Research
Open Access
Existence results for impulsive fractional q-difference equations with anti-periodic boundary conditions
Received: 13 September 2015
Accepted: 4 January 2016
Published: 16 January 2016
Abstract
This paper studies a Caputo type anti-periodic boundary value problem of impulsive fractional q-difference equations involving a q-shifting operator of the form \({}_{a}\Phi_{q}(m) = qm + (1-q)a\). Concerning the existence of solutions for the given problem, two theorems are proved via Schauder’s fixed point theorem and the Leray-Schauder nonlinear alternative, while the uniqueness of solutions is established by means of Banach’s contraction mapping principle. Finally, we discuss some examples illustrating the main results.
Keywords
quantum calculus impulsive fractional q-difference equations existence uniqueness fixed point theoremMSC
26A33 39A13 34A371 Introduction
The quantum calculus (calculus without limits or q-calculus) is concerned with difference operators and allows one to deal with sets of nondifferentiable functions. Quantum difference operators appear in several branches of mathematics, such as orthogonal polynomials, basic hyper-geometric functions, combinatorics, the calculus of variations, mechanics, and the theory of relativity. For the fundamental concepts of quantum calculus, we refer the reader to the text by Kac and Cheung [1].
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 [2–15] and the references cited therein.
In [16] and [17], the authors studied the problems involving nonlinear impulsive \(q_{k}\)-difference equations. In [18], some new concepts of fractional quantum calculus were introduced in terms of a q-shifting operator \({}_{a}\Phi_{q}(m) = qm + (1-q)a\), and existence results for initial value problems of impulsive fractional q-difference equations were obtained.
In this paper we study the following anti-periodic boundary value problem of an impulsive fractional q-difference equation: where \(0=t_{0}< t_{1}<\cdots<t_{m}<t_{m+1}=T\), \({}_{t_{k}}^{c}D^{\alpha_{k}}_{q_{k}}\) denotes the Caputo \(q_{k}\)-fractional derivative of order \(\alpha_{k}\) on \(J_{k}\), \(1<\alpha_{k}\leq2\), \(0< q_{k}<1\), \(J_{k}=(t_{k},t_{k+1}]\), \(J_{0}=[0,t_{1}]\), \(k=0,1,\ldots,m\), \(J=[0,T]\), \(f\in C( J\times{\mathbb{R}},{\mathbb{R}})\), \(\varphi_{k},\varphi_{k}^{*}\in C( \mathbb{R}, \mathbb{R})\), \(k=1,2,\ldots,m\), \({}_{t_{k}}I^{\beta_{k}}_{q_{k}}\), \({}_{t_{k}}I^{\gamma_{k}}_{q_{k}}\) denotes the Riemann-Liouville \(q_{k}\)-fractional integral of orders \(\beta_{k},\gamma_{k}>0\) on \(J_{k}\), \(k=0,1,2,\ldots,m-1\).
$$ \left \{ \textstyle\begin{array}{l} {}_{t_{k}}^{c}D_{q_{k}}^{\alpha_{k}}x(t)=f(t,x(t)),\quad t\in J_{k}\subseteq[0,T], t\neq t_{k}, \\ \Delta x(t_{k}):=x(t_{k}^{+})-x(t_{k})=\varphi_{k} ({}_{t_{k-1}}I_{q_{k-1}}^{\beta_{k-1}} x(t_{k}) ),\quad k=1,2,\ldots, m, \\ {}_{t_{k}}D_{q_{k}} x(t_{k}^{+})-{}_{t_{k-1}}D_{q_{k-1}} x(t_{k})=\varphi_{k}^{*} ({}_{t_{k-1}}I_{q_{k-1}}^{\gamma_{k-1}} x(t_{k}) ), \quad k=1,2,\ldots, m, \\ x(0)=-x(T),\qquad {}_{0}D_{q_{0}}x(0)=-{}_{t_{m}}D_{q_{m}}x(T), \end{array}\displaystyle \right . $$
(1.1)
The rest of the paper is organized as follows. In Section 2, we present some background material for the problem at hand and prove an auxiliary lemma for the linear variant of problem (1.1). Section 3 contains the main results which are established by means of Schauder’s fixed point theorem, the Leray-Schauder nonlinear alternative, and Banach’s fixed point theorem. Section 4 deals with some illustrative examples for the existence results obtained in Section 3.
2 Preliminaries
First of all, we recall some basic concepts of q-calculus [18].
The q-derivative of a function f on the interval \([a,b]\) is defined by and q-derivative of higher order is given by
$$ ({}_{a}D_{q}f) (t)=\frac{f(t)-f({}_{a}\Phi_{q}(t))}{(1-q)(t-a)}, \quad t \ne a \quad \text{and}\quad ({}_{a}D_{q}f) (a)=\lim _{t\to a}({}_{a}D_{q}f) (t), $$
(2.1)
$$ \bigl({}_{a}D_{q}^{k}f\bigr) (t)={}_{a}D_{q}^{k-1}({}_{a}D_{q}f) (t),\qquad \bigl({}_{a}D_{q}^{0}f\bigr) (t)=f(t), \quad k\in\mathbb{N}. $$
The q-derivatives of a product and ratio of functions f and g on \([a, b]\) are and where \(g(t)g({}_{a}\Phi_{q}(t))\neq0\).
$$\begin{aligned} {}_{a}D_{q}(fg) (t) =&f(t){}_{a}D_{q}g(t)+g \bigl({}_{a}\Phi_{q}(t)\bigr){}_{a}D_{q}f(t) \\ =&g(t){}_{a}D_{q}f(t)+f\bigl({}_{a} \Phi_{q}(t)\bigr){}_{a}D_{q}g(t) \end{aligned}$$
(2.2)
$$ {}_{a}D_{q} \biggl(\frac{f}{g} \biggr) (t)=\frac {g(t){}_{a}D_{q}f(t)-f(t){}_{a}D_{q}g(t)}{g(t)g({}_{a}\Phi_{q}(t))}, $$
(2.3)
The q-integral of a function f defined on the interval \([a, b]\) is given by with The fundamental theorem of calculus applies to the operators \({}_{a}D_{q}\) and \({}_{a}I_{q}\) as follows: If f is continuous at \(t=a\), then The formula for q-integration by parts on the interval \([a, b]\) is
$$ ({}_{a}I_{q}f) (t)= \int_{a}^{t}f(s)\, {}_{a}ds=(1-q) (t-a) \sum_{i=0}^{\infty }q^{i}f \bigl({}_{a}\Phi_{q^{i}}(t)\bigr), \quad t\in[a, b] , $$
(2.4)
$$ \bigl({}_{a}I_{q}^{k}f\bigr) (t)={}_{a}I_{q}^{k-1}({}_{a}I_{q}f) (t), \qquad \bigl({}_{a}I_{q}^{0}f\bigr) (t)=f(t), \quad k\in\mathbb{N}. $$
(2.5)
$$ ({}_{a}D_{q} {}_{a}I_{q}f) (t)=f(t). $$
(2.6)
$$ ({}_{a}I_{q}{}_{a}D_{q}f) (t)=f(t)-f(a). $$
(2.7)
$$ \int_{a}^{b}f(s){}_{a}D_{q}g(s) \, {}_{a}d_{q}s=(fg) (t) |_{a}^{b}- \int_{a}^{b}g\bigl({}_{a}\Phi _{q}(s)\bigr){}_{a}D_{q}f(s)\, {}_{a}d_{q}s. $$
(2.8)
Now we enlist some useful properties of the q-shifting operator \({}_{a}\Phi_{q}(m) = qm + (1-q)a\) as follows:
- (i)
\({}_{a}\Phi_{q}^{k}(m) = {}_{a}\Phi_{q}^{k-1} ({}_{a}\Phi_{q}(m) )\) with \({}_{a}\Phi_{q}^{0}(m)=m\), for any positive integer k.
- (ii)
\({}_{a}(n-m)_{q}^{(0)}=1\), \({}_{a}(n-m)_{q}^{(k)} = \prod_{i=0}^{k-1} (n-{}_{a}\Phi_{q}^{i}(m) )\), \(k\in \mathbb{N}\cup\{\infty\}\).
- (iii)
\({}_{a}(n-m)^{(\gamma)}_{q}=n^{(\gamma )}\prod_{i=0}^{\infty}\frac{1- {_{\frac{a}{n}}}\Phi^{i}_{q}(m/n)}{1- {_{\frac{a}{n}}}\Phi^{\gamma+i}_{q}(m/n)}\), \(\gamma\in\mathbb{R}\).
Next we turn to the definitions of the Riemann-Liouville fractional q-derivative and the q-integral on the interval \([a, b]\).
Definition 2.1
[18]
The fractional q-derivative of Riemann-Liouville type of order \(\nu\ge0\) on the interval \([a, b]\) is defined by \(({}_{a}D_{q}^{0}f)(t)=f(t)\) and where l is the smallest integer greater than or equal to ν.
$$ \bigl({}_{a}D_{q}^{\nu}f\bigr) (t)=\bigl({}_{a}D_{q}^{l}{}_{a}I_{q}^{l-\nu}f \bigr) (t), \quad \nu>0, $$
(2.9)
Definition 2.2
[18]
Let \(\alpha\geq0\) and f be a function defined on \([a, b]\). The fractional q-integral of Riemann-Liouville type is given by \(({}_{a}I_{q}^{0} f)(t)=f(t)\) and
$$ \bigl({}_{a}I_{q}^{\alpha} f\bigr) (t)=\frac{1}{\Gamma_{q}(\alpha)} \int_{a}^{t}{}_{a}\bigl(t-{}_{a} \Phi _{q}(s)\bigr)_{q}^{(\alpha-1)}f(s)\, {}_{a}d_{q}s, \quad \alpha>0, t\in[a, b]. $$
(2.10)
From [18], we have the following formulas:
$$\begin{aligned}& {}_{a}D_{q}^{\alpha}(t-a)^{\beta}= \frac{\Gamma_{q}(\beta+1)}{\Gamma _{q}(\beta-\alpha+1)}(t-a)^{\beta-\alpha}, \end{aligned}$$
(2.11)
$$\begin{aligned}& {}_{a}I_{q}^{\alpha}(t-a)^{\beta}= \frac{\Gamma_{q}(\beta+1)}{\Gamma _{q}(\beta+\alpha+1)}(t-a)^{\beta+\alpha}. \end{aligned}$$
(2.12)
Lemma 2.1
[18]
Let
\(\alpha, \beta\in\mathbb{R}^{+}\)
and
f
be a continuous function on
\([a,b]\), \(a\geq0\). The Riemann-Liouville fractional
q-integral has the following semi-group property:
$$ {}_{a}I^{\beta}_{q}{}_{a}I^{\alpha}_{q}f(t)={}_{a}I^{\alpha}_{q}{}_{a}I^{\beta }_{q}f(t)={}_{a}I^{\alpha+\beta}_{q}f(t). $$
(2.13)
Lemma 2.2
[18]
Let
f
be a
q-integrable function on
\([a, b]\). Then the following equality holds:
$$ {}_{a}D^{\alpha}_{q}{}_{a}I^{\alpha}_{q}f(t)=f(t), \quad \textit{for } \alpha>0, t\in[a,b]. $$
(2.14)
Lemma 2.3
[18]
Let
\(\alpha>0\)
and
p
be a positive integer. Then for
\(t\in[a, b]\)
the following equality holds:
$$\begin{aligned} {}_{a}I_{q}^{\alpha}{}_{a}D_{q}^{p}f(t)={}_{a}D_{q}^{p}{}_{a}I_{q}^{\alpha}f(t)- \sum_{k=0}^{p-1}\frac{(t-a)^{\alpha-p+k}}{\Gamma_{q}(\alpha +k-p+1)}{}_{a}D_{q}^{k}f(a). \end{aligned}$$
(2.15)
In this paper we first give the definition of Caputo fractional q-derivative as follows.
Definition 2.3
The fractional q-derivative of Caputo type of order \(\alpha\ge 0\) on the interval \([a, b]\) is defined by \(({}_{a} ^{c}D_{q}^{0}f)(t)=f(t)\) and where n is the smallest integer greater than or equal to α.
$$ \bigl({}_{a} ^{c}D_{q}^{\alpha} f\bigr) (t)=\bigl({}_{a}I_{q}^{n-\alpha} {}_{a}D_{q}^{n}f\bigr) (t), \quad \alpha>0, $$
(2.16)
Lemma 2.4
Let
\(\alpha>0\)
and
n
be the smallest integer greater than or equal to
α. Then for
\(t\in[a, b]\), the following equality holds:
$$ {}_{a}I_{q}^{\alpha}{}_{a}^{c}D_{q}^{\alpha}f(t)=f(t)- \sum_{k=0}^{n-1}\frac{(t-a)^{k}}{\Gamma _{q}(k+1)}{}_{a}D_{q}^{k}f(a). $$
(2.17)
Proof
From Lemma 2.3, for \(\alpha=p=m\), where m is a positive integer, we have Then by Definition 2.3, we have □
$$ {}_{a}I_{q}^{m}{}_{a}D_{q}^{m}f(t)={}_{a}D_{q}^{m}{}_{a}I_{q}^{m}f(t)- \sum_{k=0}^{m-1}\frac{(t-a)^{k}}{\Gamma_{q}(k+1)}{}_{a}D_{q}^{k}f(a) =f(t)-\sum_{k=0}^{m-1}\frac{(t-a)^{k}}{\Gamma_{q}(k+1)}{}_{a}D_{q}^{k}f(a). $$
$$ {}_{a}I_{q}^{\alpha}{}_{a} ^{c}D_{q}^{\alpha}f(t) = {}_{a}I_{q}^{\alpha}{}_{a}I_{q}^{n-\alpha}{}_{a}D_{q}^{n}f(t) ={}_{a}I_{q}^{n}{}_{a}D_{q}^{n}f(t) =f(t)-\sum_{k=0}^{n-1}\frac{(t-a)^{k}}{\Gamma_{q}(k+1)}{}_{a}D_{q}^{k}f(a). $$
Lemma 2.5
Let
\(h\in C(J,\mathbb{R})\). Then the unique solution of
is given by
where
\(\sum_{1}^{0}(\cdot)=0\).
$$ \left \{ \textstyle\begin{array}{l} {}_{t_{k}}^{c}D_{q_{k}}^{\alpha_{k}}x(t)=h(t), \quad t\in J_{k}\subseteq[0,T], t\neq t_{k}, \\ \Delta x(t_{k})=\varphi_{k} ({}_{t_{k-1}}I_{q_{k-1}}^{\beta_{k-1}} x(t_{k}) ),\quad k=1,2,\ldots, m, \\ {}_{t_{k}}D_{q_{k}} x(t_{k}^{+})-{}_{t_{k-1}}D_{q_{k-1}} x(t_{k})=\varphi_{k}^{*} ({}_{t_{k-1}}I_{q_{k-1}}^{\gamma_{k-1}} x(t_{k}) ), \quad k=1,2,\ldots, m, \\ x(0)=-x(T),\qquad {}_{0}D_{q_{0}}x(0)=-{_{t_{m}}}D_{q_{m}}x(T), \end{array}\displaystyle \right . $$
(2.18)
$$\begin{aligned} x(t) =&-\frac{1}{2}\sum_{i=1}^{m} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}}h(t_{i})+ \varphi_{i} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\beta_{i-1}}x(t_{i}) \bigr) \bigr] \\ &{}- \frac{1}{2}\sum_{i=1}^{m}(T-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}h(t_{i})+ \varphi_{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}}x(t_{i}) \bigr) \bigr\} \\ &{}- \frac{1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}}h(T)+ \biggl(t-\frac {T}{2} \biggr) \Biggl[-\frac{1}{2}\sum _{i=1}^{m} \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}h(t_{i}) \\ &{}+\varphi_{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr) \bigr\} -\frac{1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha _{m}-1}h(T) \Biggr] \\ &{}+\sum_{i=1}^{k} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}}h(t_{i})+ \varphi_{i} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\beta _{i-1}}x(t_{i}) \bigr) \bigr] \\ &{}+ \sum_{i=1}^{k}(t-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}-1}h(t_{i})+ \varphi_{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr) \bigr\} \\ &{}+ {}_{t_{k}}I_{q_{k}}^{\alpha_{k}}h(t), \end{aligned}$$
(2.19)
Proof
Applying the Riemann-Liouville fractional \(q_{0}\)-integral operator of order \(\alpha_{0}\) on both sides of the first equation of (2.18) for \(t\in J_{0}\) and using Lemma 2.4, we obtain which yields where \(C_{0}=x(0)\) and \(C_{1}= {}_{0}D_{q_{0}}x(0)\). In particular, for \(t=t_{1}\), we have For \(t\in J_{1}\), on application of the Riemann-Liouville fractional \(q_{1}\)-integral operator of order \(\alpha_{1}\) to (2.18) and using the above arguments, we get Using the impulsive conditions \(x(t_{1}^{+})=x(t_{1})+\varphi_{1} ({}_{t_{0}}I_{q_{0}}^{\beta_{0}} x(t_{1}) )\) and \({}_{t_{1}}D_{q_{1}} x(t_{1}^{+})={}_{t_{0}}D_{q_{0}} x(t_{1})+\varphi_{1}^{*} ({}_{t_{0}}I_{q_{0}}^{\gamma_{0}} x(t_{1}) )\), we obtain In a similar manner, for \(t\in J_{2}\), we have Repeating the above process, for \(t\in J_{k}\subseteq J\), \(k=0,1,2,\ldots,m\), we obtain where \(\sum_{1}^{0}(\cdot)=0\). Notice that \(x(0)=C_{0}\) and On the other hand, we have which implies \({}_{t_{0}}D_{q_{0}}x(0)=C_{1}\) and Now making use of the boundary conditions given by (2.18), we find that and Substituting the values \(C_{0}\) and \(C_{1}\) in (2.23) yields the solution (2.19). □
$$ {}_{t_{0}}I_{q_{0}}^{\alpha_{0}}{}_{t_{0}}^{c}D_{q_{0}}^{\alpha _{0}}x(t)=x(t)-x(0)- \frac{{}_{0}D_{q_{0}}x(0)}{\Gamma _{q_{0}}(2)}t={}_{t_{0}}I_{q_{0}}^{\alpha_{0}}h(t), $$
$$ x(t)=C_{0}+C_{1}t+{}_{t_{0}}I_{q_{0}}^{\alpha_{0}}h(t), $$
(2.20)
$$ x(t_{1})=C_{0}+C_{1}t_{1}+{}_{t_{0}}I_{q_{0}}^{\alpha_{0}}h(t_{1}) \quad \text{and}\quad {}_{t_{0}}D_{q_{0}}x(t_{1})=C_{1}+{}_{t_{0}}I_{q_{0}}^{\alpha_{0}-1}h(t_{1}). $$
(2.21)
$$ x(t)=x\bigl(t_{1}^{+}\bigr)+(t-t_{1}){}_{t_{1}}D_{q_{1}}x \bigl(t_{1}^{+}\bigr)+{}_{t_{1}}I_{q_{1}}^{\alpha_{1}}h(t). $$
(2.22)
$$\begin{aligned} x(t) =&C_{0}+C_{1}t+ \bigl[ {}_{t_{0}}I_{q_{0}}^{\alpha_{0}}h(t_{1})+ \varphi_{1} \bigl({}_{t_{0}}I_{q_{0}}^{\beta_{0}} x(t_{1}) \bigr) \bigr] \\ &{}+ (t-t_{1}) \bigl[ {}_{t_{0}}I_{q_{0}}^{\alpha_{0}-1}h(t_{1})+ \varphi_{1}^{*} \bigl({}_{t_{0}}I_{q_{0}}^{\gamma_{0}} x(t_{1}) \bigr) \bigr] +{}_{t_{1}}I_{q_{1}}^{\alpha_{1}} h(t). \end{aligned}$$
$$\begin{aligned} x(t) =&C_{0}+C_{1}t+ \bigl[ {}_{t_{0}}I_{q_{0}}^{\alpha_{0}}h(t_{1})+ \varphi_{1} \bigl({}_{t_{0}}I_{q_{0}}^{\beta_{0}} x(t_{1}) \bigr) \bigr]+ \bigl[ {}_{t_{1}}I_{q_{1}}^{\alpha_{1}}h(t_{2})+ \varphi_{2} \bigl({}_{t_{1}}I_{q_{1}}^{\beta_{1}} x(t_{2}) \bigr) \bigr] \\ &{}+ (t-t_{1}) \bigl[ {}_{t_{0}}I_{q_{0}}^{\alpha_{0}-1}h(t_{1})+ \varphi_{1}^{*} \bigl({}_{t_{0}}I_{q_{0}}^{\gamma_{0}} x(t_{1}) \bigr) \bigr] \\ &{}+ (t-t_{2}) \bigl[ {}_{t_{1}}I_{q_{1}}^{\alpha_{1}-1}h(t_{2})+ \varphi_{2}^{*} \bigl({}_{t_{1}}I_{q_{1}}^{\gamma_{1}} x(t_{2}) \bigr) \bigr] +{}_{t_{2}}I_{q_{2}}^{\alpha_{2}} h(t). \end{aligned}$$
$$\begin{aligned} x(t) =&C_{0}+C_{1}t+\sum _{i=1}^{k} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}}h(t_{i})+ \varphi_{i} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\beta _{i-1}}x(t_{i}) \bigr) \bigr] \\ &{}+ \sum_{i=1}^{k}(t-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}-1}h(t_{i})+ \varphi_{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr) \bigr\} \\ &{}+ {}_{t_{k}}I_{q_{k}}^{\alpha_{k}}h(t), \end{aligned}$$
(2.23)
$$\begin{aligned} x(T) =&C_{0}+C_{1}T+\sum_{i=1}^{m} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}}h(t_{i})+ \varphi_{i} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\beta _{i-1}}x(t_{i}) \bigr) \bigr] \\ &{}+ \sum_{i=1}^{m}(T-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}-1}h(t_{i})+ \varphi_{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr) \bigr\} \\ &{}+ {}_{t_{m}}I_{q_{m}}^{\alpha_{m}}h(T). \end{aligned}$$
$${}_{t_{k}}D_{q_{k}}x(t) = C_{1}+\sum _{i=1}^{k} \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}-1}h(t_{i})+ \varphi_{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr) \bigr\} + {}_{t_{k}}I_{q_{k}}^{\alpha_{k}-1}h(t), $$
$${}_{t_{m}}D_{q_{m}}x(T) = C_{1}+\sum _{i=1}^{m} \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}-1}h(t_{i})+ \varphi_{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr) \bigr\} + {}_{t_{m}}I_{q_{m}}^{\alpha_{m}-1}h(T). $$
$$\begin{aligned} C_{0} =&-\frac{1}{2}C_{1}T-\frac{1}{2}\sum _{i=1}^{m} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}}h(t_{i})+ \varphi_{i} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\beta_{i-1}}x(t_{i}) \bigr) \bigr] \\ &{}- \frac{1}{2}\sum_{i=1}^{m}(T-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}h(t_{i})+ \varphi_{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}}x(t_{i}) \bigr) \bigr\} - \frac{1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}}h(T) \end{aligned}$$
$$ C_{1} = -\frac{1}{2}\sum_{i=1}^{m} \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}-1}h(t_{i})+ \varphi_{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr) \bigr\} - \frac{1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}-1}h(T). $$
3 Main results
Let \(\mathit{PC}(J, \mathbb{R})\) = {\(x: J\rightarrow\mathbb{R}: x(t)\) is continuous everywhere except for some \(t_{k}\) at which \(x(t_{k}^{+})\) and \(x(t_{k}^{-})\) exist and \(x(t_{k}^{-})=x(t_{k})\), \(k=1,2,\ldots,m\)}. Observe that \(\mathit{PC}(J, \mathbb{R})\) is a Banach space equipped with the norm \(\|x\|_{\mathit{PC}}=\sup\{|x(t)| : t\in J\}\).
In view of Lemma 2.5, we define an operator \(\mathcal{A}:\mathit{PC}(J,\mathbb{R})\to \mathit{PC}(J,\mathbb{R})\) by where
\(p\in\{\alpha_{0},\ldots,\alpha_{m},\alpha_{0}-1,\ldots,\alpha _{m}-1,\beta_{0},\ldots,\beta_{m-1},\gamma_{0},\ldots,\gamma_{m-1}\}\), \(q\in\{q_{0},\ldots,q_{m}\}\), \(a\in\{t_{0},\ldots,t_{m}\}\), and \(u\in \{t,t_{1},t_{2},\ldots,t_{m},T\}\).
$$\begin{aligned} \mathcal{A}x(t) =&-\frac{1}{2}\sum _{i=1}^{m} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}}f \bigl(t_{i},x(t_{i})\bigr)+\varphi_{i} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\beta_{i-1}}x(t_{i}) \bigr) \bigr] \\ &{}- \frac{1}{2}\sum_{i=1}^{m}(T-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}f\bigl(t_{i},x(t_{i}) \bigr)+\varphi _{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}}x(t_{i}) \bigr) \bigr\} \\ &{}- \frac{1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}}f \bigl(T,x(T)\bigr)+ \biggl(t-\frac{T}{2} \biggr) \Biggl[-\frac{1}{2} \sum_{i=1}^{m} \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1} f\bigl(t_{i},x(t_{i})\bigr) \\ &{}+\varphi_{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr) \bigr\} -\frac{1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha _{m}-1}f \bigl(T,x(T)\bigr) \Biggr] \\ &{}+\sum_{i=1}^{k} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}}f \bigl(t_{i},x(t_{i})\bigr)+\varphi_{i} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\beta _{i-1}}x(t_{i}) \bigr) \bigr] \\ &{}+ \sum_{i=1}^{k}(t-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}-1}f\bigl(t_{i},x(t_{i}) \bigr)+\varphi_{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}}x(t_{i}) \bigr) \bigr\} \\ &{}+ {}_{t_{k}}I_{q_{k}}^{\alpha_{k}}f\bigl(t,x(t)\bigr), \end{aligned}$$
(3.1)
$$ {}_{a}I_{q}^{p} f\bigl(u,x(u)\bigr)= \frac{1}{\Gamma_{q}(p)} \int_{a}^{u}{}_{a}\bigl(u-{}_{a} \Phi _{q}(s)\bigr)_{q}^{(p-1)}f\bigl(s,x(s)\bigr)\, {}_{a}d_{q}s, $$
For computational convenience, we set
$$\begin{aligned}& \Omega_{1}= \frac{3}{2}\sum _{i=1}^{m+1}\frac{(t_{i}-t_{i-1})^{\alpha _{i-1}}}{\Gamma_{q_{i-1}}(\alpha_{i-1}+1)} +\frac{3}{2}\sum _{i=1}^{m}\frac{(T-t_{i})(t_{i}-t_{i-1})^{\alpha_{i-1}-1}}{ \Gamma_{q_{i-1}}(\alpha_{i-1})} \\& \hphantom{\Omega_{1}={}}{}+\frac{T}{4}\sum_{i=1}^{m+1} \frac{(t_{i}-t_{i-1})^{\alpha _{i-1}-1}}{\Gamma_{q_{i-1}}(\alpha_{i-1})}, \end{aligned}$$
(3.2)
$$\begin{aligned}& \Omega_{2}= \frac{3}{2}m M_{1}+ \frac{3}{2}M_{2}\sum_{i=1}^{m}(T-t_{i})+ \frac{T}{4}m M_{2}. \end{aligned}$$
(3.3)
Now we present our first existence result for the problem (1.1), which is based on the Schauder fixed point theorem.
Theorem 3.1
Assume that
Then the anti-periodic boundary value problem (1.1) has at least one solution on
J
if
- (H1):
-
there exist continuous functions \(a(t)\), \(b(t)\), and nonnegative constants \(M_{1}\), \(M_{2}\) such thatwith \(\sup_{t\in J}|a(t)|=a_{1}\), \(\sup_{t\in J}|b(t)|=b_{1}\), and$$ \bigl\vert f(t,x)\bigr\vert \leq a(t)+b(t)\vert x\vert , \quad (t,x)\in J\times\mathbb{R} $$(3.4)$$ \bigl\vert \varphi_{k}(x)\bigr\vert \leq M_{1}, \qquad \bigl\vert \varphi_{k}^{*}(x)\bigr\vert \leq M_{2},\quad \forall x\in\mathbb{R}, k=1,2,\ldots,m. $$(3.5)
$$ b_{1}\Omega_{1}< 1. $$
(3.6)
Proof
Let us define a closed ball \(B_{R}=\{x\in \mathit{PC}(J,\mathbb{R}) : \|x\|_{\mathit{PC}}\leq R\}\) with where \(a_{1}\), \(b_{1}\) are defined in (H1) and \(\Omega_{1}\), \(\Omega_{2}\) are, respectively, given by (3.2) and (3.3). Clearly \(B_{R}\) is a bounded, closed, and convex subset of \(\mathit{PC}(J,\mathbb{R})\). Now we show that the operator \(\mathcal{A}:\mathit{PC}(J,\mathbb{R})\to \mathit{PC}(J,\mathbb{R})\) defined by (3.1) has a fixed point in the following two steps.
$$ R>\frac{a_{1}\Omega_{1}+\Omega_{2}}{1-b_{1}\Omega_{1}}, $$
Step 1. \(\mathcal{A}:B_{R}\to B_{R}\).
For any \(x\in B_{R}\), using (2.12), we have which implies \(\|\mathcal{A}x\|_{\mathit{PC}}\leq R\). Therefore, \(\mathcal{A}:B_{R}\to B_{R}\).
$$\begin{aligned} \bigl\vert \mathcal{A}x(t)\bigr\vert \leq&\frac{1}{2}\sum _{i=1}^{m} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}} \bigl\vert f\bigl(t_{i},x(t_{i})\bigr)\bigr\vert +\bigl\vert \varphi_{i}\bigl({}_{t_{i-1}}I_{q_{i-1}}^{\beta _{i-1}}x(t_{i}) \bigr)\bigr\vert \bigr] \\ &{}+ \frac{1}{2}\sum_{i=1}^{m}(T-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}\bigl\vert f \bigl(t_{i},x(t_{i})\bigr)\bigr\vert +\bigl\vert \varphi_{i}^{*}\bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr)\bigr\vert \bigr\} \\ &{}+ \frac{1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}}\bigl\vert f\bigl(T,x(T)\bigr)\bigr\vert +\frac {T}{2} \Biggl[ \frac{1}{2}\sum_{i=1}^{m} \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1} \bigl\vert f \bigl(t_{i},x(t_{i})\bigr)\bigr\vert \\ &{}+\bigl\vert \varphi_{i}^{*}\bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr)\bigr\vert \bigr\} +\frac {1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}-1} \bigl\vert f\bigl(T,x(T)\bigr)\bigr\vert \Biggr] \\ &{}+\sum_{i=1}^{k}\bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}} \bigl\vert f\bigl(t_{i},x(t_{i})\bigr)\bigr\vert +\bigl\vert \varphi_{i} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\beta_{i-1}}x(t_{i}) \bigr)\bigr\vert \bigr] \\ &{}+ \sum_{i=1}^{k}(t-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}-1}\bigl\vert f \bigl(t_{i},x(t_{i})\bigr)\bigr\vert +\bigl\vert \varphi_{i}^{*}\bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr) \bigr\vert \bigr\} \\ &{}+ {}_{t_{k}}I_{q_{k}}^{\alpha_{k}}\bigl\vert f \bigl(t,x(t)\bigr)\bigr\vert \\ \leq&\frac{1}{2}\sum_{i=1}^{m} \bigl[\bigl(a_{1}+b_{1}\|x\| _{\mathit{PC}}\bigr){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}}1(t_{i}) +M_{1}\bigr] \\ &{}+ \frac{1}{2}\sum_{i=1}^{m}(T-t_{i}) \bigl\{ \bigl(a_{1}+b_{1}\|x\| _{\mathit{PC}}\bigr){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}1(t_{i}) +M_{2} \bigr\} \\ &{}+ \frac{1}{2}\bigl(a_{1}+b_{1}\|x \|_{\mathit{PC}}\bigr){}_{t_{m}}I_{q_{m}}^{\alpha _{m}}1(T) \\ &{}+\frac{T}{2} \Biggl[\frac{1}{2}\sum _{i=1}^{m} \bigl\{ \bigl(a_{1}+b_{1} \|x\|_{\mathit{PC}}\bigr){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1} 1(t_{i})+M_{2} \bigr\} \\ &{}+\frac{1}{2}\bigl(a_{1}+b_{1}\|x \|_{\mathit{PC}}\bigr){}_{t_{m}}I_{q_{m}}^{\alpha _{m}-1}1(T) \Biggr] \\ &{}+\sum_{i=1}^{m}\bigl[\bigl(a_{1}+b_{1} \|x\| _{\mathit{PC}}\bigr){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}}1(t_{i})+M_{1} \bigr] \\ &{}+ \sum_{i=1}^{m}(T-t_{i}) \bigl\{ \bigl(a_{1}+b_{1}\|x\| _{\mathit{PC}}\bigr){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}1(t_{i})+M_{2} \bigr\} \\ &{}+ \bigl(a_{1}+b_{1}\|x\|_{\mathit{PC}}\bigr){}_{t_{m}}I_{q_{m}}^{\alpha_{m}}1(T) \\ =&\frac{3}{2}\sum_{i=1}^{m+1} \frac{(t_{i}-t_{i-1})^{\alpha _{i-1}}}{\Gamma_{q_{i-1}}(\alpha_{i-1}+1)}\bigl(a_{1}+b_{1}\|x\|_{\mathit{PC}}\bigr)+ \frac {3}{2}mM_{1} \\ &{}+ \frac{3}{2}\sum_{i=1}^{m}(T-t_{i}) \biggl\{ \frac {(t_{i}-t_{i-1})^{\alpha_{i-1}-1}}{\Gamma_{q_{i-1}}(\alpha _{i-1})}\bigl(a_{1}+b_{1}\|x\|_{\mathit{PC}}\bigr) +M_{2} \biggr\} \\ &{}+ \frac{T}{4}\sum_{i=1}^{m+1} \frac{(t_{i}-t_{i-1})^{\alpha _{i-1}-1}}{\Gamma_{q_{i-1}}(\alpha_{i-1})}\bigl(a_{1}+b_{1}\|x\|_{\mathit{PC}}\bigr)+ \frac {T}{4}M_{2}m \\ =&a_{1}\Omega_{1}+\Omega_{2}+b_{1} \|x\|_{\mathit{PC}}\Omega_{1} \leq R, \end{aligned}$$
Step 2. The operator \(\mathcal{A}:\mathit{PC}(J,\mathbb{R})\to \mathit{PC}(J,\mathbb{R})\) is completely continuous on \(B_{R}\).
Let \(\sup_{(t,x)\in J\times B_{R}}|f(t,x)|=F_{1}\). For any \(\tau_{1}, \tau_{2}\in J_{k}\), \(k=0,1,\ldots,m\), with \(\tau_{1}<\tau_{2}\), we have which is independent of x and tends to zero as \(\tau_{2}-\tau_{1}\to 0\). Therefore \(\mathcal{A}\) is equicontinuous. Thus \(\mathcal{A}B_{R}\) is relatively compact as \(\mathcal{A}B_{R}\subset B_{R}\) is uniformly bounded. In view of the continuity of f, \(\varphi_{k}\) and \(\varphi_{k}^{*}\), \(k=1,2,\ldots,m\), it is clear that the operator \(\mathcal{A}\) is continuous. Hence the operator \(\mathcal{A}:\mathit{PC}(J,\mathbb{R})\to \mathit{PC}(J,\mathbb{R})\) is completely continuous on \(B_{R}\). Applying the Schauder fixed point theorem, we deduce that the operator \(\mathcal{A}\) has at least one fixed point in \(B_{R}\). This shows that the problem (1.1) has at least one solution on J. □
$$\begin{aligned} \bigl\vert \mathcal{A}x(\tau_{2})-\mathcal{A}x(\tau_{1}) \bigr\vert \leq&\vert \tau_{2}-\tau _{1}\vert \Biggl[ \frac{1}{2}\sum_{i=1}^{m+1} \frac{(t_{i}-t_{i-1})^{\alpha _{i-1}-1}}{\Gamma_{q_{i-1}}(\alpha_{i-1})} F_{1}+\frac{mM_{2}}{2} \Biggr] \\ &{}+\vert \tau_{2}-\tau_{1}\vert \sum _{i=1}^{k} \biggl[\frac{(t_{i}-t_{i-1})^{\alpha _{i-1}-1}}{\Gamma_{q_{i-1}}(\alpha_{i-1})}F_{1}+M_{2} \biggr] \\ &{}+\frac{F_{1}}{\Gamma_{q_{k}}(\alpha_{k})}\biggl\vert \int_{t_{k}}^{\tau _{2}}{}_{t_{k}}(\tau_{2}-{}_{t_{k}} \Phi_{q_{k}})_{q_{k}}^{(\alpha _{k}-1)}\, {}_{t_{k}}d_{q_{k}}s \\ &{}- \int_{t_{k}}^{\tau_{1}}{}_{t_{k}}(\tau_{1}-{}_{t_{k}} \Phi _{q_{k}})_{q_{k}}^{(\alpha_{k}-1)}\, {}_{t_{k}}d_{q_{k}}s \biggr\vert , \end{aligned}$$
In the next existence result, we make use of Leray-Schauder’s nonlinear alternative.
Lemma 3.1
(Nonlinear alternative for single valued maps) [19]
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 a continuous, compact (that is, \(F(\overline{U})\)
is a relatively compact subset of
C) map. Then either
- (i)
F has a fixed point in U̅, or
- (ii)
there is a \(u\in\partial U\) (the boundary of U in C) and \(\theta\in(0,1)\) with \(u=\theta F(u)\).
In the sequel, we set
$$\begin{aligned}& \Omega_{3}=\frac{3}{2}\sum _{i=1}^{m}\frac{(t_{i}-t_{i-1})^{\beta _{i-1}}}{\Gamma_{q_{i-1}}(\beta_{i-1}+1)}, \end{aligned}$$
(3.7)
$$\begin{aligned}& \Omega_{4}=\frac{3}{2}\sum _{i=1}^{m}\frac {(T-t_{i})(t_{i}-t_{i-1})^{\gamma_{i-1}}}{\Gamma_{q_{i-1}}(\gamma_{i-1}+1)} +\frac{T}{4}\sum _{i=1}^{m}\frac{(t_{i}-t_{i-1})^{\gamma _{i-1}}}{\Gamma_{q_{i-1}}(\gamma_{i-1}+1)}. \end{aligned}$$
(3.8)
Theorem 3.2
Assume that
Then the problem (1.1) has at least one solution
J.
- (H2):
-
there exist a continuous nondecreasing function \(\psi:[0,\infty)\to(0,\infty)\), a continuous function \(p:J\to \mathbb{R}^{+}\) with \(p^{*}=\sup_{t\in J}|p(t)|\) and constants \(M_{3},M_{4}>0\) such thatand$$ \bigl\vert f(t,x)\bigr\vert \leq p(t)\psi\bigl(\vert x\vert \bigr), \quad \forall(t,x)\in J\times \mathbb{R} $$(3.9)$$ \bigl\vert \varphi_{k}(x)\bigr\vert \leq M_{3}\vert x \vert ,\qquad \bigl\vert \varphi_{k}^{*}(x)\bigr\vert \leq M_{4}\vert x\vert ,\quad \forall x\in R, k=1,\ldots,m; $$(3.10)
- (H3):
Proof
We shall show that the operator \(\mathcal{A}\) defined by (2.5) has a fixed point. To accomplish this, for a positive number ρ, let \(B_{\rho} = \{x \in \mathit{PC}(J,\mathbb{R}): \|x\|_{\mathit{PC}} \le \rho\}\) denote a closed ball in \(\mathit{PC}(J,\mathbb{R})\). Then for \(x\in B_{\rho}\), \(t\in J\), and using (2.12), we have which implies that \(\|\mathcal{A}x\|_{\mathit{PC}}\leq K\).
$$\begin{aligned} \bigl\vert \mathcal{A}x(t)\bigr\vert \leq&\frac{1}{2}\sum _{i=1}^{m} \bigl[p^{*}\psi(\rho ){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}}1(t_{i}) +\rho M_{3} {}_{t_{i-1}}I_{q_{i-1}}^{\beta_{i-1}}1(t_{i}) \bigr] \\ &{}+ \frac{1}{2}\sum_{i=1}^{m}(T-t_{i}) \bigl\{ p^{*}\psi(\rho ){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}1(t_{i}) +\rho M_{4} {}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}}1(t_{i}) \bigr\} \\ &{}+ \frac{1}{2}p^{*}\psi(\rho){}_{t_{m}}I_{q_{m}}^{\alpha _{m}}1(T)+ \frac{T}{2} \Biggl[\frac{1}{2}\sum_{i=1}^{m} \bigl\{ p^{*}\psi (\rho){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1} 1(t_{i}) \\ &{}+\rho M_{4} {}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}}1(t_{i}) \bigr\} +\frac {1}{2}p^{*}\psi(\rho){}_{t_{m}}I_{q_{m}}^{\alpha_{m}-1}1(T) \Biggr] \\ &{}+\sum_{i=1}^{m} \bigl[p^{*}\psi( \rho){}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}}1(t_{i})+\rho M_{3} {}_{t_{i-1}}I_{q_{i-1}}^{\beta _{i-1}}1(t_{i}) \bigr] \\ &{}+ \sum_{i=1}^{m}(T-t_{i}) \bigl\{ p^{*}\psi(\rho ){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}1(t_{i})+ \rho M_{4} {}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}}1(t_{i}) \bigr\} \\ &{}+ p^{*}\psi(\rho){}_{t_{m}}I_{q_{m}}^{\alpha_{m}}1(T) \\ =&p^{*}\psi(\rho)\Omega_{1}+\rho M_{3}\Omega_{3}+ \rho M_{4} \Omega_{4}:=K, \end{aligned}$$
To show that the operator \(\mathcal{A}\) maps bounded sets into equicontinuous sets of \(\mathit{PC}(J,\mathbb{R})\), we take \(\tau_{1}, \tau_{2}\in J_{k}\) for some \(k\in\{0, 1,2,\ldots, m\}\) with \(\tau_{1}<\tau_{2}\) and \(x\in B_{\rho}\). Then we have which tends to zero independent of x as \(\tau_{1}\rightarrow \tau_{2}\). Thus, by the Arzelá-Ascoli theorem, the operator \(\mathcal{A}: \mathit{PC}(J,\mathbb{R})\rightarrow \mathit{PC}(J,\mathbb{R})\) is completely continuous.
$$\begin{aligned}& \bigl\vert \mathcal{A}x(\tau_{2})-\mathcal{A}x(\tau_{1}) \bigr\vert \\& \quad \leq \vert \tau_{2}-\tau_{1}\vert \Biggl[ \frac{p^{*}\psi(\rho)}{2}\sum_{i=1}^{m+1} \frac{(t_{i}-t_{i-1})^{\alpha_{i-1}-1}}{\Gamma _{q_{i-1}}(\alpha_{i-1})} +\frac{\rho M_{4}}{2}\sum_{i=1}^{m} \frac{(t_{i}-t_{i-1})^{\gamma _{i-1}}}{\Gamma_{q_{i-1}}(\gamma_{i-1}+1)} \Biggr] \\& \qquad {}+\vert \tau_{2}-\tau_{1}\vert \sum _{i=1}^{k} \biggl[\frac{(t_{i}-t_{i-1})^{\alpha _{i-1}-1}}{\Gamma_{q_{i-1}}(\alpha_{i-1})}p^{*}\psi(\rho)+ \rho M_{4}\frac{(t_{i}-t_{i-1})^{\gamma_{i-1}}}{\Gamma_{q_{i-1}}(\gamma _{i-1}+1)} \biggr] \\& \qquad {}+\frac{p^{*}\psi(\rho)}{\Gamma_{q_{k}}(\alpha_{k})}\biggl\vert \int _{t_{k}}^{\tau_{2}}{}_{t_{k}}( \tau_{2}-{}_{t_{k}}\Phi_{q_{k}})_{q_{k}}^{(\alpha _{k}-1)} \, {}_{t_{k}}d_{q_{k}}s - \int_{t_{k}}^{\tau_{1}}{}_{t_{k}}(\tau_{1}-{}_{t_{k}} \Phi _{q_{k}})_{q_{k}}^{(\alpha_{k}-1)}\, {}_{t_{k}}d_{q_{k}}s \biggr\vert , \end{aligned}$$
Finally, for \(\lambda\in(0,1)\), let \(x=\lambda\mathcal{A}x\). Then, as in the first step, we can get which can alternatively be written as In view of (H3), there exists N such that \(\|x\|_{\mathit{PC}}\neq N\). We define \(\mathcal{U} = \{x \in \mathit{PC}(J,\mathbb{R}) : \|x\|_{\mathit{PC}} < N\}\). Note that the operator \(\mathcal{A} :\overline{\mathcal{U}} \to \mathit{PC}\) is continuous and completely continuous. From the choice of \(\mathcal{U}\), there is no \(x \in\partial\mathcal{U}\) such that \(x=\lambda {\mathcal{A}}x\) for some \(\lambda\in(0,1)\). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.1), we deduce that \({\mathcal{A}}\) has a fixed point \(x \in \overline{\mathcal{U}}\) which is a solution of the problem (1.1) on J. This completes the proof. □
$$ \|x\|_{\mathit{PC}}\leq p^{*}\psi\bigl(\|x\|_{\mathit{PC}}\bigr)\Omega_{1}+ \|x\|_{\mathit{PC}} M_{3}\Omega_{3}+\|x\|_{\mathit{PC}} M_{4} \Omega_{4}, $$
$$ \frac{(1-M_{3}\Omega_{3}-M_{4}\Omega_{4})\|x\|_{\mathit{PC}}}{p^{*}\psi(\Vert x\Vert _{\mathit{PC}})\Omega_{1}}\leq 1. $$
In the last theorem, we apply Banach’s contraction principle to establish the uniqueness of solutions for the problem (1.1).
Theorem 3.3
Assume that there exist a function
\(\mathcal{W}(t)\in C(J,\mathbb{R}^{+})\)
with
\(W=\sup_{t\in J}|\mathcal{W}(t)|\)
and positive constants
\(M_{5}\), \(M_{6}\)
such that
and
for
\(k=1,2,\ldots,m\). If
then the problem (1.1) has a unique solution on
J.
$$ \bigl\vert f(t,x)-f(t,y)\bigr\vert \leq\mathcal{W}(t)\vert x-y\vert , \quad \forall(t,x)\in J\times\mathbb{R} $$
(3.12)
$$ \begin{aligned} &\bigl\vert \varphi_{k}(x)-\varphi_{k}(y)\bigr\vert \leq M_{5}\vert x-y\vert , \\ &\bigl\vert \varphi_{k}^{*}(x)- \varphi_{k}^{*}(y)\bigr\vert \leq M_{6}\vert x-y\vert , \quad x,y\in\mathbb{R}, \end{aligned} $$
(3.13)
$$ W\Omega_{1}+M_{5}\Omega_{3}+M_{6} \Omega_{4} < 1, $$
(3.14)
Proof
For any \(x,y\in \mathit{PC}(J,\mathbb{R})\), we have which yields By (3.14), we conclude that \(\mathcal{A}\) is a contraction. Thus, by Banach’s contraction mapping principle, the problem (1.1) has a unique solution on J. This completes the proof. □
$$\begin{aligned}& \bigl\vert \mathcal{A}x(t)-\mathcal{A}y(t)\bigr\vert \\& \quad \leq \frac{1}{2}\sum_{i=1}^{m} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}}\bigl\vert f(t_{i},x)-f(t_{i},y)\bigr\vert +M_{5}{}_{t_{i-1}}I_{q_{i-1}}^{\beta_{i-1}} \vert x-y\vert (t_{i}) \bigr] \\& \qquad {} + \frac{1}{2}\sum_{i=1}^{m}(T-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}\bigl\vert f(t_{i},x)-f(t_{i},y)\bigr\vert +M_{6}{}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}} \vert x-y\vert (t_{i}) \bigr\} \\& \qquad {} + \frac{1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}} \bigl\vert f(T,x)-f(T,y)\bigr\vert +\frac {T}{2} \Biggl[ \frac{1}{2}\sum_{i=1}^{m} \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1} \bigl\vert f(t_{i},x)-f(t_{i},y)\bigr\vert \\& \qquad {} +M_{6}{}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}} \vert x-y\vert (t_{i}) \bigr\} +\frac {1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}-1} \bigl\vert f(T,x)-f(T,y)\bigr\vert \Biggr] \\& \qquad {} +\sum_{i=1}^{k} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}}\bigl\vert f(t_{i},x)-f(t_{i},y)\bigr\vert +M_{5}{}_{t_{i-1}}I_{q_{i-1}}^{\beta _{i-1}} \vert x-y\vert (t_{i}) \bigr] \\& \qquad {} + \sum_{i=1}^{k}(t-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}-1}\bigl\vert f(t_{i},x)-f(t_{i},y)\bigr\vert +M_{6}{}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}} \vert x-y\vert (t_{i}) \bigr\} \\& \qquad {} + {}_{t_{k}}I_{q_{k}}^{\alpha_{k}}\bigl\vert f(t,x)-f(t,y)\bigr\vert \\& \quad \leq (W\Omega_{1}+M_{5}\Omega_{3}+M_{6} \Omega_{4})\|x-y\|_{\mathit{PC}}, \end{aligned}$$
$$ \|\mathcal{A}x-\mathcal{A}y\|_{\mathit{PC}}\leq(W\Omega_{1}+M_{5} \Omega _{3}+M_{6}\Omega_{4})\|x-y \|_{\mathit{PC}}. $$
4 Examples
In this section, we present three examples to illustrate our results.
Example 4.1
Consider the following anti-periodic boundary value problem for impulsive Caputo fractional q-difference equations:
$$ \left \{ \textstyle\begin{array}{l} {}_{t_{k}}^{c}D_{\frac{1}{k^{2}-3k+4}}^{\frac{k+3}{k+2}}x(t)=2t^{2}+1+\frac {1}{8}\sin^{2}t\frac{x^{2}(t)}{1+|x(t)|},\quad t\in[0,2]\setminus\{t_{1},t_{2},t_{3}\}, \\ \Delta x(t_{k})= \frac{k}{k+1}e^{1- ({}_{t_{k-1}}I_{\frac{1}{k^{2}-5k+8}}^{\frac {2k-1}{2}}x(t_{k}) )^{2}},\quad t_{k}=\frac{k}{2}, k=1,2,3, \\ {}_{t_{k}}D_{\frac{1}{k^{2}-3k+4}}x(t_{k}^{+})-{}_{t_{k-1}}D_{\frac{1}{k^{2}-5k+8}} x(t_{k}) \\ \quad = k^{2}\cos (\log (1+\vert {}_{t_{k-1}}I_{\frac {1}{k^{2}-5k+8}}^{\frac{2k+5}{2}}x(t_{k})\vert ) ), \quad t_{k}=\frac{k}{2}, \\ x(0)=-x(2),\qquad {}_{0}D_{\frac{1}{4}}x(0)=-{}_{\frac{3}{2}}D_{\frac{1}{4}}x(2). \end{array}\displaystyle \right . $$
(4.1)
Here \(\alpha_{k}=(k+3)/(k+2)\), \(q_{k}=1/(k^{2}-3k+4)\), \(k=0,1,2,3\), \(\beta_{k-1}=(2k-1)/2\), \(\gamma_{k-1}=(2k+5)/2\), \(t_{k}=k/2\), \(k=1,2,3\), \(m=3\), \(T=2\). With the given information, it is found that \(\Omega_{1}=7.575532753\). Also, we have With \(B=\sup_{t\in[0,2]}|(1/8)\sin^{2}t|=1/8\), we obtain \(B\Omega_{1}=0.9469415941<1\). Thus all the conditions of Theorem 3.1 are satisfied. Therefore, by the conclusion of Theorem 3.1, the problem (4.1) has at least one solution on \([0,2]\).
$$\begin{aligned}& \bigl\vert f(t,x)\bigr\vert =\biggl\vert 2t^{2}+1+ \frac{1}{8}\sin^{2}t\frac{x^{2}}{1+|x|}\biggr\vert \leq2t^{2}+1+\frac{1}{8}\sin^{2}t|x|, \\& \bigl\vert \varphi_{k}(y)\bigr\vert =\biggl\vert \frac{k}{k+1}e^{1-y^{2}}\biggr\vert \leq e,\qquad \bigl\vert \varphi_{k}^{*}(z)\bigr\vert = \bigl\vert k^{2}\cos\bigl( \log\bigl(1+\vert z\vert \bigr)\bigr)\bigr\vert \leq9, \quad k=1,2,3. \end{aligned}$$
Example 4.2
Consider the anti-periodic impulsive boundary value problem of fractional q-difference equations given by
$$ \left \{ \textstyle\begin{array}{l} {}_{t_{k}}^{c}D_{\frac{1}{k^{2}-4k+6}}^{\frac{k^{2}+5}{k^{2}+3}}x(t)=\frac {1}{(2+t)^{2}} (\log_{e} (\frac{|x(t)|}{4}+2 ) )^{2},\quad t\in[0,5/3]\setminus\{t_{1},\ldots,t_{4}\}, \\ \Delta x(t_{k})= \frac{1}{11+k}\sin ({}_{t_{k-1}}I_{\frac{1}{k^{2}-6k+11}}^{\frac {2+(-1)^{k-1}}{2}}x(t_{k}) ), \quad t_{k}=\frac{k}{3}, k=1,\ldots,4, \\ {}_{t_{k}}D_{\frac{1}{k^{2}-4k+6}}x(t_{k}^{+})-{}_{t_{k-1}}D_{\frac{1}{k^{2}-6k+11}} x(t_{k}) = \frac{1}{3+k} {}_{t_{k-1}}I_{\frac {1}{k^{2}-6k+11}}^{\frac{4+(-1)^{k-1}}{2}}x(t_{k}), \quad t_{k}=\frac {k}{3}, \\ x(0)=-x (\frac{5}{3} ),\qquad {}_{0}D_{\frac{1}{6}}x(0)=-{_{\frac{4}{3}}}D_{\frac{1}{6}}x (\frac{5}{3} ). \end{array}\displaystyle \right . $$
(4.2)
Here \(\alpha_{k}=(k^{2}+5)/(k^{2}+3)\), \(q_{k}=1/(k^{2}-4k+6)\), \(k=0,1,2,3,4\), \(\beta_{k-1}=(2+(-1)^{k-1})/2\), \(\gamma_{k-1}=(4+(-1)^{k-1})/2\), \(t_{k}=k/3\), \(k=1,2,3,4\), \(m=4\), \(T=5/3\). Using the above data, we find that \(\Omega_{1}=6.316994013\), \(\Omega_{3}=2.358729544\), \(\Omega_{4}=0.6481929403\), and Setting \(\psi(x)=(x/4)+2\), \(p^{*}=\sup_{t\in[0,5/3]}|1/(2+t)^{2}|=1/4\), \(M_{3}=1/12\), and \(M_{4}=1/4\), we find that \(M_{3}\Omega_{3}+M_{4}\Omega_{4}=0.3586090304<1\). Also, there exists a constant N such that \(N>12.80927819\) satisfying (3.11). Clearly the hypothesis of Theorem 3.2 holds true. Thus the conclusion of Theorem 3.2 implies that the problem (4.2) has at least one solution on \([0,5/3]\).
$$\begin{aligned}& \bigl\vert f(t,x)\bigr\vert =\biggl\vert \frac{1}{(2+t)^{2}}\biggl( \log_{e}\biggl(\frac {\vert x\vert }{4}+2\biggr)\biggr)^{2}\biggr\vert \leq\frac{1}{(2+t)^{2}}\biggl(\frac{\vert x\vert }{4}+2\biggr), \\& \bigl\vert \varphi_{k}(y)\bigr\vert =\frac{1}{11+k}\vert \sin y\vert \leq\frac{1}{12}\vert y\vert ,\qquad \bigl\vert \varphi_{k}^{*}(z)\bigr\vert =\frac{1}{3+k}\vert z\vert \leq \frac{1}{4}\vert z\vert ,\quad k=1,2,3,4. \end{aligned}$$
Example 4.3
Consider the following impulsive anti-periodic problem of a fractional q-difference equation:
$$ \left \{ \textstyle\begin{array}{l} {}_{t_{k}}^{c}D_{\frac{1}{k^{2}-5k+8}}^{\frac{k^{2}+k+3}{k^{2}+2}}x(t)=\frac {e^{-t}\sin(2t+1)}{t^{2}+30}\frac{x^{2}(t)+2|x(t)|}{1+|x(t)|}+\frac{1}{2},\quad t\in[0,3/2]\setminus\{t_{1},\ldots,t_{5}\}, \\ \Delta x(t_{k})= \frac{7k}{25}\tan^{-1} ({}_{t_{k-1}}I_{\frac{1}{k^{2}-7k+14}}^{\frac{2k+1}{2}}x(t_{k}) )+\frac{2}{3}, \quad t_{k}=\frac{k}{4}, k=1,\ldots,5, \\ {}_{t_{k}}D_{\frac{1}{k^{2}-5k+8}}x(t_{k}^{+})-{}_{t_{k-1}}D_{\frac{1}{k^{2}-7k+14}} x(t_{k})= \frac{\vert {}_{t_{k-1}}I_{\frac{1}{k^{2}-7k+14}}^{\frac {2k^{2}-4k+3}{2}}x(t_{k})\vert }{5k (1+\vert {}_{t_{k-1}}I_{\frac {1}{k^{2}-7k+14}}^{\frac{2k^{2}-4k+3}{2}}x(t_{k}) \vert )}+\frac{3}{4}, \quad t_{k}=\frac{k}{4}, \\ x(0)=-x (\frac{3}{2} ),\qquad {}_{0}D_{\frac{1}{8}}x(0)=-{_{\frac{5}{4}}}D_{\frac{1}{8}}x (\frac{3}{2} ). \end{array}\displaystyle \right . $$
(4.3)
Here \(\alpha_{k}=(k^{2}+k+3)/(k^{2}+2)\), \(q_{k}=1/(k^{2}-5k+8)\), \(k=0,1,2,3,4,5\), \(\beta_{k-1}=(2k+1)/2\), \(\gamma_{k-1}=(2k^{2}-4k+3)/2\), \(t_{k}=k/4\), \(k=1,2,3,4,5\), \(m=5\), \(T=3/2\). With the given values, we find that \(\Omega_{1}=5.173430458\), \(\Omega_{3}=0.2141916028\), and \(\Omega_{4}=1.375385103\). Also, we have It is easy to see that \(W=1/15\). Hence, \(W\Omega_{1}+M_{5}\Omega_{3}+M_{6}\Omega_{4}=0.9198406284<1\). Thus all the conditions of Theorem 3.3 are satisfied. Hence it follows by the conclusion of Theorem 3.3 that the problem (4.3) has a unique solution on \([0,3/2]\).
$$\begin{aligned}& \bigl\vert f(t,x_{1})-f(t,x_{2})\bigr\vert \leq \biggl\vert \frac{2e^{-t}\sin(2t+1)}{t^{2}+30} \biggr\vert |x_{1}-x_{2}|, \\& \bigl\vert \varphi_{k}(y_{1})-\varphi_{k}(y_{2}) \bigr\vert =\frac{7k}{25} \bigl\vert \tan ^{-1}y_{1}- \tan^{-1}y_{2} \bigr\vert \leq\frac{7}{5}|y_{1}-y_{2}|, \\& \bigl\vert \varphi_{k}^{*}(z_{1})-\varphi_{k}^{*}(z_{2}) \bigr\vert =\frac{1}{5k} \biggl\vert \frac {|z_{1}|}{1+|z_{1}|}- \frac{|z_{2}|}{1+|z_{2}|} \biggr\vert \leq\frac{1}{5}|z_{1}-z_{2}|, \quad k=1,2,3,4,5. \end{aligned}$$
Declarations
Acknowledgements
The project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant no. 14-130-36-RG. The authors, therefore, acknowledge with thanks DSR’s technical and financial support.
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)
Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok
(3)
Department of Mathematics, University of Ioannina
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