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

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.


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. e 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][9][10][11]. We refer to [12][13][14][15][16][17][18][19] for a more detailed study on fixed point theory and its applications in metric spaces.
Let X ≠ ∅, and τ: X ⟶ X be a self-mapping. e element a ∈ X is called a fixed point of τ if τ(a) � a. Consider the autonomous system _ y(r) � Ay(r), r ≥ 0, where A is a linear operator on a Hilbert space H. e solutions of such a system lead us to a class of linear and bounded mappings, called a semigroup. A family S � S(s): s ≥ 0 { } of linear and bounded operators is called a semigroup if it satisfies the following two conditions: e system becomes more difficult if the operator A depends on time, i.e., when A is replaced by A(t) 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 � L(s, r) { } of linear and bounded operators is said to be an evolution family, if the following hold: (1) L(s, s) � I, for all s ≥ 0 (2) L(s, t)L(t, r) � L(s, r), for all s ≥ t ≥ r ≥ 0 Remark 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. e study of fixed points for semigroups is studied by many mathematicians, such as Suzuki [21,22] and Buthinah et al. [23]. ey 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.

Preliminaries
In this article, we will frequently use the following notations: (1) By R, R + , N, and Z + , we will denote the set of all reals, nonnegative reals, natural numbers, and nonnegative integers, respectively. (2) e semigroup, evolution family, and its subfamily will be denoted by S, L, and L s , respectively. (3) e set of all common fixed points of the semigroup, evolution family, and its subfamily will be denoted and defined as Fix(G) � ∩ s≥0 Fix(G(s)), Fix(L) � ∩ s≥r≥0 Fix(L(s, r)), and Fix(L s ) � ∩ s≥0 Fix (L(s, 0)), respectively. (4) By C, we will denote a closed and convex subset of the Hilbert space H.
We denote by Fix(τ) the set of all fixed points of τ. If τ is a nonexpansive self-map on C, then Fix(τ) is nonempty, see [25].
For a fixed ζ in C and ε ∈ (0, 1), there is a unique point e nonexpansiveness of T ensures that the map is a contraction.
In 1967, Browder [26] provided the following result for self-mappings. Theorem 1. Let τ be a self-mapping on C and c n ∈ (0, 1) be a sequence such that lim n⟶∞ c n � 0. en, for a fixed ζ in C, the sequence 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 L s of nonexpansive evolution operators on Hilbert spaces. Such a family needs not be a semigroup. e following example will illustrate this fact.

Example 1.
e family defined by L � L(s, r) � { (r + 1/s + 1): s ≥ r ≥ 0} is clearly an evolution family acting on R + . Since L(s, s) � 1 (the identity on R + ), and However, if we put a condition as given in Remark 1, then such a family becomes a semigroup.

Main Results
In this section, we will present our main results. e following lemma states that the set of all common fixed points of a semigroup can be represented on the closed unit interval in place of R + . Proof.
is obvious, and we will show the reverse inclusion. Let u ∈ ∩ 0≤s≤1 Fix(G(s)), then G(s)u � 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 at is, u ∈ ∩ s≥0 Fix(G(s)). us, we conclude that

□
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. Fix(G) � ∩ s≥0 Fix(G(s)) � Fix(G(α)) ∩ Fix(G(β)), where α and β are positive such that α/β is irrational. Now, using Lemma 1 and eorem 2, we have the following corollary.
Lemma 1 can be extended to a subfamily Proof. Since it is obvious that we will again prove the reverse inclusion. Let then L(s, 0)u � u for all 0 ≤ s ≤ q. Now, since any s ≥ 0 can be written as s � nq + ϱ, for some n ∈ Z + and some 0 ≤ ϱ ≤ q, we have Hence, □ is completes the proof. e Opial condition holds on every Hilbert space, given as follows.
Proposition 1 (see [10]). If β n is sequence in H, converging to a point a ∈ H in a weak sense, then 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. Fix(L s ) ≠ ∅. Let c n ∈ (0, 1) and s n ≥ 0 be two sequences of real numbers with the property that lim n⟶∞ s n � lim n⟶∞ (c n /s n ) � 0, (e.g., s n � (1/n) and c n � (1/n 2 )).
en, for a fixed ζ in C, the sequence ζ n � c n ζ + 1 − c n L s n , 0 ζ n , where n ∈ N, (15) converges to an element of Fix(L(s, 0)) nearest to ζ in a strong sense.
Proof. Let s be a point in Fix(L s ) nearest to ζ. From we find that erefore, ζ n and L(s n , 0)ζ n both are bounded. Let ζ n i be any arbitrary subsequence of ζ n , then there exists a subsequence of ζ n i (say ζ n i j ) which converges to x in a weak sense. Our claim is that x ∈ Fix(L s ). For this, put ω j � ζ n i j , z j � c n i j , q � [t/t j ], and t j � r n i j , for n ∈ N. Fix r > 0. One writes for j ∈ N. In above inequality, the first and last terms tend to zero as j ⟶ ∞, so By the Opial condition and Proposition 1, we get L(s, 0)x � x, and therefore, x ∈ Fix(L s ).
Lastly, we will show that ω j converges to s in a strong sense. From we conclude that Journal of Mathematics at is, Since ζ is nearest to s, we can write 〈ζ − s, x − s〉 ≤ 0, for j ∈ N. We see that ω j converges to s in a strong sense.
As ω j � ζ n i is arbitrary, we obtain that ζ n 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 Remark 1. So, the results in [21] become a special case of this paper.

Example and Open Problem
where c n (v): � π 0 x(s)sin(ns)ds. 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 D(A) of all x ∈ H such that v and _ v are absolutely continuous, € v ∈ H and v(0) � v(π) � 0. Now, consider the nonautonomous Cauchy problem where z(·) ∈ H, and the function h: R + ⟶ [1, ∞) is nonexpansive on R + and obeys the periodicity condition, i.e., h(t + q) � h(t) for all t ∈ R + for some q ≥ 1. Let It is obvious that the solution x(·) of the above Cauchy problem will satisfy the evolution property: where We can find ] ≥ 0 such that the function t↦e ]t ‖u(t)‖ is bounded on R + . In fact, we have On the other hand, Hence, Using eorem 3.2 in [27], we have ω 0 (L) ≤ − 1/2M, where M ≥ 1 and ω 0 (L) is the growth bound of the family L, and see [27] for further details. is shows that the evolution family is nonexpansive on H, so eorem 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 Lemma 2 and eorem 3 can be generalized for the whole periodic and then for general evolution families?

Conclusion
e 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.
ese results can open the way for researchers to prove such convergence for the whole evolution family of operators on a Hilbert space.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request. 4 Journal of Mathematics