# Common Fixed Points of a Subfamily of a Nonexpansive Periodic Evolution Family on a Strictly Convex Banach Space

Authors: Gul Rahmat; Tariq Shah; Muhammad Sarwar; Hassen Aydi; Habes Alsamir
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this study, we establish some results for strong convergence of a sequence to a common fixed point of a subfamily of a nonexpansive and periodic evolution family of bounded linear operators acting on a closed and bounded subsetJ of a strictly convex Banach space X. In fact, we generalized the results from semigroups of the operator to an evolution family of operators.


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## Body

## 1. Introduction

Consider the autonomous Cauchy problem(1)Ω˙a=ZΩa,a≥0,Ω0=Ω0,where Z is a matrix of order α with complex entries. The solution of such systems leads to the idea of a semigroup. Note that a family T=Ta:a≥0 of mappings on X is referred to a semigroup if it satisfies these two conditions: (i) T0=I and (ii) Ta+s=TaTs for all a,s≥0.Similarly, consider the following nonautonomous system:(2)Ω˙r=ZaΩa+eiϱaI,a≥0,Ω0=Ω0,where Za is a matrix of order α. The solution of such families leads to the idea of an evolution family. A family V=Va,s:a≥s≥0 of mappings on X is said to be an evolution family if it satisfies the following two conditions:(i)
Va,a=I, for all a≥0(ii)
Va,sVs,l=Va,l, for all a≥s≥l≥0The family is strongly continuous ifVa,sτ is strongly continuous for all τ∈X.Also,V is known to be ζ-periodic if there exists ζ≥0, such that Va+ζ,s+ζ=Va,s for all a≥s≥0.The following lemma shows that an evolution family is a generalized form of a semigroup.Remark 1.
An evolution family becomes a semigroup if it is periodic of every numberζ≥0. If we define Ta=Va,0, then clearly T0=V0,0=I and Ta+s=Va+s,0=Va+s,sVs,0=TaTs, which show that the evolution family becomes a semigroup.
Also, a semigroup is an evolution family; if we takeVt,s:=Tt−s, then Vt,t:=T0=I and Vt,sVs,r=Tt−sTs−r=Tt−r=Vt,r, for all t≥s≥r≥0.Remark 2.
For a periodic evolution family with periodζ>0, the following relation holds:(3)Umζ,0=Uζ,0m,∀m∈N.
In fact,(4)Umζ,0=Umζ,m−1ζUm−1ζ,m−2ζ,…,Uζ,0=Uζ,0Uζ,0,…,Uζ,0=Umζ,0.


## 2. Preliminaries

In this study, we will use the following notations. The set of real numbers, nonnegative real numbers, and natural numbers are denoted byℝ, ℝ+ and ℕ, respectively. We represent a strictly convex Banach space by X, and by J, we mean a closed convex and bounded subset of X. By V (respectively, L), we will denote a ζ periodic and nonexpansive evolution family (respectively, subfamily).The set of all fixed points of the mappingVd,b and the family V=Va,s:a≥s≥0 are represented by ℨVd,b and ℨV, respectively, where ℨV=∩a≥s≥0ℨVVd,b.Definition 1.
A mapT is said to be nonexpansive on J, if for all τ,ν∈J, we have Tτ−Tν≤τ−ν.
IfT is nonexpansive on J, then it has a fixed point when X is uniformly convex [1–3].
It is well-known that ifV is uniformly convex and J is bounded; then, L=Va,0:a≥0 has a common fixed point [2, 4]. Bruck [5] proved the next theorem.Theorem 1.
LetX be a Banach space and J be a closed convex subset of X. Assume that J is bounded, separable, and compact in a weak sense, satisfying the property of fixed point such that for nonexpansive operators, the property of fixed points holds in J. Then, for any commuting family M of nonexpansive operators on J, the set of fixed points of M is a nonempty nonexpansive retract of J.This theorem yields that ifJ is weakly compact, or bounded and separable, and satisfying the fixed point property, then the family L=Va,0:a≥0 has a common fixed point. Many researchers have studied the convergence of nonexpansive semigroups to a fixed point [4, 6–9].Theorem 2.
(Shioji and Takahashi [10]). Let ℌ be a Hilbert space and J be a closed convex bounded subset of ℌ. Suppose that Va,0:a≥0 is a family of continuous and nonexpansive mappings on J in a strong sense. Let βm and rm be sequences of real numbers satisfying limm⟶∞βm=0 and limm⟶∞rm=∞, where 0<βm<1 and rm>0. Let y∈J be fixed and τm be a sequence in J defined by(5)τm=1−βmmζ∫0mζVs,0τmds+βmy,for m∈N. Then, the sequence τm converges to z∈ℨL in a strong sense.The next result was proved by Suzuki [11].Theorem 3.
Considerℌ, J, and Va,0:a≥0 be as in (10). Let 0<βm<1 and rm>0 with limm⟶∞am=limm⟶∞βm/am=0 and assume that y∈J is fixed and τm is a sequence in J defined by(6)τm=1−βmVa,0τm+βmy,for m∈ℕ. Then, τm converges to a fixed point of Va,0:a≥0 in a strong sense.Since the real sequencesam converge to 0 and ∞, we conclude that am is a constant sequence. These kinds of convergence theorems are considered to be a fixed point of Va,0:a≥0.In this study, we generalize the results given in [6] to an evolution family. For other recent studies of fixed points of an evolution family, refer to [12–14]. The study of fixed points has a large number of applications in differential equations, integral equations, and numerical equations. For more details, refer [15–17]. We also denote by X∗ the dual space of X. Also, we know that if τ+ν/2<1 for all τ,ν∈X with τ=ν=1 and τ≠ν, then X is called strictly convex.Lemma 1.
IfX is a Banach space, then the following are equivalent:(i)
X is strictly convex(ii)
μτ+1−μν<1 for all μ∈0,1 and τ,ν∈X with τ=ν=1 and τ≠ν(iii)
Ifτ=ν=μτ+1−μν for some μ∈0,1, then τ=ν
If for eachϒ>0, there exists Θ>0, such that τ+ν/2<1−Θ for all τ,ν∈X with τ=ν=1 and τ−ν≥ϒ; then, X is called uniformly convex. A uniformly convex Banach space X is strictly convex, if for every τ∈Xwithτ=1,limr⟶0τ+rν−τ/r exists and is uniform in ν∈X with ν=1; then, the norm of X is called Fréchet differentiable. A Banach space X has the Opial property [18] if for each weakly convergent sequence τm in X with a weak limit w, liminfm⟶∞τm−w<liminfm⟶∞τm−ν for all ν∈X with ν≠w. Every finite dimensional Banach space, ℓp spaces, and Hilbert spaces have the Opial property. Every weakly compact convex subset of X has a normal structure if it satisfies the Opial property [19]. Also, we know that if every separable Banach space has the Opial property, then they can be equivalently renormed [5].


## 3. Main Results

Here, we will present our main result.Proposition 1.
Assume thatJ is a closed convex subset of X. Assume that ξ∞>0 and Vr,0:r∈0,ξ∞ is a family of operators on J, such that the following conditions hold:(i)
For everyr∈0,ξ∞, Vr,0 is nonexpansive(ii)
There isξm, a strictly increasing sequence in 0,ξ∞, such that ξ1=0,ξm converges to ξ∞ and weakly continuous mappings r↦Vr,0τ on ξm,ξm+1 for all τ∈J and m∈ℕ.
Let(7)∩r∈0,ξ∞ℨVa,0≠∅.Then,(8)∩a∈0,ξ∞ℨVa,0=ℨM,where M is a nonexpansive mapping on J, such that(9)Mτ=1ξ∞∫0ξ∞Vs,0τds,for all τ∈J.Proof.
Fixg∈X∗. Then, the functions r↦gVr,0τ from ξm,ξm+1 into ℝ are continuous on ξm,ξm+1 for τ∈J and m∈ℕ. So, the functions r↦gVa,0τ from 0,ξ∞ into ℝ are measurable for τ∈J. Also, Va,0τ:a∈0,ξ∞ is separable for each τ∈J. Fix p∈∩a∈0,ξ∞ℨVa,0. Since(10)Va,0τ=Va,0τ−Va,0p+p≤Va,0τ−Va,0p+p≤τ−w+w,for τ∈J and a∈0,ξ∞, the mappings r↦Va,0τ for all τ∈J are Bochner integrable and M is well-defined. Using the separation theorem, we can easily prove that M is a mapping on J. Recall that(11)Mτ−Mν=1ξ∞∫0ξ∞Vs,0τ−Vs,0νds,≤1ξ∞∫0ξ∞Vs,0τ−Vs,0νds,≤1ξ∞∫0ξ∞τ−νds=τ−ν,for τ,ν∈J. So, M is a nonexpansive mapping on J. It is obvious that ∩a∈0,ξ∞ℨVa,0⊂Va,0. Suppose that ω∈ℨM∖∩a∈0,ξ∞ℨVa,0. Then, there is a1∈0,ξ∞, such that Ta1ω≠ω. Fix φ∈X∗ with(12)φ=1,φTa1ω−ω=Ta1ω−ω.
For somen∈ℕ, a1∈ξn,ξn+1, so by assumption (ii), there exists a2∈a1,ξm+1, such that(13)φTa1ω−ω>12Ta1ω−ω,for all a∈0,ζ. Define the nonexpansive mappings M1 and M2 on J by(14)M1τ=1ζ∫0ζVs,0τds,M2τ=1ξ∞−ζ∫0ξ∞Vs,0τds,for all τ∈J. We note that(15)Mτ=ζξ∞M1τ+ξ∞−ζξ∞M2τ,for all τ∈J. We have(16)φM1ω−Mω=φ1ζ∫0ζVs,0ωds−ω=φ1ζ∫0ζVs,0ω−ωds=1ζ∫0ζφVs,0ω−ωds≥1ζ∫0ζ12V0,0ω−ωds=12V0,0ω−ω>0.
Hence,(17)φM2ω−Mω=ζξ∞−ζφMω−M1ω<0.
Therefore,M1ω≠M2ω. Fix p∈∩r∈0,ξ∞ℨVa,0. Then, M1p=M2p=p. We have(18)ω−p=Mω−p=ζξ∞M1ω+ξ∞−ζξ∞M2ω−p≤ζξ∞M1ω−p+ξ∞−ζξ∞M2ω−p=ζξ∞M1ω−M1p+ξ∞−ζξ∞M2ω−M2p≤ζξ∞ω−p+ξ∞−ζξ∞ω−p=ω−p,and hence,(19)Mω−p=M1ω−p=M2ω−p.
This is a contradiction to the strict convexity ofX. So, ℨM⊂∩a∈0,ξ∞ℨVa,0.Corollary 1.
Assume thatJ is a closed convex subset of X. Let L=Va,0:a≥0 be a subfamily of a nonexpansive and ζ be a periodic evolution family V on J. Assume that ∩m=1∞ℨVa,0≠∅. Let ψm be a positive sequence for all m∈ℕ with ∑m=1∞ψm=1. Let M be a nonexpansive mapping on J defined by(20)Mτ=∑m=1∞ψmVa,0τ,for τ∈J. Then, ℨM=∩m=1∞ℨVa,0 holds.Proof.
Letψm be a strictly increasing sequence in 0,1 defined by ψ1=0 and(21)ψm=∑j=1m−1ψj,for m∈ℕ with m≥2. We note that limm⟶∞ψm=1. Define a family Va,0;a∈0,1 of nonexpansive mappings as follows: if ξm≤r<ξm+1, then(22)Va,0τ=Vm,0τ,for all τ∈J. So, we note that(23)Mτ=∑m=1∞ψmVs,0τ=∑m=1∞∫ξmξm+1Vs,0τds=∫01Vs,0τds,for all τ∈J and(24)∩m=1∞ℨVm,0=∩a∈0,1ℨVa,0.
Direct consequence of Proposition1 leads to the next proposition; in fact, here, we use the intervals 0,ξ in place of ξm,ξm+1, and the proof is similar with Proposition 1.Proposition 2.
Assume thatJ is a closed convex subset of X. Let ξ>0 and Va,0:a∈0,ξ be a subfamily of operators on J satisfying the following conditions:(i)
Va,0 is nonexpansive for every a∈0,ξ(ii)
The mappingsa↦Va,0τ are weakly continuous on 0,ξ for all τ∈J.Let(25)∩a∈0,ξℨVa,0≠∅.Then,(26)∩a∈0,ξℨVa,0=ℨM,where M is a nonexpansive mapping on J, such that(27)Mτ=1ξ∫0ξVs,0τds,for all τ∈J.Now, we are going to prove the following.Theorem 4.
Assume thatJ is a closed convex subset of X and V=Va,s:a≥s≥0 is a continuous evolution family of nonexpansive mappings on J in a strong sense. Assume that(28)∩a≥0ℨVa,0≠∅.
Define a nonexpansive mappingM on J by(29)Mτ=1ζ∫0ζVs,0τds,for all τ∈J. Then,(30)∩a≥0ℨVa,0=ℨMholds.Proof.
It is clear that∩a≥0ℨVa,0⊂ℨM. For convenience, let τ∈∩a≥0ℨVa,0; then, τ∈ℨVa,0 for all a≥0. We have(31)Va,0τ=τ∀a≥0.
Now,Vs,0τ=τ for s∈0,ζ. Hence, for every s∈0,ζ/2, we have(32)∫0ζVs,0τds=∫0ζτds=τs|0ζ=ζτ.
This implies that(33)1ζ∫0ζVs,0τds=τ.
Now by definition ofM, we have(34)Mτ=τ,hence,(35)∩a≥0ℨVa,0⊂ℨM.
Also by Proposition2, we know that(36)∩a∈0,ζ/2ℨVa,0=ℨM.
Now, for anya∈ℝ+, there are m∈ℕ∪0 and u∈0,ζ/2, such that a=u+mζ/2. We have(37)Va,0τ=Vmζ+ψ,0τ=Vmζ+ψ,mζVmζ,0τ=Vψ,0Vζ,0mτ=τ.
Thus,τ is a fixed point of Va,0:a≥0, and hence,(38)∩a≥0ℨVa,0=∩a∈0,ζ/2ℨVa,0=ℨM.
This completes the proof.Theorem 5.
Assume thatJ is a closed convex subset of X. Let Vna,0:a≥0 be a subfamily of a periodic, nonexpansive evolution family Vna,s:a≥s≥0. Let Up:p∈ℕ be a sequence with nonexpansive mappings on J. Let(39)∩n=1∞∩a∈ℝ+ℨVna,0∩∩p=1∞ℨUp≠∅.Letam, um, ψm, and ηm be real sequences, such that 0≤am<um,ψm>0, and ηm>0 for all m∈ℕ and ∑m=1∞ψm=1+∑m=1∞ηm=1. Let M be a nonexpansive mapping on J defined by(40)Mτ=∑m=1∞ψmum−am∫amumVns,0τds+∑m=1∞ηmUpτ,for all τ∈J. Then,(41)∩n=1∞∩a∈ℝ+ℨVna,0∩∩p=1∞ℨUp=ℨMholds.We know that if every nonexpansive mapping onD has a fixed point for each bounded closed convex subset D of J, then a closed convex subset J of a Banach space X has the fixed point property for nonexpansive mappings. If a Banach space X is uniformly convex, then X has the fixed point property [1, 2]. Also, every compact convex subset with a normal structure has the fixed point property in a weak sense [3].From Theorem5, we obtain Corollary 2.Corollary 2.
AssumeJ, X, Vna,0:a≥0:n∈ℕ, Un:n∈ℕ, an, un, βn, and ηn are as in Theorem 5. Suppose that J has the fixed point property and is compact in a weak sense, and(42)Vms,0∘Vna,0=Vna∘Vns,Um∘Un=Un∘Um,Um∘Vnr,0=Vna,0∘Um,for all s,a∈ℝ+ and m,n∈ℕ. Let M be a nonexpansive mapping on J defined as in Theorem 5. Then,(43)∩n=1∞∩a∈ℝ+ℨVna,0∩∩n=1∞ℨUn=ℨM≠∅holds.We get Theorem6 from [20], see also [21].Theorem 6.
Assume thatJ is a compact convex subset of X. Assume that Va,0:a≥0 is an evolution family of nonexpansive mappings on J in a strong sense. Suppose τm>0 for all m∈ℕ is a sequence in J defined by(44)τm+1=βmζ∫0ζVs,0τmds+1−βmτm,for m∈ℕ and τ1∈J, where βm∈0,1 is a sequence satisfying ∑m=1∞βm=∞ and limsupm⟶∞βm<1. Then, the sequence τm converges to z∈ℨL in a strong sense.We get Theorem7 from [22, 23].Theorem 7.
LetX be a Banach space and let one of the following hold:(i)
X has the Opial property and is strictly convex or(ii)
The norm ofX is Fréchet differentiable and X is uniformly convex.Recall thatX is a Banach space and let J be a weakly compact convex subset of X. Let Va,0:a≥0 be a strongly continuous subfamily of an evolution family of nonexpansive mappings on J. Let τn be a sequence in J defined by τ1∈J and(45)τn+1=βζ∫0ζVs,0τnds+1−βτn,for n∈ℕ, where β∈0,1. Then, the sequence τm converges to z∈ℨL in a weak sense.So, ifV is a nonexpansive mapping and β∈0,1, then(46)τ↦1−βVa,0τ+βuis a contractive mapping. Using the Banach fixed point theorem [8], each of such operators has a unique fixed point. The following theorem is obtained from the results of [24, 25], see also [26].Theorem 8.
LetVa,0:a≥0 be a family of strongly continuous nonexpansive mappings on a closed and bounded subset J of a Hilbert space ℌ. Let u∈J be fixed and τm be a sequence in J defined by(47)τm=1−βmζ∫0ζVs,0τmds+βmu,for m∈ℕ, where βm∈0,1limm⟶∞βm=0. Then, the sequence τm converges to z∈ℨL in a strong sense.Theorem 9.
LetJ, L=Va,0:a≥0, X, and ζ be as in Theorem 8. Let τm>0 for all m∈ℕ be a sequence in J defined by(48)τm+1=1−βmζ∫0ζVs,0τmds+βmu,for m∈ℕ and τ1∈J, where βm∈0,1 is a sequence satisfying the following:(49)limm⟶∞βm=0,∑m=1∞βm=∞,∑m=1∞βm+1−βm≤∞.Then, the sequenceτm converges to z∈ℨL in a strong sense.All the above results are related to the iteration method for strong convergence of a sequence to a fixed point. We are concluding this section with the following remark.Remark 3.
Using Remark1, the work performed in [6], Theorem 3.5, Proposition 3.1, and Corollary 3.3, becomes special cases of our study.


## 4. Conclusion

The fixed point theory has a lot of applications in science. In this study, we proved results concerning the strong convergence of a sequence to a fixed point of a subfamily of a periodic evolution family onX. Sometimes, it is too much difficult to find the fixed point of a semigroup or an evolution family. For such problem, the construction of the function Mτ=1/ξ∞∫0ξ∞Vs,0τds for all τ∈J is useful to find the fixed point of the given family. As perspectives of a new work, it would be interesting to generalize these results for periodic and nonperiodic general evolution families. This setting will be more applicable and fruitful for researchers working in this field or others related to the fixed point theory.



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