# Approximating Common Fixed Points of an Evolution Family on a Metric Space via Mann Iteration

Authors: Liang Luo; Rizwan Ullah; Gul Rahmat; Saad Ihsan Butt; Muhammad Numan
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this article, we present the set of all common fixed points of a subfamily of an evolution family in terms of intersection of all common fixed points of only two operators from the family; that is, for subsetM of L, we have FM=FYϱ1,0∩FYϱ2,0, where ϱ1 and ϱ2 are positive and ϱ1/ϱ2 is an irrational number. Furthermore, we approximate such common fixed points by using the modified Mann iteration process. In fact, we are generalizing the results from a semigroup of operators to evolution families of operators on a metric space.


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## Body

## 1. Introduction

The theory of fixed points has a lot of applications in mathematics. The existence of a solution for a system is equivalent to the existence of a fixed point for a particular map associated with the system. Fixed point theory helps us for assurance of existence and uniqueness of solutions of differential equations, integral equations, algebraic system, and numerical equations. For further study of fixed point theory and applications, we refer to [1]. Since 1960, an extensive amount of work has been done in the field of fixed point theory. Belluce and Kirk [2, 3], Browder and Bruck [4, 5], Lim [6], and DeMarr [7] have discussed fixed point for contractions and nonexpansive mappings on Banach spaces. However, in [8], the occurring of common fixed points of nonexpansive strongly continuous semigroups has been discussed via employing asymptotic type approach.A family of bounded linear mappingsS=St:t≥0 is called a semigroup if it satisfies the following two conditions: S0=I and Sx+y=SxSy, for all x and y≥0. Such families arise from the solution of the autonomous system:(1)J˙r=Ar,J0=ξ.The study of semigroups has a big role in the field of difference and differential equations. Although it is very important, still cannot be applied for the nonautonomous system:(2)J˙r=ArJr,J0=ξ.For such a system, we need the idea of evolution family. A familyL=Yr,l;r≥l≥0 of bounded linear mappings on a metric space ℧ is said to be an evolution family if it satisfies the following two conditions:(i)
Yr,r=I(ii)
Yr,sYs,l=Yr,l for all r≥s≥l≥0Such families are the generalized form of a semigroup.Therefore, in our study, we investigate the common fixed points of an evolution family (instead of semigroup) for a hyperbolic metric space. For some recent published work on this topic, we refer to [9–11]. In fact, in [9], the following result is proved.Theorem 1.
LetM=Yϱ1,0:ϱ1≥0 be a subfamily of an evolution family of nonexpansive and strongly continuous operators on a metric space ℧. If ϱ1andϱ2 are two positive real numbers with restriction that ϱ1/ϱ2 is an irrational number, then(3)AFPSYϱ1,0∩AFPSYϱ2,0=AFPSM,where AFPS mean approximate fixed point sequence of the operators.In [9], authors discussed the approximate fixed point sequences, but this paper is about convergence to a common fixed point of an evolution family by using the Mann iteration process. Also, we present the set of all common fixed points by the intersection of all common fixed points of only two operators from the same family. Therefore, we mention that these are two different approaches for fixed points.In [12], Khamsi and Buthinah discussed the fixed points of a semigroup on a metric space. Khamsi and Kozlowski [13, 14] presented fixed points for pointwise nonexpansive and pointwise contraction mapping in modular function spaces. Furthermore, the existence of fixed points for asymptotic and pointwise contractions and asymptotic and pointwise nonexpansive mappings has been discussed by Kirk and Xu in [15]. These results have been extended by Hussain and Khamsi [16] to metric spaces. In [17], Khamsi discussed the existence of nonlinear semigroups and gave some applications to differential equations in Musielak–Orlicz spaces.


## 2. Definitions and Preliminary Results

In this section, we present some definition which will be utilized in main results of this paper. Throughout the paper, we use the following notations for simplifications and to avoid repetitions:M1: ℝ,ℝ+,ℚc,ℚ,andℤ denotes the set of real, positive real, irrational, rational, and integer numbers, respectivelyM2: ℧,d denotes a metric spaceM3: ℧,d,H denotes a hyperbolic metric spaceM4: ϱ1,ϱ2∈Δ⟹ϱ1,ϱ2∈ℝ+ with the restriction that ϱ1/ϱ2∈ℚcDefinition 1 (see [18]).
℧,d is said to be convex metric space if for all r,q in ℧, there exist a unique metric segment r,q∈F, and for any two points p∈r,q and ζ∈0,1, such that p=1−ζr+ζq, and the following holds:(4)dr,p=ζdr,q,dp,q=1−ζdr,q,where F is the set of all line segments between the points of ℧.Definition 2 (see [19]).
A convex℧,d is called hyperbolic metric space ℧,d,H if for all c,r,q in ℧ and ζ∈0,1, and the following condition holds:(5)dζc+1−ζr,ζc+1−ζq≤1−ζdr,q.
A subsetD of ℧,d,H is convex if for all r, q in D, and the line segment r,q belongs to D.Definition 3 (see [12]).
A℧,d,H is uniformly convex if for all ν∈℧, and for all positive real numbers ξ and ε, the following inequalities hold:(6)δξ,ε=inf1−1ξd12r+12q,ν;dr,ν≤ξ,dq,ν≤ξ,dr,q≥ξε>0.
The concept of convexity in a uniform sense was developed initially only for Banach spaces (see [20]), but latter on in [21], it was extended to metric spaces. For further details, see [19, 22, 23].Definition 4 (see [12]).
A℧,d,H is called strictly convex when we have(7)dν,r=dν,q=dν,ζr+1−ζq,for any ν,r,q∈℧ and 0<ζ<1, and then r=q.Lemma 1 (see [14, 23]).
Let℧,d be uniformly convex. Also, let Cn in ℧ be a sequence of nonempty, nonincreasing, convex, closed, and bounded sets if r∈℧ and(8)0<l=limn⟶∞dr,Cn<∞.Ifrn∈Cn is with dr,rn⟶l, then rn is a Cauchy sequence.In℧,d,H, if there exist any nonincreasing sequence of sets which are nonempty, convex, closed, bounded, and also their intersection is nonempty, such property is called the property P. Moreover ℧,d,H is called complete if it has the property P. Similarly if ℧,d is complete and uniformly convex, then it satisfies the property P; for more details, we refer to [14, 23, 24].Definition 5 (see [12]).
LetD is a nonempty subset of ℧,d and S=Sx,x≥0 be a semigroup on D. The set of all common fixed points of the family S is define as(9)FS=∩x≥0FSx.Proposition 1 (see [12]).
IfD is a nonempty and closed subset of a ℧,d, the family S=Sx:D⟶ℜ;x≥0 be a semigroup of continuous operators. Moreover, if C¯=0,+∞, then(10)FS=∩x∈CFSx.Proposition 2 (see [12]).
LetD be a nonempty and closed subset of a ℧,d and S=Sx:D⟶ℜ:x≥0 be a semigroup on D. If C¯=0,+∞, then there exist ϱ1,ϱ2∈Δ as defined in M4, such that(11)∩x∈CFSx=FSϱ1∩FSϱ2.If we combine Propositions1 and 2, we get following theorem.Theorem 2 (see [12]).
LetD be a nonempty and closed subset of a ℧,d, and S=Sx:D⟶ℜ;x≥0 be a semigroup. Then, for any two ϱ1,ϱ2∈Δ as defined in M4, we have(12)FS=∩x≥0FSx=FSϱ1∩FSϱ2.Definition 6.
An operatorΓ:D⟶D is said to be nonexpansive if for any r,q∈D, the following inequality:(13)dΓr,Γq≤dr,q,holds. Moreover, s is said to be fixed point of Γ if for some s∈D and Γs=s. We denote byFΓwhich is the set of all fixed points of the operator Γ.Definition 7.
An evolution familyL=Yr,l;r≥l≥0 of bounded linear operators on a ℧,d is called nonexpansive, continuous, and periodic if the following holds:(i)
dYr,lu,Yr,lv≤du,v,∀r≥l≥0, and u,v∈℧(ii)
limr−l⟶0Yr,lu=u,∀u∈℧(iii)
There existp≥0 such that Yr+p,l+p=Yr,l for all r≥l≥0, respectively
The present paper deals with the study of exploring fixed points for a subfamily of evolution family. LetM=Yx,0;x≥0 be a subfamily of the family L=Yr,s;r≥s≥0 on a metric space ℧, that is, ϱ1 and ϱ2 periodic, where ϱ1,ϱ2∈Δ as defined in M4.
First, we define the set of all common fixed points of the above familiesL and M.(14)FL=∩r≥s≥0FYr,s,FM=∩r≥0FYr,0.
The following lemma will be useful in the proof of our main results.Lemma 2 (see [5]).
LetD⊆℧ which is strictly convex ℧,d,H. If Γ1,Γ1:D⟶℧ are nonexpansive operators such that FΓ1∩FΓ2≠∅, then for each 0<ω<1 and for nonexpansive mapping Υ:D⟶℧ such that Υs=ωΓ1s+1−ωΓ2s for s∈D, we have FΥ=FΓ1∩FΓ2.In the rest of the paper, we will denote the evolution family byL, the subfamily of the evolution family by M, and the nonempty closed and convex subset of the metric space unless stated otherwise by D.


## 3. Methodology

In this paper, we provide results concerning the fixed points of a subfamily of a nonexpansive evolution family of operators. In fact, such results are presented in [12] for semigroups of nonexpansive bounded linear operators. So, in this paper, we use the methodology as used in [12]. Such results are new, and we do not find in the existing literature.


## 4. Main Results

In this section, we will present our main results concerning the fixed points of an evolution family of bounded linear operators in terms of intersection of sets of fixed points of two operators from the family.Proposition 3.
LetL=Yr,l;r≥l≥0 be an evolution family on D. If C⊂0,+∞ such that C¯=0,+∞, then(15)FL=∩r≥l≥0FYr,l=∩c≥d≥0∈CFYc,d.Proof.
ClearlyFL⊂∩c,d∈CFYc,d. We only need to prove that(16)∩c,d∈CFYc,d⊂FY.
Lets∈∩c,d∈CFYc,d which means ∪c,ds=s for all c≥d≥0, and c,d∈D. We need to show that Yr,ls=sfor allr≥l≥0. As C¯=0,+∞, then by denseness property, there exist sequences cn,dn∈C such that cn⟶x and dn⟶y and cn≥dn. Now, consider Ycn,dns=s for all s∈ℜ.
Now taking limit on both sides, that is,limn⟶∞Ycn,dns=limn⟶∞s for all s∈ℜ gives Yr,ls=sfor allr≥l≥0. Hence, s∈FL, ∩c,d∈CFYc,d⊂FL. We get(17)FL=∩r≥l≥0FYr,l=∩c,d∈CFYc,d.Proposition 4 (see [25]).
IfC⊂ℝ is nonempty and additive subgroup, then either C¯=ℝ or for nonnegative number k, we have C=k⋅ℤ=kn,n∈ℤ. If there exist two real numbers ϱ1,ϱ2∈Δ as defined in M4, then the set(18)Cϱ1,ϱ2=nϱ1+mϱ2;n,m∈ℤ,is dense in ℝ. Moreover, if W=Cϱ1,ϱ2∩0,+∞, then W¯=0,+∞(W is dense in 0,+∞).Proposition 5.
LetM=Yr,0:r≥0 be subfamily of evolution family from D into ℧. If ϱ1,ϱ2∈ℝ+ such that these numbers are also the periods of the evolution family, then(19)∩r∈CFYr,0=FYϱ1,0∩FYϱ2,0,where C=nϱ1+mϱ2;n,m∈ℤ∩0,+∞.Proof.
Asϱ1 and ϱ2 are in C, therefore ∩r∈CFYr,0⊂FYϱ1,0∩FYϱ2,0. We need only to prove that(20)FYϱ1,0∩FYϱ2,0⊂∩r∈CFYr,0.
Lets∈FYϱ1,0∩FYϱ2,0, then Yϱ1,0s=s and Yϱ2,0s=s. Let r∈D, then r=nϱ1+mϱ2.
Consider(21)Yr,0s=Ynϱ1+mϱ2,0s=Ynϱ1,0Ymϱ2,0=Ynϱ1,0Ynϱ2,0s=Ynϱ1,0s=s.
Hence,FYϱ1,0∩FYϱ2,0⊂∩r∈CFYr,0, where we use the property U0,0 which is the identity-map and consider r=nϱ1−mϱ2, where either n or m is negative. Assume that r+mϱ2=nϱ1, then(22)Yr,0s=Yr,0Ymϱ2,0s=Yr,0Ymϱ2,0s=Yr+mϱ2,0s=Ynϱ1,0s=Ynϱ1,0s=s.
Hence,s∈FYr,0 for any r∈C, and this completes the proof.Theorem 3 (see [26]).
LetM=Yr,0:r≥0 be subfamily of the evolution family on D. If ϱ1,ϱ2∈Δ as defined in M4, then(23)FM=FYϱ1,0∩FYϱ2,0.In particular, we have(24)FM=FY2,0∩FY3,0=FYπ,0∩FYe,0.Theorem 4.
LetM=Yr,0:r≥0 be subfamily of evolution family from D into ℧ which is continuous and nonexpansive. If ϱ1,ϱ2∈Δ as defined in M4, then(25)FM=FζYϱ1,0+1−ζYϱ2,0,where ζ∈0,1.Proof.
Let us considerr∈FM=FYϱ1,0∩FYϱ2,0. This ensures that Yϱ1,0r=r and Yϱ2,0r=r. Now,(26)ζYϱ1,0+1−ζYϱ2,0r=ζYϱ1,0r+1−ζYϱ2,0r=ζr+1−ζr=r.
Hence,r∈FζYϱ1,0+1−ζYϱ2,0.
Again considerr∈FζYϱ1,0+1−ζYϱ2,0 which implies ζYϱ1,0+1−ζYϱ2,0r=r or ζYϱ1,0r+1−ζYϱ2,0r=r.
If we takelimζ⟶0 and limζ⟶1, it gives Yϱ2,0r=r and Yϱ1,0r=r, respectively. Hence, r∈FYϱ1,0∩FYϱ2,0=FM. This completes the proof.


## 5. Approximating of Fixed Points of an Evolution Family

In this section, we discuss the behavior of Mann iteration process by two operators, using previous results. With the support of these iterations, we will approximate the common fixed points of subfamily of an evolution family.Definition 8.
LetT:D⟶D, which is nonexpansive mapping and 0<ζ<1. Also let MT,ζ represent the Mann iteration process produced by the constant ζ and the mapping T, which is given by following iteration formula:(27)sn+1=ζTsn+1−ζsn,where s1 is chosen arbitrarily in D.Lemma 3 (see [14, 23]).
Let℧,d,H is uniformly convex metric space and v∈0,+∞ such that(28)limsupn⟶∞dsn,h≤v,limsupn⟶∞dtn,h≤v,limn⟶∞dh,ζsn+1−ζtn=v,for some 0<ζ<1. Then,(29)limn⟶∞dsn,tn=0.Lemma 4.
Let℧,d,H is uniformly convex and T be a nonexpansive mapping defined from D to D. Let 0<ζ<1 and sn be a sequence defined by (27). Suppose ω∈FT. Then, limn⟶∞dsn,ω exists.Proof.
Let us consider for alln≥1,(30)dsn+1,ω=dζTsn+1−ζsn,ω≤ζdTsn,ω+1−ζdsn,ω=ζdTsn,Tω+1−ζdsn,ω≤ζdsn,ω+1−ζdsn,ω=dsn,ω.
Since the sequencedsn,ω≥0 and is nonincreasing, it is bounded below, and hence it is convergent. Next, we define asymptotic radius of a sequence.Definition 9 (see [12]).
LetD be a nonempty subset ℧,d and sn be a bounded sequence in ℧. Define r.,sn:D⟶0,∞ by(31)rs,sn=limsupn⟶∞ds,sn.
The asymptotic radius of the sequencesn with respect to D is denoted by ρD and is defined by the following:(32)ρD=infrs,sn:s∈D,where ρ is used to represent the asymptotic radius of the sequence sn with respect to ℧. Let ξ∈D such that(33)rξ,sn=rD,xn=minrs,sn:s∈D.
Then,ξ is an asymptotic center of sequence sn with respect to D.
LetAD,sn represent the collection of all asymptotic centers of sn with respect to D. When D=℧, we use the notation Asn. Generally, the set AD,sn of a bounded sequence sn contains more than one point. It can be empty too; for more details, we refer to [23].
Kuczumow [27] and Lim [6] offered an approach to weak convergence, which they called Δ-convergence. In the case of CAT0 spaces, this approach was very effective and successful.Definition 10.
LetCAT0 be a bounded sequence in ℧,d. This sequence is called Δ-converge to s∈℧ if and only if for every subsequence un of sn, s is the unique asymptotic center of this subsequence. When sn is Δ-converge to s, we denote by sn⟶Δs.
Next, we discuss the iteration scheme (27) for nonexpansive mappings, and particularly, we will investigate the Δ-convergence of hyperbolic spaces which are uniformly convex.Lemma 5 (see [23, 28]).
If℧,d,H is uniformly convex, then with respect to D, there exist a unique asymptotic center for each bounded sequence sn∈℧.Lemma 6 (see [23, 28]).
If℧,d,H is uniformly convex and sn∈D is a bounded sequence such that Asn=t and rsn=ρ and let tm∈D is a sequence such that limm⟶∞rtm,sn=ρ, then limm⟶∞tm=t.In uniformly convex Banach spaces, D. Gohde gave the demi-closed principle [29]; a similar concept is discussed in next theorem.Lemma 7.
Let℧,d,H be complete and uniformly convex and T:D⟶D is a nonexpansive mapping on D. If sn∈D be a sequence such that limn⟶∞dsn,Tsn=0 and if s∈D is the asymptotic center of the sequence sn with respect to D, then s∈T, in certain, if T has approximate fixed point sequence sn∈D, such that sn⟶Δs, then s∈FT.In order to study the iterative behavior defined by (27), we give the following necessary theorem.Theorem 5 (see [28]).
Let℧,d,H be uniformly convex and T be as defined in Lemma 7, which has a fixed point. If 0<ζ<1, s1∈D, and sn is generated by (27), then(34)limn⟶∞dTsn,sn=0,and T has an approximate fixed point sequence sn.The condition thatT has a fixed point that can be removed if we assume that D is bounded (see [23]). Now, we study the Δ-convergence of the Mann iteration process given in (27).Theorem 6.
Let℧,d,H be uniformly convex and T be as defined in Lemma 7, which has a fixed point. If   0<ζ<1, s1∈D, and sn is generated by (27), then sn is Δ-converge to s∈FT.Proof.
The fixed set of operatorT is convex and closed, as given in [23]. Applying Theorem 5, T has approximate fixed point sequence sn, which implies limn⟶∞dTsn,sn=0, if the sequence sn has a unique asymptotic center s. Then, using Lemma 7, s∈FT. Now, we prove that sn is Δ-converge to s. Suppose that the sequence sn has a subsequence sni and suppose this subsequence has the unique asymptotic center z. Again since T has an approximate fixed point sequence sni, we conclude that z∈FT.
Hence,(35)limsupni⟶∞dsni,z≤limsupni⟶∞dsni,s.
Sinces,z∈FT, we get(36)limsupni⟶∞dsni,z=limn⟶∞dsn,z,limsupni⟶∞dsni,s=limn⟶∞dsn,s.
Sincesni has the unique asymptotic center s, s=z. This shows that snΔ-converge to s.
The combination of Theorems3 and 6 gives the result, which is discussed in the next theorem.Theorem 7.
Let℧,d,H is complete and uniformly convex with D be a subset of ℧. If M=Yr,0:r≥0 is a subfamily of an evolution family on D which is continuous and nonexpansive, if ϱ1,ϱ2∈Δ as defined in M4 and 0<α,β<1 are such that α+β≤1, and also if s∈D and sn∈D be a sequence defined by(37)sn+1=α+βαYϱ1,0sn+βYϱ2,0sn+1−α−βsn,where n≥1, then the sequence sn is Δ-converge to a common fixed point of M=Yr,0;r≥0.Proof.
Set(38)S=αα+βYϱ1,0+βα+βYϱ2,0.
Observe thatS:D⟶D is nonexpansive. Clearly we have FM⊂FS. Let s1∈D and sn is the sequence produced by (37). Then,(39)sn+1=ζSsn+1−ζsn,where ζ=α+β∈0,1. It means that the sequence sn is the same as produced by (27) for the operator S. According to Theorem 6, snΔ-converge to some s∈FS. By Theorem 3, we know that FM=FS; hence, s∈FM which completes the proof.
We conclude this section by the following important remark.Remark 1.
If we put the condition of periodicity on the familyL for every positive real number, then it becomes a semigroup. So, in this case, Propositions 3.1, 3.2, and Theorems 3.2 and 4.3 in [12] become subcases of our work.


## 6. Conclusion

In this paper, we established the results concerning the fixed points of a subfamily of an evolution family on a metric space. It is more difficult to derive or find the set of all common fixed points for a family than to obtain the set of all common fixed points only for two maps. So, in this paper, we provide results which make it easy to present such set of fixed points in terms of only two maps from the same family. Also, we approximated the common fixed points for subfamily of evolution family using the well-known Mann iteration process. In future, we are trying to generalize these results for periodic evolution families which seem to be more applicable and fruitful for researchers working in this field or others related to fixed point theory. Recently, we completed more projects in this area, and we hope to be publish them in near future (see [30–32]).



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