PERIODICITY OF NON-HOMOGENEOUS TRAJECTORIES FOR NON-INSTANTANEOUS IMPULSIVE HEAT EQUATIONS

. In this article, we introduce a non-instantaneous impulsive operator associated with the heat semigroup and give some basic properties. We derive 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.


Introduction
Non-instantaneous differential equations are used to characterize evolution processes 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 u t (t, y) = ∆u(t, y) + f (t, y), y ∈ Ω, t ∈ [s i−1 , t i ], δu(t i , y) = I i u(t i , y) + c i (y), y ∈ Ω, u(t, y) = B i (t)u(t + i , y), y ∈ Ω, t ∈ (t i , s i ], u(0, y) = ξ(y), y ∈ Ω, (1.1) where i ∈ N + , δu(t i , y) := u(t + i , y) − u(t i , y), ∆ := n i=1 ∂ 2 ∂y 2 i denotes the Laplace operator and Ω ⊆ R n is an open set.The sequences {s i } i∈N + and {t i } i∈N + satisfy s 0 = 0 and and f ∈ C(I, X); here L(X) is the set of bounded linear operators on X.In addition, we suppose B i (t + i ) = E, where E is the identity map.Let z(t)(y) := u(t, y), g(t)(y) := f (t, y), κ i (y) := c i (y), and then we may transform the non-instantaneous impulsive heat equation (1.1) into the abstract non-instantaneous impulsive evolution equation 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).

Preliminaries
The bounded piecewise continuous function space with values in a Banach space X is defined as Recall that the fundamental solution of the heat equation is Note that Φ is singular at the point (0, 0).For each t > 0, A semigroup of bounded linear operators (H(t)) t≥0 on X defined by Proof.For t = 0, the conclusion is obvious.For each t > 0, we have It is well known that the solution of Clearly, any solution of A function z(t) is called a mild solution of (1.2), if it satisfies the integral equation where 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 y) for t ∈ I and every y ∈ Ω.

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.
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.

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 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, To guarantee the boundedness of the solution, we introduce the following assumption: (A6) βγ < 1.Then we set , Clearly, for any fixed point t, the function M is bounded.
The proof is complete.
Corollary 4.4.For p ∈ N + , we have The above corollary follows directly from Theorem 4.3.Proof.Using Theorems 3.1 and 4.1, we obtain We now prove that {z(aT )} a∈N is a Cauchy sequence in L 1 (Ω).Indeed, for any fixed natural numbers a > b, using Corollary 4.4, we obtain When a and b are large enough, we have z(aT ) − z(bT ) → 0. Therefore, {z(aT )} a∈N is a Cauchy sequence in L 1 (Ω), so the sequence {z(aT )} a∈N is convergent in L 1 (Ω), and we put 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)g(ω)dω + r(0,t) j=1 G(t, s j )B j (s j )κ j Let a → +∞ and using the fact that lim a→+∞ z(aT ) = z * = ẑ(0), we obtain ẑ(T ) = ẑ(0).