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On the existence of solutions for a multi-singular pointwise defined fractional q-integro-differential equation

Abstract

By using the Caputo type and the Riemann–Liouville type fractional q-derivative, we investigate the existence of solutions for a multi-term pointwise defined fractional q-integro-differential equation with some boundary value conditions. In fact, we give some results by considering different conditions and using some classical fixed point techniques and the Lebesgue dominated convergence theorem.

Introduction

It is known that the subject of q-difference equations was introduced by Jackson in 1910 [1]. After that, some researchers studied q-difference equations [220]. On the other hand, many modern works on integro-differential equations by using different views and fractional derivatives have been published recently, and young researchers could use the main idea of the works for their works (see, for example, [2150]).

In 2012, Ahmad et al. studied the existence and uniqueness of solutions for the fractional q-difference equation \({}^{c}D_{q}^{ \alpha }u(t)= T ( t, u(t) ) \) with boundary conditions \(\alpha _{1} u(0) - \beta _{1} D_{q} u(0) = \gamma _{1} u(\eta _{1})\) and \(\alpha _{2} u(1) - \beta _{2} D_{q} u(1) = \gamma _{2} u(\eta _{2})\), where \(\alpha \in (1, 2]\), \(\alpha _{i}\), \(\beta _{i}\), \(\gamma _{i}\), \(\eta _{i}\) are real numbers for \(i=1,2\) and \(T \in C(J \times \mathbb{R}, \mathbb{R})\) [6]. In 2013, Zhao et al. reviewed the q-integral problem \((D_{q}^{\alpha }u)(t) + f(t, u(t) )=0\) with boundary conditions \(u(1)=\mu I_{q}^{\beta }u(\eta ) \) and \(u(0)=0\) for almost all \(t \in (0,1)\), where \(q \in (0,1)\), \(\alpha \in (1, 2]\), \(\beta \in (0, 2]\), \(\eta \in (0,1)\), μ is a positive real number, \(D_{q}^{\alpha }\) is the q-derivative of Riemann–Liouville and real-valued continuous map u defined on \(I \times [0, \infty )\) [15]. In 2014, Ahmad et al. investigated the problem

$$ {}^{c}D^{\beta }_{q} \bigl( {}^{c}D^{\gamma }_{q} + \lambda \bigr) u(t) = p f \bigl(t, u(t)\bigr) + k I_{q}^{\xi }g\bigl(t, u(t)\bigr), $$

with boundary conditions \(\alpha _{1} u(0) - \beta _{1} (t^{(1-\gamma )} D_{q} u(0)) |_{t=0}= \sigma _{1} u(\eta _{1})\) and \(\alpha _{2} u(1) + \beta _{2} D_{q} u(1)= \sigma _{2} u(\eta _{2})\), where \(t, q \in [0,1]\), \({}^{c}D_{q}^{\beta }\) is the fractional Caputo q-derivative, \(0 < \beta \), \(\gamma \leq 1\), \(I_{q}^{\xi }(\cdot) \) denotes the Riemann–Liouville integral with \(\xi \in (0, 1)\), f and g are given continuous functions, λ and p, k are real constants, \(\alpha _{i}, \beta _{i}, \sigma _{i}\in \mathbb{R}\) and \(\eta _{i} \in (0, 1)\) for \(i=1,2\) [5]. In 2017, Wang considered the existence of uniqueness and nonexistence of positive solution for fractional differential equations \(D_{0^{+}}^{\sigma }x(t) + f(t, x(t))=0\) for \(t \in (0,1)\) under conditions the \(x(0) = x'(0) = \cdots = x^{(n-2)} (0) =0\) and \(D_{0^{+}}^{\alpha }x(1) = \int _{0}^{b} \mu (t) D_{0^{+}}^{\beta }x(t) \,\mathrm{d}t\), where \(n-1< \sigma \leq n\), \(n \geq 3\), \(\alpha \in (0,1)\),

$$ \varGamma (\sigma - \alpha ) \int _{0}^{b} \mu (t) t^{\sigma - \beta -1} \,\mathrm{d}t < \varGamma (\sigma - \beta ), $$

\(b \in (0, 1]\), \(D_{0^{+}}^{\sigma }\), \(D_{0^{+}}^{\alpha }\), \(D_{0^{+}} ^{\beta }\) are the standard Riemann–Liouville derivatives, \(f : (0,1) \times [0, \infty ) \to [0, \infty )\) is continuous and \(\mu (t) \in L^{1} ([0,1])\) is nonnegative [51]. Also, in 2018 he investigated the existence and multiplicity of positive solutions for the fractional differential equation \(D_{0^{+}}^{\sigma }x(t) + f(t, x(t))=0 \) for \(t \in (0,1)\) under the conjugate type integral boundary conditions \(x(0) = x'(0) = \cdots = x^{(n-2)} (0) =0\) and \(D_{0^{+}}^{\alpha }x(1) = \int _{0}^{b} \mu (t) D_{0^{+}}^{\beta }x(t) \,\mathrm{d}V(t)\), where \(D_{0^{+}}^{\alpha }\), \(D_{0^{+}}^{ \beta }\) are the standard Riemann–Liouville derivatives, \(n \geq 3\), \(\alpha \in (0,1)\), \(0 \leq \beta < \sigma -1\), \(b \in (0, 1]\), \(f(t, x)\) may be singular at \(t=0, 1\) and \(x=0\), \(\mu (t) \in L^{1} [0,1] \cap C(0,1)\) is nonnegative, \(\int _{0}^{b} \mu (t) t^{\sigma - \beta -1} \,\mathrm{d}V(t)\) denotes the Riemann–Stieltjes integral, in which V has bounded variation [52].

In 2019, Samei et al. reviewed the existence of solutions for some multi-term q-integro-differential equations with non-separated and initial boundary conditions [12]. Also, Ntouyas et al. [10], by applying definitions of the fractional q-derivative of the Caputo type and the fractional q-integral of the Riemann–Liouville type, studied the existence and uniqueness of solutions for multi-term nonlinear fractional q-integro-differential equations under some boundary conditions

$$ {}^{c}D_{q}^{\alpha } x(t) = w \bigl( t, x(t), ( \varphi _{1} x) (t), ( \varphi _{2} x) (t), {}^{c}D_{q}^{ \beta _{1}} x(t), {}^{c}D_{q}^{\beta _{2}} x(t), \ldots , {}^{c}D_{q}^{ \beta _{n}}x(t) \bigr). $$

In 2020, Liang et al. investigated the existence of solutions for nonlinear problems regular and singular fractional q-differential equation

$$ {}^{c}D_{q}^{\alpha }f(t) = w \bigl(t, f(t), f'(t), {}^{c}D_{q}^{ \beta }f(t) \bigr), $$

with conditions \(f(0) = c_{1} f(1)\), \(f'(0)= c_{2} {}^{c}D_{q}^{ \beta } f (1)\), and \(f^{(k)}(0) = 0\) for \(2\leq k \leq n-1\), here \(n-1 < \alpha < n\) with \(n\geq 3\), \(\beta , q , c_{1}\in (0,1)\), \(c_{2} \in (0, \varGamma _{q} (2- \beta ))\), function w is an \(L^{\kappa }\)-Carathéodory, \(w(t, x_{1}, x_{2}, x_{3})\) may be singular, and \({}^{c}D_{q}^{\alpha }\) is the fractional Caputo type q-derivative [17]. Also, they discussed the existence of solutions for the fractional q-derivative inclusions

$$ {}^{c}D_{q}^{\alpha }x(t) \in F \bigl( t, x(t), x'(t), {}^{c}D_{q} ^{\beta }x(t) \bigr), $$

\(x(0) + x'(0) + {}^{c}D_{q}^{\beta }x(0) = \int _{0}^{\eta _{1}} x(s) \,\mathrm{d}s \), and \(x(1) + x'(1) + {}^{c}D_{q}^{\beta }x(1) = \int _{0}^{\eta _{2}} x(s) \,\mathrm{d}s\) for any t in I and \(q, \eta _{1}, \eta _{2}, \beta \in (0,1)\), where F maps \(I\times \mathbb{R}^{3} \) into \(2^{\mathbb{R}}\) is a compact-valued multifunction and \({}^{c}D_{q}^{\alpha }\) is the fractional Caputo type q-derivative operator of order \(\alpha \in (1, 2]\), and

$$ \varGamma _{q} (2- \beta ) \bigl(\eta ^{2} \nu - \nu ^{2} \eta - \eta ^{2} + \nu ^{2} + 4\eta - 2\nu -2\bigr) + 2(1-\eta ) \neq 0 $$

such that \(\alpha -\beta >1\) [14]. Similar results have been presented in other studies [12, 13, 19, 20, 37].

By using the main idea of [41, 42, 53], we are going to investigate the multi-singular fractional q-integro-differential pointwise defined equation

$$ D_{q}^{\alpha } u(t)= \omega \bigl(t, u(t), u'(t), D_{q}^{\beta _{1}} u(t), I_{q}^{\beta _{2}} u(t) \bigr) $$
(1)

under two distinct boundary conditions

$$ \begin{aligned} &u'(0) = u(a),\qquad u(1) =\int _{0}^{b} u(r) \,\mathrm{d}r, \quad \alpha \in [2,3), \\ &u'(0) = u(a),\qquad u(1) = \int _{0}^{b} u(r) \,\mathrm{d}r, \quad \alpha \in [3,\infty ), \end{aligned} $$
(2)

and \(u^{(j)}(0)=0\) for \(j=2,\ldots ,[\alpha ]-1\), where \(t \in \overline{J}=[0,1]\), \(u \in \mathcal{B}=C^{1}(\overline{J})\), α, \(\beta _{1}\), \(\beta _{2}\) belong to \([2,\infty )\), \(J=(0,1)\), \((1, \infty )\), \(a, b \in J\), \(D_{q}^{\alpha }\) is the Caputo fractional q-derivative of order α, and \(\omega : \overline{J} \times \mathbb{R}^{4} \to \mathbb{R}\) is a function such that \(\omega (t, \cdot, \cdot, \cdot, \cdot)\) is singular at some points \(t\in \overline{J}\).

Preliminaries

Here, we recall some basic notion, lemmas, and theorems which are used in the subsequent sections. Let \(q \in (0,1)\) and \(a \in \mathbb{R}\). Define \([a]_{q}=\frac{1-q^{a}}{1-q}\) [1]. The power function \((x-y)_{q}^{n}\) with \(n \in \mathbb{N}_{0} \) is defined by \((x-y)_{q}^{(n)}= \prod_{k=0}^{n-1} (x - yq^{k})\) for \(n\geq 1\) and \((x-y)_{q}^{(0)}=1\), where x and y are real numbers and \(\mathbb{N}_{0} := \{ 0\} \cup \mathbb{N}\) [2]. Also, for \(\alpha \in \mathbb{R}\) and \(a \neq 0\), we have

$$ (x-y)_{q}^{(\alpha )}= x^{\alpha }\prod _{k=0}^{\infty }\bigl(x-yq^{k}\bigr) \big/ \bigl(x - yq^{\alpha + k}\bigr). $$

If \(y=0\), then it is clear that \(x^{(\alpha )}= x^{\alpha }\) (Algorithm 1). The q-gamma function is given by \(\varGamma _{q}(z) = (1-q)^{(z-1)} / (1-q)^{z -1}\), where \(z \in \mathbb{R} \backslash \{0, -1, -2, \ldots \}\) [1]. Note that \(\varGamma _{q} (z+1) = [z]_{q} \varGamma _{q} (z)\). The value of q-gamma function is \(\varGamma _{q}(z)\) for input values q and z with counting the number of sentences n in summation by simplifying analysis (see Tables 13). For this design, we prepare a pseudo-code description of the technique for estimating q-gamma function of order n, which is shown in Algorithm 2. The q-derivative of function f is defined by \((D_{q} f)(x) = \frac{f(x) - f(qx)}{(1- q)x}\) and \((D_{q} f)(0) = \lim_{x \to 0} (D_{q} f)(x)\), which is shown in Algorithm 3 [2]. Also, the higher order q-derivative of a function f is defined by \((D_{q}^{n} f)(x) = D_{q}(D_{q}^{n-1} f)(x)\) for all \(n \geq 1\), where \((D_{q}^{0} f)(x) = f(x)\) [2, 3]. The q-integral of a function f defined on \([0,b]\) is defined by

$$ I_{q} f(x) = \int _{0}^{x} f(s) \,\mathrm{d}_{q} s = x(1- q) \sum_{k=0} ^{\infty } q^{k} f\bigl(x q^{k}\bigr) $$

for \(0 \leq x \leq b\), provided the series is absolutely convergent [2, 3]. The q-derivative of function f is defined by \((D_{q} f)(x) = \frac{f(x) - f(qx)}{(1- q)x}\) and \((D_{q} f)(0) = \lim_{x \to 0} (D_{q} f)(x)\), which is shown in Algorithm 3 [2, 3]. If \(a \in [0, b]\), then

$$ \int _{a}^{b} f(u) \,\mathrm{d}_{q} u = (1-q) \sum_{k=0}^{\infty } q ^{k} \bigl[ b f\bigl(b q^{k}\bigr) - a f\bigl(a q^{k}\bigr) \bigr], $$

whenever the series exists [2, 3]. The operator \(I_{q}^{n}\) is given by \((I_{q}^{0} h)(x) = h(x) \) and \((I_{q}^{n} h)(x) = (I_{q} (I_{q}^{n-1} h)) (x)\) for \(n \geq 1\) and \(g \in C([0,b])\) [2, 3]. It has been proved that \((D_{q} (I_{q} f))(x) = f(x) \) and \((I_{q} (D_{q} f))(x) = f(x) - f(0)\) whenever f is continuous at \(x =0\) [2, 3]. The fractional Riemann–Liouville type q-integral of the function f on J for \(\alpha \geq 0\) is defined by \((I_{q}^{0} f)(t) = f(t) \) and

$$ \bigl(I_{q}^{\alpha }f\bigr) (t) = \frac{1}{\varGamma _{q}(\alpha )} \int _{0}^{t} (t- qs)^{(\alpha - 1)} f(s) \,\mathrm{d}_{q}s $$

for \(t \in J\) and \(\alpha >0\) [9]. Also, the Caputo fractional q-derivative of a function f is defined by

$$ \begin{aligned}[b] \bigl( {}^{c}D_{q}^{\alpha }f \bigr) (t) & = \bigl( I_{q}^{[\alpha ]-\alpha }\bigl( D_{q}^{[\alpha ]} f\bigr) \bigr) (t) \\ & = \frac{1}{\varGamma _{q} ([\alpha ]-\alpha )} \int _{0} ^{t} (t- qs)^{ ([\alpha ]-\alpha -1 )} \bigl( D_{q}^{[ \alpha ]} f \bigr) (s) \,\mathrm{d}_{q}s, \end{aligned} $$
(3)

where \(t \in J\) and \(\alpha >0\) ([9]). It has been proved that \(( I_{q}^{\beta } (I_{q}^{\alpha } f)) (x) = ( I_{q}^{ \alpha + \beta } f) (x)\) and \((D_{q}^{\alpha } (I_{q}^{\alpha } f) ) (x)= f(x)\), where \(\alpha , \beta \geq 0\) ([9]). By using Algorithm 2, we can calculate \((I_{q}^{\alpha }f)(x)\), which is shown in Algorithm 4.

Algorithm 1
figurea

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

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 \((I_{q}^{\alpha}f)(x)\)

Table 1 Some numerical results for calculation of \(\varGamma _{q}(x)\) with \(q=\frac{1}{8}\), which is constant, for \(x=9.5, 65, 110, 780\) in Algorithm 2
Table 2 Some numerical results for calculation of \(\varGamma _{q}(x)\) with \(q=\frac{1}{8}, \frac{1}{2}, \frac{4}{5}, \frac{8}{9}\) for \(x=9.5\) of Algorithm 2
Table 3 Some numerical results for calculation of \(\varGamma _{q}(x)\) with \(q=\frac{1}{8}, \frac{1}{2}, \frac{4}{5}, \frac{8}{9}\) for \(x=110\) of Algorithm 2

We say f is multi-singular when it is singular at more than one point t. Also, we say that \(D_{q}^{\alpha } u(t) + h(t)=0\) is a pointwise defined equation on if there exists a set \(E \subset \overline{J}\) such that the measure of \(E^{c}\) is zero and the equation holds on E. In this paper, we use \(\|\cdot\|_{1}\), \(\|\cdot\|\) and \(\Vert w \Vert _{*} = \max \{\| w\|, \|w'\| \} \) as the norm of \(\overline{\mathcal{L}} =L^{1}(\overline{J})\), the sup norm \(\overline{\mathcal{A}}=C(\overline{J})\), and the norm of \(\overline{ \mathcal{B}}= C^{1}(\overline{J})\), respectively. Let Ψ be the family of nondecreasing functions \(\psi : [0, \infty ) \to [0,\infty )\) such that \(\sum_{n=1}^{\infty } \psi ^{n}(t) < \infty \) for all \(t> 0\) [54]. One can check that \(\psi (t)< t\) for all \(t>0\) [54]. Let \(T : \mathcal{X} \to \mathcal{X}\) and \(\alpha : \mathcal{X} \times \mathcal{X} \to [0,\infty )\) be two maps. Then T is called an α-admissible map whenever \(\alpha ( x, y) \geq 1\) implies \(\alpha (Tx,Ty) \geq 1\) [55]. Let \((\mathcal{X}, \rho )\) be a complete metric space, \(\psi \in \varPsi \), and \(\alpha : \mathcal{X} \times \mathcal{X} \to [0, \infty )\) be a map. A self-map \(T : \mathcal{X} \to \mathcal{X}\) is called an α-ψ-contraction whenever \(\alpha ( s, t) \rho ( Ts, Tt ) \leq \psi ( \rho ( s, t))\) for all \(s, t \in \mathcal{X}\) [55]. We need the following results.

Lemma 1

([56])

Suppose that\(0< n-1\leq \alpha < n\)and\(u \in \overline{\mathcal{A}} \cap \overline{\mathcal{L}}\). Then\(I_{q}^{\alpha } D_{q}^{\alpha } u(t)= u(t)+ \sum_{i=0}^{n-1} c_{i} t ^{i}\)for some constants\(c_{i} \in \mathbb{R}\).

Lemma 2

([55])

Let\((\mathcal{X}, \rho )\)be a complete metric space, \(\psi \in \varPsi \), \(\alpha : \mathcal{X} \times \mathcal{X} \to [0,\infty )\)be a map, and\(T : \mathcal{X} \to \mathcal{X}\)be anα-admissibleα-ψ-contraction. ThenThas a fixed point wheneverTis continuous and there exists\(x_{0} \in \mathcal{X}\)such that\(\alpha ( x_{0}, Tx_{0}) \geq 1\).

Main results

First, we state and prove the following key results.

Lemma 3

Let\(\alpha \geq 2\), \(a, b \in J\), and\(v_{0} \in L^{1}(\overline{J})\). Then\(v(t)= \int ^{1}_{0} G_{q}(t,s) v_{0}(s) \,\mathrm{d}s\)is a solution for the pointwise defined problem\(D_{q}^{\alpha }u(t) + v_{0}(t) = 0\)with boundary conditions (2), where

$$ G_{q}(t,s)= G_{q}^{0}(t, s) + \frac{1}{ 1 - b } \int _{0}^{b} G_{q} ^{0}(t,s) \,\mathrm{d}t $$
(4)

and

$$ G_{q}^{0}(t,s) = \textstyle\begin{cases} \frac{1}{\varGamma _{q}(\alpha ) } [ (1 - qs)^{(\alpha - 1)} - ( t- qs)^{( \alpha - 1)} & \\ \quad {} + (1- t) (a- qs)^{(\alpha -1)} ], & 0\leq s \leq t \leq 1, s\leq a, \\ \frac{( 1 -qs)^{(\alpha - 1)} - ( t - qs)^{(\alpha -1)}}{\varGamma _{q}( \alpha )}, & 0\leq a\leq s \leq t \leq 1, \\ \frac{(1-qs)^{(\alpha - 1)} + (1-t)( a- qs)^{(\alpha -1)}}{\varGamma _{q}( \alpha )}, & 0\leq t \leq s \leq a \leq 1, \\ \frac{(1 - qs)^{(\alpha - 1)}}{\varGamma _{q}(\alpha )},& 0\leq t \leq s \leq 1, a \leq s. \end{cases} $$

Proof

Let \(E \subset \overline{J}\) be such that the equation \(D_{q}^{\alpha }u(t) + v_{0}(t) = 0\) holds for all \(t\in E\) and the measure of \(E^{c}\) is zero. Choose \(v \in \overline{\mathcal{A }}\cap \overline{ \mathcal{L}}\) such that \(v =v_{0}\) on E. If \(v_{0} \in \overline{\mathcal{ A }}\) is a solution for the pointwise defined problem, then we put \(v(t) = -D_{q}^{\alpha } u_{0}(t)\) for all \(t \in \overline{J}\). Note that \(v \in \overline{\mathcal{A}} \cap \overline{ \mathcal{L}}\) and \(v =v_{0}|_{E}\). Also, we have

$$\begin{aligned} I_{q}^{\alpha }\bigl(v_{0}(t)\bigr) &= \frac{1}{\varGamma _{q}(\alpha )} \int _{0} ^{t} (t - qs)^{(\alpha - 1)} v_{0}(s) \,\mathrm{d}_{q}s \\ & = \frac{1}{\varGamma _{q}( \alpha )} \biggl[ \int _{[0,t] \cap E} ( t -qs)^{( \alpha - 1)} v_{0}(s) \,\mathrm{d}_{q}s \\ &\quad {} + \int _{[0,t] \cap E^{c}} ( t -qs)^{(\alpha - 1)} v_{0}(s) \,\mathrm{d}_{q}s \biggr] \\ & = \frac{1}{\varGamma _{q}( \alpha )} \int _{[0,t] \cap E} ( t - qs)^{( \alpha -1)} v(s) \,\mathrm{d}_{q}s \\ & = \frac{1}{\varGamma _{q}(\alpha )} \biggl[ \int _{[0,t] \cap E} ( t - qs)^{( \alpha - 1)} v(s) \,\mathrm{d}_{q}s \\ &\quad {} + \int _{[0,t]\cap E^{c}} ( t - qs)^{(\alpha - 1)} v(s) \,\mathrm{d}_{q}s \biggr] \\ & = \frac{1}{\varGamma _{q}(\alpha )} \int _{0}^{t} ( t - qs )^{(\alpha -1)} v(s) \,\mathrm{d}_{q}s = I_{q}^{\alpha }\bigl(v(t) \bigr) \end{aligned}$$

for each \(t\in E\). Let \(t\in E^{c} \backslash \{0\}\). Choose \(\{ t_{n}\}\) in E such that \(t_{n} \to t^{-}\). Hence,

$$\begin{aligned} I_{q}^{\alpha } \bigl( v_{0}(t)\bigr) & = \frac{1}{\varGamma _{q}(\alpha )} \int _{0}^{t} ( t -qs)^{(\alpha -1)} v_{0}(s) \,\mathrm{d}_{q}s \\ & = \lim_{n\to \infty } \frac{1}{\varGamma _{q}(\alpha )} \int _{0}^{t _{n}} (t_{n} - qs)^{(\alpha - 1)} v_{0}(s) \,\mathrm{d}_{q}s = \lim _{n\to \infty } I_{q}^{\alpha } \bigl( v_{0} (t_{n}) \bigr) \\ & = \lim_{n\to \infty } I_{q}^{\alpha } \bigl( v ( t_{n} ) \bigr) = \lim_{ n\to \infty } \frac{1}{\varGamma _{q}(\alpha )} \int _{0}^{t_{n}} (t _{n} - q s)^{(\alpha -1)} v(s) \,\mathrm{d}_{q}s \\ & = \frac{1}{\varGamma _{q}(\alpha )} \int _{0}^{t} (t - qs )^{(\alpha -1)} v_{0}(s) \,\mathrm{d}_{q}s =I_{q}^{\alpha } \bigl( v(t)\bigr). \end{aligned}$$

If \(t=0 \in E^{c}\), then \(I_{q}^{\alpha }( v_{0}(t)) = I_{q}^{\alpha } ( v(t))= 0\), and so \(I_{q}^{\alpha }( v_{0}(t)) = I_{q}^{\alpha } ( w(t))\) for all \(t \in \overline{J}\). Thus, \(I_{q}^{\alpha } ( D_{q} ^{\alpha }u(t)) = I_{q}^{\alpha }( - v_{0}(t))\) for each \(t \in \overline{J}\) whenever \(D_{q}^{\alpha }u(t) + v_{0}(t) = 0\) for \(t \in E\). Hence, \(I_{q}^{\alpha }(D^{\alpha }u(t)) = I_{q}^{\alpha }( - v(t))\) on . By employing the boundary conditions and Lemma 1, we can conclude that

$$ u(t)= - \frac{1}{ \varGamma _{q}(\alpha )} \int ^{t}_{0} ( t -qs)^{(\alpha - 1)} v(s) + c_{0} + c_{1} t. $$

Since \(u'(0)= u(a)\), \(c_{1} = - I_{q}^{\alpha }v(a)\), and so \(u(t)= -I_{q}^{\alpha }v(t) + c_{0} - I_{q}^{\alpha }v(a)\). Hence,

$$ \int ^{b}_{0} u(s) \,\mathrm{d}s = u(1)= - I_{q}^{\alpha }v(1) + c_{0} - I_{q}^{\alpha }v(a). $$

So \(c_{0}= \int ^{b}_{0} u(s) \,\mathrm{d}s + I_{q}^{\alpha }v(1) + I _{q}^{\alpha }v(a)\). Thus,

$$ u(t) = -I_{q}^{\alpha }v(t) - t I_{q}^{\alpha }v(a)+ \int ^{b}_{0} u(s) \,\mathrm{d}_{q}s + I_{q}^{\alpha }v(1) + I_{q}^{\alpha }v(a). $$

Put \(h(t) = -I_{q}^{\alpha }v(t) + (1- t) I_{q}^{\alpha }v(a) + I_{q} ^{\alpha }v(1)\). Then we get

$$ u(t)= h(t)+ \int ^{b}_{0} u(s) \,\mathrm{d}s. $$
(5)

We consider two cases. If \(t\geq a\), then

$$\begin{aligned} h(t) & = - \frac{1}{\varGamma _{q}(\alpha )} \biggl[ \int ^{a}_{0} ( t -qs)^{( \alpha - 1)} v(s) \,\mathrm{d}_{q} s \\ &\quad {} + \int ^{t}_{a} ( t -qs)^{(\alpha - 1)} v(s) \,\mathrm{d} _{q}s \biggr] \\ &\quad {} - \frac{t}{\varGamma _{q}(\alpha )} \int ^{a}_{0} (a- qs)^{( \alpha - 1)} v(s) \,\mathrm{d}_{q}s \\ &\quad {} + \frac{1}{\varGamma _{q}( \alpha )} \biggl[ \int ^{a}_{0} ( 1 - qs)^{( \alpha - 1)} v(s) \,\mathrm{d}_{q}s \\ &\quad {} + \int ^{t}_{a} ( 1 - qs)^{(\alpha - 1)} v(s) \,\mathrm{d} _{q}s + \int ^{1}_{t} ( 1 - qs)^{(\alpha - 1)} v(s) \,\mathrm{d}_{q}s \biggr] \\ &\quad {} + \frac{1}{\varGamma _{q}(\alpha )} \int ^{a}_{0} ( a- qs )^{( \alpha - 1)} v(s) \,\mathrm{d}_{q}s \\ & = \frac{1}{ \varGamma _{q}(\alpha )} \int ^{a}_{0} \bigl[ (1 - sq)^{( \alpha -1)} -(t - qs)^{(\alpha -1)} \\ &\quad {} + (1-t) (a - qs)^{(\alpha -1)} \bigr]v(s) \,\mathrm{d}_{q}s \\ &\quad {} + \frac{1}{\varGamma _{q}(\alpha )} \int ^{t}_{a} \bigl[ ( 1 -qs)^{( \alpha -1)} -( t -qs)^{(\alpha -1)} \bigr] v(s) \,\mathrm{d}_{q}s \\ &\quad {} + \frac{1}{ \varGamma _{q}(\alpha )} \int ^{1}_{t} (1-qs)^{ ( \alpha -1)} v(s) \,\mathrm{d}_{q}s. \end{aligned}$$
(6)

If \(t\leq a\), then we have

$$\begin{aligned} h(t) &= -\frac{1}{\varGamma _{q}(\alpha )} \int ^{t}_{0}( t - qs)^{(\alpha - 1)} v(s) \,\mathrm{d}_{q}s \\ &\quad {} - \frac{t}{\varGamma _{q}(\alpha )} \biggl[ \int ^{t}_{0} ( a - qs)^{( \alpha - 1)} v(s) \,\mathrm{d}_{q}s \\ &\quad {} + \int ^{a}_{t} (a- qs)^{(\alpha - 1)} v(s) \,\mathrm{d}_{q}s \biggr] \\ &\quad {} + \frac{1}{\varGamma _{q}(\alpha )} \biggl[ \int ^{t}_{0} ( 1 -qs)^{( \alpha - 1)} v(s) \,\mathrm{d}_{q}s \\ &\quad {} + \int ^{a}_{t} ( 1 -qs)^{(\alpha - 1)} v(s) \,\mathrm{d}_{q}s + \int ^{1}_{a} ( 1 -qs)^{(\alpha - 1)} v(s) \,\mathrm{d}_{q}s \biggr] \\ &\quad {} + \frac{1}{\varGamma _{q}(\alpha )} \biggl[ \int ^{t}_{0} (a - qs)^{( \alpha - 1)} v(s) \,\mathrm{d}_{q}s \\ &\quad {} + \int ^{a}_{t} (a - qs)^{(\alpha - 1)} v(s) \,\mathrm{d}_{q}s \biggr] \\ & = \frac{1 }{\varGamma _{q}(\alpha )} \int ^{t}_{0} \bigl[ ( 1 -qs)^{( \alpha -1)} -( t -qs)^{(\alpha -1)} \\ &\quad {} + (1- t ) (a- qs)^{(\alpha -1)}+ \bigr] v(s) \,\mathrm{d}_{q}s \\ &\quad {} + \frac{1 }{\varGamma _{q}(\alpha )} \int ^{a}_{t} \bigl[ ( 1 -qs)^{( \alpha -1)}- t (a- qs)^{(\alpha -1)} \\ &\quad {} + (a- qs)^{(\alpha -1)} \bigr] v(s) \,\mathrm{d}_{q}s \\ &\quad {} + \frac{1}{ \varGamma _{q}(\alpha )} \int ^{1}_{a} ( 1-qs)^{( \alpha - 1)} v(s) \,\mathrm{d}_{q}s. \end{aligned}$$
(7)

Thus, equations (6) and (7) imply that \(h(t) = \int _{0}^{1} G_{q}^{0}(t,s) v(s) \,\mathrm{d}_{q}s\), and by entering \(h(t)\) in equation (5), we see that

$$ u(t) = \int _{0}^{1} G_{q}^{0}(t,s) v(s) \,\mathrm{d}_{q}s + \int _{0} ^{b} u(s) \,\mathrm{d}s. $$

This implies that

$$\begin{aligned} \int _{0}^{b} u(t) \,\mathrm{d}t & = \int _{0}^{b} \int _{0}^{1} G_{q}^{0}(t,s) v(s) \,\mathrm{d}_{q}s \,\mathrm{d}t + \int _{0}^{b} \int _{0}^{b} u(s) \,\mathrm{d}s \,\mathrm{d}t \\ & = \int _{0}^{1} \biggl[ \int _{0}^{b} G_{q}^{0}(t,s) \,\mathrm{d}t \biggr] v(s) \,\mathrm{d}_{q}s + b \int _{0}^{b} u(s) \,\mathrm{d}s. \end{aligned}$$

Thus, \((1 - b ) \int _{0}^{b} u(t) \,\mathrm{d}t = \int _{0}^{1} [ \int _{0}^{b} G_{q}^{0}(t,s) \,\mathrm{d}t ] v(s) \,\mathrm{d} _{q}s\), and so

$$ \int _{0}^{b} u(t) \,\mathrm{d}t = \int _{0}^{1} \frac{1}{ 1-b} \biggl[ \int _{0} ^{b} G_{q}^{0} (t, s) \,\mathrm{d}t \biggr] v(s) \,\mathrm{d}_{q}s. $$

Hence,

$$\begin{aligned} u(t) &= \int _{0}^{1} G_{q}^{0}(t,s) v(s) \,\mathrm{d}_{q}s + \int _{0}^{1} \frac{1}{ 1 -b } \biggl[ \int _{0}^{b} G_{q}^{0}(t,s) \,\mathrm{d}t \biggr] v(s) \,\mathrm{d}_{q}s \\ & = \int _{0}^{1} \biggl[ G_{q}^{0}(t,s) + \frac{1}{ 1 - b } \int _{0} ^{b} G_{q}^{0}(t,s) \,\mathrm{d}t \biggr] v(s) \,\mathrm{d}_{q}s \\ & = \int _{0}^{1} G_{q}(t,s) v(s) \,\mathrm{d}_{q}s = \int _{0}^{1} G _{q}(t,s) v_{0}(s) \,\mathrm{d}_{q}s. \end{aligned}$$

This completes the proof. □

Lemma 4

Let\(G_{q}(t,s)\)be given in Lemma 3. Then

$$ 0 \leq G_{q}(t,s) \leq A_{1}(\alpha , b) (1-qs)^{(\alpha - 1)}, $$

\(\vert \frac{\partial G_{q}}{\partial t}(t,s) \vert \leq A_{2}( \alpha , b) (1-qs)^{(\alpha - 1)}\), where

$$\begin{aligned} A_{1}(\alpha , b) & = \frac{3}{ (1 - b) \varGamma _{q}(\alpha ) }, \\ A_{2}(\alpha , b) & = \frac{2}{ (1 - b ) \varGamma _{q}(\alpha -1)}, \end{aligned}$$

and finally

$$ 0 \leq \frac{(1-qs)^{(\alpha - 1)}}{\varGamma _{q}(\alpha )} \biggl[ - t + \frac{ 2 -b^{2}}{ 2 (1 - b)} \biggr] \leq G_{q}(t,s) $$
(8)

for\(t, s \in \overline{J}\).

Proof

We consider some cases. If \(0 \leq s \leq t \leq 1\) and \(s \leq a\), then \((a - qs)^{\alpha - 1} \geq t(a - qs)^{(\alpha - 1)}\) and \((1 - qs)^{( \alpha - 1)} \geq ( t - qs)^{(\alpha - 1)}\). Hence,

$$ (1 - qs)^{(\alpha - 1)} + (1-t) ( a -qs)^{(\alpha - 1)} -(t - qs)^{( \alpha - 1)} \geq 0 $$

and so \(G_{q}^{0}(t,s) \geq 0\). Thus, \(G_{q}(t,s) \geq 0\). In other cases, the proof is easy. One can see that \(G_{q}^{0}(t,s) \leq 3 (1- qs)^{(\alpha -1)}\) for each \(t,s \in \overline{J}\), and so

$$\begin{aligned} G_{q}(t,s) & \leq 3( 1 - q s)^{( \alpha -1)} + \frac{1}{( 1 -b) \varGamma _{q}(\alpha ) } \int _{0}^{b} 3 ( 1 -qs)^{(\alpha -1)} \,\mathrm{d}t \\ & = 3( 1 - qs)^{(\alpha -1)} + \frac{ 3b ( 1 - qs)^{ (\alpha -1)}}{ (1- b) \varGamma _{q}(\alpha ) } \\ & = \frac{3( 1 -qs)^{(\alpha -1) }}{ (1 - b) \varGamma _{q}(\alpha )} = A _{1}(\alpha , b) (1- qs)^{(\alpha -1)}. \end{aligned}$$

From q-Green function \(G_{q}^{0}(t,s)\) be given in Lemma 3, since

$$ \frac{\partial G_{q}^{0}(t,s)}{\partial t} = \textstyle\begin{cases} \frac{- ( t - qs)^{(\alpha -2)} -(a - qs)^{(\alpha -1)} }{ \varGamma _{q}( \alpha - 1)},& 0\leq s \leq t \leq 1, s\leq a, \\ \frac{ - ( t -qs)^{(\alpha -2)}}{\varGamma _{q}(\alpha - 1)},& 0 \leq s \leq a \leq t \leq 1, \\ \frac{-( a- qs)^{(\alpha -1)}}{\varGamma _{q}(\alpha )},& 0\leq t \leq s \leq a \leq 1, \\ 0, & 0\leq t \leq s \leq 1, a\leq s, \end{cases} $$

we have

$$ \biggl\vert \frac{\partial G_{q}^{0}(t,s)}{\partial t} \biggr\vert \leq \frac{(t-qs)^{( \alpha -1)} +(a- qs)^{ (\alpha -1)}}{\varGamma _{q}( \alpha - 1)} \leq \frac{ 2 (1 - qs)^{(\alpha -1)}}{ \varGamma _{q}(\alpha - 1)}, $$

and so

$$\begin{aligned} \biggl\vert \frac{\partial G_{q} (t,s)}{\partial t} \biggr\vert & \leq \frac{ 2 (1-qs)^{ \alpha - 1}}{\varGamma _{q}(\alpha - 1)} + \frac{ 2 b (1 -qs)^{(\alpha - 1)}}{ 1 - b} \\ & =\frac{ 2 ( 1 -qs)^{(\alpha -1)}}{\varGamma _{q}(\alpha - 1)} \biggl[ 1 + \frac{b}{1 -b} \biggr] \\ & = \frac{2(1-qs)^{(\alpha -1)}}{(1 -b) \varGamma _{q}(\alpha - 1)} = A _{2}(\alpha , b) (1-qs)^{(\alpha -1)}. \end{aligned}$$

If \(0< s< t< 1\) and \(s\leq a\), then \(t - st >0\), and so \(t ( 1 -qs) - s + t>0\). Hence, \(s- t < (1- qs)t\) and \(t(1-qs) > t-qs \). Since \(t<1\) and \(\alpha \geq 2\),

$$ \biggl( \frac{1-qs}{t-qs} \biggr)^{(\alpha -1)} > \biggl( \frac{1}{t} \biggr)^{(\alpha -1)} > \frac{1}{t}, $$

and so

$$\begin{aligned} &(1 - qs)^{(\alpha -1)} - ( t-qs)^{(\alpha -1)}+ (1- t) (a-qs)^{( \alpha - 1)} \\ &\quad > (1-qs)^{(\alpha -1)} -t(1-qs)^{(\alpha -1)} + ( 1 - t) ( a- qs)^{( \alpha -1)} \\ &\quad = (1 - t) \bigl((1- qs)^{(\alpha -1)} +(a-qs)^{(\alpha -1)} \bigr) \\ &\quad \geq (1-t) (1-qs)^{(\alpha -1)}. \end{aligned}$$

Thus, \(G_{q}^{0}(t,s) > (1-t) (1-qs)^{(\alpha -1)}\). If \(0< s\leq a < t <1\), then

$$\begin{aligned} -(t-qs)^{(\alpha -1)} +( 1 - qs)^{(\alpha -1)} &> - t ( 1-qs)^{( \alpha -1)}+ (1-qs)^{(\alpha -1)} \\ & =(1-t) (1-qs)^{(\alpha -1)}, \end{aligned}$$

and so \(G_{q}^{0}(t,s) > \frac{ (1- t)(1-qs)^{(\alpha -1)}}{ \varGamma _{q}(\alpha )}\). Hence,

$$\begin{aligned} G_{q}(t,s) &\geq \frac{1}{\varGamma _{q}(\alpha )} \biggl[ (1- t) (1 - qs)^{( \alpha -1)} + \frac{1}{1 -b} \int _{0}^{b} ( 1 - t) ( 1 -qs)^{(\alpha -1)} \,\mathrm{d}t \biggr] \\ & = \frac{1}{\varGamma _{q}(\alpha )} \biggl[(1-t) (1 - qs)^{ (\alpha -1)} + \frac{(1-qs)^{(\alpha -1)}}{ 1 - b} \biggl( b-\frac{b^{2}}{2} \biggr) \biggr] \\ & = \frac{(1 - qs)^{(\alpha -1)}}{\varGamma _{q} ( \alpha )} \biggl[ - t + \frac{ 2 -b^{2}}{2( 1 - b)} \biggr] \geq 0, \end{aligned}$$

and so inequality (8) holds. □

Consider the self-map \(T : \overline{\mathcal{B}} \to \overline{ \mathcal{B}}\) defined by

$$ T_{u}(t) = \int _{0}^{1} G_{q}(t,s) \omega \bigl(s, u(s), u'(s), D_{q} ^{\beta _{1}} u(s), I_{q}^{\beta } u(s) \bigr) \,\mathrm{d}s, $$
(9)

where \(G_{q}(t,s)\) is the q-Green function in Lemma 3. By applying Lemma 3, one can easily see that the fractional q-integro-differential equation (1) has a solution if and only if T has a fixed point.

Here, we provide our first result about the existence of solutions for problem (1).

Theorem 5

Assume that the mapTis defined by equation (9) and\(\omega : \overline{J} \times \overline{\mathcal{A}}^{4} \to \mathbb{R}\)is a singular function at some points\(t \in \overline{J}\), \(\mu _{1}, \ldots , \mu _{4} \in \overline{\mathcal{L}}\)are some nonnegative real-valued maps. Then fractional differential pointwise defined equation (1) under boundary conditions (2) has a solution whenever the following assumptions hold:

  1. (1)

    The functionωsatisfies the contraction condition

    $$ \bigl\vert \omega ( t, u_{1}, \ldots , u_{4}) -\omega (t, v_{1}, \ldots , v _{4}) \bigr\vert \leq \sum _{i=1}^{4} \mu _{i}(t) \Vert u_{i} - v_{i} \Vert $$

    for all\(u_{1}, \ldots , u_{4}, v_{1},\ldots ,v_{4} \in \overline{ \mathcal{B}}\)and\(t \in \overline{J}\).

  2. (2)

    There exist a natural number\(k_{0}\), some functions\(\gamma _{1}, \ldots , \gamma _{k_{0}} \in \overline{\mathcal{L}}\), \(\varTheta _{1}, \ldots , \varTheta _{k_{0}} : \mathbb{R}^{4} \to [0,\infty )\), nonnegative maps\(\gamma _{1}, \ldots , \gamma _{k_{0}}\), and nonnegative and nondecreasing maps in their all components\(\varTheta _{1}, \ldots , \varTheta _{k_{0}}\)such that

    $$ \bigl\vert \omega ( t,u_{1},\ldots , u_{4}) \bigr\vert \leq \sum_{i=1}^{k _{0}} \gamma _{i}(t) \varTheta _{i} (u_{1}, \ldots , u_{4}) $$

    for all\((u_{1},\ldots , u_{4}) \in \overline{\mathcal{B}}^{4}\)and\(t\in \overline{J}\)and\(\lim_{w\to \infty } \frac{ \varTheta _{i} (w,w,w,w)}{w} = \eta _{0}\), where\(\eta _{0}\)is a nonnegative real number with

    $$ 0 \leq \eta _{0} \leq \frac{m_{0}}{M(\alpha , b) \sum_{i =1}^{k_{0}} \Vert \gamma _{i} \Vert + \delta _{0}} $$

    for some\(\delta _{0} > 0\), \(M(\alpha , b)= \max \{ A_{1}(\alpha , b), A_{2}(\alpha , b)\}\), and

    $$ m_{0}= \min \bigl\{ 1, \varGamma _{q}(2 -\beta _{1}), \varGamma _{q}(\beta _{2} +1) \bigr\} . $$
  3. (3)

    We have

    $$ \tau (\alpha , b)= \biggl[\hat{\mu }_{1} + \hat{\mu }_{2} + \frac{ \hat{\mu }_{3}}{ \varGamma _{q}( 2 -\beta _{1})} + \frac{ \hat{\mu }_{4}}{ \varGamma _{q}(\beta _{2} + 1 )} \biggr] M( \alpha , b) < 1, $$

    where\(\hat{\mu }_{i}= \int _{0}^{1} (1-qs)^{(\alpha - 1)}\mu _{i}(s) \,\mathrm{d}_{q}s = \varGamma _{q}(\alpha ) I_{q}^{\alpha }\mu _{i}(1)\).

Proof

Let \(u_{1}, u_{2} \in \overline{\mathcal{B}}\) and t belong to . Then we obtain

$$\begin{aligned} \bigl\vert T_{ u_{1}}(t) - T_{u_{2}}(t) \bigr\vert &\leq \int _{0}^{1} G_{q}(t,s) \bigl\vert \omega \bigl(u_{1}(s), u'_{1}(s), D_{q}^{\beta _{1}} u_{1} (s), I _{q}^{\beta _{2}} u_{1}(s) \bigr) \\ &\quad {} - \omega \bigl( u_{2} (s), u'_{2}(s), D_{q}^{\beta _{1}} u_{2}(s), I_{q}^{\beta _{2}} u_{2} (s) \bigr) \bigr\vert \,\mathrm{d}_{q}s \\ & \leq \int _{0}^{1} A_{1}(\alpha , b) ( 1 -qs)^{(\alpha -1)} \bigl( \mu _{1}(s) \Vert u_{1} - u_{2} \Vert + \mu _{2}(s) \bigl\Vert u'_{1} - u'_{2} \bigr\Vert \\ &\quad {} + \mu _{3} (s) \bigl\Vert D_{q}^{\beta _{1}} u_{1} - D_{q}^{\beta _{1}} u _{2} \bigr\Vert - \mu _{4}(s) \bigl\Vert I_{q}^{\beta _{2}} u_{1} - I_{q}^{\beta _{2}} u _{2} \bigr\Vert \bigr) \,\mathrm{d}_{q}s. \end{aligned}$$

Since

$$\begin{aligned} \bigl\vert I_{q}^{\beta _{2}} u(t) \bigr\vert & \leq \frac{1}{ \varGamma _{q}(\beta _{2})} \int _{0}^{t} (t-qs)^{(\beta _{2})} \bigl\vert u(s) \bigr\vert \,\mathrm{d}_{q}s \\ & \leq \frac{ \Vert u \Vert }{ \varGamma _{q}(\beta _{2})} \biggl( \frac{1}{\beta _{2}} \bigl[ (t-qs)^{(\beta _{2})}|_{0}^{t} \bigr] \biggr) = \frac{ \Vert u \Vert }{ \varGamma _{q}(\beta _{2} + 1)} t^{\beta _{2}}, \end{aligned}$$

\(\varGamma _{q}(\beta _{2} + 1) \|I_{q}^{\beta _{2}} u\| \leq \| u\|\). Hence,

$$ \varGamma _{q} (\beta _{2} + 1) \bigl\Vert I_{q}^{\beta _{2}} u_{1} - I_{q}^{\beta _{2}} u_{2} \bigr\Vert = \varGamma _{q} (\beta _{2} + 1) \bigl\Vert I_{q}^{\beta _{2}}(u_{1} - u_{2}) \bigr\Vert \leq \Vert u_{1} - u_{2} \Vert . $$

Similarly, one can conclude that \(\varGamma _{q}( 2 - \beta _{1})\|D_{q} ^{\beta _{1}} u_{1} - D_{q}^{\beta _{1}} u_{2}\| \leq \| u_{1} - u_{2} \|\) for each \(u_{1}, u_{2} \in \overline{\mathcal{B}}\). Therefore

$$\begin{aligned} \bigl\vert T_{u_{1}}(t) - T_{u_{2}}(t) \bigr\vert &\leq A_{1}(\alpha , b) \int _{0}^{1} \biggl[\mu _{1} (s) \Vert u_{1} - u_{2} \Vert + \mu _{2} (s) \bigl\Vert u'_{1} - u'_{2} \bigr\Vert \\ &\quad {} + \mu _{3} (s) \frac{ \Vert u'_{1} - u'_{2} \Vert }{ \varGamma _{q}(2 - \beta _{1})} + \mu _{4}(s) \frac{ \Vert u_{1} - u_{2} \Vert }{ \varGamma _{q}(\beta _{2} + 1)} \biggr] ( 1 -qs)^{(\alpha -1)} \,\mathrm{d}_{q}s \\ & = A_{1}(\alpha , b) \int _{0}^{1} \biggl[ \biggl( \mu _{1}(s)+ \frac{ \mu _{4} (s) }{ \varGamma _{q}(\beta _{2} + 1 )} \biggr) ( 1 -qs)^{(\alpha -1)} \Vert u_{1} - u_{2} \Vert \\ &\quad {} + \biggl( \mu _{2} (s) + \frac{ \mu _{3}(s)}{ \varGamma _{q}(2- \beta _{1})} \biggr) ( 1 -qs)^{(\alpha -1)} \bigl\Vert u'_{1} - u'_{2} \bigr\Vert \biggr] \,\mathrm{d}_{q}s \\ & \leq A_{1}(\alpha , b) \Vert u_{1} - u_{2} \Vert _{*} \int _{0}^{1} \biggl[ \mu _{1} (s) + \mu _{2} (s) \\ &\quad {} + \frac{ \mu _{3}(s)}{ \varGamma _{q}( 2 -\beta _{1})} + \frac{ \mu _{4}(s)}{ \varGamma _{q}( \beta _{2} + 1)} \biggr]( 1 -qs)^{(\alpha -1)} \,\mathrm{d}_{q}s \\ & = A_{1}(\alpha , b) \biggl[ \hat{\mu }_{1} + \hat{ \mu }_{2} + \frac{ \hat{\mu }_{3}}{ \varGamma _{q}(2 - \beta _{1})} + \frac{\hat{\mu }_{4}}{ \varGamma _{q}( \beta _{2} + 1)} \biggr] \Vert u_{1} - u_{2} \Vert _{*}. \end{aligned}$$
(10)

Also, we have

$$\begin{aligned} \bigl\vert T'_{u_{1}}(t) - T'_{u_{2}}(t) \bigr\vert &\leq \int _{0}^{1} \biggl\vert \frac{ \partial G_{q}(t,s)}{ \partial t} \biggr\vert | \omega \bigl( s, u_{1}(s), u_{1}(s), D_{q}^{\beta _{1}} u_{1}(s), I_{q}^{\beta _{2}} u_{1}(s) \bigr) \\ &\quad {} - \omega \bigl(s, u_{2}(s), u_{2}(s), D_{q}^{\beta _{1}} u_{2}(s), I_{q}^{\beta _{2}} u_{2}(s)\bigr) | \,\mathrm{d}_{q}s \\ & \leq \int _{0}^{1} A_{2}(\alpha , b) ( 1 -qs)^{(\alpha -1)} \bigl[ \mu _{1} (s) \Vert u_{1} - u_{2} \Vert + \mu _{2}(s) \bigl\Vert u'_{1} - u'_{2} \bigr\Vert \\ &\quad {} + \mu _{3} (s) \bigl\Vert D_{q}^{\beta _{1}} u_{1} -D_{q}^{\beta _{1}} u _{2} \bigr\Vert + \mu _{1}(s) \bigl\Vert I_{q}^{\beta _{2}} u_{1} - I_{q}^{\beta _{2}} u _{2} \bigr\Vert \bigr] \,\mathrm{d}_{q}s \\ & \leq A_{2}(\alpha , b) \int _{0}^{1} \biggl[ \mu _{1}(s) \Vert u_{1} - u _{2} \Vert + \mu _{2} (s) \bigl\Vert u'_{1} - u'_{2} \bigr\Vert \\ &\quad {} + \mu _{3} (s) \frac{ \Vert u'_{1} - u'_{2} \Vert }{ \varGamma _{q}( 2 - \beta _{1})} + \mu _{4}(s) \frac{ \Vert u_{1} - u_{2} \Vert }{\varGamma _{q}(\beta _{2} + 1)} \biggr] (1-qs)^{(\alpha -1) } \,\mathrm{d}_{q}s \\ & = A_{2}(\alpha , b) \biggl[ \hat{\mu }_{1} + \hat{ \mu }_{2} + \frac{ \hat{\mu }_{3}}{\varGamma _{q}(2 - \beta _{1})} + \frac{\hat{\mu }_{4}}{ \varGamma _{q}( \beta _{2} + 1)} \biggr] \Vert u_{1} - u_{2} \Vert _{*}. \end{aligned}$$
(11)

By using (10) and (11), we obtain

$$\begin{aligned} \Vert T_{u_{1}} - T_{u_{2}} \Vert _{*} & = \max \bigl\{ \Vert T_{u_{1}} - T _{u_{2}} \Vert , \bigl\Vert T'_{u_{1}} - T'_{u_{2}} \bigr\Vert \bigr\} \\ & \leq M(\alpha , b) \biggl[\hat{\mu }_{1} + \hat{\mu }_{2} + \frac{ \hat{\mu }_{3}}{ \varGamma _{q}(2 - \beta _{1})} + \frac{ \hat{\mu }_{4}}{ \varGamma _{q}(\beta _{2} + 1)} \biggr] \Vert u_{1} - u_{2} \Vert _{*}. \end{aligned}$$

Hence, \(\| T_{u_{1}} - T_{u_{2}} \|_{*} \to 0\) as \(\| u_{1} - u_{2} \|_{*} \to 0\), and so T is continuous. Since \(\eta _{0} M(\alpha , b) \sum_{i=1}^{k_{0}} \| \gamma _{i}\| < m_{0}\), we can choose \(\varepsilon _{0} > 0\) such that

$$ (\eta _{0} + \varepsilon _{0}) M(\alpha , b) \sum _{i=1}^{k_{0}} \Vert \gamma _{i} \Vert < m_{0}. $$

Since \(\frac{\varTheta (w,w,w,w)}{w} \to \eta _{0}\) as \(w \to \infty \), there exists \(r = \Delta (\varepsilon _{0})>0\) such that \(\frac{\varTheta ( w, w, w, w)}{w} < \eta _{0} + \varepsilon _{0}\) for all \(w \geq \Delta (\varepsilon _{0})\). So

$$ \varTheta (w,w,w,w) < (\eta _{0} + \varepsilon _{0})w $$
(12)

for \(w\geq \Delta (\varepsilon _{0})\). Put \(B_{r} = \{ u\in \overline{ \mathcal{B}} : \| u\|_{*} < r \}\) and define \(\alpha : \overline{ \mathcal{B}}^{2} \to [0, \infty )\) by \(\alpha (u, v) = 1\) whenever \(u, v \in B_{r}\) and \(\alpha (u, v)=0\) otherwise. If \(\alpha (u, v) \geq 1\), then \(\|u\|_{*} \) and \(\|v\|_{*} \) are less than r. Let \(t \in \overline{J}\). Then we obtain

$$\begin{aligned} \bigl\vert T_{u}(t) \bigr\vert & \leq \int _{0}^{1} G_{q}(t,s) \bigl\vert \omega \bigl( s, u(s), u'(s), D_{q}^{\beta _{1}} u(s), I_{q}^{\beta _{2}} u(s) \bigr) \bigr\vert \,\mathrm{d}_{q}s \\ & \leq A_{1}(\alpha , b) \int _{0}^{1} (1 -qs)^{(\alpha - 1)} \\ &\quad{} \times \sum_{i=1}^{k_{0}} \gamma _{i} (s) \varTheta _{i} \bigl(u(s), u'(s), D_{q}^{\beta _{1}} u(s), I_{q}^{\beta _{2}} u(s) \bigr) \,\mathrm{d}_{q}s \\ & \leq A_{1}(\alpha , b) \sum_{i=1}^{k_{0}} \int _{0}^{1} (1 -qs)^{( \alpha - 1) } \gamma _{i} (s) \\ &\quad{} \times \varTheta _{i} \biggl( \Vert u \Vert , \bigl\Vert u'_{1} \bigr\Vert , \frac{ \Vert u' \Vert }{ \varGamma _{q}( 2 -\beta _{1})}, \frac{ \Vert u \Vert }{ \varGamma _{q}(\beta _{2} + 1)} \biggr) \,\mathrm{d}_{q}s \\ & \leq A_{1}(\alpha , b) \sum_{i=1}^{k_{0}} \varTheta _{i} \biggl( r, r, \frac{r}{ \varGamma _{q}(2-\beta _{1})}, \frac{ r}{\varGamma _{q}(\beta _{2} + 1)} \biggr) \\ &\quad{} \times \int _{0}^{1} \gamma _{i}(s) \sup ( 1 -qs)^{(\alpha - 1)} \,\mathrm{d}_{q}s \\ & \leq A_{1}(\alpha , b) \sum_{i=1}^{k_{0}} \varTheta _{i} \biggl( \frac{r}{ m_{0}}, \frac{r}{m_{0}}, \frac{r}{ m_{0}}, \frac{r}{m_{0}} \biggr) \Vert \gamma _{i} \Vert _{1}, \end{aligned}$$

where

$$ m_{0} = \min \bigl\{ 1, \varGamma _{q}(\beta _{2} +1), \varGamma _{q}(2 - \beta _{1}) \bigr\} = \min \bigl\{ \varGamma _{q}(\beta _{2} +1), \varGamma _{q}(2- \beta _{1}) \bigr\} . $$

Since \(r >m_{0} r\), by using (12) we obtain

$$ \varTheta _{i} \biggl( \frac{r}{m_{0}}, \frac{r}{m_{0}}, \frac{r}{m_{0}}, \frac{r}{m _{0}} \biggr)< (\eta _{0} + \varepsilon _{0}) \frac{r}{m_{0}}, $$

and so

$$\begin{aligned} \bigl\vert T_{u}(t) \bigr\vert & \leq A_{1}( \alpha , b) \sum_{i=1}^{k_{0}} \frac{r}{m _{0}} (\eta _{0} + \varepsilon _{0}) \Vert \gamma _{i} \Vert _{1} \\ & = (\eta _{0} + \varepsilon _{0}) \biggl[ \frac{A_{1}(\alpha , b) \sum_{i=1}^{k_{0}} \Vert \gamma _{i} \Vert _{1} }{m_{0}} \biggr]r < r. \end{aligned}$$

Hence, \(\|T_{u}\| \leq r\). Also, one can conclude that

$$\begin{aligned} \bigl\vert T'_{u}(t) \bigr\vert & \leq \int _{0}^{1} \biggl\vert \frac{\partial G_{q}( t,s)}{ \partial t} \biggr\vert \bigl\vert \omega \bigl( s, u(s), u'(s), D_{q}^{\beta _{1}} u(s), I_{q}^{\beta _{2}} u(s) \bigr) \bigr\vert \,\mathrm{d}_{q}s \\ & \leq A_{2}(\alpha , b) \int _{0}^{1} ( 1 -qs)^{(\alpha - 1)} \\ &\quad{} \times \sum_{i=1}^{k_{0}} \gamma _{i}(s) \varTheta _{i} \bigl( u(s), u'(s), D_{q}^{\beta _{1}} u(s), I_{q}^{\beta _{2}} u(s) \bigr) \,\mathrm{d}_{q}s \\ & \leq A_{2}(\alpha , b) \Biggl[\sum_{i=1}^{k_{0}} \varTheta _{i} \biggl( r, r, \frac{ r}{ \varGamma _{q}(2-\beta _{1})}, \frac{r}{\varGamma _{q}(\beta _{2} +1)} \biggr) \Biggr] \\ &\quad{} \times \int _{0}^{1} (1-qs)^{(\alpha - 1)} \gamma _{i}(s) \,\mathrm{d}_{q}s \\ &\leq A_{2}(\alpha , b) \sum_{i=1}^{k_{0}} \varTheta _{i} \biggl( \frac{r}{m _{0}},\frac{r}{m_{0}}, \frac{ r}{m_{0}}, \frac{ r }{ m_{0}} \biggr) \int _{0}^{1} \Vert \gamma _{i} \Vert _{1} \\ & \leq (\eta _{0} + \varepsilon _{0}) \biggl[ \frac{ A_{2}(\alpha , b) \sum_{i=1}^{k_{0}} \Vert \gamma _{i} \Vert _{1} }{m_{0}} \biggr] r < r \end{aligned}$$

and \(\|T_{u}\|_{*} = \max \{ \|T_{u}\|,\|T'_{u}\|_{*} \} \leq r\). This implies that \(T_{u} \) and so \(T_{v} \in B_{r}\), that is, \(\alpha ( T _{u}, T_{v}) \geq 1\). Thus, T is α-admissible. Since \(B_{r} \neq \emptyset \), there exists \(u_{0} \in B_{r}\) such that \(T_{u_{0}} \in B_{r}\). Hence, \(\alpha ( u_{0}, T_{u_{0}} ) \geq 1\). Put \(\psi (t) = \tau (\alpha , b) t\) for each \(t \in [0, \infty )\), here \(\tau (\alpha , b) <1\). Since

$$ \sum_{n=1}^{\infty } \psi ^{n}(t) = \sum_{n=1}^{\infty } \tau (\alpha , b)^{n} t = \biggl( \frac{ \tau (\alpha , b) }{1 - \tau (\alpha , b) } \biggr) t < \infty $$

and \(\psi : [0,\infty ) \to [0,\infty )\) is nondecreasing, we get \(\psi \in \varPsi \). Note that

$$\begin{aligned} \bigl\vert T_{u}(t)- T_{v}(t) \bigr\vert & \leq \int _{0}^{1} G_{q}(t,s) \bigl\vert \omega \bigl(s, u(s), u'(s), D_{q}^{ \beta _{1}} u(s), I_{q}^{\beta _{2}} u(s) \bigr) \\ &\quad {} - \omega \bigl( s, v(s), v'(s), D_{q}^{ \beta _{1}} v(s), I _{q}^{\beta _{2}} v (s) \bigr) \bigr\vert \,\mathrm{d}_{q}s \\ & \leq A_{1}(\alpha , b) \biggl[ \hat{\mu }_{1} + \hat{ \mu }_{2} + \frac{ \hat{\mu }_{3}}{ \varGamma _{q}( 2 - \beta _{1})} + \frac{\hat{\mu }_{4}}{ \varGamma _{q}(\beta _{2} +1)} \biggr] \Vert u - v \Vert _{*}, \end{aligned}$$

and so

$$ \Vert T_{u} - T_{v} \Vert \leq A_{1}(\alpha , b) \biggl[ \hat{\mu }_{1} + \hat{\mu }_{2} + \frac{ \hat{\mu }_{3}}{ \varGamma _{q}(2 - \beta _{1})} + \frac{ \hat{\mu }_{4}}{ \varGamma _{q}( \beta _{2} +1)} \biggr] \Vert u - v \Vert _{*}. $$
(13)

Also,

$$\begin{aligned} \bigl\vert T'_{u}(t)- T'_{v}(t) \bigr\vert & \leq \int _{0}^{1} \biggl\vert \frac{ \partial G _{q} (t,s)}{ \partial t} \biggr\vert \bigl\vert \omega \bigl(s, u(s), u'(s), D _{q}^{\beta _{1}} u(s), I_{q}^{\beta _{2}} u(s) \bigr) \\ &\quad {} - \omega \bigl( s, v(s), v'(s), D_{q}^{\beta _{1}} v(s), I_{q} ^{\beta _{2}} v(s) \bigr) \bigr\vert \,\mathrm{d}_{q}s \\ & \leq A_{2}(\alpha , b) \biggl[\hat{\mu }_{1} + \hat{ \mu }_{2} + \frac{ \hat{\mu }_{3}}{ \varGamma _{q}(2- \beta _{1})} + \frac{\hat{\mu }_{4}}{ \varGamma _{q}(\beta _{2} + 1)} \biggr] \Vert u - v \Vert _{*}. \end{aligned}$$

This implies

$$ \bigl\Vert T'_{u}- T'_{v} \bigr\Vert \leq A_{2}(\alpha , b) \biggl[ \hat{\mu }_{1} + \hat{\mu }_{2} + \frac{ \hat{\mu }_{3} }{ \varGamma _{q}(2 -\beta _{1}) } + \frac{ \hat{\mu }_{4} }{ \varGamma _{q}(\beta _{2} +1) } \biggr] \Vert u - v \Vert _{*}. $$
(14)

Thus, equations (13) and (14) imply that

$$\begin{aligned} \Vert T_{u}- T_{v} \Vert _{*} & \leq M(\alpha , b) \biggl[ \hat{\mu }_{1} + \hat{\mu }_{2} + \frac{ \hat{\mu }_{3} }{ \varGamma _{q}( 2 -\beta _{1})} + \frac{ \hat{ \mu }_{4}}{ \varGamma _{q}(\beta _{2} +1)} \biggr] \Vert u - v \Vert _{*} \\ & = \tau (\alpha , b) \Vert u - v \Vert _{*} = \psi \bigl( \Vert u - v \Vert _{*} \bigr) \end{aligned}$$

for u and v in \(B_{r}\). Hence, \(\alpha (u, v) \|T_{u} - T_{v}\| _{*} \leq \psi ( \rho (u, v))\) for each \(u, v \in \overline{ \mathcal{B}}\). By using Lemma 2, T has a fixed point, which is a solution for problem (1). □

Theorem 6

Letωbe a real-valued function on\(\overline{J} \times \overline{ \mathcal{A}}^{4}\). Then the pointwise defined problem (1) with boundary conditions (2) has a solution whenever the following assumptions hold:

  1. (1)

    There exist natural numbers\(k_{1}\), some maps\(\varTheta _{1}, \ldots , \varTheta _{k_{1}}:\mathbb{R}^{4} \to \mathbb{R}\)which are nondecreasing in their all components, \(\varTheta _{i}(w, w, w, w) \geq 0 \)for all\(w\geq 0\)and\(\frac{ \varTheta _{i} (w, w, w, w)}{w} \to \eta _{i}\)as\(w \to 0^{+}\)for some\(\eta _{i} \in [0, 1) \) (\(i=1, \ldots , k_{1}\)), and there are some nonnegative real-valued functions\(\mu _{1}, \ldots , \mu _{k_{1}} : \overline{J} \to [0, \infty )\)such that

    $$\begin{aligned} &\bigl\vert \omega (t, u_{1},u_{2}, u_{3}, u_{4}) - \omega (t, v_{1}, v_{2}, v _{3}, v_{4}) \bigr\vert \\ &\quad \leq \sum_{i=1}^{k_{1}} \mu _{i} (t) \varTheta _{i}( u_{1} - v_{1}, u _{2}- v_{2}, u_{3}- v_{3}, u_{4}- v_{4}) \end{aligned}$$

    for all\(u_{1}, \ldots , u_{4}, v_{1},\ldots , v_{4} \in \overline{ \mathcal{B} }\)with\(u_{j} \geq v_{j} \geq 0\) (\(j=1,\ldots ,4\)) and\(t \in \overline{J}\).

  2. (2)

    If\(M(\alpha , b)= \max \{ A_{1}(\alpha , b), A_{2}(\alpha , b)\}\), then

    $$ M(\alpha , b) \sum_{i=1}^{k_{1}} \bigl\Vert (1-qt)^{(\alpha -1)} \mu _{i} \bigr\Vert _{1} \leq 1. $$

Proof

Since \(\lim_{w \to 0^{+}} \frac{\varTheta _{i}(w, w, w, w)}{ w} = \eta _{i} < 1\) for \(i=1,\ldots , k_{1}\), for each \(\varepsilon _{i} > 0\) there exist \(\delta _{i} = \delta (\varepsilon _{i}) > 0\) such that \(\frac{w}{m_{0}} \in (0, \delta _{i})\) implies

$$ \varTheta _{i} \biggl( \frac{w}{m_{0}}, \ldots , \frac{w}{ m_{0}} \biggr) \leq (\eta _{i} + \varepsilon _{i}) \frac{w}{m_{0}}, $$

where \(m_{0} = \min \{ \varGamma _{q}(2 -\beta _{1}), \varGamma _{q} (\beta _{2} + 1)\}\). Let \(\varepsilon ^{0}_{i}\) be such that \(\eta _{i} + \varepsilon ^{0}_{i}< 1\) and \(\delta ^{0}_{i} = \delta ( \varepsilon ^{0}_{i})\). Put \(\eta = \max \{ \eta _{1}, \ldots , \eta _{k_{1}}\}\), \(\varepsilon _{0} = \min \{ \varepsilon ^{0}_{1}, \ldots , \varepsilon ^{0}_{k_{1}}\}\), and \(\delta = \min \{ \delta ^{0}_{1} , \ldots , \delta ^{0}_{ k_{1}}, \varepsilon _{0} \}\). Thus, \(\eta + \varepsilon _{0} < 1\) and

$$ \varTheta _{i} \biggl(\frac{w}{ m_{0}}, \ldots , \frac{w}{m_{0}} \biggr) < ( \eta _{i} + \varepsilon _{0}) \frac{ w}{m_{0}} $$

for \(\frac{w}{ m_{0}} \in (0, \delta )\) and \(1 \leq i \leq k_{1}\). Also,

$$ \varTheta _{i} \biggl(\frac{ \delta }{ m_{0}}, \ldots , \frac{ \delta }{ m _{0}} \biggr) < ( \eta + \varepsilon _{0} ) \frac{ \delta }{ m_{0}} m _{0} = (\eta + \varepsilon _{0} ) \delta \leq (\eta + \varepsilon _{0}) \varepsilon _{0}. $$

Now, we define the map \(\alpha : \overline{\mathcal{B}} \times \overline{ \mathcal{B}} \to [ 0,\infty )\) by \(\alpha (u, v) =1\) whenever \(\| u - v\|_{*} \leq \delta \) and \(\alpha (u, v)=0\) otherwise. If \(\alpha (u, v) \geq 1\), then \(\| u - v\|_{*} \leq \delta \), and so

$$\begin{aligned} \bigl\vert T_{u}(t) - T_{v}(t) \bigr\vert & \leq \int _{0}^{1} G_{q}(t,s) \bigl\vert \omega \bigl(s, u(s), u'(s), D_{q}^{\beta _{1}} u(s), I_{q}^{\beta _{2}} u( s) \bigr) \\ &\quad {} - \omega \bigl( s, v(s), v'(s), D_{q}^{\beta _{1}} v(s), I_{q} ^{\beta _{2}} v(s) \bigr) \bigr\vert \,\mathrm{d}_{q}s \\ &\leq \int _{0}^{1} G_{q}(t,s) \sum _{i=1}^{k_{1}} \mu _{i}(s) \\ &\quad{} \times \bigl\vert \varTheta _{i} \bigl( (u - v) (s), (u - v)'(s), D_{q} ^{\beta _{1}}(u - v) (s), I_{q}^{\beta _{2}} (u - v) (s) \bigr) \bigr\vert \,\mathrm{d}_{q}s \\ & \leq A_{1}(\alpha , b) \int _{0}^{1} (1-qs)^{(\alpha -1)} \sum _{i=1} ^{k_{1}} \mu _{i}(s) \\ &\quad{} \times \bigl\vert \varTheta _{i} \bigl( \Vert u -v \Vert , \bigl\Vert ( u - v)' \bigr\Vert , \bigl\Vert D _{q}^{\beta _{1}}( u - v) \bigr\Vert , \bigl\Vert I_{q}^{\beta _{2}} (u - v) \bigr\Vert \bigr) \bigr\vert \,\mathrm{d}_{q}s \\ & \leq A_{1}(\alpha , b) \sum_{i=1}^{k_{1}} \biggl( \int _{0}^{1} (1 - qs)^{( \alpha -1)} \mu _{i}(s) \,\mathrm{d}_{q}s \biggr) \\ &\quad{} \times \varTheta _{i} \biggl(\delta , \delta , \frac{ \delta }{ \varGamma _{q}( 2 -\beta _{1})}, \frac{ \delta }{\varGamma _{q}(\beta _{2} + 1)} \biggr) \\ &\leq A_{1}(\alpha , b) \sum_{i=1}^{k_{1}} \biggl( \int _{0}^{1} (1-qs)^{( \alpha -1)} \mu _{i}(s) \,\mathrm{d}_{q}s \biggr) \\ &\quad{} \times \varTheta _{i} \biggl(\frac{\delta }{ m_{0}}, \frac{\delta }{m_{0}}, \frac{\delta }{m_{0}}, \frac{\delta }{m_{0}} \biggr) \\ & \leq A_{1}(\alpha , b) \sum_{i=1}^{k_{1}} \bigl\Vert (1-qt)^{(\alpha -1)} \mu _{i} \bigr\Vert _{1} ( \eta +\varepsilon _{0}) \delta \\ &\leq A_{1}(\alpha , b) \sum_{i=1}^{k_{1}} \bigl\Vert (1-qt)^{(\alpha -1)} \mu _{i} \bigr\Vert _{1} \delta \leq \delta . \end{aligned}$$
(15)

Hence, \(\|T_{u} - T_{v}\|_{*} \leq \delta \), which implies \(\alpha (T _{u}, T_{v})=1\). If

$$ \lambda = M(\alpha , b) \sum_{i=1}^{k_{1}} \bigl\Vert (1-qt)^{(\alpha - 1)} \mu _{i} \bigr\Vert _{1} (\eta + \varepsilon _{0}), $$

then by using the assumption we get \(\lambda <1\). If \(\psi (t) = \lambda t\), then \(\psi \in \varPsi \). If \(\|u - v\| \leq \delta \), then

$$ \Vert T_{u}- T_{v} \Vert \leq A_{1}(\alpha , b) \sum_{i=1}^{k_{1}} \bigl\Vert (1-qt)^{( \alpha -1)} \mu _{i} \bigr\Vert _{1} ( \eta + \varepsilon _{0}) \Vert u - v \Vert _{*} \leq \lambda \Vert u - v \Vert _{*} $$
(16)

and

$$ \bigl\Vert T'_{u} - T'_{v} \bigr\Vert \leq A_{2}(\alpha , b) \sum_{i=1}^{ k_{1}} \bigl\Vert (1 - qt)^{(\alpha -1)} \mu _{i} \bigr\Vert _{1} (\eta + \varepsilon _{0}) \Vert u - v \Vert _{*} \leq \lambda \Vert u - v \Vert _{*}. $$
(17)

Thus, from inequality (16), we have \(\|T_{u} - T_{v}\|_{*} \leq \lambda \|u - v\|_{*}= \psi (\| u - v\|_{*})\) and so \(\alpha ( u, v) \|T_{u} - T_{v}\|_{*} \leq \psi ( \|u - v\|_{*})\) for each \(u, v\in \overline{\mathcal{B}}\). Let \(\varepsilon > 0\) be given. Since \(\varTheta _{i} (\frac{w}{m_{0}}, \ldots , \frac{w}{m_{0}}) \to 0\) as \(w \to 0^{+} \), for each \(i=1, \ldots , k_{1}\), there exists \(\delta _{i} > 0\) such that \(\varTheta _{i} ( \frac{w}{ m_{0}}, \ldots , \frac{w}{ m_{0} }) < \frac{\varepsilon }{k'}\) for all \(0 < w \leq \delta _{i}\), where

$$ \lambda ' = \Biggl[M(\alpha , b) \sum _{i=1}^{k_{1}} \bigl\Vert (1-qt)^{(\alpha -1)} \mu _{i} \bigr\Vert _{1} \Biggr] + 1. $$

If \(\delta = \min \{\delta _{i}: 1\leq i \leq k_{1} \}\), then \(\varTheta _{i} (\frac{w}{m_{0}}, \ldots , \frac{w}{m_{0}}) < \frac{ \varepsilon }{\lambda '}\) for all \(w \in (0, \delta ]\) and \(i=1, \ldots , k_{1}\). If \(u_{n} \to u\), then there exists a natural number \(n_{0}\) such that \(\|u_{n} - u\|_{*} < \delta \) for all \(n \geq n_{0}\). Hence,

$$\begin{aligned} &\bigl\vert T_{u_{n}}(t) - T_{u}(t) \bigr\vert \\ &\quad \leq \int _{0}^{1} G_{q}(t,s) \bigl\vert \omega \bigl(s, u_{n}(s), u'_{n}(s), D_{q}^{\beta _{1}} u_{n}(s), I_{q}^{\beta _{2}} u_{n}(s)\bigr) \\ &\qquad {} - \omega \bigl(s, u(s), u'(s), D_{q}^{\beta _{2}} u(s), I_{q} ^{\beta _{2}} u(s) \bigr) \bigr\vert \,\mathrm{d}s \\ &\quad \leq \int _{0}^{1} G_{q}(t,s) \sum _{i=1}^{k_{1}} \mu _{i}(s) \\ &\qquad{} \times \varTheta _{i} \bigl( \Vert u_{n} -u \Vert , \bigl\Vert (u_{n} - u)' \bigr\Vert , \bigl\Vert D _{q}^{\beta _{1}}( u_{n} -u) \bigr\Vert , \bigl\Vert I_{q}^{\beta _{2}} (u_{n}- u) \bigr\Vert \bigr) \,\mathrm{d}s \\ &\quad \leq A_{1}(\alpha , b) \sum_{i=1}^{k_{1}} \biggl[ \varTheta _{i} \biggl( \delta , \delta , \frac{\delta }{ \varGamma _{q}( 2- \beta _{1})}, \frac{ \delta }{\varGamma _{q}(\beta _{2} +1)} \biggr) \\ &\qquad{} \times \int _{0}^{1} (1-qs)^{(\alpha -1)} \mu _{i}(s) \,\mathrm{d}_{q}s \biggr] \\ & \quad \leq A_{1}(\alpha , b) \sum_{i=1}^{k_{1}} \biggl[ \varTheta _{i} \biggl( \frac{ \delta }{ m_{0}}, \frac{ \delta }{m_{0}}, \frac{\delta }{ m _{0}}, \frac{\delta }{ m_{0}} \biggr) \\ &\qquad{} \times \int _{0}^{1} (1-qs)^{(\alpha -1)} \mu _{i} (s) \,\mathrm{d}_{q}s \biggr] \\ & \quad \leq A_{1}(\alpha , b) \frac{\varepsilon }{k'} \sum _{i=1}^{k_{1}} \bigl\Vert (1 - qt)^{(\alpha -1)} \mu _{i} \bigr\Vert _{1} < \varepsilon \end{aligned}$$
(18)

for \(t \in \overline{J}\) and \(n \geq n_{0}\). By repeating the similar method, we obtain

$$\begin{aligned} \bigl\vert T'_{u_{n}}(t) - T'_{u}(t) \bigr\vert & \leq \int _{0}^{1} \biggl\vert \frac{ \partial G_{q} (t,s)}{\partial t} \biggr\vert \bigl\vert \omega \bigl(s, u_{n}(s), u'_{n}(s), D_{q}^{\beta _{1}} u_{n}(s), I_{q}^{\beta _{2}} u_{n}(s) \bigr) \\ &\quad {} - \omega \bigl( s, u(s), u'(s), D_{q}^{\beta } u(s), I_{q} ^{\beta _{2}} u(s) \bigr) \bigr\vert \,\mathrm{d}_{q}s \\ & \leq A_{2}(\alpha , b) \frac{\varepsilon }{k'} \sum _{i=1}^{k_{1}} \bigl\Vert ( 1 -qt)^{(\alpha -1)} \mu _{i} \bigr\Vert _{1} < \varepsilon \end{aligned}$$
(19)

for all t in and \(n \geq n_{0}\). Now, (18) and (19) imply that \(\|T_{u_{n}} - T_{u} \| \leq \varepsilon \) and \(\|T'_{u_{n}} - T'_{u} \| \leq \varepsilon \), respectively, for \(n \geq n_{0}\). Thus,

$$ \Vert T_{u_{n}} - T_{u} \Vert _{*} = \max \bigl\{ \Vert T_{u_{n}} - T_{u} \Vert , \bigl\Vert T'_{u _{n}} - T'_{u} \bigr\Vert \bigr\} \leq \varepsilon $$

for \(n \geq n_{0}\), and so \(T_{u_{n}} \to T_{u}\) as \(u_{n} \to u\). Indeed, T is continuous. Now, we show that there exists \(u_{0} \in \overline{\mathcal{B}}\) such that \(\alpha (u_{0} , T_{u_{0}}) = 1\). In this way, we have to show that \(\|T_{u_{0}}- u_{0}\| \leq \delta \) for some \(u_{0}\in \overline{\mathcal{B}}\). Let \(r_{0} > 0\) be a fixed real number. Since \(\varTheta _{i}(w, w, w, w) \to 0\) as \(w \to 0^{+}\), for each \(\varepsilon > 0\), there exists \(n = n(\varepsilon )\) such \(0 < w \leq \frac{ r_{0}}{n}\) implies \(\varTheta _{i}( w, w, w, w) \leq \varepsilon \) for all \(1 \leq i \leq k_{1}\). Hence, \(\varTheta _{i}(\frac{ r_{0}}{ n }, \ldots , \frac{r_{0}}{n}) \leq \varepsilon \). Put

$$ M = \max \biggl\{ \frac{1}{\varGamma _{q}( 2 - \beta _{1})}, \frac{1}{\varGamma _{q}(\beta _{2} +1) }, 1 \biggr\} = \max \biggl\{ \frac{1}{ \varGamma _{q}(2- \beta _{1})}, \frac{ 1}{ \varGamma _{q}(\beta _{2} +1)} \biggr\} $$

and choose \(\varepsilon _{M}\) such that

$$ \sum_{i=1}^{k_{1}} \bigl\Vert (1-qt)^{(\alpha -1)} \mu _{i} \bigr\Vert _{1} \varepsilon _{M} M(\alpha , b) < \delta . $$

Take \(n_{1} = n(\varepsilon _{M})\) and choose a natural number \(n_{2}\) such that

$$ \sum_{i=1}^{k_{1}} \bigl\Vert ( 1 -qt)^{(\alpha -1)} \mu _{i} \bigr\Vert _{1} \varepsilon _{M} M(\alpha , b) \leq \delta - \frac{1}{n_{2}}. $$

If \(n_{0} = \max \{n_{1}, n_{2}\}\), then \(\varTheta _{i} ( \frac{M}{n _{0}}, \ldots , \frac{M}{n_{0}} ) \leq \varepsilon _{M}\) for \(1=1, \ldots , k_{1}\). Define

$$ u_{0} (t)= \textstyle\begin{cases} 0,& t \leq \frac{1}{n_{0} +1}, \\ \frac{6 n_{0}^{2}}{6 n_{0}^{2} + 5n_{0} +2} \\ \quad {} \times [ \frac{t^{3}}{3} - \frac{2n_{0} +1}{2n_{0} (n_{0} +1)} t^{2} + \frac{t}{n_{0} (n_{0} +1)} ] +\frac{1}{n_{0} +2},& \frac{1}{n_{0} +1} < t < \frac{1}{n_{0}}, \\ \frac{1}{n_{0}}, & \frac{1}{n_{0}} \leq t. \end{cases} $$

One can easily see that \(u_{0} \in \overline{\mathcal{A}}\) and \(u(t) \in [0, \frac{1}{n_{0}} ]\), \(u_{0} ( \frac{1}{n _{0} + 1} )= 0\) and

$$ u'_{0} (t)= \textstyle\begin{cases} 0, & t \leq \frac{1}{n_{0} +1}, \\ \frac{6 n_{0}^{2}}{6 n_{0}^{2} + 5n_{0} +2} [ t^{2} - \frac{2n _{0} +1}{n_{0} (n_{0} +1)} t + \frac{1}{n_{0} (n_{0} +1)} ], & \frac{1}{n_{0} +1} < t < \frac{1}{n_{0}}, \\ 0, & \frac{1}{n_{0}} \leq t. \end{cases} $$

Hence, \(u'_{0} \) belongs to \(\overline{\mathcal{A}}\) and \(u'_{0}(\frac{1}{n _{0}+ 1} )= u'_{0}(\frac{1}{n_{0}}) = 0\). Thus, \(u_{0} \in \overline{ \mathcal{B}}\). Also, we have

$$\begin{aligned} u'_{0}(t) & \leq \frac{6 n_{0}^{2}}{6 n_{0}^{2} + 5n_{0} +2} \biggl( \frac{1}{n ^{2}_{0}} + \frac{1}{n^{4}_{0}(n_{0} +1)} - \frac{ 2n_{0} +1}{ 2n_{0} (n_{0}+1)^{2}} \biggr) \\ & \leq 1 \times \frac{1}{n_{0}} \times \biggl( \frac{1}{n_{0}} + \frac{1}{n ^{3}_{0}(n_{0} +1)} - \frac{2n_{0} }{ (n_{0}+1)^{2}} \biggr) \leq \frac{1}{n _{0}}, \end{aligned}$$

and so \(n_{0} \|u_{n_{0}} \|_{*} \leq 1\). This implies that

$$\begin{aligned} &\bigl\vert T_{u_{0}}(t) - u_{0}(t) \bigr\vert \\ &\quad = \biggl\vert \int _{0}^{1} G_{q}(t,s) \omega \bigl( s, u_{0}(s), u'_{0}(s), D_{q}^{\beta _{1}} u_{0}(s), I_{q}^{\beta _{2}} u_{0}(s) \bigr) \,\mathrm{d}_{q}s - u_{0}(s) \biggr\vert \\ &\quad \leq \int _{0}^{1} G_{q}(t,s) \bigl\vert \omega \bigl(s, u_{0}(s), u'_{0}(s), D_{q}^{\beta } u_{0}(s), I_{q}^{\beta _{2}} u_{0}(s) \bigr) \,\mathrm{d}_{q}s \bigr\vert + \frac{1}{n_{0}} \\ &\quad \leq A_{1}(\alpha , b) \int _{0}^{1} ( 1 -qs)^{(\alpha -1)} \\ &\qquad{} \times \Biggl[ \sum_{i=1}^{k_{1}} \mu _{i}(s) \varTheta _{i} \biggl( \Vert u_{0} \Vert , \bigl\Vert u'_{0} \bigr\Vert , \frac{ \Vert u'_{0} \Vert }{ \varGamma _{q}(2 - \beta _{1})}, \frac{ \Vert u_{0} \Vert }{ \varGamma _{q}(\beta _{2} +1)} \biggr) \Biggr] \,\mathrm{d}_{q}s + \frac{1}{n_{0}} \\ &\quad \leq A_{1} (\alpha , b) \sum_{i=1}^{k_{1}} \biggl[ \biggl( \int _{0} ^{1} (1-qs)^{(\alpha -1)} \mu _{i}(s) \,\mathrm{d}_{q}s \biggr) \\ &\qquad{} \times \varTheta _{i} \biggl( \frac{r_{0}}{n_{0}}, \frac{r_{0}}{n _{0}}, \frac{r_{0}}{n_{0}}, \frac{r_{0}}{n_{0}} \biggr) \biggr] + \frac{1}{n _{0}} \\ &\quad \leq A_{1}(\alpha , b) \Biggl[\sum_{i=1}^{k_{1}} \bigl\Vert ( 1 -qt)^{( \alpha -1)} \mu _{i} \bigr\Vert \Biggr] \varepsilon _{M} + \frac{1}{n_{0}} \leq \delta \end{aligned}$$
(20)

for \(t \in \overline{J}\), and so \(\|T_{u_{0}} -u_{0} \| \leq \delta \). By using a similar method, we get

$$\begin{aligned} \bigl\vert \bigl( T_{u_{0}}(t) - u_{0}(t) \bigr)' \bigr\vert &= \bigl\vert T'_{u_{0}}(t) - u'_{0}(t) \bigr\vert \\ & \leq \int _{0}^{1} \biggl\vert \frac{\partial G_{q}}{\partial t} (t,s) \biggr\vert \bigl\vert \omega \bigl( s, u_{0}(s), u'_{0}(s), D_{q}^{\beta _{1}} u _{0} (s), I_{q}^{\beta _{2}} u_{0}(s) \bigr) \,\mathrm{d}_{q}s \bigr\vert + \frac{1}{n_{0}} \\ & \leq A_{2}(\alpha , b) \Biggl[ \sum_{i=1}^{k_{1}} \bigl\Vert (1-qt)^{( \alpha -1)} \mu _{i} \bigr\Vert \Biggr] \varepsilon _{M} + \frac{1}{n_{0}} \leq \delta , \end{aligned}$$
(21)

and so \(\|(T_{u_{0}} -u_{0})' \| \leq \delta \). Hence, from equations (20) and (21), we obtain \(\|T_{u_{0}} - u_{0} \|_{*} = \max \{\|T_{u_{0}} - u_{0} \|, \|(T_{u_{0}} - u_{0})' \| \} \leq \delta \). Thus, \(\alpha (u_{0}, T_{u_{0}}) = 1\). Now, by using Lemma 2, the map T has a fixed point, which is a solution for the multi-singular fractional q-problem (1). □

Now, we present our final result.

Theorem 7

Assume that\(\omega : \overline{J} \times \overline{ \mathcal{B} } ^{4} \to [0, \infty ]\)is such that\(\omega ( t, u_{1}, u_{2}, u_{3}, u_{4}) < \infty \)for all\(u_{1}, u_{2}, u_{3}, u_{4}\in \overline{ \mathcal{B}}\)and\(t\in E\), where\(E^{c}\)is a null subset of, that is, the measure of\(E^{c}\)is zero, the map\(\omega (t, u_{1}, u_{2}, u_{3}, u_{4})\)is continuous with respect to the components\(u_{1}\), \(u_{2}\), \(u_{3}\), and\(u_{4}\)for all\(t\in E\). Then the pointwise defined problem (1) with boundary conditions (2) has a solution whenever the following assumptions hold:

  1. (1)

    There exist a natural number\(n_{1}\geq 1\)and some maps\(\mu _{1}, \ldots , \mu _{n_{1}} : \overline{J} \to [0,\infty )\)such that\(\mu _{1}, \ldots , \mu _{n_{1}} \in \overline{\mathcal{L}}\), the maps\(F_{1}, \ldots , F_{n_{1}} : \mathbb{R}^{4}\to [0,\infty )\)and\(\varOmega : \mathbb{R}^{4} \to [0,\infty )\)so that

    $$\begin{aligned}& \Vert \varOmega \Vert _{1}^{*} =\sup _{x \in \overline{\mathcal{L}} } \int _{0} ^{1} \varOmega \bigl( u(t), u(t), u(t), u(t) \bigr) \,\mathrm{d}t < \infty , \\& \Vert F_{i} \Vert _{\infty } = \sup _{w \in \mathbb{R}} \bigl\{ F_{i}( w, w, w, w)\bigr\} < \infty \end{aligned}$$

    for\(i=1,\ldots , n_{1}\)and

    $$ \bigl\vert \omega ( t, u_{1}, u_{2}, u_{3}, u_{4}) \bigr\vert \leq \sum _{i=1}^{n_{1}} \mu _{i}(t) F_{i} (u_{1}, u_{2}, u_{3}, u_{4}) + \varOmega (u_{1}, u_{2}, u_{3}, u_{4}) $$

    for\(u_{1}, \ldots , u_{4} \in \overline{\mathcal{B}}\)and\(t \in \overline{J}\).

  2. (2)

    There exist some maps\(\psi :\mathbb{R}^{4} \to [0,\infty )\)and\(h : \overline{J} \to [0,\infty )\)such that\(\| h \|_{1}^{L} = \varGamma _{q}(\alpha ) I_{q}^{\alpha }h(1) < \infty \)and\(h(t) \psi (u _{1}, \ldots , u_{4}) \leq \omega (t, u_{1}, \ldots , u_{4})\)for\(u_{1}, \ldots , u_{4} \in \overline{\mathcal{B}}\)and\(t \in \overline{J}\).

  3. (3)

    There exist\(\gamma _{1}, \gamma _{2}, \gamma _{3}, \gamma _{4} \in \overline{\mathcal{L}}\)and\(\phi : [0, \infty ) \to [0, \infty )\)such that

    $$ M( \alpha , b) \sum_{i=1}^{4} \Vert \gamma _{i} \Vert _{1} < 1, $$

    \(\phi _{m_{0}} \in \varPsi \)and

    $$ \bigl\vert \omega ( t, u_{1}, \ldots , u_{4}) - w(t, v_{1}, \ldots , v _{4}) \bigr\vert \leq \sum _{i=1}^{4} \gamma _{i}(t) \phi \bigl( \Vert u_{i} - v _{i} \Vert \bigr) $$

    for all\((u_{1}, \ldots , u_{4})\)and\((v_{1}, \ldots , v_{4}) \in \overline{ \mathcal{B}}^{4}\)with\(\| u_{i}\|, \|v_{i}\| \in [\delta _{1}, \delta _{2}]\), where\(\phi _{\lambda }(z):= \phi (\frac{z}{\lambda })\)for all\(\lambda \in (0, \infty )\),

    $$ \Vert \psi \Vert _{m}:= \min \bigl\{ \psi (u_{1}, \ldots , u_{4}): (u_{1}, \ldots , u_{4}) \in \mathbb{R}^{4} \bigr\} , $$

    \(2\delta _{1} \varGamma _{q}(\alpha ) (1- \alpha ) \leq \| \psi \|_{m} \|h \|_{1}^{L} ( 4 - \alpha ^{2} - 2\alpha )\)and

    $$ \delta _{2} \geq M(\alpha ,b) \Biggl( \sum _{i=1}^{ n_{1}} \Vert F_{i} \Vert _{\infty } \Vert \mu _{i} \Vert _{1} + \Vert \varOmega \Vert _{1}^{*} \Biggr). $$

Proof

Let \(\{u_{n}\}\) be a sequence such that \(\|u_{n} - u\|_{*} \to 0\). Then \(u_{n} \to u\) and \(u'_{n} \to u'\). By using the inequalities \(\varGamma _{q}(2 - \beta _{1}) \|D_{q}^{\beta _{1}}( u_{n} - u) \| \leq \|(u _{n} -u)' \| \) and

$$ \varGamma _{q}(\beta _{2} + 1) \bigl\Vert I_{q}^{\beta _{2}}( u_{n} -u) \bigr\Vert \leq \Vert u _{n} -u \Vert , $$

we get \(D_{q}^{\beta } u_{n} \to D_{q}^{\beta } u\) and \(I_{q}^{\beta _{2}} u_{n} \to I_{q}^{\beta _{2}} u\). Since \(\omega (t, u_{1}, \ldots , u_{4})\) is continuous with respect to \(u_{1}, \ldots , u_{4}\) for all \(t \in E\), we can conclude that

$$ \omega \bigl( t, u_{n}, u'_{n}, D_{q}^{\beta _{1}} u_{n}, I_{q}^{\beta _{2}} u_{n} \bigr) \to \omega \bigl( t, u, u', D_{q}^{\beta _{1}} u, I_{q}^{\beta _{2}} u \bigr) $$

for \(t\in E\). Let \(u \in \overline{\mathcal{B}}\) be given and \(t \in \overline{J}\). Then we have

$$\begin{aligned} \bigl\vert T_{u}(t) \bigr\vert & \leq A_{1}( \alpha , b) \Biggl[ \int _{0}^{1} ( 1 -qs)^{ \alpha -1} \Biggl( \Biggl[ \sum_{i=1}^{n_{1}} \mu _{i}(s) F_{i} \bigl( u(s), u'(s), D_{q}^{\beta _{1}} u(s), I_{q}^{\beta _{2}} u(s) \bigr) \Biggr] \\ &\quad {} + \varOmega \bigl( u(s), u'(s), D_{q}^{\beta _{1}} u(s), I_{q} ^{\beta _{2}} u(s) \bigr) \Biggr) \,\mathrm{d}_{q}s \Biggr]. \end{aligned}$$

If \(u_{M}(s) := \max \{u(s),u'(s), D_{q}^{\beta _{1}} u(s), I_{q}^{ \beta _{2}} u(s)\}\), then \(u_{M} \in \overline{\mathcal{A}}\), and so

$$\begin{aligned} \bigl\vert T_{u}(t) \bigr\vert & \leq A_{1}( \alpha , b) \Biggl[ \sum_{i=1}^{n_{1}} F_{i} \biggl( \Vert u \Vert , \bigl\Vert u' \bigr\Vert , \frac{ \Vert u' \Vert }{ \varGamma _{q}( 2 - \beta _{1})}, \frac{ \Vert u \Vert }{ \varGamma _{q}(\beta _{2} + 1)} \biggr) \Biggr] \int _{0}^{1} \mu _{i}(s) \,\mathrm{d}s \\ &\quad {} + \int _{0}^{1} \varOmega \bigl( u_{M}(s), u_{M}(s), u_{M}(s), u_{M}(s)\bigr) \,\mathrm{d}s \\ &\leq A_{1}(\alpha , b) \Biggl[ \sum_{i=1}^{n_{1} } \Vert F_{i} \Vert _{\infty } \Vert \mu _{i} \Vert + \Vert \varOmega \Vert _{1}^{*} \Biggr]. \end{aligned}$$

Similarly, one can see that \(|T'_{u}(t)| \leq A_{2}(\alpha , b) [\sum_{i=1}^{n_{1} } \|F_{i}\|_{\infty } \| \mu _{i}\| + \|\varOmega \|_{1} ^{*} ]\). Thus,

$$ \Vert T_{u} \Vert _{*} \leq M( \alpha , b) \Biggl[ \sum_{i=1}^{n_{1}} \Vert F_{i} \Vert _{\infty } \Vert \mu _{i} \Vert + \Vert \varOmega \Vert _{1}^{*} \Biggr] < \infty $$
(22)

for all \(u \in \overline{\mathcal{B}}\). By using the Lebesgue dominated convergence theorem, we conclude that

$$\begin{aligned} T_{u_{n}}(t) & = \int _{0}^{1} G_{q}(t,s) \omega \bigl(s, u_{n}(s), u _{n}'(s), D_{q}^{\beta _{1}} u_{n}(s), I_{q}^{\beta _{2}} u_{n}(s) \bigr) \,\mathrm{d}_{q}s \\ & \to \int _{0}^{1} G_{q}(t,s) \omega \bigl( s, u(s), u'(s), D _{q}^{\beta _{1}} u(s), I_{q}^{\beta _{2}} u(s) \bigr) \,\mathrm{d}_{q}s = T_{u}(t) \end{aligned}$$

for t belonging to , and so the self-map T on \(\overline{\mathcal{B}}\) is continuous. Define the map \(\alpha : \overline{ \mathcal{B}}^{2} \to [0,\infty )\) by \(\alpha (u, v)=1\) whenever \(\| u\|_{*}, \| v\|_{*} \in [\delta _{1}, \delta _{2}]\), \(\alpha (u, v)=0\), otherwise. If \(\alpha (u, v) \geq 1\), then \(\| u\|_{*}, \| v\| _{*} \in [\delta _{1}, \delta _{2}]\), and so

$$\begin{aligned} \bigl\vert T_{u}(t) \bigr\vert & = \biggl\vert \int _{0}^{1} G_{q}(t,s) \omega \bigl(s, u(s), u'(s), D_{q}^{\beta _{1}} u(s), I_{q}^{\beta _{2}} u(s) \bigr) \,\mathrm{d}_{q}s \biggr\vert \\ & \geq \int _{0}^{1} \frac{ (1-qs)^{b-1}}{ \varGamma _{q}(\alpha ) } \biggl[-t + \frac{ 2 -b^{2}}{ 2 ( 1 - b)} \biggr] h(s) \\ &\quad{} \times \psi \bigl[ u(s), u'(s), D_{q}^{\beta _{1}} u(s), I_{q} ^{\beta _{2}} u(s) \bigr] \,\mathrm{d}_{q}s \\ &\geq \Vert \psi \Vert _{m} \biggl[\frac{-t}{ \varGamma _{q}(\alpha )} \int _{0} ^{1} (1- qs)^{ \alpha - 1} h(s) \,\mathrm{d}_{q}s \\ &\quad {} + \frac{ 2 -b^{2}}{ 2 (1 - b) } \int _{0}^{1} (1-qs)^{(b-1)} h(s) \,\mathrm{d}_{q}s \biggr] \\ &\geq \Vert \psi \Vert _{m} \Vert h \Vert _{1}^{L} \biggl[ \frac{1}{ \varGamma _{q}( \alpha )} + \frac{ 2 -b^{2}}{ 2 (1 - b)} \biggr] \\ & = \Vert \psi \Vert _{m} \Vert h \Vert _{1}^{L} \biggl[\frac{4 - b^{2} - 2 b}{2 \varGamma _{q}(\alpha ) (1- b)} \biggr] \end{aligned}$$

for \(t \in \overline{J}\). Thus, \(2 \varGamma _{q}(\alpha ) (1 - b) \|T _{u}\| \geq \|\psi \|_{m} \|h\|_{1}^{L} (4 - b^{2} - 2b) \), and so

$$ \Vert T_{u} \Vert _{*} := \max \bigl\{ \Vert T_{u} \Vert , \bigl\Vert T'_{u} \bigr\Vert \bigr\} \geq \frac{ \Vert \psi \Vert _{m} \Vert h \Vert _{1}^{L} (4 - b^{2} - 2b)}{2 \varGamma _{q}( \alpha ) ( 1 - b)} \geq \delta _{1}. $$

By using (22), we obtain

$$ \Vert T_{x} \Vert _{*} \leq M(\alpha , b) \Biggl( \sum_{i=1}^{n_{1}} \Vert F_{i} \Vert _{\infty } \Vert \mu _{i} \Vert _{1} + \Vert \varOmega \Vert _{1}^{*} \Biggr) \leq \delta _{2}, $$

and so \(\alpha (T_{u}, F_{v}) \geq 1\). If \(u_{0} \in [\delta _{1}, \delta _{2}]\), then it is easy to check that \(\alpha (T_{u_{0}}, u_{0}) \geq 1\). Let \(u, v \in [\delta _{1}, \delta _{2}]\). Then

$$\begin{aligned} &\alpha (u, v) \bigl\vert T_{u}(t) - T_{v}(t) \bigr\vert \\ &\quad \leq \int _{0}^{1} G _{q}(t,s) \bigl\vert \omega \bigl( s, u(s), u'(s), D_{q}^{\beta _{1}} u(s), I _{q}^{\beta _{2}} u(s) \bigr) \\ &\qquad {} - \omega \bigl( s, v(s), v'(s), D_{q}^{\beta _{1}} v(s), I_{q} ^{\beta _{2}} v(s) \bigr) \bigr\vert \,\mathrm{d}_{q}s \\ &\quad \leq A_{1}(\alpha , b) \int _{0}^{1} (1-qs)^{(\alpha -1)} \bigl( \gamma _{1}(s) \phi \bigl( \vert u - v \vert \bigr) + \gamma _{2}(s) \phi \bigl( \bigl\vert u' - v' \bigr\vert \bigr) \\ &\qquad {} + \gamma _{3}(s) \phi \bigl( \bigl\vert D_{q}^{\beta _{1}} u - D_{q}^{ \beta _{1}} v \bigr\vert \bigr) + \gamma _{4}(s) \phi \bigl( \bigl\vert I_{q}^{\beta _{2}} u- I_{q}^{\beta _{2}} v \bigr\vert \bigr) \bigr) \,\mathrm{d}_{q}s \\ & \quad \leq A_{1}(\alpha , b) \int _{0}^{1} (1-qs)^{(\alpha -1)} \biggl( \gamma _{1}(s) \phi \bigl( \Vert u- v \Vert \bigr) + \gamma _{2}(s) \phi \bigl( \bigl\Vert u' - v' \bigr\Vert \bigr) \\ &\qquad {} + \gamma _{3}(s) \phi \biggl( \frac{ \Vert u' - v' \Vert }{ \varGamma _{q}(2 - \beta _{1})} \biggr) + \gamma _{4}(s) \phi \biggl( \frac{ \Vert u- v \Vert }{\varGamma _{q}(\beta _{2}+ 1)} \biggr) \biggr) \,\mathrm{d}_{q}s \\ &\quad \leq A_{1}(\alpha , b) \int _{0}^{1} (1-qs)^{(\alpha -1)} \phi \biggl( \frac{ \Vert u- v \Vert _{*}}{ m_{0}} \biggr) \Biggl[ \sum_{i=1}^{4} \gamma _{i}(s) \Biggr] \,\mathrm{d}_{q}s \\ &\quad \leq A_{1}(\alpha , b) \phi \biggl( \frac{ \Vert u- v \Vert _{*}}{ m_{0}} \biggr) \sum_{i=1}^{4} \int _{0}^{1} ( 1 -qs)^{(\alpha -1)} \gamma _{i}(s) \,\mathrm{d}_{q}s \\ &\quad \leq \sum_{i=1}^{4} \Vert \gamma _{i} \Vert _{1} \phi \biggl( \frac{ \Vert u - v \Vert _{*}}{ m_{0}} \biggr) \leq \phi _{\gamma _{0}}\bigl( \Vert u - v \Vert _{*} \bigr). \end{aligned}$$
(23)

Similarly, we conclude that

$$ \alpha (u, v) \bigl\Vert T'_{u} - T'_{v} \bigr\Vert \leq \phi _{m_{0}} \bigl( \Vert u - v \Vert _{*}\bigr). $$
(24)

Now, (23) and (24) imply that \(\alpha (u , v) \| T _{u} - T_{v} \| \) and \(\alpha ( u, v) \| T'_{u} - T'_{v} \|_{*} \) are less than or equal to \(\phi _{ m_{0}}( \|u - v\|_{*})\) for all \(u, v\in \overline{\mathcal{B}}\). By using Theorem 2, the self-map T has a fixed point, which is a solution for problem (1). □

Examples and algorithms for the problem

Here, we provide some examples to illustrate our main results. In this way, we give a computational technique for checking problem (1). We need to present that a simplified analysis could be executed on values of the q-gamma function. To this aim, we consider a pseudo-code description of the method for calculation of the q-gamma function of order n in Algorithms 2, 3, 4, and 5 (for more details, see the link https://en.wikipedia.org/wiki/Q-gamma_function).

Algorithm 5
figuree

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

Example 1

Consider the following pointwise defined problem, similar to (1):

$$ D_{q}^{\frac{7}{2}} u(t) + \frac{1}{96 \sqrt{\pi } [ g(t)]^{ \frac{1}{3}}} \bigl( \Vert u \Vert + \bigl\Vert u' \bigr\Vert + \bigl\Vert D_{q}^{\frac{2}{3}} u \bigr\Vert + \bigl\Vert I _{q}^{\frac{1}{2}} u \bigr\Vert \bigr) = 0 $$
(25)

with boundary conditions \(u'(0)=u(\frac{5}{6})\), \(u(1)=\int _{0}^{ \frac{1}{2}} u(s) \,\mathrm{d}s\), and \(u''(0)=0\), where \(g(t) = 0\) whenever \(t \in \overline{J} \cap Q\), \(g(t) = t\) whenever \(t \in J \cap Q^{c}\). Let \(\alpha = \frac{7}{2}\), \(\beta _{1}= \frac{2}{3}\), \(\beta _{2} = \frac{1}{3}\), \(a= \frac{5}{6}\), and \(b= \frac{1}{2}\). Then we have

$$\begin{aligned} M(\alpha , b) &= \max \bigl\{ A_{1}(\alpha , b), A_{2}(\alpha , b)\bigr\} \\ & = \max \biggl\{ \frac{3}{(1-b) \varGamma _{q}(\alpha )}, \frac{2}{(1-b) \varGamma _{q}(\alpha -1)} \biggr\} \\ &= \max \biggl\{ \frac{3}{( 1 -\frac{1}{2}) \varGamma _{q}(\frac{7}{2})}, \frac{2}{( 1- \frac{1}{2}) \varGamma _{q}(\frac{5}{2})} \biggr\} \end{aligned}$$
(26)

and

$$ m_{0} = \min \bigl\{ \varGamma _{q}( 2 -\beta _{1}), \varGamma _{q}(\beta _{2} +1) \bigr\} = \min \biggl\{ \varGamma _{q} \biggl( \frac{4}{3} \biggr), \varGamma _{q} \biggl(\frac{3}{2} \biggr) \biggr\} . $$
(27)

Tables 4 and 5 show the values of \(M(\alpha , b)\) in equation (26) and \(m_{0}\) in equation (27), respectively. Put

$$ \omega \bigl( t, u(t), u'(t), D_{q}^{\beta _{1}} u(t), I_{q}^{\beta _{2}} u(t)\bigr) := \frac{1}{96 \sqrt{\pi } [ g(t)]^{\frac{1}{3}}} \bigl( \Vert u \Vert + \bigl\Vert u' \bigr\Vert + \bigl\Vert D_{q}^{\frac{2}{3}} u \bigr\Vert + \bigl\Vert I_{q}^{\frac{1}{2}} u \bigr\Vert \bigr), $$

\(\mu _{1}(t)= \mu _{2}(t) =\mu _{3}(t) =\mu _{4}(t) = \mu (t)= \frac{1}{96\sqrt{ \pi } t^{\frac{1}{3}}}\), \(\gamma _{1}(t) = \gamma _{2}(t) = \gamma _{3}(t) =\gamma _{4}(t) =\gamma (t)= \frac{1}{96\sqrt{\pi } t^{\frac{1}{3}}}\) and \(\varTheta _{i} (u_{1}, u_{2}, u_{3}, u_{4} ):= \|u_{i}\|\) for \(i=1,\ldots ,4\). Then \(\|\mu \|_{1} = \|\gamma \|_{1} = \frac{1}{96\sqrt{ \pi }(1-\frac{1}{3})} = \frac{1}{64\sqrt{\pi } }\),

$$\begin{aligned} &\bigl\vert \omega ( t, u_{1}, u_{2}, u_{3}, u_{4}) - \omega (t, v_{1}, v_{2}, v _{3}, v_{4} ) \bigr\vert \\ &\quad = \frac{1}{96\sqrt{\pi } ( g(t) )^{ \frac{1}{3}}} \sum_{i=1}^{4} \Vert u_{i} \Vert - \Vert v_{i} \Vert \\ &\quad \leq \frac{1}{ 96 \sqrt{\pi } (g(t))^{ \frac{1}{3}}} \sum_{i=1} ^{4} \Vert u_{i} - v_{i} \Vert \\ &\quad = \frac{1}{96\sqrt{\pi } t^{\frac{1}{3}}} \sum_{i=1}^{4} \Vert u _{i} - v_{i} \Vert \\ &\quad = \sum_{i=1}^{4} \mu (t) \Vert u_{i} - v_{i} \Vert , \end{aligned}$$

and

$$\begin{aligned} \bigl\vert \omega ( t, u_{1}, u_{2}, u_{3}, u_{4} ) \bigr\vert &= \frac{1}{96\sqrt{ \pi } (g(t))^{\frac{1}{3}} } \sum_{i=1}^{4} \Vert u_{i} \Vert \\ & = \frac{1}{96\sqrt{\pi }t^{\frac{1}{3}}} \sum_{i=1}^{4} \Vert u_{i} \Vert \\ & = \sum_{i=1}^{4} \gamma (t) \varTheta _{i}( u_{1}, u_{2}, u_{3}, u_{4} ) \end{aligned}$$

for all \((u_{1}, u_{2}, u_{3}, u_{4})\) and \((v_{1}, v_{2}, v_{3}, v _{4}) \in \overline{\mathcal{B}}^{4}\) and \(t \in \overline{J}\). On the other hand, we have \(\lim_{w \to \infty } \frac{\varTheta _{i}(w, w, w, w)}{w} = 1\), and by equations (26) and (27), we get

$$ \varLambda = \frac{m_{0}}{ M(\alpha , b) \sum_{i=1}^{4} \Vert \gamma _{i}(t) \Vert _{1}} = \frac{m_{0}}{ M(\alpha , b) \sum_{i=1}^{4} \Vert \gamma (t) \Vert _{1}} > 1. $$
(28)

Table 6 shows the values of Λ. Choose \(\delta _{0}>0\) such that \(m_{0} \geq M( \alpha , b) \sum_{i=1}^{4} \| \gamma \|_{1}+ \delta _{0}\). Since

$$ \hat{\mu }= \int _{0}^{1} (1-qs)^{(\frac{7}{2}-1)} \mu (s) \,\mathrm{d}_{q}s \leq \int _{0}^{1} \mu (s) \,\mathrm{d}s = \Vert \mu \Vert _{1} = \frac{1}{64 \sqrt{\pi }}, $$

we obtain

$$\begin{aligned} \tau (\alpha , b) & = \biggl[\hat{\mu } + \hat{\mu } + \frac{ \hat{\mu }}{\varGamma _{q}(2 - \beta _{1})} + \frac{ \hat{\mu }}{ \varGamma _{q}(\beta _{2} +1)} \biggr] M( \alpha , b) \\ & \leq \frac{1}{64 \sqrt{\pi }} \biggl[1+ 1 + \frac{1}{ \varGamma _{q}( \frac{4}{3}) } + \frac{1}{ \varGamma _{q}(\frac{3}{2})} \biggr] M( \alpha , b) < 1. \end{aligned}$$
(29)

Table 7 shows the values of \(\tau (\alpha , b)\). Now, by using Theorem 5, the pointwise defined problem (25) has a solution.

Table 4 Some numerical results of \(M(\alpha , b)\) in equation (26) in Example 1 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{8}{9}\)
Table 5 Some numerical results of \(m_{0}\) in equation (27) in Example 1 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{8}{9}\)
Table 6 Some numerical results of \(\varLambda >1\) in equation (28) in Example 1 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{8}{9}\)
Table 7 Some numerical results of \(\tau (\alpha , b)<1\) in equation (29) in Example 1 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{8}{9}\). Note that \((a)=\varGamma _{q}(\beta _{2}+1)\), \((b)= \varGamma _{q}(2-\beta _{1})\), and \((c) = M(\alpha , b)\)

Example 2

Consider the fractional q-integro-differential equation

$$ D_{q}^{\frac{5}{2}} u(t) + \frac{0.09}{ t^{\frac{1}{4}} ( t - \frac{1}{3})^{ \frac{1}{8}}} \biggl[1 - \biggl( \frac{3}{5} \biggr)^{ \frac{2}{5} ( u(t) + u'(t) + D_{q}^{ \frac{1}{4}} u(t) + I_{q} ^{\frac{1}{5}} u(t) ) } \biggr]= 0 $$
(30)

for \(t\in \overline{J}\), with boundary condition \(u'(0) =u( \frac{1}{2})\) and \(u(1)= \int ^{ \frac{1}{7}}_{0} u(s) \,\mathrm{d}s\). Put \(\alpha = \frac{5}{2}\), \(\beta _{1}=\frac{1}{4}\), \(\beta _{2}= \frac{1}{5}\), \(a = \frac{1}{2}\), \(b= \frac{1}{7}\), and \(k_{1}=1\). Note that

$$\begin{aligned} M(\alpha , b) &= \max \biggl\{ \frac{3}{(1-b) \varGamma _{q}(\alpha )}, \frac{2}{(1-b) \varGamma _{q}(\alpha -1)} \biggr\} \\ & = \max \biggl\{ \frac{3}{ ( 1 -\frac{1}{7}) \varGamma _{q}( \frac{5}{2})}, \frac{2}{ (1 -\frac{1}{7}) \varGamma _{q}( \frac{3}{2})} \biggr\} . \end{aligned}$$
(31)

Table 8 shows the values of \(M(\alpha , b)\) in equation (31). Define

$$ \omega (t, u_{1}, \ldots , u_{4})= \frac{ 0.09}{ t^{\frac{1}{4}} ( t - \frac{1}{3} )^{\frac{1}{8}}} \biggl[ 1 - \biggl(\frac{3}{5} \biggr)^{\frac{2}{5} (u_{1} + \cdots + u_{4})} \biggr], $$

\(\mu (t)= \frac{0.09}{ t^{\frac{1}{4}} ( t -\frac{1}{3})^{\frac{1}{8}}}\), and

$$ \varTheta (u_{1}, \ldots , u_{4})=1 - \biggl( \frac{3}{5} \biggr)^{ \frac{2}{5} ( u_{1}+ \cdots + u_{4})}. $$

One can easily see that Θ is nondecreasing in all its components and \(\varTheta (w, w, w, w) \geq 0\) for all \(w \geq 0\). Assume that \((u_{1}, \ldots , u_{4})\) and \((v_{1}, \ldots , v_{4})\) belong to \(\overline{\mathcal{B}}^{4}\) and \(u_{i} \geq v_{i} \geq 0\) for \(i = 1, \ldots , 4\). Since \(( \frac{3}{5} )^{v_{i}} \geq (\frac{3}{5})^{u_{i}}\),

$$ \biggl( \frac{3}{5} \biggr)^{v_{i}} \biggl[ \biggl( \frac{3}{5} \biggr) ^{v_{i}} - \biggl( \frac{3}{5} \biggr)^{u_{i}} \biggr] \leq \biggl(\frac{3}{5} \biggr)^{v_{i}} - \biggl( \frac{3}{5} \biggr) ^{u_{i}}, $$

and so

$$ \biggl(\frac{3}{5} \biggr)^{v_{i}} \biggl[ \biggl( \frac{3}{5} \biggr) ^{v_{i}} - \biggl(\frac{3}{5} \biggr)^{u_{i}} \biggr] \leq \biggl(\frac{3}{5} \biggr)^{v_{i}} \biggl[1 - \biggl(\frac{3}{5} \biggr)^{u_{i} - v_{i}} \biggr]. $$

Thus,

$$ \biggl[ 1 - \biggl(\frac{3}{5} \biggr)^{u_{i} } \biggr] - \biggl[ 1 - \biggl(\frac{3}{5} \biggr)^{v_{i}} \biggr] \leq 1 - \biggl(\frac{3}{5} \biggr)^{ u_{i} -v_{i}} . $$

By replacing \(u_{i}\), \(v_{i}\) with \(\frac{2}{5}\sum_{i=1}^{4} u_{i}\), \(\frac{2}{5} \sum_{i=1}^{4} v_{i}\), respectively, we get

$$ \biggl[1 - \biggl(\frac{3}{5} \biggr)^{\frac{2}{5} \sum _{i=1}^{4} u _{i} } \biggr] - \biggl[1 - \biggl(\frac{3}{5} \biggr)^{ \frac{2}{5} \sum _{i=1}^{4} v_{i}} \biggr] \leq 1 - \biggl(\frac{3}{5} \biggr)^{\frac{2}{5} (\sum _{i=1}^{4} u_{i} - v_{i})}. $$

Hence, \(\varTheta (u_{1}, \ldots , u_{4}) - \varTheta (v_{1}, \ldots , v _{4}) \leq \varTheta (u_{1} - v_{1}, \ldots , u_{4} - v_{4})\). On the other hand, we have

$$ \omega ( t, u_{1}, \ldots , u_{4}) - \omega (t, v_{1}, \ldots , v_{4}) \leq \mu (t) \varTheta (u_{1} - v_{1}, \ldots , u_{4} - v_{4}) $$

and

$$ \lim_{w \to 0^{+}} \frac{\varTheta (w, w, w, w)}{w} = \lim \limits _{w \to 0^{+}} \frac{1 - (\frac{3}{5})^{4\times \frac{2}{5} w}}{w} = - 4 \biggl(\frac{2}{5} \biggr) \biggl[ \ln \biggl( \frac{3}{5} \biggr) \biggr] = 0.81 < 1. $$

Now, by using Theorem 6, the fractional q-integro-differential pointwise defined equation (30) has a solution.

Table 8 Some numerical results of \(M(\alpha , b)\) in equation (31) in Example 2 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{8}{9}\)

Example 3

Consider the fractional q-integro-differential equation

$$ D_{q}^{\frac{5}{2}} u(t) + \frac{0.01}{g(t)} F \bigl( u(t), u'(t), D _{q}^{\frac{1}{3}} u(t), I_{q}^{\frac{2}{3}} u(t) \bigr)+ 2 = 0\quad (t \in \overline{J}) $$
(32)

with boundary condition \(u'(0) = u(\frac{1}{2})\) and \(u(1)= \int ^{\frac{4}{5}}_{0} u(s) \,\mathrm{d}s \), where \(g: \overline{J} \to [0,\infty )\) is defined by \(g(t) =0\) whenever \(t \in Q \cap \overline{J}\) and \(g(t) = \sqrt{t}\) whenever \(t \in Q^{c} \cap \overline{J}\) and the map \(F: \mathbb{R}^{4} \to [0,\infty )\) is defined by

$$ F(u_{1}, \ldots , u_{4}) = \textstyle\begin{cases} \frac{1}{2} \sum_{i=1}^{4} \frac{ \Vert u_{i} \Vert }{ 1 + \Vert u_{i} \Vert }, & u _{1}, \ldots , u_{4} \in [0, 29], \\ \vert \sin (u_{1} + \cdots + u_{4}) \vert , & u_{1}, \ldots , u_{4} \in (-\infty , 0), \\ \frac{-29}{48} ( \frac{1 }{4} \sum_{i=1}^{4} u_{i} - 24 ), & u_{1}, \ldots , u_{4} \in [29, 30], \\ 0,& \text{otherwise}. \end{cases} $$

Put \(\alpha = \frac{5}{2}\), \(\beta _{1}=\frac{1}{3}\), \(\beta _{2}= \frac{2}{3}\), \(a= \frac{1}{2}\), and \(b= \frac{4}{5}\). Then we have

$$\begin{aligned} M(\alpha , b) & = \max \biggl\{ \frac{3}{( 1 -b) \varGamma _{q}(\alpha )}, \frac{2}{( 1 -b) \varGamma _{q}(\alpha -1) } \biggr\} \\ & = \max \biggl\{ \frac{3}{ ( 1 -\frac{4}{5}) \varGamma _{q}( \frac{5}{2})}, \frac{2}{(1 -\frac{4}{5}) \varGamma _{q}(\frac{3}{2})} \biggr\} \end{aligned}$$
(33)

and

$$ m_{0} = \min \bigl\{ \varGamma _{q}(2- \beta _{1}), \varGamma _{q}( \beta _{2} + 1) \bigr\} =\min \biggl\{ \varGamma _{q} \biggl( \frac{5}{3} \biggr), \varGamma _{q} \biggl(\frac{5}{3} \biggr) \biggr\} = \varGamma _{q} \biggl( \frac{5}{3} \biggr). $$
(34)

Tables 9 and 10 show the values of \(M(\alpha , b)\) and \(m_{0}\) in equations (33) and (34), respectively. By simple checking, we can see that \(\|F\|_{\infty } = 1\). Let \(n_{1}=1\). Define the maps \(\psi ( u_{1}, \ldots , u_{4}):= F( u_{1}, \ldots , u_{4})\), \(\mu (t)= h(t)= \gamma (t) := 0.01t\), and \(\phi (t) =\frac{1}{2} t\). If \(\varOmega (u_{1}, \ldots , u _{4})=2\),

$$ \omega ( t, u_{1}, \ldots , u_{4})= \frac{0.01}{g(t)} T(u_{1}, \ldots , u _{4})+ 2, $$

and \([\delta _{1} , \delta _{2}] =[0, 29]\), then \(\omega (t ,u_{1}, u _{2}, u_{3}, u_{4}) < \infty \) for \(u_{1}, \ldots , u_{4} \in \overline{ \mathcal{B}}\) and \(t\in E: = Q^{c}\cap \overline{J}\), \(\omega ( t, u _{1}, u_{2}, u_{3}, u_{4})\) is continuous with respect to the components \(u_{1}\), \(u_{2}\), \(u_{3}\), and \(u_{4}\) for all \(t\in E\),

$$ \omega ( t, u_{1}, \ldots , u_{4} )= \frac{1}{g(t)} F( u_{1}, \ldots , u _{4})+ 2 \geq h(t) \psi ( u_{1}, \ldots , u_{4}) $$

and

$$\begin{aligned} \bigl\vert \omega ( t, u_{1} , \ldots , u_{4}) - \omega (t, v_{1}, \ldots , v _{4}) \bigr\vert & \leq \sum_{i=1}^{4} \frac{1}{2} \times \frac{0.01}{ \sqrt{t}} \Vert u_{i} - v_{i} \Vert \\ & = \sum_{i=1}^{4} \gamma (t) \phi \bigl( \Vert u_{i} - v_{i} \Vert \bigr) \end{aligned}$$

for all \((u_{1}, \ldots , u_{4})\), \((v_{1}, \ldots , v_{4})\in \overline{ \mathcal{B}}^{4}\) and \(t \in \overline{J}\). Note that \(\phi _{m_{0}} \in \varPsi \),

$$\begin{aligned}& M(\alpha , b) \sum_{i=1}^{4} \Vert \gamma _{i} \Vert _{1} = 4 \times 0.005 \times M(\alpha , b) = 0.02 \times M(\alpha , b) < 1, \end{aligned}$$
(35)
$$\begin{aligned}& \frac{ \Vert \psi \Vert _{m} \Vert h \Vert _{1}^{L} (4- \alpha ^{2} - 2\alpha ) }{2 \varGamma _{q}(\alpha ) (1 - \alpha )} \geq 0 = \delta _{1}, \end{aligned}$$
(36)

and

$$ M(\alpha , b) \bigl( \Vert F \Vert _{\infty } \Vert \mu \Vert _{1} + \Vert \varOmega \Vert _{1} ^{*} \bigr) \leq (1 \times 0.02 + 2 )M(\alpha , b) \leq 29 = \delta _{2}. $$
(37)

Table 10 shows the values of equation (37). Now, by using Theorem 7, the fractional q-integro-differential pointwise defined equation (32) has a solution.

Table 9 Some numerical results of \(M(\alpha , b)\) in equation (33) in Example 3 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{8}{9}\)
Table 10 Some numerical results of \(M(\alpha , b)\) in equation (33) in Example 3 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{8}{9}\). Note that \((a) = m_{0}\), \((b) = 0.02 \times M(\alpha , b)\), and \((c) =2.02 \times M(\alpha , b)\)

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Acknowledgements

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

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

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MSC

  • 34A08
  • 34B16
  • 39A13

Keywords

  • Multi-singular
  • Pointwise defined
  • Caputo q-derivation
  • q-integro-differential