Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 30, pp. 1–22. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.30 PROPERTIES OF THE SOLUTIONS TO PERIODIC CONFORMABLE NON-AUTONOMOUS NON-INSTANTANEOUS IMPULSIVE DIFFERENTIAL EQUATIONS YUANLIN DING, KUI LIU Abstract. In this article, we study the properties of solutions to periodic non- autonomous conformable non-instantaneous impulsive differential equations. We use a conformable Cauchy matrix and obtain some basic properties of the periodic solution to the homogeneous and non-homogeneous problems. We consider the periodicity of solutions to nonlinear problem via a fixed theorem. 1. Introduction Hernández and O’Regan [12] establish a non-instantaneous impulsive differen- tial equation model depending on the current state and duration of action, that de- scribes phenomena in engineering, physics, biology, and many other fields. With the development of research, there are many publications studying the existence, sta- bility, controllability, and periodicity of solutions for non-instantaneous impulsive differential equations; see for example [1, 5, 7, 9, 11, 14, 15, 16, 18, 20, 22, 23, 24, 25]. Motivated by the results based on conformable derivatives, [2, 3, 4, 6, 8, 10, 13, 17, 19, 21], we consider the homogeneous linear conformable non-autonomous problem Dsl τ β(t) = α(t)β(t), t ∈ (sl, tl+1], l ∈ N0 := {0, 1, 2, . . . }, 0 < τ < 1, β(t+l ) = (E+ Pl)β(t − l ), l ∈ N := {1, 2, . . . }, β(t) = δl(t)β(t + l ), t ∈ (tl, sl], l ∈ N, β(s+l ) = β(s−l ), l ∈ N, β(c) = βc ∈ Rn, (1.1) 2020 Mathematics Subject Classification. 34A37, 34C25. Key words and phrases. Impulsive differential equation; conformable derivative; non-instantaneous; periodic solution. ©2024. This work is licensed under a CC BY 4.0 license. Submitted February 9, 2024. Published April 15, 2024. 1 2 Y. DING, K. LIU EJDE-2024/30 and the nonhomogeneous linear conformable non-autonomous problem Dsl τ β(t) = α(t)β(t) + ζ(t), t ∈ (sl, tl+1], l ∈ N0 := {0, 1, 2, . . . }, 0 < τ < 1, β(t+l ) = (E+ Pl)β(t − l ) +Ql, l ∈ N := {1, 2, . . . }, β(t) = δl(t)β(t + l ), t ∈ (tl, sl], l ∈ N, β(s+l ) = β(s−l ), l ∈ N, β(c) = βc ∈ Rn, (1.2) and the nonlinear conformable non-autonomous problem Dsl τ β(t) = α(t)β(t) + η(t, β(t)), t ∈ (sl, tl+1], l ∈ N0 := {0, 1, 2, . . . }, 0 < τ < 1, β(t+l ) = (E+ Pl)β(t − l ) +Ql, l ∈ N := {1, 2, . . . }, β(t) = δl(t)β(t + l ), t ∈ (tl, sl], l ∈ N, β(s+l ) = β(s−l ), l ∈ N, β(c) = βc ∈ Rn. (1.3) The sequences {tl}l∈N0 and {sl}l∈N0 satisfy c = s0 ≤ · · · ≤ tl < sl < tl+1 for l ∈ N. Set A = ∪∞ l=1(sl, tl+1], B = ∪∞ l=1(tl, sl], α(·) : A → Rn×n, δl(·) : B → Rn×n, ζ(·) : A → Rn and η : A × Rn → Rn are continuous functions. Also, Pl ∈ Rn×n, Ql ∈ Rn with Pl+a = Pl and Ql+a = Ql. What’s more, E denotes the unit matrix. Note that for each l ∈ N, the sequences tl, sl satisfy tl+a = tl + ϑ and sl+a = sl + ϑ where a ∈ N means the number of impulsive points of a periodic interval (c, c+ ϑ) and ϑ is a fixed positive number. This article consists of 6 sections. Section 2 presents basic theory and the con- formable Cauchy matrix. Section 3 shows the properties of the conformable Cauchy matrix. Section 4 studies the stability and periodicity of the solution of (1.1). Section 5 proves the existence and boundedness of the periodic solution of (1.2). Section 6 proves the existence and uniqueness of periodic solutions of (1.3) using Brouwer’s fixed point theorem. 2. Preliminaries Set C = [c,+∞) and PC(C,Rn) := { β : C → Rn : β ∈ C ( (tl, tl+1],Rn ) , there exists β(t−l ) and β(t+l ), l = 1, 2, . . . with β(t−l ) = β(tl) } , where C ( (tl, tl+1],Rn ) denotes the space of all continuous functions from (tl, tl+1] into Rn endowed with the norm ∥β∥ = supt∈C |β(t)|. We introduce PCϑ(C,Rn) = {β ∈ PC(C,Rn) : β(t) = β(t+ ϑ), t ∈ C}. We denote a vector a = (a1, . . . , an) ⊤ ∈ Rn with its norm ∥a∥ = ∑n i=1 |ai| and a matrix b ∈ Rn×n with its norm ∥b∥ = max∥y∥=1 ∥bβ∥. Definition 2.1 ([3, Definition 2.1]). The conformable derivative with lower index c of a function r : C → R is defined as Dc τβ(t) = lim ε→0 β(t+ ε(t− c)1−τ )− β(t) ε , t > c, 0 < τ < 1, Dc τβ(c) = lim t→c+ Dc τβ(t). Remark 2.2. For t > c we note that the conformable derivative Dc τβ(t) exists if and only if y is differentiable at t and Dc τβ(t) = (t− c)1−τy′(t). EJDE-2024/30 IMPULSIVE DIFFERENTIAL EQUATIONS 3 Definition 2.3 ([3, Notation]). The conformable integral of a function r : C → R is written as Jcτβ(t) = ∫ t c β(s)dτ (s, c) = ∫ t c (s− c)τ−1β(s)ds, t ≥ c, 0 < τ < 1, if c = 0, then we write dτ (s, c) as dτ (s). Lemma 2.4. Let β : A → Rn be a continuous function. A solution β ∈ C(A,Rn) of the linear problem Dsl τ β(t) = α(t)β(t), 0 < τ < 1, β(s) = βs, t > s ≥ c. has the form β(t) = Φ(t, s)βs, where Φ(·, ·) is the Cauchy matrix of Dsl τ β(t) = α(t)β(t). We set ∥Φ(t, s)∥ ≤ e ∫ t s ∥α(θ)∥(θ−sl) τ−1dθ for sl ≤ s ≤ t ≤ tl+1, and ϕl = maxt∈(sl,tl+1] ∥α(t)∥. Lemma 2.5. The solution β(·, s, βs) ∈ PC(C,Rn) of (1.1) with β(s) = βs has the form β(t, s, βs) = Λ(t, s)βs, t ≥ c, in which Λ(t, s) =  Φ(t, s), t, s ∈ (sl, tl+1], l = 0, 1, 2, . . . ; δl(t)δ −1 l (s), t, s ∈ (tl, sl], l = 1, 2, . . . ; δς(c,t)(t) ∏ς(c,t) l=ς(c,s)+2[(E+ Pl)Φ(tl, sl−1)δl−1(s − l−1)](E+ Pς(c,s)+1) ×Φ(tς(c,s)+1, s), s ∈ (sς(c,s), tς(c,s)+1], t ∈ (tς(c,t), sς(c,t)]; Φ(t, sς(c,t))δς(c,t)(sς(c,t)) ∏ς(c,t) l=ς(c,s)+1[(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)] ×δ−1 ς(c,s)(s), s ∈ (tς(c,s), sς(c,s)], t ∈ (sς(c,t), tς(c,t)+1]; Φ(t, sς(c,t))δς(c,t)(sς(c,t)) ∏ς(c,t) l=ς(c,s)+2[(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)] ×(E+ Pς(c,s)+1)Φ(tς(c,s)+1, s), s ∈ (sς(c,s), tς(c,s)+1], t ∈ (sς(c,t), tς(c,t)+1]; δς(c,t)(t) ∏ς(c,t) l=ς(c,s)+1[(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,s)(s), s ∈ (tς(c,s), sς(c,s)], t ∈ (tς(c,t), sς(c,t)], (2.1) when ς(c, t) = ς(c, s) and ς(c, t) denotes the number of the impulsive points which belong to (c, t). ∏ς(c,t)−1 l=ς(c,s) = E. Proof. We consider 4 cases. Case 1. No impulsive point between t and s. (i) For t, s ∈ (sl, tl+1], l = 0, 1, 2, . . . , ς(c, t), we have β(t) = Φ(t, sl)β(sl), and β(s) = Φ(s, sl)β(sl), so β(t) = Φ(t, s)β(s). We obtain Λ(t, s) = Φ(t, s). (ii) For t, s ∈ (tl, sl], l = 1, 2, . . . , ς(c, t), we have β(t) = δl(t)β(t + l ) and β(s) = δl(s)β(t + l ) 4 Y. DING, K. LIU EJDE-2024/30 so Λ(t, s) = δl(t)δ −1 l (s), t, s ∈ (tl, sl]. (iii) For any s ∈ (tl, sl] and any t ∈ (sl, tl+1], we have β(t) = Φ(t, sl)β(s + l ) = Φ(t, sl)β(s − l ) = Φ(t, sl)δl(sl)δ −1 l (s)β(s), so Λ(t, s) = Φ(t, sl)δl(sl)δ −1 l (s). Case 2. One impulsive point between time t and s. (i) For any s ∈ (sl−1, tl] and t ∈ (tl, sl], we have β(t) = δl(t)β(t + l ) = δl(t)(E+ Pl)β(t − l ) = δl(t)(E+ Pl)Φ(tl, s)β(s), so Λ(t, s) = δl(t)Φ(tl, s). (ii) For any s ∈ (sl−1, tl] and any t ∈ (sl, tl+1], we have β(t) = Φ(t, sl)β(s + l ) = Φ(t, sl)β(s − l ) = Φ(t, sl)δl(sl)β(t + l ) = Φ(t, sl)δl(sl)(E+ Pl)β(t − l ) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, s)β(s), so Λ(t, s) = Φ(t, sl)δl(sl)(E+ Pl)β(t − l )Φ(tl, s). (iii) For any s ∈ (tl−1, sl−1] and any t ∈ (tl, sl], we have β(t) = δl(t)β(t + l ) = δl(t)(E+ Pl)β(t − l ) = δl(t)(E+ Pl)Φ(tl, sl−1)β(s + l−1) = δl(t)(E+ Pl)Φ(tl, sl−1)β(s − l−1) = δl(t)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)δ −1 l−1(s)β(s), so Λ(t, s) = δl(t)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)δ −1 l−1(s). (iv) For any s ∈ (tl−1, sl−1] and any t ∈ (sl, tl+1], we have β(t) = Φ(t, sl)β(s + l ) = Φ(t, sl)β(s − l ) = Φ(t, sl)δl(sl)β(t + l ) = Φ(t, sl)δl(sl)(E+ Pl)β(t − l ) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)β(s + l ) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)β(s − l ) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)δ −1 l−1(s)β(s), so Λ(t, s) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)δ −1 l−1(s). Case 3. Two impulsive points between t and s. (i) For any s ∈ (sl−2, tl−1] and any t ∈ (tl, sl], we have β(t) = δl(t)β(t + l ) EJDE-2024/30 IMPULSIVE DIFFERENTIAL EQUATIONS 5 = δl(t)(E+ Pl)β(t − l ) = δl(t)(E+ Pl)Φ(tl, sl−1)β(s + l−1) = δl(t)(E+ Pl)Φ(tl, sl−1)β(s − l−1) = δl(t)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)β(t + l−1) = δl(t)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)β(t − l−1) = δl(t)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)Φ(tl−1, s)β(s), so Λ(t, s) = δl(t)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)Φ(tl−1, s). (ii) For any s ∈ (sl−2, tl−1] and any t ∈ (sl, tl+1], we have β(t) = Φ(t, sl)β(s + l ) = Φ(t, sl)β(s − l ) = Φ(t, sl)δl(sl)β(t + l ) = Φ(t, sl)δl(sl)(E+ Pl)β(t − l ) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)β(s + l−1) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)β(s − l−1) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)β(t + l−1) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)β(t − l−1) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)Φ(tl−1, s)β(s), so Λ(t, s) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)Φ(tl−1, s). (iii) For any s ∈ (tl−2, sl−2] and any t ∈ (tl, sl], we have β(t) = δl(t)β(t + l ) = δl(t)(E+ Pl)β(t − l ) = δl(t)(E+ Pl)Φ(tl, sl−1)β(s + l−1) = δl(t)(E+ Pl)Φ(tl, sl−1)β(s − l−1) = δl(t)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)β(t + l−1) = δl(t)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)β(t − l−1) = δl(t)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)Φ(tl−1, sl−2)β(s + l−2) = δl(t)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)Φ(tl−1, sl−2)β(s − l−2) = δl(t)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)Φ(tl−1, sl−2)δl−2(sl−2)δ −1 l−2(s)β(s), so Λ(t, s) = δl(t)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)Φ(tl−1, sl−2)δl−2(sl−2)δ −1 l−2(s). (iv) For any s ∈ (tl−2, sl−2] and any t ∈ (sl, tl+1], we have β(t) = Φ(t, sl)β(s + l ) = Φ(t, sl)β(s − l ) 6 Y. DING, K. LIU EJDE-2024/30 = Φ(t, sl)δl(sl)(E+ Pl)β(t − l ) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)β(s + l−1) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)β(s − l−1) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)β(t + l−1) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)β(t − l−1) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)Φ(tl−1, sl−2)β(s + l−2) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)Φ(tl−1, sl−2)β(s − l−2) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1) × Φ(tl−1, sl−2)δl−2(sl−2)δ −1 l−2(s)β(s), so Λ(t, s) = Φ(t, sl)δl(sl)(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)(E+ Pl−1)Φ(tl−1, sl−2) × δl−2(sl−2)δ −1 l−2(s). Case 4. Many impulsive points between t and s. (i) For any s ∈ (tς(c,s), sς(c,s)] and any t ∈ (sς(c,t), tς(c,t)+1], we have Λ(t, s) = Φ(t, sς(c,t))δς(c,t)(sς(c,t)) × ς(c,t)∏ l=ς(c,s)+1 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,s)(s). (ii) For any s ∈ (sς(c,s), tς(c,s)+1] and any t ∈ (tς(c,t), sς(c,t)], we have Λ(t, s) = δς(c,t)(t) ς(c,t)∏ l=ς(c,s)+2 [(E+ Pl)Φ(tl, sl−1)δl−1(s − l−1)] × (E+ Pς(c,s)+1)Φ(tς(c,s)+1, s). (iii) For any s ∈ (sς(c,s), tς(c,s)+1] and any t ∈ (sς(c,t), tς(c,t)+1], we have Λ(t, s) = Φ(t, sς(c,t))δς(c,t)(sς(c,t)) ς(c,t)∏ l=ς(c,s)+2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)] × (E+ Pς(c,s)+1)Φ(tς(c,s)+1, s). (iv) For any s ∈ (tς(c,s), sς(c,s)] and any t ∈ (tς(c,t), sς(c,t)], we have Λ(t, s) = δς(c,t)(t) ς(c,t)∏ l=ς(c,s)+1 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,s)(s). EJDE-2024/30 IMPULSIVE DIFFERENTIAL EQUATIONS 7 Summarizing, we can write Λ(t, s) =  Φ(t, s), t, s ∈ (sl, tl+1], l = 0, 1, 2, . . . ; δl(t)δ −1 l (s), t, s ∈ (tl, sl], l = 1, 2, . . . ; δς(c,t)(t) ∏ς(c,t) l=ς(c,s)+2[(E+ Pl)Φ(tl, sl−1)δl−1(s − l−1)] ×(E+ Pς(c,s)+1)Φ(tς(c,s)+1, s), s ∈ (sς(c,s), tς(c,s)+1], t ∈ (tς(c,t), sς(c,t)]; Φ(t, sς(c,t))δς(c,t)(sς(c,t)) ∏ς(c,t) l=ς(c,s)+1[(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)] ×δ−1 ς(c,s)(s), s ∈ (tς(c,s), sς(c,s)], t ∈ (sς(c,t), tς(c,t)+1]; Φ(t, sς(c,t))δς(c,t)(sς(c,t)) ∏ς(c,t) l=ς(c,s)+2[(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)] ×(E+ Pς(c,s)+1)Φ(tς(c,s)+1, s), s ∈ (sς(c,s), tς(c,s)+1], t ∈ (sς(c,t), tς(c,t)+1]; δς(c,t)(t) ∏ς(c,t) l=ς(c,s)+1[(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,s)(s), s ∈ (tς(c,s), sς(c,s)], t ∈ (tς(c,t), sς(c,t)]. □ Definition 2.6. A function β(·, βc) ∈ PC(C,Rn) is ϑ-periodic if β(t, βc) = β(t + ϑ, βc) for all t ≥ c. Definition 2.7. System (1.1) is exponentially stable if there exist constants λ1 ≥ 1 and λ2 < 0 such that ∥Λ(t, s)∥ ≤ λ1e λ2(t−s), c ≤ s ≤ t. 3. Basic properties of Λ(·, ·) Set κ = supl≥1 ∥E + Pl∥, µ = supl≥0 (tl+1−sl) τ τ , ξ = supl≥1 max(tl,sl] ∥δl(t)∥, ϕ = maxl≥0 ϕl. We use the following assumptions: (A1) α(t+ ϑ) = α(t) for t ∈ A; (A2) δl+a(t+ ϑ) = δl(t) for t ∈ B. Theorem 3.1. When c ≤ s ≤ t, we have ∥Λ(t, c)∥ ≤ eϕµ+ς(c,t)(lnκ+ln ξ+ϕµ), or ∥Λ(t, c)∥ ≤ eς(c,t)(lnκ+ln ξ+ϕµ). Proof. For t ∈ (sω(c,t), tω(c,t)+1], we have ∥Λ(t, c)∥ ≤ ∥Φ(t, sς(c,t))δς(c,t)(sς(c,t)) ς(c,t)∏ l=2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)] × (E+ P1)Φ(t1, c)∥ ≤ e ϕ τ (t−sς(c,t)) τ ξ(κeϕµξ)ς(c,t)−1κeϕµ ≤ eϕµ(κeϕµξ)ς(c,t) = eϕµ+ς(c,t)(lnκ+ln ξ+ϕµ). 8 Y. DING, K. LIU EJDE-2024/30 For t ∈ (tω(c,t), sω(c,t)], we have ∥Λ(t, c)∥ ≤ ∥δς(c,t)(t) ς(c,t)∏ l=2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)](E+ P1)Φ(t1, c)∥ ≤ ξ(κeϕµξ)(ς(c,t)−1)κeϕµ ≤ (κeϕµξ)ς(c,t) = eς(c,t)(lnκ+ln ξ+ϕµ). □ Theorem 3.2. If c ≤ s < u < t, then Λ(t, s) = Λ(t, u)Λ(u, s). Proof. By the form of (2.1), when s ∈ (sς(c,s), tς(c,s)+1] and t ∈ (sς(c,t), tς(c,t)+1], we have Λ(t, s) = Φ(t, sς(c,t))δς(c,t)(sς(c,t)) ς(c,t)∏ l=ς(c,u)+2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)](E+ Pς(c,u)+1) × Φ(tς(c,u)+1, u)Φ(u, sς(c,u))δς(c,u)(sς(c,u)) ς(c,u)∏ l=ς(c,s)+2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)] × (E+ Pς(c,u)+1)Φ(tς(c,u)+1, u) = Λ(t, u)Λ(u, s), u ∈ (sς(a,u), tς(a,u)+1], Λ(t, s) = Φ(t, sς(c,t))δς(c,t)(sς(c,t)) ς(c,t)∏ l=ς(c,u)+1 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,u)(u) × δς(c,u)(u) ς(c,u)∏ l=ς(c,s)+2 [(E+ Pl)Φ(ul, sl−1)δl−1(sl−1)](E+ Pς(c,s)+1)Φ(uς(c,s)+1, s) = Λ(t, u)Λ(u, s), u ∈ (tς(a,u), sς(a,u)]. When s ∈ (tς(c,s), sς(c,s)] and t ∈ (sς(c,t), tς(c,t)+1], we have Λ(t, s) = Φ(t, sς(c,t))δς(c,t)(sς(c,t)) ς(c,t)∏ l=ς(c,u)+2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)] × (E+ Pς(c,u)+1)Φ(tς(c,u)+1, u)Φ(u, sς(c,u))δς(c,u)(sς(c,u)) × ς(c,u)∏ l=ς(c,s)+1 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,s)(s) = Λ(t, u)Λ(u, s), u ∈ (sς(c,u), tς(c,u)+1], Λ(t, s) = Φ(t, sς(c,t))δς(c,t)(sς(c,t)) ς(c,t)∏ l=ς(c,u)+1 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,u)(u) EJDE-2024/30 IMPULSIVE DIFFERENTIAL EQUATIONS 9 × δς(c,u)(u) ς(c,u)∏ l=ς(c,s)+1 [(E+ Pl)Φ(ul, sl−1)δl−1(sl−1)]δ −1 ς(c,s)(s) = Λ(t, u)Λ(u, s), u ∈ (tς(c,u), sς(c,u)]. When s ∈ (sς(c,s), tς(c,s)+1] and t ∈ (tς(c,t), sς(c,t)], we have Λ(t, s) = δς(c,t)(t) ς(c,t)∏ l=ς(c,u)+2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)](E+ Pς(c,u)+1)Φ(tς(c,u)+1, u) × Φ(u, sς(c,u))δς(c,u)(sς(c,u)) ς(c,u)∏ l=ς(c,s)+2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)] × (E+ Pς(c,u)+1)Φ(tς(c,u)+1, u) = Λ(t, u)Λ(u, s), u ∈ (sς(c,u), tς(c,u)+1], Λ(t, s) = δς(c,t)(t) ς(c,t)∏ l=ς(c,u)+1 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,u)(u) × δς(c,u)(u) ς(c,u)∏ l=ς(c,s)+2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)](E+ Pς(c,s)+1)Φ(tς(c,s)+1, s) = Λ(t, u)Λ(u, s), u ∈ (tς(c,u), sς(c,u)]. When s ∈ (tς(c,s), sς(c,s)] and t ∈ (tς(c,t), sς(c,t)], we have Λ(t, s) = δς(c,t)(t) ς(c,t)∏ l=ς(c,u)+2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)](E+ Pς(c,u)+1)Φ(tς(c,u)+1, u) × Φ(u, sς(c,u))δς(c,u)(sς(c,u)) ς(c,u)∏ l=ς(c,s)+1 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,s)(s) = Λ(t, u)Λ(u, s), u ∈ (sς(c,u), tς(c,u)+1], Λ(t, s) = δς(c,t)(t) ς(c,t)∏ l=ς(c,u)+1 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,u)(u) × δς(c,u)(u) ς(c,u)∏ l=ς(c,s)+1 [(E+ Pl)Φ(ul, sl−1)δl−1(sl−1)]δ −1 ς(c,s)(s) = Λ(t, u)Λ(u, s), u ∈ (tς(c,u), sς(c,u)]. □ Theorem 3.3. If (A1) and (A2) hold, then Λ(·+ ϑ, ·+ ϑ) = Λ(·, ·), N ∈ N. 10 Y. DING, K. LIU EJDE-2024/30 Proof. Equation (2.1) implies that for s ∈ (sς(c,s), tς(c,s)+1] and t ∈ (sς(c,t), tς(c,t)+1], we have Λ(t+ ϑ, s+ ϑ) = Φ(t+ ϑ, sς(c,t+ϑ))δς(c,t+ϑ)(sς(c,t+ϑ)) × ς(c,t+ϑ)∏ l=ς(c,s+ϑ)+2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)](E+ Pς(c,s+ϑ)+1)Φ(tς(c,s+ϑ)+1, s+ ϑ) = Φ(t, sς(c,t))δς(c,t)(sς(c,t)) × ς(c,t)∏ l=ς(c,s)+2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)](E+ Pς(c,s)+1)Φ(tς(c,s)+1, s) = Λ(t, s). When s ∈ (tς(c,s), sς(c,s)] and t ∈ (sς(c,t), tς(c,t)+1], we have Λ(t+ ϑ, s+ ϑ) = Φ(t+ ϑ, sς(c,t+ϑ))δς(c,t+ϑ)(sς(c,t+ϑ)) × ς(c,t+ϑ)∏ l=ς(c,s+ϑ)+1 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,s+ϑ)(s+ ϑ) = Φ(t, sς(c,t))δς(c,t)(sς(c,t)) ς(c,t)∏ l=ς(c,s)+1 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,s)(s) = Λ(t, s). When s ∈ (sς(c,s), tς(c,s)+1] and t ∈ (tς(c,t), sς(c,t)], we have Λ(t+ ϑ, s+ ϑ) = δς(c,t+ϑ)(t+ ϑ) ς(c,t+ϑ)∏ l=ς(c,s+ϑ)+2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)] × (E+ Pς(c,s+ϑ)+1)Φ(tς(c,s+ϑ)+1, s+ ϑ) = δς(c,t)(t) ς(c,t)∏ l=ς(c,s)+2 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)](E+ Pς(c,s)+1)Φ(tς(c,s)+1, s) = Λ(t, s). When s ∈ (tς(c,s), sς(c,s)] and t ∈ (tς(c,t), sς(c,t)], we have Λ(t+ ϑ, s+ ϑ) = δς(c,t+ϑ)(t+ ϑ) ς(c,t+ϑ)∏ l=ς(c,s+ϑ)+1 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,s+ϑ)(s+ ϑ) = δς(c,t)(t) ς(c,t)∏ l=ς(c,u)+1 [(E+ Pl)Φ(tl, sl−1)δl−1(sl−1)]δ −1 ς(c,s)(s) = Λ(t, s). EJDE-2024/30 IMPULSIVE DIFFERENTIAL EQUATIONS 11 □ Theorem 3.4. Suppose that (A1) and (A2) hold. Then Λ(·+Nϑ, c) = Λ(·, c)[Λ(c+ ϑ, c)]N , N ∈ N. Proof. From Theorems 3.2 and 3.3, we have Λ(t+Nϑ, c) = Λ(t+Nϑ, c+Nϑ)Λ(c+Nϑ, c) = Λ(t+Nϑ, c+Nϑ)Λ(c+Nϑ, c+ (N − 1)ϑ)Λ(c+ (N − 1)ϑ, c) = Λ(t, c) N−1∏ l=0 Λ(c+ (N − l)ϑ, c)Λ(c+ (N − l − 1)ϑ, c) = Λ(t, c)[Λ(c+ ϑ, c)]N , N ∈ N. □ 4. Homogeneous linear problem Theorem 4.1. If (A1) and (A2) hold, then one of the following 2 items is satisfied: (i) Equation (1.1) has the trivial ϑ-periodic solution if and only if rank(E − Λ(c+ ϑ, c)) = n. (ii) Equation (1.1) has at least one nontrivial ϑ-periodic solution if and only if rank(E− Λ(c+ ϑ, c)) < n. Proof. (i) If rank(E−Λ(c+ϑ, c)) = n, then the only solution of (E−Λ(c+ϑ, c))β = 0 should be zero solution, which means the solution of (1.1) are trivial solution. (ii) If rank(E − Λ(c + ϑ, c)) < n, (E − Λ(c + ϑ, c))β = 0 has a nonzero solution which means (1.1) has ϑ-periodic nontrivial solution. □ Theorem 4.2. If (A1) holds, then lim t−s→∞ ς(s, t) t− s = a ϑ . Proof. For s ∈ [mϑ, (m+ 1)ϑ] and t ∈ [nϑ, (n+ 1)ϑ] for m ≤ n, we have (n−m− 1)ϑ ≤ t− s ≤ (n+ 1−m)ϑ, and (n−m− 1)a ≤ ς(s, t) ≤ (n+ 1−m)a. Hence, (n−m− 1)a (n+ 1−m)ϑ ≤ ς(s, t) t− s ≤ (n+ 1−m)a (n−m− 1)ϑ . It is obvious that t− s → ∞ if and only if n−m → ∞. So, a ϑ ≤ lim t−s→∞ ς(s, t) t− s ≤ a ϑ , and lim t−s→∞ ς(s, t) t− s = a ϑ . □ Theorem 4.3. If lnκ+ ln ξ + ϕµ < 0, then system (1.1) is exponentially stable. 12 Y. DING, K. LIU EJDE-2024/30 Proof. Combining Theorems 3.1 and 4.2, with ε ∈ (c, a ϑ ) and t ∈ (sl, tl+1], we obtain ∥Λ(t, c)∥ ≤ eϕµ+ς(c,t)(lnκ+ln ξ+ϕµ) ≤ eϕµ+( a ϑ−ε)(lnκ+ln ξ+ϕµ)t, in which λ1 = eϕµ ≥ 1 and λ2 = ( aϑ − ε)(lnκ+ ln ξ + ϕµ) < 0. For t ∈ (tl, sl], we obtaini ∥Λ(t, c)∥ ≤ eς(c,t)(lnκ+ln ξ+ϕµ) ≤ e( a ϑ−ε)(lnκ+ln ξ+ϕµ)t, in which λ1 = 1 and λ2 = ( aϑ − ε)(lnκ + ln ξ + ϕµ) < 0. From Definition 2.7, it follows that system (1.1) is exponentially stable. □ Theorem 4.4. If a nontrivial solution β(t, βc) of (1.1) is ϑ-periodic, then lnκ + ln ξ + ϕµ ≥ 0. Proof. β(c+ϑ) = Λ(c+ϑ, c)βc implies β(c) = Λ(c+ϑ, c)βc. Hence, ∥Λ(c+ϑ, c)∥ ≥ 1 and eς(c,t)(lnκ+ln ξ+ϕµ) ≥ 1. Then, there is lnκ+ ln ξ + ϕµ ≥ 0. □ Corollary 4.5. Suppose that (A1) and (A2) hold. If β(t) is periodic and exponen- tially stable, then β(t) = 0. Example 4.6. Consider (1.1) and let τ = 1/2, s0 = 0, sl = l, tl = l− 1 2 , l = 1, 2, . . . , ϑ = 1, a = 1, βc = (1, 0)⊤. Set α(t) = ( 3 2 (t− sς(a,t) 0 0 t− sς(a,t) ) , Pl = ( − 9 10 0 0 − 9 10 ) , δl(t) = ( 1 2 + t−tl 2(sl−tl) 0 0 −1 + t−tl sl−tl ) . So Φ(t, s) = ( e(t−sl) 3/2−(s−sl) 3/2 0 0 e 2 3 (t−sl) 3/2− 2 3 (s−sl) 3/2 ) , t, s ∈ (sl, tl+1]. Next, Λ(t, 0) = ( e(t−sς(a,t)) 3/2 0 0 e 2 3 (t−sς(a,t)) 3/2 )( 1 0 0 0 ) × ς(a,t)∏ l=1 ( 1 10 0 0 1 10 )( e(tl−sl−1) 3/2 0 0 e 2 3 (tl−sl−1) 3/2 )( 1 0 0 0 ) = ( e(t−sl) 3/2 ( 1 10e √ 2/4)ς(a,t) 0 0 0 ) . Then Λ(t, 0)βc = ( e(t−sς(a,t)) 3/2 ( 1 10e √ 2/4)ς(a,t) 0 ) and lim t→∞ ∥Λ(t, 0)βc∥ ≤ e(t−sς(a,t)) 3/2 ( 1 10 e √ 2/4)ς(a,t) ≤ e √ 2 4 +( √ 2 4 −ln 10)t → 0, so β is exponentially stable. EJDE-2024/30 IMPULSIVE DIFFERENTIAL EQUATIONS 13 Also, lnκ+ln ξ+ϕµ < 0 and Theorem 4.3 is verified. Furthermore, β(t+1) ̸= β(t) and E− Λ(1, 0) = ( 1− 1 10e √ 2/4 0 0 1 ) , det(E− Λ(1, 0)) ̸= 0, Equation (1.1) has only the trivial 1-periodic solution. 5. Nonhomogeneous linear problem Theorem 5.1. The solution of (1.2) has the form β(t) = Λ(t, c)βc + ς(c,t)−1∑ l=0 ∫ tl+1 sl Λ(t, s)ζ(s)(s− sl) τ−1ds + ∫ t sς(c,t) Λ(t, s)ζ(s)(s− sς(c,t)) τ−1ds+ ς(c,t)∑ l=1 Λ(t, sl)δl(sl)Ql. Proof. For t ∈ [s0, t1], using the variation of constants method, one has β(t) = Λ(t, c)βc + ∫ t 0 Λ(t, s)ζ(s)sτ−1ds. If it holds for t ∈ (sς(c,t)−1, tς(c,t)], one has β(t) = Λ(t, c)βc + ς(c,t)−2∑ l=0 ∫ tl+1 sl Λ(t, s)ζ(s)(s− sl) τ−1ds + ∫ t sς(c,t)−1 Λ(t, s)ζ(s)(s− sς(c,t)) τ−1ds+ ς(c,t)−1∑ l=1 Λ(t, sl)δl(sl)Ql, and for t ∈ (tς(c,t), sς(c,t)], we obtain β(t) = δς(c,t)(t)(E+ Pς(c,t))β(t − ς(c,t)) + δς(c,t)(t)Qς(c,t) = δς(c,t)(t)(E+ Pς(c,t))[Λ(t − ς(c,t), c)βc + ς(c,t)−1∑ l=0 ∫ tl+1 sl Λ(t−ς(c,t), s)ζ(s)(s− sl) τ−1ds + ς(c,t)−1∑ l=1 Λ(t−ς(c,t), sl)δl(sl)Ql] + δς(c,t)(t)Qς(c,t). Next, for t ∈ (sς(c,t), tς(c,t)+1], we have β(t) = Λ(t, sς(c,t))β(sς(c,t)) + ∫ t sς(c,t) Λ(t, s)ζ(s)(s− sς(c,t)) τ−1ds = Λ(t, sς(c,t))δς(c,t)(sς(c,t))(E+ Pς(c,t))Λ(t − ς(c,t), c)βc + ς(c,t)−1∑ l=0 ∫ tl+1 sl Λ(t, sς(c,t))δς(c,t)(sς(c,t))(E+ Pς(c,t))Λ(t − ς(c,t), s)ζ(s)(s− sl) τ−1ds 14 Y. DING, K. LIU EJDE-2024/30 + ς(c,t)−1∑ l=1 Λ(t, sς(c,t))δς(c,t)(sς(c,t))(E+ Pς(c,t))Λ(t − ς(c,t), sl)δl(sl)Ql + Λ(t, sς(c,t))δς(c,t)(sς(c,t))Qς(c,t) + ∫ t sς(c,t) Λ(t, s)ζ(s)(s− sς(c,t)) τ−1ds+ Λ(t, sς(c,t))δς(c,t)(sς(c,t))Qς(c,t) = Λ(t, c)βc + ς(c,t)−1∑ l=0 ∫ tl+1 sl Λ(t, s)ζ(s)(s− sl) τ−1ds + ∫ t sς(c,t) Λ(t, s)ζ(s)(s− sς(c,t)) τ−1ds+ ς(c,t)∑ l=1 Λ(t, sl)δl(sl)Ql. By the mathematical induction method, we obtain β(t) = Λ(t, c)βc + ς(c,t)−1∑ l=0 ∫ tl+1 sl Λ(t, s)ζ(s)(s− sl) τ−1ds + ∫ t sς(c,t) Λ(t, s)ζ(s)(s− sς(c,t)) τ−1ds+ ς(c,t)∑ l=1 Λ(t, sl)δl(sl)Ql. □ Now we introduce the following following assumption: (A3) ζ(t+ ϑ) = ζ(t), for t ∈ A. Theorem 5.2. Suppose that (A1)–(A3) hold. If the solution of (1.2) is bounded, then it is a ϑ-solution. Proof. Let β̃(t) be a bounden solution of (1.2). Then β̃(c+ nϑ) is bounded. Using Theorems 3.2 and 3.3, one obtains β̃(c+ (n+ 1)ϑ) = Λ(c+ (n+ 1)ϑ, c)βc + (n+1)a−1∑ l=0 ∫ tl+1 sl Λ(c+ (n+ 1)ϑ, s)ζ(s)(s− sl) τ−1ds + (n+1)a∑ l=1 Λ(c+ (n+ 1)ϑ, sl)δl(sl)Ql = Λ(c+ (n+ 1)ϑ, c+ nϑ) [ Λ(c+ nϑ, c)βc + na−1∑ l=0 ∫ tl+1 sl Λ(c+ nϑ, s)ζ(s)(s− sl) τ−1ds + na∑ l=1 Λ(c+ nϑ, sl)δl(sl)Ql ] + (n+1)a−1∑ l=na ∫ tl+1 sl Λ(c+ (n+ 1)ϑ, s)ζ(s)(s− sl) τ−1ds + (n+1)a∑ l=na+1 Λ(c+ (n+ 1)ϑ, sl)δl(sl)Ql = Λ(c+ ϑ, c)β̃(c+ nϑ) + a−1∑ l=0 ∫ tl+1 sl Λ(c+ (n+ 1)ϑ, s+ nϑ)ζ(s+ nϑ)(s− sl) τ−1ds EJDE-2024/30 IMPULSIVE DIFFERENTIAL EQUATIONS 15 + a∑ l=1 Λ(c+ ϑ, sl)δl(sl)Ql = Λ(c+ ϑ, c)β̃(c+ nϑ) + a−1∑ l=0 ∫ tl+1 sl Λ(c+ ϑ, s)ζ(s)(s− sl) τ−1ds + a∑ l=1 Λ(c+ ϑ, sl)δl(sl)Ql = Λ(c+ ϑ, c)β̃(c+ nϑ) + Λc, where Λc = ∑a−1 l=0 ∫ tl+1 sl Λ(c + ϑ, s)ζ(s)(s − sl) τ−1ds + ∑a l=1 Λ(c + ϑ, sl)δl(sl)Ql. Hence, β̃(c+ nϑ) = Λa(c+ ϑ, c)β̃(c) + a−1∑ l=0 Λl(c+ ϑ, c)Λc. Then, β̃(t) is a ϑ-periodic solution that needs to be proven. If β̃(t) is not the ϑ-periodic of (1.2), then we can not find a βc ∈ Rn such that (E− Λ(c+ ϑ, c))βc = Λc. By Fredholm alternative, we can find a Z ∈ Rn such that (E− Λ⊤(c+ ϑ, c))Z = 0, ⟨Λc,Z⟩ ̸= 0. Since (E − Λ⊤(c + ϑ, c))Z = 0, with each n ∈ N, we have [Λn(c + ϑ, c))]⊤Z = Z. Also, ⟨β̃(c+ nϑ),Z⟩ = ⟨Λa(c+ ϑ, c)β̃(c) + a−1∑ l=0 Λl(c+ ϑ, c)Λc,Z⟩ = ⟨β̃(c), [Λa(c+ ϑ, c)]⊤Z⟩+ a−1∑ l=0 ⟨Λc, [Λ l(c+ ϑ, c)]⊤Z⟩ = ⟨β̃(c),Z⟩+ a⟨Λc,Z⟩ → ∞, as a → ∞, which contradicts the boundness of β̃(t). So β̃(t) is a ϑ-periodic solution of (1.2). □ Now we introduce the following assumptions: (A4) det(E− Λ(c+ ϑ, c)) ̸= 0; (A5) det(E− Λ(c+ ϑ, c)) = 0. Theorem 5.3. If (A1)–(A4) hold, then (1.2) has a ϑ-periodic solution with βc = (E− Λ(c+ ϑ, c))−1Λc. Proof. det(E− Λ(c+ ϑ, c)) ̸= 0 implies βc = (E− Λ(c+ ϑ, c))−1Λc to satisfy β(c + ϑ, βc) = βc. With β̃(t) = β(t + ϑ) and the properties of Λ(t, s), we know β̃(t) = β(t + ϑ) is the solution of (1.2) with β̃(c) = β(c + ϑ) = βc. The uniqueness of the solution implies β̃(t) = β(t), i.e. β(t+ ϑ, βc) = β(t, βc). □ 16 Y. DING, K. LIU EJDE-2024/30 We study the system Dsl τ χ(t) = −α⊤(t)χ(t), t ∈ (sl, tl+1], l ∈ N0 := {0, 1, 2, . . . }, 0 < τ < 1, χ(t+l ) = −(E+ P⊤ l )−1P⊤ l χ(t−l ), l ∈ N := {1, 2, . . . }, χ(t) = (δ⊤l (t))−1χ(t+l ), t ∈ (tl, sl], l ∈ N, χ(s+l ) = χ(s−l ), l ∈ N. (5.1) Theorem 5.4. If β(t) is the solution of (1.1), and χ(t) the solution of (5.1), then ⟨β(t), χ(t)⟩ is a constant. Proof. For t ∈ (sl, tl+1], we have Dsl τ ⟨β(t), χ(t)⟩ = ⟨Dsl τ β(t), χ(t)⟩+ ⟨β(t),Dsl τ χ(t)⟩ = ⟨α(t)β(t), χ(t)⟩+ ⟨β(t),−α⊤χ(t)⟩ = ⟨β(t), α(t)⊤χ(t)⟩+ ⟨β(t),−α(t)⊤χ(t)⟩ = 0. For s ∈ (tl, sl], we have ⟨β(t), χ(t)⟩ = ⟨δl(t)β(t+l ), (δ ⊤ l (t))−1χ(t+l )⟩ = ⟨β(t+l ), χ(t + l )⟩. For t = tl, we have ⟨β(t+l ), χ(t + l )⟩ = ⟨(E+ Pl)β(tl), [E− (E+ P⊤ l )−1P⊤ l ]χ(tl)⟩ = ⟨(E+ Pl)β(tl), (E+ P⊤ l )−1χ(tl)⟩ = ⟨β(tl), χ(tl)⟩. Hence, ⟨β(t), s(t)⟩ is a constant. □ Theorem 5.5. If (A1)–(A3), (A5) hold, then (1.2) has a ϑ-periodic solution if and only if ⟨χc,Λc⟩ = 0, where χc is the initial value of the ϑ-solution of (5.1). Proof. Equation (1.2) has a ϑ-periodic solution if and only if there exists βc such that (E− Λ(c+ ϑ, c))βc = Λc. Then ⟨χc,Λc⟩ = ⟨χc, (E− Λ(c+ ϑ, c))βc⟩ = ⟨(E− Λ(c+ ϑ, c))⊤χc, βc⟩ = ⟨(E− Λ⊤(c+ ϑ, c))χc, βc⟩ = ⟨0, βc⟩ = 0. □ Example 5.6. let τ = 1/2, s0 = 0, sl = l, tl = l − 1 2 for l = 1, 2, . . . , ϑ = 1, and a = 1. Set α(t), Pl, δl(t) as in Example 4.6 and let ζ(t) = ( t− sl 0 ) , t ∈ (sl, tl+1], Ql = ( 1 1 ) . So Λc = a−1∑ l=0 ∫ tl+1 sl Λ(c+ ϑ, s)ζ(s)(s− sl) τ−1ds+ a∑ l=1 Λ(c+ ϑ, sl)δl(sl)Ql = ( 5 3 − 2 3e − √ 2/4 0 ) . EJDE-2024/30 IMPULSIVE DIFFERENTIAL EQUATIONS 17 Then E− Λ(1, 0) = ( 1− 1 10e √ 2/4 0 0 1 ) , βc = (E− Λ(1, 0))−1Λc = ( 5 3− 2 3 e − √ 2/4 1− 1 10 e √ 2/4 0 ) , Λ(t, s) = ( e(t−sς(c,t)) 3/2 0 0 e 2 3 (t−sς(c,t)) 3/2 )( 1 0 0 0 ) × ς(c,t)∏ l=ς(c,s)+2 ( 1 10 0 0 1 10 )( e(tl−sl−1) 3/2 0 0 e 2 3 (tl−sl−1) 3/2 )( 1 0 0 0 )( 1 10 0 0 1 10 ) × ( e(tς(c,s)+1−sς(c,s)) 3/2−(s−sς(c,s)) 3/2 0 0 e 2 3 (tς(c,s)+1−sς(c,s)) 3/2− 2 3 (s−sς(c,s)) 3/2 ) = ( e(t−sς(c,t)) 3/2−(s−sς(c,s)) 3/2 ( 1 10e √ 2/4)ς(c,t)−ς(c,s) 0 0 0 ) . Then β(t) = Λ(t, c)βc + ς(c,t)−1∑ l=0 ∫ tl+1 sl Λ(t, s)ζ(s)(s− sl) τ−1ds + ∫ t sς(c,t) Λ(t, s)ζ(s)(s− sς(c,t)) τ−1ds+ ς(c,t)∑ l=1 Λ(t, sl)δl(sl)Ql = ( e(t−sς(c,t)) 3/2 ( 1 10e √ 2/4)ς(c,t) 0 0 0 )( 5 3− 2 3 e − √ 2/4 1− 1 10 e √ 2/4 0 ) + ς(c,t)−1∑ l=0 ∫ tl+1 sl ( e(t−sς(c,t)) 3/2−(s−sl) 3/2 ( 1 10e √ 2/4)ς(c,t)−1−l(s− sl) 1/2 0 ) ds + ∫ t sς(c,t) ( e(t−sς(c,t)) 3/2−(s−sς(c,t)) 3/2 (s− sς(c,t)) 1/2 0 ) ds + ς(c,t)∑ l=1 ( e(t−sς(c,t)) 3/2 ( 1 10e √ 2/4)ς(c,t)−l 0 ) = ( 5 3− 2 3 e − √ 2/4 1− 1 10 e √ 2/4 e(t−sς(c,t)) 3/2 + 2 3e (t−sς(c,t)) 3/2 − 2 3 0 ) . So β(t+ 1, 0, β0) = β(t, 0, β0), β(t, 0, β0) is a 1-periodic solution, and ∥β(t)∥ ≤ 5 3e √ 2/4 − 2 3 1− 1 10e √ 2/4 + 2 3 e √ 2/4 − 2 3 . 18 Y. DING, K. LIU EJDE-2024/30 6. Nolinear problem In this section, we study the ϑ-periodic solution of (1.3), using the following assumptions: (A6) for t ∈ A and β ∈ Rn, η(t+ ϑ, β) = η(t, β); (A7) for t ∈ A and β ∈ Rn, there is a η > 0 such that ∥η(t, β)∥ ≤ η. We study the system Dsl τ β(t) = α(t)β(t) + η(t, β(t)), β(sl−1) = βl−1, t ∈ (sl−1, tl], 0 < τ < 1, l ∈ N0, β0 = β(c), whose solution is β(t) = Λ(t, sl)βl−1 + ∫ t sl Λ(t, s)η(s, β(s))(s− sl) τ−1ds. (6.1) We set the mapping Pl(βl−1) := δl(sl) ◦ ((E+ Pl) ◦ β(tl) +Ql). (6.2) Equality (6.1) implies ∥β(tl)∥ ≤ e ϕ τ (tl−sl−1) τ ∥βl−1∥+ η ϕ (e ϕ τ (tl−sl−1) τ − 1); and (6.2) implies ∥Pl(βl−1)∥ ≤ κξe ϕ τ (tl−sl−1) τ ∥βl−1∥+ κξ η ϕ (e ϕ τ (tl−sl−1) τ − 1) + κQ, where Q = maxl∈N ∥Ql∥. Then we construct the operator P := Pa ◦ Pa−1 ◦ · · · ◦ P1, and set ϕl = e ϕ τ (tl−sl−1) τ , and ϱ = κξ. Theorem 6.1. If (A7) holds, then ∥P (β0)∥ ≤ ϱa a∏ l=1 ϕl∥β0∥+ η ϕ a−1∑ l=1 { ϱa−j+1 a−1∏ j=l ϕa . . . ϕj+1(ϕj − 1) } + ηϱ ϕ (ϕa − 1) + [ a∑ l=2 ϱa−j+1 a∏ j=l ϕa . . . ϕj + 1 ] κQ. (6.3) Proof. For l = 1, we have ∥β1∥ ≤ ϱϕ1∥β0∥+ ϱη ϕ (ϕ1 − 1) + κQ. If (6.3) is satisfied with l = a− 1, then for l = a, we have ∥βa∥ ≤ ϱϕa∥βa−1∥+ ηϱ ϕ (ϕa − 1) + κQ ≤ ϱϕa { ϱa−1 a−1∏ l=1 ϕl∥β0∥+ η ϕ a−2∑ l=1 { ϱa−j a−2∏ j=l ϕa−1 . . . ϕj+1(ϕj − 1) } EJDE-2024/30 IMPULSIVE DIFFERENTIAL EQUATIONS 19 + ηϱ ϕ (ϕa−1 − 1) + [ a−1∑ l=2 ϱa−j a−1∏ j=l ϕa−1 . . . ϕj + 1 ] κQ } + ηϱ ϕ (ϕa − 1) + κQ = ϱa a∏ l=1 ϕl∥β0∥+ { η ϕ a−2∑ l=1 [ ϱa−j+1 a−2∏ j=l ϕaϕa−1 . . . ϕj+1(ϕj − 1) ] + ηϱ2 ϕ ϕa(ϕa−1 − 1) } + [ a−1∑ l=2 ϱa−j+1 a−1∏ j=l ϕaϕa−1 . . . ϕjκQ+ ϱϕaκQ ] + ηϱ ϕ (ϕa − 1) + κQ = ϱa a∏ l=1 ϕl∥β0∥+ η ϕ a−1∑ l=1 { ϱa−j+1 a−1∏ j=l ϕa . . . ϕj+1(ϕj − 1) } + a∑ l=2 ϱa−j+1 a∏ j=l ϕa . . . ϕjκQ+ ηϱ ϕ (ϕa − 1) + κQ = ϱa a∏ l=1 ϕl∥β0∥+ η ϕ a−1∑ l=1 { ϱa−j+1 a−1∏ j=l ϕa . . . ϕj+1(ϕj − 1) } + [ a∑ l=2 ϱa−j+1 a∏ j=l ϕa . . . ϕj + 1 ] κQ+ ηϱ ϕ (ϕa − 1). □ Theorem 6.2. If (A1)–(A3), (A6), (A7) hold, then (1.3) has a ϑ-periodic solution if and only if P has a fixed point. Proof. Sufficiency: If P has a fixed point β0, there is P (β0) := Pa ◦ Pa−1 ◦ · · · ◦ P1(β0) = Λ(c+ ϑ, c)β0 + a−1∑ l=0 ∫ tl+1 sl Λ(c+ ϑ, s)η(s, β(s))(s− sl) τ−1ds + a∑ l=1 Λ(c+ ϑ, sl)δl(sl)Ql. The above equality implies βc = β(c+ ϑ). Next, we show that β(·+ ϑ) = β(·). For t = t̃+Nϑ, Theorems 3.2, 3.3 and 3.4 imply that β(t) = β(t̃+Nϑ) = Λ(t̃+Nϑ, c)β(c) = [Λ(t̃+ ϑ, t̃)]NΛ(t̃, c)β(c), and β(t+ ϑ) = β(t+ (N + 1)ϑ) = Λ(t̃+ (N + 1)ϑ, c)βc = [Λ(t̃+ ϑ, t̃)]N+1Λ(t̃, c)βc = [Λ(t̃+ ϑ, t̃)]NΛ(t̃+ ϑ, c)βc = [Λ(t̃+ ϑ, t̃)]Nβ(t̃+ ϑ) = [Λ(t̃+ ϑ, t̃)]NΛ(t̃+ ϑ, ϑ)β(ϑ) = [Λ(t̃+ ϑ, t̃)]NΛ(t̃, c)βc, 20 Y. DING, K. LIU EJDE-2024/30 then β(t+ ϑ) = β(t). Necessity: if β(t) is a ϑ-periodic solution of (1.3), then P (β0) = β0 and β0 is a fixed point of P . □ Theorem 6.3. Suppose that (A1)–(A3), (A6), (A7) hold. If ρ := ϱa a∏ l=1 ϕl < 1, then (1.3) has at least one ϑ-periodic solution and ∥β0∥ ≤ ω := ρ̃ 1−ρ , where ρ̃ = η ϕ a−1∑ l=1 { ϱa−j+1 a−1∏ j=l ϕa . . . ϕj+1(ϕj − 1) } + ηϱ ϕ (ϕa − 1) + [ a∑ l=2 ϱa−j+1 a∏ j=l ϕa . . . ϕj + 1 ] κQ. Proof. ∥β0∥ ≤ ρ̃ 1−ρ and (6.3) imply ∥P (β0)∥ ≤ ρ∥β0∥+ ρ̃ ≤ ρ̃ 1− ρ . Then P : B(0, ω) → B(0, ω). Obviously, P is continuous. Next, Brouwer fixed point theorem implies there is a β0 ∈ B(0, ω) such that P (β0) = β0. □ Example 6.4. Consider (1.3) and let τ = 1 2 , s0 = 0, sl = l, tl = l − 1 2 , for l = 1, 2, . . . , ϑ = 1, a = 1, βc = (0.1, 0)⊤. Set α(t) = ( 3 2 (t− sς(a,t) 0 0 t− sς(a,t) ) , Pl = ( − 9 10 0 0 − 9 10 ) , δl(t) = ( 1 2 + t−tl 2(sl−tl) 0 0 −1 + t−tl sl−tl ) . Set α(t), Pl, δl(t) as in Example 4.6 and let η(t, β) = ( (t− sl) cosβ 0 ) , t ∈ (sl, tl+1], Ql = ( 1 1 ) . Then, η = 1/2, ϱ = κξ = 1/10, ρ = ϱϕ1 = e 3 √ 2 8 10 < 1 and ρ̃ = ηϱ ϕ (ϕa − 1) + κQ = √ 2e 3 √ 2 8 − √ 2 + 3 30 . Thus, (1.3) has at least one periodic solution and 0.1 = ∥β0∥ < ρ̃ 1−ρ = 0.16. Acknowledgments. This work was partially supported by the National Natural Science Foundation of China (11661016), by the Guizhou Provincial Basic Research Program (Natural Science) (No. QKHJC-ZK[2024]YB067), by the Guizhou Univer- sity introduced talent research project (2022) 70, and by the Basic research project of Guizhou University[2023]39. EJDE-2024/30 IMPULSIVE DIFFERENTIAL EQUATIONS 21 References [1] S. Abbas, M. Benchohra; Uniqueness and Ulam stabilities results for partial fractional dif- ferential equations with not instantaneous impulses, Applied Mathematics and Computation, 257 (2015), 190-198. [2] A. A. Abdelhakim, J. A. T. Machado; A critical analysis of the conformable derivative, Nonlinear Dynamics, 95 (2019), 3063-3073. [3] T. Abdeljawad; On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279 (2015), 57-66. [4] M. Abul-Ez, M. Zayed, A. Youssef, M. De la Sen; On conformable fractional Legendre poly- nomials and their convergence properties with applications, Alexandria Engineering Journal, 59 (2020), 5231-5245. [5] E. Alvarez, A. Gómez, M. Pinto; (ω, c)-periodic functions and mild solutions to abstract frac- tional integro-differential equations, Electronic Journal of Qualitative Theory of Differential Equations, 2018 (2018) No. 16, 1-8. [6] M. Ayata, O. Ozkan; A new application of conformable Laplace decomposition method for fractional Newell-Whitehead-Segel equation, AIMS Mathematics, 5 (2020), 7402-7412. [7] L. Bai, J. J. Nieto; Variational approach to differential equations with not instantaneous impulses, Applied Mathematics Letters, 73 (2017), 44-48. [8] M. Bohner, V. F. Hatipoǧlu; Dynamic cobweb models with conformable fractional derivatives, Nonlinear Analysis: Hybrid Systems, 32 (2019), 157-167. [9] P. Chen, Y. Li, H. Yang; Perturbation method for nonlocal impulsive evolution equations, Nonlinear Analysis-Hybrid Systems, 8 (2013), 22-30. [10] W. Chung; Fractional Newton mechanics with conformable fractional derivative, Journal of Computational and Applied Mathematics, 290 (2015), 150-158. [11] V. Colao, L. Muglia, H. Xu; An existence result for a new class of impulsive functional differential equations with delay, Journal of Mathematical Analysis and Applications, 441 (2016), 668-683. [12] E. Hernández, D. O’Regan; On a new class of abstract impulsive differential equations, Porceedings of the American Mathematical Society, 141 (2013), 1641-1649. [13] M. Li, J. Wang, D. O’Regan; Existence and Ulam’s stability for conformable fractional differ- ential equations with constant coefficients, Bulletin of the Malaysian Mathematical Sciences Society, 42 (2019), 1791-1812. [14] K. Liu, J. Wang, D. O’Regan, M. Fečkan; A new class of (ω, c)-periodic non-instantaneous impulsive differential equations, Mediterranean Journal of Mathematics, 17 (2020), 155. [15] M. Pierri, H. R. Henŕıquez, A. Prokczyk; Global solutions for abstract differential equations with non-instantaneous impulses, Mediterranean Journal of Mathematics, 34 (2016), 1685- 1708. [16] M. Pierri, D. O’Regan, V. Rolnik; Existence of solutions for semi-linear abstract differen- tial equations with not instantaneous impulses, Applied Mathematics and Computation, 219 (2013), 6743-6749. [17] A. G. Talafha, S. M. Alqaraleh, M. Al-Smadi, S. Hadid, S. Momani; Analytic solutions for a modified fractional three wave interaction equations with conformable derivative by unified method, Alexandria Engineering Journal, 59 (2020), 3731-3739. [18] Y. Tian, J. Wang, Y. Zhou; Almost periodic solutions for a class of non-instantaneous impulsive differential equations, Quaestiones Mathematicae, 42 (2019), 885-905. [19] F. Usta, M. Z. Sarıkaya; The analytical solution of Van der Pol and Lienard differential equa- tions within conformable fractional operator by retarded integral inequalities, Demonstratio Mathematica, 52 (2019), 204-212. [20] J. Wang, A. G. Ibrahim, D. O’Regan, Y. Zhou; Controllability for noninstantaneous impulsive semilinear functional differential inclusions without compactness, Indagationes Mathemati- cae, 29 (2018), 1362-1392. [21] G. Xiao, J. Wang; Representation of solutions of linear conformable delay differential Equa- tions, Applied Mathematics Letters, 117 (2021), 107088. [22] D. Yang, J. Wang; Non-instantaneous impulsive fractional-order implicit differential equa- tions with random effects, Stochastic Analysis and Applications, 35 (2017), 719-741. 22 Y. DING, K. LIU EJDE-2024/30 [23] P. Yang, J. Wang, M. Fečkan; Boundedness, periodicity, and conditional stability of nonin- stantaneous impulsive evolution equations, Mathematical Methods in the Applied Sciences, 43 (2020), 5905-5926. [24] P. Yang, J. Wang, M. Fečkan; Periodic nonautonomous differential equations with nonin- stantaneous impulsive effects, Mathematical Methods in the Applied Sciences, 42 (2019), 3700-3720. [25] P. Yang, J. Wang, D. O’Regan; Periodicity of non-homogeneous trajectories for non- instantaneous impulsive heat equations, Electronic Journal of Differential Equations, 2020 (2020) No. 18, 1-7. Yuanlin Ding Department of Mathematics, School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China Email address: yldingmath@126.com Kui Liu (corresponding author) Department of Mathematics, School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China Email address: liuk@gzu.edu.cn 1. Introduction 2. Preliminaries 3. Basic properties of (,) 4. Homogeneous linear problem 5. Nonhomogeneous linear problem 6. Nolinear problem Acknowledgments References