Upper and lower solution method for Hilfer fractional evolution equations with nonlocal conditions

Abstract

This paper is concerned with the existence of extremal mild solutions for Hilfer fractional evolution equations with nonlocal conditions in an ordered Banach space E. By employing the method of lower and upper solutions, the measure of noncompactness, and Sadovskii’s fixed point theorem, we obtain the existence of extremal mild solutions for Hilfer fractional evolution equations with noncompact semigroups. Finally, an example is provided to illustrate the feasibility of our main results.

1 Introduction

In recent years, many authors began to consider Hilfer fractional differential equations, see [1,2,3,4,5,6,7]. Presently, Hilfer fractional evolution equations have also been widely dealt with by many scholars. In [2], Gu and Trujillo investigated a class of Hilfer fractional evolution equations and established the existence results of mild solutions to such issues, and then Furati et al. [8] considered an initial value problem for a class of Hilfer fractional differential equations.

Later, the nonlocal problems have had better effects in applications than the initial problem, many contributions have been made in applications of fractional evolution equations with nonlocal conditions, see [7, 9, 10] and the references therein. For example, Liang and Yang [11] investigated the exact controllability of the nonlocal Cauchy problem for the fractional integro differential evolution equations in Banach spaces E:

$$ \textstyle\begin{cases} D^{q}x(t)+Ax(t)=f(t,x(t),Gx(t))+Bu(t),\quad t\in J, \\ x(0)=\sum_{k=1}^{m}c_{k}x(t_{k}), \end{cases} $$

where $D^{q}$ denotes the Caputo fractional derivative of order $q\in (0,1)$, $-A:D(A)\subset E\rightarrow E$ is the infinitesimal generator of a $C_{0}$-semigroup $T(t)$ ($t\geq 0$) of uniformly bounded linear operators, B is a linear bounded operator; f is a given function and the operator is given by

$$ Gx(t)= \int _{0}^{t}K(t,s)x(s)\,ds. $$

Over the past year, some recent papers investigated the existence of mild solutions for Hilfer fractional evolution equations with nonlocal conditions. In [3], Min Yang et al. studied the existence and uniqueness of mild solutions to the following Hilfer fractional evolution equations:

$$ \textstyle\begin{cases} D^{\nu ,\mu }_{0+}[u(t)-h(t,u(t))]=Au(t)+f(t,u(t)),\quad t\in J'=(0,b], \\ I^{(1-\nu )(1-\mu )}_{0+}[u(0)-h(0,u(0))]-g(u)=u_{0}, \end{cases} $$

with the associated $C_{0}$-semigroup being compact or not, where $D^{\nu ,\mu }_{0+}$ denotes the Hilfer fractional derivative of order μ and type ν, $0\leq \nu \leq 1$, $0<\mu <1$. In [5], Ahmed et al. studied the existence of mild solutions for Hilfer fractional stochastic integro-differential equations of the form

$$ \textstyle\begin{cases} D^{\nu ,\mu }_{0+}[u(t)+F(t,v(t))]+Au(t)=\int _{0}^{t} G(s,\eta (s))\,d \omega (s),\quad t\in J:=(0,b], \\ I^{(1-\nu )(1-\mu )}_{0+}u(0)-g(u)=u_{0}, \end{cases} $$

where $(t,v(t))=(t,u(t),u(b_{1}(t))),\ldots ,u(b_{m}(t)))$ and $(t,\eta (t))=(t,u(t),u(a_{1}(t))),\ldots ,u(a_{n}(t)))$, $D^{\nu , \mu }_{0+}$ denotes the Hilfer fractional derivative $0\leq \nu \leq 1$, $0<\mu <1$, −A is the infinitesimal generator of an analytic semigroup of bounded linear operators $S(t),t\geq 0$ on a separable Hilbert space.

In [6], Ahmed et al. studied the existence and controllability results for nonlinear delay Hilfer fractional differential equation with impulsive condition of the form

$$ \textstyle\begin{cases} D^{\nu ,\mu }_{0+}u(t)=Au(t)+f(t,u(\gamma _{1}(t)),\int _{0}^{t}h(t,s)g(s,u( \gamma _{2}(s)))\,ds),\quad t\in J=(0,b], t\neq t_{k}, \\ \Delta u(t_{k})=I_{k}(u(t_{k}^{-})), \quad k=1,2,\ldots ,m, \\ I^{(1-\nu )(1-\mu )}_{0+}u(0)=u_{0}, \end{cases} $$

where $D^{\nu ,\mu }_{0+}$ is the Hilfer fractional derivative, A is the infinitesimal generator of a $C_{0}$-semigroup $T(t)$ on E.

On the other hand, by employing the method of lower and upper solutions to study the existence of an extremal mild solution for a class of fractional evolution equation is an interesting issue, which has been the focus of attention in [9, 10, 12,13,14]. In [14], Chen and Li used the monotone iterative method and lower and upper solutions method to discuss the existence and uniqueness of mild solutions for a class of semilinear evolution equations with nonlocal conditions in an ordered Banach space E:

$$ \textstyle\begin{cases} u'(t)+Au(t)=f(t,u(t)),\quad t\in J=[0,b], \\ u(0)=\sum_{k=1}^{p} c_{k} u(t_{k})+u_{0}, \end{cases} $$

where $A:D(A)\subset E\rightarrow E$ is a closed linear operator and −A generates a $C_{0}$-semigroup $T(t)(t\geq 0)$ on E, $f\in C(J \times E,E)$, $J=[0,b]$, $b>0$ is a constant, $0< t_{1}< t_{2}<\cdots <t _{b}$, $p\in \mathbb{N}$, $c_{k}$ are real numbers, $c_{k}\neq 0, k=1,2, \ldots ,p, u_{0}\in E$.

In [15], Vikram Singh et al. investigated the existence and uniqueness of mild solutions for Sobolev type fractional impulsive differential systems with nonlocal conditions

$$ \textstyle\begin{cases} ^{c}D^{\beta }[Bu(t)]=Au(t)+f(t,u(t),\int _{0}^{t}K(t,s,u(s))\,ds),\quad t \in J=[0,a], t\neq t_{j}, \\ \Delta u|_{t=t_{j}}=I_{j}(u(t_{j})), \quad j=1,2,\ldots ,m,m\in \mathbb{N}, \\ {}^{L}D^{1-\beta }[Tu(0)]=u_{0}+g(u(t)), \end{cases} $$

where $^{c}D^{q}$, $^{L}D^{q}$ denote Caputo and Riemann–Liouville fractional order derivatives of order $q\in (0,1)$, respectively, by applying the monotone iterative technique coupled with the method of lower and upper solutions.

However, as far as we know, there have been few applicable results on the existence and uniqueness of solutions to the Hilfer fractional evolution equations by applying the monotone iterative technique and the method of upper and lower solutions. So far we have not seen relevant papers that study Hilfer fractional evolution equations with nonlocal problems by applying the monotone iterative technique and the method of lower and upper solutions. Motivated by these facts, in this work, we use the fixed point theorem combined with monotone iterative technique to discuss the existence of extremal mild solutions for Hilfer fractional evolution equations with nonlocal conditions

$$ \textstyle\begin{cases} D^{\nu ,\mu }_{0+}u(t)+Au(t)=f(t,u(t),Gu(t)), & t\in (0,b], \\ I^{(1-\nu )(1-\mu )}_{0+}u(0)=u_{0}+\sum_{i=1}^{m} \lambda _{i} u(\tau _{i}), & \tau _{i}\in (0,b], \end{cases} $$

(1.1)

where $D^{\nu ,\mu }_{0+}$ denotes the Hilfer fractional derivative of order μ and type ν, which will be given in the next section, $0\leq \nu \leq 1$, $\frac{1}{2}<\mu <1$, the state $u(\cdot )$ takes value in a Banach space E with norm $\|\cdot \|$ and $-A:D(A)\subset E\rightarrow E$ is the infinitesimal generator of a $C_{0}$-semigroup $\{T(t)\}_{t\geq 0}$ of uniformly bounded linear operators in E. $J=[0,b](b>0)$, $J'=(0,b]$, $f:J'\times E \times E\rightarrow E$ are given functions satisfying some assumptions, $u_{0}\in E$ and $\tau _{i}(i=1,2, \ldots ,m)$ are prefixed points satisfying $0<\tau _{1}\leq \cdots \leq \tau _{m}<b$, and $\lambda _{i}$ are real numbers. Here the nonlocal condition $I^{1-\gamma }_{0+}u(0)=u_{0}+\sum_{i=1}^{m} \lambda _{i} u(\tau _{i})$ can be applied in a physical problem with better effect than the initial condition $I^{1-\gamma }_{0+}u(0)=u_{0}$. The operator G is given by

$$ Gu(t)= \int _{0}^{t}K(t,s)u(s)\,ds $$

(1.2)

is a Volterra integral operator with integral kernel $K\in C(\nabla , \mathbb{R^{+}})$, $\nabla =\{(t,s):0\leq s\leq t\leq b\}$. Throughout this paper, we always assume that

$$ K_{0}=\sup_{t\in J} \int _{0}^{t} K(t,s)\,ds. $$

As far as we know, the nonlocal condition can have a better effect than the initial condition $u(0)=u_{0}$ in physics application. In this article, the nonlocal function $g(u)$ can be given by $g(u)=\sum_{i=1}^{m} \lambda _{i} u(\tau _{i})$, we only assume that $\lambda _{i}(i=1,2,\ldots ,m)$ satisfy condition $(F1)$ (see in Sect. 2) without the compactness of nonlocal function. Firstly, we introduce the definition of mild solutions of problem (1.1), and then we prove the existence of extremal mild solutions of problem (1.1) by employing Sadovskii’s fixed point theorem. What is more, an existence result without using the noncompactness measure condition is obtained in ordered and weakly sequentially complete Banach spaces, which is very useful in application.

Our work is organized as follows: In Sect. 2, we review some essential facts and introduce some notations. In Sect. 3, we state and prove the existence of mild solutions for Hilfer fractional differential system (1.1). Finally, in Sect. 4, an example is given to illustrate the effectiveness of the abstract results.

2 Preliminaries

Throughout this paper, by $C(J,E)$ and $C(J',E)$ we denote the spaces of all continuous functions from J to E and $J'$ to E, respectively. Let E be an ordered Banach space with the norm $\|\cdot \|$ and partial order ≤, whose positive cone $P=\{x\in E: x\geq \theta \}$ is normal with normal constant N.

Let $\gamma =\nu +\mu -\nu \mu$, then $1-\gamma =(1-\nu )(1-\mu )$, define $C_{1-\gamma }(J,E)= \{u\in C(J',E):t^{1-\gamma }u(t) \in C(J,E) \}$. Clearly, $C_{1-\gamma }(J,E)$ is a Banach space with the norm $\|u\|_{\gamma }=\sup_{t\in J'}|t^{1-\gamma }u(t)|$. And $C_{1-\gamma }(J,E)$ is also an ordered Banach space with the partial order ≤ induced by the positive cone $P'=\{u\in C_{1-\gamma }(J,E)|u(t) \geq \theta , t\in J\}$, which is also normal with the same normal constant N.

For the convenience of discussion, we recall some definitions and basic results on fractional calculus; for more details, see [2,3,4, 8, 16].

Definition 2.1

The Riemann–Liouville fractional integral of order α of a function $f:[0,\infty )\rightarrow R$ is defined as

$$ I_{0+}^{\alpha }f(t)=\frac{1}{\varGamma (\alpha )} \int _{0}^{t}(t-s)^{ \alpha -1}f(s) \,ds,\quad t>0,\alpha >0, $$

provided the right-hand side is point-wise defined on $[0,\infty )$.

Definition 2.2

The Riemann–Liouville derivative of order α with the lower limit zero for a function $f:[0,\infty )\rightarrow R$ can be written as

$$ D^{\alpha }_{0^{+}}f(t)=\frac{1}{\varGamma (n-\alpha )}\, \frac{d^{n}}{\,dt ^{n}} \int _{0}^{t}\frac{f(s)}{(t-s)^{\alpha +1-n}}\,ds,\quad t>0,n-1< \alpha < n. $$

Definition 2.3

The Caputo fractional derivative of order α for a function $f:[0,\infty )\rightarrow R$ can be written as

$$ ^{c}D^{\alpha }_{0^{+}}f(t)=D^{\alpha }_{0^{+}} \Biggl[f(t)-\sum_{k=0} ^{n-1} \frac{t^{k}}{k!}f^{(k)}(0) \Biggr],\quad t>0, n-1< \alpha < n, $$

where $n = [\alpha ] + 1$ and $[\alpha ]$ denotes the integer part of α.

Definition 2.4

(Hilfer fractional derivative see [1])

The generalized Riemann–Liouville fractional derivative of order $0\leq \nu \leq 1$ and $0<\mu <1$ with lower limit a is defined as

$$ D^{\nu ,\mu }_{a+}f(t)=I^{\nu (1-\mu )}_{a+} \frac{d}{\,dt}I^{(1-\nu )(1- \mu )}_{a+}f(t) $$

for functions such that the expression on the right-hand side exists.

Remark 2.1

• (i) If $\nu =0$, $0<\mu <1$, and $a=0$, the Hilfer fractional derivative corresponds to the classical Riemann–Liouville fractional derivative

  $$ D^{0,\mu }_{0+}f(t)=\frac{d}{\,dt}I^{1-\mu }_{0+}f(t)=D_{0+}^{\mu }f(t). $$

• (ii) If $\nu =1$, $0<\mu <1$, and $a=0$, the Hilfer fractional derivative corresponds to the classical Caputo fractional derivative

  $$ D^{1,\mu }_{0+}f(t)=I^{1-\mu }_{0+}\, \frac{d}{\,dt}f(t)={}^{c}D^{\mu }_{0^{+}}f(t). $$

Remark 2.2

The Hilfer fractional derivative is considered as an interpolator between the Riemann–Liouville and Caputo derivatives.

Remark 2.3

For $0<\mu <1$, the Laplace transformation of Hilfer fractional derivatives is given by

$$ \mathcal{L}\bigl[D^{\mu ,\nu }_{0+}f(x)\bigr](\lambda )=\lambda ^{\mu } \mathcal{L}\bigl[f(x)\bigr](\lambda )-\lambda ^{\nu (\mu -1)} \bigl(I_{0+}^{(1-\nu )(1- \mu )}f\bigr) (0+), $$

where $(I_{0+}^{(1-\nu )(1-\mu )}f)(0+)$ is the Riemann–Liouville fractional integral of order $(1-\nu )(1-\mu )$ in the limits as $t\rightarrow 0+$, and

$$ \mathcal{L}\bigl[f(x)\bigr](\lambda )= \int _{0}^{\infty }e^{-\lambda x}f(x)\,dx. $$

(2.1)

The symbol $\alpha (\cdot )$ is the Kuratowski noncompactness measure defined on a bounded subset Ω of E. For any $\varOmega \subset C(J,E)$ and $t\in J$, set $\varOmega (t)=\{u(t) : u\in B\}\subset E$. If B is bounded in $C(J,E)$, then $\varOmega (t)$ is bounded in E, and $\alpha (\varOmega (t))\leq \alpha (\varOmega )$. As is well known, the Kuratowski measure of noncompactness has the following properties.

Lemma 2.1

([17])

Let$B\subset C(J,E)$be bounded and equicontinuous, then$\overline{co}B\subset C(J,E)$is also bounded and equicontinuous.

Lemma 2.2

([18])

LetEbe a Banach space, and let$D\subset E$be bounded. Then there exists a countable set$D_{0} \subset D$such that$\alpha (D) \leq 2\alpha (D_{0})$.

Lemma 2.3

([19])

LetEbe a Banach space, and let$\varOmega \subset C(J,E)$be equicontinuous and bounded, then$\alpha (\varOmega (t))$is continuous onJ, and$\alpha (\varOmega )=\max_{t\in J} \alpha (\varOmega (t))$.

Lemma 2.4

([20])

Let$\varOmega =\{u_{n}\}_{n=1}^{\infty }\subset C(J,E)$be a bounded and countable set, and there exists a function$m\in L^{1}(J,R^{+})$such that, for every$n\in N$,

$$ \bigl\Vert u_{n}(t) \bigr\Vert \leq m(t), \quad \textit{a.e. }t \in J. $$

Then$\alpha (\varOmega (t))$is Lebesgue integral onJ, and

$$ \alpha \biggl( \biggl\{ \int _{J} u_{n}(t)\,dt: n\in \mathbb{N} \biggr\} \biggr) \leq 2 \int _{J} \alpha \bigl(\varOmega (t)\bigr)\,dt. $$

Based on Lemma 2.12 in paper [2], we give the following the lemma.

Lemma 2.5

Assume that −Ais the infinitesimal generator of a$C_{0}$-semigroup$\{T(t)\}_{t\geq 0}$of uniformly bounded linear operators inE. If$f\in C_{1-\gamma }(J,E)$for any$u\in C_{1-\gamma }(J,E)$, a functionuis a solution of the equation

$$ \textstyle\begin{cases} D^{\nu ,\mu }_{0+}u(t)+Au(t)=f(t,u(t),Gu(t)),\quad t\in J', \\ I^{1-\gamma }_{0+}u(0)=u_{0}, \end{cases} $$

(2.2)

if and only ifusatisfies the following integral equation:

$$\begin{aligned} u(t) &=S_{\nu ,\mu }(t)u_{0}+ \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds, \end{aligned}$$

where

$$ S_{\nu ,\mu }(t)=I_{0+}^{\nu (1-\mu )}K_{\mu }(t), \qquad K_{\mu }(t)=\mu \int _{0}^{\infty }\sigma t^{\mu -1}\xi _{\mu }(\sigma )T\bigl(t^{\mu }\sigma \bigr)u_{0}\,d \sigma , $$

(2.3)

the function$\xi _{\mu }$is the function of Wright type

$$ \xi _{\mu }(\sigma )=\frac{1}{\pi \mu }\sum _{n=1}^{\infty }(-\sigma )^{n-1} \frac{ \varGamma (n\mu +1)}{n!}\sin (n\pi \mu ),\quad \sigma \in (0,\infty ). $$

Lemma 2.6

([2])

Assume thatAgenerates a$C_{0}$-semigroup$\{T(t)\}_{t \geq 0}$of uniformly bounded linear operators inEand$T(t)$is continuous in the uniform operator topology for$t>0$. That is, there exists$M\geq 1$such that$\sup_{t\in [0,+\infty )}|T(t)|\leq M$. Then the operators$S_{\nu ,\mu }(t)$and$K_{\mu }(t)$have the following properties.

• (i) For any fixed$t\geq 0$, $\{S_{\nu ,\mu }(t)\}_{t>0}$and$\{K_{\mu }(t)\}_{t>0}$are linear operators, and for any$u\in E$,

  $$ \bigl\Vert S_{\nu ,\mu }(t)u \bigr\Vert \leq \frac{Mt^{\gamma -1}}{\varGamma (\gamma )} \Vert u \Vert , \qquad \bigl\Vert K_{\mu }(t)u \bigr\Vert \leq \frac{Mt^{\mu -1}}{\varGamma (\mu )} \Vert u \Vert . $$

• (ii) The operators$S_{\nu ,\mu }(t)$and$K_{\mu }(t)$are strongly continuous for all$t\geq 0$.

• (iii) If$T(t)(t\geq 0)$is an equicontinuous semigroup, then$S_{\nu ,\mu }(t)$and$K_{\mu }(t)$are equicontinuous inEfor$t>0$.

In view of [2], from Lemma 2.6, we adopt the following definition of mild solution of system (2.2).

Definition 2.5

A function $u\in C_{1-\gamma }(J,E)$ is said to be a mild solution of (2.2) if $u_{0}\in E$, the integral equation

$$\begin{aligned} u(t) &=S_{\nu ,\mu }(t)u_{0}+ \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds \end{aligned}$$

is satisfied for all $t\in J'$.

Next, we present a useful lemma that plays an important role in our main results.

Lemma 2.7

Suppose thatAis the infinitesimal generator of a$C_{0}$-semigroup$\{T(t)\}_{t\geq 0}$of uniformly bounded linear operators inEfor$0\leq \nu \leq 1$, $0<\mu <1$, then

$$ D^{\nu ,\mu }_{0+} \bigl(S_{\nu ,\mu }(t)u_{0} \bigr)=A \bigl(S_{\nu , \mu }(t)u_{0} \bigr) $$

and

$$\begin{aligned} &D^{\nu ,\mu }_{0+} \biggl( \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds \biggr) \\ &\quad =A \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds+f\bigl(t,u(t),Gu(t)\bigr). \end{aligned}$$

(2.4)

Proof

Let $\lambda >0$, we consider the one-sided stable probability density as follows:

$$ \varpi _{\mu }(\sigma )=\frac{1}{\pi }\sum _{n=1}^{\infty }(-1)^{n-1} \sigma ^{-\mu n-1}\frac{\varGamma (n\mu +1)}{n!}\sin (n\pi \mu ),\quad \sigma \in (0,\infty ), $$

whose Laplace transform is given by

$$ \int _{0}^{\infty }e^{-\lambda \sigma }\varpi _{\mu }(\sigma )\,d\sigma =e ^{-\lambda ^{\mu }}, \quad \mu \in (0,1). $$

(2.5)

Then, using (2.5), we have

$$\begin{aligned} \bigl(\lambda ^{\mu }I-A\bigr)^{-1}u &= \int _{0}^{\infty }e^{-\lambda ^{\mu }s}T(s)u _{0}\,ds= \int _{0}^{\infty }\mu t^{\mu -1}e^{-(\lambda t)^{\mu }}T \bigl(t^{ \mu }\bigr)u\,dt \\ &= \int _{0}^{\infty } \int _{0}^{\infty }e^{-(\lambda t \sigma )}\mu t ^{\mu -1}\varpi _{\mu }(\sigma )W\bigl(t^{\mu } \bigr)u\,d\sigma \,dt \\ &=\mu \int _{0}^{\infty } \int _{0}^{\infty }e^{-\lambda \theta } \frac{ \theta ^{\mu -1}}{\sigma ^{\mu }}\varpi _{\mu }(\sigma )T \biggl( \frac{ \theta ^{\mu }}{\sigma ^{\mu }} \biggr)u\,d\theta \,d\sigma \\ &= \int _{0}^{\infty }e^{-\lambda \tau } \biggl[\mu \int _{0}^{\infty }\frac{ \tau ^{\mu -1}}{\sigma ^{\mu }}\varpi _{\mu }(\sigma )T \biggl(\frac{\tau ^{\mu }}{\sigma ^{\mu }} \biggr)u\,d\sigma \biggr]d \tau \\ &= \int _{0}^{\infty }e^{-\lambda t} \biggl[\mu \int _{0}^{\infty }\frac{t ^{\mu -1}}{\sigma ^{\mu }}\varpi _{\mu }(\sigma )T \biggl(\frac{t^{\mu }}{ \sigma ^{\mu }} \biggr)u\,d\sigma \biggr]\,dt \\ &= \int _{0}^{\infty }e^{-\lambda t} \biggl[\mu \int _{0}^{\infty }\sigma t^{\mu -1}\xi _{\mu }(\sigma )T\bigl(t^{\mu }\sigma \bigr)u\,d\sigma \biggr] \,dt \\ &= \int _{0}^{\infty }e^{-\lambda t}K_{\mu }(t)u\,dt, \end{aligned}$$

(2.6)

where $\xi _{\mu }$ is a probability density function defined on $(0,\infty )$ such that

$$ \xi _{\mu }(\sigma )=\frac{1}{\mu }\sigma ^{-1-\frac{1}{\mu }} \varpi _{\mu }\bigl(\sigma ^{-\frac{1}{\mu }}\bigr)\geq 0. $$

Since the Laplace inverse transform of $\lambda ^{\nu (\mu -1)}$ is

$$ \mathcal{L}^{-1}\bigl(\lambda ^{\nu (\mu -1)} \bigr)= \textstyle\begin{cases} \frac{t^{\nu (1-\mu )-1}}{\varGamma (\nu (1-\mu ))},&0< \nu \leq 1, \\ \delta (t), & \nu =0, \end{cases} $$

(2.7)

where $\delta (t)$ is the delta function.

From (2.6), (2.7), and the Laplace transform, it is obvious to see that

$$\begin{aligned} \mathcal{L}\bigl(S_{\nu ,\mu }(t)u_{0}\bigr) &=\mathcal{L}\bigl(I_{0+}^{\nu (1-\mu )}K _{\mu }(t)u_{0} \bigr) \\ &=\mathcal{L} \biggl(\frac{t^{\nu (1-\mu )-1}}{\varGamma (\nu (1-\mu ))} *K _{\mu }(t)u_{0} \biggr) \\ &=\mathcal{L} \bigl(\mathcal{L}^{-1}\bigl(\lambda ^{\nu (\mu -1)} \bigr) *K_{ \mu }(t)u_{0} \bigr) \\ &=\lambda ^{\nu (\mu -1)}\bigl(\lambda ^{\mu }I -A \bigr)^{-1}u_{0}, \end{aligned}$$

(2.8)

where ∗ denotes the convolution of functions. By Remark 2.2, we obtain

$$\begin{aligned} \mathcal{L}\bigl( D^{\nu ,\mu }_{0+} \bigl[S_{\nu ,\mu }(t)u_{0}\bigr]\bigr) &=\lambda ^{ \mu }\mathcal{L}\bigl(S_{\nu ,\mu }(t)u_{0}\bigr)- \lambda ^{\nu (\mu -1)}u_{0} \\ &=\lambda ^{\mu } \bigl[\lambda ^{\nu (\mu -1)}\bigl(\lambda ^{\mu }I-A\bigr)^{-1} \bigr]u_{0}-\lambda ^{\nu (\mu -1)}u_{0} \\ &=\lambda ^{\nu (\mu -1)}\bigl(\lambda ^{\mu }I-A \bigr)^{-1} \bigl[\lambda ^{\mu }-\bigl( \lambda ^{\mu }-A\bigr) \bigr]u_{0} \\ &=\lambda ^{\nu (\mu -1)}\bigl(\lambda ^{\mu }I-A \bigr)^{-1} \bigl[\lambda ^{\mu }- \lambda ^{\mu }+A \bigr]u_{0} \\ &=\lambda ^{\nu (\mu -1)}\bigl(\lambda ^{\mu }I-A \bigr)^{-1}Au_{0} \\ &=A\lambda ^{\nu (\mu -1)}\bigl(\lambda ^{\mu }I-A \bigr)^{-1}u_{0}. \end{aligned}$$

(2.9)

Combining (2.8) and (2.9) yields

$$ D^{\nu ,\mu }_{0+}\bigl[S_{\nu ,\mu }(t)u_{0} \bigr]=A\bigl[S_{\nu ,\mu }(t)u_{0}\bigr]. $$

Similarly, we have

$$ \mathcal{L} \biggl( \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds \biggr)= \mathcal{L}\bigl(K_{\mu }(t)\bigr) \cdot \mathcal{L} \bigl(f\bigl(t,u(t),Gu(t)\bigr) \bigr), $$

(2.10)

and

$$\begin{aligned} &\mathcal{L} \biggl(D^{\nu ,\mu }_{0+} \biggl[ \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds \biggr] \biggr) \\ &\quad =\lambda ^{\mu }\mathcal{L} \biggl( \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds \biggr)-\lambda ^{\nu (\mu -1)}\cdot 0 \\ &\quad =\lambda ^{\mu }\mathcal{L}\bigl(K_{\mu }(t)\bigr) \cdot \mathcal{L} \bigl(f\bigl(t,u(t),Gu(t)\bigr) \bigr) \\ &\quad =\lambda ^{\mu }\bigl(\lambda ^{\mu }I-A \bigr)^{-1}\cdot \mathcal{L} \bigl(f\bigl(t,u(t),Gu(t)\bigr) \bigr) \\ &\quad =\bigl(\lambda ^{\mu }I-A+A\bigr) \bigl(\lambda ^{\mu }I-A\bigr)^{-1}\cdot \mathcal{L} \bigl(f \bigl(t,u(t),Gu(t)\bigr) \bigr) \\ &\quad =A\bigl(\lambda ^{\mu }I-A\bigr)^{-1}\cdot \mathcal{L} \bigl(f\bigl(t,u(t),Gu(t)\bigr) \bigr)+\mathcal{L} \bigl(f \bigl(t,u(t),Gu(t)\bigr) \bigr). \end{aligned}$$

(2.11)

Thus, it follows from (2.10) and (2.11) that

$$\begin{aligned} &D^{\nu ,\mu }_{0+} \biggl[ \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds \biggr] \\ &\quad =A \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds+f\bigl(t,u(t),Gu(t)\bigr), \end{aligned}$$

(2.12)

which completes the proof of Lemma 2.7. □

For the convenience of discussion, we assume the following:

• (H0) Assume that A generates a $C_{0}$-semigroup $\{T(t)\}_{t\geq 0}$ of uniformly bounded linear operators in E and $T(t)$ is continuous in the uniform operator topology for $t>0$. That is, there exists $M\geq 1$ such that $\sup_{t\in [0,+\infty )}\|T(t)\|\leq M$.

• (H1) $\lambda _{i}>0 (i=1,2,\ldots ,m)$ and $\sum_{i=1}^{m} \lambda _{i}<\frac{\varGamma (\gamma )}{Mb^{\gamma -1}}$.

In view of [14] and [11], we present the following lemma.

Lemma 2.8

Assume that (H0) and (H1) hold. For any$u\in C_{1-\gamma }(J)$such that$f(\cdot ,u,Gu)\in C_{1-\gamma }(J)$, problem (1.1) has a unique mild solution$u\in C_{1-\gamma }(J)$given by

$$\begin{aligned} u(t) ={}&S_{\nu ,\mu }(t)\overline{\varTheta } u_{0}+\sum_{i=1}^{m}\lambda _{i}S_{\nu ,\mu }(t)\overline{\varTheta } \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds \\ &{}+ \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds, \end{aligned}$$

(2.13)

where$\overline{\varTheta }= [I-\sum_{i=1}^{m}\lambda _{i}S_{\nu , \mu }(\tau _{i}) ]^{-1}$.

Proof

By assumption (H0), we have

$$ \Biggl\Vert \sum_{i=1}^{m} \lambda _{i} S_{\nu ,\mu }(t) \Biggr\Vert \leq \sum _{i=1}^{m} \vert \lambda _{i} \vert \cdot \bigl\Vert S_{\nu ,\mu }(t) \bigr\Vert \leq \sum _{i=1}^{m} \vert \lambda _{i} \vert \frac{Mb^{\gamma -1}}{\varGamma (\gamma )}< 1. $$

By operator spectrum theorem, the operator $\overline{\varTheta }:= (I-\sum_{i=1}^{m} \lambda _{i} S_{\nu ,\mu }(\tau _{i})) )^{-1}$ exists and is bounded. Furthermore, by Neumann’s expression, we obtain

$$ \Vert \overline{\varTheta } \Vert \leq \sum_{i=0}^{\infty } \Biggl\Vert \sum_{i=1}^{m} \lambda _{i} S_{\nu ,\mu }(\tau _{i}) \Biggr\Vert ^{n}=\frac{1}{1- \Vert \sum_{i=1}^{m} \lambda _{i} S_{\nu ,\mu }(\tau _{i}) \Vert }\leq \frac{1}{1-\frac{Mb^{ \gamma -1}}{\varGamma (\gamma )}\sum_{i=1}^{m} \lambda _{i}}. $$

According to Definition 2.5, a solution of system (2.2) can be expressed by

$$\begin{aligned} u(t) &=S_{\nu ,\mu }(t)I^{1-\gamma }_{0+}u(0)+ \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds. \end{aligned}$$

(2.14)

Next, we substitute $t=\tau _{i}$ into (2.13), and by applying $\lambda _{i}$ to both sides of (2.13), we have

$$\begin{aligned} \lambda _{i}u(\tau _{i}) &=\lambda _{i}S_{\nu ,\mu }(\tau _{i})I^{1-\gamma }_{0+}u(0)+ \lambda _{i} \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s) f\bigl(s,u(s),Gu(s)\bigr)\,ds. \end{aligned}$$

(2.15)

Thus, we have

$$\begin{aligned} \begin{aligned} I^{1-\gamma }_{0+}u(0) &= u_{0}+\sum _{i=1}^{m} \lambda _{i}u(\tau _{i}) \\ &=u_{0}+\sum_{i=1}^{m} \lambda _{i}S_{\nu ,\mu }(\tau _{i})I^{1-\gamma }_{0+}u(0)+ \sum_{i=1}^{m}\lambda _{i} \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds \\ &=u_{0}+\sum_{i=1}^{m} \lambda _{i}S_{\nu ,\mu }(\tau _{i})I^{1-\gamma }_{0+}u(0)+ \sum_{i=1}^{m}\lambda _{i} \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds. \end{aligned} \end{aligned}$$

Since $I-\sum_{i=1}^{m}\lambda _{i}S_{\nu ,\mu }(\tau _{i})$ has a bounded inverse operator Θ̅, it implies

$$\begin{aligned} &I^{1-\gamma }_{0+}u(0) \\ &\quad = \Biggl[I-\sum _{i=1}^{m}\lambda _{i}S_{\nu , \mu }( \tau _{i}) \Biggr]^{-1} \Biggl(u_{0}+\sum _{i=1}^{m}\lambda _{i} \int _{0} ^{\tau _{i}}K_{\mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds \Biggr) \\ &\quad =\overline{\varTheta } u_{0}+\sum_{i=1}^{m} \lambda _{i} \int _{0}^{\tau _{i}}\overline{\varTheta } K_{\mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds. \end{aligned}$$

(2.16)

Submitting (2.1) to (2.14), we obtain that (2.13). It implies that u is also a solution of the integral of Eq. (2.13) when u is a solution of system (2.12).

The necessity has been proved. Next, we will prove its sufficiency. Applying $I^{1-\gamma }_{0+}$ to both sides of (2.12), and by Lemma 2.7, we have

$$\begin{aligned} I^{1-\gamma }_{0+}u(t) ={}&I^{1-\gamma }_{0+} \Biggl(S_{\nu ,\mu }(t)\overline{ \varTheta } u_{0}+\sum _{i=1}^{m}\lambda _{i}S_{\nu ,\mu }(t) \overline{ \varTheta } \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s) f\bigl(s,u(s),Gu(s)\bigr)\,ds \\ &{}+ \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds \Biggr). \end{aligned}$$

Therefore, we have

$$\begin{aligned} &\lim_{t\rightarrow 0}I^{1-\gamma }_{0+}u(t) \\ &\quad =\lim_{t\rightarrow 0}I ^{1-\gamma }_{0+}S_{\nu ,\mu }(t) \overline{\varTheta } u_{0}+\sum_{i=1} ^{m}\lambda _{i}\lim_{t\rightarrow 0}I^{1-\gamma }_{0+}S_{\nu ,\mu }(t) \overline{ \varTheta } \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s) f\bigl(s,u(s),Gu(s)\bigr)\,ds \\ &\quad =I^{1-\gamma }_{0+}(\lim_{t\rightarrow 0}S_{\nu ,\mu }(t) (\overline{ \varTheta } u_{0})+I^{1-\gamma }_{0+}\lim _{t\rightarrow 0}S_{\nu ,\mu }(t) \sum_{i=1}^{m} \lambda _{i}\overline{\varTheta } \int _{0}^{\tau _{i}}K_{ \mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds \\ &\quad =I^{1-\gamma }_{0+} \biggl(\frac{\overline{\varTheta } u_{0}}{\varGamma ( \gamma )}t^{\gamma -1} \biggr)+I^{1-\gamma }_{0+} \biggl(\frac{\sum_{i=1} ^{m}\lambda _{i}\overline{\varTheta }\int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s) f(s,u(s),Gu(s))\,ds}{\varGamma (\gamma )}t^{\gamma -1} \biggr) \\ &\quad =\overline{\varTheta } u_{0}+\sum_{i=1}^{m} \lambda _{i}\overline{\varTheta } \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds. \end{aligned}$$

(2.17)

Substituting $t=\tau _{i}$ into (2.12), we have

$$\begin{aligned} \begin{aligned} u(\tau _{i}) ={}&S_{\nu ,\mu }(\tau _{i})\overline{ \varTheta } u_{0}+\sum_{i=1} ^{m}\lambda _{i}S_{\nu ,\mu }(\tau _{i})\overline{\varTheta } \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s) f\bigl(s,u(s),Gu(s)\bigr)\,ds \\ &{}+ \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds. \end{aligned} \end{aligned}$$

Then we obtain

$$\begin{aligned} &u_{0}+ \sum_{i=1}^{m} \lambda _{i}u(\tau _{i}) \\ &\quad = u_{0}+\sum _{i=1}^{m} \lambda _{i}S_{\nu ,\mu }( \tau _{i})\overline{\varTheta } u_{0}+\sum _{i=1} ^{m}\lambda _{i}\sum _{i=1}^{m}\lambda _{i}S_{\nu ,\mu }( \tau _{i})\overline{ \varTheta } \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s) f\bigl(s,u(s),Gu(s)\bigr)\,ds \\ &\qquad {}+\sum_{i=1}^{m}\lambda _{i} \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds \\ &\quad = \Biggl(I+\sum_{i=1}^{m}\lambda _{i}S_{\nu ,\mu }(\tau _{i})\overline{ \varTheta } \Biggr) \Biggl(u_{0}+\sum_{i=1}^{m} \lambda _{i} \int _{0}^{\tau _{i}}K _{\mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds \Biggr) \\ &\quad = \Biggl(\overline{\varTheta }^{-1}+\sum _{i=1}^{m}\lambda _{i}S_{\nu , \mu }( \tau _{i}) \Biggr) \Biggl(\overline{\varTheta } u_{0}+\sum _{i=1}^{m} \lambda _{i} \overline{\varTheta } \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds \Biggr) \\ &\quad = \overline{\varTheta } u_{0}+\sum_{i=1}^{m} \lambda _{i}\overline{\varTheta } \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds. \end{aligned}$$

(2.18)

It follows from (2.16) and (2.17) that $I^{1-\gamma }_{0+}u(0)=u_{0}+ \sum_{i=1}^{m} \lambda _{i} u(\tau _{i})$.

Next, by using $D^{\nu ,\mu }_{0+}$ to both sides of (2.12) and Lemma 2.9, we have

$$\begin{aligned} D^{\nu ,\mu }_{0+}u(t)={}&D^{\nu ,\mu }_{0+} \Biggl[S_{\nu ,\mu }(t)\overline{ \varTheta } u_{0}+\sum _{i=1}^{m}\lambda _{i}S_{\nu ,\mu }(t) \overline{ \varTheta } \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s) f\bigl(s,u(s),Gu(s)\bigr)\,ds \\ &{}+ \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds \Biggr] \\ ={}&D^{\nu ,\mu }_{0+} \Biggl[S_{\nu ,\mu }(t)\overline{\varTheta } u_{0}+ \sum_{i=1}^{m} \lambda _{i}S_{\nu ,\mu }(t)\overline{\varTheta } \int _{0} ^{\tau _{i}}K_{\mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds \Biggr] \\ &{}+D^{\nu ,\mu }_{0+} \biggl[ \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds \biggr] \\ ={}& \Biggl[\overline{\varTheta } u_{0}+\sum _{i=1}^{m}\lambda _{i}\overline{ \varTheta } \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s) f\bigl(s,u(s),Gu(s)\bigr)\,ds \Biggr]D^{\nu ,\mu }_{0+} \bigl[S_{\nu ,\mu }(t) \bigr] \\ &{}+D^{\nu ,\mu }_{0+} \biggl[ \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr) \biggr] \\ ={}& \Biggl[\overline{\varTheta } u_{0}+\sum _{i=1}^{m}\lambda _{i}\overline{ \varTheta } \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s) f\bigl(s,u(s),Gu(s)\bigr)\,ds \Biggr]AS_{\nu ,\mu }(t) \\ &{}+A \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds+f\bigl(t,u(t),Gu(t)\bigr) \\ ={}&A \Biggl(S_{\nu ,\mu }(t)\overline{\varTheta } u_{0}+\sum _{i=1}^{m} \lambda _{i}S_{\nu ,\mu }(t) \overline{\varTheta } \int _{0}^{\tau _{i}}K_{ \mu }(\tau _{i}-s) f\bigl(s,u(s),Gu(s)\bigr)\,ds \\ &{}+ \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds \Biggr)+f\bigl(t,u(t),Gu(t)\bigr) \\ ={}&Au(t)+f\bigl(t,u(t),Gu(t)\bigr). \end{aligned}$$

Hence,

$$ D^{\nu ,\mu }_{0+}u(t)=Au(t)+f\bigl(s,u(t),Gu(t)\bigr). $$

This proof is completed. □

From Lemma 2.8, we adopt the following definition of a mild solution of problem (1.1).

Definition 2.6

A function $u\in C_{1-\gamma }(J,E)$ is said to be a mild solution of problem (1.1) if it satisfies the operator equation

$$\begin{aligned} u(t) ={}&S_{\nu ,\mu }(t)\overline{\varTheta } u_{0}+\sum_{i=1}^{m}\lambda _{i}S_{\nu ,\mu }(t)\overline{\varTheta } \int _{0}^{\tau _{i}}K_{\mu }(\tau _{i}-s)f\bigl(s,u(s),Gu(s)\bigr)\,ds \\ &{}+ \int _{0}^{t}K_{\mu }(t-s)f \bigl(s,u(s),Gu(s)\bigr)\,ds, \quad t\in J', \end{aligned}$$

(2.19)

where the operators $S_{\nu ,\mu }(t)$ and $K_{\mu }(t)$ are given by (2.3).

Definition 2.7

A $C_{0}$-semigroup $\{T(t)\}_{t\geq 0}$ in E is said to be positive if the order inequality $T(t)x\geq \theta$ holds for each $x\geq \theta$, $x\in E$, and $t\geq 0$.

Remark 2.4

For any $C\geq 0$, $-(A+CI)$ also generates a $C_{0}$-semigroup $S(t)=e^{-Ct}T(t)(t\geq 0)$ on E. And $S(t)(t\geq 0)$ is a positive $C_{0}$-semigroup if $T(t)(t\geq 0)$ is a positive $C_{0}$-semigroup. For details, see [18, 21].

For $u\in E$, we define two families $\{S_{\nu ,\mu }^{*}(t)\}_{t \geq }$ and $\{K_{\mu }^{*}(t)\}_{t\geq 0}$ of operators by

$$ S_{\nu ,\mu }^{*}(t)u=I_{0+}^{\nu (1-\mu )}K_{\mu }^{*}(t)u, \qquad K_{\mu }^{*}(t)u=\mu \int _{0}^{\infty }\sigma t^{\mu -1}\xi _{\mu }( \sigma )S\bigl(t^{\mu }\sigma \bigr)u\,d\sigma , $$

where $\xi _{\mu }(\sigma )$ is given by (2.3).

Since $T(t)(t\geq 0)$ is positive, by Remark 2.4, it is easy to know that $S(t)(t\geq 0)$ is also positive. And by the definition of $\xi _{\mu }(\sigma )$, the operators $S_{\nu ,\mu }^{*}(t)$ and $~K_{\mu }^{*}(t)$ are also positive for all $t\geq 0$.

To prove our main result, for any $C>0$, we consider the following system:

$$ \textstyle\begin{cases} D^{\nu ,\mu }_{0+}u(t)+(A+CI)u(t)=f(t,u(t),Gu(t))+Cu(t), \quad t\in (0,b], \\ I^{(1-\nu )(1-\mu )}_{0+}u(0)=u_{0}+\sum_{i=1}^{m} \lambda _{i} u(\tau _{i}), \quad \tau _{i}\in (0,b]. \end{cases} $$

(2.20)

First, we assume the following:

• (F0) For any $C\geq 0$, $-(A+CI)$ also generates a $C_{0}$-semigroup $S(t)=e^{-Ct}T(t)(t\geq 0)$ on E and $S(t)$ is continuous in the uniform operator topology for $t>0$. That is, there exists $M^{*} \geq 1$ such that $\sup_{t\in [0,+\infty )}|S(t)|\leq M^{*}$.

• (F1) $\lambda _{i}>0 (i=1,2,\ldots ,m)$ and $\sum_{i=1}^{m} \lambda _{i}<\frac{\varGamma (\gamma )}{M^{*}b^{\gamma -1}}$.

By assumption (F1), we have

$$ \Biggl\Vert \sum_{i=1}^{m} \lambda _{i} S_{\nu ,\mu }^{*}(t) \Biggr\Vert \leq \frac{M ^{*}b^{\gamma -1}}{\varGamma (\gamma )}\sum_{i=1}^{m} \lambda _{i}< 1. $$

By operator spectrum theorem, the operator $I-\sum_{i=1}^{m} \lambda _{i} S_{\nu ,\mu }^{*}(\tau _{i}))$ has a bounded inverse operator

$$ \varTheta := \Biggl(I-\sum_{i=1}^{m} \lambda _{i} S_{\nu ,\mu }^{*}(\tau _{i})\Biggr) )^{-1}. $$

Furthermore, by Neumann’s expression, Θ̅ can be expressed by

$$ \varTheta =\sum_{i=0}^{\infty } \Biggl(\sum _{i=1}^{m} \lambda _{i} S_{\nu , \mu }^{*}(\tau _{i}) \Biggr)^{n}. $$

By the positivity of $C_{0}$-semigroup $S(t)(t\geq 0)$, it is easy to know that $S_{\nu ,\mu }^{*}(t)$ is positive, we have

$$ \varTheta u=\sum_{i=0}^{\infty } \Biggl(\sum _{i=1}^{m} \lambda _{i} S_{ \nu ,\mu }^{*}(\tau _{i}) \Biggr)^{n} u\geq u\geq \theta ,\quad \forall u \geq \theta . $$

So, Θ is a positive operator, and

$$ \Vert \varTheta \Vert \leq \sum_{i=0}^{\infty } \Biggl\Vert \sum_{i=1}^{m} \lambda _{i} S _{\nu ,\mu }^{*}(\tau _{i}) \Biggr\Vert ^{n}=\frac{1}{1- \Vert \sum_{i=1}^{m} \lambda _{i} S_{\nu ,\mu }^{*}(\tau _{i}) \Vert }\leq \frac{1}{1-\frac{M^{*}b^{ \gamma -1}}{\varGamma (\gamma )}\sum_{i=1}^{m} \lambda _{i}}. $$

In view of Lemma 2.8, we present the following lemma.

Lemma 2.9

Assume that (F0) and (F1) hold. For any$u\in C_{1-\gamma }(J)$such that$f(\cdot ,u,Gu)\in C_{1-\gamma }(J)$, problem (2.20) has a unique mild solution$u\in C_{1-\gamma }(J)$given by

$$\begin{aligned} u(t) ={}&S_{\nu ,\mu }^{*}(t)\overline{\varTheta } u_{0}+\sum_{i=1}^{m} \lambda _{i}S_{\nu ,\mu }^{*}(t)\overline{\varTheta } \int _{0}^{\tau _{i}}K _{\mu }^{*}( \tau _{i}-s)\bigl[f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\ &{}+ \int _{0}^{t}K_{\mu }^{*}(t-s) \bigl[f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds, \end{aligned}$$

(2.21)

where$\overline{\varTheta }= [I-\sum_{i=1}^{m}\lambda _{i}S_{\nu , \mu }^{*}(\tau _{i}) ]^{-1}$.

From Lemma 2.9 and Definition 2.7, we state the following definition of a mild solution of problem (2.20).

Definition 2.8

A function $u\in C_{1-\gamma }(J,E)$ is said to be a mild solution of problem (2.20) if, for any $u\in C_{1-\gamma }(J,E)$, the integral equation

$$\begin{aligned} u(t) ={}&S_{\nu ,\mu }^{*}(t)\overline{\varTheta } u_{0}+\sum_{i=1}^{m} \lambda _{i}S_{\nu ,\mu }^{*}(t)\overline{\varTheta } \int _{0}^{\tau _{i}}K _{\mu }^{*}( \tau _{i}-s)\bigl[f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\ &{}+ \int _{0}^{t}K_{\mu }^{*}(t-s) \bigl[f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \end{aligned}$$

is satisfied.

In the following, we will state some lemmas whose proofs are similar to those of the paper [2]. Here, we omit it.

Lemma 2.10

Under assumption (F0), the operators$S_{\nu ,\mu }^{*}(t)$and$K_{\mu }^{*}(t)$have the following properties:

• (i) For any fixed$t>0$, $\{K_{\mu }^{*}(t)\}_{t>0}$and$\{S_{\nu , \mu }^{*}(t)\}_{t>0}$are linear operators, and for any$u\in E$,

  $$ \bigl\Vert K_{\mu }^{*}(t) \bigr\Vert \leq \frac{M^{*}t^{\mu -1}}{\varGamma (\mu )}, \qquad \bigl\Vert S_{\nu ,\mu }^{*}(t) \bigr\Vert \leq \frac{M^{*}t^{\gamma -1}}{\varGamma (\gamma )}. $$

• (ii) The operators$\{K_{\mu }^{*}(t)\}_{t>0}$and$\{S_{\nu ,\mu } ^{*}(t)\}_{t>0}$are strongly continuous for$t>0$.

• (iii) If$S(t)(t\geq 0)$is an equicontinuous semigroup, then$S_{\nu ,\mu }^{*}(t)$and$K_{\mu }^{*}(t)$are equicontinuous inEfor$t>0$.

To end this section, we state a fixed point theorem, which plays a major role in the proof of our main results.

Lemma 2.11

(Sadovskii’s fixed point theorem)

LetDbe a convex, closed, and bounded subset of a Banach spaceEand$Q:D\rightarrow D$be a condensing map. ThenQhas one fixed point inD.

Lemma 2.12

([22])

Let$a\geq 0$, $\mu >0$, $c(t)$, and$u(t)$be the nonnegative locally integrable functions on$0\leq t< T<+\infty$such that

$$ u(t)\leq c(t)+a \int _{0}^{t}(t-s)^{\mu -1}u(s)\,ds, $$

then

$$ u(t)\leq c(t)+ \int _{0}^{t} \Biggl[\sum _{n=1}^{\infty }\frac{(a\varGamma ( \mu ))^{n}}{\varGamma (n\mu )}(t-s)^{n\mu -1}c(s) \Biggr]\,ds, \quad 0\leq t< T. $$

3 Main results

In this section, we discuss the existence of extremal mild solutions for problem (1.1).

Definition 3.1

An abstract function $u\in C_{1-\gamma }(J,E)$ is called a solution of problem (1.1) if $u(t)$ satisfies all the equalities of (1.1).

Definition 3.2

If the function $v_{0}\in C_{1-\gamma }(J,E)$ satisfies

$$ \textstyle\begin{cases} D^{\nu ,\mu }_{0+}v_{0}(t) +Av_{0}(t)\leq f(t,v_{0}(t),Gv_{0}(t)), \quad t\in J, \\ I^{1-\gamma }_{0+}v_{0}(0)\leq u_{0}+\sum_{i=1}^{m} \lambda _{i} v_{0}( \tau _{i}), \end{cases} $$

(3.1)

then $v_{0}$ is said to be a lower solution of problem (1.1). If all the inequalities in (3.1) are reversed, then $v_{0}$ is called an upper solution of problem (1.1).

Theorem 3.1

Assume thatEis an ordered Banach space, its positive conePis normal, and −Agenerates a positive$C_{0}$-semigroup$\{T(t)\}_{t \geq 0}$onE, $f\in C(J\times E\times E, E)$, and$u_{0}\in E$. If problem (1.1) has a lower solution$v_{0}\in C_{1-\gamma }(J,E)$and an upper solution$w_{0}\in C_{1-\gamma }(J,E)$with$v_{0}\leq w_{0}$. Suppose also that conditions (F0), (F1) and the following conditions are satisfied:

• (F2) There exists a constant$C>0$satisfying

  $$ f(t,u_{2},v_{2})-f(t,u_{1},v_{1}) \geq -C(u_{2}-u_{1}) $$

  for$\forall \in J$, and$v_{0}(t)\leq u_{1}\leq u_{2}\leq w_{0}(t)$, $Gv_{0}(t)\leq v_{1}\leq v_{2}\leq Gw_{0}(t)$.

• (F3) There exists a constant$L>0$satisfying

  $$ \alpha \bigl(\bigl\{ f(t,u_{n},v_{n})\bigr\} \bigr)\leq L\bigl(\alpha \bigl(\{u_{n}\}\bigr)+\alpha \bigl(\{v _{n}\}\bigr)\bigr) $$

  for$\forall t\in J$, and increasing or decreasing monotonic sequences$\{u_{n}\}\subset [v_{0}(t),w_{0}(t)]$and$\{v_{n}\}\subset [Gv_{0}(t),Gw _{0}(t)]$.

• (F4) Let$v_{n}=Qv_{n-1}$, $w_{n}=Qw_{n-1}$, $n=1,2,\ldots$ , such that the sequences$v_{n}(0)$and$w_{n}(0)$are convergent.

Then problem (1.1) has minimal and maximal mild solutions$\underline{u}$andu̅between$v_{0}$and$w_{0}$, which can be obtained by using the monotone iterative procedure starting from$v_{0}$and$w_{0}$respectively.

Proof

Since $C>0$, problem (1.1) can be written as system (2.20). By (2.21), we can define operator $Q:[v_{0},w_{0}]\to C_{1-\gamma }(J,E)$ as follows:

$$\begin{aligned} (Qu) (t) ={}&S_{\nu ,\mu }^{*}(t)\varTheta u_{0}+\sum_{i=1}^{m}\lambda _{i}S _{\nu ,\mu }^{*}(t)\varTheta \int _{0}^{\tau _{i}}K_{\mu }^{*}( \tau _{i}-s)\bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\ &{}+ \int _{0}^{t}K_{\mu }^{*}(t-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds, \quad t\in J'. \end{aligned}$$

(3.2)

Since f is continuous, it is easily seen that the map $Q:[v_{0},w _{0}]\to C_{1-\gamma }(J,E)$ is continuous. And by Lemma 2.9, the mild solutions of problem (1.1) are equivalent to the fixed points of the operator Q. We will divide the proof in the following steps.

Step 1. We show that $Q:[v_{0},w_{0}]\rightarrow C_{1-\gamma }(J,E)$ is an increasing monotone operator.

In fact, for $\forall t\in J'$, $v_{0}(t)\leq u\leq v\leq w_{0}$, by assumptions (F2) and (F3), we have

$$\begin{aligned} f\bigl(s,v_{0}(s),Gv_{0}(s)\bigr)+Cv_{0}(s) \leq f\bigl(s,w_{0}(s),Gw_{0}(s)\bigr)+Cw_{0}(s). \end{aligned}$$

So

$$\begin{aligned} &\int _{0}^{t}K_{\mu }^{*}(t-s) \bigl[f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds\\ &\quad \leq \int _{0} ^{t}K_{\mu }^{*}(t-s) \bigl[f\bigl(s,v(s),Gv(s)\bigr)+Cv(s)\bigr]\,ds. \end{aligned}$$

Thus, from (3.2) we have $Qu\leq Qv$.

Step 2. We show that $v_{0}\leq Qv_{0}$ and $Qw_{0}\leq w_{0}$. Let $h(t)= D^{\nu ,\mu }_{0+}v_{0}(t)+Av_{0}(t)+Cv_{0}(t)$, $h\in C_{1- \gamma }(J,E)$, and $h(t)\leq f(t,v_{0}(t),Gv_{0}(t))+Cv_{0}(t)$, $t \in J'$. By Definitions 2.7 and 3.2, we have

$$\begin{aligned} v_{0}(t) ={}&S_{\nu ,\mu }^{*}(t)v_{0}(0)+ \int _{0}^{t}K_{\mu }^{*}(t-s)h(s) \,ds \\ \leq {}&S_{\nu ,\mu }^{*}(t)\varTheta u_{0}+\sum _{i=1}^{m}\lambda _{i}S _{\nu ,\mu }^{*}(t)\varTheta \int _{0}^{\tau _{i}}K_{\mu }^{*}( \tau _{i}-s) \bigl[f\bigl(s,v_{0}(s),Gv_{0}(s) \bigr)+Cv_{0}(s)\bigr]\,ds \\ &{}+ \int _{0}^{t}K_{\mu }^{*}(t-s) \bigl[f\bigl(s,v_{0}(s),Gv_{0}(s)\bigr)+Cv_{0}(s) \bigr]\,ds \\ ={}&Qv_{0}(t), \quad t\in J'. \end{aligned}$$

It implies that $v_{0}\leq Qv_{0}$. Similarly, it can prove that $Qw_{0}\leq w_{0}$. Thus, $Q:[v_{0},w_{0}]\to [v_{0},w_{0}]$ is a continuous increasing monotone operator.

Now, we define two sequences $\{v_{n}\}$ and $\{w_{n}\}$ in $[v_{0},w_{0}]$ by the iterative scheme

$$ v_{n}=Qv_{n-1},\qquad w_{n}=Qw_{n-1}, \quad n=1,2,\dots . $$

(3.3)

Then, from the monotonicity of Q, we have

$$ v_{0}\leq v_{1}\leq v_{2}\leq \cdots \leq v_{n}\leq \cdots \leq w_{n} \leq \cdots \leq w_{2}\leq w_{1}\leq w_{0}. $$

(3.4)

Step 3. We prove that $\{v_{n}\}$ and $\{w_{n}\}$ are convergent in $J'$.

Let $B=\{v_{n}: n\in \mathbb{N}\}$ and $B_{0}=\{v_{n-1}: n\in \mathbb{N}\}$. Then $B=Q(B_{0})$. From $B_{0}=B\cup \{v_{0}\}$ it follows that $\alpha (B_{0}(t))=\alpha (B(t))$ for $t\in J'$. Let $\varphi (t):=\alpha (B(t))$, $t\in J'$, we will show that $\varphi (t) \equiv 0$ in $J'$.

For $t\in J'$, by (1.2) and Lemma 2.4, we get

$$\begin{aligned} \alpha \bigl(G(B_{0}) (t)\bigr) &=\alpha \biggl( \biggl\{ \int _{0}^{t}K(t,s)v_{n-1}(s)\,ds: n \in \mathbb{N} \biggr\} \biggr) \\ &\leq 2K_{0} \int _{0}^{t}\alpha \bigl(B_{0}(s) \bigr)\,ds \\ &=2K_{0} \int _{0}^{t} \varphi (s)\,ds, \end{aligned}$$

therefore

$$ \int _{0}^{t}\alpha \bigl(G(B_{0}) (s)\bigr)\,ds\leq 2bK_{0} \int _{0}^{t} \varphi (s)\,ds. $$

For $t\in J'$, from (3.2), using Lemma 2.2, assumptions (F3) and (F4), we have

$$\begin{aligned} \varphi (t) ={}&\alpha \bigl(B(t)\bigr)=\alpha \bigl(Q(B_{0}) (t) \bigr) \\ ={}&\alpha \Biggl( \Biggl\{ S_{\nu ,\mu }^{*}(t)\varTheta u_{0}\\ &{}+\sum_{i=1} ^{m} \lambda _{i}S_{\nu ,\mu }^{*}(t)\varTheta \int _{0}^{\tau _{i}}K_{ \mu }^{*}( \tau _{i}-s) \bigl[f\bigl(s,v_{n-1}(s),Gv_{n-1}(s) \bigr)+Cv_{n-1}(s)\bigr]\,ds \\ &{}+ \int _{0}^{t}K_{\mu }^{*}(t-s) \bigl[f\bigl(s,v_{n-1}(s),Gv_{n-1}(s)\bigr)+Cv_{n-1}(s) \bigr]\,ds \Biggr\} \Biggr) \\ \leq{}& \frac{M^{*}b^{\gamma -1}}{\varGamma (\gamma )}\\ &{}\times\alpha \Biggl( \Biggl\{ \varTheta u_{0}+ \sum_{i=1}^{m}\lambda _{i} \varTheta \int _{0}^{\tau _{i}}K_{ \mu }^{*}( \tau _{i}-s) \bigl[f\bigl(s,v_{n-1}(s),Gv_{n-1}(s) \bigr)+Cv_{n-1}(s)\bigr]\,ds \Biggr\} \Biggr) \\ &{}+\frac{2M^{*}b^{\mu -1}}{\varGamma (\mu )} \int _{0}^{t}\alpha \bigl( \bigl\{ f \bigl(s,v_{n-1}(s),Gv_{n-1}(s)\bigr)+Cv_{n-1}(s) \bigr\} \bigr)\,ds \\ \leq {}&\frac{M^{*}b^{\gamma -1}}{\varGamma (\gamma )}\alpha \bigl( \bigl\{ v _{n}(0) \bigr\} \bigr)+\frac{2M^{*}b^{\mu -1}(L+2bLK_{0}+C)}{\varGamma ( \mu )} \int _{0}^{t}\alpha \bigl(B_{0}(s) \bigr)\,ds \\ \leq {}&\frac{2M^{*}b^{\mu -1}(L+2bLK_{0}+C)}{\varGamma (\mu )} \int _{0} ^{t}\varphi (s)\,ds. \end{aligned}$$

Hence, by Lemma 2.12, $\varphi (t)\equiv 0$ in J. So, for any $t\in J$, $\{v_{n}(t)\}$ is precompact and $\{v_{n}(t)\}$ has a convergent subsequence. And by the monotonicity of (3.3), we prove that $\{v_{n}(t)\}$ itself is convergent, i.e., $\lim_{n\rightarrow \infty }v_{n}(t)=\underline{u}(t)$, $t\in J$. Similarly, $\lim_{n\rightarrow \infty }w_{n}(t)=\overline{u}(t)$, $t\in J$.

Evidently, $\{v_{n}(t)\}\in C_{1-\gamma }(J,E)$, so $\underline{u}(t)$ is bounded integrable on J. For any $t\in J$,

$$\begin{aligned} v_{n}(t) ={}&Q(v_{n-1}) \\ ={}&S_{\nu ,\mu }^{*}(t)\varTheta u_{0}+\sum _{i=1}^{m}\lambda _{i}S_{ \nu ,\mu }^{*}(t) \varTheta \int _{0}^{\tau _{i}}K_{\mu }^{*}( \tau _{i}-s) \bigl[f\bigl(s,v _{n-1}(s),Gv_{n-1}(s) \bigr)+Cv_{n-1}(s)\bigr]\,ds \\ &{}+ \int _{0}^{t}K_{\mu }^{*}(t-s) \bigl[f\bigl(s,v_{n-1}(s),Gv_{n-1}(s)\bigr)+Cv_{n-1}(s) \bigr] \,ds. \end{aligned}$$

(3.5)

If $n\to \infty$ in (3.5), by the Lebesgue dominated convergence theorem, we obtain

$$\begin{aligned} \underline{u}(t) ={}&Q\bigl(\underline{u}(t)\bigr)\\ ={}&S_{\nu ,\mu }^{*}(t) \varTheta u _{0}+\sum_{i=1}^{m} \lambda _{i}S_{\nu ,\mu }^{*}(t)\varTheta \int _{0}^{ \tau _{i}}K_{\mu }^{*}( \tau _{i}-s) \bigl[f\bigl(s,\underline{u}(s),G \underline{u}(s) \bigr)+C\underline{u}(s)\bigr]\,ds \\ &{}+ \int _{0}^{t}K_{\mu }^{*}(t-s) \bigl[f\bigl(s,\underline{u}(s),G\underline{u}(s)\bigr)+C \underline{u}(s)\bigr] \,ds. \end{aligned}$$

Thus, we have $\underline{u}(t)\in C_{1-\gamma }(J,E)$ and $\underline{u}=Q\underline{u}$. In a similar way, we can prove that there exists $\overline{u}(t)\in C_{1-\gamma }(J,E)$ such that $\overline{u}=Q\overline{u}$. Combining this with the monotonicity of (3.4), we see that $v_{0}\leq \underline{u}\leq \overline{u} \leq w_{0}$, which implies that $\underline{u}$ and u̅ are the minimal and maximal mild solutions of problem (1.1) in $[v_{0},w _{0}]$. □

Remark 3.1

If we replace positive cone P is normal by positive cone P is regular, then the conclusion in Theorem 3.1 is also valid. For more details, see [14].

As a supplement to Theorem 3.1, we further discuss the existence of mild solutions for problem (1.1) in a weakly sequentially complete Banach space, we only need to verify that conditions (F1) and (F2) are satisfied.

Corollary 3.1

Assume thatEis an ordered and weakly sequentially complete Banach space, its positive conePis normal, and −Agenerates a positive$C_{0}$-semigroup$\{T(t)\}_{t\geq 0}$onE, $f\in C(J\times E \times E, E)$, and$u_{0}\in E$. If problem (1.1) has a lower solution$v_{0}\in C_{1-\gamma }(J,E)$and an upper solution$w_{0}\in C_{1- \gamma }(J,E)$with$v_{0}\leq w_{0}$. Suppose also that conditions (F0)–(F4) are satisfied. Then problem (1.1) has minimal and maximal mild solutions$\underline{u}$andu̅between$v_{0}$and$w_{0}$, which can be obtained by a monotone iterative procedure starting from$v_{0}$and$w_{0}$, respectively.

Proof

In view of Theorem 3.1, if E is weakly sequentially complete, then conditions (F3) and (F4) hold automatically. And by Theorem 2.2 in [23], any monotonic and order bounded sequence is precompact. By the monotonicity of (3.4), it is easy to see that $v_{n}(t)$ and $w_{n}(t)$ are convergent on J. Thus, $v_{n}(0)$ and $w_{n}(0)$ are convergent, i.e., condition (F4) holds. For $t\in J$, let $\{u_{n}\} \subset [v_{0}(t),w_{0}(t)]$ and $\{v_{n}\}\subset [Gv_{0},Gw_{0}(t)]$ be two increasing or decreasing sequences. By (F2), $\{f(t,u_{n},v _{n})+Cx_{n}\}$ is an ordered monotonic and ordered bounded sequence in E. Then $\alpha (\{f(t,u_{n},v_{n})+Cx_{n}\})=0$, (F3) holds, and by Theorem 3.1 our conclusion is valid. □

Theorem 3.2

Assume thatEis an ordered Banach space, its positive conePis normal, and −Agenerates a positive and equicontinuous$C_{0}$-semigroup$\{T(t)\}_{t\geq 0}$onE, $f\in C(J\times E \times E, E)$, and$u_{0}\in E$. If problem (1.1) has a lower solution$v_{0}\in C_{1-\gamma }(J,E)$and an upper solution$w_{0}\in C_{1- \gamma }(J,E)$with$v_{0}\leq w_{0}$. Suppose also that conditions (F0)–(F3) are satisfied and

• (F5) There exists a nonnegative constant$L_{1}$with

  $$ \frac{2M^{*}b^{\mu }( L_{1}+2bL_{1}K_{0}+C)}{\varGamma (\mu )} \biggl[\frac{(b ^{\gamma -1}-\varGamma (\gamma ))M^{*}\sum_{i=1}^{m}\lambda _{i}+\varGamma ( \gamma )}{\varGamma (\gamma )(1-M^{*}\sum_{i=1}^{m}\lambda _{i})} \biggr]< 1 $$

  such that

  $$ \alpha \bigl(\bigl\{ f(t,u_{n},v_{n})\bigr\} \bigr)\leq L_{1}\bigl(\alpha \bigl(\{u_{n}\}\bigr)+\alpha \bigl( \{v _{n}\}\bigr)\bigr) $$

  for$\forall t\in J$, and an equicontinuous countable set$\{u_{n}\} \subset [v_{0}(t),w_{0}(t)]$, $\{v_{n}\}\subset [Gv_{0}(t),Gw_{0}(t)]$.

Then problem (1.1) has a minimal mild solution$\underline{u}$and maximal mild solutionu̅in$[v_{0},w_{0}]$, and

$$ v_{n}(t)\rightarrow \underline{u}(t), \qquad w_{n}(t) \rightarrow \overline{u}(t),\quad (n\rightarrow +\infty ), t\in J, $$

where$v_{n}(t)=Qv_{n-1}(t)$, $w_{n}(t)=Qw_{n-1}(t)$, which satisfy

$$ v_{0}(t)\leq v_{1}(t)\leq \cdots v_{n}(t) \leq \cdots \underline{u}(t) \leq \overline{u}(t)\leq \cdots \leq w_{n}(t)\leq \cdots w_{1}(t) \leq w_{0}(t),\quad \forall t\in J. $$

Proof

From the proof of Theorem 3.1, we know that $Q : [v_{0},w_{0}]\rightarrow [v_{0},w_{0}]$ is continuous. First, we will prove that $Q:[v_{0},w _{0}]\rightarrow C(J,E)$ is an equicontinuous operator. Since $T(t)(t\geq 0)$ is an equicontinuous $C_{0}$-semigroup, and $S(t)(t\geq 0)$ is also an equicontinuous $C_{0}$-semigroup, by the normality of the cone P, there exists $\overline{M}>0$ such that

$$ \bigl\Vert f\bigl(t,u(t),Gu(t)\bigr)+Cu(t) \bigr\Vert \leq \overline{M}, \quad u\in [v_{0},w_{0}]. $$

For any $u\in C_{1-\gamma }(J,E)$, let $y(t)=t^{1-\gamma }u(t)$, for $t_{1}=0$, $0< t_{2}\leq b$, we get

$$\begin{aligned} & \bigl\Vert y(t_{2})-y(0) \bigr\Vert \\ &\quad \leq \bigl\Vert t_{2}^{1-\gamma }S_{\nu ,\mu }^{*}(t _{2}) \bigr\Vert (\varTheta u_{0})+\sum _{i=1}^{m}\lambda _{i}\varTheta \bigl\Vert t_{2} ^{1-\gamma }S_{\nu ,\mu }^{*}(t_{2}) \bigr\Vert \int _{0}^{\tau _{i}}K_{\mu }^{*}( \tau _{i}-s) \\ &\qquad {}\times \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds\\ &\qquad {}+t_{2}^{1-\gamma } \biggl\Vert \int _{0} ^{t_{2}}K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \biggr\Vert \\ &\quad \leq \bigl\Vert t_{2}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{2}) \bigr\Vert (\varTheta u_{0})+\overline{M}\sum _{i=1}^{m}\lambda _{i}\varTheta \bigl\Vert t_{2}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{2}) \bigr\Vert \int _{0}^{\tau _{i}}K_{\mu }^{*}( \tau _{i}-s)\,ds\\ &\qquad {}+ \overline{M} \biggl\Vert \int _{0}^{t_{2}}t_{2}^{1-\gamma }K_{\mu }^{*}(t _{2}-s)\,ds \biggr\Vert \\ &\quad \rightarrow 0, \quad \text{as } t_{2}\rightarrow t_{1}=0. \end{aligned}$$

For $0< t_{1} < t_{2} \leq b$, by (3.2), we get that

$$\begin{aligned} & \bigl\Vert y(t_{2})-y(t_{1}) \bigr\Vert \\ &\quad \leq \bigl\Vert t_{2}^{1-\gamma }(Qu) (t_{2})-t_{1} ^{1-\gamma }(Qu) (t_{1}) \bigr\Vert \\ &\quad \leq \bigl\Vert t_{2}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{2})-t_{1}^{1- \gamma }S_{\nu ,\mu }^{*}(t_{1}) \bigr\Vert (\varTheta u_{0})+ \bigl\Vert t_{2} ^{1-\gamma }S_{\nu ,\mu }^{*}(t_{2})-t_{1}^{1-\gamma }S_{\nu ,\mu } ^{*}(t_{1}) \bigr\Vert \\ &\qquad {}\times \sum_{i=1}^{m}\lambda _{i}\varTheta \int _{0}^{\tau _{i}}K_{\mu } ^{*}(\tau _{i}-s) \bigl[ f\bigl(s,u(s),Gu(s) \bigr)+Cu(s)\bigr]\,ds \\ &\qquad {}+ \int _{0}^{t_{2}}t_{2}^{1-\gamma }K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\ &\qquad {}- \int _{0}^{t_{1}}t_{1}^{1-\gamma }K_{\mu }^{*}(t_{1}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\ &\quad \leq \bigl( \bigl\Vert t_{2}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{2})-t_{2} ^{1-\gamma }S_{\nu ,\mu }^{*}(t_{1}) \bigr\Vert \\ &\qquad {}+ \bigl\Vert t_{2}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{1})-t_{1}^{1-\gamma }S _{\nu ,\mu }(t_{1}) \bigr\Vert \bigr) (\varTheta u_{0})+ \bigl\Vert t_{2}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{2})-t_{1}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{1}) \bigr\Vert \\ &\qquad {}\times \sum_{i=1}^{m}\lambda _{i}\varTheta \int _{0}^{\tau _{i}}K_{\mu } ^{*}(\tau _{i}-s) \bigl[ f\bigl(s,u(s),Gu(s) \bigr)+Cu(s)\bigr]\,ds \\ &\qquad {}+ \biggl\Vert \int ^{t_{2}}_{t_{1}}t_{2}^{1-\gamma }K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \biggr\Vert \\ &\qquad {}+ \biggl\Vert \int ^{t_{1}}_{0}t_{2}^{1-\gamma }K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\ &\qquad {}- \int ^{t_{1}}_{0}t_{1}^{1-\gamma }K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \biggr\Vert \\ &\qquad {}+ \biggl\Vert \int ^{t_{1}}_{0}t_{1}^{1-\gamma }K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\ &\qquad {}- \int ^{t_{1}}_{0}t_{1}^{1-\gamma }K_{\mu }^{*}(t_{1}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \biggr\Vert \\ &\quad =J_{1}+J_{2}+J_{3}+J_{4}+J_{5}+J_{6}, \end{aligned}$$

where

$$\begin{aligned}& J_{1}= \bigl( \bigl\Vert t_{2}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{2})-t_{2}^{1- \gamma }S_{\nu ,\mu }^{*}(t_{1}) \bigr\Vert \bigr) (\varTheta u_{0}), \\& J_{2}= \bigl( \bigl\Vert t_{2}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{1})-t_{1}^{1- \gamma }S_{\nu ,\mu }^{*}(t_{1}) \bigr\Vert \bigr) (\varTheta u_{0}), \\& J_{3}= \bigl\Vert t_{2}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{2})-t_{1}^{1- \gamma }S_{\nu ,\mu }^{*}(t_{1}) \bigr\Vert \sum_{i=1}^{m}\lambda _{i} \varTheta \int _{0}^{\tau _{i}}K_{\mu }^{*}( \tau _{i}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds, \\& J_{4}= \biggl\Vert \int ^{t_{2}}_{t_{1}}t_{2}^{1-\gamma }K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \biggr\Vert , \\& J_{5}= \biggl\Vert \int ^{t_{1}}_{0}t_{2}^{1-\gamma }K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\& \hphantom{J_{5}=}{}- \int ^{t_{1}}_{0}t_{1}^{1-\gamma }K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \biggr\Vert , \\& J_{6}= \biggl\Vert \int ^{t_{1}}_{0}t_{1}^{1-\gamma }K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\& \hphantom{J_{6}=}{}- \int ^{t_{1}}_{0}t_{1}^{1-\gamma }K_{\mu }^{*}(t_{1}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \biggr\Vert . \end{aligned}$$

Here we calculate

$$ \bigl\Vert t_{2}^{1-\gamma }(Qu) (t_{2})-t_{1}^{1-\gamma }(Qu) (t_{1}) \bigr\Vert \leq \sum_{i=1}^{6} \Vert J_{i} \Vert . $$

Therefore, it is not difficult to see that $\|J_{i}\|$ tends to 0, when $t_{2}-t_{1}\rightarrow 0$, $i=1,2,\ldots ,6$.

For $J_{1}$, by Lemma 2.10, we get

$$\begin{aligned} J_{1}&= \bigl( \bigl\Vert t_{2}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{2})-t_{2}^{1- \gamma }S_{\nu ,\mu }^{*}(t_{1}) \bigr\Vert \bigr) (\varTheta u_{0})\\ &\leq \bigl\Vert t_{2}^{1-\gamma } \bigl(S_{\nu ,\mu }^{*}(t_{2})-S_{\nu ,\mu }(t_{1}) \bigr) \bigr\Vert (\varTheta u_{0})\rightarrow 0, \quad \text{as } t_{2}\rightarrow t_{1}. \end{aligned}$$

For $J_{2}$, by Lemma 2.10, we get

$$\begin{aligned} J_{2} &= \bigl( \bigl\Vert t_{2}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{1})-t_{1} ^{1-\gamma }S_{\nu ,\mu }^{*}(t_{1}) \bigr\Vert \bigr) (\varTheta u_{0}) \\ &\leq \frac{M^{*}b^{\gamma -1}}{\varGamma (\gamma )} \bigl\Vert t_{2}^{1- \gamma }-t_{1}^{1-\gamma } \bigr\Vert \Vert \varTheta u_{0} \Vert \\ &\leq \frac{M^{*}b^{\gamma -1}}{\varGamma (\gamma )} \bigl\Vert (t_{2}-t_{1})^{1- \gamma } \bigr\Vert \Vert \varTheta u_{0} \Vert \rightarrow 0, \quad \text{as } t_{2}\rightarrow t_{1}. \end{aligned}$$

For $J_{3}$, by Lemma 2.10, we have

$$\begin{aligned} J_{3} &=\sum_{i=1}^{m} \lambda _{i}\varTheta \bigl\Vert t_{2}^{1-\gamma }S_{ \nu ,\mu }^{*}(t_{1})-t_{1}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{1}) \bigr\Vert \int _{0}^{\tau _{i}}K_{\mu }^{*}( \tau _{i}-s)\bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\ &\leq \frac{\overline{M}\sum_{i=1}^{m} \vert \lambda _{i} \vert }{1-M^{*} \sum_{i=1}^{m} \vert \lambda _{i} \vert } \bigl\Vert t_{2}^{1-\gamma }S_{\nu , \mu }^{*}(t_{1})-t_{1}^{1-\gamma }S_{\nu ,\mu }^{*}(t_{1}) \bigr\Vert \int _{0}^{\tau _{i}} K_{\mu }^{*}( \tau _{i}-s)\,ds \\ &\rightarrow 0, \quad \text{as } t_{2}\rightarrow t_{1}. \end{aligned}$$

For $J_{4}$, by Lemma 2.10, we have

$$\begin{aligned} J_{4} &= \biggl\Vert \int ^{t_{2}}_{t_{1}}t_{2}^{1-\gamma }K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \biggr\Vert \\ &\leq \overline{M} \int ^{t_{2}}_{t_{1}}t_{2}^{1-\gamma }K_{\mu }^{*}(t _{2}-s)\,ds \\ &\rightarrow 0, \quad \text{as } t_{2}\rightarrow t_{1}. \end{aligned}$$

For $J_{5}$, by Lemma 2.10, we have

$$\begin{aligned} J_{5} ={}& \biggl\Vert \int ^{t_{1}}_{0}t_{2}^{1-\gamma }K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\ &{}- \int ^{t_{1}}_{0}t_{1}^{1-\gamma }K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \biggr\Vert \\ \leq{}& \frac{2M^{*}}{\varGamma (\mu )} \int ^{t_{1}}_{0} \bigl[t_{2}^{1- \gamma }(t_{2}-s)^{\mu -1}-t_{1}^{1-\gamma }(t_{1}-s)^{\mu -1} \bigr] \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds. \end{aligned}$$

Noting that

$$\begin{aligned} &\int ^{t_{1}}_{0} \bigl[t_{2}^{1-\gamma }(t_{2}-s)^{\mu -1}-t_{1}^{1- \gamma }(t_{1}-s)^{\mu -1} \bigr] \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\ &\quad \leq \int ^{t_{1}}_{0}t_{2}^{1-\gamma }(t_{2}-s)^{\mu -1} \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \end{aligned}$$

and

$$ \int ^{t_{1}}_{0} \bigl[t_{2}^{1-\gamma }(t_{2}-s)^{\mu -1}-t_{1}^{1- \gamma }(t_{1}-s)^{\mu -1} \bigr] \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds $$

exists, and by the Lebesgue dominated convergence theorem, we have

$$\begin{aligned} &\int ^{t_{1}}_{0} \bigl[t_{2}^{1-\gamma }(t_{2}-s)^{\mu -1}-t_{1}^{1- \gamma }(t_{1}-s)^{\mu -1} \bigr] \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\ &\quad \rightarrow 0, \quad \text{as } t_{2}\rightarrow t_{1}. \end{aligned}$$

It is easy to see that $\lim_{t_{2}\rightarrow t_{1}}J_{5}=0$.

For $J_{6}$, by Lemma 2.10, we have

$$\begin{aligned} J_{6} ={}& \biggl\Vert \int ^{t_{1}}_{0}t_{1}^{1-\gamma }K_{\mu }^{*}(t_{2}-s) \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\ &{}- \int ^{t_{1}}_{0}t_{1}^{1-\gamma }K_{\mu }^{*}(t_{1}-s)f \bigl(s,u(s)\bigr)\,ds \biggr\Vert \\ \leq {}&\bigl\Vert K_{\mu }^{*}(t_{2}-s)-K_{\mu }^{*}(t_{1}-s) \bigr\Vert \int ^{t_{1}}_{0}t_{1}^{1-\gamma } \bigl[ f\bigl(s,u(s),Gu(s)\bigr)+Cu(s)\bigr]\,ds \\ \rightarrow {}&0, \quad \text{as }t_{2}\rightarrow t_{1}. \end{aligned}$$

In conclusion,

$$ \bigl\Vert y(t_{2})-y(t_{1}) \bigr\Vert \leq \bigl\Vert t_{2}^{1-\gamma }(Qu) (t_{2})-t_{1} ^{1-\gamma }(Qu) (t_{1}) \bigr\Vert \rightarrow 0, $$

as $t_{2}\rightarrow t_{1}$, i.e.,

$$ \bigl\Vert (Qu) (t_{2})-(Qu) (t_{1}) \bigr\Vert _{\gamma }\rightarrow 0, \quad \text{as } t_{2} \rightarrow t_{1}, $$

which means that $Q:[v_{0},w_{0}]\rightarrow [v_{0},w_{0}]$ is equicontinuous.

So, for any $D \subset [v_{0},w_{0}]$, $Q(D) \subset [v_{0},w_{0}]$ is bounded and equicontinuous. Therefore, by Lemma 2.2, there exists a countable set $D_{0}=\{u_{n}\} \subset D$ such that

$$ \alpha \bigl(Q(D)\bigr)\leq 2\alpha \bigl(Q(D_{0}) \bigr). $$

(3.6)

For $t \in J$, by the definition of the operator Q, we have

$$\begin{aligned} &\alpha \bigl(Q\bigl(D_{0}(t)\bigr)\bigr)\\ &\quad = \alpha \Biggl( \Biggl\{ S_{\nu ,\mu }^{*}(t)\varTheta u_{0}+\sum _{i=1}^{m}\lambda _{i}S_{\nu ,\mu }^{*}(t) \varTheta \int _{0} ^{\tau _{i}}K_{\mu }^{*}( \tau _{i}-s) \bigl[f\bigl(s,v_{n-1}(s),Gv_{n-1}(s) \bigr) \\ &\qquad{} +Cv_{n-1}(s)\bigr]\,ds+ \int _{0}^{t}K_{\mu }^{*}(t-s) \bigl[f\bigl(s,v_{n-1}(s),Gv_{n-1}(s)\bigr)+Cv _{n-1}(s)\bigr]\,ds \Biggr\} \Biggr) \\ &\quad \leq \frac{2(M^{*})^{2}\sum_{i=1}^{m}\lambda _{i}b^{\mu +\gamma -2} ( L_{1}+2bL_{1}K_{0}+C)}{\varGamma (\gamma )\varGamma (\mu )(1-M^{*}\sum_{i=1} ^{m})} \int _{0}^{\tau _{i}}\alpha \bigl(D_{0}(s) \bigr)\,ds \\ &\qquad {}+\frac{2M^{*}b^{\mu -1} ( L_{1}+2bL_{1}K_{0}+C)}{\varGamma (\mu )} \int _{0}^{t}\alpha \bigl(D_{0}(s) \bigr)\,ds \\ &\quad \leq \frac{2(M^{*})^{2}\sum_{i=1}^{m}\lambda _{i}b^{\mu +\gamma -1} ( L_{1}+2bL_{1}K_{0}+C)}{\varGamma (\gamma )\varGamma (\mu )(1-\sum_{i=1}^{m} \lambda _{i})}\alpha (D)+\frac{2M^{*}b^{\mu }( L_{1}+2bL_{1}K_{0}+C)}{ \varGamma (\mu )}\alpha (D) \\ &\quad \leq \frac{2M^{*}b^{\mu }( L_{1}+2bL_{1}K_{0}+C)}{\varGamma (\mu )} \biggl[\frac{b^{\gamma -1}M^{*}\sum_{i=1}^{m}\lambda _{i}}{\varGamma ( \gamma )(1-\sum_{i=1}^{m}\lambda _{i})}+1 \biggr]\alpha (D) \\ &\quad = \frac{2M^{*}b^{\mu }( L_{1}+2bL_{1}K_{0}+C)}{\varGamma (\mu )} \biggl[\frac{(b ^{\gamma -1}-\varGamma (\gamma ))M^{*}\sum_{i=1}^{m}\lambda _{i}+\varGamma ( \gamma )}{\varGamma (\gamma )(1-M^{*}\sum_{i=1}^{m}\lambda _{i})} \biggr] \alpha (D). \end{aligned}$$

Since $Q(D_{0})$ is bounded and equicontinuous, we know from Lemma 2.3 that

$$ \alpha \bigl(Q(D_{0})\bigr)=\max_{t\in I}\alpha \bigl(Q(D_{0}) (t)\bigr). $$

And by (3.6), we have

$$ \alpha \bigl(Q(D)\bigr)\leq \eta \alpha (D), $$

where

$$ \eta = \frac{2M^{*}b^{\mu }( L_{1}+2bL_{1}K_{0}+C)}{\varGamma (\mu )} \biggl[\frac{(b^{\gamma -1}-\varGamma (\gamma ))M^{*}\sum_{i=1}^{m}\lambda _{i}+\varGamma (\gamma )}{\varGamma (\gamma )(1-M^{*}\sum_{i=1}^{m}\lambda _{i})} \biggr]< 1. $$

Thus, $Q : [v_{0},w_{0}]\rightarrow [v_{0},w_{0}]$ is a condensing operator. By Lemma 2.11, our conclusion is valid. □

4 Applications

In this section, we present an example that illustrates the applicability of our main results.

Example 4.1

We consider the following fractional partial differential equation:

$$ \textstyle\begin{cases} D^{\nu ,\mu }_{0+}u(t,x)=\sum_{|\alpha |\leq 2m}a_{\alpha }D ^{\alpha }_{x} u(t,x)+f(t,x, u(t,x),Gu(t,x)), \quad (t,x)\in J\times \varOmega , \\ I^{(1-\nu )(1-\mu )}_{0+}u(0,x)=u_{0}+\sum_{i=1}^{m}\lambda _{i}u(\tau _{i},x), \end{cases} $$

(4.1)

where $D^{\nu ,\mu }_{0+}$ is the Hilfer fractional derivative, $0\leq \nu \leq 1$, $0<\mu <1$, $t\in J=[0,b]$, $\lambda _{i}\neq 0$, $i=1,2, \ldots ,m$, integer $\mathbb{N}\geq 1$, $\varOmega \subset \mathbb{R} ^{N}$ is a bounded domain with a sufficiently smooth boundary ∂Ω, $f:J\times E\times E\rightarrow E$ is continuous and

$$ D^{\alpha }_{x}= \biggl(\frac{\partial }{\partial x_{1}} \biggr)^{\alpha _{1}} \biggl(\frac{\partial }{\partial x_{2}} \biggr)^{\alpha _{2}} \cdots \biggl(\frac{\partial }{\partial x_{n}} \biggr)^{\alpha _{n}}, $$

$\alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})$ is an n-dimensional multi-index, $|\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}$, coefficient function $a_{\alpha }(x)\in C^{2m}(\overline{ \varOmega })$.

Let $E=L^{p}(\varOmega )$ with $1< p<\infty$, $P=\{u\in L^{p}(\varOmega )\}:u(x) \geq 0,\mbox{ q.e. }x\in \varOmega \}$, and define the operator $A:D(A)\subset E \rightarrow E$ as follows:

$$ D(A)=W^{2m,p}\cap W^{m,p}_{0}(\varOmega ),\qquad Au= \sum_{|\alpha |\leq 2m}a_{\alpha }D^{\alpha }_{x}u. $$

Then E is a Banach space, P is a normal cone of E, and −A generates a positive $C_{0}$-semigroup $T(t)(t\geq 0)$ in E (see [21]). Let $f(t,u(t),Gu(t))=f(t,x, u(t,x),Gu(t,x))$, $u_{0}=u_{0}( \cdot )$, then problem (4.1) can be written as abstract (1.1).

Theorem 4.1

If the following conditions are satisfied:

• (H1) Let$u_{0}(x)\geq 0$, $x\in \varOmega$, and there exists a function$w=w(t,x)\in C_{1-\gamma }(J\times \varOmega )$such that

  $$ \textstyle\begin{cases} D^{\nu ,\mu }_{0+}u(t,x)\geq \sum_{|\alpha |\leq 2m}a_{\alpha }D^{\alpha }_{x} u(t,x)+f(t,x, u(t,x),Gu(t,x)), \\ I^{(1-\nu )(1-\mu )}_{0+}u(0,x)=u_{0}+\sum_{i=1}^{m}\lambda _{i}u(\tau _{i},x). \end{cases} $$

  (4.2)

• (H2) There exists a constant$M>0$such that

  $$ f(t,x,u_{2},v_{2})-f(t,x,u_{1},v_{1}) \geq -M(u_{2}-u_{1}) $$

  for any$t\in J$, and$0\leq u_{1}\leq u_{2}\leq w(t,x)$, $0\leq v_{1} \leq v_{2}\leq Gw(t,x)$.

• (H3) $\lambda _{i}>0$ ($i=1,2,\ldots ,m$) and$\sum_{i=1}^{m} \lambda _{i}<\frac{\varGamma (\gamma )}{M^{*}b^{\gamma -1}}$.

• (H4) There exists a constant$L>0$such that

  $$ \alpha \bigl(\bigl\{ f(t,u_{n},v_{n})\bigr\} \bigr)\leq L\bigl(\alpha \bigl(\{u_{n}\}\bigr)+\alpha \bigl(\{v _{n}\}\bigr)\bigr) $$

  for$\forall t\in J$, and increasing or decreasing monotonic sequences$\{u_{n}\}\subset [v_{0}(t),w_{0}(t)]$and$\{v_{n}\}\subset [Gv_{0}(t),Gw _{0}(t)]$.

Then problem (4.1) has minimal and maximal mild solutions between 0 and$w(x,t)$, which can be obtained by a monotone iterative procedure starting from 0 and$w(t)$, respectively.

Proof

Assumption (H1) implies that $v_{0}\equiv 0$ and $w_{0}\equiv w(x,t)$ are lower and upper solutions of problem (4.1), respectively, and from (H2), it is easy to verify that all conditions (F1)–(F3) are satisfied under the constant $M=1$. So our conclusion follows from Theorem 3.1. □

5 Conclusions

The purpose of this paper was to obtain existence results of mild solutions for a class of evolution equations with Hilfer fractional derivative. The method is inspired by using the fixed point theorem combined with the method of lower and upper solutions, some existence result of mild solutions for Hilfer fractional evolution equations with nonlocal conditions has been obtained. Here, we do not require that $C_{0}$-semigroup $\{T(t)\} _{t\geq 0}$ is compact.