Periodic BVPs for fractional order impulsive evolution equations

Abstract

In this paper, we study periodic BVPs for fractional order impulsive evolution equations. The existence and boundedness of piecewise continuous mild solutions and design parameter drift for periodic motion of linear problems are presented. Furthermore, existence results of piecewise continuous mild solutions for semilinear impulsive periodic problems are showed. Finally, an example is given to illustrate the results.

MSC:34B05, 34G10, 47D06.

1 Introduction

In order to describe dynamics of populations subject to abrupt changes as well as other evolution processes such as harvesting, diseases, and so forth, many researchers have used impulsive differential systems to describe the model since the last century. For a wide-ranging bibliography and exposition on this important object see for instance the monographs of [1–4] and the papers [5–12].

Fractional differential equations appear naturally in fields such as viscoelasticity, electrical circuits, nonlinear oscillation of earthquakes etc. In particular, impulsive fractional evolution equations are used to describe many practical dynamical systems in many evolutionary processes models. Recently, Wang et al. [13] discussed Cauchy problems and nonlocal problems for impulsive fractional evolution equations involving the Caputo fractional derivative. However, periodic boundary value problems (BVPs for short) for impulsive fractional evolution equations have not been studied extensively.

In this paper we study the periodic BVPs for impulsive fractional evolution equations. Firstly, we discuss periodic BVPs for impulsive fractional evolution equations:

$$\{\begin{array}{l}{}^{c}D_{0,t}^{q}x(t)=Ax(t)+f(t),q\in (0,1),t\in J=[0,T],t\ne t_k,\\ x(0)=x(T),\\ x(t_k^{+})=x(t_k^{-})+y_k,k=1,2,\dots ,\delta ,\end{array}$$

(1)

in Banach space X, where ${}^{c}D_{0,t}^q$ is the Caputo fractional derivative of order q with the lower limit zero, $A:D(A)\subseteq X\to X$ is the generator of a $C_0$-semigroup $\{S(t),t\ge 0\}$ on a Banach space X, $f:J\to X$ is continuous, $x(0)$ and $y_k$ are the elements of X, $0=t_0<t_1<t_2<\cdots <t_{\delta }<t_{\delta +1}=T$, and $x(t_k^{+})={lim}_{h\to 0^{+}}=x(t_k+h)$ and $x(t_k^{-})=x(t_k)$ represent respectively the right and left limits of $x(t)$ at $t=t_k$.

Secondly, we design parameter drift for the above periodic motion. We study the following impulsive periodic BVPs with parameter perturbations:

$$\{\begin{array}{l}{}^{c}D_{0,t}^{q}x(t)=Ax(t)+f(t)+p(t,x(t),\xi ),t\in J,t\ne t_k,\\ x(0)=x(T),\\ x(t_k^{+})=x(t_k^{-})+y_k,k=1,2,\dots ,\delta ,\end{array}$$

(2)

where p is a given function and $\xi \in \mathrm{\Lambda }=(-\tilde{\xi },\tilde{\xi })$ is a small parameter perturbation that may be caused by some adaptive control algorithms or parameter drift.

Finally, we consider semilinear impulsive periodic problems:

$$\{\begin{array}{l}{}^{c}D_{0,t}^{q}x(t)=Ax(t)+f(t,x(t)),t\in J,t\ne t_k,\\ x(0)=x(T),\\ x(t_k^{+})=x(t_k^{-})+I_k(x(t_k^{-})),k=1,2,\dots ,\delta ,\end{array}$$

(3)

where $f:J\times X\to X$ is continuous and $I_k:X\to X$ is continuous.

The rest of this paper is organized as follows. In Section 2, the existence and boundedness of the operator $B={[I-\mathcal{T}(T)]}^{-1}$ are given. In Section 3, the existence and boundedness of PC-mild solutions and the design parameter drift for such a periodic motion are presented. In Section 4, existence results of PC-mild solutions for impulsive periodic problems are showed. Finally, an example is presented to illustrate the theory.

2 Existence and boundedness of operator $B={[I-\mathcal{T}(T)]}^{-1}$

Suppose $T>0$, let $J=[0,T]$. Let $M={sup}_{t\ge 0}\parallel S(t)\parallel <\mathrm{\infty }$. We denote by $C(J,X)$ the Banach space of all continuous functions from J into X with the norm ${\parallel x\parallel }_C=sup\{\parallel x(t)\parallel :t\in J\}$. We also introduce the set of functions $PC(J,X)$ = {$x:J\to X:x$ is continuous at $t\in J\setminus \{t_1,t_2,\dots ,t_{\delta }\}$, and x is continuous from the left and has right-hand limits at $t\in \{t_1,t_2,\dots ,t_{\delta }\}$}. Endowed with the norm

$${\parallel x\parallel }_{PC}=max\{\underset{t\in J}{sup}\parallel x(t+0)\parallel ,\underset{t\in J}{sup}\parallel x(t-0)\parallel \},$$

it is easy to see $(PC(J,X),{\parallel \cdot \parallel }_{PC})$ is a Banach space.

For measurable functions $l:J\to R$, define the norm ${\parallel l\parallel }_{L^p(J,R)}={({\int }_J{|l(t)|}^{p}dt)}^{\frac{1}{p}}$, $1\le p<\mathrm{\infty }$. We denote by $L^p(J,R)$ the Banach space of all Lebesgue measurable functions l with ${\parallel l\parallel }_{L^p(J,R)}<\mathrm{\infty }$.

Definition 2.1 ([14])

The fractional integral of order γ with the lower limit zero for a function f is defined as

$$I_{0,t}^{\gamma }f(t)=\frac{1}{\mathrm{\Gamma }(\gamma )}{\int }_0^t\frac{f(s)}{{(t-s)}^{1-\gamma }}ds,t>0,\gamma >0,$$

provided the right side is point-wise defined on $[0,\mathrm{\infty })$, where $\mathrm{\Gamma }(\cdot )$ is the gamma function.

Definition 2.2 ([14])

The Riemann-Liouville derivative of order γ with the lower limit zero for a function $f:[0,\mathrm{\infty })\to R$ can be written as

$${}^{L}D_{0,t}^{\gamma }f(t)=\frac{1}{\mathrm{\Gamma }(n-\gamma )}\frac{d^n}{d{t}^n}{\int }_0^t\frac{f(s)}{{(t-s)}^{\gamma +1-n}}ds,t>0,n-1<\gamma <n.$$

Definition 2.3 ([14])

The Caputo derivative of order γ for a function $f:[0,\mathrm{\infty })\to R$ can be written as

$${}^{c}D_{0,t}^{\gamma }f(t){=}^L{D}_{0,t}^{\gamma }(f(t)-\sum _{k=0}^{n-1}\frac{t^k}{k!}{f}^{(k)}(0)),t>0,n-1<\gamma <n.$$

Remark 2.4 If f is an abstract function with values in X, then the integrals which appear in Definitions 2.1 and 2.2 are taken in Bochner’s sense.

As in our previous work [13], by a PC-mild solution of (1) we mean the function $x\in PC(J,X)$ satisfying

$$\begin{array}{rcl}x(t)& =& \mathcal{T}(t)x(0)+\sum _{i=1}^k\mathcal{T}(t-t_i)y_i\\ +{\int }_0^t{(t-s)}^{q-1}\mathcal{S}(t-s)f(s)ds,t\in (t_i,t_{i+1}],k=1,2,\dots ,\delta ,\end{array}$$

and $x(0)=x(T)$, where

$$\mathcal{T}(t)={\int }_0^{\mathrm{\infty }}{\xi }_q(\theta )S\left(t^q\theta \right)d\theta ,\mathcal{S}(t)=q{\int }_0^{\mathrm{\infty }}\theta {\xi }_q(\theta )S\left(t^q\theta \right)d\theta$$

and

$$\begin{array}{c}{\xi }_q(\theta )=\frac{1}{q}{\theta }^{-1-\frac{1}{q}}{\vartheta }_q\left({\theta }^{-\frac{1}{q}}\right)\ge 0,\hfill \\ {\vartheta }_q(\theta )=\frac{1}{\pi }\sum _{n=1}^{\mathrm{\infty }}{(-1)}^{n-1}{\theta }^{-qn-1}\frac{\mathrm{\Gamma }(nq+1)}{n!}sin(n\pi q),\theta \in (0,\mathrm{\infty }),\hfill \end{array}$$

here ${\xi }_q$ is a probability density function defined on $(0,\mathrm{\infty })$, that is,

$${\xi }_q(\theta )\ge 0,\theta \in (0,\mathrm{\infty })\text{and}{\int }_0^{\mathrm{\infty }}{\xi }_q(\theta )d\theta =1.$$

Lemma 2.5 (see Lemma 2.9 [15])

The operator has the following properties:

• (i) For any fixed $t\ge 0$, $\mathcal{T}(t)$ and $\mathcal{S}(t)$ are linear and bounded operators, i.e., for any $x\in X$, $\parallel \mathcal{T}(t)x\parallel \le M\parallel x\parallel$ and $\parallel \mathcal{S}(t)x\parallel \le \frac{M}{\mathrm{\Gamma }(q)}\parallel x\parallel$.

• (ii) Both $\{\mathcal{S}(t),t\ge 0\}$ and $\{\mathcal{T}(t),t\ge 0\}$ are strongly continuous.

• (iii) For every $t>0$, $\mathcal{T}(t)$ and $\mathcal{S}(t)$ are compact operators if $S(t)$ is compact.

Suppose that here the bounded operator $B:X\to X$ exists given by

$$B={[I-\mathcal{T}(T)]}^{-1}.$$

(4)

We present sufficient conditions for the existence and boundedness of the operator B.

Lemma 2.6 (see Theorem 3.3 and Remark 3.4 [16])

The operator B defined in (4) exists and is bounded, if one of the following three conditions holds:

• (i) $S(t)$ is compact for each $t>0$ and the homogeneous linear nonlocal problem

  $$\{\begin{array}{l}{}^{c}D_{0,t}^{q}x(t)=Ax(t),t\in J,\\ x(0)=x(T),\end{array}$$

  has no non-trivial PC-mild solutions.

• (ii) If $\parallel \mathcal{T}(T)\parallel <1$, then the operator $I-\mathcal{T}(T)$ is invertible and ${[I-\mathcal{T}(T)]}^{-1}\in L_b(X)$.

• (iii) If $\parallel S(t)\parallel <1$ for $t\in (0,T]$, then $\mathcal{T}(nT)\to 0$ as $n\to \mathrm{\infty }$ and the operator $I-\mathcal{T}(T)$ is invertible and ${[I-\mathcal{T}(T)]}^{-1}\in L_b(X)$.

3 Linear impulsive periodic problems and robustness

In this section, we consider the existence of PC-mild solutions of (1) and of design parameter drift for (2).

We list the following assumptions.

(HA): $A:D(A)\to X$ is the infinitesimal generator of a $C_0$-semigroup $\{S(t),t\ge 0\}$.

(HF): $f:J\to X$ is strongly measurable for $t\in J$ and there exist a constant $q_1\in (0,q)$ and a real-valued function $h(\cdot )\in L^{\frac{1}{q_1}}(J,R)$ such that $\parallel f(t)\parallel \le h(t)$, for each $t\in J$.

(HB): The operator B defined in (4) exists and is bounded.

We first give an existence theorem of PC-mild solutions of (1).

Theorem 3.1 Assume that (HA), (HF), and (HB) are satisfied. Then (1) has a PC-mild solution given by

$$\begin{array}{rcl}{x}_T(t)& =& \mathcal{T}(t)Bz+\sum _{i=1}^k\mathcal{T}(t-t_i)y_i\\ +{\int }_0^t{(t-s)}^{q-1}\mathcal{S}(t-s)f(s)ds,t\in (t_i,t_{i+1}],k=1,2,\dots ,\delta ,\end{array}$$

(5)

where

$$z=\sum _{i=1}^{\delta }\mathcal{T}(t-t_i)y_i+{\int }_0^T{(T-s)}^{q-1}\mathcal{S}(T-s)f(s)ds.$$

(6)

Further, we have

$$\parallel x_T(t)\parallel \le M(M{L}_1+1)[\sum _{i=1}^{\delta }\parallel y_i\parallel +\frac{T^{q-q_1}H}{\mathrm{\Gamma }(q){(\frac{q-q_1}{1-q_1})}^{1-q_1}}],$$

(7)

where $L_1=\parallel B\parallel$ and $H={\parallel h\parallel }_{L^{\frac{1}{q_1}}(J,R)}$.

Proof We consider an impulsive Cauchy problem,

$$\{\begin{array}{l}{}^{c}D_{0,t}^{q}x(t)=Ax(t)+f(t),t\in J,t\ne t_k,\\ x(0)=Bz:=\overline{x},z\text{ defined in (6)},\\ x(t_k^{+})=x(t_k^{-})+y_k,k=1,2,\dots ,\delta .\end{array}$$

(8)

It follows from the expression of the initial value $x(0)=Bz:=\overline{x}$ that the mild solution $x_T(\cdot )$ of (8) corresponding to the initial value $\overline{x}$ must be the PC-mild solution of (1).

For the estimation of $x_T(\cdot )$, by Lemma 2.5,

$$\begin{array}{rcl}\parallel x_T(t)\parallel & \le & \parallel \mathcal{T}(t)Bz\parallel +\sum _{i=1}^k\parallel \mathcal{T}(t-t_i)\parallel \parallel y_i\parallel +{\int }_0^t{(t-s)}^{q-1}\parallel \mathcal{S}(t-s)f(s)\parallel ds\\ \le & M\parallel B\parallel \parallel z\parallel +M\sum _{i=1}^k\parallel y_i\parallel +\frac{M}{\mathrm{\Gamma }(q)}{\int }_0^t{(t-s)}^{q-1}h(s)ds\\ \le & M\parallel B\parallel (\sum _{i=1}^{\delta }\parallel \mathcal{T}(t-t_i)\parallel \parallel y_i\parallel +{\int }_0^T{(T-s)}^{q-1}\parallel \mathcal{S}(T-s)f(s)\parallel ds)\\ +M\sum _{i=1}^{\delta }\parallel y_i\parallel +\frac{M}{\mathrm{\Gamma }(q)}{\int }_0^t{(t-s)}^{q-1}h(s)ds\\ \le & M^2\parallel B\parallel (\sum _{i=1}^{\delta }\parallel y_i\parallel +\frac{1}{\mathrm{\Gamma }(q)}{\int }_0^T{(T-s)}^{q-1}h(s)ds)\\ +M\sum _{i=1}^{\delta }\parallel y_i\parallel +\frac{M}{\mathrm{\Gamma }(q)}{\int }_0^t{(t-s)}^{q-1}h(s)ds\\ \le & M^2\parallel B\parallel [\sum _{i=1}^{\delta }\parallel y_i\parallel +\frac{1}{\mathrm{\Gamma }(q)}{\left({\int }_0^T{(T-s)}^{\frac{q-1}{1-q_1}}ds\right)}^{1-q_1}{({\int }_0^T{h}^{\frac{1}{q_1}}(s)ds)}^{q_1}]\\ +[M\sum _{i=1}^{\delta }\parallel y_i\parallel +\frac{M}{\mathrm{\Gamma }(q)}{\left({\int }_0^t{(t-s)}^{\frac{q-1}{1-q_1}}ds\right)}^{1-q_1}{({\int }_0^t{h}^{\frac{1}{q_1}}(s)ds)}^{q_1}]\\ \le & M(M{L}_1+1)[\sum _{i=1}^{\delta }\parallel y_i\parallel +\frac{T^{q-q_1}H}{\mathrm{\Gamma }(q){(\frac{q-q_1}{1-q_1})}^{1-q_1}}],\end{array}$$

where $L_1=\parallel B\parallel$ and $H={\parallel h\parallel }_{L^{\frac{1}{q_1}}(J,R)}$. The desired results are obtained. □

Remark 3.2 In Theorem 3.1, we replace (HB) by $\{S(t),t\ge 0\}$; it is a compact semigroup and

$$\{\begin{array}{l}{}^{c}D_{0,t}^{q}x(t)=Ax(t),t\in J,\\ x(0)=x(T),\end{array}$$

has no non-trivial mild solutions. Then one can use the Fredholm alternative theorem to derive that the operator equation $[I-\mathcal{T}(T)]x(0)=z$ has a unique solution $x(0)={[I-\mathcal{T}(T)]}^{-1}z:=\overline{x}$. Thus, the PC-mild solution of (1) is unique.

Define $P{C}_T(J,X)=\{x\in PC(J,X):x(0)=x(T)\}$ with ${\parallel x\parallel }_{P{C}_T}=sup\{\parallel x(t)\parallel :t\in J\}$ for $x\in P{C}_T(J,X)$. It can be seen that endowed with the norm ${\parallel \cdot \parallel }_{P{C}_T}$, $P{C}_T(J,X)$ is a Banach space. Denote

$$\begin{array}{c}{\mathcal{S}}_{\rho }=\{x\in P{C}_T(J,X):{\parallel x\parallel }_{P{C}_T}\le \rho \},\hfill \\ \mathcal{B}(x,{\rho }_1)=\{x\in P{C}_T(J;X):{\parallel x-x_T\parallel }_{P{C}_T}\le {\rho }_1\},\hfill \end{array}$$

where

$$\begin{array}{c}\rho =M(M{L}_1+1)[\sum _{i=1}^{\delta }\parallel y_i\parallel +\frac{T^{q-q_1}H}{\mathrm{\Gamma }(q){(\frac{q-q_1}{1-q_1})}^{1-q_1}}+\frac{T^q}{\mathrm{\Gamma }(q+1)}\underset{|\xi |\le \tilde{\xi }}{sup}\chi (\xi )],\hfill \\ {\rho }_1=\frac{M(M{L}_1+1)T^q}{\mathrm{\Gamma }(1+q)}\underset{|\xi |\le \tilde{\xi }}{sup}\chi (\xi ),\hfill \end{array}$$

and χ is a nonnegative function.

We introduce the assumption (HP):

(HP1): $p:J\times {\mathcal{S}}_{\rho }\times \mathrm{\Lambda }\to X$ is measurable in t.

(HP2): There exists a nonnegative function ϖ such that ${lim}_{\xi \to 0}\varpi (\xi )=\varpi (0)=0$ and for any $t\in J$, $x,y\in {\mathcal{S}}_{\rho }$, and $\xi \in \mathrm{\Lambda }$, and we have

$$\parallel p(t,x,\xi )-p(t,y,\xi )\parallel \le \varpi (\xi )\parallel x-y\parallel .$$

(HP3): There exists a nonnegative function χ such that ${lim}_{\xi \to 0}\chi (\xi )=\chi (0)=0$, and for any $t\in J$, $x\in {\mathcal{S}}_{\rho }$, and $\xi \in \mathrm{\Lambda }$, we have

$$\parallel p(t,x,\xi )\parallel \le \chi (\xi ).$$

By a PC-mild solution of (2), we mean the function $x\in PC(J,X)$ satisfying

$$\begin{array}{rcl}x(t)& =& \mathcal{T}(t)x(0)+\sum _{i=1}^k\mathcal{T}(t-t_i)y_i\\ +{\int }_0^t{(t-s)}^{q-1}\mathcal{S}(t-s)[f(s)+p(s,x(s),\xi )]ds,k=1,2,\dots ,\delta ,\end{array}$$

and $x(0)=x(T)$.

The following result shows that given a periodic motion we can design periodic motion controllers that are robust with respect to parameter drift.

Theorem 3.3 Let (HA), (HF), (HB), and (HP) hold. Then there is a ${\xi }_0\in (0,\tilde{\xi })$ such that, for $|\xi |\le {\xi }_0$, (2) has a PC-mild solution $x_T^{\xi }$ satisfying

$${\parallel x_T^{\xi }-x_T\parallel }_{P{C}_T}\le {\rho }_1$$

and ${lim}_{\xi \to 0}{x}_T^{\xi }(t)=x_T(t)$ uniformly on $t\in J$ where $x_T$ is the mild solution of (1).

Proof By (HB), one can choose

$$x_0=B[z+{\int }_0^T{(T-s)}^{q-1}\mathcal{S}(T-s)p(s,x(s),\xi )ds]\in X$$

to be fixed. Consider the map on $\mathcal{B}(x_T,{\rho }_1)$ given by

$$(\mathcal{O}x)(t)=\mathcal{T}(t)x_0+\sum _{i=1}^k\mathcal{T}(t-t_i)y_i+{\int }_0^t{(t-s)}^{q-1}\mathcal{S}(t-s)[f(s)+p(s,x(s),\xi )]ds.$$

Obviously, $\mathcal{O}x\in P{C}_T(J,X)$.

By the assumption (HP), we can choose a ${\xi }_0\in (0,\tilde{\xi })$ such that

$$\frac{M(M{L}_1+1)T^q}{\mathrm{\Gamma }(1+q)}\underset{|\xi |\le {\xi }_0}{sup}\chi (\xi )\le {\rho }_1$$

and

$$\eta =\frac{M(M{L}_1+1)T^q}{\mathrm{\Gamma }(1+q)}\underset{|\xi |\le {\xi }_0}{sup}\varpi (\xi )<1.$$

For $\xi \in (-{\xi }_0,{\xi }_0)$ and $x,y\in \mathcal{B}(x_T,{\rho }_1)$, one can verify that

$${\parallel \mathcal{O}x-x_T\parallel }_{P{C}_T}\le \frac{M(M{L}_1+1)T^q}{\mathrm{\Gamma }(1+q)}\underset{|\xi |\le {\xi }_0}{sup}\chi (\xi )\le {\rho }_1$$

(9)

and

$${\parallel \mathcal{O}x-\mathcal{O}y\parallel }_{P{C}_T}\le \eta {\parallel x-y\parallel }_{P{C}_T}.$$

This implies that is a contraction mapping on $\mathcal{B}(x_T,{\rho }_1)$. Then, has a unique fixed point $x_T^{\xi }\in \mathcal{B}(x_T,{\rho }_1)$ given by

$$x_T^{\xi }(t)=\mathcal{T}(t)x_0+{\int }_0^t{(t-s)}^{q-1}\mathcal{S}(t-s)[f(s)+p(s,x_T^{\xi }(s),\xi )]ds,$$

(10)

which is just the PC-mild solution of (2).

From the expressions (9) and (10), one can get ${\parallel x_T^{\xi }-x_T\parallel }_{P{C}_T}\le {\rho }_1$. It is easy to see that ${lim}_{\xi \to 0}{x}_T^{\xi }(t)=x_T(t)$ uniformly on $t\in J$. □

4 Semilinear impulsive periodic problems

We impose the following assumptions.

• (HF1): $f:J\times X\to X$ is continuous and there exist a constant $q_1\in (0,q)$ and a real-valued function $L_f(\cdot )\in L^{\frac{1}{q_1}}(J,R^{+})$ such that $\parallel f(t,x)-f(t,y)\parallel \le L_f(t)\parallel x-y\parallel$ for all $x,y\in X$.

• (HF2): $f:J\times X\to X$ is continuous and maps a bounded set into a bounded set.

• (HF3): For each $x_0\in X$, there exists a constant $r>0$ such that

  $$M(M{L}_1+1)(\delta M_I+\frac{T^q}{\mathrm{\Gamma }(1+q)}\underset{s\in J,\varphi \in Y_{\mathrm{\Gamma }}}{sup}\parallel f(s,\varphi (s))\parallel )\le r,$$

  where

  $$Y_{\mathrm{\Gamma }}=\{\varphi \in PC(J,X):\parallel \varphi \parallel \le r\text{ for }t\in J\}.$$

• (HI1): $I_k:X\to X$ is continuous and there exists a constant $L_I>0$ such that $\parallel I_k(x)-I_k(y)\parallel \le L_I\parallel x-y\parallel$ for all $x,y\in X$, $k=1,2,\dots ,\delta$.

• (HI2): $I_k:X\to X$ is continuous and there exists a constant $M_I>0$ such that $\parallel I_k(x)\parallel \le M_I$, for all $x\in X$, $k=1,2,\dots ,\delta$.

Theorem 4.1 Let (HA), (HB), (HI1), and (HF1) be satisfied. Then (3) has a unique PC-mild solution on J provided that

$$M(M{L}_1+1)(\delta L_I+\frac{T^{\ast }}{\mathrm{\Gamma }(q)})<1,T^{\ast }={\left[\left(\frac{1-q_1}{q-q_1}\right)T^{\frac{q-q_1}{1-q_1}}\right]}^{1-q_1}{\parallel L_f\parallel }_{L^{\frac{1}{q_1}}(J,R^{+})}.$$

(11)

Proof Consider

$$\{\begin{array}{l}{}^{c}D_{0,t}^{q}x(t)=Ax(t)+f(t,x(t)),t\in J,t\ne t_k,\\ x(0)=B{z}_1,\\ x(t_k^{+})=x(t_k^{-})+I_k(x(t_k^{-})),k=1,2,\dots ,\delta ,\end{array}$$

(12)

where

$$z_1=\sum _{i=1}^{\delta }\mathcal{T}(t-t_i)I_i\left(x\left(t_i^{-}\right)\right)+{\int }_0^T{(T-s)}^{q-1}\mathcal{S}(T-s)f(s,x(s))ds,$$

(13)

and we define an operator Q on $PC(J,X)$:

$$\begin{array}{rcl}(Qx)(t)& =& \mathcal{T}(t)B{z}_1+\sum _{i=1}^k\mathcal{T}(t-t_i)I_i\left(x\left(t_i^{-}\right)\right)\\ +{\int }_0^t{(t-s)}^{q-1}\mathcal{S}(t-s)f(s,x(s))ds,t\in (t_i,t_{i+1}],k=1,2,\dots ,\delta .\end{array}$$

(14)

Clearly, Q is well defined on $PC(J,X)$ due to our assumptions.

Then, we only need to show that Q is a contraction on $PC(J,X)$.

In general, for each $t\in (t_k,t_{k+1}]$, $k=1,2,\dots ,\delta$ we have

$$\begin{array}{c}\parallel (Qx)(t)-(Qy)(t)\parallel \hfill \\ \le M^2{L}_1{L}_I\sum _{i=1}^{\delta }\parallel x\left(t_i^{-}\right)-y\left(t_i^{-}\right)\parallel +M{L}_I\sum _{i=1}^k\parallel x\left(t_i^{-}\right)-y\left(t_i^{-}\right)\parallel \hfill \\ +\frac{M}{\mathrm{\Gamma }(q)}(M{L}_1{\int }_0^T{(T-s)}^{q-1}{L}_f(s)\parallel x(s)-y(s)\parallel ds\hfill \\ +{\int }_0^t{(t-s)}^{q-1}{L}_f(s)\parallel x(s)-y(s)\parallel ds)\hfill \\ \le M{L}_I(M{L}_1+1)\delta {\parallel x-y\parallel }_{PC}\hfill \\ +\frac{M{\parallel x-y\parallel }_{PC}}{\mathrm{\Gamma }(q)}(M{L}_1{\int }_0^T{(T-s)}^{q-1}{L}_f(s)ds+{\int }_0^t{(t-s)}^{q-1}{L}_f(s)ds)\hfill \\ \le M{L}_I(M{L}_1+1)\delta {\parallel x-y\parallel }_{PC}\hfill \\ +\frac{M{\parallel x-y\parallel }_{PC}}{\mathrm{\Gamma }(q)}[M{L}_1{\left({\int }_0^T{(T-s)}^{\frac{q-1}{1-q_1}}ds\right)}^{1-q_1}{\parallel L_f\parallel }_{L^{\frac{1}{q_1}}(J,R^{+})}\hfill \\ +{\left({\int }_0^t{(t-s)}^{\frac{q-1}{1-q_1}}ds\right)}^{1-q_1}{\parallel L_f\parallel }_{L^{\frac{1}{q_1}}([0,t_{k+1}],R^{+})}]\hfill \\ \le M(M{L}_1+1)(\delta L_I+\frac{T^{\ast }}{\mathrm{\Gamma }(q)}){\parallel x-y\parallel }_{PC}.\hfill \end{array}$$

Hence, the condition (11) allows us to conclude, in view of the Banach contraction mapping principle, that Q has a unique fixed point $x\in PC(J,X)$, which is just the unique PC-mild solution of (3). □

Theorem 4.2 Suppose that (HA), (HB), (HI2), and (HF2) and (HF3) are satisfied. Then for every $x_0\in X$, (3) has at least a PC-mild solution on J.

Proof Consider the mapping

$$Q:PC(J,X)\to PC(J,X)$$

by

$$(Qv)(t)=(Q_{1}v)(t)+(Q_{2}v)(t),$$

where

$$(Q_{1}v)(t)=\mathcal{T}(t)B{z}_1+{\int }_0^t{(t-s)}^{q-1}\mathcal{S}(t-s)f(s,v(s))ds,t\in J\setminus \{t_1,t_2,\dots ,t_{\delta }\};$$

(15)

$z_1$ is defined in (13) and

$$(Q_{2}v)(t)=\{\begin{array}{ll}0,& t\in [0,t_1],\\ {\sum }_{i=1}^k\mathcal{T}(t-t_i)I_i(v(t_i^{-})),& t\in (t_k,t_{k+1}],k=1,\dots ,\delta .\end{array}$$

(16)

For each $t\in (t_k,t_{k+1}]$, $v\in Y_{\mathrm{\Gamma }}$,

$$\begin{array}{rcl}\parallel (Qv)(t)\parallel & \le & \parallel (Q_{1}v)(t)\parallel +\parallel (Q_{2}v)(t)\parallel \\ \le & M(M{L}_1+1)(\delta M_I+\frac{T^q}{\mathrm{\Gamma }(1+q)}\underset{s\in J,\varphi \in Y_{\mathrm{\Gamma }}}{sup}\parallel f(s,\varphi (s))\parallel ).\end{array}$$

Thus, we see that $Q:Y_{\mathrm{\Gamma }}\to Y_{\mathrm{\Gamma }}$.

Just like the proof in our previous work [13], one can prove that Q is a continuous mapping from $Y_{\mathrm{\Gamma }}$ to $Y_{\mathrm{\Gamma }}$ and it is a compact operator. Now, Schauder’s fixed point theorem implies that Q has a fixed point, which gives rise to a PC-mild solution. □

5 Example

We consider impulsive fractional differential equations with periodic boundary conditions,

$$\{\begin{array}{l}{}^{c}D_{0,t}^{\frac{1}{2}}x(t,y)=\frac{{\partial }^2}{\partial y^2}x(t,y)+\frac{|x(t,y)|}{\gamma (1+9e^t)(1+|x(t,y)|)},y\in (0,\pi ),t\in [0,\frac{1}{2})\cup (\frac{1}{2},1],\\ x(t,0)=x(t,\pi )=0,\\ x(0,y)=x(1,y),y\in (0,\pi ),\\ x({\frac{1}{2}}^{+})=x({\frac{1}{2}}^{-})+\frac{|x({\frac{1}{2}}^{-},y)|}{\gamma (9+t)(1+|x({\frac{1}{2}}^{-},y)|)},y\in (0,\pi ),\end{array}$$

(17)

in $X=L^2(0,\pi )$ where $\gamma >0$ will be chosen later.

Define $Ax=-\frac{{\partial }^2}{\partial y^2}x$ for $x\in D(A)$ where $D(A)=\{x\in X:\frac{\partial x}{\partial y},\frac{{\partial }^{2}x}{\partial y^2}\in X\text{ and }x(0)=x(\pi )=0\}$. Then A is the infinitesimal generator of a $C_0$-semigroup $\{T(t),t\ge 0\}$ in $L^2(0,\pi )$. Moreover, $T(\cdot )$ is also compact and $\parallel T(t)\parallel \le e^{-t}\le 1$, $t\ge 0$. By the Fredholm alternative theorem, ${[I-\mathcal{T}(1)]}^{-1}$ exists and is bounded where $\mathcal{T}(\cdot )$ is defined in Section 2.

Define $f(t,x(t))(y)=\frac{|x(t,y)|}{\gamma (1+9e^t)(1+|x(t,y)|)}$, $t\in [0,\frac{1}{2})\cup (\frac{1}{2},1]$, $x\in X$, $y\in (0,\pi )$ and $f:[0,1]\times X\to X$ is a continuous function, $\parallel f(t,x_1)-f(t,x_2)\parallel \le L_f\parallel x_1-x_2\parallel$, with $L_f=\frac{1}{10\gamma }\in L^{\frac{1}{4}}([0,1],R^{+})$. Define $I_1(x({\frac{1}{2}}^{-}))(y)=\frac{|x({\frac{1}{2}}^{-},y)|}{\gamma (9+t)(1+|x({\frac{1}{2}}^{-},y)|)}$, $x\in X$, $y\in (0,\pi )$. $\parallel I_1(x_1)-I_1(x_2)\parallel \le L_I\parallel x_1-x_2\parallel$ for all $x_1,x_2\in X$ with $L_I=\frac{1}{10\gamma }$.

Moreover,

$$M(M{L}_1+1)(\delta L_I+\frac{T^{\ast }}{\mathrm{\Gamma }(q)})=\parallel {[I-\mathcal{T}(1)]}^{-1}\parallel (\frac{1}{10\gamma }+\frac{\sqrt[4]{27}}{10\gamma \mathrm{\Gamma }(\frac{1}{2})})<\frac{2}{5\gamma }\parallel {[I-\mathcal{T}(1)]}^{-1}\parallel .$$

Thus, one can choose $\gamma >\frac{2}{5}\parallel {[I-\mathcal{T}(1)]}^{-1}\parallel$ such that (11) holds. Therefore, (17) has a unique PC-mild solution on $[0,1]$.