Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 18, pp. 1–7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu PERIODICITY OF NON-HOMOGENEOUS TRAJECTORIES FOR NON-INSTANTANEOUS IMPULSIVE HEAT EQUATIONS PENG YANG, JINRONG WANG, DONAL O’REGAN Communicated by Giovanni Molica Bisci Abstract. In this article, we introduce a non-instantaneous impulsive opera- tor associated with the heat semigroup and give some basic properties. We de- rive an abstract formula for the solutions to non-instantaneous impulsive heat equations. Also we show the existence and uniqueness of the non-homogeneous periodic trajectory. 1. Introduction Non-instantaneous differential equations are used to characterize evolution pro- cesses in pharmacotherapy and ecological systems. This type of impulsive equations was introduced in [4] their basic theory can be found in [1, 2, 3, 4, 6, 7, 8, 9, 10]. Motivated by [4, 5, 8], we study periodicity of non-homogeneous trajectories for the non-instantaneous impulsive heat equation with Dirichlet boundary conditions ut(t, y) = ∆u(t, y) + f(t, y), y ∈ Ω, t ∈ [si−1, ti], δu(ti, y) = Iiu(ti, y) + ci(y), y ∈ Ω, u(t, y) = Bi(t)u(t+i , y), y ∈ Ω, t ∈ (ti, si], u(0, y) = ξ(y), y ∈ Ω, (1.1) where i ∈ N+, δu(ti, y) := u(t+i , y) − u(ti, y), ∆ := ∑n i=1 ∂2 ∂y2i denotes the Laplace operator and Ω ⊆ Rn is an open set. The sequences {si}i∈N+ and {ti}i∈N+ satisfy s0 = 0 and si−1 < ti < si < ti+1 < · · · for any i ∈ N+, and limi→+∞ ti = +∞. Let I = ∪∞i=1[si−1, ti] and J = ∪∞i=1(ti, si]. Assume that X = L1(Rn), Ii, Bi(·) ∈ L(X), ci(y), ξ(y) ∈ X, and f ∈ C(I, X); here L(X) is the set of bounded linear operators on X. In addition, we suppose Bi(t + i ) = E, where E is the identity map. Let z(t)(y) := u(t, y), g(t)(y) := f(t, y), κi(y) := ci(y), and then we may transform the non-instantaneous impulsive heat equation (1.1) into the abstract 2010 Mathematics Subject Classification. 35K05. Key words and phrases. Non-homogeneous periodic trajectory; heat equation; non-instantaneous impulsive. c©2020 Texas State University. Submitted July 25, 2019. Published February 13, 2020. 1 2 P. YANG, J. WANG, D. O’REGAN EJDE-2020/?? non-instantaneous impulsive evolution equation z′(t) = ∆z(t) + g(t), t ∈ [si−1, ti], δz(ti) = Iiz(ti) + κi, z(t) = Bi(t)z(t + i ), t ∈ (ti, si], z(0) = z0. (1.2) Thus, it is sufficient to show the existence and uniqueness of the inhomogeneous periodic trajectory of (1.2) to study the same problem for (1.1). 2. Preliminaries Let Ξ := {tk; k ∈ N+}, R+ = I ∪ J, PC(R+, X) := { z : R+ \ Ξ→ X is continuous, z(ti) = z(t−i ) and z(ti) 6= z(t+i ) } . The bounded piecewise continuous function space with values in a Banach space X is defined as BPC(R+, X) := { z ∈ PC(R+, X), sup t∈R+ ‖z(t)‖ <∞ } endowed with the norm ‖z‖BPC := supt∈R+ ‖z(t)‖. Recall that the fundamental solution of the heat equation is Φ(x, t) = { 1 (4πt)n/2 exp ( − |x|2/(4t) ) , x ∈ Rn, t > 0, 0, x ∈ Rn, t < 0. Note that Φ is singular at the point (0, 0). For each t > 0,∫ Rn Φ(x, t)dx = 1. A semigroup of bounded linear operators (H(t))t≥0 on X defined by (H(t)ξ)(y) = 1 (4πt)n/2 ∫ Rn e− |y−s|2 4t ξ(s)ds, t > 0; H(0) = E is called the heat semigroup generated by ∆. Lemma 2.1. For each t ≥ 0, ‖H(t)‖L(X) ≤ 1. Proof. For t = 0, the conclusion is obvious. For each t > 0, we have ‖H(t)‖L(X) = sup ‖ξ‖≤1 ‖ 1 (4πt)n/2 ∫ Rn e − |y−s|2 4t ξ(s)ds‖ ‖ξ‖ ≤ sup ‖ξ‖≤1 1 (4πt)n/2 ∫ Rn e − |y−s|2 4t ds‖ξ‖ ‖ξ‖ = 1. � It is well known that the solution of zt(t) = ∆z(t), t > τ with z(τ) = zτ , is z(t) = S(t, τ)zτ , where S(t, τ) = H(t− τ). EJDE-2020/18 NON-INSTANTANEOUS IMPULSIVE HEAT EQUATIONS 3 Definition 2.2. A non-instantaneous impulsive operator G(·, ·) : Π := {(t, s) ∈ R+ × I : s ≤ t} → L(X) is defined as G(t, s) =  Si(t, s), if t, s ∈ [si−1, ti], Sk(t, sk−1)Bk−1(sk−1)(E + Ik−1) × ∏k−1 j=i+1{Sj(tj , sj−1)Bj−1(sj−1)(E + Ij−1)}Si(ti, s), if si−1 ≤ s ≤ ti < · · · < sk−1 ≤ t ≤ tk, Bk(t)(E + Ik) ∏k j=i+1{Sj(tj , sj−1)Bj−1(sj−1)(E + Ij−1)}Ui(ti, s), if si−1 ≤ s ≤ ti < · · · < tk < t ≤ sk, where Si(t, τ) := S(t, τ)|t,τ∈[si−1,ti]. Note thatG(t, s) = E if t = s andG(t+i , s) = (E+Ii)G(ti, s) andBi(si)G(t+i , s) = G(si, s). Clearly, any solution of z′(t) = ∆z(t), t ∈ [si−1, ti], δz(ti) = Iiz(ti) + κi, z(t) = Bi(t)z(t + i ), t ∈ (ti, si], z(0) = z0, has the form z(t) = G(t, 0)z0 for t ≥ 0. A function z(t) is called a mild solution of (1.2), if it satisfies the integral equation z(t) = G(t, 0)z0 + ∫ t 0 G(t, ω)g̃(ω)dω + r(0,t)∑ j=1 G(t, sj)Bj(sj)κj , (2.1) where g̃(t) = { g(t), t ∈ I, 0, t ∈ J. The function z(·) is also called the inhomogeneous trajectory of equation (1.1). Now we present the periodic conditions that will be used in the rest of the paper. (A1) There exists a m ∈ N+ such that Bi+m(t + T ) = Bi(t) for t ∈ (ti, si] and i ∈ N+. (A2) Ii+m = Ii for i ∈ N+. (A3) si+m = si + T for i ∈ N and ti+m = ti + T for i ∈ N+. (A4) ci+m(y) = ci(y) for i ∈ N+ and every y ∈ Ω. (A5) f(t+ T, y) = f(t, y) for t ∈ I and every y ∈ Ω. 3. Basic properties for group G Let r(s, t) be the number of impulsive points in the interval (s, t). Note r(0, T ) = m. Theorem 3.1. For any s ∈ I and t ∈ R+, we have ‖G(t, s)‖ ≤ (βγ)r(s,t), where β = supi≥1 supt∈(ti,si] ‖Bi(t)‖ and γ = supi≥1 ‖E + Ii‖. Proof. Using Definition 2.2 and ‖H(t)‖L(X) ≤ 1, Following a process similar to that in [9, Theorem 3.1] we obtain the desired result. � 4 P. YANG, J. WANG, D. O’REGAN EJDE-2020/?? Theorem 3.2 ([9, Theorem 3.3]). If s ≤ u ≤ t and u, s ∈ I, then G(t, s) = G(t, u)G(u, s). Theorem 3.3 ([9, Theorem 3.2]). If (A1)–(A4) are satisfied, then G(·+T, ·+T ) = G(·, ·). From Theorems 3.2 and 3.3, we have the following result. Corollary 3.4. For any t ∈ R+ and p ∈ N , G(t+ pT, 0) = [G(t, 0)][G(T, 0)]p. 4. Inhomogeneous periodic trajectory In this section, we establish the existence and uniqueness of the inhomogeneous periodic trajectory for (1.1). Theorem 4.1 (see [9, Theorem 4.3]). If (A3) holds, then lim t−s→∞ r(s, t) t− s = m T . Remark 4.2. Theorem 4.1 shows that for an arbitrary ε, with 0 < ε < m T , there exists J > 0, and for t− s > J , ∣∣r(s, t) t− s − m T ∣∣ < ε. To guarantee the boundedness of the solution, we introduce the following as- sumption: (A6) βγ < 1. Then we set M := (βγ)( m T −ε)J lnβγ ‖g‖BPC + βc ∑ sj∈Ω4 (βγ)( m T −ε)(t−sj), Ω1 := {ω | t− ω ≤ J}, Ω2 := {ω | t− ω > J}, Ω3 := {sj | t− sj ≤ J}, Ω4 := {sj | t− sj > J}. Clearly, for any fixed point t, the function M is bounded. Theorem 4.3. Suppose (A1)–(A5) hold. For any p ∈ N+, the solution of (1.2) satisfies z((p+ 1)T ) = G(T, 0)z(pT ) + bm, where bm := ∫ T 0 G(T, ω)g̃(ω)dω + r(0,t)∑ j=1 G(t, sj)Bj(sj)κj . Proof. From (2.1), and Theorems 3.2 and 3.3, and Corollary 3.4 one has z((p+ 1)T ) = G((p+ 1)T, 0)ξ(y) + ∫ (p+1)T 0 G((p+ 1)T, ω)g̃(ω)dω + (p+1)m∑ j=1 G((p+ 1)T, sj)Bj((p+ 1)T )cj = G((p+ 1)T, pT ) [ G(pT, 0)z0 + ∫ pT 0 G(pT, ω)g̃(ω)dω EJDE-2020/18 NON-INSTANTANEOUS IMPULSIVE HEAT EQUATIONS 5 + pm∑ j=1 G(pT, sj)Bj(sj)cj ] + ∫ (p+1)T pT G((p+ 1)T, ω)g̃(ω)dω + (p+1)m∑ j=pm+1 G((p+ 1)T, sj)Bj((p+ 1)T )cj = G(T, 0)z(pT ) + ∫ T 0 G((p+ 1)T, ω + pT )g̃(ω)dω + m∑ j=1 G((p+ 1)T, sj+pm)Bj+pm((p+ 1)T )cj+pm = G(T, 0)z(pT ) + ∫ T 0 G(T, ω)g̃(ω)dω + m∑ j=1 G(T, sj)Bj(T )cj = G(T, 0)z(pT ) + bm. The proof is complete. � Corollary 4.4. For p ∈ N+, we have z(pT ) = [G(T, 0)]pz0 + p−1∑ i=0 [G(T, 0)]ibm. The above corollary follows directly from Theorem 4.3. Theorem 4.5. Suppose (A1)–(A6) hold. Then (1.2) has a unique T -periodic in- homogeneous trajectory belonging to BPC(R+, L 1(Ω)). Proof. Using Theorems 3.1 and 4.1, we obtain ‖z‖BPC = sup t∈R+ ‖G(t, 0)z0 + ∫ t 0 G(t, ω)g̃(ω)dω + r(0,t)∑ j=1 G(t, sj)Bj(sj)κj‖ ≤ sup t∈R+ ‖G(t, 0)‖‖z0‖+ sup t∈R+ ∫ t 0 ‖G(t, ω)‖dω‖g‖BPC + sup t∈R+ r(0,t)∑ j=1 ‖G(t, sj)‖‖Bj(sj)‖‖κj‖ ≤ sup t∈R+ (βγ)r(0,t)‖z0‖+ sup t∈R+ ∫ t 0 (βγ)r(ω,t)dω‖g‖BPC + sup t∈R+ βc r(0,t)∑ j=1 (βγ)r(sj ,t) ≤ sup t∈R+ (βγ)r(0,t)‖z0‖+ ∫ Ω1 (βγ)r(ω,t)dω‖g‖BPC + ∫ Ω2 (βγ)r(ω,t)dω‖g‖BPC + βc ∑ sj∈Ω3 (βγ)r(sj ,t) + βc ∑ sj∈Ω4 (βγ)r(sj ,t) ≤ ‖z0‖+ J‖g‖BPC + ∫ Ω2 (βγ)( m T −ε)(t−ω)dω‖g‖BPC + r(0, J)βc + βc ∑ sj∈Ω4 (βγ)( m T −ε)(t−sj) 6 P. YANG, J. WANG, D. O’REGAN EJDE-2020/?? ≤ ‖z0‖+ J‖g‖BPC + (βγ)( m T −ε)J lnβγ ‖g‖BPC − (βγ)( m T −ε)t lnβγ ‖g‖BPC + r(0, J)βc+ βc ∑ sj∈Ω4 (βγ)( m T −ε)(t−sj) ≤ ‖z0‖+ J‖g‖BPC − 1 lnβγ ‖g‖BPC + r(0, J)βc+M = ‖z0‖+ (J − 1 lnβγ )‖g‖BPC + r(0, J)βc+M. We now prove that {z(aT )}a∈N is a Cauchy sequence in L1(Ω). Indeed, for any fixed natural numbers a > b, using Corollary 4.4, we obtain ‖z(aT )− z(bT )‖ = ‖([G(T, 0)]a − [G(T, 0)]b)z0 + a−1∑ i=b [G(T, 0)]ibm‖ ≤ [(βγ)ar(0,T ) + (βγ)br(0,T )]‖z0‖+ a−1∑ i=b (βγ)ir(0,T )‖bm‖ ≤ [(βγ)am + (βγ)bm]‖z0‖+ a−1∑ i=b (βγ)im(‖g‖BPC +mβc) = [(βγ)am + (βγ)bm]‖z0‖+ (‖g‖BPC +mβc) (βγ)bm(1− (βγ)a−b) 1− βγ . When a and b are large enough, we have ‖z(aT ) − z(bT )‖ → 0. Therefore, {z(aT )}a∈N is a Cauchy sequence in L1(Ω), so the sequence {z(aT )}a∈N is conver- gent in L1(Ω), and we put z∗ := lim a→+∞ z(aT ) ∈ L1(Ω). Take now z∗ as the initial value, and we will prove that the inhomogeneous trajectory ẑ(t) = G(t, 0)z∗ + ∫ t 0 G(t, ω)g̃(ω)dω + r(0,t)∑ j=1 G(t, sj)Bj(sj)κj is T -periodic. Using Theorem 4.3, we obtain ‖ẑ(T )− z((a+ 1)T )‖ = ‖G(T, 0)(z∗ − z(aT ))‖ ≤ (βγ)r(0,T )‖z∗ − z(aT )‖ = (βγ)m‖z∗ − z(aT )‖. Let a→ +∞ and using the fact that lima→+∞ z(aT ) = z∗ = ẑ(0), we obtain ẑ(T ) = ẑ(0). Therefore, ẑ(t) is T -periodic. Next, we prove the uniqueness of the inhomogeneous T -periodic trajectory. Let ẑ1 and ẑ2 be two T -periodic trajectories of (1.1) with initial values ẑ10 and ẑ20, and we obtain ‖ẑ1 − ẑ2‖ = ‖G(t, 0)(ẑ10 − ẑ20)‖ ≤ (βγ)r(0,t)‖ẑ10 − ẑ20‖. EJDE-2020/18 NON-INSTANTANEOUS IMPULSIVE HEAT EQUATIONS 7 Then, using Theorem 4.1 and (A6), we have lim t→+∞ ‖ẑ1 − ẑ2‖ ≤ lim t→+∞ (βγ)( m T −ε)t‖ẑ10 − ẑ20‖ = 0. From the periodicity of ẑ1 and ẑ2, we obtain ẑ1 − ẑ2 = 0. That is ẑ1(t) = ẑ2(t) for t ∈ R+. � Acknowledgments. This work was supported by the National Natural Science Foundation of China (11661016), by the Training Object of High Level and Innova- tive Talents of Guizhou Province ((2016)4006), and by the Major Research Project of Innovative Group in Guizhou Education Department ([2018]012). References [1] R. Agarwal, D. O’Regan, S. Hristova; Monotone iterative technique for the initial value problem for differential equations with non-instantaneous impulses, Appl. Math. 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Comput., 219 (2013), 6743-6749. [8] J. Wang, M. Fečkan; Non-instantaneous impulsive differential equations, IOP Publishing, 2018. [9] P. Yang, J. Wang, M. Fečkan; Periodic nonautonomous differential equations with noninstan- taneous impulsive effects, Math. Meth. Appl. Sci., 42 (2019), 3700-3720. [10] D. Yang, J. Wang, D. O’Regan; On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses, C. R. Acad. Sci. Paris, Ser. I., 356 (2018), 150-171. Peng Yang Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China Email address: pyangmath@126.com Jinrong Wang (corresponding author) Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China. School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China Email address: jrwang@gzu.edu.cn Donal O’Regan School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland Email address: donal.oregan@nuigalway.ie 1. Introduction 2. Preliminaries 3. Basic properties for group G 4. Inhomogeneous periodic trajectory Acknowledgments References