorem 4 can be applicable for such a family and can help for the uniqueness and existence of a solution for the above system.


## 5. Conclusion

The idea of an evolution family is more general than the semigroups. In [13], Suzuki proved a strong convergence of Krasnoselskii and Mann type sequence to a common fixed point of a nonexpansive semigroup of operators on a Banach space H. We generalized the results to a subfamily of an evolution family, and these results can open the way for researchers to prove such convergence for the whole evolution family of operators on a Banach space.



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*Source: 3007572-2022-10-13.xml*