Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 118, pp. 1–13. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.118 TOPOLOGICAL PROPERTIES OF THE SOLUTION SET AND T-CONTROLLABILITY FOR SECOND-ORDER NEUTRAL EVOLUTION EQUATIONS WITH DELAY TIANTIAN WANG, HONGXIA FAN Abstract. In this article we study the topological properties of the mild solution set and T- controllability for a class of second-order neutral evolution equations with finite delay. Firstly, we establish that the set of mild solution is nonempty and compact, via Darbo-Sadovskii’s fixed point theorem with sufficient conditions provided to guarantee its Rδ-property. Secondly, we show the trajectory-controllability of the equation, under an invertibility assumption on the control operator. Finally, with an example, we illustrate the applicability of the abstract results. 1. Introduction The topological properties of solution sets of deterministic problems play an important role in qualitative research. In recent years, the topological structure of differential equation solution sets (including contractibility, acyclic, AR, Rδ-set, etc.) has attracted the attention of many scholars. Among them, Rδ-set has very important research value. In 1890, Peano [33] established that the Cauchy initial value problem x′(t) = f(t, x(t)), t ∈ [0, T ], x(0) = x0, where f : [0, T ]× Rn → Rn is a continuous function, has local solutions although the uniqueness property does not hold in general. This discovery became a driving force in studying the structure of solution sets. At the same time, Peano also proved that, in the case n = 1, the solution set is nonempty, compact and connected in the standard topology of real line, for t in some neighbourhood of 0. In 1942, Aronszajn [3] discovered a more precise characterization of the solution set, and proved that the solution set considered by Peano for the Cauchy problem is an Rδ-set. Since then, the topological structure and geometric structure of solution sets have been widely studied by many scholars [2, 13, 18, 19, 20, 21, 22, 23, 24, 40]. The Rδ-property is a core aspect in the study of the topological structure of solution sets for differential equations, integral equations and differential inclusions. Rδ-set is the intersection of a sequence of decreasing compact sequences. It is nonempty, compact, connected and may not be a single point solution set. From the algebraic topology point of view, an Rδ-set has the same homology group as a space of points. Chen et al. [13] discussed the Rδ-property of solution sets for nonlinear delay evolution inclusions on compact and noncompact intervals, respectively. By exploiting Krasnoselskii’s fixed point Theorem, Hoa et al. [22] investigated the topological structure of mild solution sets of the following fractional order neutral evolution equations CDq[x(t)− h(t, x(t), xt)] = Ax(t) + f(t, x(t), xt), t > 0, x0(t) = ξ(t), t ∈ [−r, 0], 2020 Mathematics Subject Classification. 93B05, 47J35, 93C10. Key words and phrases. Rδ-set; topological properties; trajectory-controllability; neutral evolution equations. ©2025. This work is licensed under a CC BY 4.0 license. Submitted July 9, 2025. Published December 18, 2025. 1 2 T. WANG, H. FAN EJDE-2025/118 there CDq is the Caputo fractional derivative of order q ∈ (0, 1). The functions f, g : R+×E×C0. xt : [−r, 0] → E is defined by xt(s) = x(t+ s) for all s ∈ [−r, 0]. Recently, Jiang et al. studied the topological properties of the solution set for Riemann-Liouville fractional order evolution equations and Hilfer fractional order evolution equations, and obtained the approximate controllability results of the studied systems under the assumption that the corresponding linear systems are approximately controllable, see articles [23, 24]. In 2025, Yang et al. [40] studied topological properties of solution sets for Caputo fractional order evolution inclusions. For a comprehensive analysis of the topological properties of solution sets, we refer interested readers to [2, 18, 19, 20, 21]. For the study of the equations, we should not only discuss the development characteristics and dynamic properties of the equation itself, such as regularity, stability and so on, but also to study the dynamic behavior of the equations under the action of external factors to achieve the desired state. Thus, controllability is a thought-provoking and interesting problem. As a cornerstone of mathematical control theory, the controllability concept was first formally introduced by Kalman (1963) in his seminal work [26]. Since then, there have been many results about controllability research [39, 37, 1]. Trajectory controllability (T-controllability) was proposed by Chalishajar in [8] to establish a more robust form of control for dynamic systems. In T-controllability problems, we look for a control that steers the system along a prescribed trajectory, rather than a control simply steering a given initial state to desired final state. Hence, T-controllability is a stronger concept of control- lability. The research on T-controllability of the system is helpful to reveal the control rule of the system on the specific trajectory. In today’s era of swift technological development, many systems such as aerospace systems, robot systems and intelligent transportation systems have put forward higher requirements for precise trajectory control. Therefore, a growing number of scholars have turned their focus to this issue. Chalishajar [8] studied the T-controllability in both finite and infinite dimensional abstract nonlinear integro-differential systems. A few years back, Muslim and his team [31] have used Gronwall’s inequality to establish the T-controllability of nonlinear fractional order differential equations. The research team led by Chalishajar [9] studied T-controllability of neutral stochastic integro-differential equations with mixed fractional Brownian motion. In 2018, Muslim et al. [32] discussed the exact controllability and T-controllability of the following second-order evolution equations with impulses and deviated arguments x′′(t) = A(t)x(t) +Bu(t) + f(t, x(a(t)), x[h(x(t), t)]), t ∈ J = [0, T ], t ̸= tk, x(0) = x0, x′(0) = y0, t ∈ J, a(t) ≤ t, ∆x(tk) = I1k(x(tk)), ∆x′(tk) = I2k(x(tk)), k = 1, 2, . . .m, where x(t) is the state function, the control function u ∈ L2(J, U), where U is a Hilbert space known as the control space. A is the infinitesimal generator of a strongly continuous cosine family of linear operators {C(t) : t ∈ R} on X. The nonlinear function f : J ×X ×X → X. Recently, Chalishajar [11] discussed the existence and T-controllability of Hilfer fractional or- der neutral stochastic differential equations with deviated arguments. In [12], by applying the stochastic analysis technique and Banach fixed point Theorem, it is concluded that conformable fractional order stochastic integro-differential systems with infinite delay has a mild solution, in addition, the results of T-controllability of the system are proved. In 2024, Kasinathan et al. [29] obtained the T-controllability for the time-variant impulsive neutral stochastic functional integro-differential equations. For more details on T-controllability readers can refer the articles [7, 10, 14, 15, 16, 28]. Neutral differential equations can describe dynamical systems with memory and delayed feed- back. The study of second-order neutral evolution equations can not only deepen the understand- ing of the nature of delay dynamical systems, but also provide a mathematical tool for solving complex problems in engineering, biological and social sciences. There are many interesting results on neutral evolution equations have been gained in [25, 30, 34, 35]. EJDE-2025/118 TOPOLOGICAL PROPERTIES OF THE SOLUTION SET AND T-CONTROLLABILITY 3 To the best of our knowledge, the existing literature has scarcely addressed both the topolog- ical properties of the mild solution set and T-controllability for second-order neutral evolution equations with delay. Inspired by these previous studies, this paper investigates the following second-order neutral evolution equations in Banach space X, d dt [x′(t)−G(t, xt)] = Ax(t) + Bu(t) + F (t, xt), t ∈ J = [0, T ], x(t) = φ(t), x′(0) = x0, t ∈ [−τ, 0], (1.1) where the state function x(t) ∈ X, A : D(A) ⊂ X → X is the infinitesimal generator of strongly continuous cosine family {C(t) : t ∈ R}. The control function u(·) takes in Banach space U. B is a bounded linear operator from U to X. The nonlinear functions F,G : J × C([−τ, 0];X) → X satisfy certain conditions. φ ∈ C([−τ, 0];X) and xt ∈ C([−τ, 0];X) is defined by xt(θ) = x(t + θ) for each θ ∈ [−τ, 0]. The main contributions and novelties of this paper are listed below. (i) Up to now, there are no results on the topological properties of mild solution sets for second-order neutral evolution equations. (ii) There are no papers on the T-controllability of second-order neutral evolution equations with finite delay. (iii) Through example, we demonstrate the application of our results. The rest of this article is organized as follows. In section 2, we introduce notation and preliminaries needed to establish our main results. Section 3 is devoted to the compactness and Rδ-property of the mild solution set for equation (1.1). Results on T-controllability of equation (1.1) are presented in Section 4. Finally, Section 5 provides an example demonstrating the effectiveness of our theoretical results. 2. Preliminaries In this section, we first recall some useful definitions and provide preliminary results. Let (X, ∥ · ∥) and (K, ∥ · ∥K) be two Banach spaces. L2(J,X) denote the Banach space of all square integrable functions from J into X with the norm ∥x∥L2 = ( ∫ T 0 ∥x(t)∥2dt) 1 2 and C(J,X) denote the Banach space of all continuous functions from J into X with the super norm ∥x∥C = supt∈J ∥x(t)∥. Set Ct = C([−τ, t];X) represent the Banach space of all continuous functions from [−τ, t] into X with the super norm ∥x∥Ct = supθ∈[−τ,t] ∥x(θ)∥, t ∈ J . The space of all bounded linear operators from X into K is denoted by L(X,K). We use the same notation ∥ · ∥ for the norm of L(X,K). Particulary, L(X) will denote L(X,X). Let V and Z be two metric spaces. For convenience, we introduce following symbols. P(V) = {D ⊆ V : D is nonempty}, Pb(V) = {D ⊆ V : D is nonempty and bounded}, Pf (V) = {D ⊆ V : D is nonempty and closed}, K(V) = {D ⊆ V : D is nonempty and compact (convex)}. The graph of set-valued map f : V → Pf (Z) is represented by G(f) = {(v, z) ∈ V×Z : z ∈ f(v)}. If ∆ is subset of Z, then we denote the complete pre-image of ∆ by f−1(∆) = {v ∈ V : f(v)∩∆ ̸= ∅}. If G(f) is closed in V× Z, then f is referred to as closed. Definition 2.1 ([27]). Let f : V → Pf (Z) be a set-valued map. (i) If D ⊂ V is compact, f(D) is relatively compact, then f is said to be quasicompact. (ii) For x0 ∈ V, if for any neighborhood O(f(x0)) of f(x0), there exists a neighborhood O(x0) of x0 such that f(x) ⊂ O(f(x0)) for all x ∈ O(x0), then f is called upper semicontinuous (u.s.c. for short). If V ∩ Z ̸= ∅, then v ∈ V ∩ Z is said to be a fixed point of f if v ∈ f(v). The set of all fixed points of f is denoted by Fix(f). 4 T. WANG, H. FAN EJDE-2025/118 Definition 2.2 ([17]). D ⊂ P(V) is referred to as contractible if there exist a point x0 ∈ D and a continuous map f : D × [0, 1] → D such that f(x, 0) = x and f(x, 1) = x0, for every x ∈ D. Definition 2.3 ([17]). D ⊂ P(V) is called to be an Rδ-set if D = ∩∞ n=1Dn, where {Dn}∞n=1 is a decreasing sequence of nonempty and compact contractible sets. Lemma 2.4 ([6]). Let ℏ : V → X be continuous. Assume that ℏ is a proper map, that is to say for D ⊂ K(X), ℏ−1(D) is compact. Moreover, suppose that there exists a sequence {ℏn}∞n=1 of mapping from V into X such that (i) {ℏn}∞n=1 converges uniformly to ℏ in V and ℏn is proper. (ii) For fixed y0 ∈ X and any y ∈ O(y0), then equation ℏn(x) = y admits exactly only solution, where O(y0) is the neighborhood of y0. Then, ℏ−1(y0) is an Rδ-set. Let us recall the definition and properties of the measure of noncompactness. Set D be a bounded subset of X. The Hausdorff measure of noncompactness is expressed as α(D) = inf{ε : D has a finite ε-net}. Lemma 2.5 ([4]). Let D,D1 ⊂ X be bounded, then the following properties are satisfied: (i) D is precompact if and only if α(D) = 0; (ii) α(D) = α(D̄) = α(coD), where D̄ and coD are the closure and the convex hull of D, respectively; (iii) α(D) ≤ α(D1) when D ⊂ D1; (iv) α(D +D1) ≤ α(D) + α(D1), where D +D1 = {x+ y : x ∈ D, y ∈ D1}; (v) α(D ∪D1) ≤ max {α(D), α(D1)}; (vi) α(λD) ≤ |λ|α(D) for any λ ∈ R. For each W ⊂ C(J,X), we define ∫ t 0 W (s)ds = ∫ t 0 v(s)ds, where W (s) = {v(s) : v ∈W} ⊂ X. Lemma 2.6 ([4]). IfW ⊂ C(J,X) is bounded and equicontinuous, then t→ α(W (t)) is continuous on J and α(W ) = max t∈J α(W (t)), α( ∫ t 0 W (s)ds) ≤ ∫ t 0 α(W (s))ds, for all t ∈ J. Lemma 2.7 ([4]). Let {vn}∞n=1 be a sequence of Bochner integrable functions from J to X with ∥vn(t)∥ ≤ m(t) for almost all t ∈ J and every n ≥ 1, where m(t) ∈ L(J,R+), then the function ψ(t) = α({vn}∞n=1) ∈ L(J,R+) and satisfies α({ ∫ t 0 vn(s)ds}) ≤ 2 ∫ t 0 ψ(s)ds. Remark 2.8 ([41]). If W ⊂ C(J,X) is bounded and equicontinuous, then coW ⊂ C(J,X) is also bounded and equicontinuous, where coW stands for the convex hull of W . Lemma 2.9 ([5]). For every bounded subset D of X and ε > 0, there exists a sequence {wn}∞n=1 ⊂ D satisfying α(D) ≤ 2α({wn}∞n=1) + ε. Definition 2.10 ([4]). For D∗ ⊂ D, if there exists a positive constant γ < 1 such that α(f(D∗)) ≤ γα(D∗), then f : D ⊂ X → X is said that α-contractive. Lemma 2.11 ([41]). Supposed that D be a closed and convex subset of Banach space X, F : D → D be a continuous map and F(D) be bounded. For every bounded subset D0 ⊂ D, set F1(D0) = F(D0), Fn(D0) = F(coFn−1(D0)), n = 2, 3, . . . . If there are a constant 0 ≤ γ < 1 and a positive integer n0 to such that, for any bounded subset D0 ⊂ D, α(Fn0(D0)) ≤ γα(D0), then there exists a D∗ ⊂ D satisfying α(F(D∗)) = 0. EJDE-2025/118 TOPOLOGICAL PROPERTIES OF THE SOLUTION SET AND T-CONTROLLABILITY 5 Lemma 2.12 (Darbo-Sadovskii’s fixed point Theorem [4]). Supposed that D be a bounded, closed and convex subset of X, and the continuous map F : D → D be a α-contraction. Then Fix(F) is nonempty on D. Next, we review some facts about strongly continuous cosine family. Definition 2.13 ([36]). A one parameter family {C(t) : t ∈ R} ⊂ L(X) is called a strongly continuous cosine family if (i) C(s+ t) + C(s− t) = 2C(s)C(t), for all s, t ∈ R; (ii) C(0) = I; (iii) C(t)x is continuous in t on R for each fixed x ∈ X. Define the associated sine family {S(t) : t ∈ R} by S(t)x = ∫ t 0 C(s)xds, x ∈ X, t ∈ R. The infinitesimal generator of a strongly continuous cosine family {C(t) : t ∈ R} is the operator A : D(A) ⊂ X → X defined by Ax = d2 dt2 C(t)x |t=0, x ∈ D(A), where D(A) = {x ∈ X : C(·)x ∈ C2(R,X)}. Define E = {x ∈ X : C(·)x ∈ C1(R,X)}. The properties of the strong continuous cosine family are given below. Lemma 2.14 ([36]). If A : D(A) ⊂ X → X is the infinitesimal generator of strongly continuous cosine family {C(t) : t ∈ R} and {S(t) : t ∈ R} is the associated sine family family, then (i) there exist constants M ≥ 0 and µ ≥ 0, such that ∥C(t)∥ ≤Meµ|t|, and ∥S(t)∥ ≤Meµ|t|; (ii) if x ∈ E, then S(t)x ∈ D(A) and d dtC(t)x = AS(t)x; (iii) if x ∈ D(A), then C(t)x ∈ D(A) and d2 dt2C(t)x = AC(t)x = C(t)Ax; (iv) if x ∈ E, then S(t)x ∈ D(A) and d2 dt2S(t)x = AS(t)x. We know from the Lemma 2.14(i) that {C(t) : t ∈ J} and {S(t) : t ∈ J} are uniformly bounded, that is, there are positive numbers M1, M2 such that ∥C(t)∥ ≤M1, ∥S(t)∥ ≤M2, t ∈ J. Now, we give the concept of mild solution of equation (1.1) and the definition of T-controllability of equation (1.1). Definition 2.15. A function x ∈ CT is said to be a mild solution of equation (1.1) if for u ∈ L2(J,U) it satisfies x(t) = C(t)φ(0) + S(t)(x0 −G(0, φ)) + ∫ t 0 C(t− s)G(s, xs)ds + ∫ t 0 S(t− s)(Bu(s) + F (s, xs))ds, t ∈ J, (2.1) and x(t) = φ(t), t ∈ [−τ, 0]. Let Φ be the set of all functions z(·) defined on J such that z(0) = φ(0), z(T ) = xT , z ′(0) = x0, and z(·) is twice continuously differentiable, where xT represents desired final state. The set of all feasible trajectories for equation (1.1) is denoted by Φ. Definition 2.16. The equation (1.1) is said to be trajectory controllable (T-controllable) on J if for any z ∈ Φ, there exists a control function u ∈ L2(J,U) such that the mild solution x(·) of equation (1.1) satisfies x(t) = z(t) a.e. on J . 6 T. WANG, H. FAN EJDE-2025/118 3. Topological properties of the mild solution set To obtain our main results, we need to impose the following hypotheses: (H1) B : L2(J,U) → L2(J,X) is a bounded linear operator, that is, there is a positive constant b such that ∥B∥ ≤ b. (H2) The nonlinear function F : J × C0 → X satisfies the following conditions: (i) F (t, ·) : C0 → X is continuous for t ∈ J and F (·, x) : J → X is measurable for each x ∈ C0. (ii) There exist a function mF ∈ L2(J,R+) and a constant m1 > 0 such that ∥F (t, x)∥ ≤ mF (t) +m1∥x∥C0 , for all x ∈ C0, a.e. t ∈ J . (iii) There exists a constant L1 > 0 such that for any bounded subset D ⊂ C0 and a.e. t ∈ J , α(F (t,D)) ≤ L1 sup θ∈[−τ,0] α(D(θ)). (H3) The nonlinear function G : J × C0 → X satisfies the following conditions: (i) G(t, ·) : C0 → X is continuous for t ∈ J and G(·, x) : J → X is measurable for each x ∈ C0. (ii) There exist a non-negative function mG ∈ L2(J,R) and a constant m2 > 0 such that ∥G(t, x)∥ ≤ mG(t) +m2∥x∥C0 , for all x ∈ C0, a.e. t ∈ J . In particular, when t = 0, mG(t) = 0. (iii) There exists a constant L2 > 0 such that for any bounded subset D ⊂ C0 and a.e. t ∈ J , α(G(t,D)) ≤ L2 sup θ∈[−τ,0] α(D(θ)). (H4) {C(t) : t ∈ J} and {S(t) : t ∈ J} are uniformly continuous. For the sake of simplicity, we denote Ξ(u) = {x ∈ C(J,X): x satisfies (2.1)}. Theorem 3.1. If hypotheses (H1)–(H4) hold, then for u ∈ L2(J,U), Ξ(u) is nonempty and compact provided (M1m2 +M2m1)T < 1. Proof. We define the operator Γ : C(J,X) → C(J,X) by (Γx)(t) = C(t)φ(0) + S(t)(x0 −G(0, φ)) + ∫ t 0 C(t− s)G(s, xs)ds + ∫ t 0 S(t− s)(Bu(s) + F (s, xs))ds, t ∈ J. (3.1) It is clear to see that x ∈ Ξ(u) is equivalent to x ∈ Fix(Γ). The proof is divided into 4 steps. Step 1. There exists r > 0 such that Γ maps the set Br := {x ∈ C(J,X) : ∥x∥C ≤ r} to itself. In fact, for any x ∈ Br and ρ ∈ J , one has ∥x∥C0 = sup θ∈[−τ,0] ∥x(ρ+ θ)∥ ≤ sup s∈[−τ,0] ∥x(s)∥+ sup s∈[0,ρ] ∥x(s)∥ ≤ ∥φ∥C0 + sup s∈[0,ρ] ∥x(s)∥. (3.2) EJDE-2025/118 TOPOLOGICAL PROPERTIES OF THE SOLUTION SET AND T-CONTROLLABILITY 7 In addition, for any x ∈ Br and t ∈ J , from hypotheses (H1)–(H3), (3.1), (3.2) and well-known Hölder inequality, we have ∥(Γx)(t)∥ ≤M1∥φ∥C0 +M2(∥x0∥+ ∥G(0, φ)∥) +M1 ∫ t 0 ∥G(s, xs)∥ds +M2 ∫ t 0 ∥Bu(s)∥ds+M2 ∫ t 0 ∥F (s, xs)∥ds ≤M1∥φ∥C0 +M2∥x0∥+M2m2∥φ∥C0 +M1 √ T∥mG∥L2(J,R) +M2b √ T∥u∥L2(J,U) +M2 √ T∥mF ∥L2(J,R+) + (M1m2 +M2m1)T∥φ∥C0 + (M1m2 +M2m1)Tr ≤ r. (3.3) This shows that Γ(Br) ⊆ Γ(Br). Step 2. The map Γ is continuous on Br. Let {xn}∞n=1 ⊆ Br and limn→∞ xn = x in Br. According to hypotheses (H2)(i) and (H3)(i), we can infer that lim n→∞ F (t, (xn)t) = F (t, xt), lim n→∞ G(t, (xn)t) = G(t, xt), t ∈ J. Consequently, for t ∈ J , from hypotheses (H2)(ii), (H3)(ii) and the Lebesgue dominated conver- gence Theorem, we can prove that ∥(Γxn)(t)− (Γx)(t)∥ ≤M1 ∫ t 0 ∥G(s, (xn)s)−G(s, xs)∥ds+M2 ∫ t 0 ∥F (s, (xn)s)−F (s, xs)∥ds→ 0, as n→ ∞. Therefore, we can affirm that the map Γ is continuous on Br. Step 3. We prove that Γ is a compact operator. To this goal, let 0 ≤ t1 < t2 ≤ T and for any x ∈ Br, from hypotheses (H2)(ii), (H3)(ii), (H4) and Hölder inequality, we have ∥(Γx)(t2)− (Γx)(t1)∥ ≤ ∥[C(t2)− C(t1)]φ(0)∥+ ∥[S(t2)− S(t1)](x 0 −G(0, φ))∥ + ∥ ∫ t2 t1 C(t2 − s)G(s, xs)ds∥+ ∥ ∫ t2 t1 S(t2 − s)F (s, xs)ds∥ + ∥ ∫ t1 0 [C(t2 − s)− C(t1 − s)]G(s, xs)ds∥ + ∥ ∫ t1 0 [S(t2 − s)− S(t1 − s)]F (s, xs)ds∥ + ∥ ∫ t2 t1 S(t2 − s)Bu(s)ds∥+ ∥ ∫ t1 0 [S(t2 − s)− S(t1 − s)]Bu(s)ds∥ ≤ ∥[C(t2)− C(t1)]φ(0)∥+ ∥[S(t2)− S(t1)](x 0 −G(0, φ))∥ + (t2 − t1)M1m2∥xs∥C0 + √ t2 − t1M1∥mG∥L2(J,R) + (t2 − t1)M2m1∥xs∥C0 + √ t2 − t1M2∥mF ∥L2(J,R+) + ∫ t1 0 ∥[C(t2 − s)− C(t1 − s)]G(s, xs)∥ds + ∫ t1 0 ∥[S(t2 − s)− S(t1 − s)]F (s, xs)∥ds + √ t2 − t1M2b∥u∥L2(J,U) + b ∫ t1 0 ∥[S(t2 − s)− S(t1 − s)]u(s)∥ds := 10∑ i=1 Ii. In fact, it suffices to prove that when t2 → t1, Ii → 0 (i = 1, 2, . . . , 10). By the strong continuity of {C(t)} and {S(t)}, we can infer that when t2 → t1, I1 → 0 and I2 → 0. In addition, limt2→t1 Ii = 0 (i = 3, 4, 5, 6, 9). According to condition (H4), we can obtain that when t2 → t1, I7 → 0, I8 → 0 8 T. WANG, H. FAN EJDE-2025/118 and I10 → 0. Therefore, when t2 → t1, it can be concluded that ∥(Γx)(t2)− (Γx)(t1)∥ → 0, then Γ(Br) is equicontinuous. Let Ω = coΓ(Br) be a bound, closed and convex set. It follows from Remark 2.8 that Ω = coΓ(Br) is also equicontinuous. Thus, the continuity and boundedness of Γ : Ω → Ω can be easily demonstrated. For any bounded subset Ω0 ⊆ Ω, set Γ1(Ω0) = Γ(Ω0), Γ n(Ω0) = Γ(coΓn−1(Ω0)), n = 2, 3, . . . . By Lemma 2.6, Lemma 2.7, Lemma 2.9, (H2)(iii) and (H3)(iii), we can deduce that, for any ε > 0, there is a set Ω1 = {zn}∞n=1 ⊆ Ω0 satisfying α(Γ1(Ω0)(t)) = α(Γ(Ω0)(t)) ≤ 2α(Γ(Ω1)(t)) + ε ≤ 4M1 ∫ t 0 L2 sup θ∈[−τ,0] α(Ω1(θ + s))ds+ 4M2 ∫ t 0 L1 sup θ∈[−τ,0] α(Ω1(θ + s))ds+ ε ≤ 4(M1L2 +M2L1)tα(Ω0) + ε. (3.4) Since ε > 0 is arbitrary, one has α(Γ1(Ω0)(t)) ≤ 4(M1L2 +M2L1)tα(Ω0). Similarly, for any ε > 0, the sequence Ω2 = {vn}∞n=1 ⊆ coΓ1(Ω0) can be found and for all t ∈ J , α(Γ2(Ω0)(t)) = α(Γ(coΓ1(Ω0))(t)) ≤ 2α(Γ(Ω2)(t)) + ε ≤ 4M1 ∫ t 0 L2 sup θ∈[−τ,0] α(Ω2(θ + s))ds+ 4M2 ∫ t 0 L1 sup θ∈[−τ,0] α(Ω2(θ + s))ds+ ε ≤ 4(M1L2 +M2L1) ∫ t 0 sup σ∈[0,s] α((coΓ1(Ω0))(σ))ds+ ε. ≤ 4(M1L2 +M2L1) ∫ t 0 4(M1L2 +M2L1)sα(Ω0)ds+ ε. ≤ [4(M1L2 +M2L1)t] 2 2! α(Ω0) + ε. (3.5) Since ε > 0 is arbitrary, we can get α(Γ2(Ω0)(t)) ≤ [4(M1L2 +M2L1)t] 2 2! α(Ω0). Using mathematical induction, one has α(Γn(Ω0)(t)) ≤ [4(M1L2 +M2L1)t] n n! α(Ω0) ≤ [4(M1L2 +M2L1)T ] n n! α(Ω0). (3.6) By the Stirling formula, there is a constant ϱ ∈ (0, 1) satisfying n! = √ 2πn( n e )ne 1 12n+ϱ . This and (3.6) imply that lim n→∞ [4(M1L2 +M2L1)T ] n n! = lim n→∞ [4(M1L2 +M2L1)T ] n √ 2πn(ne ) ne 1 12n+ϱ = 0. Therefore, there are γ ∈ (0, 1) and n ≥ 1 such that α(Γn(Ω0)(t)) ≤ γα(Ω0). Moreover, we can obtain α(Γn(Ω0)) = max t∈J α(Γn(Ω0)(t)) ≤ γα(Ω0). It follows from Lemma 2.11 that there is a Ω̂ ⊂ Ω ⊂ Br such that α(Γ(Ω̂)) = 0. EJDE-2025/118 TOPOLOGICAL PROPERTIES OF THE SOLUTION SET AND T-CONTROLLABILITY 9 Employing the generalized Ascoli-Arzela Theorem, we can obtain the compactness of Γ. Therefore, Lemma 2.12 shows that Γ has a fixed point in Br, that is, Ξ(u) is nonempty. Step 4. The fixed point set Fix(Γ) is compact. By Step 1, we can acquire the boundedness of Fix(Γ). Thus, the compactness of Γ ensures that Γ(Fix(Γ)) is relatively compact. In addition, in view of continuity Γ, we can easily see that Fix(Γ) is closed. Hence, from Fix(Γ) ⊆ Γ(Fix(Γ)), we can obtain that Fix(Γ) is relatively compact, that is, Ξ(u) is compact. This completes the proof. □ Theorem 3.2. Assume that (H1)–(H4) are satisfied, then for u ∈ L2(J,U), Ξ(u) is an Rδ-set. Proof. In view of assumptions (H2)(i), (H3)(i) and the Lasota-Yorke approximation Theorem [17], we can find two local Lipschitz sequences {Fn}∞n=1, {Gn}∞n=1 : J × C0 → X to ensure that ∥Fn(t, x)− F (t, x)∥ < εn, ∥Gn(t, x)−G(t, x)∥ < εn, t ∈ J, x ∈ C0, (3.7) where εn → 0, as n→ ∞. Hence, according to (H2)(ii) and (H3)(ii), we obtain ∥Fn(s, xs)∥ ≤ 1 +mF (s) +m1∥xs∥C0 , n = 1, 2, . . . , ∥Gn(s, xs)∥ ≤ 1 +mG(s) +m2∥xs∥C0 , n = 1, 2, . . . . (3.8) Next, we define an approximation operator as (Γnx)(t) = C(t)φ(0) + S(t)(x0 −G(0, φ)) + ∫ t 0 C(t− s)Gn(s, xs)ds + ∫ t 0 S(t− s)(Bu(s) + Fn(s, xs))ds, t ∈ J, n = 1, 2, . . . . According to (3.7) and (3.8), we can easily see that the operator Γn is well defined. For x ∈ Ξ(u), t ∈ J , one has ∥(I − Γn)(x)(t)− (I − Γ)(x)(t)∥ ≤M1 ∫ t 0 ∥Gn(s, xs)−G(s, xs)∥ds+M2 ∫ t 0 ∥Fn(s, xs)− F (s, xs)∥ds ≤ 2(M1 +M2)Tεn → 0, n→ ∞, (3.9) where I represents identical operator. It is easy to see from (3.9) that I − Γn converges to I − Γ uniformly on C(J,X). Moreover, similar to the proof of Theorem 3.1, for any y ∈ C(J,X), one may deduce that (I − Γn)(x) = y (3.10) yields at least a mild solution. Further, similar to the argument used in [38, Theorem 3.6], by applying the local Lipschitz continuity of {Fn}∞n=1 and {Gn}∞n=1, we can prove that the mild solution of equation (3.10) is unique. Next, we show that the mappings I−Γn and I−Γ are proper for given n = 1, 2, . . . . According to the Step 2 of the proof in Theorem 3.1, I − Γn is continuous. Now, for every compact set D in C(J,X), we need only confirm that (I − Γn) −1(D) is a compact set. Set (I − Γn)(K) = D. Thus, for sequence {kn}∞n=1 ⊂ K, there is a sequence {yn}∞n=1 ⊂ D satisfies (I − Γn)(kn) = yn, that is, kn(t) = yn(t) + C(t)φ(0) + S(t)(x0 −G(0, φ)) + ∫ t 0 C(t− s)Gn(s, (kn)s)ds + ∫ t 0 S(t− s)(Bu(s) + Fn(s, (kn)s))ds, t ∈ J, n = 1, 2, . . . . (3.11) Hence, from (3.8), (3.11), the boundedness of {yn}∞n=1 and Step 1 of Theorem 3.1, we can deduce that {kn}∞n=1 is bounded in C(J,X). In addition, from the compactness of D and the proof of Step 3 of Theorem 3.1, we can obtain that {kn}∞n=1 is relatively compact. Additionally, the continuity of I − Γn and the closedness of D imply that the set K is closed, it means that K is compact. Thus, I − Γn is proper. Following analogous arguments, we can acquire that I − Γ is also proper. Consequently, it can be obtained by Lemma 2.4 that Ξ(u) is an Rδ-set. The proof is completed. □ 10 T. WANG, H. FAN EJDE-2025/118 4. T-controllability In this section, we investigate the T-controllability of equation (1.1). To establish our results, we introduce the hypothesis (H5) The control operator B is invertible. Theorem 4.1. If hypotheses (H1)–(H5) hold, then second-order neutral evolution equation (1.1) is T-controllable on J . Proof. Let z(t) be the given trajectory on Φ. We are looking for a control function u(t) that satisfies z(t)− C(t)φ(0)− S(t)(x0 −G(0, φ))− ∫ t 0 C(t− s)G(s, zs)ds− ∫ t 0 S(t− s)F (s, zs)ds = ∫ t 0 S(t− s)Bu(s)ds. Taking the second time derivative of both sides with respect to t, we can obtain z′′(t)−AC(t)φ(0)−AS(t)x0 + (S(t)G(0, φ))′′ − ∫ t 0 AC(t− s)G(s, zs)ds −G′(t, zt)− ∫ t 0 AS(t− s)F (s, zs)ds− F (t, zt) = ∫ t 0 AS(t− s)Bu(s)ds+ Bu(t). (4.1) Equation (4.1) can be rewritten in the form ϖ(t) = ∫ t 0 κ(t, s)ϖ(s)ds+ϖ0(t), (4.2) where ϖ(t) = Bu(t), κ(t, s) = −AS(t− s), and ϖ0(t) is the left hand side of equation (4.1). Next, define an operator Υ : L2(J,X) → L2(J,X) by (Υϖ)(t) = ∫ t 0 κ(t, s)ϖ(s)ds. (4.3) It is easy to verify that Υ is a bounded linear operator [32]. In addition, one can demonstrate that Υn is a contraction mapping for large n. Thus, by generalized Banach contraction principle, there is a unique solution ϖ(t) of (4.2) for given ϖ0 ∈ L2(J,X). Hence, we can extract u(t) from the relation Bu(t) = ϖ(t), since the control operator B is invertible, it can be accomplished by taking the left inverse of B. Hence, equation (1.1) is T-controllable on J . □ 5. Application To illustrate the applicability of the conclusion, we consider the initial boundary value problem for second-order partial differential equation, ∂2 ∂t2 u(t, x)− ∂ ∂t g(t, ut(·, x)) = ∂2 ∂x2 u(t, x) + f(t, ut(·, x)) + ιv(t, x), t ∈ [0, 1], x ∈ [0, π], u(t, 0) = u(t, π) = 0, t ∈ [0, 1], u(t, x) = φ(t, x), t ∈ [0, 1], x ∈ [0, π], ∂ ∂t u(0, x) = x0, (5.1) where ι is a constant, ut(·, x) = u(·+t, x). Let X = U := L2(0, π). Define the operator A : D(A) ⊂ X → X as Au = ∂2u(t,x) ∂x2 , D(A) = H1 0 (0, π) ∩ H2(0, π), then A generates a uniformly bounded strongly continuous cosine family C(t) for t ≥ 0. EJDE-2025/118 TOPOLOGICAL PROPERTIES OF THE SOLUTION SET AND T-CONTROLLABILITY 11 We set ℓn = n2π2 and ξn(x) = √ 2 π sin(nπx), n ∈ N+. Obviously, {−ℓn, ξn}∞n=1 is the char- acteristic system of A, then 0 < ℓ1 ≤ ℓ2 ≤ . . . , ℓn → ∞, as n → ∞, and {ξn}∞n=1 is a standard orthogonal basis in X. Thus, the following properties are valid. (i) If u ∈ D(A), then Au = − ∞∑ n=1 ℓn⟨u, ξn⟩ξn, n ∈ N+. (ii) For every u ∈ X, the strong continuous cosine family {C(t) : t ∈ R} generated by A be defined as C(t)u = ∞∑ n=1 cos( √ ℓnt)⟨u, ξn⟩ξn, n ∈ N+, and the associated sine family {S(t) : t ∈ R} is given by S(t)u = ∞∑ n=1 sin( √ ℓnt)√ ℓn ⟨u, ξn⟩ξn, n ∈ N+. Obviously, {C(t) : t ∈ R} and {S(t) : t ∈ R} are periodic functions, then for any t ∈ R, we have ∥C(t)∥ ≤ 1 =M1, ∥S(t)∥ ≤ 1 =M2. We define u(t, x) = u(t)(x), f(t, ut(·, x)) = F (t, ut)(x), g(t, ut(·, x)) = G(t, ut)(x), v(t, x) = v(t)(x) and φ(t, x) = φ(t)(x). If function f(t, u) satisfies the following conditions: (i) f(t, ·) : C0 → X is continuous for t ∈ [0, 1] and f(·, u) : J → X is measurable for each u ∈ C0. (ii) There exist a function mF ∈ L2(J,R+) and 1 3 such that ∥f(t, u)∥ ≤ mF (t) + 1 3 ∥u∥C0 , for all u ∈ [0, π], a.e. t ∈ [0, 1]. (iii) There exists a constant L1 > 0 such that for any bounded subset D ⊂ C0 and a.e. t ∈ [0, 1] have α(f(t,D)) ≤ L1 sup θ∈[−τ,0] α(D(θ)). Based on (i)-(iii), it can be concluded that f(t, ut(·, x)) satisfies (H2). Similarly, we can verify that g(t, ut(·, x)) satisfies (H3). In addition, we define the control operator B : L2([0, 1];U) → L2([0, 1];X) by (Bv)(t) = ι(v)(t) for v ∈ L2([0, 1];U). Then hypotheses (H1) and (H5) are met. Accordingly, hypotheses (H1)–(H5) are all fulfilled. Hence, by Theorems 3.1, 3.2, and 4.1, we obtain: (i) For control u ∈ L2([0, 1];U), the mild solution set of equation (5.1) is nonempty and compact. (ii) For control u ∈ L2([0, 1];U), the mild solution set of equation (5.1) is an Rδ-set. (iii) Since ι is a constant, then equation (5.1) is T-controllable on [0,1]. Acknowledgments. 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Zhou; Existence of mild solutions for fractional delay evolution systems, Applied Mathematics and Computation, 218(2011), 357-367. [39] H. Yang; Exact controllability of abstract fractional evolution systems, Journal of Optimization Theory and Applications, 200(2024), 1239-1254. [40] H. H. Yang, H. Yang; Topological properties of the solution set for Caputo fractional evolution inclusions involving delay, Fractional Calculus and Applied Analysis, 28(2025), 1-23. [41] Y. Zhou, L. Zhang, X. H. Shen; Existence of mild solutions for fractional evolution equations, Journal of Integral Equations and Applications, 25(2013), 557-585. Tiantian Wang School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China Email address: wtt cassie@163.com Hongxia Fan (corresponding author) School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China Email address: ffls0217@126.com 1. Introduction 2. Preliminaries 3. Topological properties of the mild solution set 4. T-controllability 5. Application Acknowledgments References