# A Strong Convergence to a Common Fixed Point of a Subfamily of a Nonexpansive Evolution Family of Bounded Linear Operators on a Hilbert Space

Authors: Gul Rahmat; Masaud Khan; Muhammad Sarwar; Hassen Aydi; Eskandar Ameer
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this article, we establish some results for convergence in a strong sense to a common fixed point of a subfamily of a nonexpansive evolution family of bounded linear operators on a Hilbert space. The obtained results generalize some existing ones in the literature for semigroups of operators. An example and an open problem are also given at the end.


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## Body

## 1. Introduction

In the 19th century, the fixed point theory was started by Poincaré [1]. In the 20th century, many mathematicians, such as Brouwer [2], Schauder [3], Tarski [4], and others, developed the field. 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 fields such as mathematics, engineering, physics, economics, and many more. Fixed point theory can be used as a tool to discuss the uniqueness and existence of solutions of many problems such as integral equations [5], differential equations [6, 7], and numerical equations and algebraic systems [8–11]. We refer to [12–19] for a more detailed study on fixed point theory and its applications in metric spaces.LetX≠∅, and τ:X⟶X be a self-mapping. The element a∈X is called a fixed point of τ if τa=a. Consider the autonomous system(1)y˙r=Ayr,r≥0,y0=Ω,where A is a linear operator on a Hilbert space ℋ. The solutions of such a system lead us to a class of linear and bounded mappings, called a semigroup. A family S=Ss:s≥0 of linear and bounded operators is called a semigroup if it satisfies the following two conditions:(1)
G0=I,the identity map(2)
Gs+t=GsGtfor alls,t≥0The system becomes more difficult if the operatorA depends on time, i.e., when A is replaced by At in the above system. Such a system is called nonautonomous, and its solution leads to the concept of an evolution family. Again, a family L=Ls,r of linear and bounded operators is said to be an evolution family, if the following hold:(1)
Ls,s=I, for all s≥0(2)
Ls,tLt,r=Ls,r, for all s≥t≥r≥0Remark 1 (see [20]).
Every semigroup is an evolution family, but the converse is not true in general. In fact, if an evolution family is periodic at every period, then it becomes a semigroup.
The study of fixed points for semigroups is studied by many mathematicians, such as Suzuki [21, 22] and Buthinah et al. [23]. They proved different results concerning a strong convergence to a fixed point of a semigroup and the representation of the set of all common fixed points of the semigroups in a form of intersection of the sets of all common fixed points of only two operators from the family. Such results are of too importance in the field. Recently, such results were generalized to a subfamily of an evolution family acting on different spaces, see [20, 24].
In this paper, we will present some new results for fixed points of an evolution family of operators. We also generalize some other results from semigroups [21], to a subfamily of an evolution family.


## 2. Preliminaries

In this article, we will frequently use the following notations:(1)
Byℝ, ℝ+, ℕ, and ℤ+, we will denote the set of all reals, nonnegative reals, natural numbers, and nonnegative integers, respectively.(2)
The semigroup, evolution family, and its subfamily will be denoted byS, L, and Ls, respectively.(3)
The set of all common fixed points of the semigroup, evolution family, and its subfamily will be denoted and defined asℱixG=∩s≥0ℱixGs, ℱixL=∩s≥r≥0ℱixLs,r, and ℱixLs=∩s≥0ℱixLs,0, respectively.(4)
ByC, we will denote a closed and convex subset of the Hilbert space ℋ.A mapτ:C⟶C is nonexpansive if τx−τy≤x−y for all x,y∈C.We denote byℱixτ the set of all fixed points of τ. If τ is a nonexpansive self-map on C, then ℱixτ is nonempty, see [25].For a fixedζ in C and ɛ∈0,1, there is a unique point xɛ∈C such that xɛ=1−ɛTxε+ɛζ. We see that the map x⟶1−ɛTx+ɛζ is a contraction. Indeed,(2)1−ɛTx+ɛζ−1−ɛTy+ɛζ=1−ɛTx−Ty.The nonexpansiveness ofT ensures that the map is a contraction.In 1967, Browder [26] provided the following result for self-mappings.Theorem 1.
Letτ be a self-mapping on C and γn∈0,1 be a sequence such that limn⟶∞γn=0. Then, for a fixed ζ in C, the sequence(3)ζn=1−γnτζn+γnζ.converges to a fixed point of τ nearest to ζ in a strong sense.
In this paper, we will prove a theorem of Suzuki [21] for a subfamily Ls of nonexpansive evolution operators on Hilbert spaces. Such a family needs not be a semigroup. The following example will illustrate this fact.Example 1.
The family defined byL=Ls,r=r+1/s+1:s≥r≥0 is clearly an evolution family acting on ℝ+. Since Ls,s=1 (the identity on ℝ+), and(4)Ls,tLt,r=t+1s+1r+1t+1=r+1s+1=Ls,r,by setting r=0, we have Ls=Ls,0=s+1 which is a subfamily of L, but not a semigroup.
However, if we put a condition as given in Remark1, then such a family becomes a semigroup.


## 3. Main Results

In this section, we will present our main results. The following lemma states that the set of all common fixed points of a semigroup can be represented on the closed unit interval in place ofℝ+.Lemma 1.
LetG=Gs:s≥0 be a semigroup on a Hilbert space ℋ, then(5)ℱixG=∩s≥0ℱixGs=∩0≤s≤1ℱixGs.Proof.
The following inclusion(6)∩s≥0ℱixGs⊆∩0≤s≤1ℱixGs,is obvious, and we will show the reverse inclusion.
Letu∈∩0≤s≤1ℱixGs, then Gsu=u for all 0≤s≤1. Let s≥0, then it can be written as s=n+ϱ, for some 0≤ϱ≤1 and some n∈ℤ+.
Now, consider(7)Gsu=Gn+ϱu=GnGϱu=Gn1Gϱu=Gn1u=u.
That is,u∈∩s≥0ℱixGs. Thus, we conclude that(8)∩s≥0ℱixGs=∩0≤s≤1ℱixGs.In [22], it is proved that the set of all common fixed points of a semigroup can be represented by the intersection of only two operators from the family.Theorem 2.
LetG=Gs:s≥0 be a semigroup on a Hilbert space ℋ, then
ℱixG=∩s≥0ℱixGs=ℱixGα∩ℱixGβ, where α and β are positive such that α/β is irrational.
Now, using Lemma1 and Theorem 2, we have the following corollary.Corollary 1.
LetG=Gs:s≥0 be a semigroup on a Hilbert space ℋ, then
ℱixG=∩s≥0ℱixGs=ℱixGα∩ℱixGβ, where α,β∈0,1, such that α/β is irrational.
Lemma1 can be extended to a subfamily Ls=Ls,0:s≥0 of a periodic evolution family. See the following lemma.Lemma 2.
LetLs=Ls,0:s≥0 be a subfamily of a periodic evolution family with period q∈ℝ+, then(9)∩s≥0ℱixLs,0=∩0≤s≤qℱixLs,0.Proof.
Since it is obvious that(10)∩s≥0ℱixLs,0⊆∩0≤s≤qℱixLs,0,we will again prove the reverse inclusion.
Let(11)u∈∩0≤s≤qℱixLs,0,then Ls,0u=u for all 0≤s≤q.
Now, since anys≥0 can be written as s=nq+ϱ, for some n∈ℤ+ and some 0≤ϱ≤q, we have(12)Ls,0u=Lnq+ϱ,0u=Lnq+ϱ,nqLnq,0u=Lϱ,0Lnq,0u=Lϱ,0u=u.
Hence,(13)∩s≥0ℱixLs,0=∩0≤s≤qℱixLs,0.This completes the proof.The Opial condition holds on every Hilbert space, given as follows.Proposition 1 (see [10]).
Ifβn is sequence in ℋ, converging to a point a∈ℋ in a weak sense, then(14)liminfn⟶∞βna≤liminfn⟶∞βn⟶∞,for ally∈ℋ.
The next theorem is about the strong convergence of a sequence to a point near to the fixed point of the subfamily of an evolution family.Theorem 3.
LetLs=Ls,0:s≥0 be a subfamily of strongly continuous evolution operators on C such that ℱixLs≠∅. Let γn∈0,1 and sn≥0 be two sequences of real numbers with the property that limn⟶∞sn=limn⟶∞γn/sn=0, (e.g., sn=1/n and γn=1/n2).
Then, for a fixedζ in C, the sequence(15)ζn=γnζ+1−γnLsn,0ζn,wheren∈ℕ,converges to an element of ℱixLs,0 nearest to ζ in a strong sense.Proof.
Lets be a point in ℱixLs nearest to ζ. From(16)ζn−s=γnζ+1−γnLsn,0ζn−s≤γnζ−s+1−γnlsn,0ζn−s≤γnζ−s+1−γnζn−s,we find that(17)Lsn,0ζn−s≤ζn−s≤ζ−s,forn∈ℕ.
Therefore,ζnandLsn,0ζn both are bounded. Let ζni be any arbitrary subsequence of ζn, then there exists a subsequence of ζni (say ζnij) which converges to x in a weak sense. Our claim is that x∈ℱixLs.
For this, putωj=ζnij,zj=γnij,q=t/tj,andtj=rnij, for n∈ℕ. Fix r>0. One writes(18)ωj−Ls,0x≤∑k=0q−1Lk+1tj,0ωj−Lktj,0ωj+Lqtj,0ωj−Lqtj,0x+Lqtj,0x−Ls,0x≤qLtj,0ωj−ωj+ωj−x+Lt−qtj,0x−x=qzjLtj,0ωj−ζ+ωj−x+Lt−tjq,0x−x≤tzjtjLtj,0ωj−ζ+ωj−x+maxLtj,0x−x:0≤t≤tj,for j∈ℕ. In above inequality, the first and last terms tend to zero as j⟶∞, so(19)liminfj⟶∞ωj−Ls,0x≤liminfj⟶∞ωj−x.
By the Opial condition and Proposition1, we get Ls,0x=x, and therefore, x∈ℱixLs.
Lastly, we will show thatωj converges to s in a strong sense. From(20)ωj−Ltj,0ωj−s−Ltj,0s,ωj−s≥ωj−s2−Ltj,0ωj−Ltj,0s⋅ωj−s,zjωj−s2+1−zjωj−Ltj,0ωj−s−Ltj,0s,ωj−s=zjζ−s,ωj−s,we conclude that(21)zjωj−s2≤zjζ−s,ωj−s.
That is,(22)ωj−s2≤ζ−s,ωj−s.
Sinceζ is nearest to s, we can write ζ−s,x−s≤0,(23)ωj−s2≤ζ−s,ωj−s=ζ−s,ωj−x+ζ−s,x−s≤ζ−s,ωj−x,for j∈ℕ. We see that ωj converges to s in a strong sense. As ωj=ζni is arbitrary, we obtain that ζn converges to s in a strong sense.Remark 2.
Here, we mention that the above result is not applicable for a discontinuous family, see [21].Remark 3.
If we put the condition of periodicity of every positive real number on the evolution family, then it becomes a semigroup using Remark1. So, the results in [21] become a special case of this paper.


## 4. Example and Open Problem

Example 2.
Letℋ:=L20,π,ℂ be the Hilbert space and let S=Ss:s≥0 be a semigroup defined by(24)Ssvt=2π∑n=1∞e−sn2cnvsinnt,t∈0,π,s≥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(25)∂ut,ξ∂t=ht∂2ut,ξ∂2ξ,t>0,ξ∈0,π,ut,0=ut,π=0,t≥0,u0,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.
LetHt=∫0thtdt. It is obvious that the solution x⋅ of the above Cauchy problem will satisfy the evolution property:(26)xt=Lt,sxs,where Lt,s=SHt−Hs. See [22, Example 2.9b].
We can findν≥0 such that the function t↦eνtut is bounded on ℝ+. In fact, we have(27)∫0∞Lt,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,(28)∫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,(29)∫0∞‖Lt,0v‖2dt≤Cv22.
Using Theorem 3.2 in [27], we have ω0L≤−1/2M, where M≥1 and ω0L is the growth bound of the family L, and see [27] for further details. This shows that the evolution family is nonexpansive on ℋ, so Theorem 3 can be applicable for such a family and can help for the uniqueness and existence of a solution for the above system.
Open problem: we leave open the question whether Lemma2 and Theorem 3 can be generalized for the whole periodic and then for general evolution families?


## 5. Conclusion

The idea of an evolution family is more general than the semigroups. In [21], Suzuki proved a strong convergence to a fixed point of a nonexpansive semigroup of operators on a Hilbert space. In this paper, we generalized the results to a subfamily of an evolution family which is not a semigroup. These results can open the way for researchers to prove such convergence for the whole evolution family of operators on a Hilbert space.



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