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α-ψ-contractions and solutions of a q-fractional differential inclusion with three-point boundary value conditions via computational results

Abstract

We review the existence of solutions for a three-point nonlinear q-fractional differential equation and also its related inclusion. In this way, we use α-ψ-contractions and multifunctions. Also, we provide two examples to illustrate our main results. Finally by providing some algorithms and tables, we give some numerical computations for the results.

Introduction

The subject of q-difference equations was introduced by Jackson in 1908 and 1910 [1, 2]. Later, some researchers reviewed q-difference equations [319]. On the other hand, there was published recently much contemporary work on integro-differential equations by using different views and fractional derivatives which young researchers could use as the main idea for their work (see, for example, [2055]). It is notable that young researchers can consider this idea as a future direction for their work by using the numerical methodologies of [56, 57]. Also, they can use the idea for some applied modeling [5861].

In 2012, Ahmad et al. investigated the fractional q-difference equation \({}^{c}\mathcal{D}_{q}^{\alpha }y(t)=f(t,y(t))\) with boundary conditions \(\alpha _{1} y(0) - \beta _{1} \mathcal{D}_{q} y(0) = \gamma _{1} y( \eta _{1})\) and \(\alpha _{2} y(1) + \beta _{2}\mathcal{D}_{q} y(1) = \gamma _{2}y( \eta _{2})\), where \(0\leq t\leq 1\), \(1<\alpha \leq 2\), \({}^{c}\mathcal{D}_{q}^{\alpha }\) is the Caputo fractional q-derivative and \(\alpha _{i}, \beta _{i}, \gamma _{i} \in \mathbb{R}\) [7]. In 2015, Alsaedi et al. studied the fractional q-difference inclusion \({}^{c}\mathcal{D}_{q}^{\nu }y(t) \in \mathcal{F}(t, y(t))\) with nonlocal and sub-strip boundary conditions \(y(0) = g(y)\) and \(y(w) = b\int _{\delta }^{1} y(s) \,\mathrm{d}_{q}s\), where \(0\leq t\leq 1\), \(1 \leq \nu < 2\), \(0< w<\delta < 1\) and \({}^{c}\mathcal{D}_{q}^{\nu }\) denotes the fractional q-derivative of Caputo type of order ν [8]. In 2019, Ntouyas and Samei investigated the existence of solutions for the multi-term nonlinear fractional q-integro-difference equation

$$ {}^{c}\mathcal{D}_{q}^{\alpha }x(t) = w \bigl(t, x(t), ( \varphi _{1} x) (t), ( \varphi _{2} x) (t), {}^{c}\mathcal{D}_{q}^{\beta _{1}} x(t), {}^{c} \mathcal{D}_{q}^{\beta _{2}} x(t), \dots, {}^{c} \mathcal{D}_{q}^{ \beta _{n}} x(t) \bigr), $$

with boundary value conditions \(x(0) + a x(1) =0\) and \(x'(0) + bx'(1) =0\), where \(t \in [0,1]\), \(0< q<1\), \(1 < \alpha < 2\), \(\beta _{i} \in (0,1)\) with \(i= 1,2,\dots, n\), \(a,b \neq -1\), the maps \(\varphi _{j} \) are defined by \(( \varphi _{j} u)(t) = \int _{0}^{t} \gamma _{j} (t,s) u(s) \,\mathrm{d}_{q}s \) for \(j=1,2\) and \(w: [0,1] \times \mathbb{R}^{n+3} \to \mathbb{R}\) is a continuous mapping with respect to all variables [17].

We investigate the existence of solutions for the q-fractional differential equation

$$ {}^{c}\mathcal{D}_{q}^{\vartheta }y(t)= \varPhi \bigl( t, y(t), \mathcal{D}_{q} y(t), \mathcal{D}_{q}^{2} y(t) \bigr) $$
(1)

with three-point boundary value conditions

{y(0)=0,Dqy(0)+cDqσy(ν)+Dq2y(1)=0,Jqκy(0)+Jqκy(ν)+Jqκy(1)=0,
(2)

where \(0< q<1\), \(0< t< 1\), \({}^{c}\mathcal{D}_{q}^{\vartheta }\) denotes the fractional q-derivative of the Caputo type of order ϑ, \(\vartheta \in (2,3]\), \(0<\nu < 1\), \(1< \sigma < 2\) and \(\varPhi: [0,1] \times \mathbb{R}^{3}\) is a continuous mapping. Let \(\mathcal{J}_{q}^{\kappa }\) denote the fractional q-integral of the Riemann–Liouville type of order \(\kappa >0\). Also, we review the existence of solutions for the q-fractional differential inclusion

$$ {}^{c}\mathcal{D}_{q}^{\vartheta }y(t) \in \mathcal{G} \bigl( t, y(t), \mathcal{D}_{q} y(t), \mathcal{D}_{q}^{2} y(t) \bigr), $$
(3)

with the three-point boundary value conditions

{y(0)=0,Dqy(0)+cDqσy(ν)+Dq2y(1)=0,Jqκy(0)+Jqκy(ν)+Jqκy(1)=0,
(4)

where \(0< q<1\), \(0< t< 1\), \(\mathcal{G}: [0,1] \times \mathbb{R}^{3} \to \mathcal{P}(\mathbb{R})\) is a compact set-valued map.

The paper is organized as follows: In Sect. 2, some basic definitions and applied results are presented. In Sect. 3, we state our main existence results and used techniques in this direction. Finally by using the Algorithms 1–5, Fig. 1 and Tables 13, two illustrative examples of the corresponding existence results are given in Sect. 4.

Algorithm 1
figurea

The proposed method for calculated \((a-b)_{q}^{(\alpha )}\)

Preliminaries

Let \(0< q<1\). The q-analogue of the power function \((a_{1}-a_{2})^{n}\) with \(n\in \mathbb{N}_{0}\) defined by \((a_{1}-a_{2})^{(0)} = 1\) and \((a_{1}-a_{2})^{(n)} = \prod_{j=0}^{ n-1}(a_{1} - a_{2}q^{j})\), where \(a_{1}\), \(a_{2} \in \mathbb{R}\) and \(\mathbb{N}_{0}:= \{ 0,1,2, \ldots \}\) [2, 18]. Now, let ϑ be a real number. Then define

$$ (a_{1}-a_{2})^{(\vartheta )}=a_{1}^{\vartheta } \prod_{j=0}^{\infty } \frac{a_{1}-a_{2}q^{j}}{a_{1}-a_{2}q^{\vartheta +j}}, $$

with \(a_{1}\neq 0\). It is clear that, if \(a_{2}=0\), then \(a_{1}^{(\vartheta )}=a_{1}^{\vartheta }\) [2, 18]. For each real number ϑ, \([\vartheta ]_{q} \) is defined by \([\vartheta ]_{q}=\frac{1-q^{\vartheta }}{1-q}\) [2]. The q-Gamma function is defined by \(\varGamma _{q}( \vartheta ) = \frac{ (1-q)^{ (\vartheta -1)}}{ (1-q)^{\vartheta - 1}}\), where \(\vartheta \in \mathbb{R} \setminus \{ 0, -1, -2, \ldots \} \) [2, 18]. We show in Algorithm 2, a pseudo-code for estimating q-Gamma function. Also by definition of \([\vartheta ]_{q}\), the property \(\varGamma _{q}(\vartheta +1)=[\vartheta ]_{q}\varGamma _{q}(\vartheta )\) holds [2]. The definition of the q-derivative of a real-valued function y is given by \(( \mathcal{D}_{q}y)(t)= \frac{y(t)-y(qt)}{ (1-q)t}\) and \((\mathcal{D}_{q}y)(0) = \lim_{t\to 0}(\mathcal{D}_{q}y)(t)\) [18]. The q-derivative of higher order of a function y is given by \((\mathcal{D}_{q}^{0}y)(t) = y(t)\) and \((\mathcal{D}_{q}^{n}y)(t) = \mathcal{D}_{q}(\mathcal{D}_{q}^{n-1}y)(t)\) for all \(n\geq 1\) [2]. One can find in Algorithm 3 a pseudo-code for calculating q-derivative of a function f. The q-integral of a function y defined in the interval \([0, a_{2}]\) is given by

$$ (\mathcal{J}_{q}y) (t)= \int _{0}^{t} y(s) \,\mathrm{d}_{q}s=t(1-q) \sum_{j=0}^{\infty }y\bigl(tq^{j} \bigr)q^{j}, \bigl(t\in [0,a_{2}]\bigr) $$

such that the sum is absolutely convergent which is shown in Algorithm 5 [18]. Now, assume that \(a_{1}\in [0,a_{2}]\). In this case the q-integral of y from \(a_{1}\) to \(a_{2}\) is given by

$$\begin{aligned} \int _{a_{1}}^{a_{2}} y(t) \,\mathrm{d}_{q}t &= \mathcal{J}_{q} y(a_{2}) - \mathcal{J}_{q} y(a_{1}) = \int _{0}^{a_{2}} y(t) \,\mathrm{d}_{q}t - \int _{0}^{a_{1}} y(t) \,\mathrm{d}_{q}t \\ &= (1-q)\sum_{j=0}^{\infty } \bigl[a_{2} y\bigl(a_{2}q^{j}\bigr) - a_{1}y\bigl(a_{1}q^{j}\bigr)\bigr] q^{j} \end{aligned}$$

whenever the series exists which is shown in Algorithm 4 [18]. Similar to q-derivatives, we define the operator \(\mathcal{J}_{q}^{n}\) by \((\mathcal{J}_{q}^{0}y)(t)=y(t)\) and \((\mathcal{J}_{q}^{n}y)(t) = \mathcal{J}_{q}( \mathcal{J}_{q}^{ n-1}y)(t)\) for all \(n\geq 1\) [18]. Note that \((\mathcal{D}_{q}\mathcal{J}_{q}y)(t)=y(t)\) and if y is continuous at \(t=0\), then \((\mathcal{J}_{q}\mathcal{D}_{q}y)(t)=y(t)-y(0)\) [18]. Assume that \(\vartheta >0\) is a real number with \(n-1\leq \vartheta < n\), that is, \(n=[ \vartheta ]+1\). The Riemann–Liouville q-integral of a function \(y\in C([t_{1},t_{2}], \mathbb{R})\) is given by

$$ \mathcal{J}_{q}^{ \vartheta } y( t )= \frac{1}{\varGamma _{q} ( \vartheta )} \int _{0}^{t} (t -q \tau )^{(\vartheta -1 )} y(\tau ) \,\mathrm{d}_{q} \tau $$

whenever the integral exists [12, 14]. The Caputo q-derivative of y belongs to \(C^{(n)}([t_{1},t_{2}], \mathbb{R})\); it is defined by \({}^{c}\mathcal{D}_{q}^{ \vartheta } y(t )= \frac{1}{\varGamma _{q} (n- \vartheta )} \int _{0}^{t} (t -q\tau )^{(n- \vartheta -1)}\mathcal{D}_{q}^{(n)}y( \tau ) \,\mathrm{d}_{q} \tau \) [12, 14]. We need the following results.

Algorithm 2
figureb

The proposed method for calculated \(\varGamma _{q}(x)\)

Algorithm 3
figurec

The proposed method for calculated \((D_{q} f)(x)\)

Algorithm 4
figured

The proposed method for calculated \(\int _{a}^{b} f(r) d_{q} r\)

Algorithm 5
figuree

The proposed method for calculated \(I_{q}^{\sigma}[x]\)

Lemma 1

([11])

Let\(\vartheta _{1}, \vartheta _{2} \geq 0\)andybe a function defined on\([0,1]\). Then\((\mathcal{J}_{q}^{\vartheta _{2}} \mathcal{J}_{q}^{\vartheta _{1}} y)(t)=( \mathcal{J}_{q}^{\vartheta _{1} +\vartheta _{2} }y)(t)\)and\((\mathcal{D}_{q}^{\vartheta _{1}} \mathcal{J}_{q}^{ \vartheta _{1}} y)(t)=y(t)\).

Lemma 2

([11])

Let\(\vartheta >0\)andnbe a positive integer. Then

$$ \bigl(\mathcal{J}_{q}^{\vartheta }\mathcal{D}_{q}^{n} y\bigr) (t)=\bigl(\mathcal{D}_{q}^{n} \mathcal{J}_{q}^{\vartheta }y\bigr) (t) - \sum _{j=0}^{n-1} \frac{t^{\vartheta -n+j}}{\varGamma _{q} ( \vartheta +j-n+1)}\bigl( \mathcal{D}_{q}^{j}y\bigr) (0). $$

It is well known that a general solution for the q-fractional differential equation \({}^{c}\mathcal{D}_{q}^{ \vartheta }y( t )=0\) is given by \(y( t )=c_{0} + c_{1} t + c_{2} t^{2} + \cdots + c_{n-1}t^{n-1}\), where \(c_{0},\ldots,c_{n-1}\) are some real numbers and \(n=[ \vartheta ]+1\) [11]. Also, for every positive real number \(T^{*}\) and every continuous function y on \([0,T^{*}]\), we have \((\mathcal{J}_{q}^{ \vartheta } {}^{c}\mathcal{D}_{q}^{ \vartheta }) y(t )= y(t ) + c_{0} + c_{1}t + c_{2}t^{2} + \cdots + c_{n-1}t^{n-1}\), where \(c_{0}, \ldots, c_{n-1}\) are constants belonging to \(\mathbb{R}\) and \(n=[ \vartheta ]+1\) [11].

Let \((\mathcal{X},\|\cdot \|_{\mathcal{X}})\) be a normed space. Denote by \({\mathcal{P}}( \mathcal{X} )\), \({\mathcal{P}}_{cl}( \mathcal{X} )\), \({\mathcal{P}}_{b}( \mathcal{X} )\), \({\mathcal{P}}_{cp}( \mathcal{X} )\) and \({\mathcal{P}}_{cp, cv}( \mathcal{X} )\), the set of all subsets of \(\mathcal{X} \), the set of all closed subsets of \(\mathcal{X} \), the set of all bounded subsets of \(\mathcal{X} \) and the set of all compact subsets of \(\mathcal{X} \) and the set of all convex subsets of \(\mathcal{X} \), respectively. An element \(y^{*}\in \mathcal{X}\) is called a fixed point of a multivalued map \(\mathcal{G}: \mathcal{X} \to \mathcal{P}( \mathcal{X})\) whenever \(y^{*}\in \mathcal{G}(y^{*})\). The set of all fixed points of the multifunction \(\mathcal{G}\) is denoted by \({\mathit{{Fix}}} (\mathcal{G})\) [62]. A multifunction \(\mathcal{G} \) is called convex-valued whenever the set \(\mathcal{G}(y)\) is convex for all \(y \in \mathcal{X}\).

We say that a multifunction \(\mathcal{G}\) is an upper semi-continuous (u.s.c.) on the space \(\mathcal{X} \) whenever for each element \(y^{*} \in \mathcal{X} \), the set \(\mathcal{G}(y^{*})\in \mathcal{P}_{cl}(\mathcal{X})\) and there exists an open neighborhood \(\mathcal{N}_{0}^{*}\) of \(y^{*}\) such that \(\mathcal{G}(\mathcal{N}_{0}^{*}) \subseteq \mathcal{V}\) for all open set \(\mathcal{V}\) of \(\mathcal{X}\) containing \(\mathcal{H}(y^{*})\) [62]. A real-valued function \(y: \mathbb{R} \to \mathbb{R}\) is called upper semi-continuous whenever \(\lim \sup_{n\to \infty }y(\lambda _{n})\leq y(\lambda )\) for all sequence \(\{ \lambda _{n} \}_{n\geq 1}\) with \(\lambda _{n} \to \lambda \) [62]. Assume that \((\mathcal{X},d)\) is a metric space. The Pompeiu–Hausdorff metric \(\mathcal{H}_{d}: {\mathcal{P}}(\mathcal{X}) \times {\mathcal{P}}( \mathcal{X}) \to \mathbb{R} \cup \{\infty \}\) is defined by

$$ \mathcal{H}_{d}\bigl(M^{*}, N^{*}\bigr) = \max \Bigl\{ \sup_{m^{*} \in M^{*}}d\bigl(m^{*},N^{*} \bigr), \sup_{n^{*} \in N^{*}}d\bigl(M^{*},n^{*} \bigr)\Bigr\} , $$

where \(d(M^{*},n^{*}) = \inf_{m^{*}\in M^{*}}d(m^{*},n^{*})\) and \(d(m^{*},N^{*}) = \inf_{n^{*}\in N^{*}} d(m^{*},n^{*})\). A multivalued map \(\mathcal{G}: \mathcal{X} \to {\mathcal{P}}_{cl} (\mathcal{X})\) is called Lipschitzian with Lipschitz constant \(\beta >0\) whenever

$$ \mathcal{H}_{d}\bigl(\mathcal{G}(y_{1}), \mathcal{G}(y_{2})\bigr) \leq \beta d(y_{1},y_{2}), $$

for all \(y_{1}, y_{2} \in \mathcal{X}\) [62]. A Lipschitz map \(\mathcal{G}\) is called a contraction whenever \(\beta \in (0,1)\) [62].

A multifunction \(\mathcal{G}: [0,1] \to {\mathcal{P}}_{cl}(\mathbb{R})\) is called measurable whenever the map \(t \to d(\omega,\mathcal{G}(t ))\) is measurable for all real constants ω [6, 62]. We say that \(\mathcal{G}: [0,1]\times \mathbb{R} \to {\mathcal{P}}(\mathbb{R})\) is Caratheodory whenever \(t \mapsto \mathcal{G}(t,y)\) is measurable map for all \(y \in \mathbb{R}\) and \(y\mapsto \mathcal{G}(a,y) \) is upper semi-continuous map for almost all \(t\in [0,1]\) [6, 62]. Also, a Caratheodory multifunction \(\mathcal{G}: [0,1]\times \mathbb{R} \to \mathcal{P}(\mathbb{R})\) is said to be \(\mathcal{L}^{1}\)-Caratheodory whenever for each constant \(\mu >0\) there exists a function \(\phi _{\mu }\in \mathcal{L}^{1} ([0,1], \mathbb{R}^{+})\) such that \(\Vert \mathcal{G}(t,y) \Vert =\sup_{\tau \in [0,1]}\{|\tau |:\tau \in \mathcal{G}(t,y)\}\leq \phi _{\mu }(t )\) for all \(|y|\leq \mu \) and for almost all \(t \in [0,1]\) [6, 62]. The set of selections of the multifunction \(\mathcal{G}\) at point \(y \in C([0,1],\mathbb{R})\) is defined by \(S_{\mathcal{G},y}:=\{v\in \mathcal{L}^{1} ([0,1],\mathbb{R}): v(t ) \in \mathcal{G}(t, y(t) )\}\) for almost all \(t \in [0,1]\). It has been proved that \(S_{\mathcal{G},y}\neq \emptyset \) for all \(y\in C([0,1],\mathcal{X} )\) if \(\dim \mathcal{X} < \infty \) [62]. We say that an element \(y\in \mathcal{X}\) is an endpoint of a multifunction \(\mathcal{G}:\mathcal{X}\to \mathcal{P}(\mathcal{X})\) whenever \(\mathcal{G}y=\{ y\}\) [62]. Also, the multifunction \(\mathcal{G}\) has an approximate endpoint property whenever \(\inf_{x\in \mathcal{X}}\sup_{y\in \mathcal{G}x}d(x,y)=0\) [62].

We denote by Ψ, the family of nondecreasing functions \(\psi: [0,\infty )\to [0,\infty )\) such that \(\sum_{n=1}^{\infty }\psi ^{n}(t)<\infty \) for all \(t>0\) [63]. It is clear that \(\psi (t)< t\) for all \(t>0\) [63]. In 2012, Samet et al. introduced the notion of α-ψ-contractive mappings [63]. We say that the selfmap \(\mathcal{T}: \mathcal{X}\to \mathcal{X}\) is an α-ψ-contraction whenever \(\alpha (y_{1},y_{2}) d(\mathcal{T} y_{1}, \mathcal{T}y_{2})\leq \psi (d(y_{1}, y_{2}))\) for all \(y_{1}, y_{2}\in \mathcal{X}\) [63]. Also, the selfmap \(\mathcal{T}\) is called α-admissible whenever \(\alpha (y_{1},y_{2})\geq 1\) implies \(\alpha (\mathcal{T} y_{1},\mathcal{T}y_{2})\geq 1\) [63]. We say that \(\mathcal{X}\) has the property \((B)\) whenever for each sequence \(\{y_{n}\}\) in \(\mathcal{X}\) with \(\alpha (y_{n},y_{n+1})\geq 1\) for all \(n\geq 1\) and \(y_{n}\to y\), we have \(\alpha (y_{n},y)\geq 1\) for all n [63].

In 2013, Mohammadi et al. generalized this notion to multifunctions [64]. A multifunction \(\mathcal{G}: \mathcal{X} \to CB(\mathcal{X})\) is called α-ψ-contraction whenever

$$ \alpha (y_{1},y_{2}) \mathcal{H}_{d} ( \mathcal{G} y_{1}, \mathcal{G}y_{2}) \leq \psi \bigl(d(y_{1},y_{2})\bigr), $$

for all \(y_{1}, y_{2}\in \mathcal{X}\) [64]. Similarly, the space \(\mathcal{X}\) has the property \((C_{\alpha })\) whenever for each sequence \(\{ y_{n} \} \) in \(\mathcal{X}\) with \(\alpha (y_{n}, y_{n+1}) \geq 1\) for all \(n\in \mathbb{N}\), there exists a subsequence \(\{ y_{n_{k}} \}\) of \(\{ y_{n} \}\) such that \(\alpha (y_{n_{k}}, y)\geq 1\) for all \(k\in \mathbb{N}\). The multivalued map \(\mathcal{G}\) is α-admissible whenever for each \(y_{1}\in \mathcal{X}\) and \(y_{2}\in \mathcal{G}y_{1}\) with \(\alpha (y_{1},y_{2})\geq 1\), we have \(\alpha ( y_{2},y_{3})\geq 1\) for all \(y_{3}\in \mathcal{G}y_{2}\) [64]. We need the following results.

Theorem 3

([63])

Let\((\mathcal{X},d)\)be a complete metric space, \(\psi \in \varPsi \), \(\alpha: \mathcal{X}\times \mathcal{X} \to \mathbb{R}\)a map and\(\mathcal{T}\)anα-admissible andα-ψ-contractive selfmap on\(\mathcal{X}\)such that\(\alpha (y_{0}, \mathcal{T}y_{0} )\geq 1\), for some\(y_{0} \in \mathcal{X}\). If\(\mathcal{X}\)has the property\((B)\), then\(\mathcal{T}\)has a fixed point.

Theorem 4

([65], Krasnoselskii)

LetMbe a closed, bounded, convex and nonempty subset of a Banach space\(\mathcal{X}\). Let\(\mathcal{A}\)and\(\mathcal{B}\)be two operators such that

  1. (i)

    \(\mathcal{A}x+\mathcal{B}y\in M\)whenever\(x,y\in M\),

  2. (ii)

    \(\mathcal{A}\)is compact and continuous,

  3. (iii)

    \(\mathcal{B}\)is a contraction mapping.

Then there exists\(z\in M\)such that\(z=\mathcal{A}z+\mathcal{B}z\).

Theorem 5

([64])

Let\((\mathcal{X},d)\)be a complete metric space, \(\alpha: \mathcal{X}\times \mathcal{X} \to [0,\infty )\)a map, \(\psi \in \varPsi \)a strictly increasing map, \(\mathcal{G}: \mathcal{X} \to CB (\mathcal{X})\)anα-admissible andα-ψ-contractive multifunction and\(\alpha (y_{0},y_{1} )\geq 1\)for some\(y_{0} \in \mathcal{X}\)and\(y_{1} \in \mathcal{G}y_{0}\). If the space\(\mathcal{X}\)has the property\((C_{\alpha })\), then\(\mathcal{G}\)has a fixed point.

Theorem 6

([62])

Let\((\mathcal{X},d)\)be a complete metric space and\(\psi: [0, \infty ) \to [0, \infty )\)be an upper semi-continuous function such that\(\psi (t)< t\)and\(\liminf_{t\to \infty }(t-\psi (t))>0\)for all\(t>0\). Suppose that\(\mathcal{T}: \mathcal{X}\to CB(\mathcal{X})\)is a multifunction such that\(\mathcal{H}_{d}(\mathcal{T} y_{1},\mathcal{T}y_{2})\leq \psi (d(y_{1},y_{2}))\)for all\(y_{1},y_{2} \in \mathcal{X}\). Then\(\mathcal{T}\)has a unique endpoint if and only if\(\mathcal{T}\)has approximate endpoint property.

Main results

In this work, we consider the Banach space

$$ \mathcal{X}=\bigl\{ y(t): y(t),\mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2} y(t) \in C\bigl([0,1], \mathbb{R}\bigr) \bigr\} , $$

via the norm

$$ \Vert y \Vert =\sup_{t\in [0,1]} \bigl\vert y(t) \bigr\vert +\sup_{t\in [0,1]} \bigl\vert \mathcal{D}_{q}y(t) \bigr\vert + \sup_{t\in [0,1]} \bigl\vert \mathcal{D}_{q}^{2} y(t) \bigr\vert . $$

Lemma 7

Let\(\varphi \in C([0,1 ],\mathcal{X}) \). Then solution of theq-fractional boundary value problem

{Dqϑcy(t)=φ(t),y(0)=0,Dqy(0)+cDqσy(ν)+Dq2y(1)=0,Jqκy(0)+Jqκy(ν)+Jqκy(1)=0,

is given by

$$\begin{aligned} y(t) ={}& \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{\varGamma _{q}(\vartheta )} \varphi (\tau ) \, \mathrm{d}_{q}\tau \\ &{} + \frac{t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \varphi (\tau ) \, \mathrm{d}_{q}\tau \\ &{} +\frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \varphi (\tau ) \, \mathrm{d}_{q}\tau \\ &{} +\frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \varphi (\tau ) \, \mathrm{d}_{q}\tau \\ & {}+ \frac{t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varphi (\tau ) \, \mathrm{d}_{q}\tau, \end{aligned}$$
(5)

where\(\Delta _{1} = 2\nu ^{2-\sigma } + (1+q) \varGamma _{q} (3-\sigma )\), \(\Delta _{2} = (1+ \nu ^{\kappa +1})\varGamma _{q}(3-\sigma )\),

$$ \Delta _{3} = \biggl\vert \frac{- \varGamma _{q} (\kappa +3)(\nu ^{\kappa +1}+1) \Delta _{1} + \varGamma _{q} (3 -\sigma ) \varGamma _{q} ( \kappa +2 )(1+q) [\nu ^{\kappa +2}+1] }{\varGamma _{q} (\kappa +2 ) \varGamma _{q} (\kappa +3 ) \varGamma _{q} (3 - \sigma ) } \biggr\vert , $$
(6)

\(\Delta _{4} = (1+\nu ^{\kappa +1} )\Delta _{1} +\Delta _{3} \varGamma _{q}( \kappa +2) \varGamma _{q}(3-\sigma )\)and\(\Delta _{5} = \Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q}( \kappa +2) \neq 0\).

Proof

Let \(\hat{y}_{0}\) be a solution for the q-problem. Choose the constants \(a_{0}\), \(a_{1}\) and \(a_{2} \in \mathbb{R}\) such that

$$ \hat{y}_{0}(t) = \int _{0}^{t} \frac{(t-q\tau )^{(\vartheta -1)}}{\varGamma _{q} (\vartheta )} \varphi ( \tau ) \,\mathrm{d}_{q}\tau +a_{0} +a_{1}t+a_{2}t^{2}. $$
(7)

Thus, we have

$$\begin{aligned} &\mathcal{D}_{q}\hat{y}_{0}(t) = \int _{0}^{t} \frac{(t-q\tau )^{(\vartheta -2)}}{\varGamma _{q} (\vartheta -1)} \varphi (\tau ) \, \mathrm{d}_{q}\tau +a_{1}+a_{2}(1+q)t, \\ &{}^{c}\mathcal{D}_{q}^{\sigma }\hat{y}_{0}(t) = \int _{0}^{t} \frac{(t-q\tau )^{(\vartheta - \sigma -1)}}{\varGamma _{q} (\vartheta - \sigma )} \varphi (\tau ) \, \mathrm{d}_{q}\tau +a_{2} \frac{2t^{2-\sigma }}{\varGamma _{q} (3-\sigma )}, \\ &\mathcal{D}_{q}^{2}\hat{y}_{0}(t) = \int _{0}^{t} \frac{(t-q\tau )^{(\vartheta -3)}}{\varGamma _{q} (\vartheta -2)} \varphi (\tau ) \, \mathrm{d}_{q}\tau +a_{2}(1+q), \end{aligned}$$

and

$$\begin{aligned} \mathcal{J}_{q}^{\kappa }\hat{y}_{0}(t) ={}& \int _{0}^{t} \frac{(t-q\tau )^{(\vartheta + \kappa - 1)}}{\varGamma _{q} (\vartheta + \kappa )} \varphi (\tau ) \, \mathrm{d}_{q}\tau + a_{0} \frac{t^{\kappa }}{\varGamma _{q} (\kappa +1)} \\ &{} + a_{1} \frac{t^{\kappa +1}}{\varGamma _{q} (\kappa +2)} + a_{2} \frac{(1+q)t^{\kappa +2}}{\varGamma _{q} (\kappa +3)}. \end{aligned}$$

By using the boundary value conditions, we obtain \(a_{0}=0\),

$$\begin{aligned} a_{1} ={}& \frac{ \Delta _{1} }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varphi (\tau ) \, \mathrm{d}_{q}\tau \\ &{} - \frac{ \Delta _{4} }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \varphi (\tau ) \, \mathrm{d}_{q}\tau \\ &{} - \frac{ \Delta _{4} }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \varphi (\tau ) \, \mathrm{d}_{q}\tau \\ & {}+ \frac{ \Delta _{1} }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varphi (\tau ) \, \mathrm{d}_{q}\tau \end{aligned}$$

and

$$\begin{aligned} a_{2} ={}& \frac{ 1+\nu ^{\kappa +1} }{\Delta _{3} \varGamma _{q} (\kappa +2 ) } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta - 2 )} \varphi (\tau ) \, \mathrm{d}_{q}\tau \\ &{} - \frac{ 1 }{\Delta _{4} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta + \kappa -1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varphi (\tau ) \, \mathrm{d}_{q}\tau \\ &{} - \frac{ 1 }{\Delta _{3} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varphi (\tau ) \, \mathrm{d}_{q}\tau \\ &{} + \frac{ 1 + \vartheta ^{\kappa +1} }{\Delta _{3} \varGamma _{q} (\kappa + 2 ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta - \sigma )} \varphi (\tau ) \, \mathrm{d}_{q}\tau. \end{aligned}$$

By substituting the values of the \(a_{i}\) in (7), we obtain the q-integral equation (5). This completes the proof. □

Now, consider the operator \(\mathcal{T}:\mathcal{X}\to \mathcal{X}\) defined by

$$\begin{aligned} (\mathcal{T}y) (t) ={}& \int _{0}^{t} \frac{ (t-q\tau )^{ (\vartheta -1)} }{\varGamma _{q}(\vartheta )} \varPhi \bigl( \tau, y(\tau ), \mathcal{D}_{q} y(\tau ), \mathcal{D}_{q}^{2} y( \tau ) \bigr) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma )}{ \Delta _{3} \varGamma _{q} (3-\sigma )} \\ &{} \times \int _{0}^{1} \frac{(1 - q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \varPhi \bigl( \tau, y(\tau ), \mathcal{D}_{q} y(\tau ), \mathcal{D}_{q}^{2} y( \tau ) \bigr) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \\ &{} \times \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \varPhi \bigl( \tau, y(\tau ), \mathcal{D}_{q} y(\tau ), \mathcal{D}_{q}^{2} y( \tau ) \bigr) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \\ & {}\times \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \varPhi \bigl( \tau, y(\tau ), \mathcal{D}_{q} y(\tau ), \mathcal{D}_{q}^{2} y( \tau ) \bigr) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \\ &{} \times \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varPhi \bigl( \tau, y(\tau ), \mathcal{D}_{q} y(\tau ), \mathcal{D}_{q}^{2} y( \tau ) \bigr) \,\mathrm{d}_{q}\tau. \end{aligned}$$

It is clear that \(y_{0}\) is a solution for the problem (1) if and only if \(y_{0}\) is a fixed point of the operator \(\mathcal{T}\). Put

$$\begin{aligned} & \varXi _{1} = \frac{ 1 }{\varGamma _{q}(\vartheta +1)} + \frac{ (\Delta _{1} + \varGamma _{q} (3-\sigma )) (\nu ^{(\vartheta + \kappa ) } +1) }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} (\vartheta + \kappa + 1 ) } \\ & \phantom{\varXi _{1} =}{}+ \frac{ \Delta _{2} + \Delta _{4} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } + \frac{ (\Delta _{2} + \Delta _{4}) \nu ^{(\vartheta - \sigma )} }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) }, \\ \begin{aligned}&\varXi _{2} = \frac{ 1 }{\varGamma _{q}(\vartheta )} + \frac{ [\Delta _{1} + (1+q)\varGamma _{q} (3-\sigma ) ]( \nu ^{(\vartheta + \kappa )}+1) }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} (\vartheta + \kappa + 1 ) } \\ &\phantom{\varXi _{2} =}{} + \frac{ (1+q)\Delta _{2} + \Delta _{4} }{ \Delta _{5} \varGamma _{q} ( \vartheta -1 ) } + \frac{ [(1+q) \Delta _{2} + \Delta _{4}] \nu ^{( \vartheta - \sigma )} }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) }, \end{aligned} \end{aligned}$$
(8)
$$\begin{aligned} &\varXi _{3} = \frac{ 1 }{\varGamma _{q}(\vartheta -1)} + \frac{ (1+q)( \nu ^{(\vartheta + \kappa )}+1) }{\Delta _{3} \varGamma _{q} (\vartheta + \kappa + 1 ) } \\ & \phantom{\varXi _{3} =}{}+ \frac{ (1+q)\Delta _{2} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } + \frac{ (1+q)\Delta _{2} \nu ^{ ( \vartheta - \sigma )} }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) }, \\ \begin{aligned} &\Delta ^{(1)} = \frac{ (\Delta _{1} + \varGamma _{q} (3-\sigma )) (\nu ^{(\vartheta + \kappa ) } +1) }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} (\vartheta + \kappa + 1 ) } + \frac{ \Delta _{2} + \Delta _{4} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } \\ &\phantom{\Delta ^{(1)} =}{} + \frac{ (\Delta _{2} + \Delta _{4}) \nu ^{(\vartheta - \sigma )} }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) }, \\ &\Delta ^{(2)} = \frac{ [\Delta _{1} + (1+q)\varGamma _{q} (3-\sigma ) ] ( \nu ^{(\vartheta + \kappa )}+1) }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} (\vartheta + \kappa + 1 ) } + \frac{ (1+q) \Delta _{2} + \Delta _{4} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } \\ &\phantom{\Delta ^{(2)} =}{} + \frac{ [(1+q)\Delta _{2} + \Delta _{4}] \nu ^{ (\vartheta - \sigma )} }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) }, \\ &\Delta ^{(3)} = \frac{ (1+q)( \nu ^{(\vartheta + \kappa )}+1) }{\Delta _{3} \varGamma _{q} (\vartheta + \kappa + 1 ) } + \frac{ (1+q)\Delta _{2} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } + \frac{ (1+q)\Delta _{2} \nu ^{ ( \vartheta - \sigma )} }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) }, \end{aligned} \end{aligned}$$
(9)

and

$$ \varSigma _{1} = \Vert m \Vert \varXi _{1}, \qquad\varSigma _{2} = \Vert m \Vert \varXi _{2},\qquad \varSigma _{3} = \Vert m \Vert \varXi _{3}. $$
(10)

Now, we are ready to prove our main results.

Theorem 8

Let\(\psi \in \varPsi \), \(\chi: \mathbb{R}^{3} \times \mathbb{R}^{3}\to \mathbb{R}\)be a map and\(\varPhi: [0,1 ] \times \mathcal{X}^{3} \to \mathcal{X}\)a continuous function. Suppose that

\((H1)\):
$$\begin{aligned} &\bigl\vert \varPhi \bigl(t,x_{1}(t), y_{1}(t), z_{1}(t)\bigr) - \varPhi \bigl(t,x_{2}(t),y_{2}(t), z_{2}(t) \bigr) \bigr\vert \\ & \quad \leq \lambda \psi \bigl( \vert x_{1}-x_{2} \vert + \vert y_{1} -y_{2} \vert + \vert z_{1} -z_{2} \vert \bigr), \end{aligned}$$

for all\(x_{1}, y_{1}, z_{1}, x_{2}, y_{2}, z_{2} \in \mathcal{X}\)with

$$ \chi \bigl( \bigl(x_{1}(t),y_{1}(t), z_{1}(t) \bigr), \bigl(x_{2}(t),y_{2}(t), z_{2}(t)\bigr) \bigr) \geq 0, $$

for all\(t\in [0, 1 ]\), where\(\lambda =\frac{1}{\varXi _{1} + \varXi _{2} + \varXi _{3} }\),

\((H2)\):

There exists\(y_{0} \in \mathcal{X}\)such that

$$ \chi \bigl( \bigl(y_{0}(t), \mathcal{D}_{q}y_{0}(t), \mathcal{D}_{q}^{2}y_{0}(t)\bigr), \bigl( \mathcal{T}y_{0}(t),\mathcal{D}_{q}\bigl( \mathcal{T}y_{0}(t)\bigr), \mathcal{D}_{q}^{2} \bigl(\mathcal{T}y_{0}(t)\bigr) \bigr) \bigr)\geq 0, $$

for all\(t\in [0,1]\)and

$$ \chi \bigl( \bigl(y_{1}(t), \mathcal{D}_{q}y_{1}(t), \mathcal{D}_{q}^{2}y_{1}(t) \bigr), \bigl(y_{2}(t),\mathcal{D}_{q}y_{2}(t), \mathcal{D}_{q}^{2}y_{2}(t)\bigr) \bigr) \geq 0, $$

which implies

$$ \chi \bigl( \bigl( \mathcal{T}y_{1}(t), \mathcal{D}_{q} \bigl(\mathcal{T}y_{1}(t)\bigr), \mathcal{D}_{q}^{2} \bigl( \mathcal{T}y_{1}(t)\bigr)\bigr), \bigl(\mathcal{T}y_{2}(t), \mathcal{D}_{q}\bigl(\mathcal{T}y_{2}(t)\bigr), \mathcal{D}_{q}^{2} \bigl( \mathcal{T}y_{2}(t) \bigr) \bigr) \bigr) \geq 0 $$

for all\(t\in [0, 1]\)and\(y_{1},y_{2} \in \mathcal{X} \),

\((H3)\):

For each convergent sequence\(\{ y_{n} \}_{n\geq 1}\)in\(\mathcal{X}\)with\(y_{n} \to y\)and

$$ \chi \bigl( \bigl(y_{n}(t),\mathcal{D}_{q}y_{n}(t), \mathcal{D}_{q}^{2} y_{n}(t)\bigr), \bigl(y_{n+1}(t),\mathcal{D}_{q}y_{n+1}(t), \mathcal{D}_{q}^{2}y_{n+1}(t)\bigr) \bigr) \geq 0, $$

for allnand\(t\in [0,1 ]\), we have

$$ \chi \bigl( \bigl(y_{n}(t),\mathcal{D}_{q}y_{n}(t), \mathcal{D}_{q}^{2} y_{n}(t)\bigr), \bigl(y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2} y(t) \bigr) \bigr) \geq 0. $$
Then theq-fractional boundary value problem (1)(2) has at least one solution.

Proof

Let \(y_{1}, y_{2}\in \mathcal{X} \) be such that

$$ \chi \bigl( \bigl(y_{1}(t), \mathcal{D}_{q}y_{1}(t), \mathcal{D}_{q}^{2}y_{1}(t)\bigr), \bigl(y_{2}(t),\mathcal{D}_{q}y_{2}(t), \mathcal{D}_{q}^{2}y_{2}(t)\bigr) \bigr) \geq 0, $$

for all \(t\in [0,1]\). Then we have

$$\begin{aligned} &\bigl\vert \mathcal{T}y_{1}(t) - \mathcal{T} y_{2}(t) \bigr\vert \\ &\quad \leq \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{\varGamma _{q}(\vartheta )} \\ &\qquad{} \times \bigl\vert \varPhi \bigl( \tau, y_{1}(\tau ), \mathcal{D}_{q} y_{1}( \tau ), \mathcal{D}_{q}^{2} y_{1}(\tau ) \bigr) - \varPhi \bigl( \tau, y_{2}( \tau ), \mathcal{D}_{q} y_{2}(\tau ), \mathcal{D}_{q}^{2} y_{2}(\tau ) \bigr) \bigr\vert \,\mathrm{d}_{q}\tau \\ &\qquad{} + \frac{ \vert t \Delta _{1} -t^{2} \varGamma _{q} (3- \sigma ) \vert }{\Delta _{3} \varGamma _{q} (3- \sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \\ & \qquad{}\times \bigl\vert \varPhi \bigl( \tau, y_{1} (\tau ), \mathcal{D}_{q} y_{1}( \tau ), \mathcal{D}_{q}^{2} y_{1}(\tau ) \bigr) - \varPhi \bigl( \tau, y_{2}( \tau ), \mathcal{D}_{q} y_{2}(\tau ), \mathcal{D}_{q}^{2} y_{2}(\tau ) \bigr) \bigr\vert \,\mathrm{d}_{q}\tau \\ &\qquad{} + \frac{ \vert t^{2} \Delta _{2} - t \Delta _{4} \vert }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \\ &\qquad{} \times \bigl\vert \varPhi \bigl( \tau, y_{1}(\tau ), \mathcal{D}_{q} y_{1}( \tau ), \mathcal{D}_{q}^{2} y_{1}(\tau ) \bigr) - \varPhi \bigl( \tau, y_{2}( \tau ), \mathcal{D}_{q} y_{2}(\tau ), \mathcal{D}_{q}^{2} y_{2}(\tau ) \bigr) \bigr\vert \,\mathrm{d}_{q}\tau \\ &\qquad{} + \frac{ \vert t^{2} \Delta _{2} - t \Delta _{4} \vert }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \\ &\qquad{} \times \bigl\vert \varPhi \bigl( \tau, y_{1}(\tau ), \mathcal{D}_{q} y_{1}( \tau ), \mathcal{D}_{q}^{2} y_{1}(\tau ) \bigr) - \varPhi \bigl( \tau, y_{2}( \tau ), \mathcal{D}_{q} y_{2}(\tau ), \mathcal{D}_{q}^{2} y_{2}(\tau ) \bigr) \bigr\vert \,\mathrm{d}_{q}\tau \\ &\qquad{} + \frac{ \vert t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) \vert }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \\ & \qquad{}\times \bigl\vert \varPhi \bigl( \tau, y_{1}(\tau ), \mathcal{D}_{q} y_{1}( \tau ), \mathcal{D}_{q}^{2} y_{1}(\tau ) \bigr) - \varPhi \bigl( \tau, y_{2}( \tau ), \mathcal{D}_{q} y_{2}(\tau ), \mathcal{D}_{q}^{2} y_{2}(\tau ) \bigr) \bigr\vert \,\mathrm{d}_{q}\tau \\ &\quad \leq \frac{ 1 }{\varGamma _{q}(\vartheta +1)} \\ &\qquad{} \times \lambda \psi \bigl( \bigl\vert y_{1}(\tau )-y_{2}(\tau ) \bigr\vert + \bigl\vert D_{q}y_{1}( \tau )-D_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert D_{q}^{2}y_{1}( \tau )-D_{q}^{2}y_{2}( \tau ) \bigr\vert \bigr) \\ & \qquad{}+ \frac{ \Delta _{1} + \varGamma _{q} (3-\sigma ) }{ \Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} (\vartheta + \kappa + 1 ) } \\ & \qquad{}\times \lambda \psi \bigl( \bigl\vert y_{1} (\tau ) - y_{2}(\tau ) \bigr\vert + \bigl\vert D_{q} y_{1}(\tau )-D_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert D_{q}^{2}y_{1}( \tau )-D_{q}^{2}y_{2}(\tau ) \bigr\vert \bigr) \\ &\qquad{} + \frac{ \Delta _{2} + \Delta _{4} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } \\ & \qquad{}\times \lambda \psi \bigl( \bigl\vert y_{1}(\tau )-y_{2}(\tau ) \bigr\vert + \bigl\vert D_{q}y_{1}( \tau )-D_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert D_{q}^{2} y_{1}(\tau )-D_{q}^{2} y_{2}(\tau ) \bigr\vert \bigr) \\ & \qquad{}+ \frac{ (\Delta _{2} + \Delta _{4}) \nu ^{(\vartheta - \sigma )} }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) } \\ & \qquad{}\times \lambda \psi \bigl( \bigl\vert y_{1}(\tau )-y_{2}(\tau ) \bigr\vert + \bigl\vert D_{q}y_{1}( \tau )-D_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert D_{q}^{2}y_{1}( \tau )-D_{q}^{2}y_{2}( \tau ) \bigr\vert \bigr) \\ &\qquad{} + \frac{ (\Delta _{1} + \varGamma _{q} (3-\sigma )) \nu ^{(\vartheta + \kappa )} }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} ( \vartheta + \kappa + 1 ) } \\ & \qquad{}\times \lambda \psi \bigl( \bigl\vert y_{1} (\tau ) - y_{2} (\tau ) \bigr\vert + \bigl\vert D_{q} y_{1}(\tau )-D_{q} y_{2} (\tau ) \bigr\vert + \bigl\vert D_{q}^{2} y_{1}(\tau ) - D_{q}^{2} y_{2}(\tau ) \bigr\vert \bigr) \\ & \quad\leq \frac{ 1 }{\varGamma _{q}(\vartheta +1)} \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ & \qquad{}+ \frac{ \Delta _{1} + \varGamma _{q} (3-\sigma ) }{ \Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} (\vartheta + \kappa + 1 ) } \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ & \qquad{}+ \frac{ \Delta _{2} + \Delta _{4} }{ \Delta _{5} \varGamma _{q} ( \vartheta -1 ) } \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ & \qquad{}+ \frac{ (\Delta _{2} + \Delta _{4}) \nu ^{(\vartheta - \sigma )} }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) } \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ &\qquad{} + \frac{ (\Delta _{1} + \varGamma _{q} (3-\sigma )) \nu ^{(\vartheta + \kappa )} }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} ( \vartheta + \kappa + 1 ) } \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ &\quad = \lambda \varXi _{1} \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr). \end{aligned}$$

Similarly, we have

$$\begin{aligned} &\bigl\vert \mathcal{D}_{q} \mathcal{T}y_{1}(t) - \mathcal{D}_{q}\mathcal{T}y_{2}(t) \bigr\vert \\ &\quad \leq \frac{ 1 }{\varGamma _{q}(\vartheta )} \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ &\qquad{} + \frac{ \Delta _{1} + (1+q)\varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} (\vartheta + \kappa + 1 ) } \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ &\qquad{} + \frac{ (1+q)\Delta _{2} + \Delta _{4} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ & \qquad{}+ \frac{ [(1+q) \Delta _{2} + \Delta _{4}] \nu ^{(\vartheta - \sigma )} }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) } \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ &\qquad{} + \frac{ [\Delta _{1} + (1+q)\varGamma _{q} (3-\sigma )] \nu ^{(\vartheta + \kappa )} }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} ( \vartheta + \kappa + 1 ) } \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ &\quad= \lambda \varXi _{2} \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \end{aligned}$$

and

$$\begin{aligned} &\bigl\vert \mathcal{D}_{q}^{2} \mathcal{T}y_{1}(t) - \mathcal{D}_{q}^{2} \mathcal{T}y_{2}(t) \bigr\vert \\ & \quad\leq \frac{ 1 }{\varGamma _{q}(\vartheta -1 )} \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ &\qquad{} + \frac{ (1+q) }{\Delta _{3} \varGamma _{q} (\vartheta + \kappa + 1 ) } \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ &\qquad{} + \frac{ (1+q)\Delta _{2} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ & \qquad{}+ \frac{ (1+q) \Delta _{2} \nu ^{(\vartheta - \sigma )}}{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) } \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ & \qquad{}+ \frac{ (1+q) \nu ^{(\vartheta + \kappa )} }{\Delta _{3} \varGamma _{q} ( \vartheta + \kappa + 1 ) } \lambda \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr) \\ &\quad = \lambda \varXi _{3} \psi \bigl( \Vert y_{1}-y_{2} \Vert \bigr). \end{aligned}$$

Hence \(\Vert \mathcal{T} y_{1}(t)- \mathcal{T} y_{2} (t) \Vert \leq (\varXi _{1} +\varXi _{2} + \varXi _{3} ) \lambda \psi (\Vert y_{1}-y_{2} \Vert ) = \psi (\Vert y_{1}-y_{2} \Vert )\). Now, we define the non-negative function α on \(\mathcal{X} \times \mathcal{X}\) as follows:

α(y1,y2)={1if χ((y1(t),Dqy1(t),Dq2y1(t)),(y2(t),Dqy2(t),Dq2y2(t)))0,0otherwise,

for all \(y_{1}\), \(y_{2} \in \mathcal{X}\). Then we have \(\alpha (y_{1}, y_{2}) d ( \mathcal{T}y_{1},\mathcal{T}y_{2})\leq \psi (d(y_{1},y_{2}))\) for all \(y_{1}\), \(y_{2}\in \mathcal{X}\). This means that \(\mathcal{T}\) is an α-ψ-contractive operator. Furthermore, it is easy to check that \(\mathcal{T}\) is α-admissible and \(\alpha (y_{0}, \mathcal{T}y_{0} )\geq 1\). Besides, we suppose that \(\{ y_{n} \}_{n\geq 1}\) is a sequence that belongs to \(\mathcal{X}\) with \(y_{n} \to y\) and \(\alpha (y_{n}, y_{n+1}) \geq 1\) for all n. The definition of the non-negative function α implies that

$$ \chi \bigl( \bigl(y_{n}(t),\mathcal{D}_{q}y_{n}(t), \mathcal{D}_{q}^{2} y_{n}(t)\bigr), \bigl(y_{n+1}(t),\mathcal{D}_{q}y_{n+1}(t), \mathcal{D}_{q}^{2}y_{n+1}(t)\bigr) \bigr) \geq 0. $$

Thus, by the hypothesis, we get

$$ \chi \bigl( \bigl(y_{n}(t),\mathcal{D}_{q}y_{n}(t), \mathcal{D}_{q}^{2}y_{n}(t)\bigr), \bigl(y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2}y(t) \bigr) \bigr) \geq 0. $$

This shows that, for all n, \(\alpha (y_{n}, y) \geq 1\). Hence the Banach space \(\mathcal{X}\) has the property \((B)\). Now, Theorem 3 implies that the operator \(\mathcal{T}\) has fixed point \(y^{*}\in \mathcal{X}\) which is a solution for the q-fractional BVP (1)–(2). This completes the proof. □

Theorem 9

Let\(\varPhi:[0,1]\times \mathcal{X} \times \mathcal{X} \times \mathcal{X} \to \mathcal{X}\)be a continuous function. Suppose that:

\((H4)\):

there exists a continuous real-valued functionLon the closed interval\([0,1]\)such that

$$ \bigl\vert \varPhi (t,x_{1},y_{1}, z_{1})- \varPhi (t,x_{2},y_{2}, z_{2}) \bigr\vert \leq L(t) \bigl( \vert x_{1}-x_{2} \vert + \vert y_{1}-y_{2} \vert + \vert z_{1}-z_{2} \vert \bigr). $$

for all\(t\in [0,1]\)and\(x_{1},x_{2},y_{1},y_{2},z_{1}, z_{2}\in \mathcal{X}\),

\((H5)\):

there exist a continuous function\(\zeta: [0,1] \to \mathbb{R}^{+} \)and a nondecreasing continuous function\(\psi: [0,1] \to \mathbb{R}^{+} \)such that

$$ \bigl\vert \varPhi (t, y_{1}, y_{2}, y_{3}) \bigr\vert \leq \zeta (t) \psi \bigl( \vert y_{1} \vert + \vert y_{2} \vert + \vert y_{3} \vert \bigr), $$

for all\(t\in [0,1]\)and\(y_{1},y_{2},y_{3}\in \mathcal{X}\).

Then the fractionalq-difference equation (1) with the boundary value conditions (2) has at least one solution whenever

$$ K:= \Vert L \Vert \bigl(\Delta ^{(1)} + \Delta ^{(2)} + \Delta ^{(3)} \bigr)< 1, $$
(11)

where\(\Vert L \Vert =\sup_{t\in [0,1]} \vert L (t)\vert \)and\(\Delta ^{(1)}\), \(\Delta ^{(2)}\)and\(\Delta ^{(3)}\)are given by (10).

Proof

Put \(\Vert \zeta \Vert =\sup_{t\in [0,1]} \vert \zeta (t)\vert \) and choose a suitable positive constant ϵ such that

$$ \epsilon \geq \psi \bigl( \Vert y \Vert \bigr) \Vert \zeta \Vert \{ \varXi _{1} + \varXi _{2} + \varXi _{3} \}, $$
(12)

where the \(\Delta ^{(i)}\) are given by (10). Consider the set \(V_{\epsilon }=\{ y\in \mathcal{X}: \Vert y\Vert \leq \epsilon \}\), where ϵ is given in (12). One can check that the set \(V_{\epsilon }\) is a closed, convex, bounded and nonempty subset of the Banach space \(\mathcal{X}\). Now, consider the fractional operators \(\mathcal{T}^{(1)}\) and \(\mathcal{T}^{(2)}\) on the set \(V_{\epsilon }\) defined by

$$ \bigl(\mathcal{T}^{(1)}y\bigr) (t) = \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{\varGamma _{q}(\vartheta )} \varPhi \bigl( \tau, y(\tau ), \mathcal{D}_{q} y(\tau ), \mathcal{D}_{q}^{2} y( \tau ) \bigr) \,\mathrm{d}_{q}\tau $$

and

$$\begin{aligned} \bigl(\mathcal{T}^{(2)}y\bigr) (t) ={}& \frac{t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma )} \\ &{} \times \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \varPhi \bigl( \tau, y(\tau ), \mathcal{D}_{q} y(\tau ), \mathcal{D}_{q}^{2} y( \tau ) \bigr) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \\ &{} \times \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \varPhi \bigl( \tau, y(\tau ), \mathcal{D}_{q} y(\tau ), \mathcal{D}_{q}^{2} y( \tau ) \bigr) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \\ & {}\times \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \varPhi \bigl( \tau, y(\tau ), \mathcal{D}_{q} y(\tau ), \mathcal{D}_{q}^{2} y( \tau ) \bigr) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \\ &{} \times \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varPhi \bigl( \tau, y(\tau ), \mathcal{D}_{q} y(\tau ), \mathcal{D}_{q}^{2} y( \tau ) \bigr) \,\mathrm{d}_{q}\tau \end{aligned}$$

for all \(t\in [0,1]\). Put \(\hat{m}=\sup_{y\in \mathbb{R}} \psi (\Vert y\Vert ) \). For \(y_{1},y_{2}\in V_{\epsilon }\), we have

$$\begin{aligned} &\bigl\vert \bigl(\mathcal{T}^{(1)}y_{1} + \mathcal{T}^{(2)}y_{2}\bigr) (t) \bigr\vert \\ &\quad \leq \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{\varGamma _{q}(\vartheta )} \bigl\vert \varPhi \bigl( \tau, y_{1}(\tau ), \mathcal{D}_{q} y_{1}(\tau ), \mathcal{D}_{q}^{2} y_{1}(\tau ) \bigr) \bigr\vert \,\mathrm{d}_{q}\tau \\ & \qquad{}+ \frac{ \vert t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma ) \vert }{\Delta _{3} \varGamma _{q} (3-\sigma )} \\ & \qquad{}\times \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \bigl\vert \varPhi \bigl( \tau, y_{2}(\tau ), \mathcal{D}_{q} y_{2}(\tau ), \mathcal{D}_{q}^{2} y_{2}(\tau )\bigr) \bigr\vert \,\mathrm{d}_{q}\tau \\ &\qquad{} + \frac{ \vert t^{2} \Delta _{2} - t \Delta _{4} \vert }{\Delta _{5} } \\ &\qquad{} \times \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \bigl\vert \varPhi \bigl( \tau, y_{2}(\tau ), \mathcal{D}_{q} y_{2}(\tau ), \mathcal{D}_{q}^{2} y_{2}(\tau ) \bigr) \bigr\vert \,\mathrm{d}_{q}\tau \\ &\qquad{} + \frac{ \vert t^{2} \Delta _{2} - t \Delta _{4} \vert }{\Delta _{5} } \\ &\qquad{} \times \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \bigl\vert \varPhi \bigl( \tau, y_{2}(\tau ), \mathcal{D}_{q} y_{2}(\tau ), \mathcal{D}_{q}^{2} y_{2}(\tau ) \bigr) \bigr\vert \,\mathrm{d}_{q}\tau \\ & \qquad{}+ \frac{ \vert t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) \vert }{\Delta _{3} \varGamma _{q} (3-\sigma )} \\ &\qquad{} \times \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \bigl\vert \varPhi \bigl( \tau, y_{2}(\tau ), \mathcal{D}_{q} y_{2}(\tau ), \mathcal{D}_{q}^{2} y_{2}(\tau ) \bigr) \bigr\vert \,\mathrm{d}_{q}\tau \\ &\quad \leq \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{ \varGamma _{q}(\vartheta )} \psi \bigl( \bigl\vert y_{1}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau \\ &\qquad{} + \frac{ \vert t \Delta _{1} -t^{2} \varGamma _{q} ( 3 - \sigma ) \vert }{\Delta _{3} \varGamma _{q} (3-\sigma )} \\ & \qquad{}\times \int _{0}^{1} \frac{(1 -q\tau )^{ ( \vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \psi \bigl( \bigl\vert y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q} \tau \\ & \qquad{}+ \frac{ \vert t^{2} \Delta _{2} - t \Delta _{4} \vert }{\Delta _{5} } \\ & \qquad{}\times \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \psi \bigl( \bigl\vert y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q} \tau \\ & \qquad{}+ \frac{ \vert t^{2} \Delta _{2} - t \Delta _{4} \vert }{\Delta _{5} } \\ & \qquad{}\times \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \psi \bigl( \bigl\vert y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q} \tau \\ &\qquad{} + \frac{ \vert t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) \vert }{\Delta _{3} \varGamma _{q} (3-\sigma )} \\ & \qquad{}\times \int _{0}^{\nu }\frac{(\nu -q\tau )^{ (\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \psi \bigl( \bigl\vert y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q} \tau \\ & \quad\leq \hat{m} \Vert \zeta \Vert \biggl[ \frac{ 1 }{\varGamma _{q}(\vartheta +1)} + \frac{ (\Delta _{1} + \varGamma _{q} (3-\sigma )) (\nu ^{(\vartheta + \kappa ) } +1) }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} (\vartheta + \kappa + 1 ) } \\ & \qquad{}+ \frac{ \Delta _{2} + \Delta _{4} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } + \frac{ (\Delta _{2} + \Delta _{4}) \nu ^{(\vartheta - \sigma )} }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) } \biggr] = \hat{m} \Vert \zeta \Vert \varXi _{1}. \end{aligned}$$

Also,

$$\begin{aligned} &\bigl\vert \bigl(\mathcal{D}_{q} \mathcal{T}^{(1)}y_{1} + \mathcal{D}_{q} \mathcal{T}^{(2)}y_{2}\bigr) (t) \bigr\vert \\ &\quad \leq \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -2)} }{\varGamma _{q}(\vartheta -1)} \psi \bigl( \bigl\vert y_{1}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau \\ & \qquad{}+ \frac{ \vert \Delta _{1} -(1+q)t \varGamma _{q} (3 - \sigma ) \vert }{\Delta _{3} \varGamma _{q} (3-\sigma )} \\ & \qquad{}\times \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \psi \bigl( \bigl\vert y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q} \tau \\ &\qquad{} + \frac{ \vert (1+q)t \Delta _{2} - \Delta _{4} \vert }{\Delta _{5} } \\ &\qquad{} \times \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \psi \bigl( \bigl\vert y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q} \tau \\ & \qquad{}+ \frac{ \vert (1+q)t \Delta _{2} - \Delta _{4} \vert }{\Delta _{5} } \\ &\qquad{} \times \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \psi \bigl( \bigl\vert y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q} \tau \\ &\qquad{} + \frac{ \vert \Delta _{1} - (1+q)t \varGamma _{q} (3-\sigma ) \vert }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \\ &\qquad{} \times \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \psi \bigl( \bigl\vert y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{2}( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q} \tau \\ &\quad \leq \hat{m} \Vert \zeta \Vert \biggl[ \frac{ 1 }{\varGamma _{q}(\vartheta )} + \frac{ [\Delta _{1} + (1+q)\varGamma _{q} (3-\sigma ) ]( \nu ^{(\vartheta + \kappa )}+1) }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} (\vartheta + \kappa + 1 ) } \\ & \qquad{}+ \frac{ (1+q)\Delta _{2} + \Delta _{4} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } + \frac{ [(1+q)\Delta _{2} + \Delta _{4}] \nu ^{ (\vartheta - \sigma ) } }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) } \biggr] = \hat{m} \Vert \zeta \Vert \varXi _{2}, \end{aligned}$$

and similarly

$$\begin{aligned} &\bigl\vert \bigl(\mathcal{D}_{q}^{2} \mathcal{T}^{(1)}y_{1} + \mathcal{D}_{q}^{2} \mathcal{T}^{(2)}y_{2}\bigr) (t) \bigr\vert \\ &\quad \leq \hat{m} \Vert \zeta \Vert \biggl[ \frac{ 1 }{\varGamma _{q}(\vartheta -1)} + \frac{ (1+q)( \nu ^{(\vartheta + \kappa )}+1) }{\Delta _{3} \varGamma _{q} (\vartheta + \kappa + 1 ) } \\ &\qquad{} + \frac{ (1+q)\Delta _{2} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } + \frac{ (1+q)\Delta _{2} \nu ^{ ( \vartheta - \sigma )} }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) } \biggr] = \hat{m} \Vert \zeta \Vert \varXi _{3}. \end{aligned}$$

Hence, \(\Vert (\mathcal{T}^{(1)}y_{1} + \mathcal{T}^{(2)}y_{2})(t) \Vert \leq \epsilon \) and so \((\mathcal{T}^{(1)}y_{1} + \mathcal{T}^{ (2)}y_{2})(t) \in V_{\epsilon }\). Clearly, the continuity of the function Φ implies the continuity of the fractional operator \(\mathcal{T}^{(1)}\). Also,

$$\begin{aligned} &\bigl\vert \bigl(\mathcal{T}^{(1)}y\bigr) (t) \bigr\vert \leq \int _{0}^{t} \frac{(t-q\tau )^{(\vartheta -1)}}{\varGamma _{q} (\vartheta )} \bigl\vert \varPhi \bigl(\tau,y(\tau ), \mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y( \tau ) \bigr) \bigr\vert \, \mathrm{d}_{q}\tau \\ & \phantom{\bigl\vert \bigl(\mathcal{T}^{(1)}y\bigr) (t) \bigr\vert }\leq \frac{1}{\varGamma _{q} (\vartheta +1)} \Vert \zeta \Vert \psi \bigl( \Vert y \Vert \bigr), \\ &\bigl\vert \bigl(\mathcal{D}_{q}\mathcal{T}^{(1)}y\bigr) (t) \bigr\vert \leq \int _{0}^{t} \frac{(t-q\tau )^{(\vartheta -2)}}{\varGamma _{q} (\vartheta -1)} \bigl\vert \varPhi \bigl(\tau,y(\tau ), \mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y( \tau ) \bigr) \bigr\vert \, \mathrm{d}_{q}\tau \\ & \phantom{\bigl\vert \bigl(\mathcal{D}_{q}\mathcal{T}^{(1)}y\bigr) (t) \bigr\vert }\leq \frac{1}{\varGamma _{q} (\vartheta )} \Vert \zeta \Vert \psi \bigl( \Vert y \Vert \bigr) \end{aligned}$$

and

$$\begin{aligned} \bigl\vert \bigl(\mathcal{D}_{q}^{2} \mathcal{T}^{(1)}y\bigr) (t) \bigr\vert & \leq \int _{0}^{t} \frac{(t-q\tau )^{(\vartheta -3)}}{\varGamma _{q} (\vartheta -2)} \bigl\vert \varPhi \bigl(\tau,y(\tau ), \mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y( \tau ) \bigr) \bigr\vert \, \mathrm{d}_{q}\tau \\ & \leq \frac{1}{\varGamma _{q} (\vartheta -1 )} \Vert \zeta \Vert \psi \bigl( \Vert y \Vert \bigr), \end{aligned}$$

for all \(y\in V_{\epsilon }\). Hence,

$$ \bigl\Vert \mathcal{T}^{(1)} y \bigr\Vert \leq \biggl\{ \frac{1}{\varGamma _{q} (\vartheta +1)} + \frac{1}{\varGamma _{q} (\vartheta )} + \frac{1}{\varGamma _{q} (\vartheta -1 )} \biggr\} \Vert \zeta \Vert \psi \bigl( \Vert y \Vert \bigr). $$

This proves that the operator \(\mathcal{T}^{(1)}\) is uniformly bounded on \(V_{\epsilon }\). Now for checking the compactness of the fractional operator \(\mathcal{T}^{(1)}\) on \(V_{\epsilon }\), assume that \(t_{1},t_{2} \in [0,1]\) with \(t_{1} < t_{2}\). Then we have

$$\begin{aligned} &\bigl\vert \bigl(\mathcal{T}^{(1)}y\bigr) (t_{2}) - \bigl( \mathcal{T}^{(1)}y\bigr) (t_{1}) \bigr\vert \\ &\quad= \biggl\vert \int _{0}^{t_{2}} \frac{(t_{2}-q\tau )^{ (\vartheta -1)}}{\varGamma _{q} (\vartheta )} \varPhi \bigl( \tau, y(\tau ),\mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y( \tau )\bigr) \,\mathrm{d}\tau \\ &\qquad{} - \int _{0}^{t_{1}} \frac{(t_{1}-q\tau )^{(\vartheta -1)}}{\varGamma _{q} (\vartheta )}\varPhi \bigl( \tau,y(\tau ),\mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y(\tau )\bigr) \,\mathrm{d}\tau \biggr\vert \\ &\quad \leq \biggl\vert \int _{0}^{t_{1}} \frac{(t_{2}-q\tau )^{(\vartheta -1)} - (t_{1} -q\tau )^{(\vartheta -1)}}{ \varGamma _{q} (\vartheta )} \varPhi \bigl( \tau,y(\tau ), \mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y( \tau )\bigr) \,\mathrm{d}\tau \biggr\vert \\ &\qquad{} + \biggl\vert \int _{t_{1}}^{t_{2}} \frac{(t_{2}-q\tau )^{(\vartheta -1)}}{\varGamma _{q} (\vartheta )} \varPhi \bigl( \tau,y(\tau ),\mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y( \tau )\bigr) \,\mathrm{d}\tau \biggr\vert \\ & \quad\leq \int _{0}^{t_{1}} \frac{(t_{2}-q\tau )^{(\vartheta -1)} - (t_{1} -q\tau )^{(\vartheta -1)}}{\varGamma _{q} (\vartheta )} \bigl\vert \varPhi \bigl(\tau,y(\tau ),\mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y(\tau )\bigr) \bigr\vert \, \mathrm{d}\tau \\ & \qquad{}+ \int _{t_{1}}^{t_{2}} \frac{(t_{2}-q\tau )^{(\vartheta -1)}}{\varGamma _{q} (\vartheta )} \bigl\vert \varPhi \bigl(\tau,y(\tau ),\mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y(\tau )\bigr) \bigr\vert \, \mathrm{d}\tau \\ & \quad \leq \biggl\{ \frac{t_{2}^{\vartheta }-t_{1}^{\vartheta }- (t_{2}-t_{1})^{\vartheta }}{\varGamma _{q} (\vartheta + 1)} + \frac{(t_{2}-t_{1})^{\vartheta }}{\varGamma _{q} (\vartheta +1)} \biggr\} \Vert \zeta \Vert \psi \bigl( \Vert y \Vert \bigr). \end{aligned}$$

Thus, \(\vert (\mathcal{T}^{(1)}y)(t_{2}) - (\mathcal{T}^{(1)}y)(t_{1}) \vert \to 0\) as \(t_{2} \to t_{1}\). Also, we have

$$\begin{aligned} &\bigl\vert \bigl(\mathcal{D}_{q} \mathcal{T}^{(1)}y \bigr) (t_{2}) - \bigl( \mathcal{D}_{q} \mathcal{T}^{(1)}y\bigr) (t_{1}) \bigr\vert \\ &\quad = \biggl\vert \int _{0}^{t_{2}} \frac{(t_{2}-q\tau )^{(\vartheta -2)}}{\varGamma _{q} (\vartheta -1 )} \varPhi \bigl( \tau,y(\tau ),\mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y( \tau )\bigr) \,\mathrm{d}\tau \\ &\qquad{} - \int _{0}^{t_{1}} \frac{(t_{1}-q\tau )^{(\vartheta -2)}}{\varGamma _{q} (\vartheta -1 )} \varPhi \bigl( \tau,y(\tau ),\mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y( \tau )\bigr) \,\mathrm{d}\tau \biggr\vert \\ & \quad\leq \biggl\vert \int _{0}^{t_{1}} \frac{(t_{2}-q\tau )^{(\vartheta -2)} - (t_{1} -q\tau )^{(\vartheta -2)}}{ \varGamma _{q} (\vartheta -1 )} \varPhi \bigl( \tau,y(\tau ),\mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y( \tau )\bigr) \,\mathrm{d}\tau \biggr\vert \\ & \qquad{}+ \biggl\vert \int _{t_{1}}^{t_{2}} \frac{(t_{2}-q\tau )^{(\vartheta -2)}}{\varGamma _{q} (\vartheta -1 )} \varPhi \bigl( \tau,y(\tau ),\mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y( \tau )\bigr) \,\mathrm{d}\tau \biggr\vert \\ &\quad \leq \int _{0}^{t_{1}} \frac{(t_{2}-q\tau )^{(\vartheta -2)} - (t_{1} -q\tau )^{(\vartheta -2)}}{\varGamma _{q} (\vartheta -1 )} \bigl\vert \varPhi \bigl(\tau,y(\tau ),\mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y(\tau )\bigr) \bigr\vert \, \mathrm{d}\tau \\ &\qquad{} + \int _{t_{1}}^{t_{2}} \frac{(t_{2}-q\tau )^{(\vartheta -2)}}{\varGamma _{q} (\vartheta -1 )} \bigl\vert \varPhi \bigl(\tau,y(\tau ),\mathcal{D}_{q}y(\tau ), \mathcal{D}_{q}^{2} y(\tau )\bigr) \bigr\vert \, \mathrm{d}\tau \\ & \quad \leq \biggl\{ \frac{t_{2}^{(\vartheta -1)} -t_{1}^{(\vartheta -1)} - (t_{2}-t_{1})^{(\vartheta -1)} }{\varGamma _{q} (\vartheta )} + \frac{(t_{2}-t_{1})^{(\vartheta -1)} }{\varGamma _{q} (\vartheta )} \biggr\} \Vert \zeta \Vert \psi \bigl( \Vert y \Vert \bigr) \end{aligned}$$

and so \(\vert (\mathcal{D}_{q} \mathcal{T}^{(1)}y)(t_{2}) - ( \mathcal{D}_{q} \mathcal{T}^{(1)}y)(t_{1})\vert \to 0\) as \(t_{2} \to t_{1}\). Similarly, we can show that

$$ \bigl\vert \bigl(\mathcal{D}_{q}^{2} \mathcal{T}^{(1)}y\bigr) (t_{2}) - \bigl( \mathcal{D}_{q}^{2} \mathcal{T}^{(1)}y\bigr) (t_{1}) \bigr\vert \to 0, $$

as \(t_{2} \to t_{1}\) Hence, \(\Vert ( \mathcal{T}^{(1)}y) (t_{2}) - (\mathcal{T}^{(1)}y)(t_{1}) \Vert \) tends to zero as \(t_{2} \to t_{1}\). Thus, \(\mathcal{T}^{(1)}\) is equi-continuous and so \(\mathcal{T}^{(1)}\) is relatively compact on \(V_{\epsilon }\). Now by using the Arzela–Ascoli theorem, the fractional operator \(\mathcal{T}^{(1)} \) is compact on \(V_{\epsilon }\). Finally, we prove that \(\mathcal{T}^{(2)} \) is a contraction map. Let \(y_{1},y_{2}\in V_{\epsilon }\). Then we have

$$\begin{aligned} &\bigl\vert \bigl(\mathcal{T}^{(2)} y_{1} \bigr) (t) - \bigl(\mathcal{T}^{(2)} y_{2} \bigr) (t) \bigr\vert \\ &\quad \leq \frac{ \vert t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma ) \vert }{\Delta _{3} \varGamma _{q} (3-\sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \\ & \qquad{}\times L(\tau ) \bigl( \bigl\vert y_{1}(\tau )-y_{2} ( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}(\tau ) - \mathcal{D}_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}(\tau ) - \mathcal{D}_{q}^{2} y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau \\ &\qquad{} + \frac{ \vert t^{2} \Delta _{2} - t \Delta _{4} \vert }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \\ &\qquad{} \times L(\tau ) \bigl( \bigl\vert y_{1}(\tau )-y_{2} ( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}(\tau ) - \mathcal{D}_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}(\tau ) - \mathcal{D}_{q}^{2} y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau \\ & \qquad{}+ \frac{ \vert t^{2} \Delta _{2} - t \Delta _{4} \vert }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \\ & \qquad{}\times L(\tau ) \bigl( \bigl\vert y_{1}(\tau )-y_{2} ( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}(\tau ) - \mathcal{D}_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}(\tau ) - \mathcal{D}_{q}^{2} y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau \\ &\qquad{} + \frac{ \vert t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) \vert }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \\ &\qquad{} \times L(\tau ) \bigl( \bigl\vert y_{1}(\tau )-y_{2} ( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}(\tau ) - \mathcal{D}_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}(\tau ) - \mathcal{D}_{q}^{2} y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau. \end{aligned}$$

Also

$$\begin{aligned} &\bigl\vert \bigl( \mathcal{D}_{q} \mathcal{T}^{(2)} y_{1} \bigr) (t) - \bigl( \mathcal{D}_{q} \mathcal{T}^{(2)} y_{2} \bigr) (t) \bigr\vert \\ &\quad \leq \frac{ \vert \Delta _{1} - (1+q)t \varGamma _{q} (3-\sigma ) \vert }{\Delta _{3} \varGamma _{q} (3-\sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \\ & \qquad{}\times L(\tau ) \bigl( \bigl\vert y_{1}(\tau )-y_{2} ( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}(\tau ) - \mathcal{D}_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}(\tau ) - \mathcal{D}_{q}^{2} y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau \\ & \qquad{}+ \frac{ \vert (1+q)t \Delta _{2} - \Delta _{4} \vert }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \\ &\qquad{} \times L(\tau ) \bigl( \bigl\vert y_{1}(\tau )-y_{2} ( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}(\tau ) - \mathcal{D}_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}(\tau ) - \mathcal{D}_{q}^{2} y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau \\ &\qquad{} + \frac{ \vert (1+q)t \Delta _{2} - \Delta _{4} \vert }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \\ & \qquad{}\times L(\tau ) \bigl( \bigl\vert y_{1}(\tau )-y_{2} ( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}(\tau ) - \mathcal{D}_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}(\tau ) - \mathcal{D}_{q}^{2} y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau \\ & \qquad{}+ \frac{ \vert \Delta _{1} - (1+q)t \varGamma _{q} (3-\sigma ) \vert }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \\ & \qquad{}\times L(\tau ) \bigl( \bigl\vert y_{1}(\tau )-y_{2} ( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}(\tau ) - \mathcal{D}_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}(\tau ) - \mathcal{D}_{q}^{2} y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau \end{aligned}$$

and

$$\begin{aligned} &\bigl\vert \bigl( \mathcal{D}_{q}^{2} \mathcal{T}^{(2)} y_{1} \bigr) (t) - \bigl( \mathcal{D}_{q}^{2} \mathcal{T}^{(2)} y_{2} \bigr) (t) \bigr\vert \\ &\quad \leq \frac{ (1+q) \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \\ & \qquad{}\times L(\tau ) \bigl( \bigl\vert y_{1}(\tau )-y_{2} ( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}(\tau ) - \mathcal{D}_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}(\tau ) - \mathcal{D}_{q}^{2} y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau \\ &\qquad{} + \frac{ (1+q) \Delta _{2} }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \\ &\qquad{} \times L(\tau ) \bigl( \bigl\vert y_{1}(\tau )-y_{2} ( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}(\tau ) - \mathcal{D}_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}(\tau ) - \mathcal{D}_{q}^{2} y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau \\ &\qquad{} + \frac{ (1+q) \Delta _{2} }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \\ & \qquad{}\times L(\tau ) \bigl( \bigl\vert y_{1}(\tau )-y_{2} ( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}(\tau ) - \mathcal{D}_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}(\tau ) - \mathcal{D}_{q}^{2} y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau \\ &\qquad{} + \frac{ (1+q) \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \\ & \qquad{}\times L(\tau ) \bigl( \bigl\vert y_{1}(\tau )-y_{2} ( \tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}y_{1}(\tau ) - \mathcal{D}_{q}y_{2}(\tau ) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y_{1}(\tau ) - \mathcal{D}_{q}^{2} y_{2}( \tau ) \bigr\vert \bigr) \,\mathrm{d}_{q}\tau. \end{aligned}$$

Hence, we get

$$\begin{aligned} &\sup_{t\in [0,1]} \bigl\vert \bigl(\mathcal{T}^{(2)} y_{1} \bigr) (t) - \bigl( \mathcal{T}^{(2)} y_{2} \bigr) (t) \bigr\vert \leq \Vert L \Vert \Delta ^{(1)} \Vert y_{1}-y_{2} \Vert , \\ &\sup_{t\in [0,1]} \bigl\vert \bigl( \mathcal{D}_{q} \mathcal{T}^{(2)} y_{1} \bigr) (t) - \bigl( \mathcal{D}_{q} \mathcal{T}^{(2)} y_{2} \bigr) (t) \bigr\vert \leq \Vert L \Vert \Delta ^{(2)} \Vert y_{1}-y_{2} \Vert , \\ &\sup_{t\in [0,1]} \bigl\vert \bigl( \mathcal{D}_{q}^{2} \mathcal{T}^{(2)} y_{1} \bigr) (t) - \bigl( \mathcal{D}_{q}^{2} \mathcal{T}^{(2)} y_{2} \bigr) (t) \bigr\vert \leq \Vert L \Vert \Delta ^{(3)} \Vert y_{1}-y_{2} \Vert . \end{aligned}$$

Thus, \(\Vert \mathcal{T}^{(2)} y_{1} - \mathcal{T}^{(2)} y_{2} \Vert \leq \Vert L\Vert (\Delta ^{(1)} + \Delta ^{(2)} + \Delta ^{(3)})\Vert y_{1}-y_{2} \Vert \) or \(\Vert \mathcal{T}^{(2)} y_{1} - \mathcal{T}^{(2)} y_{2} \Vert \leq K \Vert y_{1}-y_{2}\Vert \). Hence, \(\mathcal{T}^{(2)}\) is a contraction on \(V_{\epsilon }\) with constant \(K<1\). Now by using Theorem 4, the q-fractional boundary value problem (1)–(2) has at least one solution. □

Now we investigate the existence of solutions for the fractional q-difference inclusion problem (3)–(4). A function \(y\in C_{\mathcal{X}}([0,1], \mathcal{X})\) is called a solution for the fractional q-difference inclusion (3) whenever it satisfies the boundary conditions and there exists a function \(\varTheta \in L^{1}([0,1] )\) such that \(\varTheta (t)\in \mathcal{G}(t, y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2}y(t))\) for almost all \(t\in [0,1] \) and

$$\begin{aligned} y(t) ={}& \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{\varGamma _{q}(\vartheta )} \varTheta ( \tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \varTheta (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \varTheta (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \varTheta (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varTheta (\tau ) \,\mathrm{d}_{q}\tau \end{aligned}$$

for all \(t\in [0,1] \). For each \(y\in \mathcal{X}\), define the set of selections of the operator \(\mathcal{G}\) by

$$ \mathcal{S}_{\mathcal{G},y}= \bigl\{ \varTheta \in L^{1}\bigl([0,1] \bigr): \varTheta (t) \in \mathcal{G}\bigl(t, y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2}y(t) \bigr) \text{ for all } t\in [0,1] \bigr\} . $$

Also, consider the operator \(\mathcal{N}: \mathcal{X}\to \mathcal{P}(\mathcal{X})\) defined by

$$ \mathcal{N}(y)= \bigl\{ p\in \mathcal{X}: \text{ there exists } \varTheta \in \mathcal{S}_{ \mathcal{G},y}: p(t) = w(t) \text{ for all } t\in [0,1] \bigr\} , $$
(13)

where

$$\begin{aligned} w(t) ={}& \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{\varGamma _{q}(\vartheta )} \varTheta ( \tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \varTheta (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \varTheta (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \varTheta (\tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varTheta (\tau ) \,\mathrm{d}_{q}\tau. \end{aligned}$$

Theorem 10

Let\(\mathcal{G}: [0,1] \times \mathcal{X} \times \mathcal{X} \times \mathcal{X} \to \mathcal{P}_{cp}(\mathcal{X})\)be a set-valued map. Suppose that:

\((H6)\):

The operator\(\mathcal{G}\)is integrable bounded and\(\mathcal{G}(\cdot, y_{1}, y_{2}, y_{3}): [0,1] \to \mathcal{P}_{cp} ( \mathcal{X})\)is measurable for all\(y_{1},y_{2}, y_{3}\in \mathcal{X}\).

\((H7)\):

There exist\(m\in C([0,1], [0,\infty ))\)and\(\psi \in \varPsi \)such that

$$\begin{aligned} &\mathcal{H}_{d} \bigl(\mathcal{G} (t,y_{1},y_{2}, y_{3}), \mathcal{G} \bigl(t,y'_{1}, y'_{2}, y'_{3} \bigr) \bigr) \\ &\quad\leq \frac{m(t) \lambda }{ \Vert m \Vert } \psi \bigl( \bigl\vert y_{1}-y'_{1} \bigr\vert + \bigl\vert y_{2} - y'_{2} \bigr\vert + \bigl\vert y_{3} -y'_{3} \bigr\vert \bigr), \end{aligned}$$
(14)

for all\(t\in [0,1] \)and\(y_{1}, y_{2}, y_{3},y_{1}', y_{2}', y_{3}' \in \mathcal{X}\), where\(\lambda = \frac{1}{\varXi _{1} + \varXi _{2} + \varXi _{3}}\)and the constants\(\varXi _{1} \), \(\varXi _{2} \)and\(\varXi _{3} \)are given by (8).

\((H8)\):

There exists a function\(\chi: \mathbb{R}^{3} \times \mathbb{R}^{3} \to \mathbb{R}\)such that\(\chi ( ( y_{1}, y_{2}, y_{3}), (y_{1},y'_{2}, y'_{3}) ) \geq 0\)for all\(y_{1}, y_{2}, y_{3},y_{1}', y_{2}', y_{3}' \in \mathcal{X}\).

\((H9)\):

If\(\{ y_{n} \}_{n\geq 1}\)is a sequence in\(\mathcal{X}\)with\(y_{n} \to y\)and

$$ \chi \bigl( \bigl(y_{n}(t),\mathcal{D}_{q}y_{n}(t), \mathcal{D}_{q}^{2}y_{n} (t)\bigr), \bigl(y_{n+1}(t),\mathcal{D}_{q}y_{n+1}(t), \mathcal{D}_{q}^{2}y_{n+1} (t)\bigr) \bigr) \geq 0, $$

for all\(t\in [0,1]\)and\(n\geq 1\), then there exists a subsequence\(\{ y_{n_{j}}\}_{j\geq 1} \)of\(\{ y_{n} \} \)such that

$$ \chi \bigl( \bigl(y_{n_{j}}(t), \mathcal{D}_{q}y_{n_{j}} (t), \mathcal{D}_{q}^{2}y_{n_{j}} (t) \bigr), \bigl(y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2}y(t) \bigr) \bigr) \geq 0, $$

for all\(t\in [0,1] \)and\(j\geq 1\).

\((H10)\):

There exist\(y_{0}\in \mathcal{X}\)and\(p\in \mathcal{N}(y_{0})\)such that

$$ \chi \bigl( \bigl(y_{0}(t),\mathcal{D}_{q}y_{0} (t), \mathcal{D}_{q}^{2}y_{0} (t) \bigr), \bigl(p(t), \mathcal{D}_{q}p(t), \mathcal{D}_{q}^{2}p(t) \bigr) \bigr) \geq 0, $$

for all\(t\in [0,1]\), where the operator\(\mathcal{N}:\mathcal{X} \to \mathcal{P}(\mathcal{X})\)is given by (13).

\((H11)\):

For each\(y\in \mathcal{X}\)and\(p\in \mathcal{N}(y)\)with

$$ \chi \bigl( \bigl(y(t),\mathcal{D}_{q}y (t), \mathcal{D}_{q}^{2}y (t)\bigr),\bigl(p(t), \mathcal{D}_{q}p(t), \mathcal{D}_{q}^{2} p(t)\bigr) \bigr) \geq 0, $$

there exists\(w\in \mathcal{N}(y)\)such that

$$ \chi \bigl( \bigl(p(t),\mathcal{D}_{q}p(t), \mathcal{D}_{q}^{2}p(t) \bigr),\bigl(w(t), \mathcal{D}_{q}w(t), \mathcal{D}_{q}^{2}w(t) \bigr) \bigr) \geq 0, $$
for all\(t\in [0,1]\). Then theq-fractional inclusion problem (3)(4) has a solution.

Proof

It is clear that each fixed point of the operator \(\mathcal{N}: \mathcal{X} \to \mathcal{P}(\mathcal{X})\) is a solution for the q-fractional inclusion problem (3). Since the multivalued map \(t\mapsto \mathcal{G} (t, y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2} y(t))\) is measurable and it is closed-valued for all \(y\in \mathcal{X}\), \(\mathcal{G}\) has a measurable selection and the set \(\mathcal{S}_{\mathcal{G},y}\) is not empty. We show that the subset \(\mathcal{N}(y)\) of \(\mathcal{X}\) is closed for all \(y\in \mathcal{X}\). Let \(\{ y_{n} \}_{n\geq 1}\) be a sequence in \(\mathcal{N}(y)\) converging to y. For each n, there exists \(\varTheta _{n} \in \mathcal{S}_{\mathcal{G},y}\) such that

$$\begin{aligned} y_{n}(t) ={}& \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{\varGamma _{q}(\vartheta )} \varTheta _{n} (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \varTheta _{n} (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \varTheta _{n} (\tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \varTheta _{n} (\tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varTheta _{n} (\tau ) \,\mathrm{d}_{q}\tau, \end{aligned}$$

for almost all \(t\in [0,1]\). Since \(\mathcal{G}\) has compact values, we pass into a subsequence (if necessary) to find that a subsequence \(\{ \varTheta _{n} \}_{n\geq 1}\) that converges to some \(\varTheta \in L^{1}([0,1] )\). Hence, \(\varTheta \in \mathcal{S}_{\mathcal{G},y}\) and

$$\begin{aligned} y_{n}(t) \to{}& y(t) \\ ={}& \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{\varGamma _{q}(\vartheta )} \varTheta ( \tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \varTheta (\tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \varTheta (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \varTheta (\tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varTheta (\tau ) \,\mathrm{d}_{q}\tau \end{aligned}$$

for all \(t\in [0,1]\). This shows that \(y\in \mathcal{N}(y)\) and so the operator \(\mathcal{N}\) is closed-valued. Since \(\mathcal{G}\) is a multifunction with compact values, it is easy to prove that \(\mathcal{N}(y)\) is bounded for all \(y\in \mathcal{X}\). Now, we show that the operator \(\mathcal{N}\) is an α-ψ-contractive set-valued map. For this purpose, define the non-negative function α on \(\mathcal{X} \times \mathcal{X} \) by

α(y,y)={1if χ((y(t),Dqy(t),Dq2y(t)),(y(t),Dqy(t),Dq2y(t)))0,0otherwise,

for all \(y,y'\in \mathcal{X}\). Let \(y,y'\in \mathcal{X}\) and \(p_{1} \in \mathcal{N}(y')\). Choose \(\varTheta _{1} \in \mathcal{S}_{\mathcal{G}, y'}\) such that

$$\begin{aligned} p_{1}(t) ={}& \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{\varGamma _{q}(\vartheta )} \varTheta _{1} (\tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \varTheta _{1} (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \varTheta _{1} (\tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \varTheta _{1} (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varTheta _{1} (\tau ) \,\mathrm{d}_{q}\tau, \end{aligned}$$

for all \(t\in [0,1] \). By using (14), we have

$$\begin{aligned} &\mathcal{H}_{d} \bigl(\mathcal{G} \bigl(t, y, \mathcal{D}_{q}y, \mathcal{D}_{q}^{2}y \bigr), \mathcal{G} \bigl(t,y',\mathcal{D}_{q}y', \mathcal{D}_{q}^{2}y' \bigr) \bigr) \\ & \quad \leq \frac{m(t)\lambda }{ \Vert m \Vert } \psi \bigl( \bigl\vert y-y' \bigr\vert + \bigl\vert \mathcal{D}_{q}y - \mathcal{D}_{q}y' \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y - \mathcal{D}_{q}^{2}y' \bigr\vert \bigr), \end{aligned}$$

for all \(y,y'\in \mathcal{X}\) with

$$ \chi \bigl( \bigl( y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2}y(t) \bigr), \bigl( y'(t) , \mathcal{D}_{q}y'(t), \mathcal{D}_{q}^{2} y'(t)\bigr) \bigr) \geq 0, $$

for almost all \(t\in [0,1] \). Thus, there exists \(w\in \mathcal{G}(t, y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2}y(t))\) such that

$$ \bigl\vert \varTheta _{1}(t)-w \bigr\vert \leq \frac{m(t) \lambda }{ \Vert m \Vert } \psi \bigl( \bigl\vert y(t)-y'(t) \bigr\vert + \bigl\vert \mathcal{D}_{q}y(t) - \mathcal{D}_{q}y'(t) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y(t) - \mathcal{D}_{q}^{2}y'(t) \bigr\vert \bigr). $$

Now, consider the multivalued map \(B: [0,1] \to \mathcal{P}(\mathcal{X})\) defined by

$$\begin{aligned} B(t) ={}& \biggl\{ w\in \mathcal{X}: \bigl\vert \varTheta _{1}(t)-w \bigr\vert \leq \frac{m(t) \lambda }{ \Vert m \Vert } \\ & {}\times \psi \bigl( \bigl\vert y(t)-y'(t) \bigr\vert + \bigl\vert \mathcal{D}_{q}y(t) - \mathcal{D}_{q}y'(t) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y(t) - \mathcal{D}_{q}^{2}y'(t) \bigr\vert \bigr) \biggr\} \end{aligned}$$

for all \(t\in [0,1] \). Since \(\varTheta _{1}\) and

$$ \varphi = \frac{m \lambda }{ \Vert m \Vert } \psi \bigl( \bigl\vert y-y' \bigr\vert + \bigl\vert \mathcal{D}_{q}y - \mathcal{D}_{q}y' \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y - \mathcal{D}_{q}^{2}y' \bigr\vert \bigr) $$

are measurable, the multifunction \(B(\cdot ) \cap \mathcal{G}(\cdot, y( \cdot ), \mathcal{D}_{q}y( \cdot ), \mathcal{D}_{q}^{2}y( \cdot ) )\) is measurable. Now, select \(\varTheta _{2} \) in \(\mathcal{G}(t,y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2}y(t))\) such that

$$ \bigl\vert \varTheta _{1}(t)-\varTheta _{2}(t) \bigr\vert \leq \frac{m(t) \lambda }{ \Vert m \Vert } \psi \bigl( \bigl\vert y(t)-y'(t) \bigr\vert + \bigl\vert \mathcal{D}_{q}y(t) - \mathcal{D}_{q}y'(t) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y(t) - \mathcal{D}_{q}^{2}y'(t) \bigr\vert \bigr) $$

for all \(t\in [0,1] \). Define \(p_{2} \in \mathcal{N}(u)\) by

$$\begin{aligned} p_{2}(t) ={}& \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{\varGamma _{q}(\vartheta )} \varTheta _{2} (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \varTheta _{2} (\tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \varTheta _{2} (\tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \varTheta _{2} (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varTheta _{2} (\tau ) \,\mathrm{d}_{q}\tau \end{aligned}$$

for all \(t\in [0,1] \). Let \(\sup_{t\in [0,1]}\vert m(t)\vert = \Vert m\Vert \). Then we have

$$\begin{aligned} \vert p_{1} - p_{2} \vert \leq{}& \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{\varGamma _{q}(\vartheta )} \bigl\vert \varTheta _{1} (\tau ) -\varTheta _{2} (\tau ) \bigr\vert \,\mathrm{d}_{q}\tau \\ &{} + \frac{ \vert t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma ) \vert }{\Delta _{3} \varGamma _{q} (3-\sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \bigl\vert \varTheta _{1} (\tau ) -\varTheta _{2} (\tau ) \bigr\vert \,\mathrm{d}_{q} \tau \\ & {}+ \frac{ \vert t^{2} \Delta _{2} - t \Delta _{4} \vert }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \bigl\vert \varTheta _{1} (\tau ) -\varTheta _{2} (\tau ) \bigr\vert \,\mathrm{d}_{q} \tau \\ &{} + \frac{ \vert t^{2} \Delta _{2} - t \Delta _{4} \vert }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \bigl\vert \varTheta _{1} (\tau ) -\varTheta _{2} (\tau ) \bigr\vert \,\mathrm{d}_{q} \tau \\ & {}+ \frac{ \vert t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) \vert }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \bigl\vert \varTheta _{1} (\tau ) -\varTheta _{2} (\tau ) \bigr\vert \,\mathrm{d}_{q} \tau \\ \leq{}& \frac{1}{\varGamma _{q} (\vartheta +1 )} \Vert m \Vert \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr) \biggl(\frac{p}{ \Vert m \Vert } \biggr) \\ & {}+ \frac{ \Delta _{1} + \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} (\vartheta + \kappa +1)} \Vert m \Vert \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr) \biggl( \frac{\lambda }{ \Vert m \Vert } \biggr) \\ & {}+ \frac{ \Delta _{2} + \Delta _{4} }{ \Delta _{5} \varGamma _{q}( \vartheta -1) } \Vert m \Vert \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr) \biggl( \frac{\lambda }{ \Vert m \Vert } \biggr) \\ &{} + \frac{ [\Delta _{2} + \Delta _{4}] \nu ^{ (\vartheta - \sigma ) } }{\Delta _{5} \varGamma _{q} (\vartheta -1) } \Vert m \Vert \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr) \biggl( \frac{\lambda }{ \Vert m \Vert } \biggr) \\ &{} + \frac{ [ \Delta _{1} + \varGamma _{q} (3-\sigma ) ] \nu ^{(\vartheta + \kappa ) } }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} ( \vartheta + \kappa +1) } \Vert m \Vert \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr) \biggl( \frac{\lambda }{ \Vert m \Vert } \biggr) \\ ={}& \biggl[ \frac{ 1 }{\varGamma _{q}(\vartheta +1)} + \frac{ (\Delta _{1} + \varGamma _{q} (3-\sigma )) (\nu ^{(\vartheta + \kappa ) } +1) }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} (\vartheta + \kappa + 1 ) } + \frac{ \Delta _{2} + \Delta _{4} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } \\ & {}+ \frac{ (\Delta _{2} + \Delta _{4}) \nu ^{(\vartheta - \sigma )} }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) } \biggr] \Vert m \Vert \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr) \biggl( \frac{\lambda }{ \Vert m \Vert } \biggr) \\ ={}& \varXi _{1} \Vert m \Vert \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr) \biggl( \frac{\lambda }{ \Vert m \Vert } \biggr) \\ = {}&\lambda \varXi _{1} \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr). \end{aligned}$$

Also, we have

$$\begin{aligned} \vert \mathcal{D}_{q}p_{1} - \mathcal{D}_{q}p_{2} \vert \leq{}& \biggl[ \frac{ 1 }{\varGamma _{q}(\vartheta )} + \frac{ [\Delta _{1} + (1+q)\varGamma _{q} (3-\sigma ) ]( \nu ^{(\vartheta + \kappa )}+1) }{\Delta _{3} \varGamma _{q} (3-\sigma ) \varGamma _{q} (\vartheta + \kappa + 1 ) } \\ &{} + \frac{ (1+q)\Delta _{2} + \Delta _{4} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } + \frac{ [(1+q)\Delta _{2} + \Delta _{4}] \nu ^{(\vartheta - \sigma ) } }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) } \biggr] \\ &{} \times \Vert m \Vert \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr) \biggl( \frac{\lambda }{ \Vert m \Vert } \biggr) \\ ={}& \varXi _{2} \Vert m \Vert \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr) \biggl( \frac{\lambda }{ \Vert m \Vert } \biggr) \\ = {}&\lambda \varXi _{2} \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr) \end{aligned}$$

and

$$\begin{aligned} \bigl\vert \mathcal{D}_{q}^{2} p_{1} - \mathcal{D}_{q}^{2} p_{2} \bigr\vert \leq{}& \biggl[ \frac{ 1 }{\varGamma _{q}(\vartheta -1)} + \frac{ (1+q)( \nu ^{(\vartheta + \kappa )}+1) }{\Delta _{3} \varGamma _{q} (\vartheta + \kappa + 1 ) } \\ &{} + \frac{ (1+q)\Delta _{2} }{\Delta _{5} \varGamma _{q} ( \vartheta -1 ) } + \frac{ (1+q)\Delta _{2} \nu ^{ ( \vartheta - \sigma )} }{\Delta _{5} \varGamma _{q} (\vartheta - \sigma +1 ) } \biggr] \\ &{} \times \Vert m \Vert \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr) \biggl( \frac{\lambda }{ \Vert m \Vert } \biggr) \\ = {}&\varXi _{3} \Vert m \Vert \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr) \biggl( \frac{\lambda }{ \Vert m \Vert } \biggr) \\ = {}&\lambda \varXi _{3} \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr), \end{aligned}$$

for all \(t\in [0,1] \). Hence, we obtain

$$\begin{aligned} \Vert p_{1} - p_{2} \Vert ={}& \sup_{t\in [0,1] } \bigl\vert p_{1} (t) -p_{2} (t) \bigr\vert + \sup _{t\in [0,1] } \bigl\vert \mathcal{D}_{q} p_{1}(t) - \mathcal{D}_{q} p_{2} (t) \bigr\vert \\ &{} + \sup_{t\in [0,1] } \bigl\vert \mathcal{D}_{q}^{2} p_{1}(t) - \mathcal{D}_{q}^{2} p_{2} (t) \bigr\vert \\ \leq{}& (\varXi _{1} + \varXi _{2} + \varXi _{3} ) \lambda \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr) = \psi \bigl( \bigl\Vert y-y' \bigr\Vert \bigr). \end{aligned}$$

Thus, \(\alpha (y,y') \mathcal{H}_{d} (\mathcal{N}(y),\mathcal{N}(y')) \leq \psi (\Vert y-y' \Vert ) \) holds for all \(y,y'\in \mathcal{X}\) which means that \(\mathcal{N}\) is an α-ψ-contractive set-valued map. Now, consider two functions \(y\in \mathcal{X}\) and \(y'\in \mathcal{N}(y)\) such that \(\alpha (y,y')\geq 1\). In this case,

$$ \chi \bigl( \bigl(y(t),\mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2}y(t) \bigr), \bigl(y'(t), \mathcal{D}_{q}y'(t), \mathcal{D}_{q}^{2}y'(t)\bigr) \bigr) \geq 0, $$

so there exists a function \(w\in \mathcal{N}(y')\) such that

$$ \chi \bigl( \bigl(y'(t),\mathcal{D}_{q}y'(t), \mathcal{D}_{q}^{2}y'(t)\bigr), \bigl(w(t), \mathcal{D}_{q}w(t), \mathcal{D}_{q}^{2}w(t) \bigr) \bigr) \geq 0. $$

It follows that \(\alpha (y', w) \geq 1\) and so the operator \(\mathcal{N}\) is α-admissible. Now, let \(y_{0}\in \mathcal{X}\) and \(y'\in \mathcal{N}(y_{0})\) be such that

$$ \chi \bigl( \bigl(y_{0}(t),\mathcal{D}_{q}y_{0}(t), \mathcal{D}_{q}^{2} y_{0}(t)\bigr) , \bigl(y'(t), \mathcal{D}_{q} y'(t), \mathcal{D}_{q}^{2} y'(t)\bigr) \bigr) \geq 0, $$

for all t. Then we have \(\alpha (y_{0}, y') \geq 1\). Suppose that \(\{ y_{n} \}_{n\geq 1}\) is a sequence in \(\mathcal{X}\) with \(y_{n} \to y\) and \(\alpha (y_{n}, y_{n+1}) \geq 1\) for all n. Then we get

$$ \chi \bigl( \bigl(y_{n}(t),\mathcal{D}_{q}y_{n}(t), \mathcal{D}_{q}^{2} y_{n}(t)\bigr), \bigl(y_{n+1}(t),\mathcal{D}_{q}y_{n+1}(t), \mathcal{D}_{q}^{2}y_{n+1}(t)\bigr) \bigr) \geq 0. $$

By using (H9), there exists a subsequence \(\{ y_{n_{j}}\}_{j\geq 1} \) of \(\{ y_{n} \} \) such that

$$ \chi \bigl( \bigl(y_{n_{j}}(t), \mathcal{D}_{q}y_{n_{j}} (t), \mathcal{D}_{q}^{2}y_{n_{j}} (t) \bigr), \bigl(y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2}y(t) \bigr) \bigr) \geq 0 $$

for all \(t\in [0,1] \). Thus, \(\alpha (y_{n_{j}}, y) \geq 1\) for all j. This means that the Banach space \(\mathcal{X}\) has the property \((C_{\alpha })\). Now by using Theorem 5, the map \(\mathcal{N} \) has a fixed point which is a solution for the q-fractional inclusion problem (3)–(4). □

Theorem 11

Let\(\mathcal{G}: [0,1] \times \mathcal{X} \times \mathcal{X} \times \mathcal{X} \to \mathcal{P}_{cp}(\mathcal{X})\)be a set-valued map. Suppose that:

\((H12)\):

The non-negative function\(\psi: [0,\infty )\to [0,\infty )\)is nondecreasing upper semi-continuous map such that\(\lim \inf_{t\to \infty }(t-\psi (t))>0\)and\(\psi (t)< t\)for all\(t>0\).

\((H13)\):

The operator\(\mathcal{G}: [0,1]\times \mathcal{X}^{3} \to {\mathcal{P}}_{cp}( \mathcal{X})\)is an integrable bounded multifunction such that\(\mathcal{G}( \cdot, y_{1}, y_{2}, y_{3} ): [0,1] \to {\mathcal{P}}_{cp}( \mathcal{X})\)is measurable for all\(y_{1}, y_{2}, y_{3} \in \mathcal{X}\).

\((H14)\):

There exists a non-negative function\(m\in C([0,1], [0,\infty ))\)such that

$$\begin{aligned} &\mathcal{H}_{d} \bigl(\mathcal{G}(t, y_{1}, y_{2}, y_{3} ) - \mathcal{G} \bigl(t,y'_{1}, y'_{2}, y'_{3} \bigr) \bigr) \\ &\quad \leq m(t) \lambda \psi \bigl( \bigl\vert y_{1}-y'_{1} \bigr\vert + \bigl\vert y_{2}-y'_{2} \bigr\vert + \bigl\vert y_{3}-y'_{3} \bigr\vert \bigr) \end{aligned}$$

for all\(t\in [0,1]\)and\(y_{1}, y_{2}, y_{3}, y'_{1}, y'_{2}, y'_{3} \in \mathcal{X}\), where\(\lambda = \frac{1}{\varSigma _{1} +\varSigma _{2} + \varSigma _{3} }\)and the\(\varSigma _{i}\)are given by (10).

\((H15)\):

The operator\(\mathcal{N}\)has the approximate endpoint property where\(\mathcal{N}\)is defined by (13).

Then theq-fractional inclusion problem (3)(4) has a solution.

Proof

We show that the set-valued map \(\mathcal{N}: \mathcal{X}\to {\mathcal{P}}(\mathcal{X})\) has an endpoint. First, we prove that \(\mathcal{N}(y)\) is closed for all \(y\in \mathcal{X}\). Since the map \(t\mapsto \mathcal{G}(t,y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2}y(t) ) \) is measurable and has closed values for all \(y\in \mathcal{X}\), it has a measurable selection and so \(\mathcal{S}_{\mathcal{G}, y} \neq \emptyset \) for all \(y\in \mathcal{X}\). Similar to the proof of Theorem 10, we can show that the operator \(\mathcal{N}(y)\) has closed values. Also, \(\mathcal{N}(y)\) is a bounded set for all \(y\in \mathcal{X}\) because \(\mathcal{G}\) is a compact multivalued map. Finally, one can show that \(\mathcal{H}_{d} (\mathcal{N}(y), \mathcal{N}(w))\leq \psi (\Vert y-w \Vert )\) holds. Let \(y,w\in \mathcal{X}\) and \(p_{1} \in \mathcal{N}(w)\). Choose \(\varTheta _{1} \in \mathcal{S}_{\mathcal{G}, w}\) such that

$$\begin{aligned} p_{1}(t)={}& \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{\varGamma _{q}(\vartheta )} \varTheta _{1} (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \varTheta _{1} (\tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \varTheta _{1} (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \varTheta _{1} (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varTheta _{1} (\tau ) \,\mathrm{d}_{q}\tau \end{aligned}$$

for almost all \(t\in [0,1]\). Since

$$\begin{aligned} &\mathcal{H}_{d} \bigl(\mathcal{G}\bigl(t,y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2}y(t) \bigr) - \mathcal{G}\bigl(t,w(t), \mathcal{D}_{q}w(t), \mathcal{D}_{q}^{2}w(t)\bigr) \bigr) \\ & \quad \leq m(t) \lambda \psi \bigl( \bigl\vert y(t)-w(t) \bigr\vert + \bigl\vert \mathcal{D}_{q}y(t) - \mathcal{D}_{q}w(t) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y(t) - \mathcal{D}_{q}^{2}w(t) \bigr\vert \bigr) \end{aligned}$$

for all \(t\in [0,1]\), there exists \(\bar{\sigma } \in \mathcal{G}(t,y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2}y(t) ) \) such that

$$\begin{aligned} \bigl\vert \varTheta _{1}(t) - \bar{\sigma } \bigr\vert & \leq m(t) \lambda \psi \bigl( \bigl\vert y(t) - w(t) \bigr\vert + \bigl\vert \mathcal{D}_{q}y(t) - \mathcal{D}_{q} w(t) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y(t) - \mathcal{D}_{q}^{2}w(t) \bigr\vert \bigr) \end{aligned}$$

for all \(t\in [0,1]\). Now, consider the multivalued map \(O: [0,1]\to {\mathcal{P}}(\mathcal{X})\) which is defined by

$$\begin{aligned} O(t) = {}&\bigl\{ \bar{\sigma } \in \mathcal{X}: \bigl\vert \varTheta _{1}(t)- \bar{\sigma } \bigr\vert \leq m(t) \lambda \psi \bigl( \bigl\vert y(t)-w(t) \bigr\vert \\ &{} + \bigl\vert \mathcal{D}_{q}y(t) - \mathcal{D}_{q}w(t) \bigr\vert + \bigl\vert \mathcal{D}_{q}^{2}y(t) - \mathcal{D}_{q}^{2}w(t) \bigr\vert \bigr) \bigr\} . \end{aligned}$$

By the measurability of \(\varTheta _{1}\) and \(\varphi = m \lambda \psi ( \vert y-w\vert + \vert \mathcal{D}_{q}y - \mathcal{D}_{q}w \vert + \vert \mathcal{D}_{q}^{2}y - \mathcal{D}_{q}^{2}w \vert )\), it is easy to see that the multifunction \(O( \cdot )\cap \mathcal{G}(\cdot,y(\cdot ), \mathcal{D}_{q}y( \cdot ), \mathcal{D}_{q}^{2} y(\cdot ) ) \) is measurable. Now, we choose \(\varTheta _{2}(t)\in \mathcal{G}(t,y(t), \mathcal{D}_{q}y(t), \mathcal{D}_{q}^{2}y(t)) \) such that

$$\begin{aligned} \bigl\vert \varTheta _{1}(t)-\varTheta _{2}(t) \bigr\vert \leq{}& m(t)\psi \bigl( \bigl\vert y(t)-w(t) \bigr\vert + \bigl\vert \mathcal{D}_{q}y(t) - \mathcal{D}_{q}w(t) \bigr\vert \\ & {}+ \bigl\vert \mathcal{D}_{q}^{2}y(t) - \mathcal{D}_{q}^{2} w(t) \bigr\vert \bigr) \biggl[ \frac{1}{ \varSigma _{1} + \varSigma _{2} + \varSigma _{3} } \biggr], \end{aligned}$$

for all \(t\in [0,1]\). Select \(p_{2} \in \mathcal{N}(y)\) such that

$$\begin{aligned} p_{2}(t)={}& \int _{0}^{t} \frac{ (t-q\tau )^{(\vartheta -1)} }{\varGamma _{q}(\vartheta )} \varTheta _{2} (\tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t \Delta _{1} -t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma )} \int _{0}^{1} \frac{(1 -q\tau )^{ (\vartheta + \kappa -1)}}{\varGamma _{q} (\vartheta + \kappa ) } \varTheta _{2} (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{1} \frac{(1 -q\tau )^{(\vartheta - 3)}}{ \varGamma _{q} (\vartheta -2 )} \varTheta _{2} (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta - \sigma - 1)}}{ \varGamma _{q} (\vartheta -\sigma )} \varTheta _{2} (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t \Delta _{1} - t^{2} \varGamma _{q} (3-\sigma ) }{\Delta _{3} \varGamma _{q} (3-\sigma ) } \int _{0}^{\nu }\frac{(\nu -q\tau )^{(\vartheta + \kappa - 1)}}{ \varGamma _{q} (\vartheta + \kappa )} \varTheta _{2} (\tau ) \,\mathrm{d}_{q}\tau \end{aligned}$$

for all \(t\in [0,1]\). Thus by using a similar method in the proof of Theorem 10, we get

$$\begin{aligned} \Vert p_{1}-p_{2} \Vert ={}& \sup_{t\in [0,1]} \bigl\vert p_{1}(t)-p_{2}(t) \bigr\vert +\sup _{t\in [0,1]} \bigl\vert \mathcal{D}_{q}p_{1}(t) - \mathcal{D}_{q}p_{2}(t) \bigr\vert \\ &{} + \sup_{t\in [0,1]} \bigl\vert \mathcal{D}_{q}^{2}p_{1}(t) - \mathcal{D}_{q}^{2}p_{2}(t) \bigr\vert \\ \leq{}& ( \varSigma _{1} + \varSigma _{2} + \varSigma _{3} ) \lambda \psi \bigl( \Vert y-w \Vert \bigr) = \psi \bigl( \Vert y-w \Vert \bigr). \end{aligned}$$

Hence \(\mathcal{H}_{d} (\mathcal{N}(y), \mathcal{N}(w))\leq \psi (\Vert y-w \Vert )\) for all \(y,w\in \mathcal{X}\). By using \((H15)\), we find that the multifunction \(\mathcal{N}\) has approximate endpoint property. Now by using Theorem 6, there exists \(y^{*}\in \mathcal{X}\) such that \(\mathcal{N}(y^{*})=\{ y^{*} \}\). This implies that \(y^{*}\) is a solution for the q-fractional inclusion problem (3)–(4). □

Examples

Here, we provide two examples to illustrate our main results. Also we present a simplified analysis that can be executed to calculate the value of the q-Gamma function, \(\varGamma _{q} (x)\), for input q, x and different values of n. To this aim, we consider a pseudo-code description of the method for the calculated q-Gamma function of order n in Algorithm 2 (for more details, see the link https://en.wikipedia.org/wiki/Q-gamma_function).

Example 1

Consider the q-fractional boundary value problem

$$\begin{aligned} {}^{c}\mathcal{D}_{q}^{\frac{7}{3}} y(t) ={}& \frac{t \vert \sin t \vert }{16(1 + t^{2})} + \frac{ t \vert \cos (y(t)) \vert }{16(1+ \vert \cos (y(t)) \vert ) } + \frac{ t^{2} \vert \mathcal{D}_{q} y(t) \vert }{ 16(1+ \vert \mathcal{D}_{q} y(t) \vert )} \\ &{} + \frac{ t \vert \tan ^{-1} (\mathcal{D}_{q}^{2} y(t)) \vert }{ 16(1+ 16 \vert \tan ^{-1}(\mathcal{D}_{q}^{2} y(t) ) \vert ) }, \end{aligned}$$
(15)

with the three-point boundary value conditions

{y(0)=0,Dqy(0)+cDq87y(35)+Dq2y(1)=0,Jq3y(0)+Jq3y(35)+Jq3y(1)=0,
(16)

where \(0< q<1\) and \(0< t< 1\). Put \(\vartheta = \frac{7}{3}\), \(\nu = \frac{3}{5}\) and \(\sigma = \frac{8}{7}\) belonging to \((2, 3]\), \((0,1)\) and \((1,2)\), respectively, and \(\kappa =3\). Here, \({}^{c}\mathcal{D}_{q}^{\frac{7}{3}}\) denotes the fractional q-derivative of the Caputo type of order \(\frac{7}{3}\) and \(\mathcal{J}_{q}^{3}\) denotes the fractional q-integral of the Riemann–Liouville type of order 3. Define the continuous map \(\varPhi: [0,1] \times \mathbb{R}^{3} \to \mathbb{R}\) defined by

$$\begin{aligned} \varPhi \bigl(t, x(t), y(t), z(t) \bigr) ={}& \frac{t \vert \sin t \vert }{16( 1+ t^{2})} + \frac{ t \vert \cos (x(t)) \vert }{16( 1+ \vert \cos (x(t)) \vert )} \\ &{} + \frac{ t^{2} \vert y(t) \vert }{ 16(1+ \vert y(t) \vert )}+ \frac{ t \vert \tan ^{-1} z(t) \vert }{ 16( 1 + \vert z(t) \vert )}. \end{aligned}$$

For each \(x_{1},x_{2}, y_{1},y_{2}, z_{1}. z_{2} \in \mathbb{R}\), we have

$$\begin{aligned} &\bigl\vert \varPhi \bigl(t, x_{1}(t), y_{1}(t), z_{1}(t) \bigr) - \varPhi \bigl(t, x_{2}(t), y_{2}(t), z_{2}(t) \bigr) \bigr\vert \\ &\quad \leq \frac{t}{16} \bigl( \bigl\vert \cos x_{1}(t) - \cos x_{2}(t) \bigr\vert + \bigl\vert y_{1} (t) - y_{2} (t) \bigr\vert \\ &\qquad{} + \bigl\vert \tan ^{-1}\bigl( z_{1}(t) \bigr)- \tan ^{-1}\bigl(z_{2}(t) \bigr) \bigr\vert \bigr) \\ &\quad \leq \frac{t}{16} \bigl( \bigl\vert x_{1}(t) - x_{2}(t) \bigr\vert + \bigl\vert y_{1} (t) - \mathcal{D}_{q} y_{2} (t) \bigr\vert + \bigl\vert z_{1}(t) - z_{2}(t) \bigr\vert \bigr). \end{aligned}$$

Put \(L (t) =\frac{t}{16}\) for all t. Then \(\| L \| = \frac{t}{16}\). Consider the continuous and nondecreasing function \(\psi: [0,1] \to \mathbb{R}^{+}\) defined by \(\psi (x) = x \) for all \(x \in \mathbb{R}^{+}\). Then we have

$$\begin{aligned} \varPhi \bigl(t, y(t), \mathcal{D}_{q} y(t), \mathcal{D}_{q}^{2} y(t) \bigr) & \leq \frac{t}{16} \bigl( \vert y \vert + \vert \mathcal{D}_{q} y \vert + \bigl\vert \mathcal{D}_{q}^{2} y \bigr\vert \bigr) \\ & = \frac{t}{16} \psi \bigl( \vert y \vert + \vert \mathcal{D}_{q} y \vert + \bigl\vert \mathcal{D}_{q}^{2} y \bigr\vert \bigr). \end{aligned}$$

It is clear that \(\zeta: [0,1] \to \mathbb{R}^{+} \) defined by \(\zeta = \frac{t}{10}\) is continuous function. Now by using (6), we have

$$\begin{aligned} & \Delta _{1} = 2 \biggl(\frac{3}{5} \biggr)^{2-\frac{8}{7}} + (1+q) \varGamma _{q} \biggl(3- \frac{8}{7} \biggr)= 2 \biggl(\frac{3}{5} \biggr)^{\frac{6}{7}} + (1+q) \varGamma _{q} \biggl( \frac{13}{7} \biggr), \\ &\Delta _{2} = \biggl[1 + \biggl(\frac{3}{5} \biggr)^{3 +1} \biggr] \varGamma _{q} \biggl( 3- \frac{8}{7} \biggr)= \biggl[1 + \biggl( \frac{3}{5} \biggr)^{4} \biggr] \varGamma _{q} \biggl( \frac{13}{7} \biggr), \\ &\Delta _{3} = \biggl\vert \frac{-\varGamma _{q} ( 6) [ (\frac{3}{5} )^{4}+1 ] \Delta _{1} + \varGamma _{q} (\frac{13}{7} ) \varGamma _{q} ( 5) ( 1 + q) [ (\frac{3}{5} )^{5}+1 ]}{ \varGamma _{q} (5 ) \varGamma _{q} (6 ) \varGamma _{q} ( \frac{13}{7} )} \biggr\vert , \\ &\Delta _{4} = \biggl[1+ \biggl(\frac{3}{5} \biggr)^{4} \biggr] \Delta _{1} +\Delta _{3} \varGamma _{q}(5) \varGamma _{q} \biggl( \frac{13}{7} \biggr), \\ &\Delta _{5} = \Delta _{3} \varGamma _{q} \biggl( 3 - \frac{8}{7} \biggr) \varGamma _{q}(3 +2) = \Delta _{3} \varGamma _{q} \biggl( \frac{13}{7} \biggr) \varGamma _{q}(5), \end{aligned}$$

and by applying (9), we obtain

$$\begin{aligned} \Delta ^{(1)} = {}&\frac{ (\Delta _{1} + \varGamma _{q} ( 3-\frac{8}{7} ) ) ( (\frac{3}{5} )^{(\frac{7}{3} + 3) } +1 ) }{ \Delta _{3} \varGamma _{q} ( 3 -\frac{8}{7}) \varGamma _{q} ( \frac{7}{3} + 3 + 1 ) }+ \frac{ \Delta _{2} + \Delta _{4} }{ \Delta _{5} \varGamma _{q} ( \frac{7}{3} -1 ) } \\ &{} + \frac{ (\Delta _{2} + \Delta _{4}) (\frac{3}{5} )^{ (\frac{7}{3} - \frac{8}{7} )}}{\Delta _{5} \varGamma _{q} (\frac{7}{3} - \frac{8}{7} +1 ) }, \\ ={}& \frac{ (\Delta _{1} + \varGamma _{q} ( \frac{13}{7} ) ) ( (\frac{3}{5} )^{\frac{16}{3}} +1 ) }{ \Delta _{3} \varGamma _{q} ( \frac{13}{7}) \varGamma _{q} ( \frac{19}{3} ) } + \frac{ \Delta _{2} + \Delta _{4} }{ \Delta _{5} \varGamma _{q} ( \frac{4}{3} )} + \frac{ (\Delta _{2} + \Delta _{4}) ( \frac{3}{5} )^{ \frac{25}{21}}}{\Delta _{5} \varGamma _{q} (\frac{46}{21} ) }, \\ \Delta ^{(2)} ={}& \frac{ [\Delta _{1} + (1 + q) \varGamma _{q} ( 3 -\frac{8}{7} ) ] ( (\frac{3}{5} )^{ ( \frac{7}{3} + 3 ) } + 1 ) }{ \Delta _{3} \varGamma _{q} ( 3 -\frac{8}{7} ) \varGamma _{q} (\frac{7}{3} + 3 + 1 )} + \frac{ (1 + q) \Delta _{2} + \Delta _{4} }{ \Delta _{5} \varGamma _{q} ( \frac{7}{3} -1 ) } \\ &{} + \frac{ [ ( 1 + q ) \Delta _{2} + \Delta _{4} ] (\frac{3}{5} )^{ ( \frac{7}{3} - \frac{8}{7} )} }{\Delta _{5} \varGamma _{q} (\frac{7}{3} - \frac{8}{7} +1 ) }, \\ = {}&\frac{ [\Delta _{1} + (1 + q) \varGamma _{q} ( \frac{13}{7} ) ] ( (\frac{3}{5} )^{\frac{16}{3} } + 1 ) }{ \Delta _{3} \varGamma _{q} ( \frac{13}{7} ) \varGamma _{q} (\frac{19}{3} )} + \frac{ (1 + q) \Delta _{2} + \Delta _{4} }{ \Delta _{5} \varGamma _{q} ( \frac{4}{3} ) } \\ &{} + \frac{ [ ( 1 + q ) \Delta _{2} + \Delta _{4} ] (\frac{3}{5} )^{ \frac{25}{21} } }{\Delta _{5} \varGamma _{q} (\frac{46}{21} ) }, \\ \Delta ^{(3)} ={}& \frac{(1+q) ( (\frac{3}{5} )^{ ( \frac{7}{3} + 3 ) } + 1 ) }{ \Delta _{3} \varGamma _{q} (\frac{7}{3} + 3 + 1 ) } + \frac{ (1 + q ) \Delta _{2} }{ \Delta _{5} \varGamma _{q} ( \frac{7}{3} -1 ) } + \frac{ (1 + q) \Delta _{2} (\frac{3}{5} )^{ ( \frac{7}{3} - \frac{8}{7} )} }{ \Delta _{5} \varGamma _{q} ( \frac{7}{3} - \frac{8}{7} +1 )}, \\ ={}& \frac{(1+q) ( (\frac{3}{5} )^{ \frac{19}{3} } + 1 ) }{ \Delta _{3} \varGamma _{q} ( \frac{19}{3} ) } + \frac{ (1 + q ) \Delta _{2} }{ \Delta _{5} \varGamma _{q} ( \frac{4}{3} ) } + \frac{ (1 + q) \Delta _{2} (\frac{3}{5} )^{ \frac{25}{21} } }{ \Delta _{5} \varGamma _{q} ( \frac{46}{21} )}. \end{aligned}$$

In the last section, according to Table 4, we obtain \(\Delta _{1}\approx 2.4176\), 2.7391, 3.0751, \(\Delta _{2}\approx 1.1136\), 1.0906, 1.0749, \(\Delta _{3}\approx 1.1057\), 0.4816, 0.1911, \(\Delta _{4}\approx 4.4208\), 5.3826, 6.4537, \(\Delta _{5}\approx 1.6900\), 2.885, 2.9801 for \(q=\frac{1}{7}, \frac{1}{2}\), \(\frac{7}{8}\), respectively. Also, Table 5 shows the values of the \(\Delta ^{(i)}\) as follows: \(\Delta ^{(1)}\approx 6.8888\), 5.2250, 4.3006, \(\Delta ^{(2)}\approx 7.1089\), 5.6979, 4.8565, \(\Delta ^{(3)}\approx 1.7608\), 1.4188, 1.1913 for \(q=\frac{1}{7}, \frac{1}{2}\), \(\frac{7}{8}\), respectively, and values of K in (11) for \(q=\frac{1}{7}, \frac{1}{2}\) and \(\frac{7}{8}\) are shown in Table 6 as \(K_{q_{1}}:=\Vert L\Vert (\Delta ^{(1)} + \Delta ^{(2)} + \Delta ^{(3)} ) = 0.9849 <1\), \(K_{q_{2}}:=\Vert L\Vert (\Delta ^{(1)} + \Delta ^{(2)} + \Delta ^{(3)} )= 0.7714<1\) and \(K_{q_{3}}:=\Vert L\Vert (\Delta ^{(1)} + \Delta ^{(2)} + \Delta ^{(3)} ) = 0.6468<1\), respectively (Algorithm 6). Now by using Theorem 9, the q-fractional boundary value problem (15)–(16) has a solution.

Figure 1
figure1

Numerical results of K for \(q= \frac{1}{10}, \frac{1}{2}, \frac{6}{7}\) in Example 1

Table 1 Some numerical results for calculation of \(\varGamma _{q}(x)\) with constant \(q=\frac{1}{3}\), \(x=4.5, 8.4, 12.7\) and \(n=1, 2, \ldots, 15\) of Algorithm 2
Table 2 Some numerical results for calculation of \(\varGamma _{q}(x)\) with \(q=\frac{1}{3}, \frac{1}{2}, \frac{2}{3}\), \(x=5\) and \(n=1, 2, \ldots, 35\) of Algorithm 2
Table 3 Some numerical results for calculation of \(\varGamma _{q}(x)\) with \(x=8.4\), \(q=\frac{1}{3}, \frac{1}{2}, \frac{2}{3}\) and \(n=1, 2, \ldots, 40\) of Algorithm 2
Algorithm 6
figuref

The proposed method for calculated \(\Delta _{i}\) and \(\Delta ^{(i)}\) in Example 1

Table 4 Some numerical results of the \(\Delta _{i}\) in Example 1 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{7}{8}\)
Table 5 Some numerical results for calculation of the \(\Delta ^{(i)}\) in Example 1 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{7}{8}\)
Table 6 Some numerical results of the \(\Delta ^{(i)}\) and K in Example 1 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{7}{8}\)

Example 2

Consider the q-fractional inclusion problem

$$ {}^{c}\mathcal{D}_{q}^{\frac{14}{5}} y(t) \in \biggl[ 0, \frac{t \sin ^{2} y(t)}{25(1+ 2 t^{2})} + \frac{2(t+1) \vert \cos (\mathcal{D}_{q} y(t)) \vert }{ 75 (3 + \vert \cos (\mathcal{D}_{q} y(t) ) \vert ) } + \frac{ t \vert \mathcal{D}_{q}^{2} y(t) \vert }{ 25(2 + \vert \mathcal{D}_{q}^{2} y(t) \vert )} \biggr], $$
(17)

with three-point boundary value conditions

{y(0)=0,Dqy(0)+cDq74y(16)+Dq2y(1)=0,Jq4y(0)+Jq4y(16)+Jq4y(1)=0,
(18)

where \(0< q<1\) and \(t \in [0,1]\). Put \(\vartheta =\frac{14}{5}\), \(\nu = \frac{1}{6}\) and \(\sigma =\frac{7}{4}\) belongs to \((2, 3]\), \((0,1)\) and \((1,2)\), respectively, and \(\kappa =4\). Here, \({}^{c}\mathcal{D }_{q}^{\frac{14}{5}}\) denotes the fractional q-derivative of the Caputo type and \(\mathcal{J}_{q}^{4}\) is the fractional q-integral of the Riemann–Liouville type. Now, define the set-valued map \(\mathcal{G}: [0,1] \times \mathbb{R}^{3} \to \mathcal{P}(\mathbb{R})\) by

$$ \mathcal{G} \bigl(t, x(t), y(t), z(t) \bigr)= \biggl[ 0, \frac{t \sin ^{2} x(t)}{25(1+ 2 t^{2})} + \frac{2(t+1) \vert \cos (y(t)) \vert }{75(3 + \vert \cos ( y(t) ) \vert ) } + \frac{ t \vert z(t) \vert }{ 25(2 + \vert z(t) \vert )} \biggr], $$

for all \(t \in [0,1]\). Choose the non-negative function \(m\in C([0,1], [0,\infty ))\) defined by \(m(t)=\frac{t}{25}\) for all t. Then \(\|m\|= \frac{1}{25}\). Also, consider the non-negative and nondecreasing upper semi-continuous function \(\psi: [0,\infty )\to [0,\infty )\) defined by \(\psi (t) = \frac{t}{3}\) for almost all \(t >0\). It is clear that \(\lim \inf_{t\to \infty }(t-\psi (t))>0\) and \(\psi (t)< t\) for all \(t>0\). On the other hand, by applying Eq. (8), we have

$$\begin{aligned} &\Delta _{1} = 2 \biggl(\frac{1}{6} \biggr)^{ ( 2 - \frac{7}{4} )} + (1+q) \varGamma _{q} \biggl(3 - \frac{7}{4} \biggr) = 2 \biggl(\frac{1}{6} \biggr)^{ \frac{1}{4}} + (1+q) \varGamma _{q} \biggl( \frac{5}{4} \biggr), \\ &\Delta _{2} = \biggl( 1 + \biggl( \frac{1}{6} \biggr)^{4 +1} \biggr) \varGamma _{q} \biggl( 3 - \frac{7}{4} \biggr) = \biggl( 1 + \biggl( \frac{1}{6} \biggr)^{5} \biggr) \varGamma _{q} \biggl( \frac{5}{4} \biggr), \\ &\Delta _{3} = \biggl\vert \frac{- \varGamma _{q} ( 7 ) ( ( \frac{1}{6} )^{5} + 1 ) \Delta _{1} + \varGamma _{q} ( \frac{5}{4} ) \varGamma _{q} ( 6 ) (1+q) [ ( \frac{1}{6} )^{6} +1 ] }{ \varGamma _{q} (6 ) \varGamma _{q} (7 ) \varGamma _{q} ( \frac{5}{4} ) } \biggr\vert , \\ &\Delta _{4} = \biggl( 1 + \biggl( \frac{1}{6} \biggr)^{5} \biggr) \Delta _{1} + \Delta _{3} \varGamma _{q} (6) \varGamma _{q} \biggl( \frac{5}{4} \biggr), \\ &\Delta _{5} = \Delta _{3} \varGamma _{q} \biggl( 3-\frac{7}{4} \biggr) \varGamma _{q}(4 +2) = \Delta _{3} \varGamma _{q} \biggl( \frac{5}{4} \biggr) \varGamma _{q}(6), \end{aligned}$$

and by using Eq. (9), we obtain

$$\begin{aligned} \varXi _{1} ={}& \frac{1}{\varGamma _{q} ( \frac{14}{5} +1 )} + \frac{ ( \Delta _{1} + \varGamma _{q} ( 3 - \frac{7}{4} ) ) ( ( \frac{1}{6} )^{ ( \frac{14}{5} + 4 ) } +1 ) }{ \Delta _{3} \varGamma _{q} ( 3 -\frac{7}{4} ) \varGamma _{q} ( \frac{14}{5} + 4 + 1 ) } \\ &{} + \frac{ \Delta _{2} + \Delta _{4} }{ \Delta _{5} \varGamma _{q} (\frac{14}{5} -1 ) } + \frac{ (\Delta _{2} + \Delta _{4}) ( \frac{1}{6} )^{ ( \frac{14}{5} - \frac{7}{4} ) } }{ \Delta _{5} \varGamma _{q} ( \frac{14}{5} - \frac{7}{4} +1 ) } \\ ={}& \frac{1}{\varGamma _{q} ( \frac{19}{5} )} + \frac{ ( \Delta _{1} + \varGamma _{q} ( \frac{5}{4} ) ) ( ( \frac{1}{6} )^{ ( \frac{34}{5} ) } +1 ) }{ \Delta _{3} \varGamma _{q} ( \frac{5}{4} ) \varGamma _{q} ( \frac{39}{5} ) } \\ & + \frac{ \Delta _{2} + \Delta _{4} }{ \Delta _{5} \varGamma _{q} (\frac{9}{5} ) } + \frac{ (\Delta _{2} + \Delta _{4}) ( \frac{1}{6} )^{ \frac{21}{20} }}{ \Delta _{5} \varGamma _{q} ( \frac{41}{20} )}, \\ \varXi _{2} ={}& \frac{1}{ \varGamma _{q} (\frac{14}{5} ) } + \frac{ [ \Delta _{1} + (1+q) \varGamma _{q} ( 3 - \frac{7}{4} ) ] ( ( \frac{1}{6} )^{ ( \frac{14}{5} + 4 ) } + 1 )}{ \Delta _{3} \varGamma _{q} ( 3 -\frac{7}{4} ) \varGamma _{q} ( \frac{14}{5} + 4 + 1 ) } \\ &{} + \frac{ (1+q) \Delta _{2} + \Delta _{4} }{ \Delta _{5} \varGamma _{q} ( \frac{14}{5} -1 )} + \frac{ [ ( 1 + q ) \Delta _{2} + \Delta _{4} ] ( \frac{1}{6} )^{ ( \frac{14}{5} - \frac{7}{4} ) } }{ \Delta _{5} \varGamma _{q} ( \frac{14}{5} - \frac{7}{4} +1 )} \\ = {}&\frac{1}{ \varGamma _{q} ( \frac{14}{5} ) } + \frac{ [ \Delta _{1} + (1+q) \varGamma _{q} ( \frac{5}{4} ) ] ( ( \frac{1}{6} )^{\frac{34}{5} } + 1 )}{ \Delta _{3} \varGamma _{q} ( \frac{5}{4} ) \varGamma _{q} ( \frac{39}{5} ) } \\ &{} + \frac{ (1+q) \Delta _{2} + \Delta _{4} }{ \Delta _{5} \varGamma _{q} ( \frac{9}{5} )} + \frac{ [ ( 1 + q ) \Delta _{2} + \Delta _{4} ] ( \frac{1}{6} )^{ (\frac{21}{20} ) } }{ \Delta _{5} \varGamma _{q} ( \frac{41}{20} )}, \\ \varXi _{3}={}& \frac{1}{\varGamma _{q} ( \frac{14}{5} -1 )} + \frac{ ( 1 + q) ( ( \frac{1}{6} )^{ ( \frac{14}{5} + 4 ) } + 1 ) }{ \Delta _{3} \varGamma _{q} ( \frac{14}{5} + 4 + 1 ) } \\ & {}+ \frac{ (1 + q) \Delta _{2} }{ \Delta _{5} \varGamma _{q} ( \frac{14}{5} -1 ) } + \frac{ (1+q) \Delta _{2} ( \frac{1}{6} )^{ ( \frac{14}{5} - \frac{7}{4} ) } }{ \Delta _{5} \varGamma _{q} (\frac{14}{5} - \frac{7}{4} +1 )} \\ ={}& \frac{1}{\varGamma _{q} ( \frac{9}{5} )} + \frac{ ( 1 + q) ( ( \frac{1}{6} )^{ ( \frac{34}{5} ) } + 1 ) }{ \Delta _{3} \varGamma _{q} ( \frac{39}{5} ) } + \frac{ (1 + q) \Delta _{2} }{ \Delta _{5} \varGamma _{q} ( \frac{9}{5} ) } + \frac{ (1+q) \Delta _{2} ( \frac{1}{6} )^{ \frac{21}{20} } }{ \Delta _{5} \varGamma _{q} (\frac{41}{20} )}, \end{aligned}$$

\(\varSigma _{1} = \frac{1}{25} \varXi _{1}\), \(\varSigma _{2} = \frac{1}{25} \varXi _{2}\) and \(\varSigma _{3} = \frac{1}{25} \varXi _{3}\). Table 7 shows the numerical results of the \(\varXi _{i}\) for \(q=\frac{1}{7}\), \(\frac{1}{2}\) and \(\frac{7}{8}\), respectively, as follows: \(\varXi _{1} \approx 6.4475\), 6.7585, 2.5046, \(\varXi _{2} \approx 4.3664\), 5.0262, 2.1097 and \(\varXi _{3} \approx 3.3904\), 4.1583, 1.19111 (Algorithm 7). For each \(x_{1}, x_{2}, y_{1}, y_{2}, z_{1}, z_{2}\in \mathbb{R}\), we have

$$\begin{aligned} &\mathcal{H}_{d} \bigl(\mathcal{G} \bigl(t, x_{1}(t), y_{1} (t), z_{1}(t) \bigr) - \mathcal{G} \bigl( t, x_{2}(t), y_{2} (t), z_{2}(t) \bigr) \bigr) \\ &\quad \leq \frac{t}{25}.\frac{1}{3} \bigl( \vert x_{1} - x_{2} \vert + \vert y_{1} - y_{2} \vert + \vert z_{1} - z_{2} \vert \bigr) \\ &\quad = \frac{t}{25} \psi \bigl( \vert x_{1} - x_{2} \vert + \vert y_{1} - y_{2} \vert + \vert z_{1} - z_{2} \vert \bigr) \\ &\quad \leq m(t) \psi \bigl( \vert x_{1} - x_{2} \vert + \vert y_{1} - y_{2} \vert + \vert z_{1} - z_{2} \vert \bigr) \biggl[ \frac{1}{\varSigma _{1} +\varSigma _{2} + \varSigma _{3} } \biggr]. \end{aligned}$$
Algorithm 7
figureg

The proposed method for calculated \(\Delta _{i}\) and \(\varXi _{i}\) in Example 2

Table 7 Some numerical results for calculation of the Ξ in Example 2 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{7}{8}\)

Consider the operator \(\mathcal{N}: \mathcal{X}\to \mathcal{P}(\mathcal{X})\) defined by

$$ \mathcal{N}(u)= \bigl\{ h\in \mathcal{X}: \text{ there exists } \varTheta \in \mathcal{S}_{ \mathcal{G},u} \text{ such that } h(t) = w(t) \text{ for all } t\in [0,1] \bigr\} , $$

where

$$\begin{aligned} w(t) ={}& \int _{0}^{t} \frac{ (t - q \tau )^{ ( \frac{14}{5} -1 ) } }{ \varGamma _{q} ( \frac{14}{5} ) } \varTheta (\tau ) \,\mathrm{d}_{q}\tau \\ &{} + \frac{t \Delta _{1} -t^{2} \varGamma _{q} ( 3 -\frac{7}{4} ) }{ \Delta _{3} \varGamma _{q} ( 3 -\frac{7}{4} ) } \int _{0}^{1} \frac{ ( 1 - q \tau )^{ (\frac{14}{5} + 4 - 1 ) }}{ \varGamma _{q} ( \frac{14}{5} + 4 ) } \varTheta (\tau ) \,\mathrm{d}_{q} \tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{1} \frac{ (1 - q \tau )^{ ( \frac{14}{5} - 3 ) } }{ \varGamma _{q} (\frac{14}{5} -2 ) } \varTheta (\tau ) \,\mathrm{d}_{q} \tau \\ & {}+ \frac{ t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{\frac{1}{6}} \frac{ (\frac{1}{6} -q \tau )^{ ( \frac{14}{5} - \frac{7}{4}- 1 ) } }{ \varGamma _{q} ( \frac{14}{5} -\frac{7}{4} )} \varTheta (\tau ) \,\mathrm{d}_{q} \tau \\ & {}+ \frac{ t \Delta _{1} - t^{2} \varGamma _{q} ( 3 - \frac{7}{4} ) }{ \Delta _{3} \varGamma _{q} ( 3 -\frac{7}{4} ) } \int _{0}^{\frac{1}{6}} \frac{ (\frac{1}{6} - q \tau )^{ ( \frac{14}{5} + 4 - 1 ) } }{ \varGamma _{q} ( \frac{14}{5} + 4 )} \varTheta (\tau ) \,\mathrm{d}_{q} \tau \\ ={}& \int _{0}^{t} \frac{ (t - q \tau )^{ \frac{9}{5} }}{ \varGamma _{q} ( \frac{14}{5} ) } \varTheta (\tau ) \,\mathrm{d}_{q}\tau \\ & {}+ \frac{t \Delta _{1} -t^{2} \varGamma _{q} ( \frac{5}{4} ) }{ \Delta _{3} \varGamma _{q} ( \frac{5}{4} ) } \int _{0}^{1} \frac{ ( 1 - q \tau )^{ \frac{29}{5} }}{ \varGamma _{q} ( \frac{34}{5} ) } \varTheta (\tau ) \,\mathrm{d}_{q} \tau \\ &{} + \frac{t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{1} \frac{ (1 - q \tau )^{-\frac{1}{5} } }{ \varGamma _{q} (\frac{4}{5} ) } \varTheta (\tau ) \,\mathrm{d}_{q} \tau \\ &{} + \frac{ t^{2} \Delta _{2} - t \Delta _{4} }{\Delta _{5} } \int _{0}^{\frac{1}{6}} \frac{ (\frac{1}{6} -q \tau )^{ \frac{1}{20} } }{ \varGamma _{q} ( \frac{21}{20} )} \varTheta (\tau ) \,\mathrm{d}_{q} \tau \\ &{} + \frac{ t \Delta _{1} - t^{2} \varGamma _{q} ( \frac{5}{4} ) }{ \Delta _{3} \varGamma _{q} ( \frac{5}{4} ) } \int _{0}^{\frac{1}{6}} \frac{ (\frac{1}{6} - q \tau )^{ \frac{29}{5} } }{ \varGamma _{q} ( \frac{34}{5} )} \varTheta (\tau ) \,\mathrm{d}_{q} \tau. \end{aligned}$$

Now by using Theorem 11, the q-fractional inclusion problem (17)–(18) has a solution.

Conclusion

It is important that we increase our abilities from different points of view for studying distinct fractional integro-differential equations and inclusions. In this way, we should try to use modern and new techniques in our investigations. It would be significant if we could add numerical calculations in our results. In the present work, we studied the existence of solutions for a three-point nonlinear q-fractional differential equation and its related inclusion. In this way, we used α-ψ-contractions and multifunctions. We provided two examples to illustrate our main results. Finally, by providing some algorithms and tables, we gave some numerical computations for the results.

References

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    Jackson, F.H.: On q-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910). https://doi.org/10.1017/S0080456800002751

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Acknowledgements

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

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Etemad, S., Rezapour, S. & Samei, M.E. α-ψ-contractions and solutions of a q-fractional differential inclusion with three-point boundary value conditions via computational results. Adv Differ Equ 2020, 218 (2020). https://doi.org/10.1186/s13662-020-02679-w

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MSC

  • 34A08
  • 39A12
  • 39A13

Keywords

  • α-ψ-contraction
  • q-fractional differential equation
  • Approximate endpoint property
  • Boundary value inclusion problem