Uniqueness of Solution for Impulsive Evolution Equation in Ordered Banach Spaces ^†

Abstract

This paper investigates the periodic boundary value problem for impulsive evolution equation in ordered Banach space. By applying the Poincaré mapping and monotone iterative method, we obtain the existence results of mild solutions and positive mild solutions for impulsive evolution equation. Further, we obtain the uniqueness of mild solution.

1. Introduction

The theory of impulsive evolution equation is a new and important branch of differential equation theory, which has an extensive range of application background in physics, population dynamics, ecology, chemicals, biological systems and engineering. Therefore, the research of impulsive evolution equation has attracted great enthusiasm of scholars in various fields and has been rapidly developed. One may consult [1,2,3,4,5,6,7,8,9,10] for background on numerous important theories.

During the early decades, in [11], Du et al. investigated the existence of extremal solutions to initial value problem of ordinary differential equations without impulses by using the monotone iterative technique in the presence of upper and lower solutions. And Guo and Liu constructed a new monotone iterative method for impulsive ordinary integro-differential equation, see [12] for a survey of this method. Especially, Li and Liu [13] expanded results in the aforementioned reference, they studied the existence of extremal solutions for impulsive integro-differential equation deleting the measure condition of non-compactness for impulsive functions.

Later, under the theory of semigroup of linear operators, Chen and Mu [14] discussed the existence and uniqueness of mild solutions to the initial value problem of impulsive integro-differential evolution in an abstract space. And then, under conditions that nonlinearity and impulsive functions are mixed monotone, Chen and Li [15] established a mixed monotone iterative technique in the presence of a new concept of upper and lower solutions for impulsive evolution equation

$$\left\{\begin{array}{c}{u}^{\prime }\left(t\right)+Au\left(t\right)=f(t,u\left(t\right),u\left(t\right)),t\in J,t\ne t_k,\hfill \\ {\mathsf{\Delta }u|}_{t=t_k}=I_k(u\left(t_k\right),u\left(t_k\right)),k=1,2,\dots ,m,\hfill \\ u\left(0\right)=x_0,\hfill \end{array}\right.$$

where $f\in C(J\times E\times E,E),J=[0,a],a>0$ is a constant. Li and Gou [16] demonstrated the existence of mild solutions to the first order semi-linear impulsive integro-differential evolution equation of Volterra type by using the monotone iterative technique in ordered Banach space.

However, above results were all directly assume that the equation has upper and lower solutions, but didn’t give or find specific upper and lower solutions. Currently, Li et al. [17] discussed the following abstract fractional evolution equation

$$\left\{\begin{array}{c}{}^{\mathrm{c}}D_{\mathrm{t}}^{\mathrm{q}}u\left(t\right)+Au\left(t\right)=f(t,u\left(t\right)),t\ge 0,\hfill \\ u\left(0\right)=u_0.\hfill \end{array}\right.$$

By using the monotone iterative technique and starting from the characteristics of operator semigroups, they obtained the existence result of positive S-asymptotically $\omega$-periodic mild solution.

Recently, Zhang et al. [18] investigated a new class of piecewise conformable fractional impulsive differential system with delay by Banach and Schauder fixed point theorem. Also [19] investigated the existence of solutions for first order non-linear adjoint impulsive delay dynamic system on time scale applying Banach fixed point theorem. Furthermore, ref. [20] proved some fixed point theorems for ordered contractions in partially ordered b-metric spaces and verified the existence and uniqueness of solutions to a large number of equations. Ref. [21] introduced a family of generalized equi-KKM mappings and proved a common fixed-point theorem for this family of generalized equi-KKM mappings using the Brouwer fixed-point theorem.

This paper discusses the following periodic boundary value problem of impulsive evolution equation (PBVP)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{\prime }\left(t\right)+Au\left(t\right)=f(t,u\left(t\right)),t\in J,t\ne t_k,\hfill \\ {\mathsf{\Delta }u|}_{t=t_k}=I_k\left(u\left(t_k\right)\right),k=1,2,\dots ,m,\hfill \\ u\left(0\right)=u\left(\omega \right)\hfill \end{array}\right.\end{array}$$

(1)

in an ordered Banach space E, where $A:D\left(A\right)\subset E\to E$ is a linear operator and $-A$ generates a $C_0$-semigroup $T\left(t\right)(t\ge 0)$ on E, $J=[0,\omega ],\omega >0$ is a constant, $0=t_0<t_1<t_2<\dots <t_m=t_{m+1}<\omega$, $J^{\prime }=J\setminus \{t_1,t_2,\dots ,t_m\}$. And ${\mathsf{\Delta }u|}_{t=t_k}$ denotes the jump of $u\left(t\right)$ at $t=t_k$, i.e., ${\mathsf{\Delta }u|}_{t=t_k}=u\left(t_k^{+}\right)-u\left(t_k^{-}\right)$, where $u\left(t_k^{+}\right)$ and $u\left(t_k^{-}\right)$ represent the right and left limits of $u\left(t\right)$ at $t=t_k$, respectively.

Without the assumption of upper and lower solutions, we firstly prove that the corresponding linear periodic boundary value problem has a unique positive solution $w_0$ by using the Poincaré operator [22] and contraction mapping principle. By using the monotone iterative method in $[-w_0,w_0]$, the existence of mild solutions to PBVP(1) on E is obtained. Then, we establish an accurate estimate of spectral radius for the resolvent operator, and obtain the uniqueness of mild solution. Furthermore, choosing a function related to the eigenfunction of smallest eigenvalue of operator A as a lower solution, we establish the existence result of positive mild solutions to PBVP (1).

Next Section, we deal with the linear impulsive periodic boundary value problem. Section 3 discusses the existence and uniqueness theorems for PBVP(1). In the last section, an example is given to demonstrate how to utilize our main results.

2. Preliminaries

Let $(E,\parallel ·\parallel )$ be an ordered Banach space, positive cone $K=\{x\in E:x\ge \theta \}$ ($\theta$ is the zero element of E) be normal with normal constant N. Let $C(J,E)$ denote the Banach space of all continuous E-value functions on interval J with the maximum norm ${\parallel u\parallel }_C={max}_{t\in J}\parallel u\left(t\right)\parallel$. Let $PC(J,E)=\{u:J\to E,u(t)$ is continuous at $t\ne t_k$, left continuous at $t=t_k$, and $u\left(t_k^{+}\right)$ exists, $k=1,2,\dots ,m\}$ be a Banach space with the norm ${\parallel u\parallel }_{PC}={sup}_{t\in J}\parallel u\left(t\right)\parallel$. Obviously, $C(J,E)$ and $PC(J,E)$ are ordered Banach spaces reduced by the convex cone $K_C=\{u\in E|u\left(t\right)\ge 0,t\in J\}$ and the positive cone $K_{PC}=\{u\in PC(J,E):u\left(t\right)\ge 0,t\in J\}$, respectively.

Let $A:D\left(A\right)\subset E\to E$ be a linear operator and $-A$ generate a $C_0$-semigroup $T\left(t\right)(t\ge 0)$ in E. By the exponential boundedness of $C_0$-semigroup $T\left(t\right)(t\ge 0)$, there exist constants $C\ge 1$ and $\nu \in \mathbb{R}$, such that

$$\begin{array}{c}\hfill \parallel T\left(t\right)\parallel \le C{e}^{\nu t},t\ge 0.\end{array}$$

(2)

The constant

$${\nu }_0=inf\{\nu \in \mathbb{R}|\mathrm{There}\mathrm{exists}C\ge 1,\mathrm{such}\mathrm{that}\left|\right|T\left(t\right)\left|\right|\le C{e}^{\nu t},t\ge 0\}$$

is called the growth exponent of $C_0$-semigroup $T\left(t\right)(t\ge 0)$. If ${\nu }_0<0$, then $T\left(t\right)(t\ge 0)$ is called an exponentially stable $C_0$-semigroup.

Definition 1

([23]). A $C_0$-semigroup $T\left(t\right)(t\ge 0)$ on E is said to be positive, if $T\left(t\right)x\ge \theta$ for each $x\ge \theta ,x\in E$ and $t\ge 0$.

Definition 2

([23]). A $C_0$-semigroup $T\left(t\right)(t\ge 0)$ on E is called compact, if $T\left(t\right)$ is a compact operator for every $t>0$.

If $T\left(t\right)(t\ge 0)$ is continuous in the uniform operator topology for $t>0$, then ${\nu }_0$ can also be expressed by spectral set $\sigma \left(A\right)$, i.e.,

$${\nu }_0=-inf\{\mathrm{Re}\lambda :\lambda \in \sigma \left(\mathrm{A}\right)\}.$$

Moreover, if $-A$ generates a compact $C_0$-semigroup $T\left(t\right)(t\ge 0)$, then $T\left(t\right)$ is continuous in the uniform operator topology for $t>0$. By the famous Krein-Rutmann theorem, A has smallest eigenvalue ${\lambda }_1>0$ with the positive eigenfunction $e_1$, and

$${\lambda }_1=inf\{\mathrm{Re}\lambda :\lambda \in \sigma \left(\mathrm{A}\right)\},$$

which implies that ${\nu }_0=-{\lambda }_1$. For any $\nu \in (0,|{\nu }_0\left|\right)$, we define the equivalent norm $|·|$ by

$$\left|x\right|=\underset{t\ge 0}{sup}\left|\right|e^{\nu t}T\left(t\right)x\left|\right|,$$

then $\left|\right|x\left|\right|\le \left|x\right|\le C\left|\right|x\left|\right|$. Denote the norm of $T\left(t\right)$ in E by $\left|T\right(t\left)\right|$, we have $\left|T\left(t\right)\right|\le e^{-\nu t}$.

To prove our main results, firstly, we consider the linear periodic boundary value problem

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{\prime }\left(t\right)+Au\left(t\right)=h\left(t\right),t\in J,t\ne t_k,\hfill \\ {\mathsf{\Delta }u|}_{t=t_k}=a_{k}u\left(t_k\right)+b_k,k=1,2,\dots ,m,\hfill \\ u\left(0\right)=u\left(\omega \right)\hfill \end{array}\right.\end{array}$$

(3)

A function $u\in PC(J,E)\cap C^1(J^{\prime },E)$ is called a solution of PBVP(1.1), if $u\left(t\right)$ satisfies all the equations of (1). Let

$$P{C}^1(J,E)=\{u\in PC(J,E)\cap C^1(J^{\prime },E)|u^{\prime }\left(t_k^{+}\right)\mathrm{and}{u}^{\prime }\left(t_k^{-}\right)\mathrm{exist},k=1,2,\dots ,m\}.$$

Obviously, for any $u\in P{C}^1(J,E)$, left derivative $u_{-}^{\prime }\left(t_k\right)$ of $u\left(t_k\right)$ at $t_k$ exists and $u_{-}^{\prime }\left(t_k\right)=u^{\prime }\left(t_k^{-}\right)$. Let $u^{\prime }\left(t_k\right)=u^{\prime }\left(t_k^{-}\right)$, thus $u^{\prime }\in PC(J,E)$. If $u\in PC(J,E)\cap C^1(J^{\prime },E)$ is a solution of PBVP(1.1), from the continuity of h, we can obtain that $u\in P{C}^1(J,E)$. Moreover, if $h\in C^1(J,E)$, then $u\in P{C}^1(J,E)$ satisfies all the equations of (3) is a solution of the periodic problem (3).

Lemma 1.

Let $A:D\left(A\right)\subset E\to E$ be a linear operator and $-A$ generate a positive $C_0$-semigroup $T\left(t\right)(t\ge 0)$. For any $h\in P{C}^1(J,E)$ with $h\ge 0$, if there exist constants $a_k\ge 0$ and $\theta \le b_k\in E,k=1,2,\dots ,m$, such that $\prod _{k=1}^m(1+a_k)\le e^{|{\nu }_0|\omega }$, then the linear periodic boundary value problem (3) has a unique positive solution.

Proof. 

Let $I_k\left(u\left(t_k\right)\right)=a_{k}u\left(t_k\right)+b_k$, for $t\in J_k^{\prime }=(t_k,t_{k+1}],k=0,1,\dots ,m$, we consider the initial value problem

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{\prime }\left(t\right)+Au\left(t\right)=h\left(t\right),\hfill \\ u\left(t_k^{+}\right)=u\left(t_k\right)+I_k\left(u\left(t_k\right)\right).\hfill \end{array}\right.\end{array}$$

(4)

It is well-known [23] (Chapter 4, Theorem 2.9) (theory of operator semigroup and resolvent operator) that for any $h\in P{C}^1(J,E)$, the unique solution of initial value problem (4) is given by

$$u\left(t\right)=T(t-t_k)\left(u\left(t_k\right)+I_k\left(u\left(t_k\right)\right)\right)+{\int }_{t_k}^{t}T(t-s)h\left(s\right)ds,t\in J,$$

then for $t\in J_k^{\prime },k=0,1,\dots ,m$, we have

$$u\left(t_{k+1}\right)=T(t_{k+1}-t_k)\left(u\left(t_k\right)+I_k\left(u\left(t_k\right)\right)\right)+{\int }_{t_k}^{t_{k+1}}T(t-s)h\left(s\right)ds.$$

(5)

Thus, for initial value $x_1,x_2\in E$ with $x_1\ne x_2$, we can define a Poincaré mapping

$$\mathsf{\Pi }:x\mapsto u(\omega ,x).$$

From (5), for any $\nu \in (0,|{\nu }_0\left|\right)$ and $t\in J_0^{\prime }=[0,t_1]$, it follows that

$$|u_2\left(t_1\right)-u_1\left(t_1\right)|\le |T\left(t_1\right)x_2-T\left(t_1\right)x_1|\le e^{-\nu t_1}|x_2-x_1|.$$

For $t\in J_1^{\prime }=(t_1,t_2]$,

$$\begin{array}{cc}\hfill |u_2\left(t_2\right)-u_1\left(t_2\right)|& \le \left|T(t_2-t_1)\left[u_2\left(t_1\right)+I_1\left(u_2\left(t_1\right)\right)-\left(u_1\left(t_1\right)+I_1\left(u_1\left(t_1\right)\right)\right)\right]\right|\hfill \\ \hfill & \le e^{-\nu (t_2-t_1)}{e}^{-\nu t_1}(1+a_1)|x_2-x_1|\hfill \\ \hfill & \le e^{-\nu t_2}(1+a_1)|x_2-x_1|.\hfill \end{array}$$

Continuing the above process interval by interval, for $t\in J_m^{\prime }=(t_m,\omega ]$, we have

$$|u_2\left(\omega \right)-u_1\left(\omega \right)|\le e^{-\nu \omega }\prod _{k=1}^m(1+a_k)|x_2-x_1|.$$

Therefore, we can obtain that

$$|\mathsf{\Pi }\left(x_2\right)-\mathsf{\Pi }\left(x_1\right)|\le e^{-\nu \omega }\prod _{k=1}^m(1+a_k)|x_2-x_1|.$$

Obviously, $\mathsf{\Pi }$ is a contraction mapping. According to the positivity of operator h and semigroup $T\left(t\right)(t\ge 0)$, we can deduce that the periodic problem (3) has a unique positive solution $u\in P{C}^1(J,E)$. □

Let $h\in PC(J,E)$, $a_1,a_2,\dots ,a_m\ge 0$ satisfying $\prod _{k=1}^m(1+a_k)\le e^{\nu \omega }$, we consider linear periodic boundary value problem

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{\prime }\left(t\right)+Au\left(t\right)=h\left(t\right),t\in J,t\ne t_k,\hfill \\ {\mathsf{\Delta }u|}_{t=t_k}=a_{k}u\left(t_k\right),k=1,2,\dots ,m,\hfill \\ u\left(0\right)=u\left(\omega \right),\hfill \end{array}\right.\end{array}$$

(6)

and have the following result:

Lemma 2.

Let $A:D\left(A\right)\subset E\to E$ be a liner operator and $-A$ generate an exponentially stable $C_0$-semigroup $T\left(t\right)(t\ge 0)$. Then the problem (6) has a unique mild solution $u:=\mathcal{P}h\in PC(J,E)$ and resolvent operator $\mathcal{P}:PC(J,E)\to PC(J,E)$ is a linear and bounded operator with $r\left(\mathcal{P}\right)\le {\displaystyle \frac{1}{|{\nu }_0|}}\left({\displaystyle \sum _{i=1}^m}(1+a_i)+1+{\displaystyle \sum _{i=1}^m}{\displaystyle \prod _{j=i}^m}(1+a_j)\right)$.

Proof. 

For any $t\in [0,t_1]$, consider the initial value problem

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{\prime }\left(t\right)+Au\left(t\right)=h\left(t\right),\hfill \\ u\left(0\right)=x_0,\hfill \end{array}\right.\end{array}$$

(7)

From the literature [23], problem (7) admits a unique mild solution $u\in C(J_0,E)$:

$$u\left(t\right)=T\left(t\right)x_0+{\int }_0^{t}T(t-s)h\left(s\right)ds.$$

For any $t\in (t_1,t_2]$, initial value problem

$$\left\{\begin{array}{c}{u}^{\prime }\left(t\right)+Au\left(t\right)=h\left(t\right),\hfill \\ u\left(t_1^{+}\right)=u\left(t_1\right)+a_{1}u\left(t_1\right)\hfill \end{array}\right.$$

admits a unique mild solution $u\in C(J_1,E)$:

$$u\left(t\right)=(1+a_1)T\left(t\right)x_0+(1+a_1){\int }_0^{t_1}T(t-s)h\left(s\right)ds+{\int }_{t_1}^{t}T(t-s)h\left(s\right)ds.$$

Especially, $u\left(t_2\right)=(1+a_1)T\left(t_2\right)x_0+(1+a_1){\int }_0^{t_1}T(t_2-s)h\left(s\right)ds+{\int }_{t_1}^{t_2}T(t_2-s)h\left(s\right)ds$.

For any $t\in (t_2,t_3]$, problem (6) admits a unique mild solution $u\in C(J_2,E)$:

$$\begin{array}{ccc}\hfill u\left(t\right)& =& T(t-t_2)u\left(t_2^{+}\right)+{\int }_{t_2}^{t}T(t-s)h\left(s\right)ds\hfill \\ & =& T(t-t_2)(1+a_2)u\left(t_2\right)+{\int }_{t_2}^{t}T(t-s)h\left(s\right)ds\hfill \\ & =& (1+a_1)(1+a_2)T\left(t\right)x_0+(1+a_1)(1+a_2){\int }_0^{t_1}T(t-s)h\left(s\right)ds\hfill \\ & & +(1+a_2){\int }_{t_1}^{t_2}T(t-s)h\left(s\right)ds+{\int }_{t_2}^{t}T(t-s)h\left(s\right)ds.\hfill \end{array}$$

Especially, hold

$$\begin{array}{ccc}\hfill u\left(t_3\right)& =& (1+a_1)(1+a_2)T\left(t_3\right)x_0+(1+a_1)(1+a_2){\int }_0^{t_1}T(t_3-s)h\left(s\right)ds\hfill \\ & & +(1+a_2){\int }_{t_1}^{t_2}T(t_3-s)h\left(s\right)ds+{\int }_{t_2}^{t_3}T(t_3-s)h\left(s\right)ds.\hfill \end{array}$$

Inductively, for any $t\in (t_{k-1},t_k]$, problem (6) admits a unique mild solution $u\in C(J_{k-1},E)$:

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \prod _{i=1}^{k-1}(1+a_i)T\left(t\right)x_0+\sum _{i=1}^{k-1}\prod _{j=i}^{k-1}(1+a_j){\int }_{t_{i-1}}^{t_i}T(t-s)h\left(s\right)ds+{\int }_{t_{k-1}}^{t}T(t-s)h\left(s\right)ds.\hfill \end{array}$$

For any $t\in (t_k,t_{k+1}]$, $k=0,1,\dots ,m$, initial value problem

$$\left\{\begin{array}{c}{u}^{\prime }\left(t\right)+Au\left(t\right)=h\left(t\right),\hfill \\ u\left(t_k^{+}\right)=u\left(t_k\right)+a_{k}u\left(t_k\right)\hfill \end{array}\right.$$

admits a unique mild solution $u\in C(J_k,E)$:

$$\begin{array}{ccc}\hfill u\left(t\right)& =& T(t-t_k)u\left(t_k^{+}\right)+{\int }_{t_k}^{t}T(t-s)h\left(s\right)ds\hfill \\ & =& T(t-t_k)(1+a_k)u\left(t_k\right)+{\int }_{t_k}^{t}T(t-s)h\left(s\right)ds\hfill \\ & =& (1+a_k)\prod _{i=1}^{k-1}(1+a_i)T\left(t\right)x_0+(1+a_k){\int }_{t_{k-1}}^{t_k}T(t-s)h\left(s\right)ds+{\int }_{t_k}^{t}T(t-s)h\left(s\right)ds\hfill \\ & & +(1+a_k)\sum _{i=1}^{k-1}\prod _{j=i}^{k-1}(1+a_j){\int }_{t_{i-1}}^{t_i}T(t-s)h\left(s\right)ds\hfill \\ & =& \prod _{i=1}^k(1+a_i)T\left(t\right)x_0+\sum _{i=1}^k\prod _{j=i}^k(1+a_j){\int }_{t_{i-1}}^{t_i}T(t-s)h\left(s\right)ds+{\int }_{t_k}^{t}T(t-s)h\left(s\right)ds.\hfill \end{array}$$

Thus, for any $t\in [0,\omega ]$, problem (6) admits a unique mild solution $u\in PC(J,E)$:

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \prod _{i=1}^m(1+a_i)T\left(t\right)x_0+\sum _{i=1}^m\prod _{j=i}^m(1+a_j){\int }_{t_{i-1}}^{t_i}T(t-s)h\left(s\right)ds+{\int }_{t_m}^{t}T(t-s)h\left(s\right)ds.\hfill \end{array}$$

For every $\nu \in (0,|{\nu }_0\left|\right)$, we define a new equivalent norm in E

$$\left|x\right|=\underset{t\ge 0}{sup}\parallel e^{\nu t}\prod _{i=1}^m{(1+a_i)}^{-1}T\left(t\right)x\parallel ,$$

then $C_1\parallel x\parallel \le \left|x\right|\le C_2\parallel x\parallel$, here $C_1,C_2$ are corresponding coefficients. Therefore, one has

$$\begin{array}{ccc}\hfill \left|\prod _{i=1}^m(1+a_i)T\left(t\right)x\right|& =& \underset{t\le 0}{sup}\parallel e^{\nu t}T\left(t\right)x\parallel \hfill \\ & \le & e^{-\nu t}\left|x\right|.\hfill \end{array}$$

Especially $\left|{\displaystyle \prod _{i=1}^m}(1+a_i)T\left(\omega \right)\right|\le e^{-\nu \omega }<1$. Then $I-{\displaystyle \prod _{i=1}^m}(1+a_i)T\left(\omega \right)$ has bounded inverse operator satisfying

$$\left|{\left(I-\prod _{i=1}^m(1+a_i)T\left(\omega \right)\right)}^{-1}\right|\le {\displaystyle \frac{1}{1-e^{-\nu \omega }}}.$$

From the periodic boundary value condition $u\left(0\right)=x_0=u\left(\omega \right)$, we can conclude that problem (6) admits a unique mild solution

$$\begin{array}{ccc}\hfill u\left(t\right)& =& {\left(I-\prod _{i=1}^m(1+a_i)T\left(\omega \right)\right)}^{-1}\{\prod _{i=1}^m(1+a_i){\int }_{t-\omega }^{0}T(t-s)h\left(s\right)ds\hfill \\ & & +\sum _{i=1}^m\prod _{j=i}^m(1+a_j){\int }_{t_{i-1}}^{t_i}T(t-s)h\left(s\right)ds+{\int }_{t_m}^{t}T(t-s)h\left(s\right)ds\}\hfill \\ & :=& \mathcal{P}h\left(t\right).\hfill \end{array}$$

Obviously $\mathcal{P}:PC(J,E)\to PC(J,E)$ is a bounded linear operator. In fact, for every $t\in [0,\omega ]$,

$$\begin{array}{ccc}& & \left|\right(\mathcal{P}h\left)\right(t\left)\right|\hfill \\ & \le & \left|{\left(I-\prod _{i=1}^m(1+a_i)T\left(\omega \right)\right)}^{-1}\right|·\prod _{i=1}^m(1+a_i){\int }_{t-\omega }^0\left|T(t-s)h\left(s\right)\right|ds\hfill \\ & & +\left|{\left(I-\prod _{i=1}^m(1+a_i)T\left(\omega \right)\right)}^{-1}\right|·\sum _{i=1}^m\prod _{j=i}^m(1+a_j){\int }_{t_{i-1}}^{t_i}\left|T(t-s)h\left(s\right)\right|ds\hfill \\ & & +\left|{\left(I-\prod _{i=1}^m(1+a_i)T\left(\omega \right)\right)}^{-1}\right|·{\int }_{t_m}^t\left|T(t-s)h\left(s\right)\right|ds\hfill \\ & \le & {\displaystyle \frac{1}{1-e^{-\nu \omega }}}\prod _{i=1}^m(1+a_i){\int }_{t-\omega }^0|T(t-s)h\left(s\right)|ds+{\displaystyle \frac{1}{1-e^{-\nu \omega }}}{\int }_{t_m}^t\left|T(t-s)h\left(s\right)\right|ds\hfill \\ & & +{\displaystyle \frac{1}{1-e^{-\nu \omega }}}\sum _{i=1}^m\prod _{j=i}^m(1+a_j){\int }_{t_{i-1}}^{t_i}\left|T(t-s)h\left(s\right)\right|ds\hfill \end{array}$$

$$\begin{array}{ccc}& \le & {\displaystyle \frac{1}{1-e^{-\nu \omega }}}\prod _{i=1}^m(1+a_i){\int }_{t-\omega }^0{e}^{-\nu (t-s)}{ds\left|h\right|}_C+{\displaystyle \frac{1}{1-e^{-\nu \omega }}}{\int }_{t_m}^t{e}^{-\nu (t-s)}ds{\left|h\right|}_C\hfill \\ & & +{\displaystyle \frac{1}{1-e^{-\nu \omega }}}\sum _{i=1}^m\prod _{j=i}^m(1+a_j){\int }_{t_{i-1}}^{t_i}{e}^{-\nu (t-s)}ds{\left|h\right|}_C\hfill \\ & \le & {\displaystyle \frac{1}{\nu }}\left(\prod _{i=1}^m(1+a_i)+1+\sum _{i=1}^m\prod _{j=i}^m(1+a_j)\right){\left|h\right|}_C.\hfill \end{array}$$

So, we can deduce that ${\left|\mathcal{P}h\right|}_C\le {\displaystyle \frac{1}{\nu }}\left({\displaystyle \prod _{i=1}^m}(1+a_i)+1+{\displaystyle \sum _{i=1}^m}{\displaystyle \prod _{j=i}^m}(1+a_j)\right){\left|h\right|}_C$, then

$$r\left(\mathcal{P}\right)\le \left|\mathcal{P}\right|\le {\displaystyle \frac{1}{\nu }}\left(\prod _{i=1}^m(1+a_i)+1+\sum _{i=1}^m\prod _{j=i}^m(1+a_j)\right).$$

From the arbitrariness of $\nu \in (0,|{\nu }_0\left|\right)$, we can obtain that $r\left(\mathcal{P}\right)\le {\displaystyle \frac{1}{|{\nu }_0|}}\left({\displaystyle \prod _{i=1}^m}(1+a_i)+1+{\displaystyle \sum _{i=1}^m}{\displaystyle \prod _{j=i}^m}(1+a_j)\right)$. □

3. Main Results

In this section we give and prove our main results. For $v,w\in PC(J,E)$ with $v\le w$, we use $[v,w]$ denote the order interval $\{u\in PC(J,E):v\le u\le w\}$ in $PC(J,E)$, and $\left[v\right(t),w(t\left)\right]$ denote the order interval $\{x\in E:v(t)\le x(t)\le w(t),t\in J\}$ in E. By enlarging and reducing the nonlinearity and the impulsive function, we can obtain a pair of upper and lower solutions of PBVP(1). Further, under that the nonlinearity and impulsive functions satisfy the ordered growth conditions, we can obtain the existence result of mild solutions for PBVP(1).

(H1)

There exist a constant $0<a<\nu$ and a function $h\in P{C}^1(J,E)$ with $h\ge 0$, such that for a.e. $t\in J$ and $x\in K_{PC}$,

$$\begin{array}{c}\hfill f(t,x)\le ax+h\left(t\right),f(t,-x)\ge -ax+h\left(t\right).\end{array}$$

(8)

(H2)

There exist constants $a_k\ge 0$ with ${\displaystyle \prod _{k=1}^m}(1+a_k)\le e^{\nu \omega }$, and $\theta \le b_k\in E,k=1,2,\dots ,m$, such that for every $x\in K_{PC}$,

$$\begin{array}{c}\hfill I_k\left(x\right)\le a_{k}x+b_k,I_k(-x)\ge -a_{k}x+b_k.\end{array}$$

(9)

(H3)

There exists a constant $M>0$, such that for every $x_i\in E(i=1,2)$ with $x_1\le x_2$,

$$f(t,x_2)-f(t,x_1)\ge -M(x_2-x_1),t\in J^{\prime }.$$

(H4)

For every $x_i\in E(i=1,2)$ with $x_1\le x_2$,

$$I_k\left(x_1\right)\le I_k\left(x_2\right),k=1,2,\dots ,m.$$

Theorem 1.

Let $A:D\left(A\right)\subset E\to E$ be a linear operator and $-A$ generate an exponentially stable, positive and compact $C_0$-semigroup $T\left(t\right)(t\ge 0)$ in E. If conditions (H1)–(H6) are satisfied, then PBVP (1) has at least one mild solutions on E.

Proof. 

For $h\left(t\right)\in P{C}^1(J,E)$ in (H1), we investigate the following linear periodic problem

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{\prime }\left(t\right)+(A-aI)u\left(t\right)=h\left(t\right),t\in J,t\ne t_k,\hfill \\ {\mathsf{\Delta }u|}_{t=t_k}=I_k\left(u\left(t_k\right)\right),k=1,2,\dots ,m,\hfill \\ u\left(0\right)=u\left(\omega \right),\hfill \end{array}\right.\end{array}$$

(10)

where $u\left(0\right)\in D\left(A\right)$. Since $a<\nu$, we can obtain that $-(A-aI)$ generates an exponentially stable, positive and compact $C_0$-semigroup $S\left(t\right)(t\ge 0)=e^{at}T\left(t\right)$ with $\parallel S\left(t\right)\parallel \le C{e}^{-(\nu -a)t}$. For every $x\in K_{PC}$, by Lemma 1, we can deduce that LPBVP (10) has a unique positive solution $w_0\left(t\right)\in P{C}^1(J,E)$. Choosing $w_0$ as an upper solution of PBVP (1), obviously, let $v_0\left(t\right)=-w_0\left(t\right)$, then $v_0$ is a lower solution of PBVP (1) and $v_0$ satisfies that

$$v_0^{\prime }\left(t\right)+A{v}_0\left(t\right)=h\left(t\right)+a{v}_0\left(t\right)\le f(t,v_0\left(t\right)),$$

$$\mathsf{\Delta }{v}_0{|}_{t=t_k}=a_k{v}_0\left(t_k\right)+b_k\le I_k\left(v_0\left(t_k\right)\right).$$

Now, let $M>\nu$ be a constant in (H3), we consider the periodic boundary value problem

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{\prime }\left(t\right)+Au\left(t\right)+Mu\left(t\right)=f(t,u\left(t\right))+Mu\left(t\right),t\in J^{\prime },\hfill \\ {\mathsf{\Delta }u|}_{t=t_k}=I_k\left(u\left(t_k\right)\right),k=1,2,\dots ,m,\hfill \\ u\left(0\right)=u\left(\omega \right).\hfill \end{array}\right.\end{array}$$

(11)

In light of characteristics of $T\left(t\right)(t\ge 0)$, it is easy to see that $-A-MI$ generates a positive compact $C_0$-semigroup $T_1\left(t\right)(t\ge 0)=e^{-Mt}T\left(t\right)$ and $\left|\right|T_1\left(t\right)\left|\right|\le e^{-(M+\nu )t}$. Let $h\left(t\right)=f(t,u(t\left)\right)+Mu\left(t\right)$, For every $t\in J$, from Lemma 2, the solution of PBVP (11) is given by

$$\begin{array}{c}\hfill u\left(t\right)={\mathcal{P}}_{M}h=T_1\left(t\right)B\left(h\right)+{\int }_0^t{T}_1(\omega -s)h\left(s\right)ds+\sum _{0<t_k<t}{T}_1(t-t_k)I_k\left(u\left(t_k\right)\right),\end{array}$$

where $B\left(h\right)={(I-T_1\left(\omega \right))}^{-1}\left[{\int }_0^{\omega }{T}_1(\omega -s)h\left(s\right)ds+{\displaystyle \sum _{0<t_k<\omega }}{T}_1(\omega -t_k)I_k\left(u\left(t_k\right)\right)\right]$.

In sequel, we use the monotone iterative method to find the existence of mild solutions of PBVP (1) in $[v_0,w_0]$. We define a map $F_M$ by

$$F_M\left(u\right)\left(t\right)=f(t,u\left(t\right))+Mu\left(t\right)$$

in $[v_0,w_0]$. From conditions (H3) and (H4), it follows that $F_M:[v_0,w_0]\to PC(J,E)$ is a continuously increasing operator. We define operator $Q:[v_0,w_0]\to PC(J,E)$ by

$$\begin{array}{c}\hfill \begin{array}{cc}\left(Qu\right)\left(t\right)=\hfill & {(I-T\left(\omega \right))}^{-1}{\int }_{t-\omega }^{t}T(t-s)F_M(s,u\left(s\right))ds\hfill \\ \hfill & +{(I-T\left(\omega \right))}^{-1}{\displaystyle \sum _{t-\omega <t_k<t}}T(t-t_k)I_k\left(u\left(t_k\right)\right),\hfill \end{array}\end{array}$$

then the composite map $Q={\mathcal{P}}_M\circ F_M:[v_0,w_0]\to PC(J,E)$ also is a continuously increasing operator and the mild solution of PBVP(1) is equivalent to the fixed point of Q with

$$v_0\le Q{v}_0,Q{w}_0\le w_0.$$

In fact, let $h_1\left(t\right)=F_M\left(v_0\right)\left(t\right)$, by Lemma 2, we know that

$$\begin{array}{ccc}\hfill v_0\left(t\right)& \le & T_1\left(t\right)v_0\left(0\right)+{\int }_0^t{T}_1(t-s)(f(s,v_0\left(s\right))+M{v}_0\left(s\right))ds+\sum _{0<t_k<t}{T}_1(t-t_k)\mathsf{\Delta }{v}_0{|}_{t=t_k}\hfill \\ & =& T_1\left(t\right)v_0\left(0\right)+{\int }_0^t{T}_1(t-s)h_1\left(s\right)ds+\sum _{0<t_k<t}{T}_1(t-t_k)I_k\left(v_0\left(t_k\right)\right)\hfill \\ & \le & T_1\left(t\right){(I-T\left(\omega \right))}^{-1}\left({\int }_0^{\omega }{T}_1(\omega -s)h_1\left(s\right)ds+\sum _{0<t_k<\omega }{T}_1(\omega -t_k)I_k\left(v_0\left(t_k\right)\right)\right)\hfill \\ & & +{\int }_0^t{T}_1(t-s)h_1\left(s\right)ds+\sum _{0<t_k<t}{T}_1(t-t_k)I_k\left(v_0\left(t_k\right)\right)\hfill \\ & \le & T_1\left(t\right)B\left(h_1\right)+{\int }_0^t{T}_1(t-s)h_1\left(s\right)ds+\sum _{0<t_k<t}{T}_1(t-t_k)I_k\left(v_0\left(t_k\right)\right)\hfill \\ & =& Q{v}_0\left(t\right).\hfill \end{array}$$

Hence for all $t\in J$, $v_0\left(t\right)\le \left(Q{v}_0\right)\left(t\right)$, which implies that $v_0\le Q{v}_0$. Similarly, it can be shown that $Q{w}_0\le w_0$. Therefore $Q:[v_0,w_0]\to [v_0,w_0]$ is a continuously increasing operator.

Next, we show $Q:[v_0,w_0]\to [v_0,w_0]$ is completely continuous. For $u\in [v_0,w_0]$, let

$$\begin{array}{c}\left(Wu\right)\left(t\right)={\int }_0^t{T}_1(t-s)F_M\left(u\right)\left(s\right)ds,\\ \left(Vu\right)\left(t\right)={\displaystyle \sum _{0<t_k<t}}{T}_1(t-t_k)I_k\left(u\left(t_k\right)\right),\\ B\left(u\right)={(I-T_1\left(\omega \right))}^{-1}\left(\left(Wu\right)\left(\omega \right)+\left(Vu\right)\left(\omega \right)\right).\end{array}$$

Subsequently, we prove that for a.e. $0<t\le \omega$, $\{\left(Wu\right)\left(t\right):u\in [v_0,w_0]\}$ is precompact in E. For $0<ϵ<t$ and $u\in [v_0,w_0]$, we have

$$\left(W_{ϵ}u\right)\left(t\right)=T_1\left(ϵ\right){\int }_0^{t-ϵ}{T}_1(t-s-ϵ)F_M\left(u\right)\left(s\right)ds.$$

(12)

There exists $\mathcal{M}>0$, such that $\parallel F_M\left(u\right)\left(s\right)\parallel \le \mathcal{M}$ since the cone P is normal. And the compactness of $T_1\left(ϵ\right)$ implies that $\{\left(W_{ϵ}u\right)\left(t\right):u\in [v_0,w_0]\}$ is precompact in E. Let $\overline{M}=\underset{t\in J}{sup}\parallel T_1\left(t\right)\parallel$, combining

$$\begin{array}{c}\hfill \parallel \left(Wu\right)\left(t\right)-\left(W_{ϵ}u\right)\left(t\right)\parallel {\int }_{t-ϵ}^t\parallel T_1(t-s)\parallel ·\parallel F_M\left(u\right)\left(s\right)ds\parallel \le ϵ\overline{M}\mathcal{M},\end{array}$$

with the completeness of E, we know that $\{\left(Wu\right)\left(t\right):u\in [v_0,w_0]\}$ is precompact in E.

On the other hands, for $0\le t^{\prime }\le t^{″}\le \omega$, one has

$$\begin{array}{cc}\hfill & \parallel \left(Wu\right)\left(t^{″}\right)-\left(Wu\right)\left(t^{\prime }\right)\parallel \hfill \\ \hfill =& \parallel {\int }_0^{t^{″}}{T}_1(t^{″}-s)F_M\left(u\right)ds-{\int }_0^{t^{\prime }}{T}_1(t^{\prime }-s)F_M\left(u\right)ds\parallel \hfill \\ \hfill \le & \parallel {\int }_0^{t^{\prime }}(T_1(t^{″}-s)-T_1(t^{\prime }-s)F_M\left(u\right)ds+{\int }_{t^{\prime }}^{t^{″}}{T}_1(t^{″}-s)F_M\left(u\right)ds\parallel \hfill \\ \hfill \le & \mathcal{M}{\int }_0^{t^{\prime }}\parallel T_1(t^{″}-t^{\prime }+s)-T_1\left(s\right)\parallel ds+\overline{M}\mathcal{M}(t^{″}-t^{\prime }).\hfill \end{array}$$

The compactness of $T(·)$ implies that $T_1(·)$ is compact, thus $T_1(·)$ is equicontinuous semigroup. So, we can deduce that $\parallel \left(Wu\right)\left(t^{″}\right)-\left(Wu\right)\left(t^{\prime }\right)\parallel \to 0$, as $t^{″}-t^{\prime }\to 0$. Hence $W\left([v_0,w_0]\right)$ is equicontinuous function of cluster in E.

Using the same method, we can obtain that $V\left([v_0,w_0]\right)$ is precompact and equicontinuous in E. From the representation of $B\left([v_0,w_0]\right)$, it follows that $B\left([v_0,w_0]\right)$ is precompact and equicontinuous in E. According to Arzelá-Ascoli theorem, $Q:[v_0,w_0]\to [v_0,w_0]$ is a completely continuous operator. Then, by the theory of monotone increasing operators, Q has a minimal fixed point $\underline{u}$ and a maximal fixed point $\overline{u}$ which are the minimal and the maximal mild solutions of the problem (1) in $[v_0,w_0]$, respectively. Therefore, PBVP(1) has at least one mild solutions on E. □

Next, applying the estimation of spectral radius of resolvent operators, we discuss the uniqueness of mild solutions to PBVP (1) in $[v_0,w_0]$ under that the nonlinearity and impulsive function satisfy Lipschitz conditions:

(H5)

There exists a constant $L<-{\nu }_0$, such that for a.e. $t\in J$ and $x_1,x_2\in E$ with $x_1\le x_2$,

$$f(t,x_2)-f(t,x_1)\le L(x_2-x_1).$$

(H6)

There exist constants $0\le a_k\le 1(k=1,2,\dots ,m)$, such that for any $x_1,x_2\in E$ with $x_1\le x_2$,

$$I_k\left(x_2\right)-I_k\left(x_1\right)\le a_k(x_2-x_1).$$

Theorem 2.

Let X be a Banach space, K be a normal cone in X and $-A$ generate an exponential stable and positive $C_0$-semigroup $T\left(t\right)(t\ge 0)$. If conditions (H1)–(H6) hold and

$$\prod _{i=1}^m(1+a_i)+1+\sum _{i=1}^m\prod _{j=i}^m(1+a_j)<{\displaystyle \frac{M-{\nu }_0}{M+L}},$$

(13)

then PBVP (1) has a unique mild solution $u\in PC(J,E)$ in $[v_0,w_0]$.

Proof. 

Let Q be the operator defined in Theorem 1, then $Q:[v_0,w_0]\to [v_0,w_0]$ is a continuously increasing operator with $v_0\le Q{v}_0,Q{w}_0\le w_0$. We establish iterative schemes

$$v_n=Q{v}_{n-1},w_n=Q{w}_{n-1},n=1,2,\dots ,$$

then from the monotonicity of Q, we obtain that

$$v_0\le v_1\le v_2\le \dots \le v_n\le \dots \le w_n\le \dots \le w_2\le w_1\le w_0.$$

(14)

Thus, by comparison principle, we have

$$\begin{array}{cc}\hfill 0\le & w_n\left(t\right)-v_n\left(t\right)\le \left(M+L\right)·{\mathcal{P}}_M\left(w_{n-1}\left(t\right)-v_{n-1}\left(t\right)\right).\hfill \end{array}$$

Iterating the above inequality, one has

$$0\le w_n\left(t\right)-v_n\left(t\right)\le {\left(M+L\right)}^n{\mathcal{P}}_M^n(w_0\left(t\right)-v_0\left(t\right)).$$

By means of Lemma 2,

$$r\left({\mathcal{P}}_M\right)\le {\displaystyle \frac{1}{M-{\nu }_0}}\left(\prod _{i=1}^m(1+a_i)+1+\sum _{i=1}^m\prod _{j=i}^m(1+a_j)\right).$$

Let N be the normal constant of normal cone K, then

$$\parallel w_n\left(t\right)-v_n\left(t\right)\parallel \le N\parallel {\left(M+L\right)}^n\parallel ·\parallel P_M^n\parallel ·\parallel w_0\left(t\right)-v_0\left(t\right)\parallel .$$

Since $L<-{\nu }_0$ and inequality (13), by the spectral radius formula of Gelland, we have $\underset{n\to \infty }{lim}\sqrt[n]{\parallel {\mathcal{P}}_M^n\parallel }=r\left({\mathcal{P}}_M\right)\le {\displaystyle \frac{1}{M-{\nu }_0}}\left({\displaystyle \prod _{i=1}^m}(1+a_i)+1+{\displaystyle \sum _{i=1}^m}{\displaystyle \prod _{j=i}^m}(1+a_j)\right)$. Thus, there exists $N_0$, such that

$$\parallel w_n-v_n\parallel \le N{\left(\prod _{i=1}^m(1+a_i)+1+\sum _{i=1}^m\prod _{j=i}^m(1+a_j)\right)}^n{\left({\displaystyle \frac{M+L}{M-{\nu }_0}}\right)}^n·\parallel w_0-v_0\parallel .$$

as $n\ge N_0$. Then, be similarly to the nested interval theorem, there exists unique $u^{\ast }\in {\cap }_{n=0}^{\infty }[v_n,w_n]$, such that

$$\underset{n\to +\infty }{lim}{v}_n=\underset{n\to +\infty }{lim}{w}_n=u^{\ast }.$$

In Formula (15), we can obtain that $u^{\ast }=Q{u}^{\ast }$ as $n\to +\infty$. Therefore, $u^{\ast }\in PC(J,E)$ is a unique mild solution of PBVP (1) in $[v_0,w_0]$. □

Replacing conditions (H1) and (H2) by the stricter contraction:

(H1)′

There exist a constant $0<a\le {\lambda }_1$, a function $h\in P{C}^1(J,E)$ with $h\ge 0$, and a positive constant $\sigma$, such that

$$f(t,x)\le ax+h\left(t\right),x\in K_{PC},t\in J^{\prime },$$

$$f(t,x)\ge {\lambda }_{1}x,x\ge \sigma e_1.$$

(H2)′

There exist constants $a_k\ge 0$ with ${\mathsf{\Pi }}_{k=1}^m(1+a_k)\le e^{{\lambda }_1\omega }$, and $\theta \le b_k\in E,k=1,2,\dots ,m$, such that for every $x\in K_{PC}$,

$$\begin{array}{c}\hfill \theta \le I_k\left(x\right)\le a_{k}x+b_k.\end{array}$$

Consequently, we have the following existence result of positive mild solutions:

Theorem 3.

Let K be a regenerative cone in Banach space X and $-A$ generate an exponentially stable, positive and compact $C_0$-semigroup $T\left(t\right)(t\ge 0)$. If conditions (H1)′, (H2)′, (H3) and (H4) hold, then PBVP (1) has at least one positive mild solution in E.

Proof. 

According to the proof of Theorem 1 and conditions (H1)′, (H2)′, PBVP (1) has an upper solution $w_0$ satisfying $w_0^{\prime }\left(t\right)+A{w}_0\left(t\right)=a{w}_0\left(t\right)+h\left(t\right)$, where $h\left(t\right)\ge ({\lambda }_1-a)\sigma e_1$ for all $t\ge 0$. Since K is a regenerative cone in X, $T\left(t\right)(t\ge 0)$ is a positive compact semigroup. For ${\lambda }_0>-inf\{Re\lambda :\lambda \in \sigma \left(A\right)\}$ large enough, then ${\lambda }_{0}I+A$ has positive and bounded inverse operator ${({\lambda }_{0}I+A)}^{-1}$. By $\sigma \left(A\right)\ne \varnothing$, we have

$$r\left({({\lambda }_{0}I+A)}^{-1}\right)={\displaystyle \frac{1}{dist(-{\lambda }_0,\sigma \left(A\right))}}>0.$$

From the famous Krein-Rutmann theorem, it follows that A has smallest eigenvalue ${\lambda }_1>0$ with positive eigenfunction $e_1$. Further, let $v_0\equiv \sigma e_1$, then $v_0\left(t\right)=\sigma e_1$ for all $t\ge 0$ and satisfies

$$g\left(t\right):=v_0^{\prime }\left(t\right)+A{v}_0\left(t\right)={\lambda }_1\sigma e_1\le f(t,\sigma e_1).$$

Let Q be the operator defined in Theorem 1, by the positivity of semigroup $T_1\left(t\right)(t>0)$, condition (H1)′, for any $u\ge v_0$ and $t>0$, one can obtain that

$$\begin{array}{ccc}\hfill \sigma e_1& =& v_0\left(t\right)=T_1\left(t\right)v_0\left(0\right)+{\int }_0^t{T}_1(t-s)g\left(s\right)ds+\sum _{0<t_k<t}{T}_1(t-t_k)\mathsf{\Delta }{v}_0{|}_{t=t_k}\hfill \\ & \le & T_1\left(t\right)\sigma e_1+{\int }_0^t{T}_1(t-s)F_M(t,\sigma e_1)ds+\sum _{0<t_k<t}{T}_1(t-t_k)I_k\left(\sigma e_1\right)\hfill \\ & \le & T_1\left(t\right)u_0+{\int }_0^t{T}_1(t-s)F_M(t,u\left(s\right))ds+\sum _{0<t_k<t}{T}_1(t-t_k)I_k\left(u\left(t_k\right)\right)\hfill \\ & =& Qu\left(t\right).\hfill \end{array}$$

Thus, we can obtain that ${\lambda }_1\sigma e_1$ is a lower solution of PBVP (1). Then, we use the monotone iterative method to find the mild solution of PBVP (1) only prove that $v_0\le w_0$. For any $t>0$, let $u\left(t\right)=w_0\left(t\right)-v_0\left(t\right)$, we have

$$\begin{array}{cc}\hfill & u^{\prime }\left(t\right)+Au\left(t\right)\hfill \\ \hfill =& {\left(w_0\left(t\right)-v_0\left(t\right)\right)}^{\prime }+A\left(w_0\left(t\right)-v_0\left(t\right)\right)\hfill \\ \hfill =& au\left(t\right)+a\sigma e_1+h\left(t\right)-{\lambda }_1\sigma e_1\hfill \\ \hfill =& au\left(t\right)+h\left(t\right)+(a-{\lambda }_1)\sigma e_1\hfill \\ \hfill \ge & au\left(t\right).\hfill \end{array}$$

Moreover, applying the maximum principle, we can ensure that $u\left(t\right)\ge \theta$. Therefore, we can deduce that PBVP (1) has at least one positive mild solutions on E. □

4. Example

In this section, we give an example to demonstrate the applicability of abstract results. Let $\mathsf{\Omega }\subset {\mathbb{R}}^N(N\ge 1)$ be a bounded domain with a sufficiently smooth boundary $\partial \mathsf{\Omega }$. Consider the impulsive parabolic periodic boundary value problem

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\partial }{\partial t}}u(x,t)-\iota {\displaystyle \frac{{\partial }^2}{\partial x^2}}u(x,t)=u(x,t)+{sin}^{2}t,x\in \overline{\mathsf{\Omega }},t\in [0,\pi ],t\ne t_k,\hfill \\ {\mathsf{\Delta }u|}_{t=t_k}={\displaystyle \frac{u(x,t)}{1+\left|u\right(x,t\left)\right|}},x\in \overline{\mathsf{\Omega }},k=1,2,\dots ,m,\hfill \\ {u|}_{\partial \mathsf{\Omega }}=0,\hfill \\ u(x,0)=u(x,\omega ),x\in \overline{\mathsf{\Omega }}.\hfill \end{array}\right.\end{array}$$

(15)

where $\iota >0$ is a constant.

Let $E=L^2(\mathsf{\Omega },\mathbb{R}):=L^2\left(\mathsf{\Omega }\right)$, $P=\{u\in L^2\left(\mathsf{\Omega }\right):u\left(x\right)\ge 0,a.e.x\in \mathsf{\Omega }\}$, then P is a regular cone of E. We define the operator A in E as follows:

$$D\left(A\right)=\{u\in H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right):u{|}_{\partial \mathsf{\Omega }}=0\},Au=-\iota {\displaystyle \frac{{\partial }^2}{\partial x^2}}u(x,t),$$

From [22] we know that $-A$ is a self-adjoint operator in E and generates an exponentially stable positive and analytic semigroup $T\left(t\right)(t\ge 0)$, which is contractive in E, which implies that the growth exponent of the semigroup $T\left(t\right)(t\ge 0)$ satisfies ${\nu }_0=-\iota$. Hence, $\parallel T\left(t\right)\parallel \le 1$ for every $t>0$. Moveover, A has a discrete spectrum with eigenvalues of the form $\iota n^2$ and the corresponding normalized eigenvectors are $e_n\left(x\right)=\sqrt{{\displaystyle \frac{2}{\pi }}}sin\left(nx\right)$. Define the map $f:[0,2\pi ]\times L^p(\mathsf{\Omega };\mathbb{R})\to L^p\left(\mathsf{\Omega }\right)$ and functions $I_k:L^p\left(\mathsf{\Omega }\right)\to L^p(\mathsf{\Omega };\mathbb{R})$ as

$$f(t,u)\left(x\right)=u(x,t)+{sin}^{2}t,I_k\left(u\right)\left(x\right)={\displaystyle \frac{u(x,t)}{1+\left|u\right(x,t\left)\right|}},$$

thus the problem (15) can be transformed into problem (1).

Let $a={\lambda }_1=\iota >2^{m+2}+e^{2\pi },h\left(t\right)=e^t,\sigma =1$ in (H1)′, $a_k=1$ with ${\sum }_{k=1}^m(1+a_k)\le e^{\pi },b_k=\theta$ in (H2)′ and (H4), $M=1$ in (H3), we can prove that conditions (H1)′, (H2)′, (H3) and (H4) hold. In fact, $2u(x,t)$ and ${\displaystyle \frac{u(x,t)}{1+\left|u\right(x,t\left)\right|}}$ are monotonically increasing. And we can obtain that $\sqrt{{\displaystyle \frac{2}{\pi }}}sin\left(x\right)$ is a lower solution of problem (1). By Theorem 3, it follows that the parabolic periodic boundary value problem (15) has at least one positive solutions.