## 1 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 $$t\in \overline{J}$$.

## 2 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.

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$$\alpha ( x_{0}, Tx_{0}) \geq 1$$.

## 3 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). □

## 4 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).

### 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.

### 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.

### 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.