Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 17, pp. 1–26. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu (ω, c)-PERIODIC SOLUTIONS FOR NON-INSTANTANEOUS IMPULSIVE SYSTEMS WITH UNBOUNDED TIME-VARYING COEFFICIENTS KUI LIU, MICHAL FEČKAN, DONAL O’REGAN, JINRONG WANG Abstract. In this article, we study (ω, c)-periodic solutions for non-instan- taneous impulsive systems and the time-varying coefficient A(t) is a family of unbounded linear operators. We show the existence and uniqueness of (ω, c)- periodic solutions using a fixed point theorem. An example is given to illustrate our results. 1. Introduction Non-instantaneous impulsive can characterize drug absorption, diffusion and metabolism of drugs in the body and was considered in 2013 by Hernández and O’Regan [10]. Existence and uniqueness of non-instantaneous impulsive solutions in various situations can be found in [1, 3, 9, 14, 20, 21, 24, 25, 26, 32, 33]. The so-called time-varying system refers to the system whose characteristics change with time, it is also called the variable coefficient system. The character- istic of a time-varying system is that its output waveform is not only related to the input waveform, but also to the time when the input signal is added and some results were obtained at pulse time-varying systems. In [28, 29, 30] the authors constructed corresponding impulse evolution operators, considered the appropri- ate Poincáre operator method, introduced an appropriate Gronwall inequality and studied a class of impulsive periodic systems with time-varying generation opera- tors, linear, nonlinear and integro-differential, and also the authors studied periodic PC-mild solutions. The authors in [18] studied a class of hybrid nonlinear impulse integral differential equations with time-varying generation operators and existence of a PC-mild solution is established, and existence of an optimal pair of a class of mixed pulse integral differential equations was discussed, and in [16, 17, 19] the authors constructed the corresponding evolution system generated by an opera- tor matrix, by introducing an appropriate solution of the second-order nonlinear pulse, and discussed the existence of optimal control for the second-order nonlinear pulse evolution differential equation system with unbounded operator perturba- tion, the second-order nonlinear pulse evolution differential equation system, and the Lagrange problem with second-order nonlinear mixed pulse integral differential equations. 2020 Mathematics Subject Classification. 34A37. Key words and phrases. Non-instantaneous impulsive systems; (ω, c)-periodic solutions; existence; uniqueness. ©2022. This work is licensed under a CC BY 4.0 license. Submitted December 1, 2021. Published March 4, 2022. 1 2 K. LIU, M. FEČKAN, D. O’REGAN, J. WANG EJDE-2022/17 Because of the universality of periodic phenomena the study of periodicity was considered; see [6, 8, 12, 23, 31, 34] and the references therein. The authors in [5] studied a class of (ω, c)-periodic functions containing periodic, anti-periodic, Bloch and unbounded functions and gave several properties on this class of functions, in [2] the authors studied uniqueness and existence of (ω, c)-periodic solutions for semilinear evolution equation x′ = Ax+f(t, x) in complex Banach spaces, where A is a bounded or unbounded linear operator, in [11] the authors studied existence and uniqueness of (ω, c)-periodic solutions for a class of impulsive differential systems by constructing Green functions and adjoint systems, and in [27] the authors presented existence and uniqueness results for a class of (ω, c)-periodic time varying impulsive differential equations. Recently, Wang et al. [25] introduced new linear non-instantaneous impulsive differential equations and derived the representation of the solution and the as- ymptotic stability of linear and nonlinear problems. Motivated by [2, 5, 11, 25, 27], we study (ω, c)-periodic solutions for the following homogeneous non-instantaneous impulsive system with time-varying generating operators: y′(t) = A(t)y(t), t ∈ [si−1, ti], i ∈ N+, y(t+i ) = Bi(ti)y(t−i ), i ∈ N+, y(t) = Bi(t)y(t−i ), t ∈ (ti, si], i ∈ N+, y(s+ i ) = y(s−i ), i ∈ N+, (1.1) in the Banach space X, where A(t) is a family of closed densely linear unbounded operators on X and A(t) can generate a strongly continuous family of evolution {U(t, s), t ≥ s ≥ 0}. The sequence ti, si satisfies si−1 < ti < si < ti+1 < . . . , limi→∞ ti = ∞, si is the connection point and ti is the pulse point. The symbols y(t+) := limε→0+ y(t + ε) and y(t−) := limε→0− y(t + ε) represent the right and left limits of y(s) at s = t, respectively. We set y(t−) = y(t). To help the reader understand (1.1) quickly, we give a sketch map of (1.1), see Figure 1. The black curves are corresponding to the first equation on [si−1, ti]. The second equation represents the jump at impulsive point ti. The red curves are corresponding to the third equation on (ti, si], which is the non-instantaneous impulsive action and ends at each connection point si. The fourth equation represents that y is continuous at each connection point si, which guarantee y can turn to the first differential equation of (1.1) again and again. We study the structure and existence of (ω, c)-periodic solutions for the linear nonhomogeneous non-instantaneous impulsive differential system, y′(t) = A(t)y(t) + h(t), t ∈ [si−1, ti], i ∈ N+, y(t+i ) = Bi(ti)y(t−i ) + bi, i ∈ N+, y(t) = Bi(t)y(t−i ) + bi, t ∈ (ti, si], i ∈ N+, y(s+ i ) = y(s−i ), i ∈ N+, (1.2) where h : D1 → X is continuous, where D1 = ∪∞i=1[si−1, ti], D2 = ∪∞i=1(ti, si], and ci ∈ X, i ∈ N+. EJDE-2022/17 (ω, c)-PERIODIC SOLUTIONS 3 Figure 1. Sketch map of (1.1). We also study the existence, uniqueness and stability of (ω, c)-periodic solutions for the semilinear non-instantaneous impulsive differential system, y′(t) = A(t)y(t) + f(t, y(t)), t ∈ [si−1, ti], i ∈ N+, y(t+i ) = Bi(ti)y(t−i ) + bi, i ∈ N+, y(t) = Bi(t)y(t−i ) + bi, t ∈ (ti, si], i ∈ N+, y(s+ i ) = y(s−i ) = y(si), i ∈ N+, (1.3) where f : D1 ×X → X is continuous. This work extends the results for time invariant system in [7] to the case of time variable system. We transform the study of systems into the study of Cauchy operators corresponding to systems. Here A(t) and Bi(t) are not required to be commutative when we construct the Cauchy operators corresponding to the system. It is worth mentioning that our situation is more complicated than that in the literature [7]. The existence and uniqueness of (ω, c)-periodic solutions for time- varying unbounded systems are established by fixed point theorems. To study (ω, c)-periodic solutions, we impose the following assumptions. (A1) A(·) is ω-periodic, i.e., for all t ∈ D1, A(t + ω) = A(t), where ω > 0 . For each i ∈ N+, Bi(·) ∈ Lb(X), B(·) is invertible, B−1 i (·) ∈ Lb(X), and B−1 i (·), Bi(·) is ω-periodic, i.e., B−1 i+m(t+ ω) = B−1 i (t), Bi+m(t + ω) = Bi(t), for all t ∈ D2. (A2) The time sequence si, ti are such that si+m = si + ω, ti+m = ti + ω for some fixed m, i ∈ N. (A3) c /∈ σ(S(ω, 0)). (A4) c ∈ σ(S(ω, 0)). (A5) h(·) is (ω, c)-periodic function, i.e. h(· + ω) = ch(·) for all · ∈ R+ and bi+m = cbi, i ∈ N+. (A6) For all t ∈ R+ and x ∈ X such that f(t+ ω, cx) = cf(t, x). (A7) There is a constant Lu > 0 such that ‖f(t, x1) − f(t, x2)‖ ≤ Lu‖x1 − x2‖ for all t ∈ R+ and x1, x2 ∈ X. (A8) There are constants α, γ ≥ 0 such that ‖f(t, x)‖ ≤ α+γ‖x‖ for any t ∈ R+ and x ∈ X. (A9) The resolvent R(λ,A(t)) is compact, for all t ≥ 0. (A10) (see [15, p.135]) For t ∈ [0, ω], 4 K. LIU, M. FEČKAN, D. O’REGAN, J. WANG EJDE-2022/17 (1) {A(t)}t∈[0,ω] is a stable family of generators of the evolution operator in X with stability indices M ≥ 1, $. (2) Let Y be a dense subspace of X and Y is A(t) admissible for each t ∈ [0, ω] and the family {Ã(t)}t∈[0,ω] of parts {Ã(t)} of A(t) in Y , is a stable family in Y with stability constants M̃, $̃. (3) For t ∈ [0, ω], D(A(t)) ⊃ Y,A(t) is a bounded operator from Y into X and t→ A(t) is continuous in the B(Y,X) norm ‖ · ‖Y→X . (A11) (see [4, p.158]) For t ∈ [0, ω] one has (1) The domain D(A(t)) = D is independent of t and is dense in X. (2) For t ≥ 0, the resolvent R(λ,A(t)) = (λI − A(t))−1 exists for all λ with Reλ ≤ 0, and there is a constant N independent of λ and t such that R(λ,A(t)) ≤ N(1 + |λ|)−1 for Reλ ≤ 0. (3) There exist constants L > 0 and 0 < α ≤ 1 such that ‖(A(t)A(θ))A−1(τ)‖ ≤ L|t− θ|α for t, θ, τ ∈ [0, ω]. 2. Preliminaries Let L(X) denote the space of linear operators in the Banach space X, and Lb(X) denote the space of bounded linear operators in X. Note that Lb(X) is a Banach space with the usual supremum norm. Set PC(R+, X) := {x : R+ → X|x ∈ C((ti, ti+1], X), i = 0, 1, 2, . . . , } and there exist x(t+i ), x(t−i ) with x(t−i ) = x(ti) with the norm ‖x‖PC := supt∈R+ ‖x(t)‖. Lemma 2.1 ([4, p.159]). If (A11) holds, then the Cauchy problem y′(t) = A(t)y(t), y(s) = ys ∈ X, t > s ≥ 0. has a unique evolution system {U(t, s)|0 ≤ s ≤ t ≤ ω} in X, and the solution of the Cauchy problem of the linear homogeneous development equation can be written as y(t) = U(t, s)y(s), and satisfies the following properties: (1) For all 0 ≤ s ≤ t ≤ ω,U(t, s) ∈ Lb(X), U(s, s) = I, s ≥ 0; (2) For all 0 ≤ s ≤ r ≤ t ≤ ω,U(t, r)U(r, s) = U(t, s); (3) For all 0 ≤ s ≤ t ≤ a < ∞, U(t, s) is continuous in the strong operator topology in L(X). Lemma 2.2 ([13] or [28, Lemma 2.5]). Assume (A1), (A9), (A11) hold. The evolution operator {U(t, s)|0 ≤ s ≤ t ≤ ω} has the following properties: (1) For 0 ≤ s ≤ t ≤ ω,U(t+ ω, s+ ω) = U(t, s). (2) For 0 ≤ s ≤ t ≤ ω,U(t, s) is a compact operator. Lemma 2.3 ([15, p.135, Theorem 3.1]). Let A(t), 0 ≤ t ≤ ω, be the infinitesimal generator of a C0-semigroup St(s), s ≥ 0, on X. If the family {A(t)}t∈[0,ω] satisfies condition (A10), then there exists a unique evolution system U(t, s), 0 ≤ s ≤ t ≤ ω, in X satisfying ‖U(t, s)‖ ≤M exp{$(t− s)}, for 0 ≤ s ≤ t ≤ ω. EJDE-2022/17 (ω, c)-PERIODIC SOLUTIONS 5 We consider the homogeneous non-instantaneous impulsive system (1.1) and its corresponding Cauchy problem y′(t) = A(t)y(t), t ∈ [si−1, ti], i ∈ N+, y(t+i ) = Bi(ti)y(t−i ), i ∈ N+, y(t) = Bi(t)y(t−i ), t ∈ (ti, si], i ∈ N+, y(s+ i ) = y(s−i ) = y(si), i ∈ N+, y(0) = y0. (2.1) The Cauchy problem (2.1) has a unique classical solution y ∈ PC([D1 ∪ D2, ω];X) and it can be represented by y(t; s, ys) = S(t, s)ys, t ≥ 0 where S(·, ·) : ∆ = {(t, s) ∈ R+ × D1 : t ≥ s} → X is given by the formula S(t, s) =  U(t, s), if t, s ∈ [si−1, ti], i ∈ N+ Bi(t)B −1 i (s), if t, s ∈ (ti, si], U(t, sk−1) ∏k−1 j=i+1[Bj(sj)U(tj , sj−1)]Bi(si)U(ti, s) if si−1 ≤ s ≤ ti < · · · < sk−1 ≤ t ≤ tk, Bk(t)U(tk, sk−1) ∏k−1 j=i+1[Bj(sj)U(tj , sj−1)]Bi(si)U(ti, s), if si−1 ≤ s ≤ ti < · · · < tk < t ≤ sk, U(t, sk) ∏k j=i+1[Bj(sj)U(tj , if sj−1)]Bi(si)B −1 i (s), ti < s ≤ si < · · · < sk ≤ t ≤ tk+1, Bk(t)U(tk, sk−1) ∏k−1 j=i+1[Bj(sj)U(tj , sj−1)]Bi(si)B −1 i (s), if ti < s ≤ si < · · · < tk < t ≤ sk. (2.2) Note that Bi(t)S(t−i , s) = S(t, s) and S(s, t) := S−1(t, s). We use the standard convention ∏k−1 j=i+1 = I for i+ 1 ≥ k − 1. Definition 2.4 ([5]). A function g : R → X is called (ω, c)-periodic if there is a pair (ω, c), where ω > 0 and c ∈ R\{0} such that g(t+ ω) = cg(t) for all t ∈ R. Definition 2.5 ([5]). Any solution y(t; 0, y0) of the non-instantaneous impulsive differential systems (1.1);(1.2);(1.3) is called a (ω, c)-periodic solution if y(t+ω; 0, y0) = cy(t; 0, y0), t ≥ 0. Set Ψω,c := {y ∈ PC(R, X) : cy(·) = y(· + ω)}, i.e. Ψω,c denotes the set of all piecewise continuous and (ω, c)-periodic functions. Lemma 2.6 ([2] or [11, Lemma 2.2]). y ∈ Ψω,c if and only if y(ω) = cy(0). 3. Homogeneous linear non-instantaneous impulsive systems Let α = supi≥1(ti − si−1), β = supi≥1 maxt∈(ti,si) ‖Bi(t)‖, and i(t, s) denote the number of impulsive points in (s, t). Theorem 3.1. Assume that (A10) holds. For any l̃, k ∈ N+, and l̃ ≤ k, t ∈ [sk−1, tk], s ∈ [sl̃−1, tl̃], and s < t, we obtain ‖S(t, s)‖ ≤M i(t,s)+1βi(t,s) exp($α(i(t, s) + 1)). 6 K. LIU, M. FEČKAN, D. O’REGAN, J. WANG EJDE-2022/17 For any t ∈ [tk, sk] and s ∈ [sl̃−1, tl̃], we obtain ‖S(t, s)‖ ≤M i(t,s)βi(t,s) exp($αi(t, s)). Let K = max{exp($α), 1} and N = max{βm, 1}, then for s ∈ [sl̃−1, tl̃] and any t ∈ [0, ω], we have ‖S(t, s)‖ ≤ KNM i(t,s)+1 exp($αi(t, s)). (3.1) Proof. For any t, s ∈ [sl̃−1, tl̃] and s < t, we have ‖S(t, s)‖ = ‖U(t, s)‖ ≤M exp{$(t− s)}. For any sl̃−1 ≤ s ≤ tl̃ < · · · < sk−1 < t ≤ tk, from (2.2), we have ‖S(t, s)‖ = ‖U(t, sk−1) k−1∏ j=l̃+1 [Bj(sj)U(tj , sj−1)]Bl̃(sl̃)U(tl̃, s)‖ ≤ ‖U(t, sk−1)‖ k−1∏ j=l̃+1 [‖Bj(sj)‖‖U(tj , sj−1)‖]‖Bl̃(sl̃)‖‖U(tl̃, s)‖ ≤M i(t,s)+1βi(t,s) exp($α(i(t, s) + 1)). For any sl̃−1 ≤ s ≤ tl̃< · · · (t)x(t), t ∈ (si, ti], i = 0, 1, 2, . . . , x(t+i ) = [B>i (ti)] −1x(t−i ), i = 1, 2, . . . , x(t) = [B>i (t)]−1x(t−i ), t ∈ (ti, si], i = 1, 2, . . . , x(s+ i ) = x(s−i ), i = 1, 2, . . . . (4.4) Here A>(t), B>i (t) is the adjoint operator of A(t), Bi(t), respectively. By assump- tion (A1), A>(t + ω) = A>(t), B>i+m(t + ω) = B>i (t). Let U>(·, ·) be the adjoint operator of U(·, ·) and U>(·, ·) satisfies some properties similar to U(·, ·) because of the convexity of X∗. From the reflexivity of X note A> is also an infinitesimal generating element of C0-semigroup {U>(·, ·)} in X∗ and B>i (t) ∈ Lb(X∗) . It is easy to obtain that the solution of (4.4) with the initial value x(0) = x0 is given by x(t) = [S>(t, 0)]−1x0 . (4.5) Theorem 4.3. Assume (A1), (A2), (A4) hold. Then the adjoint system (4.4) of (2.1) has l linearly independent (ω, 1 c )-periodic solutions for 1 ≤ l ≤ n. EJDE-2022/17 (ω, c)-PERIODIC SOLUTIONS 15 Proof. From (A4), we know that (cI−S(ω, 0))−1 does not exist, so we assume that the operator equation (cI − S(ω, 0))y0 = 0 has l linearly independent solutions for 1 ≤ l ≤ n. This implies that dim ker[cI − S>(ω, 0)] = dim ker[cI − S(ω, 0)]. From (4.5), we have x(t) = [S>(t, 0)]−1x0. Then x(ω) = 1 c x0 ⇐⇒ [S>(ω, 0)]−1x0 = 1 c x0 ⇐⇒ [ cI − S>(ω, 0) ] x0 = 0 ⇐⇒ x0 ∈ ker ( cI − S>(ω, 0) ) = ker(cI − S(ω, 0))>. (4.6) Therefore, dim ker(cI − S>(ω, 0)) = n− rank(cI − S>(ω, 0)) = n− rank(cI − S(ω, 0)) = l. Hence the adjoint system (4.4) has l linearly independent (ω, 1/c)-periodic solutions. � Theorem 4.4. Let y and x be the solutions of (2.1) and (4.4), respectively. Then 〈y(t), x(t)〉 is constant for t ≥ 0. Proof. Let t ∈ (si, ti+1], i = 0, 1, . . . . Then 〈y(t), x(t)〉′ = 〈y′(t), x(t)〉+ 〈y(t), x′(t)〉 = 〈A(t)y(t), x(t)〉+ 〈y(t),−A>(t)x(t)〉 = 〈y(t), A>(t)x(t)〉+ 〈y(t),−A>(t)x(t)〉 = 0. Let t ∈ (ti, si], i = 1, 2, . . . . Then 〈y(t), x(t)〉 = 〈Bi(t)y(t−i ), [B>i (t)]−1x(t−i )〉 = 〈y(t−i ), B>i (t)[B>i (t)]−1x(t−i )〉 = 〈y(t−i ), x(t−i )〉. Let t = ti, i = 1, 2, . . . , then 〈y(t+i ), x(t+i )〉 = 〈Bi(ti)y(t−i ), [B>i (ti)] −1x(t−i )〉 = 〈y(t−i ), B>i (ti)[B > i (ti)] −1x(t−i )〉 = 〈y(t−i ), x(t−i )〉. Thus, 〈x(t), y(t)〉 = 〈x(0), y(0)〉 which is a constant. � Lemma 4.5. Assume that (A1), (A2), (A4) hold. Then (1.2) has l linearly inde- pendent (ω, c)-periodic solutions if and only if∫ ω 0 〈x(τ), h̃(τ)〉X∗,Xdτ + m∑ k=1 〈x(sk), bk〉X∗,X = 0 (4.7) Proof. Let y be a (ω, c)-periodic solution of (1.2), and the initial condition y(0) = y0 satisfy (cI − S(ω, 0))y0 = ∫ ω 0 S(ω, τ)h̃(τ)dτ + i(ω,0)∑ k=1 S(ω, sk)bk. (4.8) Let x(t) be a nontrivial (ω, 1/c)-periodic solution of the adjoint system (4.4). Using (4.6), we obtain 0 = 〈 (cI − S(ω, 0))>x0, y0 〉 X∗,X = 〈 x0, (cI − S(ω, 0))y0 〉 X∗,X 16 K. LIU, M. FEČKAN, D. O’REGAN, J. WANG EJDE-2022/17 = 〈 x0, ∫ ω 0 S(ω, τ)h̃(τ)dτ + m∑ i=1 S(ω, sk)bk 〉 X∗,X = ∫ ω 0 〈x0, S(ω, τ)h̃(τ)〉X∗,Xdτ + m∑ i=1 〈 x0, S(ω, sk)bk 〉 X∗,X = ∫ ω 0 〈S>(ω, τ)x0, h̃(τ)〉X∗,Xdτ + m∑ i=1 〈 S>(ω, sk)x0, bk 〉 X∗,X = ∫ ω 0 〈x(τ), h̃(τ)〉X∗,Xdτ + m∑ i=1 〈x(sk), bk〉X∗,X . This proves the necessity part. Suppose (4.7) holds. Assume the solution x(t) is a (ω, 1/c)-periodic solution of the adjoint system (4.4) with initial condition x(0) = x0. Therefore, the solution is given by x(τ) = S>(ω, τ)x0, where aj is a nonzero constant. Using (4.7), we have 0 = ∫ ω 0 〈x(τ), h̃(τ)〉X∗,Xdτ + m∑ i=1 〈x(sk), bk〉X∗,X = ∫ ω 0 〈S>(ω, τ)x0, h̃(τ)〉X∗,Xdτ + m∑ k=1 〈S>(ω, sk)x0, bk〉X∗,X = ∫ ω 0 〈x0, S(ω, τ)h̃(τ)〉X∗,Xdτ + m∑ k=1 〈x0, S(ω, sk)bk〉X∗,X = 〈 x0, ∫ ω 0 S(ω, τ)h̃(τ)dτ + m∑ k=1 S(ω, sk)bk 〉 X∗,X = 〈 x0, (cI − S(ω, 0))y0 〉 X∗,X (4.9) Note that (4.8) and (4.9) imply that (cI − S(ω, 0))y0 = 0 is equivalent to (cI − S(ω, 0))>x0 = 0. The system (4.8) has a solution if and only if rank(cI−S(ω, 0)) = rank(cI − S(ω, 0))> = n − l. Thus, the system (1.2) has l linearly independent (ω, c)-periodic solutions. This proof is complete. � Lemma 4.6. Assume that (A10) holds. Then m∑ j=1 ‖Q(t, sj)bj‖ ≤ L$ :=  KNMm+1 exp($αm)(KNM (m+1) exp($αm) ×‖(cI − S(ω, 0))−1‖+ 1) ∑m−1 j=1 ‖bj‖, $ > 0, KNMm+1 ∑m j=1[KNMm+1‖(cI − S(ω, 0))−1‖+ 1]‖bj‖, $ ≤ 0. for any t ∈ [0, ω]. Proof. According to (4.3) and Theorem 3.1, we have m∑ j=1 ‖Q(t, sj)bj‖ EJDE-2022/17 (ω, c)-PERIODIC SOLUTIONS 17 ≤ m∑ j=1 ‖Q(t, sj)‖‖bj‖ = i(t,0)−1∑ j=1 ‖Q(t, sj)‖‖bj‖+ m∑ j=i(t,0) ‖Q(t, sj)‖‖bj‖ ≤ i(t,0)−1∑ j=1 ‖S(t, 0)(cI − S(ω, 0))−1S(ω, sj) + S(t, sj)‖‖bj‖ + m∑ j=i(t,0) ‖S(t, 0)(cI − S(ω, 0))−1S(ω, sj)‖‖bj‖ ≤ i(t,0)−1∑ j=1 [ ‖S(t, 0)‖‖(cI − S(ω, 0))−1‖‖S(ω, sj)‖+ ‖S(t, sj)‖ ] ‖bj‖ + m∑ j=i(t,0) ‖S(t, 0)‖‖(cI − S(ω, 0))−1‖‖S(ω, sj)‖‖bj‖ ≤ i(t,0)−1∑ j=1 [KNM i(t,0)+1 exp($αi(t, 0))‖(cI − S(ω, 0))−1‖KNM i(t,0)+1 × exp($αi(ω, sj)) +KNM i(t,0)+1 exp($αi(t, sj))]‖bj‖ + m∑ j=i(t,0) KNM i(t,0)+1 exp($αi(t, 0))‖(cI − S(ω, 0))−1‖KNM i(t,0)+1 × exp($αi(ω, sj))‖bj‖ = i(t,0)−1∑ j=1 [K2N2M2(i(t,0)+1) exp($α(i(t, 0) + i(ω, sj))‖(cI − S(ω, 0))−1‖ +KNM i(t,0)+1 exp($αi(t, sj))]‖bj‖ + m∑ j=i(t,0) K2N2M2(i(t,0)+1) exp($α(i(t, 0) + i(ω, sj)))‖(cI − S(ω, 0))−1‖‖bj‖. For $ > 0, m∑ j=1 ‖Q(t, sj)‖‖bj‖ ≤ i(t,0)−1∑ j=1 [K2N2M2(i(t,0)+1) exp($α(i(t, 0) + i(ω, sj))‖(cI − S(ω, 0))−1‖ +KNM i(t,0)+1 exp($αi(t, sj))]‖bj‖ + m∑ j=i(t,0) K2N2M2(i(t,0)+1) exp($α(i(t, 0) + i(ω, sj)))‖(cI − S(ω, 0))−1‖‖bj‖ 18 K. LIU, M. FEČKAN, D. O’REGAN, J. WANG EJDE-2022/17 ≤ m∑ j=1 K2N2M2(m+1) exp($α2m)‖(cI − S(ω, 0))−1‖‖bj‖ + m∑ j=1 KNMm+1 exp($αm)‖bj‖ ≤ KNMm+1 exp($αm)(KNM (m+1) exp($αm)‖(cI − S(ω, 0))−1‖+ 1) m−1∑ j=1 ‖bj‖. For $ ≤ 0, m∑ j=1 ‖Q(t, sj)‖‖bj‖ ≤ i(t,0)−1∑ j=1 [K2N2M2(i(t,0)+1) exp($α(i(t, 0) + i(ω, sj))‖(cI − S(ω, 0))−1‖ +KNM i(t,0)+1 exp($αi(t, sj))]‖bj‖ + m∑ j=i(t,0) K2N2M2(i(t,0)+1) exp($α(i(t, 0) + i(ω, sj)))‖(cI − S(ω, 0))−1‖‖bj‖ ≤ m∑ j=1 [K2N2M2(m+1)‖(cI − S(ω, 0))−1‖+KNMm+1]‖bj‖ ≤ KNMm+1 m∑ j=1 [KNMm+1‖(cI − S(ω, 0))−1‖+ 1]‖bj‖. The proof is complete. � Lemma 4.7. Assume that (A10) holds. For 0 < t < ω, we have∫ ω 0 ‖Q(t, τ)‖dτ ≤ K$ := { KNMm+1[KNMm+1 exp($αm)‖(cI − S(ω, 0))−1‖+ 1] exp($αm)ω, $ > 0, KNMm+1[KNMm+1 exp($αm)‖(cI − S(ω, 0))−1‖+ 1]ω, $ ≤ 0. Proof. According to (4.3), from (3.1), we have∫ ω 0 ‖Q(t, τ)‖dτ ≤ ∫ t 0 ‖Q(t, τ)‖dτ + ∫ ω t ‖Q(t, τ)‖dτ = ∫ t 0 ‖S(t, 0)(cI − S(ω, 0))−1S(ω, τ) + S(t, τ)‖dτ + ∫ ω t ‖S(t, 0)(cI − S(ω, 0))−1S(ω, τ)‖dτ ≤ ∫ t 0 [‖S(t, 0)‖‖(cI − S(ω, 0))−1‖‖S(ω, τ)‖+ ‖S(t, τ)‖]dτ EJDE-2022/17 (ω, c)-PERIODIC SOLUTIONS 19 + ∫ ω t ‖S(t, 0)‖‖(cI − S(ω, 0))−1‖‖S(ω, τ)‖dτ ≤ ∫ ω 0 ‖S(t, 0)‖‖(cI − S(ω, 0))−1‖‖S(ω, τ)‖dτ + ∫ t 0 ‖S(t, τ)‖dτ ≤ ∫ ω 0 KNM i(t,0)+1 exp($αi(t, 0))‖(cI − S(ω, 0))−1‖KNM i(ω,τ)+1 × exp($αi(ω, τ))dτ + ∫ t 0 KNM i(t,τ)+1 exp($αi(t, τ))dτ ≤ KNMm+1[KNMm+1 exp($αm)‖(cI − S(ω, 0))−1‖+ 1] ∫ ω 0 exp($αi(t, τ))dτ. For $ > 0,∫ ω 0 ‖Q(t, τ)‖dτ ≤ KNMm+1[KNMm+1 exp($αm)‖(cI − S(ω, 0))−1‖+ 1] ∫ ω 0 exp($αi(t, τ))dτ ≤ KNMm+1[KNMm+1 exp($αm)‖(cI − S(ω, 0))−1‖+ 1] exp($αm)ω. For $ ≤ 0,∫ ω 0 ‖Q(t, τ)‖dτ ≤ KNMm+1[KNMm+1 exp($αm)‖(cI − S(ω, 0))−1‖+ 1] ∫ ω 0 exp($αi(t, τ))dτ ≤ KNMm+1[KNMm+1 exp($αm)‖(cI − S(ω, 0))−1‖+ 1]ω. The proof is complete. � 5. Nonlinear non-instantaneous impulsive systems In this section, we apply the Banach fixed point theorem and the Schauder fixed point theorem to establish existence theorems for (ω, c)-periodic solutions of (1.3). Theorem 5.1. Assume that (A1)–(A3), (A6), (A7), (A10) hold. If 0 < LuK$ < 1, then (1.3) has a unique (ω, c)-periodic solution y ∈ Ψω,c satisfying ‖y‖PC ≤ f̃0K$ + L$ 1− LuK$ , where f̃0 = maxt∈[0,ω] |f(t, 0)|. Proof. Consider any y ∈ Ψω,c, i.e., y(·+ ω) = cy(·). From assumption (A6), f(t+ ω, y(t+ ω)) = f(t+ ω, cy(t)) = cf(t, y), t ∈ R+. Thus, f(·, y(·)) ∈ Ψω,c. From Lemma 4.2, our goal is to consider the fixed point problem y(t) = ∫ ω 0 Q(t, τ)f̃(τ, y(τ))dτ + m∑ j=1 Q(t, sj)bj . 20 K. LIU, M. FEČKAN, D. O’REGAN, J. WANG EJDE-2022/17 We consider the operator P : Ψ→ Ψ given by Py(t) = ∫ ω 0 Q(t, τ)f̃(τ, y(τ))dτ + m∑ j=1 Q(t, sj)bj . (5.1) For any x, y ∈ Ψ, we have ‖Px(t)− Py(t)‖ ≤ ∫ ω 0 ‖Q(t, τ)f̃(τ, x(τ))−Q(t, τ)f̃(τ, y(τ))‖dτ ≤ ∫ ω 0 ‖Q(t, τ)‖‖f̃(τ, x(τ))− f̃(τ, y(τ))‖dτ ≤ Lu‖x− y‖PC ∫ ω 0 Q(t, τ)dτ ≤ LuK$‖x− y‖PC . This implies that ‖Px−Py‖PC ≤ LuK$‖x−y‖PC . Since 0 < LuK$ < 1, operator P is a contraction mapping. Thus, P has a unique fixed point. Furthermore, using Lemma 4.6, we have ‖y(t)‖ = ‖Py(t)‖ ≤ ∫ ω 0 ‖Q(t, τ)‖‖f̃(τ, y(τ))‖dτ + m∑ j=1 ‖Q(t, sj)‖‖bj‖ ≤ ∫ ω 0 ‖Q(t, τ)‖‖f̃(τ, y(τ))− f̃(τ, 0) + f̃(τ, 0)‖dτ + m∑ j=1 ‖Q(t, sj)‖‖bj‖ ≤ Lu‖y‖PC ∫ ω 0 ‖Q(t, τ)‖dτ + ∫ ω 0 ‖Q(t, τ)‖‖f̃(τ, 0)‖dτ + m∑ j=1 ‖Q(t, sj)‖‖bj‖ ≤ LuK$‖y‖PC + ‖f̃0‖K$ + L$. Thus ‖y‖PC ≤ f̃0K$ + L$ 1− LuK$ . The proof is complete. � Theorem 5.2. Assume that (A1)–(A3), (A8)–(A11) hold. If 0 < γK$ < 1, then (1.3) has a (ω, c)-periodic solution y ∈ Ψω,c. Proof. Consider the operator P defined in (5.1) on Bl := {y ∈ Ω | ‖y‖ ≤ l}, and l ≥ αK$+L$ 1−γK$ . Step 1. We show that P (Bl) ⊂ Bl. For any y ∈ Bl, t ∈ [0, ω] , by Lemmas 4.6 and 4.7 and (A8), we have ‖(Py)(t)‖ ≤ ∫ ω 0 ‖Q(t, τ)‖‖f̃(τ, y(τ))‖dτ + m∑ j=1 ‖Q(t, sj)bj‖ ≤ γ ∫ ω 0 ‖Q(t, τ)‖‖y(τ)‖dτ + α ∫ ω 0 ‖Q(t, τ)‖dτ + m∑ i=1 ‖Q(t, sj)bj‖ ≤ γK$‖y‖PC + αK$ + L$ = l, which implies that ‖Py‖PC ≤ l. Thus P (Bl) ⊂ Bl for any y ∈ Bl and t ∈ [0, ω]. EJDE-2022/17 (ω, c)-PERIODIC SOLUTIONS 21 Step 2. We prove that P is continuous. Let yn be a Cauchy sequence such that yn → y (as n→∞) in Bl. Since f̃(·, yn(·))→ f̃(·, y(·)) as yn → y for any t ∈ [0, ω], we obtain ‖(Pyn)(t)− (Py)(t)‖ ≤ ∫ ω 0 ‖Q(t, τ)‖‖f̃n(τ, yn(τ)− f̃(τ, y(τ))‖dτ ≤ ‖f̃n − f̃‖PC ∫ ω 0 ‖Q(t, τ)‖dτ ≤ K$‖f̃n − f̃‖PC . Thus, P is continuous. Step 3. We show that P (Bl) is relatively compact set. Since P (Bl) ⊂ Bl, it is easy to see that P (Bl) is uniformly bounded. Next, we show that P is an equicontinuous operator. For 0 < t1 < t2 ≤ ω and y ∈ Bl, we have ‖(Py)(t2)− (Py)(t1)‖ ≤ ∫ ω 0 ‖Q(t2, τ)−Q(t1, τ)‖‖f̃(τ, y(τ))‖dτ + m∑ j=1 ‖Q(t2, sj)−Q(t1, sj)‖‖bj‖ ≤ (α+ β‖y‖) ∫ ω 0 ‖Q(t2, τ)−Q(t1, τ)‖dτ + m∑ i=1 ‖Q(t2, sj)−Q(t1, sj)‖‖bj‖. (5.2) From (4.3), we obtain ‖Q(t2, τ)−Q(t1, τ)‖ =  ‖S(t2, 0)(cI − S(ω, 0))−1S(ω, τ) + S(t2, τ) −S(t1, 0)(cI − S(ω, 0))−1S(ω, τ)− S(t1, τ)‖, if 0 < τ < t1 < t2, ‖S(t2, 0)(cI − S(ω, 0))−1S(ω, τ) −S(t1, 0)(cI − S(ω, 0))−1S(ω, τ)‖, if t1 < t2 < τ < ω. ≤  ‖(cI − S(ω, 0))−1‖‖S(ω, τ)‖‖S(t2, 0)− S(t1, 0))‖ +‖S(t2, τ)− S(t1, τ))‖, if 0 < τ < t1 < t2, ‖S(t2, 0)− S(t1, 0)‖‖(cI − S(ω, 0))−1‖‖S(ω, τ))‖, if t1 < t2 < τ < ω. (5.3) Letting t1 → t2, from (5.3) and the compactness of S(·, ·) we have Q(t2, τ)→ Q(t1, τ), as t1 → t2. Thus, we have ‖(Py)(t1)− (Py)(t2)‖ → 0 as t1 → t2. Then P is an equicontinuous operator. Now consider the approximate operator Pε on Bl as follows∫ ω 0 Q(t− ε, τ)f̃(τ, y(τ))dτ + m∑ j=1 Q(t− ε, sj)bj , t ∈ [0, ω]. (5.4) Consider K = {(Py)(t) : t ∈ [0, ω]} and Kε = S(ε, 0){(Pεy)(t) : t ∈ [0, ω]}, 0 < ε < ω. 22 K. LIU, M. FEČKAN, D. O’REGAN, J. WANG EJDE-2022/17 From Theorem 3.5 and K bounded, Kε is precompact. Next, ‖(Pεy)(t)− (Py)(t)‖ ≤ ∫ ω 0 ‖Q(t, τ)−Q(t− ε, τ)‖‖f(τ, y(τ))‖dτ + m∑ i=1 ‖Q(t, si)−Q(t− ε, si)‖‖bj‖ ≤ (α+ βl) ∫ ω 0 ‖Q(t− ε, s)−Q(t, s)‖ds + m∑ i=1 ‖Q(t− ε, si)−Q(t, si)‖‖bi‖. (5.5) Then ‖(Pεy)(t)− (Py)(t)‖ tends to zero when ε→ 0. Thus K can be approximated to an arbitrary degree of accuracy by a precompact set Kε. Hence K itself is a precompact set in X, that is, B takes a bounded set into a precompact set in X. The Arzelà-Ascoli theorem implies the compactness of B. Thus Schauder’s fixed point theorem guarantees the result. The proof is complete. � Next we present an example. Since we develop our theory mainly for infinite dimensional Banach spaces, we need consider a partial differential equation. Example 5.3. We consider the problem ∂ ∂t y(t, x) = (2 + sin 2t) ∂2 ∂x2 y(t, x) + a sin t+ b y3 2 + y2 , x ∈ (0, π), t ∈ [si−1, ti], i ∈ N+, y(t+i , x) = 2y(t−i , x), x ∈ (0, π), y(t, x) = (2− t− ti si − ti )y(t−i , x), t ∈ (ti, si], x ∈ (0, π), y(s+ i , x) = y(s−i , x) [= y(t−i , x)], x ∈ (0, π), y(t, 0) = y(t, π) = 0, t ≥ 0, (5.6) where a, b ∈ R, a 6= 0, b 6= 0, 0 = t0 = s0, ti = (2i− 1)π/2, si = iπ. Let Bi(t) = 2− t−ti si−ti , m = 1, ω = π. then Bi+m(t+ ω) = Bi+1(t+ π) = 2− t+ π − ti+1 si+1 − ti+1 = 2− t+ π − ti − π si + π − ti − π = 2− t− ti si − ti = Bi(t). Set X = L2(0, π) with a norm ‖y‖ = √∫ π 0 y2(t)dt. Define A(t)y = (2 + sin 2t) ∂2 ∂x2 y for y ∈ D(A) = { y ∈ X : ∂y ∂x , ∂2y ∂x2 ∈ X, y(0) = y(π) = 0 } . Then A(t) is the infinitesimal generator of a strongly continuous semigroup {S(t, s), t ≥ 0} in X. Indeed, we know that the sequence {√ 2/π sin kx}k∈N is an orthonormal basis of X. Thus for y0 = ∑ k∈N y0k √ 2 π sin kt ∈ X, ‖y0‖ = √∑ k∈N y2 0k, EJDE-2022/17 (ω, c)-PERIODIC SOLUTIONS 23 we have S(t, s)y0 = ∑ k∈N exp ( − k2 ( 2(t− s) + cos 2s− cos 2t 2 )) y0k √ 2 π sin kt ⇒ ‖S(t, s)y0‖ = √∑ k∈N exp(−k2(4(t− s) + cos 2s− cos 2t))y2 0k ≤ exp(−(t− s))‖y0‖ Hence M = 1 and $ = −1. Moreover, σ(S(π, 0)) = {e−2πk2 , k ∈ N}, so −1 /∈ σ(S(ω, 0)). Next, (−I − S(π, 0))−1y0 = − ∑ k∈N 1 1 + e−2πk2 y0k √ 2 π sin kx, and then ‖(−I − S(π, 0))−1y0‖ = ∥∥∑ k∈N 1 1 + e−2πk2 y0k √ 2 π sin kx ∥∥ = √∑ k∈N 1 (1 + e−2πk2)2 y2 0k and sup k∈N 1 (1 + e−2πk2)2 = 1, ‖y0‖ = √∑ k∈N y2 0k, thus ‖(−I − S(π, 0))−1‖ = sup ‖y0‖=1 ‖(−I − S(π, 0))−1y0‖ = 1, and K$ = 2π. On the other hand, we have f(t, y) = a sin t+ b y3 2+y2 , t ∈ R+. Then f(t+ π,−y) = a sin (t+ π) + b (−y) 3 2 + (−y) 2 = −(a sin t+ b y3 2 + y2 ) = −f(t, y) for t ∈ R+, so c = −1 and ‖f(t, y)‖ ≤ ‖a sin t‖+ ∥∥b y3 2 + y2 ∥∥ ≤ |a|‖ sin t‖+ |b| (∫ π 0 y6(t) (2 + y2(t))2 dt )1/2 ≤ |a| √ π/2 + |b| (∫ π 0 y2(t)dt )1/2 ≤ |a| √ π/2 + |b|‖y‖. We have α = |a| √ π/2 and γ = |b|. Then 0 < γK$ < 1 reduces to 0 < 2π|b| < 1, which holds for some suitable b: Let |b| = 1/(3π) and then 0 < γK$ = 2/3 < 1. Thus all the assumptions in Theorem 5.2 are satisfied. Accordingly, system (5.6) has a (π,−1)-periodic solution. 24 K. LIU, M. FEČKAN, D. O’REGAN, J. WANG EJDE-2022/17 6. Conclusions In this paper, time-varying development systems in infinite-dimensional space are studied for the first time. When A(t) and Bi(t) are not commutative, the Cauchy operator of the linear homogeneous system is constructed, and the study of the system is transformed into its corresponding Cauchy operator. Firstly, some properties of Cauchy operator are obtained. A sufficient and necessary condition for the existence of (ω, c)-periodic solutions for linear homogeneous systems is given. The existence of (ω, c)-periodic solutions in critical and noncritical cases for linear inhomogeneous systems is discussed. The existence of periodic solutions for nonlin- ear systems (ω, c)-periodic solutions is obtained using Banach’s fixed point theorem and Schauder’s fixed point theorem. Acknowledgments. This work was partially supported by the National Natu- ral Science Foundation of China (12161015), by the Guizhou Provincial Science and Technology Foundation ([2020]1Y002), by the Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), by the Youth Devel- opment Project of Guizhou Provincial Education Department ([2021]266), by the Academic seedling cultivation and innovation exploration special cultivation project plan: GZLGXM-17, by the Guizhou Data Driven Modeling Learning and Opti- mization Innovation Team ([2020]5016), by the Slovak Research and Development Agency under the contract No. APVV-18-0308, and by the Slovak Grant Agency VEGA No. 1/0358/20 and No. 2/0127/20. 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[34] Yang, P.; Wang, J.; Fečkan, M.; Periodic nonautonomous differential equations with non- instantaneous impulsive effects, Mathematical Methods in the Applied Sciences, 2019, 42, 3700–3720. 26 K. LIU, M. FEČKAN, D. O’REGAN, J. WANG EJDE-2022/17 Kui Liu Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China. Science College, Guizhou Institute of Technology, Guiyang, Guizhou 550025, China Email address: liuk180916@163.com Michal Fečkan Department of Mathematical Analysis and Numerical Mathematics, Faculty of Math- ematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia Email address: Michal.Feckan@fmph.uniba.sk Donal O’Regan School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland Email address: donal.oregan@nuigalway.ie Jinrong Wang (corresponding author) Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China Email address: jrwang@gzu.edu.cn 1. Introduction 2. Preliminaries 3. Homogeneous linear non-instantaneous impulsive systems 4. Nonhomogeneous linear non-instantaneous impulsive systems Case 1: c-.25ex-.25ex-.25ex-.25ex(S(,0)). Case 2: c (S(, 0)) 5. Nonlinear non-instantaneous impulsive systems 6. Conclusions Acknowledgments References