EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6147 ISSN 1307-5543 – ejpam.com Published by New York Business Global Fixed Point Theorems for a Subfamily of Non-expansive Evolution Operators Muhammad Sarwar1,3,∗, Gul Rahmat2, Sadam Hussain1, Mohammad Yousef Bani Mufarrej3, Kamaleldin Abodayeh3 1 Department of Mathematics, University of Malakand, Chakdara, 18800, Khyber Pakhtunkhwa, Pakistan 2 Department of Mathematics, Islamia College Peshawar, Khyber Pakhtunkhwa, Peshawar, Pakistan 3 Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia Abstract. In this manuscript, we study the fixed points of a specific subfamily within a non- expansive evolution family of bounded linear operators on a Hilbert space H. Employing the framework of nets and a progression algorithm, we establish several theorems concerning the strong convergence of sequences to a common fixed point of the considered subfamily. To illustrate the applicability of our results, we provide a concrete example that supports the theoretical findings. The manuscript concludes with an open problem to encourage further research. 2020 Mathematics Subject Classifications: 46A32, 46E20, 47B02, 47H10 Key Words and Phrases: Fixed Point, Hilbert Space, Nonexpansive mapping, evolution family, linear operator 1. Introduction Our primary objective in this paper is to demonstrate the convergence of an algorithm to a shared fixed point(FP) of a subfamily within a NeE family in Hilbert spaces. Initially, we delve into the significance and concept of semigroups and evolution families of bounded linear operators. Let’s consider the autonomous system{ ℧̇(t) = A℧(t), t ≥ 0 ℧(0) = ℧0, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6147 Email addresses: sarwarswati@gmail.com (M. Sarwar), gulrahmat@icp.edu (G. Rahmat), sadamgaloch@gmail.com (S. Hussain), mfarrej@psu.edu.sa (M.Y.B. Mufarrej), kamal@psu.edu.pk (K. Abodayeh) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6147 2 of 18 where A is a matrix of order m with complex entries. The solution of such a system leads to the idea of a semigroup. We recall that a family S = {S(a), a ≥ 0}, of bounded linear operators on a Hilbert space H is said to be a semigroup if it satisfies the following two conditions, S(a + b) = S(a) + S(b) and S(0) = I for all a, b ≥ 0, where I is the identity operator on H. Similarly the solution of the following non-autonomous system{ ℧̇(t) = A(t)℧(t) + eiνtI t ≥ 0 ℧(0) = ℧0, where A(t) is a matrix of order m, leads to the idea of an evolution family. A family of bounded linear operators E = {E(a, b), a ≥ b ≥ 0}, is said to be an evolution family if it satisfies E(a, b)E(b, c) = E(a, c) and E(a, a) = I for all a ≥ b ≥ c ≥ 0. The theory of fixed points finds myriad applications in mathematics, engineering sci- ences, economics, and statistics, among other fields (see, for example, [1–5]). Motivated by Baillon’s work [6], various authors have explored FP approximations for non-expansive families using different progressions and algorithms. For instance, in [7], it is demonstrated that a sequence of approximations converges to a FP of a non-expansive and non-linear mapping in Hilbert spaces. Halpern, in [8], provides an approximation of a FP of non- expansive maps. Indeed, all the aforementioned references focus on the convergence of algorithms or progressions to fixed points of non-expansive maps in Hilbert spaces. These results are then utilized to prove the convergence of algorithms to fixed points of semi- groups in Hilbert spaces, as seen in [9]. Baillon [6] states that if Z is a convex and closed subset of a Hilbert space H, and G : Z → Z is a non-expansive mapping such that FG(z) ̸= ∅, where FG(z) represents the set of all fixed points of G, then for every element z ∈ Z, the Cesaro mean ( 1 m )∑m q=1G qz converges weakly to some z ∈ FG(z). Here, if we set z = PFG(z) z for any element z ∈ Z, then PFG(z) from Z onto FG(z) is a non-expansive retraction. In 1976, Brezis and Baillon [8] proved that if G = G(s)s ≥ 0 : Z → Z is a non-expansive semigroup, then{( 1 a ) ∫ a 0 G(s)uds } s≥0 converges weakly to a common FP of G. These results have been generalized by several authors, first by Halpern in [10] and then by Wittmann in [11]. They considered the following algorithm ς0 = ς ∈ Z; ςm+1 = µm+1ς + (1− µm+1)Twm, m ≥ 0. Where {ςm} is a progression whose terms are between 0 and 1 and satisfying the following three conditions, • limm→+∞ ςm = 0; • ∑+∞ m=1 ςm = +∞; • ∑+∞ m=1 |ςm+1 − ςm| < +∞. In [11], Wittmann proved that for every ς ∈ Z, the progression ςm strongly converges to a unique FP P (ς) ∈ FG, where P : H → FG is a metric projection. Recently, in [12], M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6147 3 of 18 Kimure et al. generalized Wittmann’s result by proving that if ς1 ∈ Z, where Z is a convex and closed subset of a Hilbert space, then the following iterative scheme converges strongly to Pς , where P : Z → F is a non-expansive retraction and F = ⋂q k=1 FGk ̸= ∅. Takahashi and Shimizu, in [13], studied the strong convergence of the progression ℧m defined by the equation ℧m+1=δm℧+ (1− δ) 1 αm ∫ αm 1 G(s)℧mds, m ≥ 0, where δm is a progression in (0, 1) and αm is a progression in (0,+∞) that diverges to +∞. For other recent work in the field of fixed point theory we refer to [12, 14–19]. In this paper, we aim to generalize the results provided in [9] from a semigroup to a subfamily of a NeE equation in a Hilbert space. For same generalization we refer to [20, 21]. 2. Preliminaries In this section, we present basic definitions and results that are instrumental in proving our main results. Let Z ̸= ∅ be a closed and convex subset of a real Hilbert space H. An operator G : Z → Z is considered non-expansive if ∥Gd−Gb∥ ≤ ∥d− b∥ for all d, b ∈ Z. An operator PZ : H → Z is is said to be a projection from H onto Z if for each element ς ∈ H and PZς ∈ Z the following hold, ∥ς − PZς∥ = inf ℧∈Z ∥ς − ℧∥ := d(ς, Z), and ∥PZς − PZ℧∥ ≤ ∥ς − ℧∥. The above inequality illustrates that PZ is non-expansive, and furthermore ∥PZς − PZ℧∥2 ≤ ⟨ς − ℧, PZς − PZ℧⟩, for all ς,℧ ∈ H. Moreover, we have ⟨ς − ℧, PZς − PZ℧⟩ ≤ 0, (1) and ∥ς − PZς∥2 + ∥℧− PZ℧∥2 ≤ ∥ς − ℧∥2, for all ς ∈ H and ℧ ∈ Z. The evolution family E = {E(d, b)}d≥b≥0 is called non-expansive, continuous and pe- riodic with period q ≥ 0 if it satisfy the following conditions: • ∥E(d, b)y − E(d, b)z∥ ≤ ∥y − z∥ ∀ y, z ∈ H and d ≥ b ≥ 0; • for each z ∈ H, (d, b) → E(d, b)z is continuous. • E(a+ q, b+ q) = E(a, b) for all a ≥ b ≥ 0. Evolution family is a generalized form of a semigroup, see the following remarks. M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6147 4 of 18 Remark 1. [22] Every semigroup can be viewed as a special case of an evolution family, not every evolution family is a semigroup, particularly if it doesn’t satisfy the semigroup property. Remark 2. [22] If an evolution family repeats periodically for every positive number, then it transforms into a semigroup. In this paper, we aim to establish a convergence theorem to a FP of a subfamily Ls of a NeE family on a Hilbert space H. It’s important to note that such a family need not be a semigroup. The following example will illustrate this fact. Example 1. The family defined by L = L(s, r) = s+1 r+1 : s ≥ r ≥ 0 is evidently an evolution family acting on R+. Since L(s, s) = 1 (the identity on R+) and L(s, t)L(t, r) = ( s+ 1 t+ 1 )( t+ 1 r+ 1 ) = s+ 1 r+ 1 = L(s, r). Setting r = 0, we obtain Ls = L(s, 0) = s+ 1, which is indeed a sub-family of L. However, it is not a semigroup. Definition 1. Let Z ̸= ∅ and G : Z → Z, be an operator, then the set of all fixed points of G is denoted by FG(z) and is defined as FG(z) = {z ∈ Z : G(z) = z}. Definition 2. A mapping that is continuous from a topological space into one of its subspaces and maintains the positions of all points in that subspace is referred to as a retraction. Definition 3. Open Ball and Closed Ball: An open ball is denoted by B(Υ;K) and define as B(Υ;K) = {ζ ∈ H : ∥ζ −Υ∥ < K}, while closed ball is denoted by B(Υ;K) and define as B(Υ;K) = {ζ ∈ H : ∥ζ −Υ∥ ≤ K}. Throughout this paper we will denote by FE(d,b) and FE the set of all fixed points of the operator E(d, b) and the family E respectively, i.e. FE = ∩ d≥b≥0 FE(d,b). The following lemmas will be useful in the proof of our main results. Lemma 1. [7] Consider Z as a closed and convex subset of H, and let G : Z → Z be a non-expansive map. Then, I −G is demiclosed. Specifically, if ςm is a sequence in Z such that ςm ⇀ ς and (I −G)ςm → ℧, then (I −G)ς = ℧. Lemma 2. [23] Let ςn and ℧n be bounded sequences in a Banach space Y, and λn be a sequence in the closed interval [0, 1] satisfying the condition 0 < lim inf n→+∞ λn ≤ lim sup n→+∞ λn < 1. If the sequence ςn+1 = (1− λn)ςn + λn℧n ∀n ≥ 0, M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6147 5 of 18 satisfy lim sup n→+∞ (∥ςn − ℧n−1∥ − ∥ςn − ςn−1∥) ≤ 0. Then lim n→+∞ ∥ςn − ℧n∥ = 0. Lemma 3. [24] Consider that {bn} is a sequence in [0,+∞) such that bn+1 ≤ (1−λn)bn+ δnλn for all n, where {λn} ∈]0, 1[ and {δn} are sequences satisfiyng (a) ∑+∞ n=1 λn = +∞, and (b) lim supn→∞ λn ≤ 0 or ∑+∞ n=1 |δnλn| < +∞. Then limn→+∞ bn = 0. 3. Main Results This section is devoted to our main results. Throughout the paper Z will be a non- empty, closed, bounded and convex subset of Hilbert space H. Lemma 4. Let E = {E(z, 0)z≥0} be a subfamily of a NeE family on Z with condition ∥℧r − E(( sr )jr, 0)℧r∥2 ≤ jr r diam(Z), where ℧r = (1r ) ∑r j=1 ςj,r ∈ Z and jr ≥ 0 is an integer. For ς ∈ D and s > 0, set Ft(ς) = 1 s ∫ s 0 E(z, 0)wdz. Then, for each t ≥ 0, lim s→+∞ sup ς∈Z ∥∥E(t, 0)Fs(ς)− Fs(ς) ∥∥ = 0. Proof. Assume that s > t where t ∈ R+ is fix. Then, for r ∈ N, there exist jr ∈ N such that ( s r )jr ≤ s ≤ ( s r )jr + 1. Therefore, we have limr→+∞( sr )jr = t. By using the idea in [3] for {ςj,r}∞j,r=1 ⊆ Z and ℧r = (1r ) ∑r j=1 ςj,r ∈ Z, we have ∥℧r − η∥2 = 1 r r∑ j=1 ∥ςj,r − η∥2 − 1 r r∑ j=1 ∥ςj,r − ℧r∥2 ∀η ∈ H. Set η = E(( tr )jr, 0)℧r and ςj,r = E(( tr )j, 0)ς, for ς ∈ Z. Therefore from the given condition we have ∥∥∥∥℧r − E(( s r )jr, 0)℧r ∥∥∥∥2 ≤ jr r diam(Z). For ϵ > 0, and from jr r ≤ t s , there exists a number s1 > 0 such that 0 < jr r diam(Z) ≤ t s diam(Z) < ϵ2, ∀s ≥ s1. From the above estimation we have∥∥∥∥℧r − E(( s r )jr, 0)℧r ∥∥∥∥ < ϵ, ∀r ∈ N, ς ∈ Z. M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6147 6 of 18 One can write lim r→+∞ ℧r = lim r→+∞ 1 r r∑ j=1 E(( s r j), 0)ς = lim r→∞ 1 s r∑ j=1 s r E(( s r j), 0)ς = 1 s ∫ s 0 E(z, 0)wdz = Fs(ς), ∀ ς ∈ Z. Therefore, for each ς ∈ Z, lim r→+∞ ∥℧r − Fs(ς)∥ = 0. (2) Also we have ∥Fs(ς)− E(t, 0)Fs(ς)∥ ≤ ∥Fs(ς)− ℧r∥+ ∥∥∥∥℧r − E(( s r jr), 0)℧r ∥∥∥∥ + ∥∥∥∥E(( s r jr), 0)℧r − E(( s r jr), 0)Fs(ς) ∥∥∥∥ + ∥∥∥∥E(( s r jr), 0)Fs(ς)− E(t, 0)Fs(ς) ∥∥∥∥ ≤ 2∥℧r − Fs(ς)∥+ ∥∥∥∥℧r − E(( s r jr), 0)℧r ∥∥∥∥ + ∥∥∥∥E(( s r jr), 0)Fs(ς)− E(t, 0)Fs(ς) ∥∥∥∥ < 2∥℧r − Fs(ς)∥ + ∥∥∥∥E(( s r jr), 0)Fs(ς)− E(t, 0)Fs(ς) ∥∥∥∥+ ϵ ∀s ≥ s1, r ∈ N, ς ∈ Z. From limr→+∞( sr )jr = t, and (2) we have ∥Fs(ς)− E(t, 0)Fs(ς)∥ < ϵ ∀s ≥ s1, r ∈ N, ς ∈ Z. Therefore, for each t ∈ R+ we have lim s→+∞ sup ς∈Z ∥∥Fs(ς)− E(t, 0)Fs(ς) ∥∥ = 0. Theorem 1. Let E = {E(s, 0)}s≥0 : Z → Z be a subfamily of a NeE family such that ∥℧m − E(( t m)im, 0)℧m∥2 ≤ im m diam(Z) where ς ∈ Z, t > 0 and FE ̸= ∅. Let {λt} and {γt}0 0, and sup n≥1 { ∥ςn−1∥, ∥vn−1∥, 2∥ςn−1 − p∥ } ≤ M. Hence lim sup n→+∞ ( ∥℧n − ℧n−1∥ − ∥ςn − ςn−1∥ ) ≤ 0. Using Lemma (2) we have lim n→+∞ ∥℧n − ςn∥ = 0. So, it follows that lim n→+∞ ∥ςn+1 − ςn∥ = lim n→+∞ µn∥℧n − ςn∥. Note that ∥E(τ, 0)ςn − ςn∥ ≤ ∥∥E(τ, 0)ςn − E(τ, 0) 1 γn ∫ γn 0 E(s, 0)ςnds ∥∥ + ∥∥E(τ, 0) 1 γn ∫ γn 0 E(s, 0)ςnds− 1 γn ∫ γn 0 E(s, 0)ςnds ∥∥ + ∥∥ 1 γn ∫ γn 0 E(s, 0)ςnds− ςn ∥∥ ≤ ∥∥E(τ, 0) 1 γn ∫ γn 0 E(s, 0)ςnds− 1 γn ∫ γn 0 E(s, 0)ςnds ∥∥ +2 ∥∥ςn − 1 γn ∫ γn 0 E(s, 0)ςnds ∥∥. (9) From (8), we have∥∥∥∥ςn − 1 γn ∫ γn 0 E(s, 0)ςnds ∥∥∥∥ ≤ ∥∥ςn − ςn+1 ∥∥+ ∥∥∥∥ςn+1 − 1 γn ∫ γn 0 E(s, 0)ςnds ∥∥∥∥ ≤ ∥∥ςn − ςn+1 ∥∥+ (1− µn) ∥∥∥∥ςn − 1 γn ∫ γn 0 E(s, 0)ςnds ∥∥∥∥ M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6147 13 of 18 +ηnλn ∥∥∥∥ςn − 1 γn ∫ γn 0 E(s, 0)ςnds ∥∥∥∥+ ηn(1− λn) ∥∥∥∥ 1 γn ∫ γn 0 E(s, 0)ςnds ∥∥∥∥. Now (1− 1 + µn − ηnλn) ∥∥∥∥ςn − 1 γn ∫ γn 0 E(s, 0)ςnds ∥∥∥∥ ≤ ∥∥ςn − ςn+1 ∥∥+ ηn(1− λn) ∥∥∥∥ 1 γn ∫ γn 0 E(s, 0)ςnds ∥∥∥∥. It gives that∥∥∥∥ςn − 1 γn ∫ γn 0 E(s, 0)ςnds ∥∥∥∥ ≤ 1 µn − ηnλn [ ∥ςn − ςn+1∥+ ηn(1− λn) ∥∥∥∥ 1 γn ∫ γn 0 E(s, 0)ςnds ∥∥∥∥]. (10) From (9), (10) and Lemma (4), we have lim n→+∞ ∥E(τ, 0)ςn − ςn∥ = 0, ∀τ ≥ 0. (11) Note that the sequence {ςn} is bounded and {ςn} → ς̃ weakly. Let ς∗ = PFE(0), then there exists M > 0 such that B(ς∗,M) contains {ςn}. Also, B(ς∗,K) is E(s)-invariant for all s ≥ 0 and so, we can assume that {E(s, 0)}s≥0 is a nonexpansive family on B(ς∗,K). By making use of (11) and Lemma 1(demiclosedness principle), we have ς̃ ∈ FE and therefore lim sup n→+∞ ⟨ς∗, ςn+1 − ς∗⟩ = lim n→+∞ ⟨ς∗, ς̃ − ς∗⟩ ≤ 0. In the end, we prove that ςn → ς∗. Set ς́n = ηn(λnςn) + (1− ηn) 1 γn ∫ γn 0 E(s, 0)ςnds, which gives that ℧n = PZ [ς́n] ∀n ≥ 0. By making use of metric projection property 1, we have ⟨℧n − ς́n,℧n − ς∗⟩ = ⟨ς́n − ℧n, ς ∗ − ℧n⟩ ≤ 0. Therefore ∥℧n − ς∗∥2 = ⟨℧n − ς∗,℧n − ς∗⟩ = ⟨℧n − ς́n,℧n − ς∗⟩+ ⟨ς́n − ς∗,℧n − ς∗⟩ ≤ ⟨ς́n − ς∗,℧n − ς∗⟩ = ⟨ηn(λnςn) + (1− ηn) 1 γn ∫ γn 0 E(s, 0)ςnds− ς∗,℧n − ς∗⟩ = ⟨ηn(λnςn) + ηnλnς ∗ − ηnλnς ∗ − ηnς ∗ + ηnς ∗ +(1− ηn) 1 γn ∫ γn 0 E(s, 0)ςnds− ς∗,℧n − ς∗⟩ M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6147 14 of 18 = ⟨ηnλn(ςn − ς∗)− ηn(1− λn)ς ∗ + (1− ηn)(vn − ς∗),℧n − ς∗⟩ = ηnλn⟨ςn − ς∗,℧n − ς∗⟩ − ηn(1− λn)⟨ς∗,℧n − ς∗⟩ +(1− ηn)⟨vn − ς∗,℧n − ς∗⟩ = ηnλn∥ςn − ς∗∥∥℧n − ς∗∥ − ηn(1− λn)⟨ς∗,℧n − ς∗⟩ +(1− ηn)∥vn − ς∗∥∥℧n − ς∗∥ ≤ ηnλn∥ςn − ς∗∥∥℧n − ς∗∥ − ηn(1− λn)⟨ς∗,℧n − ς∗⟩ +(1− ηn)∥ςn − ς∗∥∥℧n − ς∗∥ = [1− (1− λn)ηn]∥ςn − ς∗∥∥℧n − ς∗∥ − ηn(1− λn)⟨ς∗,℧n − ς∗⟩ ≤ [1− (1− λn)ηn] 2 ∥ςn − ς∗∥2 + 1 2 ∥℧n − ς∗∥ − ηn(1− λn)⟨ς∗,℧n − ς∗⟩, that is, ∥ςn − ς∗∥2 ≤ [1− (1− λn)ηn]∥ςn − ς∗∥2 − 2ηn(1− λn)⟨ς∗,℧n − ς∗⟩. By the convexity of norm, we have ∥ςn+1 − ς∗∥2 ≤ (1− η)∥ςn − ς∗∥2 + η∥℧n − ς∗∥2 ≤ (1− η)∥ςn − ς∗∥2 +η ( [1− (1− λn)ηn]∥ςn − ς∗∥2 − 2ηn(1− λn)⟨ς∗, ςn − ς∗⟩ ) = [1− (1− λn)ηnη]∥ςn − ς∗∥2 − 2(1− λn)ηnη⟨ς∗,℧n − ς∗⟩. So all the conditions of Lemma 1 are satisfied. Therefore, we obtain that ςn → ς∗. If we put 1 instead of µn, ∀ n ≥ 0, in Theorem 2, then we have the following result. Corollary 1. Let E = {E(s, 0)}s≥0 : Z → Z be nonexpansive family with condition FE ̸= ∅. Let {ςn} be the sequence ςn+1 = (1− µn)ςn + µnPZ [(1− ηn) γn ∫ γn 0 E(s, 0)ςnds+ ηn(λnςn) ] , ∀n ≥ 0. (12) Where {ηn}, {µn} and {λn} are sequences in [0, 1] and {γn} is a sequence in (0,∞). Sup- pose the following axioms holds: (i) limn→+∞ ηn = 0,Σ+∞ n=0ηn = +∞ and limn→∞ λn = 1; (ii) limn→+∞ γn = ∞ and limn→+∞ γn−1 γn = 1. Then the sequence {ςn} strongly converge to a point ς∗ ∈ FE. Remark 4. The algorithm ςn+1 = (1− µn)ςn + µnPZ [ ηn(λnςn) + (1− ηn) 1 γn ∫ γn 0 E(s, 0)ςnds ] , ∀n ≥ 0, has just weak convergence. But, the sequence (8) (with λn → 1) has strong convergence. M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6147 15 of 18 Using Remark 2, we have the following remark. Remark 5. The results given in [9], become special case of our work, if an evolution family is periodic of every positive real number. 4. Example and Open Problem Example 2. Let H := L2([0, π], C) be the Hilbert space of all square integrable functions on [0, π] and S = {S(a) : a ≥ 0} be a semigroup defined by (S(a)ϑ)(t) = 2 π ∞∑ m=1 e−am2 cm(ϑ) sinmΥ, Υ ∈ [0, π], a ≥ 0, (13) where cm(ϑ) := ∫ π 0 y(s) sin(ms)ds. Obviously, S is a strongly continuous and non-expansive semigroup on H and it is generated by the linear operator A given by ϑ̈ = Aϑ. Now Consider the following non-autonomous Cauchy problem ∂℧(Υ,ζ) ∂Υ = k(Υ)∂ 2℧(Υ,ζ) ∂2ζ , Υ > 0, ζ ∈ [0, π], ℧(Υ, 0) = ℧(Υ, π) = 0, Υ ≥ 0, ℧(0, ζ) = q(ζ), where q(Υ) ∈ H, and k : R+ → [1,∞) is a non-expansive and periodic function, i.e., k(Υ + p) = k(Υ) for all Υ ∈ R+ for some p ≥ 1. LetK(Υ) = ∫ Υ 0 k(Υ)dΥ. Clearly the solution y(.) of the above Cauchy problem satisfies the evolution property: y(Υ) = E(Υ, ξ)y(ξ), (14) where E(Υ, ξ) = S(K(Υ)−K(ξ)). See [[25], Example 2.9b]. We can find ϑ ≥ 0 such that the function Υ 7→ eϑΥ∥℧(Υ)∥ is bounded on R+. In fact, we have∫ +∞ 0 ∥E(Υ, 0)ϑ∥2dΥ = 2 π ∫ +∞ 0 +∞∑ ϑ=1 c2ϑ(ϑ)e −2ϑ2K(Υ)dΥ = 2 π +∞∑ ϑ=1 c2ϑ(ϑ) ∫ +∞ 0 e−2ϑ2K(Υ)dΥ = ∥ϑ∥22 ∫ +∞ 0 e−2ϑ2K(Υ)dΥ = ∥ϑ∥22 ∫ +∞ 0 e−2K(Υ)dΥ. (15) Now consider ∫ +∞ 0 e−2K(Υ)dΥ = +∞∑ j=0 ∫ (j+1)z jz e−2K(Υ)dΥ M. Sarwar et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6147 16 of 18 = +∞∑ j=0 ∫ z 0 e−2K(jz+Υ)dΥ = +∞∑ j=0 e−2jK(z) ∫ z 0 e−2K(Υ)dΥ ≤ z +∞∑ j=0 e−2jK(z) = ze2K(z) e2K(z) − 1 := C. (16) Hence, ∫ +∞ 0 ∥E(Υ, 0)ϑ∥2dΥ ≤ C∥ϑ∥22. (17) Now, by Theorem 8 in [26], for M ≥ 1 the growth bound ω0 of the family E satisfy ω0 ≤ −1 2M , for further details see [26], obviously this shows that the evolution family is non-expansive on the Hilbert space H. Therefore both the net ςt in Theorem ?? and the sequence ςm in Theorem 2 strongly converges to a FP of a subfamily of the above NeE family. Open problem: We leave open the question whether all the main results could be generalized for the whole periodic and then for the general non-periodic evolution families in Hilbert spaces? 5. Conclusion In this work, we investigated the fixed point properties of a specific subfamily within a non-expansive evolution family of bounded linear operators on a Hilbert space H. By utilizing the framework of nets and a progression algorithm, we established several results concerning the strong convergence of sequences to a common fixed point of the consid- ered subfamily. The theoretical findings were further supported by a concrete example, demonstrating the applicability of the developed framework. We conclude by posing an open problem to inspire future research in this direction. Acknowledgements The authors M. Sarwar, M.Y.B.Mufarrej and K. Abodayeh would like to thank Prince Sultan University for paying the APC and for the support through the TAS research lab. M. Sarwar et al. / Eur. J. 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