Periodic problem for non-instantaneous impulsive partial di

We obtain a new maximum principle of the periodic solutions when the corresponding impulsive equation is linear. If the nonlinear is quasi-monotonicity, we study the existence of the minimal and maximal periodic mild solutions for impulsive partial differential equations by using the perturbation method, the monotone iterative technique and the method of upper and lower solution. We give an example in last part to illustrate the main theorem.

The monotone iterative method are important mechanism. Abbas and Benchohra [2] investigated the existence of solutions for IVP of impulsive partial hyperbolic differential equations by employing the method of lower and upper solutions and the Schauder fixed point theorem. Li and Liu [26], Guo and Liu [17] studied impulsive integro-differential equations applying the monotone iterative method. In the papers [6,7], authors considered the nonlocal evolution equations with impulses by exploiting the monotone iterative method. For more monotone iterative method, we refer to the monographs [20][21][22][23][24][25] and references there in.
Recently, E. Hernandez and D. O'Regan [18] firstly studied new non-instantaneous impulsive evolution equations which have been used to describe gradual and continuous process such as the hemodynamic equilibrium of a person, the introduction of the drugs in the bloodstream , the consequent absorption for the body and so on. Authors investigated non-instantaneous impulsive fractional differential equations in [14,29,32]. In [8], Colao and Muglia considered bounded solutions of non-instantaneous impulsive differential equations with delay. Researchers in [30,33] studied PBVP of nonlinear non-instantaneous impulsive volution equations.
However, the literature concerning the existence of periodic mild solutions to this problem is untreated by using the perturbation method and the monotone iterative technique. Inspired by the above literatures, this paper is to construct a new maximum principle for the ω-periodic solutions of the corresponding linear equation with non instantaneous impulses. By using perturbation method and monotone iterative technique, we consider the existence of the minimal and maximal periodic solutions for Eq (1.1).
The organization of this paper as follows: some definitions and preliminary facts are recalled in next section, which will be used through this paper. First, we investigate the existence of periodic mild solution for linear non-instantaneous impulsive equation, which is significant for us to prove the key conclusion. Furthermore, for linear impulsive evolution equation corresponding to Eq (1.1), we established a new maximum principle. In Section 3, our major results on the periodic mild solutions of Eq (1.1) are proposed and proved. In Section 4, we presented an example to demonstrate our abstract results.

Preliminaries
We can see more relevant the properties of the C 0 -semigroup from the monographs [5,28].
Let PC(J, E) = u : J → E | u(t) is continuous in J , and left continuous at t i , and u(t + i ) exists, i = 1, 2, · · · , m . K PC = {u ∈ PC(J, E)|u(t) ≥ θ, t ∈ J} is the positive cone and " ≤ " is the partial order induced by K PC , then PC(J, E) is an order Banach space with the norm · PC = sup t∈J u(t) and the partial order " ≤ ". K PC is normal with the same normal constant N.
We use E 1 to denote the Banach space D(A) with the graph norm · 1 = · + A · . We can find more relevant the properties of the partial and cone from the monographs [9,15].
For the linear problem in E we have got the following conclusion. Lemma 2.1. Let T (t) (t ≥ 0) generated by −A be C 0 -semigroup in Banach space E. For any g ∈ PC(J, E), y i ∈ PC(J, E), i = 1, 2, · · · , m, problem (2.1) has a unique mild solution u ∈ PC(J, E) given by 3) has a unique classical solution u ∈ C 1 (I, E) ∩ C(I, E 1 ) expressed by Let t ∈ (t i , s i ], then u(t) = y i (t), i = 1, 2, · · · , m.
Let t ∈ (s i , t i+1 ], problem (2.1) is changed into IVP of linear evolution equation (2.4) Furthermore, after calculated, the function u ∈ PC(J, E) defined by (2.2) is a mild solution of problem (2.1). Hence problem (2.1) has a unique mild solution u ∈ PC(J, E) given by (2.2). This completes the proof.
if it is a PC-mild solution of Cauchy problem (2.1) corresponding to some x 0 and u(t + ω) = u(t) for t ≥ 0.
The solution of periodic boundary value problem can expressed by Next, we show that the solution of the Eq (2.5) is a ω-periodic.
Suppose that the following conditions are satisfied: Then Eq (2.5) has ω-periodic mild solution. Proof. In J, the periodic problem (2.5) is equal to PBVP (2.7). We only prove the solution u(t) expressed by (2.8) of PBVP(2.7) is periodic.
By the condition (G1) and (2.6), we have By the assumption (Y1), we have By the conditions (G1) and (Y1), we have Therefore, we can asset the solution u(t) of PBVP(2.7) is periodic. u(t) extended by ω-periodic is the periodic mild solution of Eq (2.5).
The proof is completed. Remark 2.1. In Lemma 2.2, let T (t) (t ≥ 0) generated by −A be a positive C 0 -semigroup in an ordered Banach space E. For any g ≥ θ, and y i ≥ θ, i = 1, 2, · · · , m, then the mild solution of Eq (2.5) is a positive solution.
Remark 2.1 implies the following maximum principle:

Main results
In this section, we present and prove our major results. We state the definition of the lower and upper ω-periodic solutions of PBVP(1.2).
then v 0 is known as a lower ω-periodic solution of problem (1.2); on the contrary , if all the inequalities of (3.1) are inverse, it is called an upper solution of problem (1.2).

is a continuous and increase operator.
Step 3. The operator Q exist fixed points on interval [v 0 , w 0 ].
Denote two sequences v n and w n by v n = Q(v n−1 ), w n = Q(w n−1 ), n = 1, 2, · · · (3.8) Since the operator Q is monotonous, we have Next, we prove that v n and w n are convergent in J.
By the assumption (F2), it follows that Since image sets of f (t, v 0 (t)) and f (t, w 0 (t)) are compact sets in E by the continuity of f (t, v 0 (t)) and f (t, w 0 (t)) in compact set [0, ω], furthermore image sets are bounded. Additionally, since the cone K is normal in E, we have ∃C 1 > 0, ∀v n−1 ∈ G 0 , From the condition (H2), we get that Since the cone K is normal in E, for i = 1, 2, · · · , m, we have ∃C 2 > 0, ∀v n−1 ∈ G 0 , We divide our proof into three cases.
Step 4. we prove that u(t) and u(t) are the minimal and maximal fixed points of Q in [v 0 , w 0 ], respectively. In fact, for any u * ∈ [v 0 , w 0 ], Q(u * ) = u * , we have v 0 ≤ u * ≤ w 0 and v 1 = Q(v 0 ) ≤ Q(u * ) = u * ≤ Q(w 0 ) = w 1 . Continuing such progress, we get v n ≤ u * ≤ w n . Letting n → ∞, we get u(t) ≤ u * ≤ u(t). Therefor, u(t) and u(t) between v 0 and w 0 are the minimal and maximal ω-periodic mild solutions of PBVP (1.2), which can be obtained by iteration from v 0 and w 0 , respectively.
When the positive cone is regular, we obtain the following conclusion of existence of PBVP (1.2). Similarly, in [v 0 , w 0 ], we define the two sequences v n (t) and w n (t) by (3.8). Since conditions (F2), (H2) and (H3) are satisfied, so sequences v n (t) and w n (t) are ordered-monotonic and orderedbounded in E. Any ordered-monotonic and ordered-bounded sequence in E is convergent while the cone K is regular. Using the similar method of Theorem 3.1, we can prove that u(t) and u(t) are the minimal and maximal ω-periodic mild solutions of the problem (1.2) between v 0 and w 0 , which can be obtained by iteration from v 0 and w 0 , respectively. Corollary 3.5 Let T (t) (t ≥ 0) generated by −A be a positive C 0 -semigroup in an ordered and weakly sequentially complete Banach space E, which positive cone K is normal with the normal constant N 0 . Assume that v 0 and w 0 with v 0 (t) ≤ w 0 (t)(t ∈ J) are lower and upper solutions of problem (1.2) and the conditions (F1), (F2), (H1), (H2) and (H3) are satisfied. Then PBVP (1.2) exist minimal and maximal ω-periodic mild solutions u and u between v 0 and w 0 , which can be obtained by iteration from v 0 and w 0 .
Proof. We know that the normal cone K is regular in an ordered and weakly sequentially complete Banach space.

Application
We make an example in this section to illustrate the main theorem.
Theorem 4.2 Let the first eigenvalue of operator − 2 u be λ 1 under zero boundary conditions and ϕ 1 (x) be the corresponding positive eigenvector. Then the impulsive parabolic partial differential equation (4.1) has minimal and maximal mild solutions.
Proof. It is easy to prove that v 0 ≡ 0 and w 0 ≡ ϕ 1 are lower and upper solutions of the Eq (4.1) respectively. We can easily verify that conditions (F1), (F2) are satisfied with 1 3 < M < 1 and the conditions (H1), (H2) and (H3) are satisfied too. Therefore, by Theorem 3.1, we have that PBVP (4.1) has minimal and maximal mild solutions. Then the proof is complete.

Conclusions
In this paper, when the nonlinear of the non instantaneous impulsive evolution equation is quasimonotonicity, we have considered the existence of the minimal and maximal ω-periodic mild solutions by combining perturbation method and monotone iterative technique. The main result (Theorem 3.1) is new.