Open Access

Nonlocal boundary value problems for impulsive fractional \(q_{k}\)-difference equations

  • Bashir Ahmad1Email author,
  • Ahmed Alsaedi1,
  • Sotiris K Ntouyas1, 2,
  • Jessada Tariboon3 and
  • Faris Alzahrani1
Advances in Difference Equations20162016:124

https://doi.org/10.1186/s13662-016-0848-9

Received: 4 March 2016

Accepted: 27 April 2016

Published: 9 May 2016

Abstract

In this paper, we investigate the existence and uniqueness of solutions for a nonlocal boundary value problem of impulsive fractional \(q_{k}\)-difference equations involving a new \(q_{k}\)-shifting operator \({}_{a}\Phi_{q_{k}}(m) = q_{k} m + (1-q_{k})a\). Our main results rely on Banach’s contraction mapping principle, Leray-Schauder nonlinear alternative, and Rothe fixed point theorem. Examples illustrating the obtained results are also presented.

Keywords

quantum calculus impulsive fractional \(q_{k}\)-difference equations existence uniqueness fixed point theorem

MSC

26A33 39A13 34A37

1 Introduction

The main purpose of this manuscript is to study the existence and uniqueness of solutions for impulsive boundary value problems of fractional \(q_{k}\)-difference equations of the form
$$ \left \{ \textstyle\begin{array}{l} {}_{t_{k}}D_{q_{k}}^{\alpha_{k}}x(t)=f(t,x(t)), \quad t\in J_{k}\subseteq[0,T], t\neq t_{k}, \\ {}_{t_{k}}I_{q_{k}}^{1-\alpha_{k}}x(t_{k}^{+})-x(t_{k})=\varphi _{k} (x(t_{k}) ), \quad k=1,2,\ldots, m, \\ a{}_{t_{0}}I_{q_{0}}^{1-\alpha_{0}}x(0)=bx(T)+\sum_{l=0}^{m}c_{l} {}_{t_{l}}I^{\gamma_{l}}_{q_{l}}x(t_{l+1}), \end{array}\displaystyle \right . $$
(1.1)
where \(0=t_{0}< t_{1}<\cdots<t_{m}<t_{m+1}=T\), \({}_{t_{k}}D^{\alpha_{k}}_{q_{k}}\) denotes the Riemann-Liouville \(q_{k}\)-fractional derivative of order \(\alpha_{k}\) on \(J_{k}\), \(0<\alpha_{k}\leq1\), \(0< q_{k}<1\), \(J_{k}=(t_{k},t_{k+1}]\), \(J_{0}=[0,t_{1}]\), \(k=0,1,\ldots,m\), \(J=[0,T]\), \(f\in C( J\times {\mathbb{R}},{\mathbb{R}})\), \(\varphi_{k}\in C( \mathbb{R}, \mathbb{R})\), \(k=1,2,\ldots,m\), \({}_{t_{k}}I^{\alpha_{k}}_{q_{k}}\) denotes the Riemann-Liouville \(q_{k}\)-fractional integral of order \(\alpha_{k}>0\) on \(J_{k}\), \(a,b,c_{l}\in\mathbb{R}\), \(\gamma_{l}>0\), \(l=0,1,2,\ldots,m\).

The quantum calculus is known as the calculus without limits and provides a descent approach to deal with sets of nondifferentiable functions by considering difference operators. Quantum difference operators play an important role in several mathematical areas such as orthogonal polynomials, basic hyper-geometric functions, combinatorics, the calculus of variations, mechanics, and the theory of relativity. The book by Kac and Cheung [1] covers many fundamental aspects of the quantum calculus.

In recent years, the topic of q-calculus has attracted the attention of several researchers, and a variety of new results can be found in the papers [211] and the references therein.

In [12], the notions of \(q_{k}\)-derivative and \(q_{k}\)-integral for a function \(f:J_{k}:=[t_{k},t_{k+1}]\to{\mathbb{R}}\), were introduced, and several their properties were obtained. Also, the existence and uniqueness results for initial value problems of first- and second-order impulsive \(q_{k}\)-difference equations were studied. \(q_{k}\)-calculus analogues of some classical integral inequalities such as Hölder, Hermite-Hadamard, trapezoid, Ostrowski, Cauchy-Bunyakovsky-Schwarz, Grüss and Grüss-Čebyšev were proved in [13].

In [14], new concepts of fractional quantum calculus were defined by introducing a new q-shifting operator \({}_{a}\Phi_{q}(m) = qm + (1-q)a\). After giving the basic properties of the new q-shifting operator, the q-derivative and q-integral were defined. New definitions of the Riemann-Liouville fractional q-integral and q-difference on an interval \([a,b]\) were given, and their basic properties were discussed. As applications of the new concepts, existence and uniqueness results for first- and second-order initial value problems for impulsive fractional q-difference equations were presented. Recently, the existence of solutions for impulsive fractional q-difference equations with antiperiodic boundary conditions was discussed in [15], whereas the existence results for a nonlinear impulsive \(q_{k}\)-integral boundary value problem were obtained in [16].

In this paper, we consider a boundary value problem of impulsive fractional \(q_{k}\)-difference equations (1.1) by introducing a new \(q_{k}\)-shifting operator \({}_{a}\Phi_{q_{k}}(m) = q_{k} m + (1-q_{k})a\) and establish some existence results for the new problem. The rest of this paper is organized as follows: In Section 2, we recall some known facts about fractional \(q_{k}\)-calculus, present an auxiliary lemma, which is used to convert problem (1.1) into a fixed point problem, and a lemma dealing with useful bounds. Section 3 contains the main results, whereas some illustrative examples are presented in Section 4.

2 Preliminaries

For any positive integer k, the \(q_{k}\)-shifting operator: \({}_{a}\Phi_{q_{k}}(m) = q_{k}m + (1-q_{k})a\) [14] satisfies the relation
$$ {}_{a}\Phi_{q_{k}}^{k}(m) = {}_{a}\Phi_{q_{k}}^{k-1} \bigl({}_{a} \Phi_{q_{k}}(m) \bigr)\quad \text{with } {}_{a} \Phi_{q_{k}}^{0}(m)=m. $$
We define the power of \(q_{k}\)-shifting operator as
$$ {}_{a}(n-m)_{q_{k}}^{(0)}=1, \qquad {}_{a}(n-m)_{q_{k}}^{(k)} = \prod _{i=0}^{k-1} \bigl(n-{}_{a} \Phi_{q_{k}}^{i}(m) \bigr),\quad k\in \mathbb{N}\cup\{\infty\}. $$
If \(\gamma\in\mathbb{R}\), then
$$ {}_{a}(n-m)^{(\gamma)}_{q_{k}}=n^{(\gamma)} \prod_{i=0}^{\infty}\frac{1- {}_{\frac{a}{n}}\Phi^{i}_{q_{k}}(m/n)}{1- {}_{\frac{a}{n}}\Phi^{\gamma+i}_{q_{k}}(m/n)}, \quad n \neq0. $$
The \(q_{k}\)-derivative of a function f on interval \([a,b]\) is defined by
$$ ({}_{a}D_{q_{k}}f) (t)=\frac{f(t)-f({}_{a}\Phi _{q_{k}}(t))}{(1-q_{k})(t-a)}, \quad t \ne a\quad \text{and} \quad ({}_{a}D_{q_{k}}f) (a)=\lim _{t\to a}({}_{a}D_{q_{k}}f) (t), $$
and the \(q_{k}\)-derivative of higher order is given by
$$ \bigl({}_{a}D_{q_{k}}^{k}f\bigr) (t)={}_{a}D_{q_{k}}^{k-1}({}_{a}D_{q_{k}}f) (t),\qquad \bigl({}_{a}D_{q_{k}}^{0}f\bigr) (t)=f(t), \quad k\in\mathbb{N}. $$
The \(q_{k}\)-integral of a function f defined on the interval \([a, b]\) is given by
$$ ({}_{a}I_{q_{k}}f) (t)= \int_{a}^{t}f(s){}_{a} \,ds=(1-q_{k}) (t-a)\sum_{i=0}^{\infty}q_{k}^{i}f \bigl({}_{a}\Phi_{q_{k}^{i}}(t)\bigr),\quad t\in[a, b] $$
and
$$ \bigl({}_{a}I_{q_{k}}^{k}f\bigr) (t)={}_{a}I_{q_{k}}^{k-1}({}_{a}I_{q_{k}}f) (t),\qquad \bigl({}_{a}I_{q_{k}}^{0}f\bigr) (t)=f(t),\quad k\in\mathbb{N}. $$
The fundamental theorem of \(q_{k}\)-calculus applies to the operator \({}_{a}D_{q_{k}}\) and \({}_{a}I_{q_{k}}\) as follows:
$$ ({}_{a}D_{q_{k}}{}_{a}I_{q_{k}}f) (t)=f(t). $$
If f is continuous at \(t=a\), then
$$ ({}_{a}I_{q_{k}}{}_{a}D_{q_{k}}f) (t)=f(t)-f(a). $$
The formula of \(q_{k}\)-integration by parts on the interval \([a, b]\) is
$$ \int_{a}^{b}f(s){}_{a}D_{q_{k}}g(s){}_{a} \, d_{q_{k}}s=(fg) (t) \big|_{a}^{b}- \int _{a}^{b}g\bigl({}_{a} \Phi_{q_{k}}(s)\bigr){}_{a}D_{q_{k}}f(s){}_{a} \, d_{q_{k}}s. $$

Now we recall the definitions of the Riemann-Liouville fractional \(q_{k}\)-integral and \(q_{k}\)-difference on interval \([a, b]\).

Definition 2.1

Let \(\nu\geq0\), and let f be a function defined on \([a, b]\). The fractional \(q_{k}\)-integral of Riemann-Liouville type is given by \(({}_{a}I_{q_{k}}^{0} f)(t)=h(t)\) and
$$ \bigl({}_{a}I_{q_{k}}^{\nu}f\bigr) (t)= \frac{1}{\Gamma_{q_{k}}(\nu)} \int_{a}^{t}{}_{a}\bigl(t-{}_{a} \Phi _{q_{k}}(s)\bigr)_{q_{k}}^{(\nu-1)}f(s){}_{a} \, d_{q_{k}}s,\quad \nu>0, t\in[a, b]. $$

Definition 2.2

The fractional \(q_{k}\)-derivative of Riemann-Liouville type of order \(\nu\ge0\) on the interval \([a, b]\) is defined by \(({}_{a}D_{q_{k}}^{0}f)(t)=f(t)\) and
$$ \bigl({}_{a}D_{q_{k}}^{\nu}f\bigr) (t)= \bigl({}_{a}D_{q_{k}}^{l}{}_{a}I_{q_{k}}^{l-\nu}f \bigr) (t), \quad \nu>0, $$
where l is the smallest integer greater than or equal to ν.

Lemma 2.1

Let \(\alpha, \beta\in\mathbb{R}^{+}\), and let f be a continuous function on \([a,b]\), \(a\geq0\). The Riemann-Liouville fractional \(q_{k}\)-integral has the following semigroup property:
$$ {}_{a}I^{\beta}_{q_{k}}{}_{a}I^{\alpha}_{q_{k}}f(t)={}_{a}I^{\alpha }_{q_{k}}{}_{a}I^{\beta}_{q_{k}}f(t)={}_{a}I^{\alpha+\beta}_{q_{k}}f(t). $$

Lemma 2.2

Let f be a \(q_{k}\)-integrable function on \([a, b]\). Then
$$ {}_{a}D^{\alpha}_{q_{k}}{}_{a}I^{\alpha}_{q_{k}}f(t)=f(t) \quad \textit{for } \alpha>0, t\in[a,b]. $$

Lemma 2.3

Let \(\alpha>0\), and let p be a positive integer. Then, for \(t\in [a,b]\),
$$ {}_{a}I_{q_{k}}^{\alpha }{}_{a}D_{q_{k}}^{p}f(t)={}_{a}D_{q_{k}}^{p}{}_{a}I_{q_{k}}^{\alpha }f(t)- \sum_{k=0}^{p-1}\frac{(t-a)^{\alpha-p+k}}{\Gamma _{q_{k}}(\alpha+k-p+1)}{}_{a}D_{q_{k}}^{k}f(a). $$
From [14] we have the following formulas
$$\begin{aligned}& {}_{a}D_{q_{k}}^{\alpha}(t-a)^{\beta}= \frac{\Gamma_{q_{k}}(\beta +1)}{\Gamma_{q_{k}}(\beta-\alpha+1)}(t-a)^{\beta-\alpha}, \end{aligned}$$
(2.1)
$$\begin{aligned}& {}_{a}I_{q_{k}}^{\alpha}(t-a)^{\beta}= \frac{\Gamma_{q_{k}}(\beta +1)}{\Gamma_{q_{k}}(\beta+\alpha+1)}(t-a)^{\beta+\alpha}. \end{aligned}$$
(2.2)
In the sequel, we define \(\mathit{PC}(J,\mathbb{R})\) = {\(x:J\rightarrow\mathbb{R}\), \(x(t)\) is continuous everywhere except for some \(t_{k}\) at which \(x(t_{k}^{+})\) and \(x(t_{k}^{-})\) exist and \(x(t_{k}^{-})=x(t_{k})\), \(k=1,2,3,\ldots,m\)}. For \(\beta\in\mathbb{R}_{+}\), we introduce the space \(C_{\beta,k}(J_{k}, \mathbb{R})=\{x:J_{k}\rightarrow \mathbb{R}:(t-t_{k})^{\beta}x(t)\in C(J_{k},\mathbb{R})\}\) with the norm \(\|x\|_{C_{\beta,k}}=\sup_{t\in J_{k}}|(t-t_{k})^{\beta}x(t)|\) and \(\mathit{PC}_{\beta}\) = {\(x:J\rightarrow\mathbb{R}:\) for each \(t\in J_{k}\), \((t-t_{k})^{\beta}x(t)\in C(J_{k},\mathbb{R})\), \(k=0,1,2,\ldots,m\)} with the norm
$$\|x\|_{\mathit{PC}_{\beta}}=\max\Bigl\{ \sup_{t\in J_{k}}\bigl\vert (t-t_{k})^{\beta}x(t)\bigr\vert :k=0,1,2,\ldots,m\Bigr\} . $$
Clearly, \(\mathit{PC}_{\beta}\) is a Banach space.

Lemma 2.4

Let \(y\in AC(J,\mathbb{R})\). Then \(x \in \mathit{PC}(J,\mathbb{R})\) is a solution of
$$ \left \{ \textstyle\begin{array}{l} {}_{t_{k}}D_{q_{k}}^{\alpha_{k}}x(t)=y(t), \quad t\in J, t\neq t_{k}, \\ {}_{t_{k}}I_{q_{k}}^{1-\alpha_{k}}x(t_{k}^{+})-x(t_{k})=\varphi _{k} (x(t_{k}) ), \quad k=1,2,\ldots, m, \\ a {}_{t_{0}}I_{q_{0}}^{1-\alpha_{0}}x(0)=bx(T)+\sum_{l=0}^{m}c_{l}{}_{t_{l}}I^{\gamma_{l}}_{q_{l}}x(t_{l+1}), \end{array}\displaystyle \right . $$
(2.3)
if and only if
$$\begin{aligned} x(t) =&\frac{(t-t_{k})^{\alpha_{k}-1}}{\Gamma_{q_{k}}(\alpha_{k})} \Biggl(\prod _{j=0}^{k-1}\frac{(t_{j+1}-t_{j})^{\alpha_{j}-1}}{\Gamma _{q_{j}}(\alpha_{j})} \Biggr) \Biggl\{ \frac{b}{\Omega} \Biggl[\sum_{j=0}^{m-1} \biggl(\prod_{j< i\leq m}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha _{i})} \biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}}y(t_{j+1})+ \varphi _{j+1}\bigl(x(t_{j+1})\bigr) \bigr\} \Biggr]+ \frac{b}{\Omega }{}_{t_{m}}I_{q_{m}}^{\alpha_{m}}y(T) \\ &{}+\sum_{l=0}^{m}\frac{c_{l}(t_{l+1}-t_{l})^{\alpha_{l}+\gamma_{l}-1}}{\Omega \Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})} \Biggl[\sum_{j=0}^{l-1} \biggl(\prod _{j< i\leq l-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma _{q_{i}}(\alpha_{i})} \biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}}y(t_{j+1})+ \varphi _{j+1}\bigl(x(t_{j+1})\bigr) \bigr\} \Biggr] +\sum _{l=0}^{m}\frac{c_{l}}{\Omega}{}_{t_{l}}I_{q_{l}}^{\alpha_{l}+\gamma _{l}}y(t_{l+1}) \Biggr\} \\ &{}+\frac{(t-t_{k})^{\alpha_{k}-1}}{\Gamma_{q_{k}}(\alpha_{k})} \Biggl[\sum_{j=0}^{k-1} \biggl(\prod_{j< i\leq k-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}}y(t_{j+1})+\varphi _{j+1}\bigl(x(t_{j+1})\bigr) \bigr\} \Biggr] \\ &{}+{}_{t_{k}}I_{q_{k}}^{\alpha_{k}}y(t), \end{aligned}$$
(2.4)
where \(\sum_{a}^{b}(\cdot)=0\), \(\prod_{a}^{b}(\cdot)=1\) for \(a>b\), \(\prod_{a< a}(\cdot)=1\), and the nonzero constant Ω is defined by
$$\begin{aligned} \Omega =& a-b \Biggl(\prod_{j=0}^{m} \frac{(t_{j+1}-t_{j})^{\alpha _{j}-1}}{\Gamma_{q_{j}}(\alpha_{j})} \Biggr) \\ &{}-\sum_{l=0}^{m}\frac{c_{l}(t_{l+1}-t_{l})^{\alpha_{l}+\gamma _{l}-1}}{\Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})} \Biggl(\prod_{j=0}^{l-1}\frac{(t_{j+1}-t_{j})^{\alpha_{j}-1}}{\Gamma _{q_{j}}(\alpha_{j})} \Biggr). \end{aligned}$$
(2.5)

Proof

Applying the Riemann-Liouville fractional \(q_{0}\)-integral of order \(\alpha_{0}\) to both sides of the first equation of (2.3) for \(t\in J_{0}\), we obtain
$$\begin{aligned} {}_{t_{0}}I_{q_{0}}^{\alpha_{0}}{}_{t_{0}}D_{q_{0}}^{\alpha _{0}}x(t)={}_{t_{0}}I_{q_{0}}^{\alpha _{0}}{}_{t_{0}}D_{q_{0}}{}_{t_{0}}I_{q_{0}}^{1-\alpha _{0}}x(t)={}_{t_{0}}I_{q_{0}}^{\alpha_{0}}y(t). \end{aligned}$$
(2.6)
From Lemmas 2.1, 2.2, and 2.3 for \(t\in J_{0}\), we have
$$ x(t)=\frac{t^{\alpha_{0}-1}}{\Gamma_{q_{0}}(\alpha _{0})}{}_{t_{0}}I_{q_{0}}^{1-\alpha_{0}}x(0)+{}_{t_{0}}I_{q_{0}}^{\alpha_{0}}y(t). $$
For \(t\in J_{1}\), applying the Riemann-Liouville fractional \(q_{1}\)-integral of order \(\alpha_{1}\) again to the first equation in (2.3) and using the previous process, we get
$$ x(t)=\frac{(t-t_{1})^{\alpha_{1}-1}}{\Gamma_{q_{1}}(\alpha _{1})}{}_{t_{1}}I_{q_{1}}^{1-\alpha_{1}}x \bigl(t_{1}^{+}\bigr)+{}_{t_{1}}I_{q_{1}}^{\alpha_{1}}y(t). $$
(2.7)
The impulsive condition implies that
$$\begin{aligned} x(t) =&\frac{(t-t_{1})^{\alpha_{1}-1}}{\Gamma_{q_{1}}(\alpha_{1})} \biggl[\frac{t_{1}^{\alpha_{0}-1}}{\Gamma_{q_{0}}(\alpha_{0})} {}_{t_{0}}I_{q_{0}}^{1-\alpha_{0}}x(0)+{}_{t_{0}}I_{q_{0}}^{\alpha _{0}}y(t_{1})+ \varphi_{1}\bigl(x(t_{1})\bigr) \biggr] \\ &+ {}_{t_{1}}I_{q_{1}}^{\alpha_{1}}y(t). \end{aligned}$$
Similarly, for \(t\in J_{2}\), we have
$$\begin{aligned} x(t) =&\frac{(t-t_{2})^{\alpha_{2}-1}}{\Gamma_{q_{2}}(\alpha_{2})} \biggl[\frac{(t_{2}-t_{1})^{\alpha_{1}-1}}{\Gamma_{q_{1}}(\alpha_{1})} \biggl(\frac {t_{1}^{\alpha_{0}-1}}{\Gamma_{q_{0}}(\alpha_{0})} {}_{t_{0}}I_{q_{0}}^{1-\alpha_{0}}x(0)+{}_{t_{0}}I_{q_{0}}^{\alpha _{0}}y(t_{1})+ \varphi_{1}\bigl(x(t_{1})\bigr) \biggr) \\ &{}+ {}_{t_{1}}I_{q_{1}}^{\alpha_{1}}y(t_{2})+ \varphi_{2}\bigl(x(t_{2})\bigr) \biggr]+{}_{t_{2}}I_{q_{2}}^{\alpha_{2}}y(t). \end{aligned}$$
Repeating this process for \(t\in J_{k}\subseteq J\), \(k=0,1,2,\ldots,m\), we obtain
$$\begin{aligned} x(t) =&\frac{(t-t_{k})^{\alpha_{k}-1}}{\Gamma_{q_{k}}(\alpha_{k})} \Biggl(\prod _{j=0}^{k-1}\frac{(t_{j+1}-t_{j})^{\alpha_{j}-1}}{\Gamma _{q_{j}}(\alpha_{j})} \Biggr) \bigl({}_{t_{0}}I_{q_{0}}^{1-\alpha_{0}}x(0) \bigr) \\ &{}+\frac{(t-t_{k})^{\alpha_{k}-1}}{\Gamma_{q_{k}}(\alpha_{k})} \Biggl[\sum_{j=0}^{k-1} \biggl(\prod_{j< i\leq k-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}}y(t_{j+1})+\varphi _{j+1}\bigl(x(t_{j+1})\bigr) \bigr\} \Biggr] \\ &{} +{}_{t_{k}}I_{q_{k}}^{\alpha_{k}}y(t). \end{aligned}$$
(2.8)
In particular, for \(t=T\), we get
$$\begin{aligned} x(T) =& \Biggl(\prod_{j=0}^{m} \frac{(t_{j+1}-t_{j})^{\alpha _{j}-1}}{\Gamma_{q_{j}}(\alpha_{j})} \Biggr) \bigl({}_{t_{0}}I_{q_{0}}^{1-\alpha_{0}}x(0) \bigr) \\ &{}+ \Biggl[\sum_{j=0}^{m-1} \biggl(\prod _{j< i\leq m}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}}y(t_{j+1})+\varphi _{j+1}\bigl(x(t_{j+1})\bigr) \bigr\} \Biggr] +{}_{t_{m}}I_{q_{m}}^{\alpha_{m}}y(T). \end{aligned}$$
Taking the Riemann-Liouville fractional \(q_{l}\)-integral of order \(\gamma_{l}\) on (2.8) from \(t_{l}\) to \(t_{l+1}\) and using (2.2), we have
$$\begin{aligned} {}_{t_{l}}I_{q_{l}}^{\gamma_{l}}x(t_{l+1}) =& \frac{(t_{l+1}-t_{l})^{\alpha _{l}+\gamma_{l}-1}}{\Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})} \Biggl(\prod_{j=0}^{l-1} \frac{(t_{j+1}-t_{j})^{\alpha_{j}-1}}{\Gamma _{q_{j}}(\alpha_{j})} \Biggr) \bigl({}_{t_{0}}I_{q_{0}}^{1-\alpha_{0}}x(0) \bigr) \\ &{}+\frac{(t_{l+1}-t_{l})^{\alpha_{l}+\gamma_{l}-1}}{\Gamma_{q_{l}}(\alpha _{l}+\gamma_{l})} \Biggl[\sum_{j=0}^{l-1} \biggl(\prod_{j< i\leq l-1}\frac {(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}}y(t_{j+1})+ \varphi _{j+1}\bigl(x(t_{j+1})\bigr) \bigr\} \Biggr]+{}_{t_{l}}I_{q_{l}}^{\alpha_{l}+\gamma _{l}}y(t_{l+1}). \end{aligned}$$
By the boundary condition of (2.3) we find that
$$\begin{aligned} {}_{t_{0}}I_{q_{0}}^{1-\alpha_{0}}x(0) =&\frac{b}{\Omega} \Biggl[\sum_{j=0}^{m-1} \biggl(\prod _{j< i\leq m}\frac{(t_{i+1}-t_{i})^{\alpha _{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}}y(t_{j+1})+ \varphi _{j+1}\bigl(x(t_{j+1})\bigr) \bigr\} \Biggr]+ \frac{b}{\Omega }{}_{t_{m}}I_{q_{m}}^{\alpha_{m}}y(T) \\ &{}+\sum_{l=0}^{m}\frac{c_{l}(t_{l+1}-t_{l})^{\alpha_{l}+\gamma_{l}-1}}{\Omega \Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})} \Biggl[\sum_{j=0}^{l-1} \biggl(\prod _{j< i\leq l-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha _{i})} \biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}}y(t_{j+1})+ \varphi _{j+1}\bigl(x(t_{j+1})\bigr) \bigr\} \Biggr]+\sum _{l=0}^{m}\frac{c_{l}}{\Omega }{}_{t_{l}}I_{q_{l}}^{\alpha_{l}+\gamma_{l}}y(t_{l+1}). \end{aligned}$$
Substituting the value of \({}_{t_{0}}I_{q_{0}}^{1-\alpha_{0}}x(0)\) into (2.8) yields (2.4). The converse follows by direct computation. This completes the proof. □

Lemma 2.5

Assume that all conditions of Lemma  2.4 hold. In addition, assume that \(\sup_{t\in J}|y(t)|=N_{1}\) and there exists a constant \(N_{2}\) such that \(|\varphi_{k}(x)|\leq N_{2}\) for \(k=1,2,\ldots,m\) and \(x\in\mathbb{R}\). Then the following inequality holds:
$$ \bigl\vert x(t)\bigr\vert \leq\Psi_{1}N_{1}+ \Psi_{2}N_{2} $$
(2.9)
for all \(t\in J\), where
$$\begin{aligned} \Psi_{1} =& \Biggl(\prod_{j=0}^{m} \frac{(t_{j+1}-t_{j})^{\alpha _{j}-1}}{\Gamma_{q_{j}}(\alpha_{j})} \Biggr) \Biggl\{ \frac{|b|}{|\Omega |} \Biggl[\sum _{j=0}^{m} \biggl(\prod _{j< i\leq m}\frac {(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr)\frac {(t_{j+1}-t_{j})^{\alpha_{j}}}{\Gamma_{q_{j}}(\alpha_{j}+1)} \Biggr] \\ &{}+\sum_{l=0}^{m}\frac{|c_{l}|(t_{l+1}-t_{l})^{\alpha_{l}+\gamma _{l}-1}}{|\Omega|\Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})} \Biggl[\sum_{j=0}^{l-1} \biggl(\prod _{j< i\leq l-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma _{q_{i}}(\alpha_{i})} \biggr)\frac{(t_{j+1}-t_{j})^{\alpha_{j}}}{\Gamma _{q_{j}}(\alpha_{j}+1)} \Biggr] \\ &{}+\sum_{l=0}^{m}\frac{|c_{l}|}{|\Omega|} \frac{(t_{l+1}-t_{l})^{\alpha _{l}+\gamma_{l}}}{\Gamma_{q_{l}}(\alpha_{l}+\gamma_{l}+1)} \Biggr\} \\ &{}+\sum_{j=0}^{m} \biggl(\prod _{j< i\leq m}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr)\frac{(t_{j+1}-t_{j})^{\alpha_{j}}}{\Gamma_{q_{j}}(\alpha_{j}+1)} \end{aligned}$$
and
$$\begin{aligned} \Psi_{2} =& \Biggl(\prod_{j=0}^{m} \frac{(t_{j+1}-t_{j})^{\alpha _{j}-1}}{\Gamma_{q_{j}}(\alpha_{j})} \Biggr) \Biggl\{ \frac{|b|}{|\Omega |}\sum _{j=0}^{m-1} \biggl(\prod _{j< i\leq m}\frac {(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \\ &{}+\sum_{l=0}^{m}\frac{|c_{l}|(t_{l+1}-t_{l})^{\alpha_{l}+\gamma _{l}-1}}{|\Omega|\Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})}\sum _{j=0}^{l-1} \biggl(\prod _{j< i\leq l-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma _{q_{i}}(\alpha_{i})} \biggr) \Biggr\} \\ &{}+\sum_{j=0}^{m-1} \biggl(\prod _{j< i\leq m}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr). \end{aligned}$$

Proof

For any \(t\in J_{k}\), we have
$$\begin{aligned} \bigl\vert x(t)\bigr\vert \leq&\frac{(t-t_{k})^{\alpha_{k}-1}}{\Gamma_{q_{k}}(\alpha_{k})} \Biggl(\prod _{j=0}^{k-1}\frac{(t_{j+1}-t_{j})^{\alpha_{j}-1}}{\Gamma _{q_{j}}(\alpha_{j})} \Biggr) \Biggl\{ \frac{|b|}{|\Omega|} \Biggl[\sum_{j=0}^{m-1} \biggl(\prod_{j< i\leq m}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha _{i})} \biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}}\bigl\vert y(t_{j+1})\bigr\vert +\bigl\vert \varphi _{j+1} \bigl(x(t_{j+1})\bigr)\bigr\vert \bigr\} \Biggr]+\frac{\vert b\vert }{\vert \Omega \vert }{}_{t_{m}}I_{q_{m}}^{\alpha_{m}} \bigl\vert y(T)\bigr\vert \\ &{}+\sum_{l=0}^{m}\frac{\vert c_{l}\vert (t_{l+1}-t_{l})^{\alpha_{l}+\gamma _{l}-1}}{\vert \Omega \vert \Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})} \Biggl[\sum_{j=0}^{l-1} \biggl(\prod _{j< i\leq l-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma _{q_{i}}(\alpha_{i})} \biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}}\bigl\vert y(t_{j+1})\bigr\vert +\bigl\vert \varphi _{j+1} \bigl(x(t_{j+1})\bigr)\bigr\vert \bigr\} \Biggr] +\sum _{l=0}^{m}\frac{\vert c_{l}\vert }{\vert \Omega \vert }{}_{t_{l}}I_{q_{l}}^{\alpha_{l}+\gamma _{l}} \bigl\vert y(t_{l+1})\bigr\vert \Biggr\} \\ &{}+\frac{(t-t_{k})^{\alpha_{k}-1}}{\Gamma_{q_{k}}(\alpha_{k})} \Biggl[\sum_{j=0}^{k-1} \biggl(\prod_{j< i\leq k-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}}\bigl\vert y(t_{j+1})\bigr\vert +\bigl\vert \varphi _{j+1} \bigl(x(t_{j+1})\bigr)\bigr\vert \bigr\} \Biggr] \\ &{}+{}_{t_{k}}I_{q_{k}}^{\alpha_{k}}\bigl\vert y(t)\bigr\vert \\ \leq&\frac{(T-t_{m})^{\alpha_{m}-1}}{\Gamma_{q_{m}}(\alpha_{m})} \Biggl(\prod_{j=0}^{m-1} \frac{(t_{j+1}-t_{j})^{\alpha_{j}-1}}{\Gamma _{q_{j}}(\alpha_{j})} \Biggr) \Biggl\{ \frac{\vert b\vert }{\vert \Omega \vert } \Biggl[\sum _{j=0}^{m-1} \biggl(\prod _{j< i\leq m}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha _{i})} \biggr) \\ &{}\times \bigl\{ N_{1}{}_{t_{j}}I_{q_{j}}^{\alpha_{j}}1+N_{2} \bigr\} \Biggr]+\frac {\vert b\vert }{\vert \Omega \vert }N_{1}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}}1 +\sum_{l=0}^{m}\frac{\vert c_{l}\vert (t_{l+1}-t_{l})^{\alpha_{l}+\gamma _{l}-1}}{\vert \Omega \vert \Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})} \\ &{}\times \Biggl[\sum_{j=0}^{l-1} \biggl( \prod_{j< i\leq l-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \bigl\{ N_{1}{}_{t_{j}}I_{q_{j}}^{\alpha_{j}}1+N_{2} \bigr\} \Biggr] +\sum_{l=0}^{m} \frac{\vert c_{l}\vert }{\vert \Omega \vert }N_{1}{}_{t_{l}}I_{q_{l}}^{\alpha _{l}+\gamma_{l}}1 \Biggr\} \\ &{}+\frac{(T-t_{m})^{\alpha_{m}-1}}{\Gamma_{q_{m}}(\alpha_{m})} \Biggl[\sum_{j=0}^{m-1} \biggl(\prod_{j< i\leq m-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \bigl\{ N_{1}{}_{t_{j}}I_{q_{j}}^{\alpha_{j}}1+N_{2} \bigr\} \Biggr] +N_{1}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}}1 \\ =& \Biggl(\prod_{j=0}^{m} \frac{(t_{j+1}-t_{j})^{\alpha_{j}-1}}{\Gamma _{q_{j}}(\alpha_{j})} \Biggr) \Biggl\{ \frac{\vert b\vert }{\vert \Omega \vert } \Biggl[\sum _{j=0}^{m-1} \biggl(\prod _{j< i\leq m}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha _{i})} \biggr) \\ &{}\times \biggl\{ N_{1}\frac{(t_{j+1}-t_{j})^{\alpha_{j}}}{\Gamma _{q_{j}}(\alpha_{j}+1)}+N_{2} \biggr\} \Biggr]+\frac{\vert b\vert }{\vert \Omega \vert }N_{1}\frac {(T-t_{m})^{\alpha_{m}}}{\Gamma_{q_{m}}(\alpha_{m}+1)} +\sum _{l=0}^{m}\frac{\vert c_{l}\vert (t_{l+1}-t_{l})^{\alpha_{l}+\gamma _{l}-1}}{\vert \Omega \vert \Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})} \\ &{}\times \Biggl[\sum_{j=0}^{l-1} \biggl( \prod_{j< i\leq l-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \biggl\{ N_{1}\frac{(t_{j+1}-t_{j})^{\alpha_{j}}}{\Gamma_{q_{j}}(\alpha _{j}+1)}+N_{2} \biggr\} \Biggr] \\ &{}+\sum_{l=0}^{m}\frac{\vert c_{l}\vert }{\vert \Omega \vert }N_{1} \frac{(t_{l+1}-t_{l})^{\alpha _{l}+\gamma_{l}}}{\Gamma_{q_{l}}(\alpha_{l}+\gamma_{l}+1)} \Biggr\} + \Biggl[\sum_{j=0}^{m-1} \biggl(\prod_{j< i\leq m}\frac {(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \\ &{}\times \biggl\{ N_{1}\frac{(t_{j+1}-t_{j})^{\alpha_{j}}}{\Gamma _{q_{j}}(\alpha_{j}+1)}+N_{2} \biggr\} \Biggr] +N_{1}\frac{(T-t_{m})^{\alpha_{m}}}{\Gamma_{q_{m}}(\alpha_{m}+1)} \\ \leq& \Psi_{1}N_{1}+\Psi_{2}N_{2}. \end{aligned}$$
This completes the proof. □

3 Main results

In view of Lemma 2.4, we define the operator \(\mathcal{L}:\mathit{PC}(J,\mathbb{R})\to \mathit{PC}(J,\mathbb{R})\) by
$$\begin{aligned} \mathcal{L}x(t) =&\frac{(t-t_{k})^{\alpha_{k}-1}}{\Gamma_{q_{k}}(\alpha _{k})} \Biggl(\prod _{j=0}^{k-1}\frac{(t_{j+1}-t_{j})^{\alpha_{j}-1}}{\Gamma _{q_{j}}(\alpha_{j})} \Biggr) \Biggl\{ \frac{b}{\Omega} \Biggl[\sum_{j=0}^{m-1} \biggl(\prod_{j< i\leq m}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha _{i})} \biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha _{j}}f \bigl(t_{j+1},x(t_{j+1})\bigr)+\varphi_{j+1} \bigl(x(t_{j+1})\bigr) \bigr\} \Biggr]+\frac {b}{\Omega}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}}f \bigl(T,x(T)\bigr) \\ &{}+\sum_{l=0}^{m}\frac{c_{l}(t_{l+1}-t_{l})^{\alpha_{l}+\gamma_{l}-1}}{\Omega \Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})} \Biggl[\sum_{j=0}^{l-1} \biggl(\prod _{j< i\leq l-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma _{q_{i}}(\alpha_{i})} \biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha _{j}}f \bigl(t_{j+1},x(t_{j+1})\bigr)+\varphi_{j+1} \bigl(x(t_{j+1})\bigr) \bigr\} \Biggr] +\sum _{l=0}^{m}\frac{c_{l}}{\Omega}{}_{t_{l}}I_{q_{l}}^{\alpha_{l}+\gamma _{l}}f \bigl(t_{l+1},x(t_{l+1})\bigr) \Biggr\} \\ &{}+\frac{(t-t_{k})^{\alpha_{k}-1}}{\Gamma_{q_{k}}(\alpha_{k})} \Biggl[\sum_{j=0}^{k-1} \biggl(\prod_{j< i\leq k-1}\frac{(t_{i+1}-t_{i})^{\alpha _{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha _{j}}f \bigl(t_{j+1},x(t_{j+1})\bigr)+\varphi_{j+1} \bigl(x(t_{j+1})\bigr) \bigr\} \Biggr] +{}_{t_{k}}I_{q_{k}}^{\alpha_{k}}f \bigl(t,x(t)\bigr), \end{aligned}$$
(3.1)
where
$$ {}_{a}I_{q}^{p} f\bigl(u,x(u)\bigr)= \frac{1}{\Gamma_{q}(p)} \int_{a}^{u}{}_{a}\bigl(u-{}_{a} \Phi _{q}(s)\bigr)_{q}^{(p-1)}f\bigl(s,x(s) \bigr){}_{a}\, d_{q}s, $$
\(a\in\{t_{0},t_{1},\ldots,t_{m}\}\), \(q\in\{q_{0},q_{1},\ldots,q_{m}\}\), \(p\in\{\alpha_{0},\alpha_{1},\ldots,\alpha_{m},\alpha_{0}+\gamma _{0},\alpha_{1}+\gamma_{1},\ldots,\alpha_{m}+\gamma_{m}\}\), \(u\in\{t,t_{1},t_{2},\ldots,t_{m},T\}\).

Now we present our first result, which deals with the existence and uniqueness of solutions for problem (1.1) and is based on the Banach contraction mapping principle.

Theorem 3.1

Assume that there exist a function \(\mathcal{M}\in C(J,\mathbb{R}^{+})\) and a positive constant \(M_{2}\) such that
(H1): 

\(|f(t,x)-f(t,y)|\leq\mathcal{M}(t)|x-y|\) and \(|\varphi_{k}(x)-\varphi _{k}(y)|\leq M_{2}|x-y|\) for \(t\in J\), \(x,y\in\mathbb{R}\) and \(k=1,2,\ldots,m\).

Then problem (1.1) has a unique solution on J if
$$ (M_{1}\Psi_{1}+M_{2} \Psi_{2})T^{\beta}< 1, $$
(3.2)
where \(M_{1}=\sup_{t\in J}|\mathcal{M}(t)|\), the constants \(\Psi_{1}\), \(\Psi_{2}\) are defined in Lemma  2.5, and \(\beta>0\).

Proof

Consider the operator \(\mathcal{L}:\mathit{PC}(J,\mathbb{R})\to \mathit{PC}(J,\mathbb{R})\) defined by (3.1) and show that \(\mathcal{L}\in \mathit{PC}_{\beta}\). For this, let \(\tau_{1},\tau_{2}\in J_{k}\). Then we have
$$\begin{aligned}& \bigl\vert (\tau_{1}-t_{k})^{\beta}\mathcal{L}x( \tau_{1})-(\tau _{2}-t_{k})^{\beta} \mathcal{L}x(\tau_{2})\bigr\vert \\& \quad \leq \biggl\vert \frac{(\tau_{1}-t_{k})^{\beta+\alpha_{k}-1}-(\tau_{2}-t_{k})^{\beta +\alpha_{k}-1}}{\Gamma_{q_{k}}(\alpha_{k})}\biggr\vert K_{x} \\& \qquad {} +\bigl\vert (\tau_{1}-t_{k})^{\beta}{}_{t_{k}}I_{q_{k}}^{\alpha_{k}}f \bigl(\tau_{1}, x(\tau_{1})\bigr)-(\tau_{2}-t_{k})^{\beta}{}_{t_{k}}I_{q_{k}}^{\alpha_{k}}f \bigl(\tau_{2}, x(\tau_{2})\bigr)\bigr\vert , \end{aligned}$$
where
$$\begin{aligned} K_{x} :=& \Biggl(\prod_{j=0}^{k-1} \frac{(t_{j+1}-t_{j})^{\alpha _{j}-1}}{\Gamma_{q_{j}}(\alpha_{j})} \Biggr) \Biggl\{ \frac{\vert b\vert }{\vert \Omega \vert } \Biggl[\sum _{j=0}^{m-1} \biggl(\prod _{j< i\leq m}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha _{i})} \biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha _{j}}\bigl\vert f\bigl(t_{j+1},x(t_{j+1})\bigr)\bigr\vert +\bigl\vert \varphi_{j+1}\bigl(x(t_{j+1})\bigr)\bigr\vert \bigr\} \Biggr]+\frac{\vert b\vert }{\vert \Omega \vert }{}_{t_{m}}I_{q_{m}}^{\alpha_{m}}\bigl\vert f\bigl(T,x(T)\bigr)\bigr\vert \\ &{}+\sum_{l=0}^{m}\frac{\vert c_{l}\vert (t_{l+1}-t_{l})^{\alpha_{l}+\gamma _{l}-1}}{\vert \Omega \vert \Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})} \Biggl[\sum_{j=0}^{l-1} \biggl(\prod _{j< i\leq l-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma _{q_{i}}(\alpha_{i})} \biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha _{j}}\bigl\vert f\bigl(t_{j+1},x(t_{j+1})\bigr)\bigr\vert +\bigl\vert \varphi_{j+1}\bigl(x(t_{j+1})\bigr)\bigr\vert \bigr\} \Biggr] \\ &{}+\sum_{l=0}^{m}\frac{\vert c_{l}\vert }{\vert \Omega \vert }{}_{t_{l}}I_{q_{l}}^{\alpha_{l}+\gamma _{l}} \bigl\vert f\bigl(t_{l+1},x(t_{l+1})\bigr)\bigr\vert \Biggr\} \\ &{}+ \Biggl[\sum_{j=0}^{k-1} \biggl(\prod _{j< i\leq k-1}\frac {(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha _{j}}\bigl\vert f\bigl(t_{j+1},x(t_{j+1})\bigr)\bigr\vert +\bigl\vert \varphi_{j+1}\bigl(x(t_{j+1})\bigr)\bigr\vert \bigr\} \Biggr]. \end{aligned}$$
(3.3)
As \(\tau_{1}\to\tau_{2}\), we get \(\vert (\tau_{1}-t_{k})^{\beta}\mathcal{L}x(\tau_{1})-(\tau _{2}-t_{k})^{\beta}\mathcal{L}x(\tau_{2})\vert \to 0\) for each \(k=0,1,\ldots,m\). Thus, \(\mathcal{L}x(t)\in \mathit{PC}_{\beta}\).
Now we define the ball \(B_{r}=\{x\in \mathit{PC}_{\beta}(J,\mathbb{R}) : \|x\|_{\mathit{PC}_{\beta}}\leq r\}\). We will show that \(\mathcal{L}B_{r}\subset B_{r}\). Let \(\sup_{t\in J}|f(t,0)|=A_{1}\), \(\max\{|\varphi(0)| : k=1,\ldots,m\}=A_{2}\) and choose a constant r such that
$$ r\geq \frac{(A_{1}\Psi_{1}+A_{2}\Psi_{2})T^{\beta}}{1-(M_{1}\Psi_{1}+M_{2}\Psi _{2})T^{\beta}}. $$
Then, for any \(x\in B_{r}\) and \(t\in J\), we have
$$ (t-t_{k})^{\beta}\bigl\vert \mathcal{L}x(t) \bigr\vert \leq \frac{(t-t_{k})^{\beta+\alpha _{k}-1}}{\Gamma_{q_{k}}(\alpha_{k})}K_{x} +(t-t_{k})^{\beta}{}_{t_{k}}I_{q_{k}}^{\alpha_{k}} \bigl\vert f\bigl(t,x(t)\bigr)\bigr\vert , $$
(3.4)
where \(K_{x}\) is given by (3.3). Using the inequalities
$$\begin{aligned}& \bigl\vert f(s,x)\bigr\vert \leq \bigl\vert f(s,x)-f(s,0)\bigr\vert + \bigl\vert f(s,0)\bigr\vert \leq M_{1}r+A_{1}, \\& \bigl\vert \varphi(x)\bigr\vert \leq \bigl\vert \varphi(x)-\varphi(0)\bigr\vert +\bigl\vert \varphi(0)\bigr\vert \leq M_{2}r+A_{2} \end{aligned}$$
in (3.4) for \(x\in B_{r}\) and \(s\in J\) and the computational details of Lemma 2.5, together with
$$\begin{aligned} K_{x} \leq& \Biggl(\prod_{j=0}^{k-1} \frac{(t_{j+1}-t_{j})^{\alpha _{j}-1}}{\Gamma_{q_{j}}(\alpha_{j})} \Biggr) \Biggl\{ \frac{|b|}{|\Omega|} \Biggl[\sum _{j=0}^{m-1} \biggl(\prod _{j< i\leq m}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha _{i})} \biggr) \\ &{}\times \biggl\{ (M_{1}r+A_{1}) \biggl( \frac{(t_{j+1}-t_{j})^{\alpha _{j}}}{\Gamma(\alpha_{j}+1)} \biggr)+(M_{2}r+A_{2}) \biggr\} \Biggr] \\ &{}+(M_{1}r+A_{1}) \frac{|b|}{|\Omega|} \biggl( \frac{(T-t_{m})^{\alpha_{m}}}{\Gamma(\alpha _{m}+1)} \biggr) \\ &{}+\sum_{l=0}^{m}\frac{|c_{l}|(t_{l+1}-t_{l})^{\alpha_{l}+\gamma _{l}-1}}{|\Omega|\Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})} \Biggl[\sum_{j=0}^{l-1} \biggl(\prod _{j< i\leq l-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma _{q_{i}}(\alpha_{i})} \biggr) \\ &{}\times \biggl\{ (M_{1}r+A_{1}) \biggl( \frac{(t_{j+1}-t_{j})^{\alpha _{j}}}{\Gamma(\alpha_{j}+1)} \biggr)+(M_{2}r+A_{2}) \biggr\} \Biggr] + \sum_{l=0}^{m}\frac{|c_{l}|}{|\Omega|} \frac{(t_{l+1}-t_{l})^{\alpha _{l}+\gamma_{l}}}{\Gamma_{q_{l}}(\alpha_{l}+\gamma_{l}+1)} \Biggr\} \\ &{}+ \Biggl[\sum_{j=0}^{k-1} \biggl(\prod _{j< i\leq k-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \biggl\{ (M_{1}r+A_{1}) \biggl(\frac{(t_{j+1}-t_{j})^{\alpha_{j}}}{\Gamma (\alpha_{j}+1)} \biggr)+(M_{2}r+A_{2}) \biggr\} \Biggr], \end{aligned}$$
we obtain
$$\begin{aligned} (t-t_{k})^{\beta}\bigl\vert \mathcal{L}x(t)\bigr\vert \leq&(t-t_{k})^{\beta} \bigl(\Psi _{1}(M_{1}r+A_{1})+ \Psi_{2}(M_{2}r+A_{2}) \bigr) \\ \leq& r(\Psi_{1}+M_{1})T^{\beta}+( \Psi_{1}A_{1}+\Psi_{2}A_{2})T^{\beta} \\ \leq& r. \end{aligned}$$
This implies that \(\|\mathcal{L}x\|_{\mathit{PC}_{\beta}}\leq r\) and, consequently, \(\mathcal{L}B_{r}\subset B_{r}\).
For all \(x,y\in \mathit{PC}_{\beta}(J,\mathbb{R})\) and \(t\in J\), as in Lemma 2.5, we get
$$ \bigl\vert \mathcal{L}x(t)-\mathcal{L}y(t)\bigr\vert \leq (M_{1} \Psi_{1}+M_{2}\Psi_{2})\|x-y\|_{\mathit{PC}_{\beta}}. $$
Multiplying both sides of this inequality by \((t-t_{k})^{\beta}\) for each \(t\in J_{k}\), we have
$$\begin{aligned} (t-t_{k})^{\beta}\bigl\vert \mathcal{L}x(t)-\mathcal{L}y(t) \bigr\vert \leq& (t-t_{k})^{\beta}(M_{1} \Psi_{1}+M_{2}\Psi_{2})\|x-y\|_{\mathit{PC}_{\beta}} \\ \leq& T^{\beta}(M_{1}\Psi_{1}+M_{2} \Psi_{2})\|x-y\|_{\mathit{PC}_{\beta}}, \end{aligned}$$
which leads to \(\|\mathcal{L}x-\mathcal{L}y\|_{\mathit{PC}_{\beta}}\leq T^{\beta}(M_{1}\Psi_{1}+M_{2}\Psi_{2})\|x-y\|_{\mathit{PC}_{\beta}}\). In view of condition (3.2), it follows by the Banach contraction mapping principle that the operator \(\mathcal{L}\) is a contraction. Hence, \(\mathcal{L}\) has a fixed point, which is a unique solution of problem (1.1) on J. □

The next existence result is based on Leray-Schauder’s nonlinear alternative.

Lemma 3.1

(Nonlinear alternative for single valued maps [17])

Let E be a Banach space, C a closed, convex subset of E, U an open subset of C, and \(0\in U\). Suppose that \(F:\overline{U}\to C\) is continuous and compact (that is, \(F(\overline{U})\) is a relatively compact subset of C) map. Then either
  1. (i)

    F has a fixed point in , or

     
  2. (ii)

    there are \(u\in\partial U\) (the boundary of U in C) and \(\theta\in(0,1)\) with \(u=\theta F(u)\).

     

Theorem 3.2

Assume that
(H2): 
there exist continuous nondecreasing functions \(Q,V:[0,\infty)\to(0,\infty)\) and a continuous function \(p:J\to \mathbb{R}^{+}\) such that
$$ \bigl\vert f(t,x)\bigr\vert \leq p(t)Q\bigl(\vert x\vert \bigr)\quad \textit{and}\quad \bigl\vert \varphi_{k}(x)\bigr\vert \leq V\bigl( \vert x\vert \bigr) $$
(3.5)
for all \((t,x)\in(J\times\mathbb{R})\) and \(k=1,2,\ldots,m\);
(H3): 
there exists a constant \(M^{*}>0\) such that such that
$$ \frac{M^{*}}{(p^{*}Q(M^{*})\Psi_{1}+V(M^{*})\Psi_{2})T^{\beta}}> 1, $$
(3.6)
where \(p^{*}=\sup_{t\in J}|p(t)|\), \(\beta>0\), and the constants \(\Psi_{1}\), \(\Psi_{2}\) are defined in Lemma  2.5.
Then problem (1.1) has at least one solution on J.

Proof

First, we show that the operator \({\mathcal {L}}\) defined by (3.1) maps bounded sets (balls) into bounded sets in \(\mathit{PC}_{\beta}\). To accomplish this, for a positive number ρ, let \(B_{\rho} = \{x \in \mathit{PC}_{\beta}: \|x\|_{\mathit{PC}_{\beta}} \le \rho\}\) be a ball in \(\mathit{PC}_{\beta}\). Then, for \(x\in B_{\rho}\) and \(t\in J\), using the method of proof used in Lemma 2.5, we obtain
$$\bigl\vert \mathcal{L}x(t)\bigr\vert \leq \frac{(t-t_{k})^{\alpha_{k}-1}}{\Gamma_{q_{k}}(\alpha _{k})}K_{x}+{}_{t_{k}}I_{q_{k}}^{\alpha_{k}} \bigl\vert f\bigl(t,x(t)\bigr)\bigr\vert , $$
where \(K_{x}\) is defined by (3.3). From (H2) we have
$$\begin{aligned} K_{x} \leq& \Biggl(\prod_{j=0}^{k-1} \frac{(t_{j+1}-t_{j})^{\alpha _{j}-1}}{\Gamma_{q_{j}}(\alpha_{j})} \Biggr) \Biggl\{ \frac{|b|}{|\Omega|} \Biggl[\sum _{j=0}^{m-1} \biggl(\prod _{j< i\leq m}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha _{i})} \biggr) \\ &{}\times \biggl\{ p^{*}Q(\rho) \biggl(\frac{(t_{j+1}-t_{j})^{\alpha _{j}}}{\Gamma(\alpha_{j}+1)} \biggr)+V(\rho) \biggr\} \Biggr]+p^{*}Q(\rho) \frac{|b|}{|\Omega|} \biggl(\frac{(T-t_{m})^{\alpha_{m}}}{\Gamma(\alpha _{m}+1)} \biggr) \\ &{}+\sum_{l=0}^{m}\frac{|c_{l}|(t_{l+1}-t_{l})^{\alpha_{l}+\gamma _{l}-1}}{|\Omega|\Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})} \Biggl[\sum_{j=0}^{l-1} \biggl(\prod _{j< i\leq l-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma _{q_{i}}(\alpha_{i})} \biggr) \\ &{}\times \biggl\{ p^{*}Q(\rho) \biggl(\frac{(t_{j+1}-t_{j})^{\alpha _{j}}}{\Gamma(\alpha_{j}+1)} \biggr)+V(\rho) \biggr\} \Biggr] +\sum_{l=0}^{m}\frac{|c_{l}|}{|\Omega|} \frac{(t_{l+1}-t_{l})^{\alpha _{l}+\gamma_{l}}}{\Gamma_{q_{l}}(\alpha_{l}+\gamma_{l}+1)} \Biggr\} \\ &{}+ \Biggl[\sum_{j=0}^{k-1} \biggl(\prod _{j< i\leq k-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})} \biggr) \biggl\{ p^{*}Q(\rho) \biggl(\frac{(t_{j+1}-t_{j})^{\alpha_{j}}}{\Gamma (\alpha_{j}+1)} \biggr)+V(\rho) \biggr\} \Biggr], \end{aligned}$$
and thus
$$\bigl\vert \mathcal{L}x(t)\bigr\vert \le p^{*}Q(\rho)\Psi_{1}+V( \rho)\Psi_{2}. $$
Therefore, \((t-t_{k})^{\beta}|\mathcal{L}x(t)|\leq (t-t_{k})^{\beta}(p^{*}Q(\rho)\Psi_{1}+V(\rho)\Psi_{2})\), which means that \(\|\mathcal{L}x\|_{\mathit{PC}_{\beta}}\leq T^{\beta}(p^{*}Q(\rho)\Psi_{1}+V(\rho)\Psi_{2})\).

Next we show that \(\mathcal{L}\) maps bounded sets into equicontinuous sets of \(\mathit{PC}_{\beta}\).

Letting \(\tau_{1}, \tau_{2}\in J_{k}\) for some \(k\in\{0, 1,2,\ldots, m\}\) with \(\tau_{1}<\tau_{2}\) and \(x\in B_{\rho}\), where \(B_{\rho}\) is a ball in \(\mathit{PC}_{\beta}\), we have
$$\begin{aligned} \bigl\vert \mathcal{L}x(\tau_{2})-\mathcal{L}x( \tau_{1})\bigr\vert \leq& \biggl\vert \frac{(\tau_{2}-t_{k})^{\alpha_{k}-1}-(\tau_{1}-t_{k})^{\alpha _{k}-1}}{\Gamma_{q_{k}}(\alpha_{k})}\biggr\vert K_{x} \\ &{}+\bigl\vert {}_{t_{k}}I_{q_{k}}^{\alpha_{k}}f\bigl( \tau_{2}, x(\tau _{2})\bigr)-{}_{t_{k}}I_{q_{k}}^{\alpha_{k}}f \bigl(\tau_{1}, x(\tau_{1})\bigr)\bigr\vert \\ \leq&\biggl\vert \frac{(\tau_{2}-t_{k})^{\alpha_{k}-1}-(\tau_{1}-t_{k})^{\alpha _{k}-1}}{\Gamma_{q_{k}}(\alpha_{k})}\biggr\vert K_{x} \\ &{}+ p^{*}Q(\rho)\biggl\vert \frac{(\tau_{2}-t_{k})^{\alpha_{k}}-(\tau _{1}-t_{k})^{\alpha_{k}}}{\Gamma_{q_{k}}(\alpha_{k}+1)}\biggr\vert . \end{aligned}$$
(3.7)
As \(\tau_{1}\rightarrow\tau_{2}\), the right-hand side of inequality (3.7) tends to zero independently of x, that is,
$$ \bigl\vert (\tau_{2}-t_{k})^{\beta}\mathcal{L}x( \tau_{2})-(\tau _{1}-t_{k})^{\beta} \mathcal{L}x(\tau_{1})\bigr\vert \to 0 \quad \text{as } | \tau_{2}-\tau_{1}|\to0. $$
Therefore, by the Arzelà-Ascoli theorem, \(\mathcal{L}:\mathit{PC}_{\beta}\rightarrow \mathit{PC}_{\beta}\) is completely continuous.
Our result will follow from the Leray-Schauder nonlinear alternative once we show the boundedness of the set of all solutions to the equation \(x(t)=\lambda\mathcal{L} x(t)\) for \(0<\lambda<1\). Let x be a solution. For any \(t\in J\) and \(x\in \mathit{PC}_{\beta}\), following the method of proof used in the first step together with condition (H2), we get
$$ \Vert x\Vert _{\mathit{PC}_{\beta}}\leq \bigl(p^{*}Q\bigl(\Vert x\Vert _{\mathit{PC}_{\beta}}\bigr)\Psi_{1}+V\bigl(\Vert x\Vert _{\mathit{PC}_{\beta}}\bigr)\Psi _{2}\bigr)T^{\beta}. $$
In consequence, we have
$$ \frac{\|x\|_{\mathit{PC}_{\beta}}}{(p^{*}Q(\|x\|_{\mathit{PC}_{\beta}})\Psi_{1}+V(\|x\| _{\mathit{PC}_{\beta}})\Psi_{2})T^{\beta}}\leq 1. $$
By condition (H3) there exists \(M^{*}\) such that \(\|x\|_{\mathit{PC}_{\beta}}\neq M^{*}\). We define \(U = \{x \in \mathit{PC}_{\beta} : \|x\|_{\mathit{PC}_{\beta}} < M^{*}\}\). Note that the operator \(\mathcal{L} :\overline{U} \to \mathit{PC}_{\beta}\) is continuous and completely continuous. By the choice of U there is no \(x \in\partial U\) such that \(x=\lambda{\mathcal {L}}x\) for some \(\lambda\in(0,1)\). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.1) we deduce that \({\mathcal {L}}\) has a fixed point \(x \in\overline{U}\), which is a solution of problem (1.1) on J. This completes the proof. □

A key to prove the final result is based on the following fixed point theorem.

Lemma 3.2

[18]

Suppose that \(A:\bar{\Omega}\rightarrow E\) is a completely continuous operator. Suppose that one of the following condition is satisfied:
  1. (i)

    (Altman) \(\|Ax-x\|^{2}\geq\|Ax\|^{2}-\| x\|^{2} \) for all \(x\in\partial\Omega\),

     
  2. (ii)

    (Rothe) \(\|Ax\|\leq\|x\|\) for all \(x\in\partial\Omega\),

     
  3. (iii)

    (Petryshyn) \(\|Ax\|\leq\|Ax-x\|\) for all \(x\in\partial\Omega\).

     
Then \(\deg(I-A,\Omega,\theta) = 1\), and hence A has at least one fixed point in Ω.

Theorem 3.3

Assume that
(H4): 
the continuous functions \(f:J\times\mathbb{R}\to \mathbb{R}\) and \(\varphi_{k}:\mathbb{R}\to\mathbb{R}\), \(k=1,2,\ldots,m\), satisfy
$$ \lim_{x\rightarrow0} \frac{f(t,x)}{x}=0 \quad \textit{and}\quad \lim _{x\rightarrow0}\frac{\varphi_{k}(x)}{x}=0, \quad k=1,2,\ldots,m. $$
(3.8)
Then problem (1.1) has at least one solution on J.

Proof

Let \(x\in \mathit{PC}_{\beta}\). Taking ε sufficiently small, we can choose two positive constants \(\delta_{1}\) and \(\delta_{2}\) such that \(|f(t,x)|<\varepsilon|x|\) whenever \(\|x\|_{\mathit{PC}_{\beta}}<\delta_{1}\) and \(\varphi_{k}(x)<\varepsilon|x|\) whenever \(\|x\|_{\mathit{PC}_{\beta}}<\delta_{2}\) for \(k=1,2,\ldots,m\). Setting \(\delta=\min\{\delta_{1},\delta_{2}\}\), we define the open ball \(B_{\delta}=\{x\in \mathit{PC}_{\beta} : \|x\|_{\mathit{PC}_{\beta}}<\delta\}\). As in Theorem 3.2, it is clear that the operator \(\mathcal{L}:\mathit{PC} \to \mathit{PC}\) is completely continuous. For any \(x\in \partial B_{\delta}\), we have
$$\begin{aligned} \bigl\vert \mathcal{L}x(t)\bigr\vert =&\frac{(t-t_{k})^{\alpha_{k}-1}}{\Gamma_{q_{k}}(\alpha_{k})}\Biggl(\prod _{j=0}^{k-1}\frac{(t_{j+1}-t_{j})^{\alpha_{j}-1}}{\Gamma_{q_{j}}(\alpha _{j})}\Biggr) \Biggl\{ \frac{\vert b\vert }{\vert \Omega \vert } \Biggl[\sum_{j=0}^{m-1} \biggl(\prod_{j< i\leq m}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma_{q_{i}}(\alpha _{i})}\biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}} \bigl\vert f\bigl(t_{j+1},x(t_{j+1})\bigr)\bigr\vert +\bigl\vert \varphi_{j+1}\bigl(x(t_{j+1})\bigr) \bigr\vert \bigr\} \Biggr] +\frac{\vert b\vert }{\vert \Omega \vert }{}_{t_{m}}I_{q_{m}}^{\alpha_{m}} \bigl\vert f\bigl(T,x(T)\bigr)\bigr\vert \\ &{}+\sum_{l=0}^{m}\frac{\vert c_{l}\vert (t_{l+1}-t_{l})^{\alpha_{l}+\gamma _{l}-1}}{\vert \Omega \vert \Gamma_{q_{l}}(\alpha_{l}+\gamma_{l})} \Biggl[\sum_{j=0}^{l-1} \biggl(\prod _{j< i\leq l-1}\frac{(t_{i+1}-t_{i})^{\alpha_{i}-1}}{\Gamma _{q_{i}}(\alpha_{i})}\biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}} \bigl\vert f\bigl(t_{j+1},x(t_{j+1})\bigr)\bigr\vert +\bigl\vert \varphi_{j+1}\bigl(x(t_{j+1})\bigr) \bigr\vert \bigr\} \Biggr] +\sum_{l=0}^{m} \frac{\vert c_{l}\vert }{\vert \Omega \vert }{}_{t_{l}}I_{q_{l}}^{\alpha_{l}+\gamma _{l}}\bigl\vert f \bigl(t_{l+1},x(t_{l+1})\bigr)\bigr\vert \Biggr\} \\ &{}+\frac{(t-t_{k})^{\alpha_{k}-1}}{\Gamma_{q_{k}}(\alpha_{k})} \Biggl[\sum_{j=0}^{k-1} \biggl(\prod_{j< i\leq k-1}\frac{(t_{i+1}-t_{i})^{\alpha _{i}-1}}{\Gamma_{q_{i}}(\alpha_{i})}\biggr) \\ &{}\times \bigl\{ {}_{t_{j}}I_{q_{j}}^{\alpha_{j}} \bigl\vert f\bigl(t_{j+1},x(t_{j+1})\bigr)\bigr\vert +\bigl\vert \varphi_{j+1}\bigl(x(t_{j+1})\bigr) \bigr\vert \bigr\} \Biggr] +{}_{t_{k}}I_{q_{k}}^{\alpha_{k}}\bigl\vert f \bigl(t,x(t)\bigr)\bigr\vert \\ \leq& (\Psi_{1}+\Psi_{2})\varepsilon \vert x\vert . \end{aligned}$$
Setting \(\varepsilon\leq(\Psi_{1}+\Psi_{2})^{-1}\), we deduce that
$$ |\mathcal{L}x|\leq|x|. $$
Multiplying both sides of this inequality by \((t-t_{k})^{\beta}\), we have \(\|\mathcal{L}x\|_{\mathit{PC}_{\beta}}\leq \|x\|_{\mathit{PC}_{\beta}}\). It follows from Lemma 3.2(ii) that problem (1.1) has at least one solution on J. □

4 Examples

In this section, we present three examples to illustrate our results.

Example 4.1

Consider the following nonlocal boundary value problem for impulsive fractional q-difference equations:
$$ \left \{ \textstyle\begin{array}{l} {}_{t_{k}}D_{ (\frac{k^{2}+2}{k^{2}+3} )}^{ (\frac {k+1}{k+2} )}x(t)= (\frac{\cos^{2}t+e^{-t}}{60} ) (\frac{x^{2}(t)+|x(t)|}{|x(t)|+1} )+\frac{3}{4},\quad t\in[0,4/3]\setminus t_{k}, \\ {}_{t_{k}}I_{ (\frac{k^{2}+2}{k^{2}+3} )}^{ (\frac{1}{k+2} )}x(t_{k}^{+})-x(t_{k})=\frac{1}{16\pi k}\sin(|\pi x(t_{k})|), \quad t_{k}=\frac{k}{3}, k=1,2,3, \\ \frac{1}{2} {}_{0}I_{\frac{2}{3}}^{\frac{1}{2}}x(0)=\frac{2}{3}x (\frac {4}{3} )+\sum_{l=0}^{3} (\frac{l^{2}+l+1}{l^{2}+2l+2} ){}_{t_{l}}I_{ (\frac{l^{2}+2}{l^{2}+3} )} ^{ (\frac{2l+1}{l+3} )}x(t_{l+1}). \end{array}\displaystyle \right . $$
(4.1)
Here \(\alpha_{k}=(k+1)/(k+2)\), \(q_{k}=(k^{2}+2)/(k^{2}+3)\), \(\gamma_{k}=(2k+1)/(k+3)\), \(c_{k}=(k^{2}+k+1)/(k^{2}+2k+2)\), \(k=0,1,2,3\), \(a=1/2\), \(b=2/3\), \(T=4/3\), \(t_{k}=k/3\), \(k=1,2,3\). With the given values, we find that \(\Omega=-2.102954268\), \(\Psi_{1}=4.421252518\), and \(\Psi_{2}=6.317984153\). Also, we have
$$ \bigl\vert f(t,x)-f(t,y)\bigr\vert \leq\frac{\cos^{2}t+e^{-t^{2}}}{30}\vert x-y\vert \leq \frac{1}{15}\vert x-y\vert $$
and
$$ \bigl\vert \varphi_{k}(x)-\varphi_{k}(y)\bigr\vert \leq \frac{1}{16}\vert x-y\vert ,\quad k=1,2,3, $$
which suggests that (H1) is satisfied with \(M_{1}=1/15\) and \(M_{2}=1/16\). Further, there exists \(\beta=1\) such that \((M_{1}\Psi_{1}+M_{2}\Psi_{2})T^{\beta}=0.9194989033<1\). Thus, all the conditions of Theorem 3.1 hold. Therefore, by the conclusion of Theorem 3.1, problem (4.1) has a unique solution on \([0, 4/3]\).

Example 4.2

Consider the problem of impulsive fractional q-difference equations given by
$$ \left \{ \textstyle\begin{array}{l} {}_{t_{k}}D_{ (\frac{k^{2}+k+2}{k^{2}+k+3} )}^{ (\frac {k^{2}+2k+2}{k^{2}+3k+3} )}x(t)=\frac{e^{-3t^{2}}}{10+t^{2}}\log ^{2}_{e} (\frac{|x(t)|}{10}+2 ),\quad t\in[0,5]\setminus t_{k}, \\ {}_{t_{k}}I_{ (\frac{k^{2}+k+2}{k^{2}+k+3} )}^{ (\frac{k+1}{k^{2}+3k+3} )}x(t_{k}^{+})-x(t_{k})=\frac {x^{2}(t_{k})}{50(|x(t_{k})|+1)}+\frac{1}{5k}, \quad t_{k}=k, k=1,2,3,4, \\ \frac{2}{3} {}_{0}I_{\frac{2}{3}}^{\frac{1}{3}}x(0)=\frac{3}{4}x (5 )+\sum_{l=0}^{4} (\frac{l+3}{l^{2}+3l+4} ){}_{t_{l}}I_{ (\frac{l^{2}+l+2}{l^{2}+l+3} )} ^{ (\frac{l^{2}+2l+1}{l+2} )}x(t_{l+1}). \end{array}\displaystyle \right . $$
(4.2)
Here \(\alpha_{k}=(k^{2}+2k+2)/(k^{2}+3k+3)\), \(q_{k}=(k^{2}+k+2)/(k^{2}+k+3)\), \(\gamma_{k}=(k^{2}+2k+1)/(k+2)\), \(c_{k}=(k+3)/(k^{2}+3k+4)\), \(k=0,1,2,3,4\), \(a=2/3\), \(b=3/4\), \(T=5\), \(t_{k}=k\), \(k=1,2,3,4\). With this data, we find that \(\Omega=-0.8144800590\), \(\Psi_{1}=6.521521011\), and \(\Psi_{2}=4.376841316\). Further, we have
$$ \bigl\vert f(t,x)\bigr\vert =\biggl\vert \frac{e^{-3t^{2}}}{10+t^{2}} \log^{2}_{e} \biggl(\frac {|x|}{10}+2 \biggr)\biggr\vert \leq\frac{e^{-3t^{2}}}{10+t^{2}} \biggl(\frac{|x|}{10}+2 \biggr) $$
and
$$ \bigl\vert \varphi_{k}(x)\bigr\vert =\frac{x^{2}}{50(|x|+1)}+ \frac{1}{5k}\leq\frac {|x|}{50}+\frac{1}{5},\quad k=1,2,3,4. $$
Setting \(Q(x)=(x/10)+2\), \(V(x)=(x/50)+(1/5)\), \(p^{*}=1/10\), and \(\beta=1\), there exists a constant \(M^{*}>46.13262248\) satisfying (3.6). Thus, the hypothesis of Theorem 3.2 is satisfied. In consequence, the conclusion of Theorem 3.2 applies, and problem (4.2) has at least one solution on \([0,5]\).

Example 4.3

Consider the problem of impulsive fractional q-difference equations given by
$$ \left \{ \textstyle\begin{array}{l} {}_{t_{k}}D_{ (\frac{k^{2}+2k+2}{2k^{2}+2k+3} )}^{ (\frac {2k^{2}+k+3}{3k^{2}+2k+4} )}x(t)=\frac{2t}{3t+1} (\sin x(t)-x(t) )e^{x^{2}(t)\cos^{4} x(t)},\quad t\in[0,5/4]\setminus t_{k}, \\ {}_{t_{k}}I_{ (\frac{k^{2}+2k+2}{2k^{2}+2k+3} )}^{ (\frac{k^{2}+k+1}{3k^{2}+2k+4} )}x(t_{k}^{+})-x(t_{k})=\frac {kx^{4}(t_{k})+2kx^{2}(t_{k})}{\log(|x^{3}(t_{k})|+2)}, \quad t_{k}=k, k=1,2,3,4, \\ \frac{3}{4} {}_{0}I_{\frac{2}{3}}^{\frac{1}{4}}x(0)=\frac{4}{5}x (\frac {5}{4} )+\sum_{l=0}^{4} (\frac{2l^{2}+3l+1}{3l^{2}+2l+2} ){}_{t_{l}} I_{ (\frac{l^{2}+2l+2}{2l^{2}+2l+3} )} ^{ (\frac{2l+1}{2} )}x(t_{l+1}). \end{array}\displaystyle \right . $$
(4.3)
Here \(\alpha_{k}=(2k^{2}+k+3)/(3k^{2}+2k+4)\), \(q_{k}=(k^{2}+2k+2)/(2k^{2}+2k+3)\), \(\gamma_{k}=(2k+1)/2\), \(c_{k}=(2k^{2}+3k+1)/(3k^{2}+2k+2)\), \(k=0,1,2,3,4\), \(a=3/4\), \(b=4/5\), \(T=5/4\), \(t_{k}=k/4\), \(k=1,2,3,4\). With this data, we find that \(|\Omega|=2.037343386\neq0\). The functions \(f(t,x)=((2t)/(3t+1))(\sin x-x)e^{x^{2}\cos^{4}x}\) and \(\varphi_{k}(x)=(kx^{4}+2kx^{2})/(\log(|x^{3}|+2))\), \(k=1,2,3,4\), satisfy
$$ \lim_{x\rightarrow0}\frac{f(t,x)}{x}=\lim_{x\rightarrow 0} \frac{2t}{3t+1} \biggl(\frac{\sin x}{x}-1 \biggr)e^{x^{2}\cos^{4} x}=0 $$
and
$$ \lim_{x\rightarrow0}\frac{\varphi_{k}(x)}{x}=\lim_{x\rightarrow 0} \frac{kx^{3}+2kx}{\log(|x^{3}|+2)}=0,\quad k=1,2,3,4. $$
Thus, condition (H4) of Theorem 3.3 holds. Therefore, by applying Theorem 3.3 we conclude that problem (4.3) has at least one solution on \([0,5/4]\).

Declarations

Acknowledgements

This paper was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University. The authors, therefore, acknowledge with thanks DSR for technical and financial support. The authors thank the reviewers for their useful comments.

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)
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University
(2)
Department of Mathematics, University of Ioannina
(3)
Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok

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