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Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions

Boundary Value Problems20172017:161

https://doi.org/10.1186/s13661-017-0892-8

Received: 24 May 2017

Accepted: 23 October 2017

Published: 3 November 2017

Abstract

In this paper, we are concerned with the existence and uniqueness of solutions for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary condition. Our results are based on the Banach contraction mapping principle and the Krasnoselskii fixed point theorem. Some examples are also given to illustrate our results.

Keywords

antiperiodic boundary value problemsimpulsivefractional integro-differential equationsexistence results

1 Introduction

Fractional differential equations appear naturally in a number of fields such as physics, chemistry, electromagnetic, engineering, control, and other branches; see [116] and the references therein. Fractional differential equations have recently gained much importance and attention. The study of fractional differential equations ranges from the theoretical aspects of the existence of solutions to the analytic and numerical methods for finding solutions.

Impulsive differential equations arising from the real world describe the dynamics of processes in which sudden discontinuous jumps occur. Such processes are naturally seen in physics, engineering, biology, and so on. Due to their significance, it is important to study the solvability of impulsive differential equations. Impulsive differential equations of fractional order have not been much studied, and many aspects of these equations are yet to be explored. The recent results on impulsive fractional differential equations can be found in [1732] and the references therein.

Recently, the boundary value problem of impulsive fractional differential equations with antiperiodic boundary conditions have been studied in the literature; see [3339]. The authors of [3638] investigated the following antiperiodic boundary value problem for impulsive differential equations of fractional order:
$$\textstyle\begin{cases} {}^{\mathrm{c}}D^{q}u(t)=f(t,u(t)), \quad t\in[0,T]\setminus \{t_{1},t_{2},\ldots,t_{m}\} , 1< q\leq2, \\ \Delta u|_{t=t_{k}}=I_{k}(u(t_{k})), \qquad \Delta u'|_{t=t_{k}}=J_{k}(u(t_{k})),\quad k=1,2,\ldots,m, \\ u(0)=-u(T),\qquad u'(0)=-u'(T), \end{cases} $$
where \({}^{\mathrm{c}}D^{q}\) is the Caputo fractional derivative of order q, \(f: [0,T]\times\mathbf{R}\to\mathbf{R}\) is continuous, \(I_{k}, J_{k}: \mathbf{R}\to\mathbf{R}\), \(\mathbf{R}=(-\infty, +\infty)\), \(0=t_{0}< t_{1}<\cdots<t_{m}<t_{m+1}=T\). By applying the Banach contraction mapping principle, Krasnoselskii fixed point theorem, Schaefer fixed point theorem, and a nonlinear alternative of the Leray-Schauder-type theorem, some existence results of solutions are obtained.
However, the existence and uniqueness of solutions to impulsive fractional differential equations for antiperiodic boundary value problems with constant coefficients seem to be rarely involved. It should be pointed out that Kilbas et al. (see (3.1.32)-(3.1.34) in [1]) obtained that the solution u of the linear fractional differential equation with constant coefficients
$$ \textstyle\begin{cases} {}^{\mathrm{c}}D^{q}u(t)+\lambda u(t)=h(t), \quad t\in[0,1], 0< q< 1, \\ u(0)=u_{0}, \end{cases} $$
(1.1)
is given by
$$u(t)=E_{q}\bigl(-t^{q}\lambda\bigr)u_{0}+ \int_{0}^{t}(t-s)^{q-1}E_{q,q} \bigl(-(t-s)^{q}\lambda \bigr)h(s)\,ds, \quad t\in[0,1], $$
where \(E_{q}\) and \(E_{q,q}\) are the so-called classical and generalized Mittag-Leffler functions.
More recently, Wang and Lin [40] studied antiperiodic boundary value problems for impulsive fractional differential equations with constant coefficients
$$ \textstyle\begin{cases} {}^{\mathrm{c}}D^{q}u(t)+\lambda u(t)=f(t,u(t)),\quad t\in J'=J\setminus \{ t_{1},t_{2},\ldots,t_{m}\}, 0< q< 1, \\ \Delta u|_{t=t_{k}}=u(t_{k}^{+})-u(t_{k}^{-})=y_{k},\quad k=1,2,\ldots,m, \\ u(0)=-u(1), \end{cases} $$
(1.2)
where \(\lambda> 0\), \(y_{k} \in\mathbf{R}\), \({}^{\mathrm{c}}D^{q}\) is the Caputo fractional derivative of order \(q \in(0, 1)\), \(f: J\times\mathbf{R}\to\mathbf{R}\), \(J=[0,1]\), and the fixed impulsive times \(t_{k}\) satisfy \(0=t_{0}< t_{1}<\cdots<t_{m}<t_{m+1}=1\). By means of fixed point theorems, some sufficient conditions on the existence and uniqueness of solutions for problem (1.2) are established under Lipschitz and nonlinear growth conditions.
Motivated by the works mentioned and many known results, in this paper, we are concerned with the existence and uniqueness of solutions for impulsive fractional integro-differential equation of mixed type with constant coefficients and antiperiodic boundary condition
$$ \textstyle\begin{cases} {}^{\mathrm{c}}D^{q}u(t)+\lambda u(t)=f(t,u(t),Tu(t),Su(t)),\quad t\in J'=J\setminus \{t_{1},t_{2},\ldots,t_{m}\}, \\ \Delta u|_{t=t_{k}}=I_{k}(u(t_{k})),\quad k=1,2,\ldots,m, \\ u(0)=-u(1), \end{cases} $$
(1.3)
where \({}^{\mathrm{c}}D^{q}\) is the Caputo fractional derivative of order \(q\in(0,1)\), \(\lambda>0\), \(0=t_{0}< t_{1}<\cdots<t_{m}<t_{m+1}=1\), \(f\in C(J\times\mathbf {R}\times\mathbf{R}\times\mathbf{R},\mathbf{R})\), \(J=[0,1]\), R is the set of real numbers, \(\Delta u|_{t=t_{k}}\) denotes the jump of \(u(t)\) at \(t=t_{k}\), that is, \(\Delta u|_{t=t_{k}}=u(t_{k}^{+})-u(t_{k}^{-})\), where \(u(t_{k}^{+})\) and \(u(t_{k}^{-})\) represent the right and left limits of \(u(t)\) at \(t=t_{k}\), respectively, T and S are the linear operators defined by
$$(Tu) (t)= \int_{0}^{t}k(t,s)u(s)\,ds\quad \mbox{and}\quad (Su) (t)= \int_{0}^{1}h(t,s)u(s)\,ds, \quad t\in J, $$
where \(k\in C(D,\mathbf{R})\), \(D= \{(t,s)\in J\times J:t\ge s \}\), and \(h\in C(J\times J,\mathbf{R})\).

At present, the concept of solutions for impulsive fractional differential equations has been argued extensively. There are some ways to consider the notion of solution for impulsive fractional differential equations; for example, see [2932]. In this paper, we adopt the formula of the solution in Lemma 2.4, which comes from [40].

This paper is arranged as follows. In Section 2, we present some definitions and preliminary lemmas. In Section 3, we establish the existence and uniqueness of solutions for the boundary value problem (1.3) by using the Banach contraction mapping principle and Krasnoselskii fixed point theorem. Some illustrated examples are presented in Section 4.

2 Preliminaries and lemmas

Let \(J_{0}=[0,t_{1}], J_{1}=(t_{1},t_{2}],\ldots, J_{m-1}=(t_{m-1},t_{m}], J_{m}=(t_{m},1]\), and
$$\begin{aligned} \mathit{PC}(J,\mathbf{R})={}&\bigl\{ u:J\rightarrow\mathbf{R}:u\in C(J_{k},\mathbf {R}), k=0,1,2,\ldots,m, \\ & u\bigl(t_{k}^{+}\bigr)\mbox{ and } u\bigl(t_{k}^{-}\bigr) \mbox{ exist}, k=1,\ldots,m, \mbox{and }u\bigl(t_{k}^{-}\bigr)= u(t_{k})\bigr\} . \end{aligned}$$
Then \(\mathit{PC}(J,\mathbf{R})\) is a Banach space with the norm \(\|u\|_{\mathit{PC}}=\sup \{|u(t)|: t\in J\}\). For a measurable function \(\mu:J\rightarrow\mathbf {R}\), define the norm
$$\|\mu\|_{L^{p}(J)}= \textstyle\begin{cases} (\int_{J}|\mu(t)|^{p}\,dt )^{\frac{1}{p}},& 1\leq p < \infty, \\ \inf_{\operatorname{mes}(\overline{J})=0}\{\sup_{t\in J\setminus\overline{J}}|u(t)|\} , & p=\infty. \end{cases} $$
Then \(L^{p}(J,\mathbf{R})\) is the Banach space of Lebesgue-measurable functions \(\mu:J\rightarrow\mathbf{R}\) with \(\|\mu\|_{L^{p}(J)}<\infty\).

Definition 2.1

([1])

The fractional integral of order α with lower limit zero for a function \(f:[0,\infty )\rightarrow\mathbf{R}\) is defined as
$$I^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}f(s)\,ds, $$
provided that the right-hand side is pointwise defined on \([0,+\infty)\).

Definition 2.2

([1])

The Caputo derivative of order α for a function \(f:[0,\infty)\rightarrow\mathbf{R}\) can be written as
$${}^{\mathrm{c}}D^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)} \biggl( \frac{d}{dt} \biggr)^{n} \int_{0}^{t}\frac{f(s)-\sum^{n-1}_{k=0}\frac {s^{k}}{k!}f^{(k)}(0)}{(t-s)^{\alpha+1-n}}\,ds, \quad t>0, n = [ \alpha] + 1, $$
where \([\alpha]\) denotes the integer part of α.

Remark 2.1

([30])

If \(f\in C^{n}[0,+\infty)\), then
$${}^{\mathrm{c}}D^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)} \int_{0}^{t}(t-s)^{n-\alpha -1}f^{(n)}(s) \,ds=I^{n-\alpha}f^{(n)}(t), \quad t>0, n = [\alpha] + 1, $$
that is, Definition 2.2 is just the usual Caputo fractional derivative. In this paper, we consider an impulsive problem, so Definition 2.2 is appropriate.

Definition 2.3

A function \(u\in \mathit{PC}(J,\mathbf{R})\) is said to be a solution of problem (1.3) if it satisfies the equation \({}^{\mathrm{c}}D^{q}u(t)+\lambda u(t)=f(t,u(t),Su(t), Tu(t))\) a.e. on \(J'\) and the conditions \(\Delta u|_{t=t_{k}}=I_{k}(u(t_{k}))\), \(k=1,\ldots,m\), and \(u(0)=-u(1)\).

Lemma 2.1

([41])

The nonnegative functions \(E_{q}\) and \(E_{q,q}\) given by
$$E_{q}(z)=\sum^{\infty}_{k=0} \frac{z^{k}}{\Gamma(qk+1)},\qquad E_{q,q}(z)=\sum^{\infty}_{k=0} \frac{z^{k}}{\Gamma(qk+q)}, $$
have the following properties:
  1. (1)
    For any \(\lambda>0\) and \(t\in J\),
    $$E_{q}\bigl(-t^{q}\lambda\bigr)\leq1, \qquad E_{q,q} \bigl(-t^{q}\lambda\bigr)\leq\frac{1}{\Gamma(q)}. $$
    Moreover,
    $$E_{q}(0)=1,\qquad E_{q,q}(0)=\frac{1}{\Gamma(q)}. $$
     
  2. (2)
    For any \(\lambda>0\) and \(t_{1}, t_{2}\in J\),
    $$\begin{aligned}& E_{q}\bigl(-{t_{2}}^{q}\lambda\bigr)\rightarrow E_{q}\bigl(-{t_{1}}^{q}\lambda\bigr) \quad \textit{as } t_{2}\rightarrow t_{1}, \\& E_{q,q}\bigl(-{t_{2}}^{q}\lambda\bigr)\rightarrow E_{q,q}\bigl(-{t_{1}}^{q}\lambda\bigr) \quad \textit{as } t_{2}\rightarrow t_{1}. \end{aligned}$$
     
  3. (3)
    For any \(\lambda>0\) and \(t_{1}, t_{2}\in J\) such that \(t_{1}\leq t_{2}\),
    $$E_{q}\bigl(-{t_{2}}^{q}\lambda\bigr)\leq E_{q}\bigl(-{t_{1}}^{q}\lambda\bigr),\qquad E_{q,q}\bigl(-{t_{2}}^{q}\lambda\bigr)\leq E_{q,q}\bigl(-{t_{1}}^{q}\lambda\bigr). $$
     

Lemma 2.2

([42])

Let M be a closed, convex, and nonempty subset of a Banach space X, and let A, B be operators such that:
  1. (1)

    \(Ax+By\in M\) whenever \(x,y\in M\).

     
  2. (2)

    A is compact and continuous.

     
  3. (3)

    B is a contraction mapping.

     
Then there exists \(z\in M\) such that \(z=Az+Bz\).

Lemma 2.3

([43])

Let X be a Banach space, and let \(J=[0,T]\). Suppose that \(W\subset \mathit{PC}(J,X)\) satisfies the following conditions:
  1. (1)

    W is a uniformly bounded subset of \(\mathit{PC}(J,X)\).

     
  2. (2)

    W is equicontinuous in \((t_{k},t_{k+1})\), \(k=0,1,\ldots ,m\), where \(t_{0}=0,t_{m+1}=T\).

     
  3. (3)

    Its t-sections \(W(t)= \{u(t):u\in W,t\in J\setminus\{t_{1},\ldots,t_{m} \} \}\), \(W(t^{+}_{k})=\{u(t^{+}_{k}):u\in W\} \), and \(W(t^{-}_{k})=\{u(t^{-}_{k}):u\in W\}\) are relatively compact subsets of X.

     
Then W is a relatively compact subset of \(\mathit{PC}(J,X)\).

Lemma 2.4

([40])

Let \(h:J\rightarrow\mathbf{R}\) be a continuous function. The function u given by
$$u(t)= \textstyle\begin{cases} \frac{-E_{q}(-\lambda)E_{q}(-t^{q}\lambda)}{1+E_{q}(-\lambda)}\sum^{m}_{i=1}\frac {y_{i}}{E_{q}(-t_{i}^{q}\lambda)}+\int_{0}^{t}(t-s)^{q-1}E_{q,q}(-(t-s)^{q}\lambda )h(s)\,ds \\ \quad {}-\frac{E_{q}(-t^{q}\lambda)}{1+E_{q}(-\lambda)}\int _{0}^{1}(1-s)^{q-1}E_{q,q}(-(1-s)^{q}\lambda)h(s)\,ds,\quad t\in J_{0}, \\ \frac{E_{q}(-t^{q}\lambda)}{1+E_{q}(-\lambda)} \{\sum^{m}_{i=1}\frac {y_{i}}{E_{q}(-t_{i}^{q}\lambda)}-\int_{0}^{1}(1-s)^{q-1}E_{q,q}(-(1-s)^{q}\lambda )h(s)\,ds \} \\ \quad {}-E_{q}(-t^{q}\lambda)\sum^{m}_{j=k+1}\frac{y_{j}}{E_{q}(-t_{j}^{q}\lambda )} \\ \quad {}+\int_{0}^{t}(t-s)^{q-1}E_{q,q}(-(t-s)^{q}\lambda)h(s)\,ds, \quad t\in J_{k}, k=1,2,\ldots,m-1, \\ \frac{E_{q}(-t^{q}\lambda)}{1+E_{q}(-\lambda)} \{\sum^{m}_{i=1}\frac {y_{i}}{E_{q}(-t_{i}^{q}\lambda)}-\int_{0}^{1}(1-s)^{q-1}E_{q,q}(-(1-s)^{q}\lambda )h(s)\,ds \} \\ \qquad {}+\int_{0}^{t}(t-s)^{q-1}E_{q,q}(-(t-s)^{q}\lambda)h(s)\,ds,\quad t\in J_{m}, \end{cases} $$
is a unique solution of the impulsive problem
$$\textstyle\begin{cases} {}^{\mathrm{c}}D^{q}u(t)+\lambda u(t)=h(t),\quad t\in J', \\ \Delta u|_{t=t_{k}}=y_{k},\quad k=1,2,\ldots,m, \\ u(0)=-u(1). \end{cases} $$
It follows from Lemma 2.4 that the solution of (1.3) can be expressed by
$$u(t)= \textstyle\begin{cases} \frac{-E_{q}(-\lambda)E_{q}(-t^{q}\lambda)}{1+E_{q}(-\lambda)}\sum^{m}_{i=1}\frac {I_{i}(u(t_{i}))}{E_{q}(-t_{i}^{q}\lambda)}+\int _{0}^{t}(t-s)^{q-1}E_{q,q}(-(t-s)^{q}\lambda)f(s,u(s),Tu(s),Su(s))\,ds \\ \quad {}-\frac{E_{q}(-t^{q}\lambda)}{1+E_{q}(-\lambda)}\int _{0}^{1}(1-s)^{q-1}E_{q,q}(-(1-s)^{q}\lambda)f(s,u(s),Tu(s),Su(s))\,ds,\quad t\in J_{0}, \\ \frac{E_{q}(-t^{q}\lambda)}{1+E_{q}(-\lambda)} \{\sum^{m}_{i=1}\frac {I_{i}(u(t_{i}))}{E_{q}(-t_{i}^{q}\lambda)}-\int _{0}^{1}(1-s)^{q-1}E_{q,q}(-(1-s)^{q}\lambda)f(s,u(s),Tu(s),Su(s))\,ds \} \\ \quad {}-E_{q}(-t^{q}\lambda)\sum^{m}_{j=k+1}\frac {I_{j}(u(t_{j}))}{E_{q}(-t_{j}^{q}\lambda)} \\ \quad +\int _{0}^{t}(t-s)^{q-1}E_{q,q}(-(t-s)^{q}\lambda)f(s,u(s),Tu(s),Su(s))\,ds, \\ \quad t\in J_{k}, k=1,2,\ldots,m-1, \\ \frac{E_{q}(-t^{q}\lambda)}{1+E_{q}(-\lambda)} \{\sum^{m}_{i=1}\frac {I_{i}(u(t_{i}))}{E_{q}(-t_{i}^{q}\lambda)}-\int _{0}^{1}(1-s)^{q-1}E_{q,q}(-(1-s)^{q}\lambda)f(s,u(s),Tu(s),Su(s))\,ds \} \\ \quad {}+\int_{0}^{t}(t-s)^{q-1}E_{q,q}(-(t-s)^{q}\lambda )f(s,u(s),Tu(s),Su(s))\,ds, \quad t\in J_{m}. \end{cases} $$

3 Main results

Theorem 3.1

Assume that conditions (H1)-(H3) hold:
(H1): 
There exist \(L_{i}(t)\in C(J,(0,+\infty))\) (\(i=1,2,3\)) such that
$$\bigl\vert f(t,u_{1},v_{1},w_{1})-f(t,u_{2},v_{2},w_{2}) \bigr\vert \leq L_{1}(t) \vert u_{1}-u_{2} \vert +L_{2}(t) \vert v_{1}-v_{2} \vert +L_{3}(t) \vert w_{1}-w_{2} \vert $$
for all \(t\in J\) and \(u_{j}, v_{j}, w_{j}\in\mathbf{R}\), \(j=1,2\).
(H2): 
There exists a constant \(L_{4}>0\) such that
$$\bigl\vert I_{k}(u)-I_{k}(v) \bigr\vert \leq L_{4} \vert u-v \vert , \quad u,v\in\mathbf{R}, k=1,2,\ldots,m. $$
(H3): 
$$\chi=\frac{3}{|1+E_{q}(-\lambda)|} \Biggl(\sum^{m}_{i=1} \frac {{L_{4}}}{|E_{q}(-t_{i}^{q}\lambda)|} +\frac{(\overline{L_{1}}+\overline{L_{2}}k_{0}+\overline{L_{3}}h_{0})}{\Gamma (q+1)} \Biggr)< 1, $$
where \(\overline{L_{j}}=\max\{L_{j}(t):t\in J\}\), \(j=1,2,3\), \(k_{0}=\max\{ |k(t,s)|:(t,s)\in D\}\), and \(h_{0}=\max\{|h(t,s)|:(t,s)\in J\times J\}\).

Then the boundary value problem (1.3) has a unique solution.

Proof

Let \(M=\sup\{|f(t,0,0,0)|:t\in J\}\), \(M'=\max \{|I_{i}(0)|: i=1,2,\ldots,m\}\), and \(B_{r}=\{u\in \mathit{PC}(J,\mathbf{R}):\|u\| _{\mathit{PC}}\leq r\}\), where
$$r\geq\frac{\sum^{m}_{i=1}\frac{M'}{|E_{q}(-t_{i}^{q}\lambda)|}+\frac{M}{\Gamma (q+1)}}{\frac{|1+E_{q}(-\lambda)|}{3}- [\sum^{m}_{i=1}\frac{{L_{4}}}{|E_{q}(-t_{i}^{q}\lambda)|}+\frac{\overline{L_{1}} +\overline{L_{2}}k_{0}+\overline{L_{3}}h_{0}}{\Gamma(q+1)} ]}. $$
Define the operator \(F:B_{r}\rightarrow \mathit{PC}(J,\mathbf{R})\) by
$$\begin{aligned} Fu(t) =&\frac{E_{q}(-t^{q}\lambda)}{1+E_{q}(-\lambda)} \Biggl\{ \sum^{m}_{i=1} \frac {I_{i}(u(t_{i}))}{E_{q}(-t_{i}^{q}\lambda)} \\ &{}- \int_{0}^{1}(1-s)^{q-1}E_{q,q} \bigl(-(1-s)^{q}\lambda \bigr)f\bigl(s,u(s),Tu(s),Su(s)\bigr)\,ds \Biggr\} \\ &{}-E_{q}\bigl(-t^{q}\lambda\bigr)\sum ^{m}_{j=k+1}\frac {I_{j}(u(t_{j}))}{E_{q}(-t_{j}^{q}\lambda)} \\ &{}+ \int_{0}^{t}(t-s)^{q-1}E_{q,q} \bigl(-(t-s)^{q}\lambda\bigr)f\bigl(s,u(s),Tu(s),Su(s)\bigr)\,ds, \\ &t\in J_{k}, k=0,1,2,\ldots,m. \end{aligned}$$
First, we show that \(F(B_{r})\subset B_{r}\). For any \(u\in B_{r}\) and \(t\in J\), by Lemma 2.1 we have
$$\begin{aligned}& \bigl\vert (Fu) (t) \bigr\vert \\& \quad \leq \bigl\vert E_{q}\bigl(-t^{q}\lambda\bigr) \bigr\vert \Biggl\vert \frac{1}{1+E_{q}(-\lambda)}\Biggl\{ \sum^{m}_{i=1} \frac{I_{i}(u(t_{i}))}{E_{q}(-t_{i}^{q}\lambda)} \\& \qquad {} - \int_{0}^{1}(1-s)^{q-1}E_{q,q} \bigl(-(1-s)^{q}\lambda \bigr)f\bigl(s,u(s),Tu(s),Su(s)\bigr)\,ds\Biggr\} -\sum^{m}_{j=k+1}\frac {I_{j}(u(t_{j}))}{E_{q}(-t_{j}^{q}\lambda)} \Biggr\vert \\& \qquad {} + \biggl\vert \int_{0}^{t}(t-s)^{q-1}E_{q,q} \bigl(-(t-s)^{q}\lambda \bigr)f\bigl(s,u(s),Tu(s),Su(s)\bigr)\,ds \biggr\vert \\& \quad \leq\frac{1}{ \vert 1+E_{q}(-\lambda) \vert }\Biggl\{ \sum^{m}_{i=1} \frac { \vert I_{i}(u(t_{i})) \vert }{ \vert E_{q}(-t_{i}^{q}\lambda) \vert }+\frac{1}{\Gamma(q)} \int _{0}^{1}(1-s)^{q-1} \bigl\vert f \bigl(s,u(s),Tu(s),Su(s)\bigr) \bigr\vert \,ds\Biggr\} \\& \qquad {} +\sum^{m}_{i=1} \frac{ \vert I_{i}(u(t_{i})) \vert }{ \vert E_{q}(-t_{i}^{q}\lambda) \vert }+\frac {1}{\Gamma(q)} \int_{0}^{t}(t-s)^{q-1} \bigl\vert f \bigl(s,u(s),Tu(s),Su(s)\bigr) \bigr\vert \,ds \\& \quad \leq\frac{1+ \vert 1+E_{q}(-\lambda) \vert }{ \vert 1+E_{q}(-\lambda) \vert }\Biggl\{ \sum^{m}_{i=1} \frac{ \vert I_{i}(u(t_{i}))-I_{i}(0) \vert +M'}{ \vert E_{q}(-t_{i}^{q}\lambda) \vert }\Biggr\} \\ & \qquad {} +\frac{1}{\Gamma(q) \vert 1+E_{q}(-\lambda) \vert } \int _{0}^{1}(1-s)^{q-1} \bigl\vert f \bigl(s,u(s),Tu(s),Su(s)\bigr)-f(s,0,0,0) \bigr\vert \,ds \\ & \qquad {} +\frac{1}{\Gamma(q) \vert 1+E_{q}(-\lambda) \vert } \int _{0}^{1}(1-s)^{q-1} \bigl\vert f(s,0,0,0) \bigr\vert \,ds \\ & \qquad {} +\frac{1}{\Gamma(q)} \int _{0}^{t}(t-s)^{q-1} \bigl\vert f \bigl(s,u(s),Tu(s),Su(s)\bigr)-f(s,0,0,0) \bigr\vert \,ds \\ & \qquad {} +\frac{1}{\Gamma(q)} \int_{0}^{t}(t-s)^{q-1} \bigl\vert f(s,0,0,0) \bigr\vert \,ds \\ & \quad \leq\frac{3}{ \vert 1+E_{q}(-\lambda) \vert }\sum^{m}_{i=1} \frac {L_{4}r+M'}{ \vert E_{q}(-t_{i}^{q}\lambda) \vert }+\frac{M}{\Gamma(q+1) \vert 1+E_{q}(-\lambda ) \vert }+\frac{M}{\Gamma(q+1)} \\ & \qquad {} +\frac{1}{\Gamma(q) \vert 1+E_{q}(-\lambda) \vert } \int _{0}^{1}(1-s)^{q-1} \bigl[L_{1}(s) \bigl\vert u(s) \bigr\vert +L_{2}(s) \bigl\vert Tu(s) \bigr\vert +L_{3}(s) \bigl\vert Su(s) \bigr\vert \bigr]\,ds \\ & \qquad {} +\frac{1}{\Gamma(q)} \int _{0}^{t}(t-s)^{q-1} \bigl[L_{1}(s) \bigl\vert u(s) \bigr\vert ++L_{2}(s) \bigl\vert Tu(s) \bigr\vert +L_{3}(s) \bigl\vert Su(s) \bigr\vert \bigr]\,ds \\ & \quad \leq\frac{3}{ \vert 1+E_{q}(-\lambda) \vert }\Biggl\{ \sum^{m}_{i=1} \frac {{L_{4}}r+M'}{ \vert E_{q}(-t_{i}^{q}\lambda) \vert }+\frac{M}{\Gamma(q+1)}\Biggr\} \\ & \qquad {} +\frac{1}{\Gamma(q) \vert 1+E_{q}(-\lambda) \vert } \int_{0}^{1}(1-s)^{q-1} ( \overline{L_{1}}r+\overline{L_{2}}k_{0}r+ \overline{L_{3}}h_{0}r)\,ds \\ & \qquad {} +\frac{1}{\Gamma(q)} \int_{0}^{t}(t-s)^{q-1}(\overline {L_{1}}r+\overline{L_{2}}k_{0}r+ \overline{L_{3}}h_{0}r)\,ds \\ & \quad \leq\frac{3}{ \vert 1+E_{q}(-\lambda) \vert }\Biggl\{ \sum^{m}_{i=1} \frac {M'}{ \vert E_{q}(-t_{i}^{q}\lambda) \vert }+\frac{M}{\Gamma(q+1)} \\ & \qquad {}+\Biggl[\sum^{m}_{i=1} \frac{{L_{4}}}{ \vert E_{q}(-t_{i}^{q}\lambda) \vert }+\frac{\overline {L_{1}}+\overline{L_{2}}k_{0}+\overline{L_{3}}h_{0}}{\Gamma(q+1)}\Biggr]r\Biggr\} \\& \quad \leq r. \end{aligned}$$
Hence \(F(B_{r})\subset B_{r}\).
Next, we show that the operator F is a contraction mapping. For any \(t\in J\) and \(u, v\in B_{r}\), we obtain
$$\begin{aligned}& \bigl\vert (Fu) (t)-(Fv) (t) \bigr\vert \\ & \quad = \Biggl\vert \frac{E_{q}(-t^{q}\lambda)}{1+E_{q}(-\lambda)} \Biggl\{ \sum ^{m}_{i=1}\frac {I_{i}(u(t_{i}))-I_{i}(v(t_{i}))}{E_{q}(-t_{i}^{q}\lambda)} \\ & \qquad {} - \int_{0}^{1}(1-s)^{q-1}E_{q,q} \bigl(-(1-s)^{q}\lambda\bigr) \bigl(f\bigl(s,u(s),Tu(s),Su(s)\bigr)-f \bigl(s,v(s),Tv(s),Sv(s)\bigr) \bigr)\,ds \Biggr\} \\ & \qquad {} -E_{q}\bigl(-t^{q}\lambda\bigr)\sum ^{m}_{j=k+1}\frac {I_{j}(u(t_{j}))-I_{j}(v(t_{j}))}{E_{q}(-t_{j}^{q}\lambda)} \\ & \qquad {} + \int_{0}^{t}(t-s)^{q-1} E_{q,q} \bigl(-(t-s)^{q}\lambda \bigr) \bigl(f\bigl(s,u(s),Tu(s),Su(s)\bigr)-f \bigl(s,v(s),Tv(s),Sv(s)\bigr)\bigr)\,ds \Biggr\vert \\ & \quad \leq \biggl(\frac{1}{1+ \vert E_{q}(-\lambda) \vert }+1 \biggr)\sum^{m}_{i=1} \frac {L_{4} \vert u(t_{i})-v(t_{i}) \vert }{ \vert E_{q}(-t_{i}^{q}\lambda) \vert }+\frac{1}{\Gamma(q) \vert 1+E_{q}(-\lambda) \vert } \\& \qquad {} \cdot \int_{0}^{1}(1-s)^{q-1} \biggl\{ L_{1}(s) \bigl\vert u(s)-v(s) \bigr\vert +L_{2}(s) \int_{0}^{s} \bigl\vert k(s,\tau) \bigr\vert \bigl\vert u(\tau)-v(\tau) \bigr\vert \,d\tau \\& \qquad {} +L_{3}(s) \int_{0}^{1} \bigl\vert h(s,\tau) \bigr\vert \bigl\vert u(\tau)-v(\tau) \bigr\vert \,d\tau \biggr\} \,ds+\frac{1}{\Gamma(q)} \int_{0}^{t}(t-s)^{q-1} \biggl\{ L_{1}(s) \bigl\vert u(s)-v(s) \bigr\vert \\& \qquad {} +L_{2}(s) \int_{0}^{s} \bigl\vert k(s,\tau) \bigr\vert \bigl\vert u(\tau)-v(\tau) \bigr\vert \,d\tau+L_{3}(s) \int _{0}^{1} \bigl\vert h(s,\tau) \bigr\vert \bigl\vert u(\tau)-v(\tau) \bigr\vert \,d\tau \biggr\} \,ds \\& \quad \leq\frac{3}{|1+E_{q}(-\lambda)|}\sum^{m}_{i=1} \frac{{L_{4}}\|u-v\| _{\mathit{PC}}}{|E_{q}(-t_{i}^{q}\lambda)|}+\frac{1}{\Gamma(q)|1+E_{q}(-\lambda)|} \\& \qquad {} \cdot \int_{0}^{1}(1-s)^{q-1} \bigl( \overline{L_{1}}\|u-v\|_{\mathit{PC}}+\overline{L_{2}}k_{0} \|u-v\|_{\mathit{PC}}+\overline {L_{3}}h_{0}\|u-v \|_{\mathit{PC}} \bigr)\,ds \\& \qquad {} +\frac{1}{\Gamma(q)} \int_{0}^{t}(t-s)^{q-1} \bigl( \overline{L_{1}}\|u-v\| _{\mathit{PC}}+\overline{L_{2}}k_{0} \|u-v\|_{\mathit{PC}}+\overline{L_{3}}h_{0}\|u-v \|_{\mathit{PC}} \bigr)\,ds \\& \quad \leq\frac{3}{|1+E_{q}(-\lambda)|} \Biggl(\sum^{m}_{i=1} \frac {{L_{4}}}{|E_{q}(-t_{i}^{q}\lambda)|}+\frac{(\overline{L_{1}}+\overline {L_{2}}k_{0}+\overline{L_{3}}h_{0})}{\Gamma(q+1)} \Biggr)\|u-v\|_{\mathit{PC}} \\& \quad =\chi\|u-v\|_{\mathit{PC}}. \end{aligned}$$
Thus \(\|Fu-Fv\|_{\mathit{PC}}\leq\chi\|u-v\|_{\mathit{PC}}\). Then from the Banach contraction mapping principle it follows that problem (1.3) has a unique solution. This completes the proof. □

Theorem 3.2

Assume that condition (H2) and the following conditions (H4)-(H5) hold:
(H4): 
There exist a function \(\mu\in L^{\frac{1}{\sigma }}(J,(0,+\infty))\) (\(0<\sigma<q<1\)) and a nondecreasing function \(\overline{\omega}\in C([0,\infty),(0,+\infty))\) such that
$$\bigl\vert f\bigl(t,u(t),Tu(t),Su(t)\bigr) \bigr\vert \leq\mu(t)\overline{ \omega}\bigl(\|u\|_{\mathit{PC}}\bigr),\quad u\in \mathit{PC}(J,\mathbf{R}), t\in J. $$
(H5): 
$$\frac{3}{|1+E_{q}(-\lambda)|} \Biggl(\frac{\|\mu\|_{L^{\frac{1}{\sigma }}(J)}}{\Gamma(q)(\frac{q-\sigma}{1-\sigma})^{1-\sigma}} \liminf_{r\rightarrow+\infty} \frac{\overline{\omega}(r)}{r}+\sum^{m}_{i=1} \frac{{L_{4}}}{|E_{q}(-t_{i}^{q}\lambda)|} \Biggr)< 1. $$
Then the boundary value problem (1.3) has at least one solution.

Proof

For \(r>0\), the set \(B_{r}=\{u\in \mathit{PC}(J,\mathbf {R}):\|u\|_{\mathit{PC}}\leq r\}\) is a bounded closed convex set in \(\mathit{PC}(J,\mathbf {R})\). Define the operators P and Q on \(B_{r}\) as
$$\begin{aligned}& (Pu) (t)= \int_{0}^{t}(t-s)^{q-1}E_{q,q} \bigl(-(t-s)^{q}\lambda \bigr)f\bigl(s,u(s),Tu(s),Su(s)\bigr)\,ds \\& \hphantom{(Pu)(t)={}}{}-\frac{E_{q}(-t^{q}\lambda)}{1+E_{q}(-\lambda)} \int _{0}^{1}(1-s)^{q-1}E_{q,q} \bigl(-(1-s)^{q}\lambda\bigr)f\bigl(s,u(s),Tu(s),Su(s)\bigr)\,ds, \\& (Qu) (t)=\frac{E_{q}(-t^{q}\lambda)}{1+E_{q}(-\lambda)}\sum^{m}_{i=1} \frac {I_{i}(u(t_{i}))}{E_{q}(-t_{i}^{q}\lambda)} -E_{q}\bigl(-t^{q}\lambda\bigr)\sum ^{m}_{j=k+1}\frac{I_{j}(u(t_{j}))}{E_{q}(-t_{j}^{q}\lambda)}. \end{aligned}$$
By (H4) and the Hölder inequality, for any \(u \in B_{r}\), we have
$$\begin{aligned}& \int_{0}^{t} \bigl\vert (t-s)^{q-1}f \bigl(s,u(s),Tu(s),Su(s)\bigr) \bigr\vert \,ds \\& \quad \leq \int_{0}^{t}\bigl|(t-s)^{q-1}\mu(s)\overline{ \omega}(r)\bigr|\,ds \\& \quad \leq \biggl( \int_{0}^{t}(t-s)^{\frac{q-1}{1-\sigma}}\,ds \biggr)^{1-\sigma} \biggl( \int_{0}^{t}\bigl(\overline{\omega}(r)\mu(s) \bigr)^{\frac{1}{\sigma}}\,ds \biggr)^{\sigma}\\& \quad \leq\frac{t^{q-\sigma}}{(\frac{q-\sigma}{1-\sigma})^{1-\sigma }}\overline{\omega}(r)\|\mu\|_{L^{\frac{1}{\sigma}}(J)} \\& \quad \leq\frac{\|\mu\|_{L^{\frac{1}{\sigma}}(J)}}{(\frac{q-\sigma}{1-\sigma })^{1-\sigma}}\overline{\omega}(r). \end{aligned}$$
Similarly, we have
$$\int_{0}^{1} \bigl\vert (1-s)^{q-1}f \bigl(s,u(s),Tu(s),Su(s)\bigr) \bigr\vert \,ds \leq\frac{\|\mu \|_{L^{\frac{1}{\sigma}}(J)}}{(\frac{q-\sigma}{1-\sigma})^{1-\sigma }}\overline{ \omega}(r). $$
Next, we show that there exists \(r_{0}>0\) with \(Pu+Qv\in B_{r_{0}}\) for \(u,v\in B_{r_{0}}\). If this were not true, then, for each \(r>0\), there would exist \(u_{r}, v_{r}\in B_{r}\) and \(t_{r}\in J\) such that \(|(Pu_{r})(t_{r})+(Qv_{r})(t_{r})|>r\). Assumption (H2) implies \(|I_{i}(u(t_{i}))|\leq|I_{i}(u(t_{i}))-I_{i}(0)|+|I_{i}(0)|\leq{L_{4}}r+M'\). Hence
$$\begin{aligned} r < & \bigl\vert (Pu_{r}) (t_{r})+(Qv_{r}) (t_{r}) \bigr\vert \\ \leq&\frac{\|\mu\|_{L^{\frac{1}{\sigma }}(J)}}{\Gamma(q)|1+E_{q}(-\lambda)|(\frac{q-\sigma}{1-\sigma})^{1-\sigma }}\overline{\omega}(r) \\ &{}+\frac{\|\mu\|_{L^{\frac{1}{\sigma}}(J)}}{\Gamma(q)(\frac {q-\sigma}{1-\sigma})^{1-\sigma}}\overline{\omega}(r) +\frac {1}{|1+E_{q}(-\lambda)|}\sum ^{m}_{i=1}\frac{{L_{4}}r+M'}{|E_{q}(-{t_{i}}^{q}\lambda )|}+\sum ^{m}_{i=1}\frac{{L_{4}}r+M'}{|E_{q}(-{t_{i}}^{q}\lambda)|} \\ =& \Biggl(\frac{\|\mu\|_{L^{\frac{1}{\sigma}}(J)}}{\Gamma(q)(\frac {q-\sigma}{1-\sigma})^{1-\sigma}}\overline{\omega}(r)+ \sum ^{m}_{i=1}\frac{{L_{4}}r+M'}{|E_{q}(-{t_{i}}^{q}\lambda)|} \Biggr) \biggl(1+ \frac {1}{|1+E_{q}(-\lambda)|} \biggr) \\ \leq&\frac{3}{|1+E_{q}(-\lambda)|} \Biggl(\frac{\|\mu\|_{L^{\frac {1}{\sigma}}(J)}}{\Gamma(q)(\frac{q-\sigma}{1-\sigma})^{1-\sigma }}\overline{\omega}(r)+ \sum ^{m}_{i=1}\frac{{L_{4}}r+M'}{|E_{q}(-{t_{i}}^{q}\lambda)|} \Biggr). \end{aligned}$$
Dividing both sides by r and taking the lower limit as \(r\rightarrow +\infty\), we obtain
$$1\leq\frac{3}{|1+E_{q}(-\lambda)|} \Biggl(\frac{\|\mu\|_{L^{\frac{1}{\sigma }}(J)}}{\Gamma(q)(\frac{q-\sigma}{1-\sigma})^{1-\sigma}}\liminf_{r\rightarrow\infty} \frac{\overline{\omega}(r)}{r} +\sum^{m}_{i=1} \frac{{L_{4}}}{|E_{q}(-t_{i}^{q}\lambda)|} \Biggr), $$
which contradicts condition (H5). Thus, there exists \(r_{0}>0\) such that \(Pu+Qv\in B_{r_{0}}\) for all \(u,v\in B_{r_{0}}\).
For all \(t\in J\) and \(u,v\in B_{r}\), we get
$$\begin{aligned}& \bigl\vert (Qu) (t)-(Qv) (t) \bigr\vert \\& \quad \leq\frac{|E_{q}(-t^{q}\lambda)|}{|1+E_{q}(-\lambda)|}\sum^{m}_{i=1} \frac{|I_{i}(u(t_{i}))-I_{i}(v(t_{i}))|}{|E_{q}\bigl(-t_{i}^{q}\lambda \bigr)|} \\& \qquad {}+ \bigl\vert E_{q}\bigl(-t^{q}\lambda\bigr) \bigr\vert \sum^{m}_{i=1}\frac {|I_{i}(u(t_{i}))-I_{i}(v(t_{i}))|}{|E_{q}(-t_{i}^{q}\lambda)|} \\ & \quad \leq\sum^{m}_{i=1}\frac{|I_{i}(u(t_{i}))-I_{i}(v(t_{i}))|}{|E_{q}(-t_{i}^{q}\lambda )|} \biggl(1+\frac{1}{|1+E_{q}(-\lambda)|} \biggr) \\ & \quad \leq\frac{3}{|1+E_{q}(-\lambda)|}\sum^{m}_{i=1} \frac {L_{4}|u(t_{i})-v(t_{i})|}{|E_{q}(-t_{i}^{q}\lambda)|} \\ & \quad \leq\frac{3}{|1+E_{q}(-\lambda)|}\sum^{m}_{i=1} \frac{{L_{4}}\|u-v\| _{\mathit{PC}}}{|E_{q}(-t_{i}^{q}\lambda)|}. \end{aligned}$$
Let \(\chi'=\frac{3}{|1+E_{q}(-\lambda)|}\sum^{m}_{i=1}\frac {{L_{4}}}{|E_{q}(-t_{i}^{q}\lambda)|}\). From (H5) we have \(0<\chi'<1\) and \(\| Qu-Qv\|_{\mathit{PC}}\leq\chi'\|u-v\|_{\mathit{PC}}\), so Q is a contraction mapping.
The continuity of f implies that the operator P is continuous. We now prove that P is a compact operator. Following the procedure used in the first part of Theorem 3.1, it follows that \(P(B_{r})\) is uniformly bounded on \(\mathit{PC}(J,\mathbf{R})\). We now show that \(P(B_{r})\) is equicontinuous on \(J_{k}\) (\(k=1,\ldots ,m\)). Let \(\Omega=J\times B_{r}\times TB_{r}\times SB_{r}\) and \(\overline {f}=\sup_{(t,u,Tu,Su)\in\Omega}|f(t,u,Tu,Su)|\). Then, for any \(t_{k}< \tau_{2}<\tau_{1}\leq t_{k+1}\), we have
$$\begin{aligned}& \bigl\vert (Pu) (\tau_{2})-(Pu) (\tau_{1}) \bigr\vert \\ & \quad \leq \biggl\vert \int_{0}^{\tau_{2}}(\tau_{2}-s)^{q-1}E_{q,q} \bigl(-(\tau_{2}-s)^{q}\lambda \bigr)f\bigl(s,u(s),Tu(s),Su(s) \bigr)\,ds \\ & \qquad {} - \int_{0}^{\tau_{1}}(\tau_{1}-s)^{q-1}E_{q,q} \bigl(-(\tau_{1}-s)^{q}\lambda \bigr)f\bigl(s,u(s),Tu(s),Su(s) \bigr)\,ds \biggr\vert \\ & \qquad {} + \biggl\vert \frac{E_{q}(-{\tau_{2}}^{q}\lambda)-E_{q}(-{\tau_{1}}^{q}\lambda )}{1+E_{q}(-\lambda)} \int_{0}^{1}(1-s)^{q-1}E_{q,q} \bigl(-(1-s)^{q}\lambda \bigr)f\bigl(s,u(s),Tu(s),Su(s)\bigr)\,ds \biggr\vert \\ & \quad \leq \biggl\vert \int_{0}^{\tau_{2}}(\tau_{2}-s)^{q-1}E_{q,q} \bigl(-(\tau_{2}-s)^{q}\lambda \bigr)f\bigl(s,u(s),Tu(s),Su(s) \bigr)\,ds \\ & \qquad {} - \int_{0}^{\tau_{2}}(\tau_{1}-s)^{q-1}E_{q,q} \bigl(-(\tau_{2}-s)^{q}\lambda \bigr)f\bigl(s,u(s),Tu(s),Su(s) \bigr)\,ds \\ & \qquad {} + \int_{0}^{\tau_{2}}(\tau_{1}-s)^{q-1}E_{q,q} \bigl(-(\tau_{2}-s)^{q}\lambda \bigr)f\bigl(s,u(s),Tu(s),Su(s) \bigr)\,ds \\ & \qquad {} - \int_{0}^{\tau_{2}}(\tau_{1}-s)^{q-1}E_{q,q} \bigl(-(\tau_{1}-s)^{q}\lambda \bigr)f\bigl(s,u(s),Tu(s),Su(s) \bigr)\,ds \\ & \qquad {} - \int_{\tau_{2}}^{\tau_{1}}(\tau_{1}-s)^{q-1}E_{q,q} \bigl(-(\tau _{1}-s)^{q}\lambda\bigr)f\bigl(s,u(s),Tu(s),Su(s) \bigr)\,ds \biggr\vert \\ & \qquad {} +\frac{ \vert E_{q}(-{\tau_{2}}^{q}\lambda)-E_{q}(-{\tau_{1}}^{q}\lambda) \vert }{\Gamma (q) \vert 1+E_{q}(-\lambda) \vert } \int_{0}^{1}(1-s)^{q-1} \bigl\vert f \bigl(s,u(s),Tu(s),Su(s)\bigr) \bigr\vert \,ds \\ & \quad \leq \int_{0}^{\tau_{2}} \bigl\vert (\tau_{2}-s)^{q-1}-( \tau_{1}-s)^{q-1} \bigr\vert \bigl\vert E_{q,q} \bigl(-(\tau _{2}-s)^{q}\lambda\bigr) \bigr\vert \overline{f}\,ds \\ & \qquad {} + \int_{0}^{\tau_{2}}(\tau_{1}-s)^{q-1} \bigl\vert E_{q,q}\bigl(-(\tau_{2}-s)^{q}\lambda \bigr)-E_{q,q}\bigl(-(\tau_{1}-s)^{q}\lambda\bigr) \bigr\vert \overline{f}\,ds \\ & \qquad {} +\frac{\overline{f}}{\Gamma(q)} \biggl\vert \int_{\tau_{2}}^{\tau_{1}}(\tau _{1}-s)^{q-1} \,ds \biggr\vert +\frac{ \vert E_{q}(-{\tau_{2}}^{q}\lambda)-E_{q}(-{\tau_{1}}^{q}\lambda) \vert \overline {f}}{\Gamma(q+1) \vert 1+E_{q}(-\lambda) \vert } \\ & \quad \leq\frac{\overline{f}}{\Gamma(q)} \biggl\vert \int_{0}^{\tau_{2}}\bigl((\tau _{2}-s)^{q-1}-( \tau_{1}-s)^{q-1}\bigr)\,ds \biggr\vert + \frac{(\tau_{1}-\tau_{2})^{q}\overline{f}}{\Gamma(q+1)}+ \frac{ \vert E_{q}(-{\tau _{2}}^{q}\lambda)- E_{q}(-{\tau_{1}}^{q}\lambda) \vert \overline{f}}{\Gamma(q+1) \vert 1+E_{q}(-\lambda) \vert } \\& \qquad {} +\overline{f} \int_{0}^{\tau_{2}}(\tau_{1}-s)^{q-1} \bigl\vert E_{q,q}\bigl(-(\tau _{2}-s)^{q}\lambda \bigr)-E_{q,q}\bigl(-(\tau_{1}-s)^{q}\lambda\bigr) \bigr\vert \,ds \\& \quad \leq\frac{(\tau_{1}-\tau_{2})^{q}+\tau_{1}^{q}-\tau_{2}^{q}}{\Gamma (q+1)}\overline{f}+\frac{(\tau_{1}-\tau_{2})^{q}\overline{f}}{\Gamma(q+1)} +\frac{ \vert E_{q}(-{\tau_{2}}^{q}\lambda)-E_{q}(-{\tau_{1}}^{q}\lambda) \vert \overline {f}}{\Gamma(q+1) \vert 1+E_{q}(-\lambda) \vert } \\& \qquad {} +\overline{f} \int_{0}^{\tau_{2}}(\tau_{1}-s)^{q-1} \bigl\vert E_{q,q}\bigl(-(\tau _{2}-s)^{q}\lambda \bigr)-E_{q,q}\bigl(-(\tau_{1}-s)^{q}\lambda\bigr) \bigr\vert \,ds. \end{aligned}$$
By Lemma 2.1(2) we know that \(E_{q,q}(-{t}^{q}\lambda)\) is continuous on \(t\in J\), and thus \(E_{q,q}(-{t}^{q}\lambda)\) is uniformly continuous on \(t\in J\). Hence, for any \(\varepsilon>0\), there is a sufficiently small \(\delta >0\) such that, for \(t_{1},t_{2}\in J\) with \(|t_{1}-t_{2}|<\delta\), we have
$$\bigl\vert E_{q,q}\bigl(-t_{1}^{q}\lambda \bigr)-E_{q,q}\bigl(-t_{2}^{q}\lambda\bigr) \bigr\vert < \frac{\varepsilon}{\tau _{2}^{\frac{q}{2-q}}}. $$
Let \(\sigma_{1}=\frac{2-q}{2(1-q)}\) and \(\sigma_{2}=\frac{2-q}{q}\). Then \(\sigma_{1}>1\), \(\sigma_{2}>1\), and \(\frac{1}{\sigma_{1}}+\frac{1}{\sigma _{2}}=1\). By the Hölder inequality we have
$$\begin{aligned}& \int_{0}^{\tau_{2}}(\tau_{1}-s)^{q-1} \bigl\vert E_{q,q}\bigl(-(\tau_{2}-s)^{q}\lambda \bigr)-E_{q,q}\bigl(-(\tau_{1}-s)^{q}\lambda\bigr) \bigr\vert \,ds \\& \quad \leq \biggl[ \int_{0}^{\tau_{2}}(\tau_{1}-s)^{(q-1)\frac{2-q}{2(1-q)}} \,ds \biggr]^{\frac{2(1-q)}{2-q}} \\& \qquad {}\cdot\biggl[ \int_{0}^{\tau_{2}}\bigl(E_{q,q}\bigl(-( \tau_{2}-s)^{q}\lambda\bigr)-E_{q,q}\bigl(-(\tau _{1}-s)^{q}\lambda\bigr)\bigr)^{\frac{2-q}{q}}\,ds \biggr]^{\frac{q}{2-q}} \\& \quad \leq \biggl[\frac{\tau_{1}^{\frac{q}{2}}-(\tau_{1}-\tau_{2})^{\frac {q}{2}}}{\frac{q}{2}} \biggr]^{\frac{2(1-q)}{2-q}}\cdot \biggl[ \int_{0}^{\tau_{2}} \biggl(\frac{\varepsilon}{\tau_{2}^{\frac{q}{2-q}}} \biggr)^{\frac {2-q}{q}}\,ds \biggr]^{\frac{q}{2-q}} \\& \quad = \biggl[\frac{2\tau_{1}^{\frac{q}{2}}-2(\tau_{1}-\tau_{2})^{\frac {q}{2}}}{q} \biggr]^{\frac{2(1-q)}{2-q}}\cdot\varepsilon, \end{aligned}$$
so \(\int_{0}^{\tau_{2}}(\tau_{1}-s)^{q-1}|E_{q,q}(-(\tau_{2}-s)^{q}\lambda )-E_{q,q}(-(\tau_{1}-s)^{q}\lambda)|\,ds\) tends to zero as \(\tau_{2}\rightarrow \tau_{1}\). Therefore, \(|(Pu)(\tau_{2})-(Pu)(\tau_{1})|\) tends to zero as \(\tau _{2}\rightarrow\tau_{1}\). This yields that P is equicontinuous on the interval \(J_{k}\).

Combining the above arguments and the PC-type Arzelà-Ascoli theorem (Lemma 2.3 in the case \(X = \mathbf{R}\)), we conclude that \(P: B_{r}\rightarrow B_{r}\) is compact and completely continuous. Then it follows from Lemma 2.2 that problem (1.3) has at least one solution. This completes the proof. □

4 Examples

In this section, we give two examples to illustrate our main results.

Example 4.1

Consider the following impulsive fractional integro-differential equation with antiperiodic boundary condition:
$$ \textstyle\begin{cases} {}^{\mathrm{c}}D^{\frac{1}{2}}u(t)+u(t)=\frac{ u(t)+1}{36(e^{t}+1)}+\frac{1}{t^{2}+15}\int^{t}_{0} \frac {u(s)}{e^{(t+3)s}}\,ds \\ \hphantom{{}^{\mathrm{c}}D^{\frac{1}{2}}u(t)+u(t)={}}{}+\frac{2}{\sqrt{t}+49}\int^{1}_{0}\frac {u(s)}{(8+t+s)^{2}}\,ds,\quad t\in[0,1]\setminus \{\frac{1}{2} \}, \\ \Delta u|_{t=\frac{1}{2}}=\frac{|u(\frac{1}{2})|}{12+|u(\frac{1}{2})|}, \\ u(0)=-u(1), \end{cases} $$
(4.1)
Let
$$\begin{aligned}& f(t,u,v,w)=\frac{u+1}{36(e^{t}+1)}+\frac{v}{t^{2}+15}+\frac {2w}{\sqrt{t}+49}, \qquad I_{k}(u)=\frac{|u|}{12+|u|}, \\& (Tu) (t)= \int^{t}_{0}e^{-(t+3)s}u(s)\,ds, \qquad (Su) (t)= \int^{1}_{0}\frac {u(s)}{(8+t+s)^{2}}\,ds. \end{aligned}$$
By direct computation, \(k_{0}=\max \{\frac{1}{e^{(t+3)s}}: 0\leq s\leq t \leq1 \}=1\) and \(h_{0}=\max \{\frac{1}{(8+t+s)^{2}}:0\leq s, t \leq1 \}=\frac{1}{64}\). For \(u_{1}, u_{2}, v_{1}, v_{2}, \omega_{1}, \omega_{2}\in\mathbf{R}\) and \(t\in J\), we have
$$\begin{aligned}& \bigl\vert f(t,u_{1},v_{1},w_{1})-f(t,u_{2},v_{2},w_{2}) \bigr\vert \\& \quad \leq\frac {1}{36(e^{t}+1)} \vert u_{1}-u_{2} \vert +\frac{1}{t^{2}+15} \vert v_{1}-v_{2} \vert + \frac{2}{\sqrt {t}+49} \vert w_{1}-w_{2} \vert , \\& \bigl\vert I_{k}(u_{1})-I_{k}(u_{2}) \bigr\vert \leq\frac{1}{12} \vert u_{1}-u_{2} \vert . \end{aligned}$$
Let
$$\begin{aligned}& L_{1}(t)=\frac{1}{36(e^{t}+1)}, \qquad L_{2}(t)= \frac{1}{t^{2}+15}, \\& L_{3}(t)=\frac{2}{\sqrt{t}+49}, \qquad L_{4}=\frac{1}{12}. \end{aligned}$$
It is easy to see that \(\overline{L_{1}}=\frac{1}{72}\), \(\overline {L_{2}}=\frac{1}{15}\), \(\overline{L_{3}}=\frac{2}{49}\), \(E_{\frac{1}{2}}(-1)=\frac{1+\frac{\pi-2}{\sqrt{\pi}}}{1+\sqrt{\pi+(\pi -2)}}\approx0.42\), \(E_{\frac{1}{2}}(-(\frac{1}{2})^{\frac{1}{2}})\approx 0.52\), \(\Gamma(\frac{3}{2})=\frac{1}{2}\sqrt{\pi}\approx0.89\),
$$\begin{aligned} \chi =&\frac{3}{|1+E_{\frac{1}{2}}(-1)|} \biggl(\frac{{L_{4}}}{|E_{\frac {1}{2}}(-(\frac{1}{2})^{\frac{1}{2}})|} +\frac{(\overline{L_{1}}+\overline{L_{2}}k_{0}+\overline{L_{3}}h_{0})}{\Gamma (\frac{3}{2})} \biggr) \\ \approx&\frac{3}{1+0.42} \biggl(\frac{\frac{1}{12}}{0.52}+\frac{\frac {1}{72}+\frac{1}{15}+\frac{2}{49}\times\frac{1}{64}}{0.89} \biggr)< 1. \end{aligned}$$
Then by Theorem 3.1 problem (4.1) has a unique solution.

Example 4.2

Consider the following impulsive antiperiodic problem:
$$ \textstyle\begin{cases} {}^{\mathrm{c}}D^{\frac{1}{2}}u(t)+u(t)= (\frac{\sqrt[3]{t+1}}{16}+\frac {1}{16\sqrt[3]{t+1}} )\frac{|u(t)|}{1+|u(t)|} +\frac{\sqrt[3]{t+1}}{16e^{t}}\sin (\int^{t}_{0}\sin(t-s)u(s)\,ds ) \\ \hphantom{{}^{\mathrm{c}}D^{\frac{1}{2}}u(t)+u(t)={}} {}+\frac{1}{16\sqrt [3]{t+1}}\cos (\int^{1}_{0}\frac{u(s)}{1+ts}\,ds ),\quad t\in [0,1]\setminus \{\frac{1}{2} \}, \\ \Delta u|_{t=\frac{1}{2}}=\frac{|u(\frac{1}{2})|}{12+|u(\frac{1}{2})|}, \\ u(0)=-u(1), \end{cases} $$
(4.2)
where
$$f(t,u,v,w)= \biggl(\frac{\sqrt[3]{t+1}}{16}+\frac{1}{16\sqrt [3]{t+1}} \biggr) \frac{|u|}{1+|u|} +\frac{\sqrt[3]{t+1}}{16e^{t}}\sin v+\frac{1}{16\sqrt[3]{t+1}}\cos w. $$
By computation we obtain
$$\begin{aligned} \begin{aligned} \bigl\vert f(t,u,Tu,Su) \bigr\vert &\leq\frac{\sqrt[3]{t+1}}{16}+ \frac{1}{16\sqrt [3]{t+1}}+\frac{\sqrt[3]{t+1}}{16}\|u\|_{\mathit{PC}}+\frac{1}{16\sqrt[3]{t+1}}\| u\|_{\mathit{PC}} \\ &= \biggl(\frac{\sqrt[3]{t+1}}{16}+\frac {1}{16\sqrt[3]{t+1}} \biggr) \bigl( \Vert u \Vert _{\mathit{PC}}+1\bigr). \end{aligned} \end{aligned}$$
Let \(\mu(t)=\frac{\sqrt[3]{t+1}}{16}+\frac{1}{16\sqrt[3]{t+1}}\), \(\sigma =\frac{1}{3}\), and \(\overline{\omega}(r)=r+1\). Then \(\liminf_{r\rightarrow\infty}\frac{\overline{\omega}(r)}{r}=1\) and \({L_{4}}=\frac {1}{12}\). Thus,
$$\biggl\{ \frac{\frac{1}{12}}{E_{\frac{1}{2}}(-(\frac{1}{2})^{\frac{1}{2}})} +\frac{[\int^{1}_{0}(\frac{\sqrt[3]{t+1}}{16}+\frac{1}{16\sqrt [3]{t+1}})^{3}\,dt ]^{\frac{1}{3}}}{ \Gamma(\frac{1}{2})(\frac{\frac{1}{2}-\frac{1}{3}}{1-\frac {1}{3}})^{1-\frac{1}{3}}} \biggr\} \frac{3}{1+E_{\frac{1}{2}}(-1)} \approx0.95< 1. $$
By Theorem 3.2 problem (4.2) has at least one solution.

5 Conclusion

In this paper, we are concerned with the existence and uniqueness of solutions for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary condition. The paper has several new features. First, we consider the impulsive fractional integro-differential equation of mixed type, that is, the nonlinear f involves linear operators T and S. The second new feature is that we studied antiperiodic boundary value problems with constant coefficients. Our results are based on the Banach contraction mapping principle and the Krasnoselskii fixed point theorem.

Declarations

Acknowledgements

The authors would like to thank the referees for their pertinent comments and valuable suggestions.

Availability of data and materials

Not applicable.

Funding

This work is supported financially by the National Natural Science Foundation of China (11501318, 11371221), the Natural Science Foundation of Shandong Province of China (ZR2015AM022, ZR2014AM032), and the China Postdoctoral Science Foundation (2017M612230).

Authors’ contributions

All authors contributed equally to the writing of this paper. The authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Qufu Normal University, Qufu, P.R. China
(2)
Department of Mathematics and Statistics, Curtin University, Perth, Australia
(3)
Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao, P.R. China

References

  1. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
  2. Podlubny, I: Fractional Differential Equations. Academic Press, New York (1999) MATHGoogle Scholar
  3. Baleanu, D, Diethelm, K, Scalas, E, Trujillo, JJ: Fractional Calculus Models and Numerical Methods. World Scientific, Boston (2012) View ArticleMATHGoogle Scholar
  4. Wang, Y, Liu, L: Uniqueness and existence of positive solutions for the fractional integro-differential equation. Bound. Value Probl. 2017, 13 (2017) View ArticleMathSciNetGoogle Scholar
  5. Wang, Y, Liu, L: Positive solutions for a class of fractional 3-point boundary value problems at resonance. Adv. Differ. Equ. 2017, 7 (2017) View ArticleMathSciNetGoogle Scholar
  6. Cui, Y: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48-54 (2016) View ArticleMATHMathSciNetGoogle Scholar
  7. Zhang, X, Liu, L, Wu, Y: The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium. Appl. Math. Lett. 37, 26-33 (2014) View ArticleMATHMathSciNetGoogle Scholar
  8. Zhang, X, Liu, L, Wu, Y: The iterative solutions of nonlinear fractional differential equations. Appl. Math. Comput. 219, 4680-4691 (2013) View ArticleMATHMathSciNetGoogle Scholar
  9. Bai, Z, Zhang, S, Sun, S, Yin, C: Monotone iterative method for fractional differential equations. Electron. J. Differ. Equ. 2016, 6 (2016) View ArticleMATHMathSciNetGoogle Scholar
  10. Bai, Z, Chen, Y, Lian, H, Sun, S: On the existence of blow up solutions for a class of fractional differential equations. Fract. Calc. Appl. Anal. 17, 1175-1187 (2014) View ArticleMATHMathSciNetGoogle Scholar
  11. Zou, Y, Cui, Y: Existence results for a functional boundary value problem of fractional differential equations. Adv. Differ. Equ. 2013, 233 (2013) View ArticleMathSciNetGoogle Scholar
  12. Cui, Y: Existence results for singular boundary value problem of nonlinear fractional differential equation. Abstr. Appl. Anal. 2011, Article ID 605614 (2011) View ArticleMATHMathSciNetGoogle Scholar
  13. Hao, X: Positive solution for singular fractional differential equations involving derivatives. Adv. Differ. Equ. 2016, 139 (2016) View ArticleMathSciNetGoogle Scholar
  14. Hao, X, Liu, L, Wu, Y: Positive solutions for nonlinear fractional semipositone differential equation with nonlocal boundary conditions. J. Nonlinear Sci. Appl. 9, 3992-4002 (2016) MATHMathSciNetGoogle Scholar
  15. Liu, X, Jia, M: Existence of solutions for the integral boundary value problems of fractional order impulsive differential equations. Math. Methods Appl. Sci. 39, 475-487 (2016) View ArticleMATHMathSciNetGoogle Scholar
  16. Agarwal, RP, Ahmad, B, Nieto, JJ: Fractional differential equations with nonlocal (parametric type) anti-periodic boundary conditions. Filomat 31, 1207-1214 (2017) View ArticleMathSciNetGoogle Scholar
  17. Mahmudov, N, Unul, S: On existence of BVP’s for impulsive fractional differential equations. Adv. Differ. Equ. 2017, 15 (2017) View ArticleMathSciNetGoogle Scholar
  18. Wang, J, Feckan, M, Zhou, Y: A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal. 19, 806-831 (2016) View ArticleMATHMathSciNetGoogle Scholar
  19. Liu, Y: Survey and new results on boundary-value problems of singular fractional differential equations with impulse effects. Electron. J. Differ. Equ. 2016, 296 (2016) View ArticleMATHMathSciNetGoogle Scholar
  20. Liu, Y: Boundary value problems of singular multi-term fractional differential equations with impulse effects. Math. Nachr. 289, 1526-1547 (2016) View ArticleMATHMathSciNetGoogle Scholar
  21. Bai, Z, Dong, X, Yin, C: Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions. Bound. Value Probl. 2016, 63 (2016) View ArticleMATHMathSciNetGoogle Scholar
  22. Xu, L, Hu, H, Qin, F: Ultimate boundedness of impulsive fractional differential equations. Appl. Math. Lett. 62, 110-117 (2016) View ArticleMATHMathSciNetGoogle Scholar
  23. Bonanno, G, Rodriguez-Lopez, R, Tersian, S: Existence of solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17, 717-744 (2014) View ArticleMATHMathSciNetGoogle Scholar
  24. Rehman, MU, Eloe, PW: Existence and uniqueness of solutions for impulsive fractional differential equations. Appl. Math. Comput. 224, 422-431 (2013) MATHMathSciNetGoogle Scholar
  25. Liu, Z, Li, X: Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 18, 1362-1373 (2013) View ArticleMATHMathSciNetGoogle Scholar
  26. Xu, Y, Liu, X: Some boundary value problems of fractional differential equations with fractional impulsive conditions. J. Comput. Anal. Appl. 19, 426-443 (2015) MATHMathSciNetGoogle Scholar
  27. Nyamoradi, N, Rodriguez-Lopez, R: On boundary value problems for impulsive fractional differential equations. Appl. Math. Comput. 271, 874-892 (2015) View ArticleMathSciNetGoogle Scholar
  28. Yang, S, Zhang, S: Impulsive boundary value problem for a fractional differential equation. Bound. Value Probl. 2016, 203 (2016) View ArticleMATHMathSciNetGoogle Scholar
  29. Shu, X, Shi, Y: A study on the mild solution of impulsive fractional evolution equations. Appl. Math. Comput. 273, 465-476 (2016) View ArticleMathSciNetGoogle Scholar
  30. Li, B, Gou, H: Existence of solutions for impulsive fractional evolution equations with periodic boundary condition. Adv. Differ. Equ. 2017, 236 (2017) View ArticleMathSciNetGoogle Scholar
  31. Wang, G, Ahmad, B, Zhang, L, Nieto, JJ: Comments on the concept of existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 19, 401-403 (2014) View ArticleMathSciNetGoogle Scholar
  32. Feckan, M, Wang, J, Zhou, Y: Response to ‘Comments on the concept of existence of solution for impulsive fractional differential equations’. Commun. Nonlinear Sci. Numer. Simul. 19, 4213-4215 (2014) View ArticleMathSciNetGoogle Scholar
  33. Wang, G, Ahmad, B, Zhang, L: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Nonlinear Anal. 74, 792-804 (2011) View ArticleMATHMathSciNetGoogle Scholar
  34. Li, X, Chen, F, Li, X: Generalized anti-periodic boundary value problems of impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 18, 28-41 (2013) View ArticleMATHMathSciNetGoogle Scholar
  35. Liu, X, Li, Y: Some antiperiodic boundary value problem for nonlinear fractional impulsive differential equations. Abstr. Appl. Anal. 2014, Article ID 571536 (2014) MathSciNetGoogle Scholar
  36. Ahmad, B, Nieto, JJ: Existence of solution for impulsive anti-periodic boundary value problems of fractional order. Taiwan. J. Math. 15, 981-993 (2011) View ArticleMATHMathSciNetGoogle Scholar
  37. Zhang, L, Wang, G: Existence of solutions for nonlinear fractional differential equations with impulses and anti-periodic boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2011, 7 (2011) View ArticleMATHMathSciNetGoogle Scholar
  38. Chen, A, Chen, Y: Existence of solutions to anti-periodic boundary value problem for nonlinear fractional differential equations with impulses. Adv. Differ. Equ. 2011, Article ID 915689 (2011) MATHMathSciNetGoogle Scholar
  39. Benchohra, M, Slimani, BA: Existence and uniqueness of solutions to impulsive fractional differential equations. Electron. J. Differ. Equ. 2009, 10 (2009) MATHMathSciNetGoogle Scholar
  40. Wang, J, Lin, Z: On the impulsive fractional anti-periodic BVP modelling with constant coefficients. J. Appl. Math. Comput. 46, 107-121 (2014) View ArticleMATHMathSciNetGoogle Scholar
  41. Wang, J, Feckan, M, Zhou, Y: Presentation of solutions of impulsive fractional Langevin equations and existence results. Eur. Phys. J. Spec. Top. 222, 1857-1874 (2013) View ArticleGoogle Scholar
  42. Kransnoselskii, MA: Two remarks on the method of successive approximations. Usp. Mat. Nauk 10, 123-127 (1955) MathSciNetGoogle Scholar
  43. Wei, W, Xiang, X, Peng, Y: Nonlinear impulsive integro-differential equations of mixed type ang optimal controls. Optimization 55, 141-156 (2006) View ArticleMATHMathSciNetGoogle Scholar

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