Skip to main content

Theory and Modern Applications

Well-posed conditions on a class of fractional q-differential equations by using the Schauder fixed point theorem

Abstract

In this paper, we propose the conditions on which a class of boundary value problems, presented by fractional q-differential equations, is well-posed. First, under the suitable conditions, we will prove the existence and uniqueness of solution by means of the Schauder fixed point theorem. Then, the stability of solution will be discussed under the perturbations of boundary condition, a function existing in the problem, and the fractional order derivative. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

1 Introduction

In many applications fractional differential equations present more accurate models of phenomena than the ordinary differential equations. Therefore they have obtained importance due to their applications in science and engineering such as, physics, chemistry, mechanics, fluid dynamic, etc. [1, 2]. Meanwhile, there have appeared many papers dealing with the existence of solutions for different types of fractional boundary value problems; see, for example, [3–19].

The quantum calculus was introduced by Jackson in 1910 [20]. Al-Salam started the fitting of the concept of q-fractional calculus [21]. Then Agarwal continued by studying certain q-fractional integrals and derivatives [22]. After it, some researchers studied q-difference equations (for more details, see [23–36]). There are also many papers dealing with the existence of solutions for q-fractional boundary value problems (see, for example, [37–49]).

Existence of solutions to fractional differential equations has received considerable interest in recent years. There are several papers dealing with the existence and uniqueness of solutions to initial and boundary value problem of fractional order in Caputo or Riemann–Liouville sense (for more details, see [50–52] and the references therein). Some authors have also investigated the existence and uniqueness solutions for a coupled system of multi-term fractional differential equations [53, 54]. However, in general, the study of well-posed conditions for fractional differential equations is less considered in the literature.

In 2015, Houas et al. [55] investigated the existence and uniqueness of solutions for \({}^{c}\mathcal{D}_{q}^{\sigma }[y](t) + w ( y(t), {}^{c} \mathcal{D}_{q}^{\varsigma }[y](t) ) = 0\), for \(t \in \overline{J}_{0} :=[0,1]\), where \(2 < \sigma \leq 3\), \(\varsigma \in J_{0}:=(0,1)\), under the initial conditions \(y(0) = y_{0}\), \(y'(0)=0\), \(y'(1) = \eta \mathcal{I}^{\zeta }y(e)\), where \({}^{c}\mathcal{D}^{\sigma }\) is the Caputo fractional derivative, \(e \in J_{0}\), w is a continuous function on \(\mathbb{R}^{2}\), and η is a real constant [55]. In [56], authors studied the existence and uniqueness of solution for the fractional differential equation \({}\mathcal{D}^{\sigma }[y](t) = w ( t, y(t), {}\mathcal{D}^{\varsigma }[y](t) )\), where \(2 < \sigma < 3\), \(\varsigma \in J_{0}\), via sum boundary conditions \(y(0) = 0\),

$$ {}\mathcal{D}_{q}^{\sigma }[y](1) = \sum _{i = 1}^{m - 2} a_{i} {} \mathcal{D}_{q}^{\varsigma }[y]( e_{i}), $$

\(y''(1) = 0\), where, \(a_{i}, e_{i} \in J_{0}\) and \({}\mathcal{D}_{q}^{\sigma }\) is the Caputo fractional derivative. In 2015, Akrami et al. [57] proved the conditions on which the following class of fractional differential equations \({}^{C}\mathcal{D}^{\sigma }[y](t) = w ( y(t), {}\mathcal{D}^{\varsigma }[y](t) )\) for \(t \in \overline{J}_{0}\) is well-posed, where \(2 < \sigma \leq 3\) and \(\varsigma \in J_{0}\), and \({}^{C}\mathcal{D}_{q}^{\sigma }\) is the Caputo fractional derivative subject to the boundary value conditions \(y(0) = y'(0)=0\), \(y(1) = ay(e)\), where \(e \in J_{0}\), \(0 \leq a <\frac{1}{e^{2}}\).

In this article, we investigate the conditions on which the fractional q-differential equation

$$ {}^{C}\mathcal{D}_{q}^{\sigma }[y](t) = w \bigl( y(t), {}^{C} \mathcal{D}_{q}^{\varsigma }[y](t) \bigr) $$
(1)

for \(t \in \overline{J}_{0}\) is well-posed, where \(2 < \sigma \leq 3\), \(\varsigma \in J_{0}\), and \({}^{C}\mathcal{D}_{q}^{\sigma }\) is the standard Caputo q-derivative subject to the boundary value conditions

$$ y(0) = y'(0) = 0,\qquad y'(1) = a y (e), $$
(2)

where \(e\in \overline{J}_{0}\) with \(0 \leq a< \frac{1}{e^{2}}\). We recall that a problem is said to be well-posed if it has a uniqueness solution and this solution depends on a parameter in a continuous way. This parameter, in the classical order differential equations, is dependent on the initial conditions and the function exists in the problem; whereas in the FDEs this dependency and the stability solution with respect to the perturbation of fractional order derivative should be taken into the account too [58].

The rest of the paper is organized as follows. We first prove the existence solution of (1) by means of the Schauder fixed point theorem on the interval \(\overline{J}_{0}\) in Sect. 3. Then, we prove the uniqueness by using the Banach contraction map theorem under a suitable condition in Sect. 3. Also, Sect. 3 is devoted to investigating the stability of solutions under the perturbations on boundary condition, the function exists in the problem and the fractional order derivative. Finally, in Sect. 4, we bring some examples to illustrate our results. Let us start with some basic preliminaries in Sect. 2 that we will use in the sequel.

2 Preliminaries and lemmas

This section is devoted to some notations and essential preliminaries that are acting as necessary prerequisites for the results of the subsequent sections. Throughout the context, we shall apply the notations of time scales calculus [59].

In fact, we consider the fractional q-calculus on the specific time scale \(\mathbb{T}_{t_{0}} = \{0 \} \cup \{ t: t=t_{0}q^{n} \} \) for \(n \in \mathbb{N}\), \(t_{0} \in \mathbb{R}\), and \(q \in (0,1)\). For brief, we shall denote \(\mathbb{T}_{t_{0}}\) by \(\mathbb{T}\). Let \(a \in \mathbb{R}\). Define \([s]_{q} =(1-q^{s})/(1-q)\) [20]. The q-factorial function \((v-w)_{q}^{(n)}\) with \(n \in \mathbb{N}_{0} := \{ 0\} \cup \mathbb{N}\) is defined by \((v-w)_{q}^{(n)}= \prod_{k=0}^{n-1} (v - wq^{k})\), and \(( v - w)_{q}^{(0)}=1\), where v, w are real numbers [23]. Also, for \(\sigma \in \mathbb{R}\) and \(s \neq 0\), we have \((v - w)_{q}^{(\sigma )} = v^{\sigma }\prod_{k=0}^{\infty }(v-wq^{k} )(v -wq^{ -(\sigma + k)})\). In the paper [60], the authors proved \((v -w)_{q}^{(\sigma +\nu )} = (v - w)_{q}^{(\sigma )} (v -q^{\sigma }w)_{q}^{( \nu )}\) and \((sv -sw)_{q}^{(\sigma )} = s^{\sigma }(v - w)_{q}^{(\sigma )}\). If \(w=0\), then it is clear that \(v^{(\sigma )}= v^{\sigma }\). The q-gamma function is given by \(\Gamma _{q} (v) = (1-q)^{1-v} (1-q)_{q}^{(v-1)}\), where \(z \in \mathbb{R} \backslash \{\ldots , -2, -1, 0\}\) [20]. Note that \(\Gamma _{q} (z+1) = [z]_{q} \Gamma _{q} (z)\) [60, Lemma 1]. For a function \(\wp : \mathbb{T} \to \mathbb{R}\), the q-derivative of ℘ is

$$ \mathcal{D}_{q} [\wp ] (v) = \frac{ \wp (v) - \wp (qv)}{(1- q) v} $$

for all \(t \in \mathbb{T} \setminus \{0\}\), and \(\mathcal{D}_{q} [\wp ](0) = \lim_{v \to 0} \mathcal{D}_{q} \wp (v)\) [23]. Also, the higher order q-derivative of the function ℘ is defined by \(\mathcal{D}_{q}^{n} \wp (v) = \mathcal{D}_{q} (\mathcal{D}_{q}^{ n-1} \wp )(v)\) for all \(n \geq 1\), where \(\mathcal{D}_{q}^{0} \wp (v) = \wp (v)\) [23]. The q-integral of the function ℘ is defined by

$$ \mathcal{I}_{q} \wp (v) = \int _{0}^{v} \wp (\xi ) {\,\mathrm{d}}_{q} \xi = v(1- q) \sum_{k=0}^{\infty } q^{k} \wp \bigl(v q^{k} \bigr) $$

for \(0 \leq v \leq b\), provided the series absolutely converges [23]. If v in \([0, b]\), then

$$ \int _{a}^{v} \wp (\xi ) { \,\mathrm{d}}_{q} \xi = \mathcal{I}_{q} \wp (v) - \mathcal{I}_{q} \wp (a) = (1-q) \sum_{k=0}^{\infty } q^{k} \bigl[ v \wp \bigl(v q^{k} \bigr) - a \wp \bigl(a q^{k} \bigr) \bigr], $$

whenever the series exists [61]. The operator \(\mathcal{I}_{q}^{n}\) is given by \(\mathcal{I}_{q}^{0} \wp (v) = \wp (v)\) and \(\mathcal{I}_{q}^{n} \wp (v) = \mathcal{I}_{q} [ \mathcal{I}_{q}^{n-1} \wp ] (v)\) for \(n \geq 1\) and \(\wp \in C([0, b])\) [23]. It has been proved that \(\mathcal{D}_{q} ( \mathcal{I}_{q} \wp )(v) = \wp (v)\), \(\mathcal{I}_{q} (\mathcal{D}_{q} \wp )(v) = \wp (v) - \wp (0)\), whenever the function ℘ is continuous at \(v=0\) [23]. The fractional Riemann–Liouville type q-integral of the function ℘ is defined by \(\mathcal{I}_{q}^{0} \wp (v) = \wp (v)\) and

$$ \mathcal{I}_{q}^{\sigma }\wp (v) = \frac{1}{\Gamma _{q}(\sigma )} \int _{0}^{v} (v- \xi )_{q}^{(\sigma - 1)} \wp (\xi ) \,\mathrm{d}_{q}\xi $$

for \(v \in [0, 1]\) and \(\sigma >0\) [24, 29]. The Caputo fractional q-derivative of the function ℘ is defined by

$$ {}^{C}\mathcal{D}_{q}^{\sigma }\wp (v) = \mathcal{I}_{q}^{[\sigma ]- \sigma } \bigl[\mathcal{D}_{q}^{[\sigma ]} \wp \bigr] (v) = \frac{1}{\Gamma _{q} ([\sigma ] - \sigma )} \int _{0}^{v} (v- \xi )_{q}^{ ( [\sigma ]-\sigma -1 )} \mathcal{D}_{q}^{[ \sigma ]} \wp (\xi ) {\,\mathrm{d}}_{q} \xi $$

for \(v \in [0,1]\) and \(\sigma >0\) [29, 62]. It has been proved that \(\mathcal{I}_{q}^{ \nu } ( \mathcal{I}_{q}^{\sigma } \wp ) (v) = \mathcal{I}_{q}^{\sigma + \nu } \wp (v)\) and \({}^{c}\mathcal{ D}_{q}^{\sigma } ( \mathcal{I}_{q}^{ \sigma } \wp ) (v)= \wp (v)\), where \(\sigma , \nu \geq 0\) [29]. The authors in [61] presented all algorithms and MATLAB lines to simplify q-factorial functions \((v -w)_{q}^{(n)}\), \((v -w)_{q}^{(\sigma )}\), \(\Gamma _{q}(v)\), \(\mathcal{I}_{q} [\wp ](v)\), and some necessary equations.

Now, we introduce some basic definitions, lemmas, and theorems which are used in the subsequent sections.

Lemma 2.1

([63])

Let \(n\in \mathbb{N}\), \(n-1<\alpha \leq n\), and \(\wp \in AC^{n}[a, b]\). Then one has \(\mathcal{I}^{\sigma }({}^{c}\mathcal{ D}_{q}^{\sigma }\wp ) (v)= \wp (v) + \sum_{i=0}^{n-1}c_{i} (v-a)^{i}\), where \(c_{0}, c_{1} ,\dots , c_{n-1}\in \mathbb{R}\).

Lemma 2.2

Let \(\sigma _{1} > \sigma _{2} > 0\). Then the formula \({}^{C}\mathcal{D}_{q}^{\sigma _{1}} ( \mathcal{I}_{q}^{\sigma _{2}} \wp ) (v) = \mathcal{I}_{q}^{\sigma _{1} - \sigma _{2}} \wp (v)\) holds almost everywhere on \(v \in [a,b]\) for \(\wp \in L_{1} [a,b]\), and it is valid at any point \(v \in [a,b]\) if \(\wp \in C([a,b], \mathbb{R})\).

Lemma 2.3

([1])

Let \(\sigma > 0\) and \(\wp \in C(0,1) \cap L^{1} (0,1)\) with a derivative of order n. Then \({}\mathcal{I}_{q}^{\sigma }( {}^{C}\mathcal{D}_{q}^{\sigma }\wp (v)) = \wp (v) + d_{0} + d_{1}t + d_{2}t^{2} + \cdots + d_{n - 1}t^{ n - 1}\) for \(d_{i} \in \mathbb{R}\) with \(i = 1, 2, \dots , n-1 \), where \(n - 1 < \sigma \leq n\).

Definition 2.4

A real function \(\wp (v)\), \(v > 0 \) is said to be in the space \(C_{r}\), \(r \in \mathbb{R}\), if there exists a real number ν (>r) such that \(\wp (v) = v^{\nu }\wp _{1}(v) \), where \(\wp _{1}(v) \in C( [0, \infty ), \infty )\).

Theorem 2.5

(Banach contraction principle, [64])

Let \(\mathcal{X}\) be a Banach space. If \(A : \mathcal{X}\to \mathcal{X}\) is the contraction map, then there exists \(x\in \mathcal{X}\) such that \(Ax=x\).

3 Main results

First, we consider the following important lemmas in our article.

Lemma 3.1

Let \(v\in AC (0,1)\) and \(2 < \sigma \leq 3\). The fractional q-differential equation

$$ {}^{C}\mathcal{D}_{q}^{\sigma }[y](t) = v(t) $$
(3)

for \(2 < \sigma \leq 3\) under conditions \(y(0) = y'(0) = 0\), \(y'(1) = a y (e)\), \(e \in J_{0}\) with \(0 \leq a <\frac{1}{e^{2}}\) has a solution

$$ y(t) = \int _{0}^{1} G_{q}(t, \xi ) v(\xi ) \,\mathrm{d}_{q}\xi + \frac{ a t^{2}}{1 - ae^{2}} \int _{0}^{1} G_{q}(e, \xi ) v(\xi ) \,\mathrm{d}_{q}\xi , $$
(4)

where

$$ G_{q}(t, \xi ) = \textstyle\begin{cases} \frac{ ( t - \xi )_{q}^{ (\sigma - 1)} - t^{2} ( 1 - \xi )_{q}^{( \alpha - 1) }}{ \Gamma _{q} ( \sigma ) },& \xi < t, \\ \frac{- t^{2} ( 1 - \xi )_{q}^{ (\sigma - 1)}}{\Gamma _{q} ( \sigma ) }, & t< \xi , \end{cases} $$
(5)

for all \(t, \xi \in \overline{J}_{0}\).

Proof

By Lemma (2.3) the solution of Eq. (3) can be written as

$$ y(t) = \int _{0}^{t} (t - \xi )_{q}^{ (\sigma - 1)} v(\xi ) \,\mathrm{d}_{q}\xi - d_{0} - d_{1}t - d_{2}t^{2}. $$

Since \(y(0) = y'(0 ) = 0 \), a simple calculation gives \(d_{0} - d_{1} = 0\), and from the boundary condition, we get \(\mathcal{I}_{q}^{\sigma }[v](1) -d_{2} = a \mathcal{I}_{q}^{\sigma }[v](e) - d_{2} a e^{2}\). Hence,

$$ d_{2} = \frac{1}{1 - a e^{2}} \bigl( \mathcal{I}_{q}^{\sigma }[v](1) - a \mathcal{I}_{q}^{\sigma }[v](e) \bigr). $$

Thus, the solution of boundary value problem (3) is

$$\begin{aligned} y(t) &= \mathcal{I}_{q}^{\sigma }[v](t) - \frac{t^{2}}{1 - a e^{2}} \bigl( \mathcal{I}_{q}^{\sigma }[v](1) - a \mathcal{I}_{q}^{\sigma }[v](e) \bigr) \\ & = \mathcal{I}_{q}^{\sigma }[v](t) - t^{2} \mathcal{I}_{q}^{\sigma }[v](1) - \frac{ae^{2}t^{2}}{1 - a e^{2}} \mathcal{I}_{q}^{\sigma }[v](1) + \frac{at^{2}}{1 - a e^{2}} \mathcal{I}_{q}^{\sigma }[v](e) \\ & = \frac{1}{\Gamma _{q} ( \sigma )} \int _{0}^{t} \bigl( ( t - \xi )_{q}^{ (\sigma - 1)} - t^{2} (1 - \xi )_{q}^{ (\sigma - 1)} \bigr) v(\xi ) \,\mathrm{d}_{q}\xi \\ &\quad {} - \frac{1}{\Gamma _{q} ( \sigma )} \int _{1}^{t} t^{2} (1 - \xi )_{q}^{( \sigma - 1)} v(\xi ) \,\mathrm{d}_{q}\xi \\ &\quad {} + \frac{a t^{2}}{(1 - a e^{2}) \Gamma _{q} ( \sigma ) } \biggl[ \int _{0}^{e} \bigl( (e - \xi )_{q}^{ (\sigma - 1)} - e^{2} (1 - \xi )_{q}^{ (\sigma - 1)} \bigr) v(\xi ) \,\mathrm{d}_{q}\xi \\ &\quad {} - \int _{e}^{t} e^{2} (1 - \xi )_{q}^{ (\sigma - 1)} v(\xi ) \,\mathrm{d}_{q}\xi \biggr] \\ & = \int _{0}^{1} G_{q}(t, \xi ) v(\xi ) \,\mathrm{d}_{q}\xi + \frac{a t^{2}}{1 - a e^{2}} \int _{0}^{1} G_{q}(e, \xi ) v(\xi ) \,\mathrm{d}_{q}\xi , \end{aligned}$$

where \(G_{q}(t, \xi )\) is defined in Eq. (5). This completes the proof. □

Now, in order to investigate the existence of solutions, we prove some properties of the function \(G_{q}(t, \xi )\).

Lemma 3.2

The functions \(G_{q}(t,\cdot) \) and \(\frac{\partial }{\partial t} G_{q}(t, \cdot)\) are integrable for each \(t \in \overline{J}_{0}\) and have the following properties:

$$ \int _{0}^{1} \bigl\vert G_{q}(t, \xi ) \bigr\vert \,\mathrm{d}_{q}\xi \leq \frac{2}{ \Gamma _{q} ( \sigma + 1)}, \int _{0}^{1} \biggl\vert \frac{ \partial }{\partial t} G_{q}(t, \xi ) \biggr\vert \,\mathrm{d}_{q} \xi \leq \frac{3}{ \Gamma _{q} ( \sigma )}. $$

Proof

Let \(t \in \overline{J}_{0}\). Then we have

$$\begin{aligned} \int _{0}^{1} \bigl\vert G_{q}(t, \xi ) \bigr\vert \,\mathrm{d}_{q}\xi &\leq \mathcal{I}_{q}^{\sigma }[I](t) + + t^{2}\mathcal{I}_{q}^{\sigma }[I](1) \\ & \leq \frac{t^{\sigma }}{\Gamma _{q} ( \sigma + 1)} + \frac{t^{2}}{\Gamma _{q} ( \sigma + 1)} \leq \frac{2}{\Gamma _{q} ( \sigma + 1)} \end{aligned}$$

and

$$\begin{aligned} \int _{0}^{1} \biggl\vert \frac{\partial }{ \partial t} G_{q}(t, \xi ) \biggr\vert \,\mathrm{d}_{q}\xi & \leq 2t \mathcal{I}_{q}^{\sigma }[I](1) + + \mathcal{I}_{q}^{\sigma -1}[I](t) \\ & \leq \frac{2t}{\Gamma _{q} ( \sigma + 1)} + \frac{t^{\sigma -1 }}{ \Gamma _{q} ( \sigma ) } \leq \frac{3}{ \Gamma _{q} ( \sigma )}. \end{aligned}$$

Hence, \(G_{q}(t, \cdot) \) and \(\frac{ \partial }{\partial t} G_{q}(t, \cdot)\) are integrable. □

Let \(C^{1} (\overline{J}_{0})\) be the class of all continuous functions. Since \({}^{C}\mathcal{D}_{q}^{\varsigma }[y](t) = \mathcal{I}_{q}^{1 - \varsigma } [y'](t)\) for \(\varsigma \in J_{0}\), the operator \({}^{C}\mathcal{D}_{q}^{\varsigma }\) is continuous for any \(y \in C^{1} (\overline{J}_{0})\). Now, for \(y \in C^{1}(\overline{J}_{0})\), we define the space

$$ \mathcal{A} = \bigl\{ y(t) : y(t) \in C^{1} (\overline{J}_{0}) \bigr\} $$

endowed with the maximum norm

$$ \Vert y \Vert = \max_{t \in \overline{J}_{0} } \bigl\vert y(t) \bigr\vert + \max_{t \in \overline{J}_{0} } \bigl\vert {}^{C}\mathcal{ D}_{q}^{\varsigma }[y](t) \bigr\vert . $$

Lemma 3.3

\((\mathcal{A}, \|\cdot\|)\) is a Banach space.

Proof

Let \(\{y_{n} \}_{n=1}^{\infty }\) be a Cauchy sequence in the space \((\mathcal{A}, \|\cdot\|)\). Obviously, \(\{y_{n} \}_{n=1}^{\infty }\) and \(\{ {}^{C}\mathcal{D}_{q}^{\varsigma }y_{n} \}_{n=1}^{\infty }\) are Cauchy sequences in the space \(C (\overline{J}_{0})\). Since \(C (\overline{J}_{0})\) is compact, \(\{y_{n} \}_{n=1}^{\infty }\) and \(\{ {}^{C}\mathcal{D}_{q}^{\varsigma }y_{n} \}_{n=1}^{\infty }\) uniformly converge to some v, \(v'\) on \(\overline{J}_{0}\). Furthermore, v, \(v'\) belong to \(C(\overline{J}_{0})\). In the following, we need to show that \(v'= {}^{C}\mathcal{D}_{q}^{\varsigma }v\). Now, by the definition of fractional integral,

$$\begin{aligned} \bigl\vert \mathcal{I}_{q}^{\varsigma } \bigl[{}^{C} \mathcal{D}_{q}^{\varsigma }[y_{n}] \bigr](t) - \mathcal{I}_{q}^{\varsigma }v'(t) \bigr\vert & \leq \mathcal{I}_{q}^{\varsigma } \bigl[ \bigl\vert {}^{C}\mathcal{D}_{q}^{\varsigma }y_{n} - v' \bigr\vert \bigr](t) \\ & \leq \frac{1}{ \Gamma _{q}(\varsigma +1)} \max_{t \in \overline{J}_{0}} \bigl\vert {}^{C}\mathcal{D}_{q}^{\varsigma }y_{n} - v' \bigr\vert . \end{aligned}$$

Therefore, using the convergence of \(\{ {}^{C}\mathcal{D}_{q}^{\varsigma }y_{n} \}_{n=1}^{\infty }\) implies that

$$ \lim_{n \to \infty } \mathcal{I}_{q}^{\varsigma } \bigl[{}^{C}\mathcal{D}_{q}^{\varsigma }[y_{n}] \bigr](t) = \mathcal{I}_{q}^{\varsigma } \bigl[v' \bigr](t) $$

uniformly on \(\overline{J}_{0}\). On the other hand, we know \(\mathcal{I}_{q}^{\varsigma }[{}^{C}\mathcal{D}_{q}^{\varsigma }[y_{n}]](t) = y_{n} \) for each \(t \in \overline{J}_{0}\) and \(\varsigma \in J_{0}\). Hence, \(\mathcal{I}_{q}^{\varsigma }[v'](t)= v\), and this means \(v'={}^{C}\mathcal{D}_{q}^{\varsigma }v\). This completes the proof. □

Remark 3.1

Lemma (2.3) implies that the solution of problem (1) coincides with the fixed point of the operator \(\mathcal{O}\) defined as

$$\begin{aligned} \mathcal{O} y(t) & = \int _{0}^{1} G_{q}(t, \xi ) w \bigl( y(t), {}^{C} \mathcal{D}_{q}^{\varsigma }[y](t) \bigr) \,\mathrm{d}_{q}\xi \\ &\quad {} + \frac{a t^{2}}{1 - a e^{2}} \int _{0}^{1} G_{q}(e, \xi ) w \bigl( y(t), {}^{C}\mathcal{D}_{q}^{\varsigma }[y](t) \bigr) \,\mathrm{d}_{q}\xi . \end{aligned}$$

3.1 Existence and uniqueness

According to the Schauder fixed point theorem, the existence result has been stated.

Theorem 3.4

Suppose that \(w: \mathbb{R}^{2} \to \mathbb{R}\) is a continuous function and there exist constants \(m_{0}, m_{1} \geq 0\), \(\beta _{0}, \beta _{1} \in J_{0}\) such that one of the following conditions is satisfied:

  1. (A1)

    There exists a nonnegative function \(\mu (t) \in \overline{J}_{0}\) such that

    $$ \bigl\vert w(y,z) \bigr\vert \leq \mu (t) + m_{0} \vert y \vert ^{\beta _{0}} + m_{1} \vert z \vert ^{\beta _{1}}. $$
    (6)
  2. (A2)

    The function w satisfies

    $$ \bigl\vert w(y,z) \bigr\vert \leq m_{0} \vert y \vert ^{\beta _{0}} + m_{1} \vert z \vert ^{\beta _{1}}. $$
    (7)

Then boundary value problem (1) has at least one solution \(y(t)\).

Proof

First, suppose that condition (A1) holds. Define the set \(\mathcal{B}\) by

$$ \mathcal{B} = \bigl\{ y(t) : \bigl\Vert y(t) \bigr\Vert \leq \delta , t \in \overline{J}_{0} \bigr\} , $$

where

$$\begin{aligned} &\delta \geq \max \biggl\{ ( 6\Delta m_{0})^{\frac{1}{1 - \beta _{0}} }, (6\Delta m_{1} )^{ \frac{1}{1 - \beta _{1}}}, 6\Delta M_{1}, \biggl( \frac{ 12 \Delta m_{0} }{ \Gamma _{q} (2 - \varsigma )} \biggr)^{ \frac{1}{1 - \beta _{0}} }, \\ &\hphantom{\delta \geq{}} \biggl( \frac{ 12 \Delta m_{1}}{ \Gamma _{q} (2 - \varsigma )} \biggr)^{\frac{1}{1 - \beta _{1}}}, \frac{ 16 a M_{1}}{ \Gamma _{q} (2 - \varsigma )( 1 - a e^{2})}, \frac{ 8 M_{2}}{ \Gamma _{q} (2 - \varsigma ) } \biggr\} , \\ &\Delta = \biggl( 1 + \frac{a}{ 1 - ae^{2}} \biggr) \frac{2}{\Gamma _{q} ( \sigma + 1 )}, \end{aligned}$$
(8)

and

$$\begin{aligned}& M_{1} = \max_{t \in \overline{J}_{0}} \biggl\{ \frac{1}{\Gamma _{q} (\sigma )} \int _{0}^{1} \bigl\vert G_{q}(t, \xi ) \mu ( \xi ) \bigr\vert \,\mathrm{d}_{q}\xi \biggr\} \\& M_{2} = \max_{t \in \overline{J}_{0} } \biggl\{ \frac{1}{\Gamma _{q} (\sigma )} \int _{0}^{1} \biggl\vert \frac{\partial }{\partial t} G_{q}(t, \xi ) \mu (\xi ) \biggr\vert \,\mathrm{d}_{q}\xi \biggr\} . \end{aligned}$$
(9)

It is clear that \(\mathcal{B}\) is a closed, bounded, and convex subset of Banach space \({}\mathcal{A}\). Here, we prove that \(\mathcal{O} : \mathcal{B} \to \mathcal{B}\). For any \(y \in \mathcal{B}\), we obtain

$$\begin{aligned} \bigl\vert \mathcal{O} y(t) \bigr\vert & \leq \int _{0}^{1} \bigl\vert G_{q} (t, \xi ) w \bigl( y(t), {}^{C}\mathcal{D}_{q}^{\varsigma }[y] (t) \bigr) \bigr\vert \,\mathrm{d}_{q}\xi \\ &\quad{} + \frac{a t^{2}}{1 - ae^{2}} \int _{0}^{1} \bigl\vert G_{q}(e, \xi ) w \bigl( y(t), {}^{C}\mathcal{D}_{q}^{\varsigma }[y](t) \bigr) \bigr\vert \,\mathrm{d}_{q}\xi \\ & \leq \int _{0}^{1} \bigl\vert G_{q}(t, \xi ) \mu (\xi ) \bigr\vert \,\mathrm{d}_{q} \xi + \bigl[ m_{0} \delta ^{\beta _{0} } + m_{1} \delta ^{\beta _{1}} \bigr] \int _{0}^{1} \bigl\vert G_{q}(t, \xi ) \bigr\vert \,\mathrm{d}_{q}\xi \\ &\quad{} + \frac{a}{1 - a e^{2}} \biggl[ \int _{0}^{1} \bigl\vert G_{q}(e, \xi ) \mu (\xi ) \bigr\vert \,\mathrm{d}_{q}\xi \\ &\quad{} + \bigl( m_{0} \delta ^{\beta _{0}} + m_{1} \delta ^{\beta _{1}} \bigr) \int _{0}^{1} \bigl\vert G_{q}(t, \xi ) \bigr\vert \,\mathrm{d}_{q}\xi \biggr] \\ & \leq \biggl( 1+ \frac{a}{1 - a e^{2}} \biggr) \biggl[ M_{1} + \frac{2}{\Gamma _{q} (\sigma + 1 ) } \bigl( m_{0} \delta ^{\beta _{0}} + m_{1} \delta ^{\beta _{1}} \bigr) \biggr] \\ & \leq \Delta \bigl[ M_{1} + \bigl( m_{0} \delta ^{\beta _{0}} + m_{1} \delta ^{\beta _{1}} \bigr) \bigr] \leq \frac{1}{2} \delta . \end{aligned}$$

Thus, for almost all \(\varsigma \in J_{0}\), we have

$$\begin{aligned} \bigl\vert {}^{C}\mathcal{D}_{q}^{\varsigma }[ \mathcal{O}y](t) \bigr\vert & = \bigl\vert \mathcal{I}_{q}^{1 - \varsigma } \bigl[\mathcal{O}y' \bigr](t) \bigr\vert \\ & \leq \frac{1}{\Gamma _{q} ( 1 - \varsigma )} \int _{0}^{t} ( t - \xi )_{q}^{(-\varsigma )} \\ &\quad{} \times \biggl( \int _{0}^{1} \biggl\vert \frac{ \partial }{\partial \xi } G_{q}(\xi , \tau ) w \bigl( \tau , y ( \tau ), {}^{C} \mathcal{ D}_{q}^{\varsigma }[y] (\tau ) \bigr) \biggr\vert \,\mathrm{d}_{q}\tau \\ &\quad{} + \frac{2a \xi }{(1 - a e^{2}) } \int _{0}^{1} \bigl\vert G_{q}(e, \tau ) w \bigl( \tau , y( \tau ), {}^{C}\mathcal{ D}_{q}^{\varsigma }[y] ( \tau ) \bigr) \bigr\vert \,\mathrm{d}_{q} \tau \biggr) \,\mathrm{d}_{q}\xi \\ & \leq \frac{1}{\Gamma _{q} ( 1 - \varsigma ) } \int _{0}^{t} ( t - q \xi )^{(-\varsigma )} \biggl[ \int _{0}^{1} \biggl\vert \frac{\partial }{\partial \xi } G_{q}(\xi , \tau ) \mu (\tau ) \biggr\vert \\ &\quad{} + \bigl( m_{0} \delta ^{\beta _{0}} + m_{1} \delta ^{\beta _{1}} \bigr) \int _{0}^{1} \biggl\vert \frac{\partial }{\partial \xi } G_{q}(\xi , \tau ) \biggr\vert \,\mathrm{d}_{q}\tau \\ &\quad{} + \frac{2a\xi }{1 - ae^{2}} \biggl( \int _{0}^{1} \bigl\vert G_{q} ( \xi , \tau )\mu (\tau ) \bigr\vert {\,\mathrm{d}}_{q}\tau \\ &\quad{} + \bigl( m_{0} \delta ^{\beta _{0}} + m_{0} \delta ^{\beta _{1}} \bigr) \int _{0}^{1} \bigl\vert G_{q}(e, \tau ) \bigr\vert { \,\mathrm{d}}_{q}\tau \biggr) \biggr] { \,\mathrm{d}}_{q}\xi \\ & \leq \frac{1}{\Gamma _{q} ( 1 - \varsigma ) } \int _{0}^{t} ( t - \xi )_{q}^{(-\varsigma )} \biggl( M_{2} + \frac{3 }{\Gamma _{q} (\sigma )} \bigl( m_{0} \delta ^{\beta _{0}} + m_{1} \delta ^{\beta _{1}} \bigr) \biggr) \,\mathrm{d}_{q}\xi \\ &\quad{} + \frac{2a}{ (1 - ae^{2}) \Gamma _{q} (1 - \varsigma ) } \int _{0}^{t} \xi (t-\xi )_{q}^{(-\varsigma )} \\ &\quad{} \times \biggl( M_{1} + \frac{2}{\Gamma _{q} (\sigma + 1) } \bigl( m_{0} \delta ^{\beta _{0}} + m_{1} \delta ^{\beta _{1}} \bigr) \biggr) \,\mathrm{d}_{q}\xi \\ & \leq \frac{1}{\Gamma _{q} ( 1 - \varsigma ) } \biggl( M_{2} + \frac{3 }{\Gamma _{q} (\sigma )} \bigl( m_{0} \delta ^{\beta _{0}} + m_{1} \delta ^{\beta _{1}} \bigr) \biggr) \frac{t^{1 - \varsigma }}{1 - \varsigma } \\ &\quad{} + \frac{2a}{ (1 - ae^{2}) \Gamma _{q} (1 - \varsigma ) } \biggl( M_{1} + \frac{2}{\Gamma _{q} (\sigma + 1) } \bigl( m_{0} \delta ^{\beta _{0}} + m_{1} \delta ^{\beta _{1}} \bigr) \biggr) \\ &\quad{} \times \frac{t^{2 - \varsigma }}{(1 - \varsigma ) \Gamma _{q}(2 - \varsigma )} \\ & \leq \frac{1}{\Gamma _{q} ( 2 - \varsigma ) } \biggl( M_{2} + \frac{3 }{\Gamma _{q} (\sigma )} \bigl( m_{0} \delta ^{\beta _{0}} + m_{1} \delta ^{\beta _{1}} \bigr) \biggr) \\ &\quad{} + \frac{2a}{ (1 - ae^{2}) \Gamma _{q} (3 - \varsigma ) } \biggl( M_{1} + \frac{2}{\Gamma _{q} (\sigma + 1) } \bigl( m_{0} \delta ^{\beta _{0}} + m_{1} \delta ^{\beta _{1}} \bigr) \biggr) \\ & \leq \frac{3 \Delta }{\Gamma _{q} ( 2 - \varsigma )} \bigl( m_{0} \delta ^{\beta _{0}} \bigr) + \frac{2a M_{1} }{(1 - ae^{2})\Gamma _{q} ( 2 - \varsigma ) } + \frac{M_{2}}{\Gamma _{q} ( 2 - \varsigma ) } \\ & \leq \frac{1}{2} \delta . \end{aligned}$$

Clearly, \(\mathcal{O}y(t)\) and \({}^{C}\mathcal{D}_{q}^{\varsigma }[\mathcal{O}y](t)\) are continuous in \(\overline{J}_{0}\). Therefore \(\mathcal{O} : \mathcal{B} \to \mathcal{B}\). In the second case, suppose that condition (A2) holds. Choose

$$ 0 < \delta \leq \min \biggl\{ \biggl( \frac{1}{4\Delta m_{0}} \biggr)^{ \frac{1}{1 - \beta _{0}}}, \biggl( \frac{1}{4\Delta m_{1}} \biggr)^{ \frac{1}{1 - \beta _{1}}}, \biggl( \frac{ \Gamma _{q} (2 - \varsigma ) }{6\Delta m_{0}} \biggr)^{ \frac{1}{1 - \beta _{0}}}, \biggl( \frac{\Gamma _{q} (2 - \varsigma ) }{6\Delta m_{1}} \biggr)^{ \frac{1}{1 - \beta _{1}} } \biggr\} . $$

Again, by a similar way, we get \(\| \mathcal{O}y \| \leq \delta \), and therefore, in this case, \(\mathcal{O} : \mathcal{B} \to \mathcal{B}\). Here, we need to show that \(\mathcal{O}\) is a completely continuous operator. First, the equicontinuity of \(\mathcal{O}\) will be shown as follows. Suppose that \(s_{1}, s_{2} \in \overline{J}_{0}\) with \(s_{1} < s_{2} \) and

$$ N_{0} = 1+\max_{ t \in \overline{J}_{0}} \bigl\{ \bigl\vert w \bigl( t, y(t), {}^{C}\mathcal{D}_{q}^{\varsigma }[y](t) \bigr) \bigr\vert : y \in \mathcal{B} \bigr\} . $$

Then

$$\begin{aligned} &\bigl\vert \mathcal{O}y (s_{1}) - \mathcal{O}y (s_{2}) \bigr\vert \\ &\quad = \biggl\vert \int _{0}^{1} \bigl( G_{q}(s_{2}, \xi ) - G_{q}(s_{1}, \xi ) \bigr) w \bigl( y(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y]( \xi ) \bigr) \,\mathrm{d}_{q}\xi \\ &\quad \quad{} + \frac{a( s_{2}^{2} - s_{1}^{2} ) }{ 1 - ae^{2}} \int _{0}^{1} G_{q}(e, \xi ) w \bigl( y(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y]( \xi ) \bigr) \,\mathrm{d}_{q}\xi \biggr\vert \\ &\quad \leq N_{0} \int _{0}^{1} \bigl\vert G_{q}(s_{2}, \xi ) - G_{q}(s_{1}, \xi ) \bigr\vert \,\mathrm{d}_{q}\xi + \frac{2 a N_{0} }{1 - a e^{2}} \bigl( s_{2}^{2} - s_{1}^{2} \bigr) \\ &\quad \leq \frac{2 a N_{0} }{1 - a e^{2}} \bigl( s_{2}^{2} - s_{1}^{2} \bigr) + \frac{N_{0}}{\Gamma _{q} ( \sigma ) } \biggl[ \int _{0}^{s_{1}} \bigl( s_{2}^{2} - s_{1}^{2} \bigr) (1 -\xi )_{q}^{(\sigma - 1)} \\ &\quad \quad{} + (s_{2} - \xi )_{q}^{(\sigma - 1)} + (s_{1} -\xi )_{q}^{( \sigma - 1)} \,\mathrm{d}_{q}\xi \\ &\quad \quad{} + \int _{s_{1}}^{s_{2}} \bigl( s_{2}^{2} - s_{1}^{2} \bigr) (1 - \xi )_{q}^{( \sigma - 1)} + (s_{2} - \xi )_{q}^{(\sigma - 1)} \,\mathrm{d}_{q} \xi \\ &\quad \quad{} + \int _{s_{2}}^{1} \bigl( s_{2}^{2} - s_{1}^{2} \bigr) (1 - \xi )_{q}^{( \sigma - 1)} \,\mathrm{d}_{q}\xi \biggr] \\ &\quad \leq \frac{ 2 a N_{0} }{1 - ae^{2})} \bigl( s_{2}^{2} - s_{1}^{2} \bigr) + \frac{N_{0}}{\Gamma _{q} ( \alpha )} \biggl[ \bigl( s_{2}^{2} - s_{1}^{2} \bigr) \int _{0}^{1} (1 - \xi )_{q}^{(\sigma - 1)} \,\mathrm{d}_{q}\xi \\ &\quad \quad{} + \int _{0}^{s_{2}} \bigl(s_{2}^{2} - \xi \bigr)_{q}^{(\sigma - 1)} \,\mathrm{d}_{q}\xi - \int _{0}^{s_{1}} \bigl(s_{1}^{2} - \xi \bigr)_{q}^{( \sigma - 1)} \,\mathrm{d}_{q}\xi \biggr] \\ & \quad \leq \frac{N_{0}}{\Gamma _{q} ( \alpha + 1)} \biggl[ s_{2}^{2} - s_{1}^{2} + s_{2}^{\sigma }- s_{1}^{\sigma }+ \frac{ 2 a(s_{2}^{2} - s_{1}^{2} )}{1- a e^{2} } \biggr] \\ &\quad \leq N_{0} \biggl[ \Delta \bigl( s_{2}^{2} - s_{1}^{2} \bigr) + \frac{ s_{2}^{\sigma }- s_{1}^{\sigma }}{ \Gamma _{q} ( \sigma + 1) } \biggr], \end{aligned}$$

and

$$\begin{aligned} &\bigl\vert {}^{C}\mathcal{D}_{q}^{\varsigma }[ \mathcal{O}y](s_{2}) - {}^{C} \mathcal{D}_{q}^{\varsigma }[ \mathcal{O}y ] (s_{2}) \bigr\vert \\ &\quad = \frac{1}{ \Gamma _{q} ( 1 - \varsigma ) } \biggl\vert \int _{0}^{s_{2}} (s_{2} - \xi )_{q}^{(-\varsigma )} \biggl( \int _{0}^{1} \frac{\partial }{ \partial \xi } G_{q}( \xi , \tau ) w \bigl( y (\tau ), {}^{C} \mathcal{D}_{q}^{\varsigma }[y]( \tau ) \bigr) \,\mathrm{d}_{q}\tau \\ &\quad \quad{} + \frac{ 2 a\xi }{1 - ae^{2}} \int _{0}^{1} G_{q}(e, \tau ) w \bigl( y(\tau ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y] (\tau ) \bigr) \biggr) \,\mathrm{d}_{q}\xi \\ &\quad \quad{} - \int _{0}^{s_{1}} (s_{1} - \xi )_{q}^{(-\varsigma )} \biggl( \int _{0}^{1} \frac{\partial }{\partial \xi } G_{q}(\xi , \tau ) w \bigl( y(\tau ), {}^{C} \mathcal{D}_{q}^{\varsigma }[y] (\tau ) \bigr) \,\mathrm{d}_{q}\tau \\ &\quad \quad{} + \frac{ 2 a \xi }{ 1 - ae^{2} } \int _{0}^{1} G_{q}(e, \tau ) w \bigl( y(\tau ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y] ( \tau ) \bigr) \,\mathrm{d}_{q}\tau \biggr) \,\mathrm{d}_{q} \xi \biggr\vert \\ &\quad \leq \frac{3N_{0} }{ \Gamma _{q} ( 1 - \varsigma ) \Gamma _{q} ( \sigma )} \biggl\vert \int _{0}^{s_{2}} (s_{2} - \xi )_{q}^{(-\varsigma )} \,\mathrm{d}_{q}\xi - \int _{0}^{s_{1}} (s_{1} - \xi )_{q}^{(- \varsigma )} \,\mathrm{d}_{q}\xi \biggr\vert \\ &\quad \quad{} + \frac{6 a N_{0} }{\Gamma _{q} ( 1 - \varsigma ) \Gamma _{q} ( \sigma ) ( 1 - ae^{2}) } \\ &\quad \quad{} \times \biggl\vert \int _{0}^{s_{2}} \xi (s_{2} - \xi )_{q}^{(- \varsigma )} \,\mathrm{d}_{q}\xi - \int _{0}^{s_{1}} \xi (s_{1} - \xi )_{q}^{(-\varsigma )} \,\mathrm{d}_{q}\xi \biggr\vert \\ &\quad \leq \frac{3N_{0}}{\Gamma _{q} ( 1 - \varsigma ) \Gamma _{q} ( \sigma )} \biggl\vert \int _{0}^{s_{1}} \bigl( (s_{2} - \xi )_{q}^{(-\varsigma )} - (s_{1} - \xi )_{q}^{(-\varsigma )} \bigr) \,\mathrm{d}_{q}\xi \\ &\quad \quad{} + \int _{s_{1}}^{s_{2}} (s_{2} - q\xi )^{(-\varsigma )} \,\mathrm{d}_{q}\xi \biggr\vert + \frac{6 aN_{0} }{ \Gamma _{q} ( 1 - \varsigma ) \Gamma _{q} ( \sigma )(1 - ae^{2})} \\ &\quad \quad{} \times \biggl\vert \int _{0}^{s_{1}} \bigl( \xi (s_{2} - \xi )_{q}^{(- \varsigma )} - \xi (S_{1} - \xi )_{q}^{(-\varsigma )} \bigr) \,\mathrm{d}_{q}\xi \\ &\quad \quad{} + \int _{s_{1}}^{s_{2}} \xi ( s_{2} - q\xi )^{(-\varsigma )} \,\mathrm{d}_{q}\xi \biggr\vert \\ &\quad \leq \frac{3N_{0} }{\Gamma _{q} ( 1 - \varsigma ) \Gamma _{q} ( \sigma ) } \bigl( s_{2}^{1 - \varsigma } - s_{1}^{ 1 - \varsigma } + 2 ( s_{2} - s_{1} )_{q}^{(1 - \varsigma )} \bigr) \\ &\quad \quad{} + \frac{6 a N_{0} }{ \Gamma _{q} ( \sigma ) ( 1 - ae^{2})} \\ &\quad \quad{} \times \biggl( \frac{ 2 s_{1} ( s_{2} - s_{1} )_{q}^{(1 - \varsigma )} }{ \Gamma _{q} ( 2 - \varsigma )} + \frac{ s_{2}^{2}- s_{1}^{1} }{ \Gamma _{q} ( 3 - \varsigma )} + \frac{2 ( s_{2} - s_{1} )_{q}^{(2 - \varsigma )} }{ \Gamma _{q} ( 3 - \varsigma ) } \biggr). \end{aligned}$$

Since the functions \(s_{2}^{2} -s_{1}^{2} \), \(d_{2}^{\sigma }- s_{1}^{\sigma }\), \(( s_{2} - s_{1} )_{q}^{(2 - \varsigma )}\), and \(s_{1} ( s_{2} - s_{1} )^{1 - \varsigma }\) are continuous, we conclude that \(\mathcal{O}y\) is an equicontinuous set. Obviously, \(\mathcal{O}y \) is uniformly bounded because \(\mathcal{O} (\mathcal{B}) \subseteq \mathcal{B}\). By means of the Arzelá–Ascoli theorem, \(\mathcal{O}\) is a compact operator. Therefore, from the Schauder fixed point theorem, the operator \(\mathcal{O}\) has a fixed point, i.e., the q-fractional boundary value problem (1) has a solution. □

In what follows, we prove the uniqueness of solution for Eq. (1) based on application of the Banach fixed point theorem.

Theorem 3.5

Let \(w : \mathbb{R}^{2} \to \mathbb{R}\) be a continuous function and let it fulfill a Lipschitz condition with respect to the first and second variables with Lipschitz constant

$$ 0 < \ell < \frac{\Gamma _{q} ( 2 - \varsigma )}{\Delta [ 3 + \Gamma _{q} ( 2 - \varsigma )] }, $$
(10)

i.e.,

$$ \bigl\vert w(y_{1},z_{1}) - w(y_{2}, z_{2}) \bigr\vert \leq \ell \bigl( \vert y_{1} - y_{2} \vert + \vert z_{1} - z_{2} \vert \bigr). $$

Then problem (1) has a unique solution.

Proof

In Theorem 3.4, we have shown that \(\mathcal{O}\) is a continuous operator and \(\mathcal{O} : \mathcal{B} \to \mathcal{B}\). Therefore, using the Banach fixed point theorem, it is sufficient to show that \(\mathcal{O}\) is a contraction mapping. For any \(y_{1}, y_{2} \in \mathcal{A}\),

$$\begin{aligned} &\bigl\vert \mathcal{O}y_{1} (t) - \mathcal{O} y_{2}(t) \bigr\vert \\ &\quad \leq \biggl\vert \int _{0}^{1} G_{q}(t, \xi ) \bigl( w \bigl( y_{1}(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y_{1}]( \xi ) \bigr) \\ &\quad \quad{} - w \bigl( y_{2}(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y_{2}]( \xi ) \bigr) \bigr) \,\mathrm{d}_{q}\xi \biggr\vert \\ &\quad \quad{} + \frac{at^{2} }{1 - a e^{2}} \biggl\vert \int _{0}^{1} G_{q}(e, \xi ) \bigl( w \bigl( y_{1}(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y_{1}]( \xi ) \bigr) \\ &\quad \quad{} - w \bigl( y_{2} (\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y_{2}] (\xi ) \bigr) \bigr) \,\mathrm{d}_{q}\xi \biggr\vert \\ &\quad \leq \ell \Vert y_{1} - y_{2} \Vert \biggl( \int _{0}^{1} \bigl\vert G_{q}(t, \xi ) \bigr\vert \,\mathrm{d}_{q}\xi + \frac{at^{2} }{1 - a e^{2}} \int _{0}^{1} \bigl\vert G_{q}(e, \xi ) \bigr\vert \,\mathrm{d}_{q}\xi \biggr) \\ &\quad \leq \ell \Delta \Vert y_{1} - y_{2} \Vert , \\ &\bigl\vert {}^{C}\mathcal{D}_{q}^{\varsigma }[ \mathcal{O}y](s_{2}) - {}^{C} \mathcal{D}_{q}^{\varsigma }[ \mathcal{O}y](s_{2}) \bigr\vert \\ &\quad = \biggl\vert \frac{1}{\Gamma _{q} ( 1 - \varsigma ) } \int _{0}^{t} (t - \xi )_{q}^{(-\varsigma )} \bigl( \mathcal{O}'y_{1} ( \xi ) - \mathcal{O}'y_{1} (\xi ) \bigr) \,\mathrm{d}_{q} \xi \biggr\vert \\ &\quad \leq \frac{1}{\Gamma _{q} ( 1 - \varsigma )} \biggl\vert \int _{0}^{t} (t - \xi )_{q}^{(-\varsigma )} \\ &\quad \quad{} \times \biggl( \int _{0}^{1} \frac{ \partial }{ \partial \xi } G_{q} (\xi , \tau ) \bigl( w \bigl( y_{1}(\tau ), {}^{C}\mathcal{D }_{q}^{\varsigma }[y_{1}]( \tau ) \bigr) \\ &\quad \quad{} - w \bigl( y_{2}(\tau ) , {}^{C}\mathcal{D}_{q}^{\varsigma }[y_{2}] (\tau ) \bigr) \bigr) \,\mathrm{d}_{q}\tau \\ &\quad \quad{} + \frac{2a\xi }{1 - a e^{2}} \int _{0}^{1} G_{q}(e, \tau ) \bigl( w \bigl( y_{1}(\tau ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y_{1}]( \tau ) \bigr) \\ &\quad \quad{} - w \bigl( y_{2}(\tau ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y_{2}]( \tau ) \bigr) \bigr) \,\mathrm{d}_{q}\tau \biggr) \,\mathrm{d}_{q} \xi \biggr\vert \\ &\quad \leq \frac{3\ell }{\Gamma _{q} ( 1 - \varsigma )\Gamma _{q} ( \sigma ) } \Vert y_{1} - y_{2} \Vert \\ &\quad \quad{} \times \biggl( \int _{0}^{t} (t - \xi )_{q}^{(-\varsigma )} \,\mathrm{d}_{q}\xi + \frac{2a }{1 - ae^{2}} \int _{0}^{t} \xi (t - \xi )_{q}^{(-\varsigma )} \,\mathrm{d}_{q}\xi \biggr) \\ &\quad \leq \frac{3 \ell \Delta }{\Gamma _{q} ( 2 - \varsigma ) } \Vert y_{1} - y_{2} \Vert . \end{aligned}$$

Therefore

$$ \Vert \mathcal{O}y_{1} - \mathcal{O}y_{2} \Vert \leq \biggl[ \Delta \ell + \frac{3 \ell \Delta }{\Gamma _{q} ( 2 - \varsigma ) } \biggr] \Vert y_{1} - y_{2} \Vert . $$

Hence, by the Banach fixed point theorem, \(\mathcal{O}\) has a unique fixed point which is a solution of problem (1). □

3.2 Stability of solution

In this section, we study the stability analysis of problem (1) under various perturbations. Dependence solution on the boundary value condition is discussed in Theorem 3.6. Stability of the solution with respect to the perturbation of w is analyzed in Theorem 3.7. Finally, the perturbation effect of fractional order derivative on the solution is studied in Lemma 3.8 and Theorem 3.9.

Theorem 3.6

Suppose that function w fulfills the conditions of Theorem 3.5, and let \(\hat{v}(t)\) be the solution of the following perturbed problem:

$$ {}^{C}\mathcal{D}_{q}^{\sigma }[y](t) = w \bigl( y(t), {}^{C} \mathcal{D}_{q}^{\varsigma }[y](t) \bigr) $$
(11)

for each \(2 < \alpha \leq 3\), \(\varsigma \in J_{0}\), on the boundary value conditions \(y(0) = \epsilon _{1}\), \(y'(0) = \epsilon _{2}\), and

$$ y(1) = a y(e) + \epsilon _{3} $$

for \(e \in J_{0}\) with \(0 \leq a < \frac{1}{e^{2}}\). Then \(\| y - \hat{v} \| = O(\epsilon )\), here \(\epsilon = \max \{ \epsilon _{1}, \epsilon _{2}, \epsilon _{3}\}\).

Proof

Similar to Lemma 2.3 the solution of problem (11) is

$$\begin{aligned} \hat{v}(t) & = \int _{0}^{1} G_{q}(t, q\xi ) w \bigl( \hat{v}(\xi ), {}^{C} \mathcal{D}_{q}^{\varsigma } \bigl[\hat{v}(\xi ) \bigr] \bigr) \,\mathrm{d}_{q} \xi \\ &\quad{} + \frac{at^{2} }{1 - a e^{2}} \int _{0}^{1} G_{q}(e, q\xi ) w \bigl( \hat{v}(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[ \hat{v}](\xi ) \bigr) \,\mathrm{d}_{q}\xi + h(t), \end{aligned}$$
(12)

where

$$ h(t) = \frac{ t^{2} }{1 - ae^{2}} \bigl( \epsilon _{1} ( a- 1 ) + \epsilon _{2} ( ae - 1 ) \bigr) + \epsilon _{2} t + \epsilon _{1}. $$

Thus,

$$\begin{aligned} \vert y - \hat{v} \vert & \leq \biggl\vert \int _{0}^{1} G_{q}(t, \xi ) \bigl[ w \bigl( y(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y]( \xi ) \bigr) - w \bigl( y(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y]( \xi ) \bigr) \bigr] \,\mathrm{d}_{q}\xi \biggr\vert \\ &\quad{} + \frac{ a t^{2} }{1 - ae^{2}} \biggl\vert \int _{0}^{1} G_{q}(e, \xi ) \bigl[ w \bigl( y(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y]( \xi ) \bigr) \\ &\quad{} - w \bigl( \hat{v}(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[ \hat{v}](\xi ) \bigr) \bigr] \,\mathrm{d}_{q}\xi \biggr\vert + \bigl\vert h(t) \bigr\vert \\ & \leq \ell \Vert y - \hat{v} \Vert \biggl( \int _{0}^{1} G_{q}(t, \xi ) { \,\mathrm{d}}_{q}\xi + \frac{ a t^{2} }{1 - ae^{2}} \int _{0}^{1} G_{q}(t, \xi ) { \,\mathrm{d}}_{q}\xi \biggr) + \bigl\vert h(t) \bigr\vert \\ & \leq \ell \Delta \Vert y - \hat{v} \Vert + \bigl\vert h(t) \bigr\vert , \end{aligned}$$

and

$$\begin{aligned} &\bigl\vert {}^{C}\mathcal{D}_{q}^{\varsigma }[y](t) - {}^{C}\mathcal{D}_{q}^{\varsigma }[\hat{v}](t) \bigr\vert \\ &\quad = \frac{ 1 }{\Gamma _{q} ( 1 - \varsigma )} \biggl\vert \int _{0}^{t} ( t - \xi )_{q}^{(-\varsigma )} \\ &\qquad {}\times \biggl( \int _{0}^{1} \frac{ \partial }{\partial \xi } G_{q}( \xi , \tau ) \bigl( w \bigl( y(\tau ), {}^{C} \mathcal{D}_{q}^{\varsigma }[y](\tau ) \bigr) - w \bigl( \hat{v}( \tau ), {}^{C} \mathcal{D}_{q}^{\varsigma }[\hat{v}]( \tau ) \bigr) \bigr) \,\mathrm{d}_{q}\tau \\ &\quad \quad{} + \frac{ 2 a\xi }{1 - ae^{2} } \int _{0}^{1} G_{q}(e, \tau ) \bigl( w \bigl( y(\tau ), {}^{C}\mathcal{D }_{q}^{\varsigma }[y]( \tau ) \bigr) \\ &\quad \quad{} - w \bigl( \hat{v}(\tau ), {}^{C}\mathcal{D}_{q}^{\varsigma }[ \hat{v}](\tau ) \bigr) \bigr) \,\mathrm{d}_{q}\tau \biggr) \,\mathrm{d}_{q}\xi \biggr\vert + \bigl\vert {}^{C} \mathcal{D}_{q}^{\varsigma }[h](t) \bigr\vert \\ &\quad \leq \frac{3 \ell }{ \Gamma _{q} ( 1 - \varsigma )\Gamma _{q} (\sigma )} \Vert y - \hat{v} \Vert \biggl( \int _{0}^{t} ( t - \xi )_{q}^{(-\varsigma )} \,\mathrm{d}_{q}\xi \\ &\quad \quad{} + \frac{ 2 a\xi }{1 - ae^{2} } \int _{0}^{t} \xi ( t - \xi )_{q}^{(- \varsigma )} \,\mathrm{d}_{q}\xi \biggr)+ \bigl\vert {}^{C}\mathcal{D}_{q}^{\varsigma }[h](t) \bigr\vert \\ &\quad \leq \frac{3 \ell \Delta }{ \Gamma _{q} ( 1 - \varsigma )} \Vert y - \hat{v} \Vert + \bigl\vert {}^{C} \mathcal{D}_{q}^{\varsigma }[h](t) \bigr\vert . \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert y - \hat{v} \Vert & \leq \frac{1}{ 1 - ( \ell \Delta + \frac{3 \ell \Delta }{\Gamma _{q} ( 2 - \varsigma )} ) } \\ &\quad{} \times \biggl( \biggl\vert \frac{ at^{2} }{1 - ae^{2}} \bigl( \epsilon _{1} ( a- 1 ) + \epsilon _{2}( ae - 1 ) \bigr) +\epsilon _{2} t + \epsilon _{1} \biggr\vert \\ &\quad{} + \biggl\vert \frac{2t^{2-\varsigma }}{(1-ae^{2}) \Gamma _{q}(3-\varsigma )} \bigl( \epsilon _{1} ( a- 1 ) + \epsilon _{2}( ae - 1 ) \bigr)+ \frac{ \epsilon _{2}}{\Gamma _{q} ( 2 - \varsigma )} t^{ 1 - \varsigma } \biggr\vert \biggr) \\ & \leq \frac{\epsilon }{1 - (\ell \Delta + \frac{3 \ell \Delta }{\Gamma _{q} ( 2 - \varsigma )} )} \\ &\quad{} \times \biggl\vert \frac{ 1 }{1 - ae^{2}} \biggl[ 1 + \frac{2}{\Gamma _{q} ( 3 - \varsigma )} \biggr] ( a+ 2ae) + 2 + \frac{1}{\Gamma _{q} ( 2 - \varsigma )} \biggr\vert . \end{aligned}$$

This completes the proof. □

Theorem 3.7

Suppose that the conditions of Theorem 3.5hold, and let \(\hat{v}(t)\) be the solution of the following perturbed problem on function w:

$$ {}^{C}\mathcal{D}_{q}^{\alpha }[y](t) = w \bigl( y(t), {}^{C} \mathcal{D}_{q}^{\varsigma }[y](t) \bigr) + \epsilon $$
(13)

for \(t \in \overline{J}_{0}\), \(2 < \alpha \leq 3\), and \(\varsigma \in J_{0}\), with the boundary conditions \(y_{0} = y'_{0} = 0\), \(y_{1} = a y(e)\) for \(e \in J_{0}\) with \(0 \leq a < \frac{1}{e^{2}}\). Then \(\| y - \hat{v} \| = O(\epsilon )\).

Proof

The solution of problem (13) is

$$\begin{aligned} \hat{v} (t)&= \int _{0}^{1} G_{q}(t, \xi ) \bigl( w \bigl( \hat{v}( \xi ), {}^{C}\mathcal{D}_{q}^{\varsigma } \bigr) + \epsilon \bigr) \,\mathrm{d}_{q}\xi \\ &\quad{} + \frac{a t^{2} }{1 - ae^{2}} \int _{0}^{1} G_{q}(e, \xi ) \bigl( w \bigl(\hat{v}(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[ \hat{v}](\xi ) \bigr) + \epsilon \bigr) \,\mathrm{d}_{q}\xi . \end{aligned}$$
(14)

Then, similar to the proof of the previous theorem

$$\begin{aligned} \vert y - \hat{v} \vert & \leq \ell \Delta \Vert y - \hat{v} \Vert + \epsilon \biggl( \int _{0}^{1} G_{q}(t, \xi ) \,\mathrm{d}_{q}\xi + \frac{ a t^{2} }{1 - ae^{2}} \int _{0}^{1} G_{q}(t, \xi ) \,\mathrm{d}_{q}\xi \biggr) \\ & \leq \Delta \Vert y - \hat{v} \Vert + \epsilon \Delta \end{aligned}$$

and

$$\begin{aligned} \bigl| {}^{C}\mathcal{D}_{q}^{\varsigma }[y](t)- {}^{C}\mathcal{D}_{q}^{\varsigma }[\hat{v}](t)\bigr| &\leq \frac{3\ell \Delta }{\Gamma _{q} ( 2 - \varsigma ) \Gamma _{q} ( \sigma )} \Vert y - \hat{v} \Vert \\ &\quad{} + \epsilon \biggl( \int _{0}^{t} ( t - \xi )_{q}^{(- \varsigma )} \,\mathrm{d}_{q}\xi + \frac{ 2 a}{1 - ae^{2}} \int _{0}^{t} \xi ( t - \xi )_{q}^{(-\varsigma )} \,\mathrm{d}_{q}\xi \biggr) \\ & \leq \frac{3\ell \Delta }{\Gamma _{q} ( 2 - \varsigma )} \Vert y - \hat{v} \Vert + \frac{3 \epsilon \Delta }{\Gamma _{q} ( 2 - \varsigma )}. \end{aligned}$$

Indeed,

$$ \Vert y - \hat{v} \Vert \leq \frac{\epsilon }{1 - (\ell \Delta + \frac{3 \epsilon \Delta }{ \Gamma _{q} ( 2 - \varsigma )} )} \biggl[ \Delta + \frac{3\Delta }{ \Gamma _{q} ( 2 - \varsigma )} \biggr]. $$

This completes the proof. □

For perturbation analysis on the fractional order of the q-derivative, we first state and prove the following lemma and then the main theorem will be discussed.

Lemma 3.8

Let \(s,t \in \overline{J}_{0}\) and \(2 < \sigma - \epsilon < \sigma \), then

$$ \int _{0}^{t} \biggl\vert \frac{s^{\sigma - 1}}{ \Gamma _{q} (\sigma )} - \frac{s^{ \sigma - \epsilon - 1}}{ \Gamma _{q} (\sigma - \epsilon )} \biggr\vert \,\mathrm{d}_{q}s = O (\epsilon ). $$

Proof

We estimate the integral as follows:

$$\begin{aligned} \int _{0}^{t} \biggl\vert \frac{s^{\sigma - 1}}{ \Gamma _{q} (\sigma ) } - \frac{s^{\sigma - \epsilon - 1} }{ \Gamma _{q} (\sigma - \epsilon )} \biggr\vert \,\mathrm{d}_{q}s &\leq \int _{0}^{t} \biggl\vert \frac{s^{\sigma - 1}}{ \Gamma _{q} (\sigma )} - \frac{ s^{\sigma - \epsilon - 1}}{ \Gamma _{q} (\sigma - \epsilon )} \biggr\vert \,\mathrm{d}_{q}s \\ &\quad{} + \int _{0}^{t} \biggl\vert \frac{s^{\sigma - \epsilon - 1}}{ \Gamma _{q} (\sigma )} - \frac{ s^{\sigma - \epsilon - 1}}{\Gamma _{q} (\sigma - \epsilon )} \biggr\vert \,\mathrm{d}_{q}s \\ & \leq \frac{1}{\Gamma _{q} (\sigma ) } \biggl[ \frac{1}{\sigma } - \frac{1}{ \alpha - \epsilon } \biggr] + \frac{1}{\sigma - \epsilon } \biggl[ \frac{1}{\Gamma _{q}(\sigma )} - \frac{1}{\Gamma _{q}(\sigma - \epsilon ) } \biggr] \\ & \leq \epsilon \biggl[ \frac{1}{ \sigma (\sigma - \epsilon ) \Gamma _{q}(\sigma )} + \frac{ \vert \Gamma _{q}(\alpha ) \vert }{ (\sigma - \epsilon ) \Gamma _{q} (\sigma ) (\sigma - \epsilon )} \biggr], \end{aligned}$$

where \(\sigma - \epsilon < \alpha <\sigma \). □

Theorem 3.9

Suppose that the conditions of Theorem 3.5hold, and let \(\hat{v}(t)\) be the solution of the following perturbed problem on fractional order derivative σ:

$$ {}^{C}\mathcal{D}_{q}^{\sigma - \epsilon } [y](t) = w \bigl( y(t), {}^{C} \mathcal{D}_{q}^{\varsigma }[y](t) \bigr), $$
(15)

for \(t \in \overline{J}_{0}\), \(2 < \sigma \leq 3\), \(\varsigma \in J_{0}\), under the boundary conditions \(y_{0} = y'_{0} = 0\), \(y_{1} = a y(e)\), \(e \in J_{0}\) with \(0 \leq a < \frac{1}{e^{2}}\) and \(2 < \sigma - \epsilon < \sigma \leq 3\). Then \(\| y - \hat{v} \| = O (\epsilon )\).

Proof

According to the above discussion, the solution of problem (15) is given by

$$\begin{aligned} \hat{v}(t) &= \int _{0}^{1} \hat{G}_{q}( t, \xi ) w \bigl( \hat{v}( \xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[ \hat{v}](\xi ) \bigr) \,\mathrm{d}_{q}\xi \\ &\quad{} + \frac{ a t^{2} }{1 - ae^{2}} \int _{0}^{1} \hat{G}_{q}( t, \xi ) w \bigl( \hat{v}(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[ \hat{v}](\xi ) \bigr) \,\mathrm{d}_{q}\xi , \end{aligned}$$
(16)

where

$$ \hat{G}_{q}(t, \xi ) = \textstyle\begin{cases} \frac{( t - \xi )_{q}^{( \sigma - \epsilon - 1)} - t^{2} ( 1 - \xi )_{q}^{( \sigma - \epsilon - 1)}}{ \Gamma _{q} (\sigma )}, & \xi < t, \\ \frac{- t^{2} ( 1 - \xi )_{q}^{( \sigma - \epsilon - 1)} }{ \Gamma _{q} ( \sigma )},& t < \xi , \end{cases} $$
(17)

for \(t, \xi \in \overline{J}_{0}\). Then

$$\begin{aligned} \vert y - \hat{v} \vert & \leq \biggl\vert \int _{0}^{1} G_{q}(t, \xi ) w \bigl( y( \xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y]( \xi ) \bigr) \,\mathrm{d}_{q}\xi \\ &\quad{} - \int _{0}^{1} \hat{G}_{q}( t, \xi ) w \bigl( \hat{v}(\xi ), {}^{C} \mathcal{D}_{q}^{\varsigma }[ \hat{v}](\varsigma ) \bigr) \,\mathrm{d}_{q}\xi \biggr\vert \\ &\quad{} + \frac{ at^{2} }{1 - ae^{2}} \biggl\vert \int _{0}^{1} G_{q}(e, \xi ) w \bigl( y(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y]( \xi ) \bigr) \,\mathrm{d}_{q}\xi \\ &\quad{} - \int _{0}^{1} \hat{G}_{q}(e, \xi ) w \bigl( \hat{v}(\xi ), {}^{C} \mathcal{D}_{q}^{\varsigma }[ \hat{v}](\xi ) \bigr) \,\mathrm{d}_{q} \xi \biggr\vert \\ & \leq \biggl| \int _{0}^{1} G_{q}(t, \xi ) \bigl( w \bigl( y(\xi ), {}^{C} \mathcal{D}_{q}^{\varsigma }[y]( \xi ) \bigr) - w \bigl( \hat{v}(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[ \hat{v}] \bigr) (\xi ) \bigr) \,\mathrm{d}_{q}\xi \biggr\vert \\ &\quad{} + \biggl\vert \int _{0}^{1} \bigl( G_{q}(e, \xi ) - \hat{G}_{q}( e, \xi ) \bigr) w \bigl(\hat{v}(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[\hat{v}] (\xi ) \bigr) \,\mathrm{d}_{q}\xi \biggr\vert \\ &\quad{} + \frac{ at^{2} }{1 - ae^{2}} \biggl( \biggl\vert \int _{0}^{1} G_{q}(e, \xi ) \bigl( w \bigl( y(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y]( \xi ) \bigr) \\ &\quad{} - w \bigl( \hat{v}(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[ \hat{v}](\xi ) \bigr) \bigr) \,\mathrm{d}_{q}\xi \biggr\vert \\ &\quad{} + \biggl\vert \int _{0}^{1} \bigl( G_{q}(e, \xi ) - \hat{G}_{q}(e, \xi ) \bigr) w \bigl( \hat{v}(\xi ), {}^{C}\mathcal{D}_{q}^{\varsigma }[\hat{v}](\xi ) \bigr) \biggr\vert \biggr) \\ & \leq \ell \Vert y - \hat{v} \Vert \int _{0}^{1} | G_{q}(t, \xi ) \,\mathrm{d}_{q}\xi + \Vert w \Vert _{\epsilon } \int _{0}^{1} \bigl\vert G_{q}(t, \xi ) - \hat{G}_{q}( t, \xi ) \bigr\vert \,\mathrm{d}_{q} \xi \\ &\quad{} + \frac{a}{1 - ae^{2}} \biggl( \ell \Vert y - \hat{v} \Vert \int _{0}^{1} \bigl\vert G_{q}(e, \xi ) \bigr\vert \,\mathrm{d}_{q}\xi \\ &\quad{} + \Vert w \Vert _{\epsilon } \int _{0}^{1} \bigl\vert G_{q}(e, \xi ) - \hat{G}_{q}( e, \xi ) \bigr\vert \,\mathrm{d}_{q} \xi \biggr) \\ & \leq \ell \Delta \Vert y - \hat{v} \Vert + \Vert w \Vert _{\epsilon } \biggl( \int _{0}^{1} \bigl\vert G_{q}(t, \xi ) - \hat{G}_{q}( t, \xi ) \bigr\vert \,\mathrm{d}_{q} \xi \\ &\quad{} + \frac{ a}{1 - ae^{2}} \int _{0}^{1} \bigl\vert G_{q}(e, \xi ) - \hat{G}_{q}( e, \xi ) \bigr\vert \,\mathrm{d}_{q} \xi \biggr), \end{aligned}$$

where

$$ \Vert w \Vert _{\epsilon }= \sup_{ 0 < \epsilon < \sigma - 2} \bigl\vert w \bigl( \hat{v}(t), {}^{C}\mathcal{D}_{q}^{\varsigma }[ \hat{v}] (t) \bigr) \bigr\vert . $$

Also, we have

$$\begin{aligned} \bigl\vert {}^{C}\mathcal{D}_{q}^{\varsigma }[y](t) - {}^{C} \mathcal{ D}_{q}^{\varsigma }[\hat{v}] (t) \bigr\vert &\leq \frac{1}{\Gamma _{q} ( 1 - \varsigma )} \biggl\vert \int _{0}^{t} ( t - \xi )_{q}^{(-\varsigma )} \\ &\quad{} \times \biggl( \int _{0}^{1} \frac{\partial }{\partial \xi } G_{q}( \xi , \tau ) w \bigl( y(\tau ), {}^{C} \mathcal{D}_{q}^{\varsigma }[y] ( \tau ) \bigr) \\ &\quad{} - \int _{0}^{1} \frac{ \partial }{ \partial \xi } \hat{G}_{q}( \xi , \tau ) w \bigl( \hat{v}(\tau ), {}^{C} \mathcal{D}_{q}^{\varsigma }[\hat{v}](\tau ) \bigr) \,\mathrm{d}_{q}\tau \biggr) \,\mathrm{d}_{q}\xi \biggr\vert \\ &\quad{} + \frac{2 a }{\Gamma _{q}( 1 - \varsigma ) ( 1 - ae^{2})} \biggl\vert \int _{0}^{t} (t - \xi )_{q}^{(-\varsigma )} \\ &\quad{} \times \biggl( \int _{0}^{1} \xi G_{q}(e, \tau ) w \bigl( y( \tau ), {}^{C}\mathcal{D}_{q}^{\varsigma }[y]( \tau ) \bigr) \\ &\quad{} - \int _{0}^{1} \xi \hat{G}(e, \tau ) w \bigl( \hat{v} (\tau ), {}^{C}\mathcal{D}_{q}^{\varsigma }[ \hat{v}](\tau ) \bigr) \,\mathrm{d}_{q}\tau \biggr) \,\mathrm{d}_{q}\xi \biggr\vert \\ & \leq \frac{\ell \Vert y - \hat{v} \Vert }{ \Gamma _{q}( 1 -\varsigma )} \int _{0}^{t} ( t - \xi )_{q}^{(-\varsigma )} \biggl( \int _{0}^{1} \biggl\vert \frac{\partial }{\partial \xi } G_{q}(\xi , \tau ) \biggr\vert \,\mathrm{d}_{q}\tau \biggr) \,\mathrm{d}_{q}\xi \\ &\quad{} + \Vert w \Vert _{\epsilon }\frac{1}{ \Gamma _{q} (1-\varsigma )} \int _{0}^{t} ( t - \xi )_{q}^{(-\varsigma )} \\ &\quad{} \times \biggl( \int _{0}^{1} \biggl\vert \frac{\partial }{\partial \xi } G_{q}(\xi , \tau ) - \frac{ \partial }{\partial \xi } \hat{G}_{q}( \xi , \tau ) \biggr\vert \,\mathrm{d}_{q} \tau \biggr) \,\mathrm{d}_{q} \xi \\ &\quad{} + \frac{ 2 a }{\Gamma _{q} ( 1-\varsigma ) ( 1 - ae^{2}) } \biggl[ \ell \Vert y - \hat{v} \Vert \\ &\quad{} \times \int _{0}^{t} \xi ( t - \xi )_{q}^{(-\varsigma )} \biggl( \int _{0}^{1} \bigl\vert G_{q}(e, \tau ) \bigr\vert \,\mathrm{d}_{q}\tau \biggr) \,\mathrm{d}_{q} \xi \\ &\quad{} + \Vert w \Vert _{\epsilon } \int _{0}^{t} \xi (t- \xi )_{q}^{(- \varsigma )} \\ &\quad{} \times \biggl( \int _{0}^{1} \bigl\vert G_{q}(e, \tau ) - \hat{G}_{q}(e, \tau ) \bigr\vert \,\mathrm{d}_{q} \tau \biggr) \,\mathrm{d}_{q}\xi \biggr] \\ & \leq \frac{3 \ell \Delta }{\Gamma _{q} ( 2 - \varsigma )} \Vert y - \hat{v} \Vert + \Vert \ell \Vert _{\epsilon } \frac{1}{\Gamma _{q} ( 1 - \varsigma )} \int _{0}^{t} \xi ( t - \xi )_{q}^{(- \varsigma )} \\ &\quad{} \times \biggl( \int _{0}^{1} \biggl\vert \frac{\partial }{\partial \xi } G_{q}(\xi , \tau ) - \frac{ \partial }{\partial \xi } \hat{G}_{q}( \xi , \tau ) \biggr\vert \,\mathrm{d}_{q} \tau \biggr) \,\mathrm{d}_{q} \xi \\ &\quad{} + \frac{2 a}{\Gamma _{q} ( 1 - \varsigma ) ( 1 - ae^{2})} \Vert w \Vert _{\epsilon } \\ &\quad{} \times \int _{0}^{t} \xi ( t - \xi )_{q}^{(-\varsigma )} \biggl( \int _{0}^{1} \bigl\vert G_{q}(e, \tau ) - \hat{G}_{q}(e, \tau ) \bigr\vert \,\mathrm{d}_{q} \tau \biggr) \,\mathrm{d}_{q}\xi . \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert y - \hat{v} \Vert & \leq \frac{1}{1 - (\ell \Delta + \frac{3 \ell \Delta }{\Gamma _{q} ( 2 -\varsigma )} ) } \biggl[ \int _{0}^{t} \bigl\vert G_{q}(t, \xi ) - \hat{G}_{q}(t, \xi ) \bigr\vert \,\mathrm{d}_{q} \xi \\ &\quad{} + \frac{a}{1 - a e^{2}} \int _{0}^{1} \bigl\vert G_{q}(e, \xi ) - \hat{G}_{q}(e, \xi ) \bigr\vert \,\mathrm{d}_{q} \xi \\ &\quad{} + \frac{1}{\Gamma _{q} ( 1 - \varsigma )} \int _{0}^{t} \xi ( t - q\xi )^{(-\varsigma )} \\ &\quad{} \times \biggl( \int _{0}^{1} \biggl\vert \frac{\partial }{\partial \xi } G_{q}(\xi , \tau ) - \frac{ \partial }{\partial \xi } \hat{G}_{q}( \xi , \tau ) \biggr\vert \,\mathrm{d}_{q} \tau \biggr) \,\mathrm{d}_{q} \xi \\ &\quad{} + \frac{2 a}{\Gamma _{q} ( 1 - \varsigma ) ( 1 - ae^{2})} \int _{0}^{t} \xi ( t - q\xi )^{(-\varsigma )} \\ &\quad{} \times \biggl( \int _{0}^{t} \bigl\vert G_{q}(e, \tau ) - \hat{G}_{q}(e, \tau ) \bigr\vert \,\mathrm{d}_{q} \tau \biggr) \,\mathrm{d}_{q}\xi \biggr]. \end{aligned}$$

According to the structure of \(G_{q}(t, \xi ) \), we know that every term of \(| G_{q}(t, \xi ) - \hat{G}_{q}(t, \xi ) |\) and

$$ \biggl\vert \frac{ \partial }{\partial \xi } G_{q}(t, \xi ) - \frac{\partial }{\partial \xi } \hat{G}_{q}(t, \xi ) \biggr\vert $$

is in the form of Eq. (15). Hence, Lemma 3.8 implies

$$\begin{aligned} & \int _{0}^{1} \bigl\vert G_{q}(t, \xi ) - \hat{G}_{q}(t, \xi ) \bigr\vert \,\mathrm{d}_{q} \xi = O (\epsilon ), \\ & \int _{0}^{1} \biggl\vert \frac{\partial }{\partial \xi } G_{q}(t, \xi ) - \frac{\partial }{\partial \xi } \hat{G}_{q}(t, \xi ) \biggr\vert = O( \epsilon ). \end{aligned}$$

Therefore, \(\| y - \hat{v} \| = O (\epsilon ) \) and the proof is complete. □

4 Some illustrative examples

Herein, we give some examples to show the validity of the main results. In this way, we give a computational technique for checking problem (1). We need to present a simplified analysis that is able to execute the values of the q-gamma function. For this purpose, we provided a pseudo-code description of the method for calculation of the q-gamma function of order n [61].

Example 4.1

Consider the problem

$$ {}^{D}\mathcal{D}_{q}^{\frac{8}{3}} [y](t) = \frac{4}{7} \bigl( y(t) \bigr)^{ \frac{1}{2} } + \frac{3}{10} \bigl( {}^{C}\mathcal{D}_{q}^{ \frac{1}{2}} [y](t) \bigr)^{\frac{1}{4}} $$
(18)

via boundary conditions \(y(0) = y'(0)=0\) and \(y(1) =\frac{14}{9} y ( \frac{3}{5} )\). Clearly, \(\sigma = \frac{8}{3} \in (2, 3]\), \(\varsigma = \frac{1}{2} \in J_{0}\), \(e=\frac{3}{5} \in \overline{J}_{0}\), and \(a=\frac{14}{9} \in [0, \frac{1}{e^{2}})\). We define \(w: \mathbb{R}^{2} \to \mathbb{R}\) by

$$ w(y, z) = \frac{4}{7} \bigl( y(t) \bigr)^{\frac{1}{2}} + \frac{3}{10} \bigl( z(t) \bigr)^{\frac{1}{4}} $$

for \(y, z \in \mathbb{R}\). Then we have

$$\begin{aligned} \bigl\vert w \bigl( y(t), {}^{C}\mathcal{D}_{q}^{\frac{1}{2}} [y](t) \bigr) \bigr\vert & = \biggl\vert \frac{4}{7} \bigl( y(t) \bigr)^{\frac{1}{2}} + \frac{3}{10} \bigl( {}^{C} \mathcal{D}_{q}^{\frac{1}{2}} [z](t) \bigr)^{\frac{1}{4}} \biggr\vert \\ & \leq \frac{4}{7} \bigl\vert \bigl( y(t) \bigr)^{\frac{1}{2}} \bigr\vert + \frac{3}{10} \bigl\vert \bigl( {}^{C} \mathcal{D}_{q}^{\frac{1}{2}} [z](t) \bigr)^{ \frac{1}{4}} \bigr\vert \\ & \leq \mu (t) + \frac{4}{7} \bigl\vert \bigl( y(t) \bigr)^{\frac{1}{2}} \bigr\vert + \frac{3}{10} \bigl\vert \bigl( {}^{C}\mathcal{D}_{q}^{\frac{1}{2}} [z](t) \bigr)^{\frac{1}{4}} \bigr\vert , \end{aligned}$$

where \(\mu (t) = \exp (t)\). We take \(m_{0}=\frac{4}{7}\), \(m_{1}= \frac{3}{10}\), \(\beta _{0}=\frac{1}{2}\), and \(\beta _{1}=\frac{1}{4}\). Also, by using Eq. (8), we obtain

$$\begin{aligned} \Delta &= \frac{2}{\Gamma _{q} ( \sigma + 1 )} \biggl[ 1 + \frac{a}{ 1 - ae^{2}} \biggr] \\ & = \frac{2}{\Gamma _{q} ( \frac{8}{3} + 1 )} \biggl[ 1 + \frac{14}{9 ( 1 - \frac{14}{25} )} \biggr] = \frac{2}{\Gamma _{q} ( \frac{11}{3} ) }\times \frac{449}{99}= \frac{898}{ 99 \Gamma _{q} ( \frac{11}{3} ) } \end{aligned}$$

and

$$\begin{aligned} \delta & \geq \max \biggl\{ ( 6\Delta m_{0})^{\frac{1}{1 - \beta _{0}} }, (6 \Delta m_{1} )^{ \frac{1}{1 - \beta _{1}}}, 6\Delta M_{1}, \biggl( \frac{ 12 \Delta m_{0} }{ \Gamma _{q} (2 - \varsigma )} \biggr)^{ \frac{1}{1 - \beta _{0}} }, \\ &\quad \biggl( \frac{ 12 \Delta m_{1}}{ \Gamma _{q} (2 - \varsigma )} \biggr)^{\frac{1}{1 - \beta _{1}}}, \frac{ 16 a M_{1}}{ \Gamma _{q} (2 - \varsigma )( 1 - a e^{2})}, \frac{ 8 M_{2}}{ \Gamma _{q} (2 - \varsigma ) } \biggr\} \\ & = \max \biggl\{ \biggl( \frac{24}{7}\Delta \biggr)^{2}, \biggl( \frac{9}{5} \Delta \biggr)^{ \frac{4}{3}}, 6\Delta M_{1}, \biggl( \frac{48 \Delta }{7 \Gamma _{q} (\frac{3}{2} ) } \biggr)^{2}, \biggl( \frac{ 18 \Delta }{ 5\Gamma _{q} (\frac{3}{2} ) } \biggr)^{\frac{4}{3} }, \\ &\quad \frac{ 5600 M_{1}}{99 \Gamma _{q} (\frac{3}{2} )}, \frac{ 8 M_{2}}{\Gamma _{q} (\frac{3}{2} ) } \biggr\} . \end{aligned}$$

Table 1 shows \(\Delta \cong 6.5573\), 4.2076, 2.6074 for \(q=\frac{1}{5}\), \(\frac{1}{2}\), \(\frac{7}{8}\), respectively. Figure 1 shows 2D graphs of Δ. Therefore, condition (A1) in Theorem 3.4 holds, and hence this problem has a solution.

Figure 1
figure 1

2D graphs of Δ for \(q = \frac{1}{5}\), \(\frac{1}{2}\), \(\frac{7}{8}\) in Example 4.1

Table 1 Numerical results of \(\Gamma _{q}(\sigma +1)\) and Δ for \(q = \frac{1}{5}\), \(\frac{1}{2}\), \(\frac{7}{8}\) in Example 4.1 (Algorithm 1)

Example 4.2

Consider the following problem:

$$ {}^{C}\mathcal{D}_{q}^{\frac{27}{11} } [y](t) = \frac{4}{5} \bigl( y(t) \bigr)^{3} + 3 \bigl( {}^{C}\mathcal{D}_{q}^{\frac{1}{8}} [y](t) \bigr)^{4} $$
(19)

under the boundary conditions \(y(0) = y'(0)=0\) and \(y(1) = \frac{1}{2} y ( \frac{2}{7} )\). Then

$$ w \bigl( y(t), {}^{C}\mathcal{D}_{q}^{\frac{2}{5}} [y](t) \bigr) \leq 4 \bigl\vert y(t) \bigr\vert ^{3} + 2 \bigl\vert {}^{C} \mathcal{D}_{q}^{\frac{2}{5}} [y](t) \bigr\vert ^{5}. $$

Clearly, \(\sigma = \frac{27}{11} \in (2, 3]\), \(\varsigma = \frac{1}{8} \in J_{0}\), \(e=\frac{2}{7} \in \overline{J}_{0}\), and \(a=\frac{19}{4} \in [0, \frac{1}{e^{2}})\). We define \(w: \mathbb{R}^{2} \to \mathbb{R}\) by

$$ w(y, z) = \frac{4}{5} \bigl( y(t) \bigr)^{3} + 3 \bigl( z(t) \bigr)^{4} $$

for \(y, z \in \mathbb{R}\). Then we have

$$\begin{aligned} \bigl\vert w \bigl( y(t), {}^{C}\mathcal{D}_{q}^{\frac{1}{8}} [y](t) \bigr) \bigr\vert & = \frac{4}{5} \bigl\vert \bigl( y(t) \bigr)^{3} \bigr\vert + 3 \bigl( {}^{C} \mathcal{D}_{q}^{\frac{1}{8}} [y](t) \bigr)^{4} \\ & \leq \frac{4}{5} \bigl\vert y(t) \bigr\vert ^{3} + 3 \bigl\vert {}^{C} \mathcal{D}_{q}^{\frac{1}{8}} [y](t) \bigr\vert ^{4}. \end{aligned}$$

We take \(m_{0}=\frac{4}{5}\), \(m_{1}= 3\), \(\beta _{0}=3\), and \(\beta _{1}=4\). Also, by using Eq. (8), we obtain

$$\begin{aligned} \Delta &= \frac{2}{\Gamma _{q} ( \sigma + 1 )} \biggl[ 1 + \frac{a}{ 1 - ae^{2}} \biggr] \\ & = \frac{2}{\Gamma _{q} ( \frac{27}{11} + 1 )} \biggl[ 1 + \frac{19}{4 ( 1 - \frac{19}{49} )} \biggr] \\ & = \frac{2}{\Gamma _{q} ( \frac{38}{11} ) }\times \frac{1051}{120} = \frac{2102}{ 120 \Gamma _{q} ( \frac{38}{11} )}. \end{aligned}$$

Table 2 shows \(\Delta \cong 1.3258\), \(9.1665\times 10^{1}\), 6.2138 for \(q=\frac{1}{5}\), \(\frac{1}{2}\), \(\frac{7}{8}\), respectively. Figure 2 shows 2D graphs of Δ. Therefore, condition (A2) in Theorem 3.4 holds, and hence this problem has a solution.

Figure 2
figure 2

2D graphs of Δ for \(q = \frac{1}{5}\), \(\frac{1}{2}\), \(\frac{7}{8}\) in Example 4.2

Table 2 Numerical results of \(\Gamma _{q}(\sigma +1)\) and Δ for \(q = \frac{1}{5}\), \(\frac{1}{2}\), \(\frac{7}{8}\) in Example 4.2 (Algorithm 1)

Example 4.3

Consider the problem

$$ {}^{C}\mathcal{D}_{q}^{\frac{12}{5}} [y](t) = \frac{1}{18} \bigl( y(t) \bigr) + \frac{1}{9} \sin \bigl( {}^{C}\mathcal{D}_{q}^{\frac{3}{7}} [y](t) \bigr) $$
(20)

with boundary conditions \(y(0) = y'(0)=0\) and \(y(1) = \frac{1}{3} y (\frac{8}{11} )\). It is clear that \(\sigma = \frac{12}{5} \in (2, 3]\), \(\varsigma = \frac{3}{7} \in J_{0}\), \(e=\frac{8}{11} \in \overline{J}_{0}\), and \(a=\frac{81}{64} \in [0, \frac{1}{e^{2}})\). We define \(w: \mathbb{R}^{2} \to \mathbb{R}\) by

$$ w(y, z) = \frac{1}{18} y(t) + \frac{1}{9} \sin \bigl( z(t) \bigr) $$

for \(y, z \in \mathbb{R}\). Then we have

$$\begin{aligned} &\bigl\vert w \bigl( y(t), {}^{C}\mathcal{D}_{q}^{\frac{3}{7}} [y](t) \bigr) - w \bigl( z(t), {}^{C}\mathcal{D}_{q}^{ \frac{3}{7}} [z](t) \bigr) \bigr\vert \\ &\quad = \biggl\vert \frac{1}{18} \bigl( y(t) \bigr) + \frac{1}{9} \sin \bigl( {}^{C} \mathcal{D}_{q}^{\frac{3}{7}} [y](t) \bigr) \\ &\quad \quad{} - \biggl( \frac{1}{18} \bigl( z(t) \bigr) + \frac{1}{9} \sin \bigl( {}^{C}\mathcal{D}_{q}^{\frac{3}{7}} [z](t) \bigr) \biggr) \biggr\vert \\ &\quad \leq \frac{1}{18} \bigl\vert y(t) - z(t) \bigr\vert \\ &\quad \quad{} + \frac{1}{9} \bigl\vert \sin \bigl( {}^{C} \mathcal{D}_{q}^{ \frac{3}{7}} [y](t) \bigr) - \sin \bigl( {}^{C}\mathcal{D}_{q}^{ \frac{3}{7}} [z](t) \bigr) \bigr\vert \\ &\quad \leq \frac{1}{18} \bigl\vert y(t) - z(t) \bigr\vert + \frac{1}{9} \bigl\vert {}^{C} \mathcal{D}_{q}^{ \frac{3}{7}} [y](t) - {}^{C}\mathcal{D}_{q}^{ \frac{3}{7}} [z](t) \bigr\vert \\ &\quad \leq \frac{1}{9} \bigl( \bigl\vert y(t) - z(t) \bigr\vert + \bigl\vert {}^{C} \mathcal{D}_{q}^{ \frac{3}{7}} [y](t) - {}^{C}\mathcal{D}_{q}^{ \frac{3}{7}} [z](t) \bigr\vert \bigr). \end{aligned}$$

We take \(\ell = \frac{1}{9}\). Table 3 shows that

$$ \frac{\Gamma _{q} ( 2 - \varsigma )}{\Delta [ 3 + \Gamma _{q} ( 2 - \varsigma )] } = 1.3522\times10 ^{-1}, 1.9197 \times10 ^{-1}, 2.7708 \times10 ^{-1} $$

for \(q=\frac{1}{5}\), \(\frac{1}{2}\), \(\frac{7}{8}\), respectively. Also, the results prove that

$$ \ell \leq \frac{\Gamma _{q} ( 2 - \varsigma )}{\Delta [ 3 + \Gamma _{q} ( 2 - \varsigma )] }. $$

Figure 3 shows 2D graphs of

$$ \frac{\Gamma _{q} ( 2 - \varsigma )}{\Delta [ 3 + \Gamma _{q} ( 2 - \varsigma )] }. $$

Also, by using Eq. (8), we obtain

$$\begin{aligned} \Delta &= \frac{2}{\Gamma _{q} ( \sigma + 1 )} \biggl[ 1 + \frac{a}{ 1 - ae^{2}} \biggr] \\ & = \frac{2}{\Gamma _{q} ( \frac{12}{5} + 1 )} \biggl[ 1 + \frac{81}{64 ( 1 - \frac{81}{121} )} \biggr] = \frac{2}{\Gamma _{q} ( \frac{17}{5} ) } \times \frac{12\text{,}361}{2560}= \frac{24\text{,}722}{ 2560 \Gamma _{q} ( \frac{38}{11} )}. \end{aligned}$$

Table 3 shows that \(\Delta = 7.3955\), 5.2090, 3.6090 for \(q=\frac{1}{5}\), \(\frac{1}{2}\), \(\frac{7}{8}\), respectively. Figure 4 shows 2D graphs of Δ. Now, by applying Eq. (10), we get

$$ \ell < \frac{\Gamma _{q} ( 2 - \varsigma )}{\Delta [ 3 + \Gamma _{q} ( 2 - \varsigma )] } = \frac{ \Gamma _{q} ( 2 - \frac{3}{7} )}{ \Delta [ 3 + \Gamma _{q} ( 2 - \frac{3}{7} ) ] } = \frac{ \Gamma _{q} ( \frac{11}{7} )}{ \Delta [ 3 + \Gamma _{q} ( \frac{11}{7} ) ] }. $$
Figure 3
figure 3

2D graphs of \(\frac{\Gamma _{q} ( 2 - \varsigma )}{\Delta [ 3 + \Gamma _{q} ( 2 - \varsigma )] }\) for \(q = \frac{1}{5}\), \(\frac{1}{2}\), \(\frac{7}{8}\) in Example 4.3

Figure 4
figure 4

2D graphs of Δ for \(q = \frac{1}{5}\), \(\frac{1}{2}\), \(\frac{7}{8}\) in Example 4.3

Table 3 Numerical results of \(\Gamma _{q}(\sigma +1)\), Δ, and \(\ell < \frac{\Gamma _{q} ( 2 - \varsigma )}{\Delta [ 3 + \Gamma _{q} ( 2 - \varsigma )] }\) for \(q = \frac{1}{5}\), \(\frac{1}{2}\), \(\frac{7}{8}\) in Example 4.2 (Algorithm 2)

Since \(0 < \ell < \frac{1}{9}< 0.263\), Theorem 3.5 implies that this problem has a unique solution.

5 Conclusion

The Schauder fixed point theorem has been applied in the research study to discuss the well-posed conditions for a class of q-fractional order boundary value problems As a result, we have proved the existence and uniqueness of solution by means of the Schauder fixed point and Banach contraction map theorems on the interval \([0,1]\). We have also studied the perturbation on boundary condition on the function exists in the right-hand side of the problem and on the fractional order. To the leading of our information, the results have never been detailed in other works [11, 12, 61] that consider the problems. In this manner, it is very apparent that the solution of the problem is stable under the small perturbation.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  2. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1993)

    MATH  Google Scholar 

  3. Rezapour, S., Imran, A., Hussain, A., Martinez, F., Etemad, S., Kaabar, M.K.A.: Condensing functions and approximate endpoint criterion for the existence analysis of quantum integro-difference FBVPs. Symmetry 13(3), 469 (2021). https://doi.org/10.3390/sym13030469

    Article  Google Scholar 

  4. Adiguzel, R.S., Aksoy, U., Karapinar, E., Erhan, I.M.: On the solutions of fractional differential equations via Geraghty type hybrid contractions. Appl. Comput. Math. 20(2), 313–333 (2021)

    Google Scholar 

  5. Adiguzel, R.S., Aksoy, U., Karapinar, E., Erhan, I.M.: Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2021, 155 (2021). https://doi.org/10.1007/s13398-021-01095-3

    Article  MathSciNet  MATH  Google Scholar 

  6. Alqahtani, B., Aydi, H., Karapinar, E., Rakocevic, V.: A solution for Volterra fractional integral equations by hybrid contractions. Mathematics 7(8), 694 (2019). https://doi.org/10.3390/math7080694

    Article  Google Scholar 

  7. Bachir, F.S., Abbas, S., Benbachir, M., Benchohra, M.: Hilfer–Hadamard fractional differential equations; existence and attractivity. Adv. Theory Nonlinear Anal. Appl. 5, 49–57 (2021)

    MATH  Google Scholar 

  8. Baitiche, Z., Derbazi, C., Benchohra, M.: ψ-Caputo fractional differential equations with multi-point boundary conditions by topological degree theory. Results Nonlinear Anal. 3, 167–178 (2020)

    Google Scholar 

  9. Baleanu, D., Jajarmi, A., Mohammadi, H., Rezapour, S.: A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative. Chaos Solitons Fractals 134, 109705 (2020). https://doi.org/10.1016/j.chaos.2020.109705

    Article  MathSciNet  Google Scholar 

  10. Karapinar, E., Fulga, A., Rashid, M., Shahid, L., Aydi, H.: Large contractions on quasi-metric spaces with an application to nonlinear fractional differential equations. Mathematics 7(5), 444 (2019). https://doi.org/10.3390/math7050444

    Article  Google Scholar 

  11. Abdeljawad, A., Agarwal, R.P., Karapinar, E., Kumari, P.S.: Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space. Symmetry 11, 686 (2019). https://doi.org/10.3390/sym11050686

    Article  MATH  Google Scholar 

  12. Adiguzel, R.S., Aksoy, U., Karapinar, E., Erhan, I.M.: On the solution of a boundary value problem associated with a fractional differential equation. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6652

    Article  MATH  Google Scholar 

  13. Karapinar, E., Fulga, A.: An admissible hybrid contraction with an Ulam type stability. Demonstr. Math. 52, 428–436 (2019). https://doi.org/10.1515/dema-2019-0037

    Article  MathSciNet  MATH  Google Scholar 

  14. Alqahtani, B., Fulga, A., Karapinar, E.: Fixed point results on δ-symmetric quasi-metric space via simulation function with an application to Ulam stability. Mathematics 6(10), 208 (2018). https://doi.org/10.3390/math6100208

    Article  Google Scholar 

  15. Brzdek, J., Karapinar, E., Petrsel, A.: A fixed point theorem and the Ulam stability in generalized dq-metric spaces. J. Math. Anal. Appl. 467, 501–520 (2018). https://doi.org/10.1016/j.jmaa.2018.07.022

    Article  MathSciNet  MATH  Google Scholar 

  16. Rezapour, S., Azzaoui, B., Tellab, B., Etemad, S., Masiha, H.P.: An analysis on the positive solutions for a fractional configuration of the Caputo multiterm semilinear differential equation. J. Funct. Spaces 2021, Article ID 6022941 (2021). https://doi.org/10.1155/2021/6022941

    Article  MathSciNet  MATH  Google Scholar 

  17. Matar, M.M., Abbas, M.I., Alzabut, J., Kaabar, M.K.A., Etemad, S., Rezapour, S.: Investigation of the p-Laplacian nonperiodic nonlinear boundary value problem via generalized Caputo fractional derivatives. Adv. Differ. Equ. 2021, 68 (2021). https://doi.org/10.1186/s13662-021-03228-9

    Article  MathSciNet  Google Scholar 

  18. Baleanu, D., Rezapour, S., Saberpour, Z.: On fractional integro-differential inclusions via the extended fractional Caputo–Fabrizio derivation. Bound. Value Probl. 2019, 79 (2019). https://doi.org/10.1186/s13661-019-1194-0

    Article  MathSciNet  Google Scholar 

  19. Baleanu, D., Etemad, S., Rezapour, S.: A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions. Bound. Value Probl. 2020, 64 (2020). https://doi.org/10.1186/s13661-020-01361-0

    Article  MathSciNet  Google Scholar 

  20. Jackson, F.H.: q-Difference equations. Am. J. Math. 32, 305–314 (1910). https://doi.org/10.2307/2370183

    Article  MATH  Google Scholar 

  21. Al-Salam, W.A.: q-Analogues of Cauchy’s formula. Proc. Am. Math. Soc. 17, 182–184 (1952–1953)

  22. Agarwal, R.P.: Certain fractional q-integrals and q-derivatives. Proc. Camb. Philos. Soc. 66, 365–370 (1965). https://doi.org/10.1017/S0305004100045060

    Article  MathSciNet  MATH  Google Scholar 

  23. Adams, C.R.: The general theory of a class of linear partial q-difference equations. Trans. Am. Math. Soc. 26, 283–312 (1924)

    MathSciNet  MATH  Google Scholar 

  24. Annaby, M.H., Mansour, Z.S.: q-Fractional Calculus and Equations. Springer, Cambridge (2012). https://doi.org/10.1007/978-3-642-30898-7

    Book  MATH  Google Scholar 

  25. Abdeljawad, T., Alzabut, J., Baleanu, D.: A generalized q-fractional Gronwall inequality and its applications to nonlinear delay q-fractional difference systems. J. Inequal. Appl. 2016, 240 (2016). https://doi.org/10.1186/s13660-016-1181-2

    Article  MathSciNet  MATH  Google Scholar 

  26. Ahmad, B., Etemad, S., Ettefagh, M., Rezapour, S.: On the existence of solutions for fractional q-difference inclusions with q-antiperiodic boundary conditions. Bull. Math. Soc. Sci. Math. Roum. 59(107(2)), 119–134 (2016). https://doi.org/10.1186/s13660-016-1181-2

    Article  MathSciNet  MATH  Google Scholar 

  27. Goodrich, C.S.: Existence of a positive solution to a class of fractional differential equations. Appl. Math. Lett. 23, 1050–1055 (2010)

    Article  MathSciNet  Google Scholar 

  28. Jafari, H., Daftardar-Gejji, V.: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Appl. Math. Comput. 180(2), 700–706 (2006). https://doi.org/10.1186/s13662-020-02766-y

    Article  MathSciNet  MATH  Google Scholar 

  29. Ferreira, R.A.C.: Nontrivial solutions for fractional q-difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Samei, M.E., Rezapour, S.: On a system of fractional q-differential inclusions via sum of two multi-term functions on a time scale. Bound. Value Probl. 2020, 135 (2020). https://doi.org/10.1186/s13661-020-01433-1

    Article  MathSciNet  Google Scholar 

  31. Rezapour, S., Samei, M.E.: On the existence of solutions for a multi-singular pointwise defined fractional q-integro-differential equation. Bound. Value Probl. 2020, 38 (2020). https://doi.org/10.1186/s13661-020-01342-3

    Article  MathSciNet  Google Scholar 

  32. Goodrich, C., Peterson, A.C.: Discrete Fractional Calculus. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25562-0

    Book  MATH  Google Scholar 

  33. Darzi, R., Agheli, B.: Existence results to positive solutions of fractional BVP with q-derivatives. J. Appl. Math. Comput. 55, 353–367 (2017)

    Article  MathSciNet  Google Scholar 

  34. Samei, M.E., Hedayati, V., Rezapour, S.: Existence results for a fraction hybrid differential inclusion with Caputo–Hadamard type fractional derivative. Adv. Differ. Equ. 2019, 163 (2019). https://doi.org/10.1186/s13662-019-2090-8

    Article  MathSciNet  MATH  Google Scholar 

  35. Kac, V., Cheung, P.: Quantum Calculus. Universitext. Springer, New York (2002). https://doi.org/10.1007/978-1-4613-0071-7-1

    Book  MATH  Google Scholar 

  36. Liang, S., Zhang, J.: Existence and uniqueness of positive solutions to m-point boundary value problem for nonlinear fractional differential equation. J. Appl. Math. Comput. 38(1), 225–241 (2012). https://doi.org/10.1186/s13661-020-01433-1

    Article  MathSciNet  MATH  Google Scholar 

  37. Bota, M.F., Karapinar, E., Mlesnite, O.: Ulam–Hyers stability results for fixed point problems via α-ψ-contractive mapping in b-metric space. Abstr. Appl. Anal. 2013, Article ID 825293 (2013). https://doi.org/10.1155/2013/825293

    Article  MathSciNet  MATH  Google Scholar 

  38. Karapinar, E., Panda, S.K., Lateef, D.: A new approach to the solution of Fredholm integral equation via fixed point on extended b-metric spaces. Symmetry 10(10), 512 (2018). https://doi.org/10.3390/sym10100512

    Article  Google Scholar 

  39. Karapinar, E., Atangana, A., Fulga, A.: Pata type contractions involving rational expressions with an application to integral equations. Discrete Contin. Dyn. Syst. 14(10), 3629–3640 (2021). https://doi.org/10.3934/dcdss.2020420

    Article  MathSciNet  MATH  Google Scholar 

  40. Aksoy, U., Karapinar, E., Erhan, I.M.: Fixed point theorems in complete modular metric spaces and an application to anti-periodic boundary value problems. Filomat 31(17), 5475–5488 (2017). https://doi.org/10.2298/FIL1717475A

    Article  MathSciNet  Google Scholar 

  41. Aksoy, U., Karapinar, E., Erhan, I.M.: Fixed points of generalized α-admissible contractions on b-metric spaces with an application to boundary value problems. J. Nonlinear Convex Anal. 17(6), 1095–1108 (2016)

    MathSciNet  MATH  Google Scholar 

  42. Aydogan, S.M., Baleanu, D., Mousalou, A., Rezapour, S.: On high order fractional integro-differential equations including the Caputo–Fabrizio derivative. Bound. Value Probl. 2018, 90 (2018). https://doi.org/10.1186/s13661-018-1008-9

    Article  MathSciNet  MATH  Google Scholar 

  43. Baleanu, D., Mousalou, A., Rezapour, S.: On the existence of solutions for some infinite coefficient-symmetric Caputo–Fabrizio fractional integro-differential equations. Bound. Value Probl. 2017, 145 (2017). https://doi.org/10.1186/s13661-017-0867-9

    Article  MathSciNet  MATH  Google Scholar 

  44. Baleanu, D., Etemad, S., Rezapour, S.: On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators. Alex. Eng. J. 59(5), 3019–3027 (2020). https://doi.org/10.1016/j.aej.2020.04.053

    Article  Google Scholar 

  45. Mohammadi, H., Kumar, S., Rezapour, S., Etemad, S.: A theoretical study of the Caputo–Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control. Chaos Solitons Fractals 144, 110668 (2021). https://doi.org/10.1016/j.chaos.2021.110668

    Article  MathSciNet  Google Scholar 

  46. Samei, M.E., Yang, W.: Existence of solutions for k-dimensional system of multi-term fractional q-integro-differential equations under anti-periodic boundary conditions via quantum calculus. Math. Methods Appl. Sci. 43(7), 4360–4382 (2020). https://doi.org/10.1002/mma.6198

    Article  MathSciNet  MATH  Google Scholar 

  47. Samei, M.E.: Existence of solutions for a system of singular sum fractional q-differential equations via quantum calculus. Adv. Differ. Equ. 2019, 163 (2019). https://doi.org/10.1186/s13662-019-2480-y

    Article  Google Scholar 

  48. Aydogan, M., Baleanu, D., Aguilar, J.F.G., Rezapour, S., Samei, M.E.: Approximate endpoint solutions for a class of fractional q-differential inclusions. Fractals 28(8), 2040029 (2020). https://doi.org/10.1142/S0218348X20400290

    Article  Google Scholar 

  49. Ahmadi, A., Samei, M.E.: On existence and uniqueness of solutions for a class of coupled system of three term fractional q-differential equations. J. Adv. Math. Stud. 13(1), 69–80 (2020)

    MathSciNet  MATH  Google Scholar 

  50. Su, X., Zhang, S.: Solution to boundary value problems for nonlinear differential equations of fractional order. Electron. J. Differ. Equ. 2009, 26 (2009)

    Article  MathSciNet  Google Scholar 

  51. Matar, M.M.: Boundary value problem for fractional integro-differential equations with nonlocal conditions. Int. J. Open Probl. Comput. Sci. Math. 3(4), 481–489 (2010)

    Article  MathSciNet  Google Scholar 

  52. Tian, Y., Zhou, Y.: Positive solutions for multipoint boundary value problem of functional differential equations. J. Appl. Math. Comput. 38, 417–427 (2012)

    Article  MathSciNet  Google Scholar 

  53. Nyamoradi, N.: Multiple positive solutions for fractional differential systems. Ann. Univ. Ferrara 58(2), 359–369 (2012)

    Article  MathSciNet  Google Scholar 

  54. Sun, S., Li, Q., Li, Y.: Existence and uniqueness of solutions for a coupled system of multiterm nonlinear fractional differential equations. Comput. Math. Appl. 64(10), 3310–3320 (2012)

    Article  MathSciNet  Google Scholar 

  55. Houas, M., Benbachir, M.: Existence solutions for three point boundary value problem for differential equations. J. Fract. Calc. Appl. 6(1), 160–174 (2015)

    MathSciNet  MATH  Google Scholar 

  56. Khan, R.A., Khan, H.: On existence of solution for multipoints boundary value problem. J. Fract. Calc. Appl. 5(2), 121–132 (2014)

    MathSciNet  Google Scholar 

  57. Akrami, M.H., Erjaee, G.H.: Existence uniqueness and well-posed conditions on a class of fractional differential equations with boundary condition. J. Fract. Calc. Appl. 6(2), 171–185 (2015). https://doi.org/10.3390/sym11050686

    Article  MathSciNet  Google Scholar 

  58. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)

    Book  Google Scholar 

  59. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhäuser, Boston (2001)

    Book  Google Scholar 

  60. Atici, F., Eloe, P.W.: Fractional q-calculus on a time scale. J. Nonlinear Math. Phys. 14(3), 341–352 (2007). https://doi.org/10.2991/jnmp.2007.14.3.4

    Article  MathSciNet  MATH  Google Scholar 

  61. Samei, M.E., Zanganeh, H., Aydogan, S.M.: Investigation of a class of the singular fractional integro-differential quantum equations with multi-step methods. J. Math. Ext. 17(1), 1–545 (2021)

    Google Scholar 

  62. Rajković, P.M., Marinković, S.D., Stanković, M.S.: Fractional integrals and derivatives in q-calculus. Appl. Anal. Discrete Math. 1, 311–323 (2007)

    Article  MathSciNet  Google Scholar 

  63. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  64. Smart, D.R.: Fixed Point Theorems. Cambridge University Press, New York (1980)

    MATH  Google Scholar 

Download references

Acknowledgements

The first and second authors were supported by Bu-Ali Sina University. The fifth author was supported by Azarbaijan Shahid Madani University. The authors express their gratitude to dear unknown referees for their helpful suggestions which improved the final version of this paper.

Funding

Not applicable.

Author information

Authors and Affiliations

Authors

Contributions

The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

Corresponding authors

Correspondence to Jehad Alzabut or Shahram Rezapour.

Ethics declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Appendix:  Supporting information

Appendix:  Supporting information

Algorithm 1
figure a

MATLAB lines for Examples 4.1 and 4.2

Algorithm 2
figure b

MATLAB codes for Example 4.3

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Samei, M.E., Ahmadi, A., Selvam, A.G.M. et al. Well-posed conditions on a class of fractional q-differential equations by using the Schauder fixed point theorem. Adv Differ Equ 2021, 482 (2021). https://doi.org/10.1186/s13662-021-03631-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-021-03631-2

MSC

Keywords