1 Introduction

In several areas of sciences, such as biology, chemistry, economics, physics, and engineering, fractional calculus and its relevant differential equations and BVPs have been used extensively [1–3]. Indeed, fractional derivatives are not only a generalization of ordinary derivatives, but also they explain dynamical behavior of various physical processes specifically and effectively (real life phenomena) in contrast to integer order derivatives. References [4–18] are available for some improvements on the fractional differential equations theory.

By virtue of developments in fractional quantum calculus (q-FC), a number of scientists and researchers [19, 20] were attracted to a study of fractional q-difference equations, beginning in the nineteenth century, and wide interest lately [21–23].

In 2007, Atici et al. [24] studied some notions in relation to fractional q-calculus on time scales. Then in 2012, Annaby and Mansour presented their investigations by publishing a book on equations and BVPs in the context of fractional q-calculus [25]. Jarad et al. [26] turned to the stability notion on q-fractional non-autonomous systems and after that, Abdeljawad et al. [27] introduced Gronwall-type inequality in q-operator settings. By combining the two above notions, Butt et al. [28] investigated Ulam stability for a Caputo delay q-difference equation by means of q-Gronwall-type inequality. Also, some fascinating insights concerning IVPs and BVPs containing q-difference equations can be found in [29–35] and the references therein. Ahmad, Nieto, Alsaedi and Al-Hutami [36] turned to the q-difference FBVP with nonlocal integral conditions and implemented an existence analysis on the solutions of the proposed q-BVP which takes the format

$$\begin{aligned} \textstyle\begin{cases} { {}^{C}\mathcal{D}}_{q}^{\rho } ({ {}^{C}\mathcal{D}}_{q}^{\beta }+b )u(t)=f(t, u(t)), \\ u(0)=d_{1}I_{q}^{\gamma -1}u(\theta )=d_{1} \int _{0}^{ \theta }\frac{(\theta -qs)^{(\gamma -2)}}{\Gamma _{q}(\gamma -1)}u(s)\,d_{q}s, \\ u(1)= d_{2}I_{q}^{\gamma -1}u(\pi )=d_{2} \int _{0}^{ \pi }\frac{(\pi -qs)^{(\gamma -2)}}{\Gamma _{q}(\gamma -1)}u(s)\,d_{q}s,\quad \gamma >2, \theta >0, \pi < 1, \end{cases}\displaystyle \end{aligned}$$
(1)

where \(f\in C([0, 1]\times \mathbb{R}, \mathbb{R})\), \(\rho , \beta \in (0, 1]\), \(q\in (0, 1)\), \(b, d_{1}, d_{2} \in \mathbb{R}\) and \({{}^{C}\mathcal{D}}_{q}^{\rho }\), \({{}^{C}\mathcal{D}}_{q}^{\beta }\) denote the q-fractional derivatives in Caputo sense of orders ρ and β.

In 2014, Ahmad et al. [37] studied the existence criteria of the following q-difference equation involving two nonlinear terms and four-point nonlocal boundary conditions:

$$\begin{aligned} \textstyle\begin{cases} { {}^{C}\mathcal{D}}_{q}^{\rho } ({ {}^{C}\mathcal{D}}_{q}^{\beta }+b )u(t)=mf(t, u(t))+nI_{q}^{\gamma }g(t, u(t)), \\ w_{1} u(0)-k_{1}(t^{1-\beta } { {}^{C}\mathcal{D}}_{q}^{1} u(0))|_{t=0}= c_{1}u(r_{1}), \quad 0< r_{1}< 1, \\ w_{2} u(1)+ k_{2} { {}^{C}\mathcal{D}}_{q}^{1} u(1)= c_{2} u(r_{2}),\quad 0< r_{2}< 1, \end{cases}\displaystyle \end{aligned}$$
(2)

in which \(f, g\in C([0, 1]\times \mathbb{R}, \mathbb{R})\), \(\rho , \beta \in (0, 1]\), \(q\in (0, 1)\), \(b, m, n, w_{1}, w_{2}, k_{1}, k_{2} \in \mathbb{R}\), \(c_{1}, c_{2} \in (0, 1)\) and \({{}^{C}\mathcal{D}}_{q}^{\rho }\), \({{}^{C}\mathcal{D}}_{q}^{\beta }\) denotes the q-fractional derivatives in Caputo sense and \(I_{q}^{\gamma }\) denotes the fractional q-integral in Riemann–Liouville sense of order \(\gamma \in (0, 1)\).

In continuation to the investigation of the q-variant of fractional problems and inspired by the aforementioned work, we aim to examine this area from another angle. Several known methods of functional analysis are used to establish required results on the existence of solutions for a class of q-difference problem. More specifically, we consider the sequential four-point Caputo fractional q-difference boundary value problem (q-CFBVP) of the format

$$ \textstyle\begin{cases} { \mathcal{D}}_{q}^{\alpha } ({ \mathcal{D}}_{q}^{\beta }u(t)-g(t, u(t)) )=f(t, u(t)), \quad t\in \mathrm{J}:=[0,T], \\ a_{1} u(0)+b_{1}{ \mathcal{D}}_{q}^{\gamma }u(0)=\lambda _{1} \int _{0}^{\eta _{1}} \frac{(\eta _{1} - qs)^{(\sigma _{1} - 1)}}{\Gamma _{q}(\sigma _{1})}u(s) \,\mathrm{d}_{q}s, \quad \eta _{1} \in (0,T), \sigma _{1}>0, \\ a_{2} u(T)+ b_{2} { \mathcal{D}}_{q}^{\gamma }u(T)= \lambda _{2} \int _{0}^{\eta _{2}} \frac{(\eta _{2} - qs)^{(\sigma _{2} - 1)}}{\Gamma _{q}(\sigma _{2})}u(s) \,\mathrm{d}_{q}s, \quad \eta _{2} \in (0,T), \sigma _{2}>0, \end{cases} $$
(3)

where \({ \mathcal{D}}_{q}^{\mu }\) is the μth-q-difference derivative in the Caputo structure with \(\mu \in \{ \gamma , \beta , \alpha \}\) such that \(0 < \alpha , \beta \leq 1\), \(0 <\gamma \leq 1\) and \(I_{q}^{\theta }\) is the θth-q-difference integral in the Riemann–Liouville structure with \(\theta > 0\) subject to \(\theta \in \{\sigma _{1}, \sigma _{2}\}\) and also \(f,g: \mathrm{J}\times \mathbb{R} \longrightarrow \mathbb{R} \) are continuous functions. \(a_{1}\), \(a_{2}\), \(b_{1}\), \(b_{2}\), \(\lambda _{1}\), \(\lambda _{2}\) are suitably chosen constants in \(\mathbb{R}^{+}\).

Regarding to the novelty of the paper, in comparison to above q-problems, our supposed sequential q-CFBVP is more general. Under the given boundary value conditions, we have used both Caputo and Riemann–Liouville q-fractional operators in four different points of domain of the unknown solution function u simultaneously, in which the linear combinations of the unknown function and its fractional derivative is corresponding to a multiple of q-Riemann–Liouville integral in two mid-points. In this paper, we have designed an extended form of Langevin equations by providing a nonlinear function g in the left-side hand of the given boundary value problem (3). Also, to prove the existence of solutions for such an applied q-problem, we shall utilize some pure notions of functional analysis based on the measure of non-compactness, condensing operators and eigenvalue of the operator, which have been used in papers limited in this regard so far and this distinguishes our research from the work of others. Moreover, we here emphasize that this paper may have useful and effective applications in physics and quantum mechanics such as Langevin systems in the context of quantum operators.

The remaining part of this paper is organized as follows: Sect. 2 is devoted to the primitive notions of q-FC. At first, in Sect. 3, we give an auxiliary lemma which provides the solution of the supposed q-CFBVP (3) and then based on the obtained integral equation, by using fixed point theorems due to Sadovski, Krasnoselskii–Zabreiko and O’Regan, we establish the existence of solutions for the q-CFBVP (3) and also for its uniqueness, we utilize the famous Banach principle. In Sect. 4, the stability criteria of Ulam–Hyers type and its generalized type are checked. Additionally, in Sect. 5, we provide two examples which ensure the usability of the results presented in Sect. 3. The manuscript is ended by our conclusions in Sect. 6.

2 Preliminaries regarding q-operators

We collect some important basic notions of q-FC in this section. For details, we refer to [19, 21, 38, 39]. Let \(q \in (0,1)\). A q-real number is denoted by \([m]_{q}\) and is defined as

$$ [m]_{q}=\frac{1-q^{m}}{1-q},\quad m \in \mathbb{R}. $$

The q-power function \((m-n)^{k}\) with \(m,n \in \mathbb{R}\) is

$$ (m-n)^{(0)}=1,\qquad (m-n)^{(k)}=\prod_{j=0}^{k-1} \bigl(m-n q^{j} \bigr),\quad k \in \mathbb{N} \cup \{ 0\}, $$

and, if \(\beta \in \mathbb{R}\), then

$$ (m-n)^{(\beta )}=m^{\beta } \prod_{i=0}^{\infty } \frac{m-n q^{i}}{m-n q^{\beta +i}}. $$

On the other side, \([c(m-n)]^{(\beta )}=c^{\beta }(m-n)^{(\beta )}\) holds for \(c\in \mathbb{R}\) and also notice that \(m^{(\beta )}=m^{\beta }\) if \(n=0\). The q-Gamma function is given by

$$ \Gamma _{q}(\alpha )=\frac{(1-q)^{(\alpha -1)}}{(1-q)^{\alpha -1}}, \quad \alpha \in \mathbb{R} \backslash \{0,-1, \dots \}, $$

and satisfies \(\Gamma _{q}(\alpha +1)=[\alpha ]_{q} \Gamma _{q}(\alpha )\).

The 1st-q-derivative of an arbitrary mapping ϕ is defined by the following rule:

$$ (D_{q} \phi ) (x)=\frac{ \phi (q x)- \phi (x)}{(1-q) x},\quad x \neq 0, $$

and for the higher orders, it becomes

$$ D_{q}^{0} \phi =\phi ,\qquad D_{q}^{k} \phi =D_{q} \bigl(D_{q}^{k-1} \phi \bigr),\quad k \in \mathbb{N} \cup \{ 0\}. $$

The 1st-q-integral of an arbitrary mapping ϕ given on the interval \([0, n]\) is given by

$$ (I_{q} \phi ) (x)= \int _{0}^{x} \phi (r) \,d_{q} r=x(1-q) \sum_{k=0}^{\infty } \phi \bigl(x q^{k} \bigr) q^{k},\quad x \in [0, n]. $$

If \(m \in [0, n]\), then

$$ \int _{m}^{n} \phi (r) \,d_{q} r= \int _{0}^{n} \phi (r) \,d_{q} r - \int _{0}^{m} \phi (r) \,d_{q} r. $$

Similarly, for the higher orders, it becomes

$$ I_{q}^{0} \phi = \phi , \qquad I_{q}^{k} \phi = I_{q} \bigl(I_{q}^{k-1} \phi \bigr),\quad k \in \mathbb{N} \cup \{ 0\}. $$

For two first order q-operators \(D_{q}\) and \(I_{q}\), we have

$$ D_{q} I_{q} \phi (x)= \phi (x). $$

Here, we assemble some definitions about such q-operators from the fractional point of view.

Definition 1

([39])

Let \(\alpha \geq 0\). The αth-q-integral of the Riemann–Liouville type for ϕ defined on \([0,\infty )\) is given by \(I_{q}^{0} \phi (t)= \phi (t)\) and

$$ I_{q}^{\alpha } \phi (t)=\frac{1}{\Gamma _{q}(\alpha )} \int _{0}^{t}(t-qr)^{( \alpha -1)} \phi (t) \,d_{q} r,\quad \alpha >0. $$

Definition 2

([19])

The Caputo αth-q-derivative for an absolutely continuous mapping ϕ is formulated by

$$ {}D_{q}^{\alpha }\phi (t)=I_{q}^{[\alpha ]-\alpha }D_{q}^{[\alpha ]} \phi (t), $$

where \([\alpha ]\) denotes the integer part of α.

For more information on the fractional q-operators, we refer the reader to [38].

Lemma 3

([19])

Let \(\alpha , \beta \in \mathbb{R}_{+}\). Then we have the following formulas:

  1. (1)

    \(I_{q}^{\alpha } I_{q}^{\beta } \phi (t)=I_{q}^{a+\beta } \phi (t)\);

  2. (2)

    \(D_{q}^{\alpha } I_{q}^{\alpha } \phi (t)=\phi (t)\).

Lemma 4

([40])

Let \(\alpha \in {\mathbb{R}}_{+}\) and \(\beta \in (-1,\infty )\). One has

$$ I_{q}^{\alpha } t^{\beta }= \frac{\Gamma _{q}(\beta +1)}{\Gamma _{q}(\alpha +\beta +1)} t^{ \alpha +\beta },\quad t>0. $$

In particular, if \(\phi \equiv 1\), then

$$ I_{q}^{\alpha }1(t)=\frac{1}{\varGamma _{q}(1+\alpha )}t^{(\alpha )}, \quad \textit{for all } t>0. $$

Lemma 5

([19])

Let \(\alpha , \sigma >0\). Then

$$ I_{q}^{\alpha } D_{q}^{\sigma } \phi (t)=D_{q}^{\sigma } I_{q}^{\alpha } \phi (t)- \sum_{j=0}^{\sigma -1} \frac{t^{\alpha -\sigma +j}}{\Gamma _{q}(\alpha +j-\sigma +1)} D_{q}^{j} \phi (0). $$

Lemma 6

([30, 40])

Let \(k -1 < \alpha < k\). Then

$$ I_{q}^{\alpha } D_{q}^{\alpha } \phi (t)= \phi (t)-\sum_{j=0}^{k-1} \frac{t^{j}}{\Gamma _{q}(j+1)} D_{q}^{j} \phi (0). $$

For the homogeneous q-difference equation \(D_{q}^{\alpha }\phi (t)=0\), the general series solution by Lemma 6 is given as \(\phi (t) = \mu _{0} + \mu _{1} t + \mu _{2} t^{2} + \cdots + \mu _{k -1} t^{k -1}\) via \(\mu _{j} \in \mathbb{R}\) and \(k = [ \alpha ] + 1\) [19]. So, we have

$$ \bigl(I_{q}^{\alpha }D_{q}^{\alpha }\phi \bigr) (t)= \phi (t) + \mu _{0} + \mu _{1} t + \mu _{2} t^{2} + \cdots + \mu _{k-1} t^{k-1}. $$

3 Results regarding existence property

In the present section, before moving to our fundamental results, we define \(\Vert \cdot \Vert \) on \(X = C(J,\mathbb{R})\) as \(\|u\|=\sup_{t \in J}|u(t)| \), which in this phase, X transforms into a Banach space. Now, in the first place, we provide the next auxiliary lemma.

Lemma 7

Let \(\psi \in C(\mathrm{J},\mathbb{R})\), \(\alpha , \beta , \gamma \in (0,1)\), \(\sigma _{1}, \sigma _{2} >0\), \(a_{1},a_{2},b_{1},b_{2},\lambda _{1},\lambda _{2} \in \mathbb{R}^{+}\) and \(g_{u}(t) = g(t,u(t))\). Then the solution of the linear sequential four-point q-CFBVP defined by

$$ \textstyle\begin{cases} { \mathcal{D}}_{q}^{\alpha } ({ \mathcal{D}}_{q}^{\beta }u(t)-g_{u}(t) )=\psi (t), \quad t\in \mathrm{J}:=[0,T], \\ a_{1} u(0)+b_{1}{ \mathcal{D}}_{q}^{\gamma }u(0)=\lambda _{1}I_{q}^{ \sigma _{1}}u(\eta _{1}), \quad 0< \eta _{1}< T, \sigma _{1}>0, \\ a_{2} u(T)+ b_{2} { \mathcal{D}}_{q}^{\gamma }u(T)= \lambda _{2}I_{q}^{ \sigma _{2}}u(\eta _{2}), \quad 0< \eta _{2}< T, \sigma _{2}>0, \end{cases} $$
(4)

is given by

$$\begin{aligned} u(t)={}& I_{q}^{\beta }g_{u}(t)+I_{q}^{\alpha +\beta } \psi (t) \\ &{}+\mu _{1}(t) \bigl[ I_{q}^{\beta +\sigma _{1}}g_{u}( \eta _{1})+I_{q}^{ \alpha +\beta +\sigma _{1}}\psi (\eta _{1}) \bigr] \\ &{}+\mu _{2}(t) \bigl[ \lambda _{2} \bigl(I_{q}^{\beta +\sigma _{2}}g_{u}( \eta _{2})+I_{q}^{\alpha +\beta +\sigma _{2}} \psi (\eta _{2})\bigr) \\ &{}-b_{2}\bigl(I_{q}^{\beta -\gamma }g_{u}(T)+I_{q}^{\alpha +\beta -\gamma } \psi (T)\bigr) -a_{2}\bigl(I_{q}^{\beta }g_{u}(T)+I_{q}^{\alpha +\beta } \psi (T)\bigr) \bigr], \end{aligned}$$
(5)

where

$$\begin{aligned} \begin{aligned} &\mu _{1}(t)= \lambda _{1} \biggl( \frac{(\Lambda - \Lambda _{2} \Lambda _{3})\Gamma _{q}(\beta +1) - t^{\beta }\Lambda _{1} \Lambda _{3}}{\Lambda \Lambda _{1} \Gamma _{q}(\beta +1)} \biggr), \\ & \mu _{2}(t) = \frac{\Lambda _{1} t^{\beta }+ \Lambda _{2} \Gamma _{q}(\beta +1)}{\Lambda \Gamma _{q}(\beta +1)}, \\ &\Lambda _{1}= a_{1}- \frac{\lambda _{1}\eta _{1}^{\sigma _{1}}}{\Gamma _{q}(\sigma _{1}+1)}, \qquad \Lambda _{2}= \frac{\lambda _{1}\eta _{1}^{\beta +\sigma _{1}}}{\Gamma _{q}(\beta +\sigma _{1}+1)}, \\ & \Lambda _{3}= a_{2}- \frac{\lambda _{2}\eta _{2}^{\sigma _{2}}}{\Gamma _{q}(\sigma _{2}+1)}, \\ & \Lambda _{4}= \frac{a_{2}T^{\beta }}{\Gamma _{q}(\beta +1)}+ \frac{b_{2}T^{\beta -\gamma }}{\Gamma _{q}(\beta -\gamma +1)} - \frac{\lambda _{2}\eta _{2}^{\beta + \sigma _{2}}}{\Gamma _{q}(\beta + \sigma _{2}+1)}, \end{aligned} \end{aligned}$$
(6)

and Λ is given by

$$\begin{aligned} \Lambda = {}& \biggl(\frac{a_{2}T^{\beta }}{\Gamma _{q}(\beta +1)}+ \frac{b_{2}T^{\beta -\gamma }}{\Gamma _{q}(\beta -\gamma +1)} - \frac{\lambda _{2}\eta _{2}^{\beta + \sigma _{2}}}{\Gamma _{q}(\beta + \sigma _{2}+1)} \biggr) \biggl(a_{1}- \frac{\lambda _{1}\eta _{1}^{\sigma _{1}}}{\Gamma _{q}(\sigma _{1}+1)} \biggr) \\ &{}+ \frac{\lambda _{1}\eta _{1}^{\beta +\sigma _{1}}}{\Gamma _{q}(\beta +\sigma _{1}+1)} \biggl(a_{2}- \frac{\lambda _{2}\eta _{2}^{\sigma _{2}}}{\Gamma _{q}(\sigma _{2}+1)} \biggr) = \Lambda _{4} \Lambda _{1} + \Lambda _{2} \Lambda _{3} \neq 0. \end{aligned}$$
(7)

Proof

By using Lemma 6, we obtain the integral equation corresponding to (4):

$$ u(t) = I_{q}^{\beta }g_{u}(t) + I_{q}^{\alpha +\beta }\psi (t)+ \frac{t^{\beta }}{\Gamma _{q}(\beta +1)}k_{0}+k_{1} ,\quad k_{0}, k_{1} \in \mathbb{R}. $$
(8)

Using the given boundary conditions in (4), we may obtain

$$ I_{q}^{\sigma _{i}}u(t) = I_{q}^{\sigma _{i}+\beta }g_{u}(t) +I_{q}^{ \sigma _{i}+\alpha +\beta }\psi (t)+k_{0} \frac{t^{\beta +\sigma _{i}}}{\Gamma _{q}(\beta +\sigma _{i}+1)}+k_{1} \frac{t^{\sigma _{i}}}{\Gamma _{q}(\sigma _{i}+1)} $$

for \(i=1,2\) and

$$ { \mathcal{D}}_{q}^{\gamma }x(t) = I_{q}^{\beta -\gamma }g_{u}(t) +I_{q}^{ \alpha +\beta -\gamma }\psi (t) +k_{0} \frac{t^{\beta -\gamma }}{\Gamma _{q}(\beta -\gamma +1)}. $$

By categorizing similar terms, we obtain the expressions

$$ \Lambda _{1}k_{1}- \Lambda _{2}k_{0}=\lambda _{1}{ \mathcal{I}}_{q}^{ \sigma _{1}+\beta }g_{u}( \eta _{1})+\lambda _{1}I_{q}^{\sigma _{1}+ \beta +\alpha }\psi (\eta _{1}) $$
(9)

and

$$\begin{aligned} \Lambda _{3}k_{1} +\Lambda _{4}k_{0} ={}&\lambda _{2} \bigl(I_{q}^{ \sigma _{2}+\beta }g_{u}( \eta _{2})+I_{q}^{\sigma _{2}+\beta +\alpha } \psi (\eta _{2}) \bigr) \\ &{}-a_{2} \bigl(I_{q}^{\beta }g_{u}(T)+I_{q}^{\beta +\alpha } \psi (T) \bigr) -b_{2} \bigl(I_{q}^{\beta -\gamma }g_{u}(T)+I_{q}^{\beta + \alpha -\gamma } \psi (T) \bigr). \end{aligned}$$
(10)

Therefore, from (9) and (10), we get

$$\begin{aligned} k_{0}={}& -\frac{\Lambda _{3}}{\Lambda } \bigl(\lambda _{1}I_{q}^{ \sigma _{1}+\beta }g_{u}( \eta _{1})+\lambda _{1}I_{q}^{\sigma _{1}+ \beta +\alpha }\psi (\eta _{1}) \bigr) \\ &{}+ \frac{\Lambda _{1}}{\Lambda } \bigl[ \lambda _{2} \bigl(I_{q}^{ \sigma _{2}+\beta }g_{u}(\eta _{2})+I_{q}^{\sigma _{2}+\beta +\alpha } \psi (\eta _{2}) \bigr) \\ &{}-a_{2} \bigl(I_{q}^{\beta }g_{u}(T)+I_{q}^{\beta +\alpha } \psi (T) \bigr) -b_{2} \bigl(I_{q}^{\beta -\gamma }g_{u}(T)+I_{q}^{\beta + \alpha -\gamma } \psi (T) \bigr) \bigr], \end{aligned}$$

and, by inserting \(k_{0}\) into (9), we obtain

$$\begin{aligned} k_{1} ={}& \frac{\Lambda - \Lambda _{2}\Lambda _{3}}{\Lambda _{1} \Lambda } \bigl( \lambda _{1}I_{q}^{\sigma _{1}+\beta }g_{u}( \eta _{1})+\lambda _{1}I_{q}^{ \sigma _{1}+\beta +\alpha }\psi (\eta _{1}) \bigr) \\ &{}+\frac{\Lambda _{2}}{\Lambda } \bigl[ \lambda _{2} \bigl(I_{q}^{\sigma _{2}+ \beta }g_{u}( \eta _{2})+I_{q}^{\sigma _{2}+\beta +\alpha }\psi (\eta _{2}) \bigr) -a_{2} \bigl(I_{q}^{\beta }g_{u}(T)+I_{q}^{\beta +\alpha } \psi (T) \bigr) \\ &{}-b_{2} \bigl(I_{q}^{\beta -\gamma }g_{u}(T)+I_{q}^{\beta +\alpha - \gamma } \psi (T) \bigr) \bigr]. \end{aligned}$$

Substituting the value of \(k_{0}\), \(k_{1}\) in (8), we get (5), which completes the proof. □

Note that, for simplicity, we set \(g(t,u(t)) = g_{u}(t)\) and \(f(t, u(t))=f_{u}(t)\) throughout the manuscript.

3.1 The first existence criterion

In this subsection, we prove an existence result for the sequential four-point q-CFBVP (3) by making use of Sadovskii’s fixed-point theorem. Before moving towards it, we would like to recall several auxiliary facts which are our main tools. X is supposed as a Banach space.

Definition 8

Consider a bounded subset M of \((X, d)\). The Kuratowski measure of non-compactness, denoted by \(\alpha (M)\), is defined by

$$ \alpha (M) := \inf \Biggl\{ \epsilon >0: \exists \text{ finitely many sets } M_{i} \text{ s.t. } M = \bigcup_{i=1}^{n} M_{i} \text{ and } D(M_{i}) \leq \epsilon \Biggr\} , $$

where \(D(M_{i}) = \sup \{ \vert u-\tilde{u} \vert : u, \tilde{u} \in M_{i} \}\).

Definition 9

([41])

Consider a bounded and continuous function \(\Phi : \operatorname{Dom}(\Phi ) \subseteq X \rightarrow X\) on X. For an arbitrary bounded set \(M \subset \operatorname{Dom}(\Phi )\), the map Φ is condensing if

$$ \alpha \bigl(\Phi (M)\bigr)< \alpha (M), $$

in which α is introduced above.

Lemma 10

([42])

Let \(\mathcal{K}_{1}, \mathcal{K}_{2}: E \subseteq X \rightarrow X\). The operator \(\mathcal{K}_{1}+\mathcal{K}_{2}\) is condensing if

  1. i.

    \(\mathcal{K}_{1}\) is k-contraction; that is, \(\forall u, v \in E\) and \(\exists k \in (0,1)\), so that

    $$ \Vert \mathcal{K}_{1} u-\mathcal{K}_{1} v \Vert \leq k \Vert u-v \Vert ; $$
  2. ii.

    \(\mathcal{K}_{2}\) is compact.

Theorem 11

([43])

Consider the bounded, closed and convex subset B of X and the condensing mapping \(\Phi : B \rightarrow B\). Then Φ has a fixed point.

From now on, we put

$$\begin{aligned} \Theta _{1}={} &\frac{T^{\alpha +\beta }}{\Gamma _{q}(\alpha +\beta +1)}+ \bigl\vert \mu _{1}(T) \bigr\vert \frac{\eta _{1}^{\alpha +\beta +\sigma _{1}}}{\Gamma _{q}(\alpha +\beta +\sigma _{1}+1)} \\ &{}+ \bigl\vert \mu _{2}(T) \bigr\vert \biggl( \lambda _{2} \frac{\eta _{2}^{\alpha +\beta +\sigma _{2}}}{\Gamma _{q}(\alpha +\beta +\sigma _{2}+1)} +b_{2} \frac{T^{\alpha +\beta -\gamma }}{\Gamma _{q}(\alpha +\beta -\gamma +1)} \\ &{}+a_{2}\frac{T^{\alpha +\beta }}{\Gamma _{q}(\alpha +\beta +1)} \biggr) \end{aligned}$$
(11)

and

$$\begin{aligned} \Theta _{2}={}&\frac{T^{\beta }}{\Gamma _{q}(\beta +1)}+ \bigl\vert \mu _{1}(T) \bigr\vert \frac{\eta _{1}^{\beta +\sigma _{1}}}{\Gamma _{q}(\beta +\sigma _{1}+1)} \\ &{}+ \bigl\vert \mu _{2}(T) \bigr\vert \biggl( \lambda _{2} \frac{\eta _{2}^{\beta +\sigma _{2}}}{\Gamma _{q}(\beta +\sigma _{2}+1)} +b_{2}\frac{T^{\beta -\gamma }}{\Gamma _{q}(\beta -\gamma +1)} +a_{2} \frac{T^{\beta }}{\Gamma _{q}(\beta +1)} \biggr). \end{aligned}$$
(12)

Theorem 12

Consider the following assertions:

\((\mathbf{B}_{1})\):

\(\exists L>0\) so that \(|f_{u}(t)-f_{v}(t)| \leq L|u(t)-v(t)|\), \(\forall t \in J\), \(u, v \in \mathbb{R}\);

\((\mathbf{B}_{2})\):

\(|f_{u}(t)| \leq \sigma (t)\) and \(|g_{u}(t)| \leq \rho (t)\), where \(\sigma , \rho \in C (J, \mathbb{R}^{+} )\).

Then the sequential four-point q-CFBVP (3) has a solution on J if \(Q:=L \Theta _{1}<1 \), by introducing \(\Theta _{1}\) as (11).

Proof

Consider a bounded, closed and convex subset \(B_{r}=\{u \in X :\|u\| \leq r\}\) of \(X = C(J, \mathbb{R})\) for a fixed constant r. With regard to Lemma 7, define \(\mathcal{K}: X \rightarrow X\) as follows:

$$\begin{aligned} \mathcal{K}u(t) ={}& I_{q}^{\beta }g_{u}(t)+I_{q}^{\alpha +\beta }f_{u}(t) \\ &{}+\mu _{1}(t) \bigl[I_{q}^{\beta +\sigma _{1}}g_{u}( \eta _{1})+I_{q}^{ \alpha +\beta +\sigma _{1}}f_{u}(\eta _{1}) \bigr] \\ &{}+\mu _{2}(t) \bigl[\lambda _{2} \bigl(I_{q}^{\beta +\sigma _{2}}g_{u}( \eta _{2})+I_{q}^{\alpha +\beta +\sigma _{2}}f_{u}( \eta _{2}) \bigr) \\ &{}-b_{2} \bigl(I_{q}^{\beta -\gamma }g_{u}(T)+I_{q}^{\alpha +\beta - \gamma }f_{u}(T) \bigr) -a_{2} \bigl(I_{q}^{\beta }g_{u}(T)+I_{q}^{ \alpha +\beta }f_{u}(T) \bigr) \bigr]. \end{aligned}$$
(13)

We split the operator \(\mathcal{K}\) on the set \(B_{r}\) into \(\mathcal{K}=\mathcal{K}_{1}+\mathcal{K}_{2}\), where

$$\begin{aligned} \mathcal{K}_{1} u(t)={}& I_{q}^{\alpha +\beta }f_{u}(t)+ \mu _{1}(t) \bigl(I_{q}^{\alpha +\beta +\sigma _{1}}f_{u}( \eta _{1}) \bigr) \\ &{}+\mu _{2}(t) \bigl[\lambda _{2} \bigl(I_{q}^{\alpha +\beta +\sigma _{2}}f_{u}( \eta _{2}) \bigr) -b_{2} \bigl(I_{q}^{\alpha +\beta -\gamma }f_{u}(T) \bigr) -a_{2} \bigl(I_{q}^{\alpha +\beta }f_{u}(T) \bigr) \bigr] \end{aligned}$$
(14)

and

$$\begin{aligned} \mathcal{K}_{2} u(t)={}& I_{q}^{\beta }g_{u}(t)+ \mu _{1}(t) \bigl(I_{q}^{ \beta +\sigma _{1}}g_{u}( \eta _{1}) \bigr) \\ &{}+\mu _{2}(t) \bigl[\lambda _{2} \bigl(I_{q}^{\beta +\sigma _{2}}g_{u}( \eta _{2}) \bigr) -b_{2} \bigl(I_{q}^{\beta -\gamma }g_{u}(T) \bigr) -a_{2} \bigl(I_{q}^{\beta }g_{u}(T) \bigr) \bigr]. \end{aligned}$$
(15)

We want to prove that the operators \(\mathcal{K}_{1}\) and \(\mathcal{K}_{2}\) follow all the assertions of Theorem 11. We proceed to implement the proof in four steps.

Step 1: \(\mathcal{K} B_{r} \subset B_{r}\)

Let us select r so that \(r \geq \|\sigma \| \Theta _{1}+\|\rho \| \Theta _{2}\), where \(\Theta _{2}\), \(\Theta _{1}\) are given by (11) and (12) and \(\Vert \sigma \Vert = \sup_{t\in J} \vert \sigma (t) \vert \) and \(\Vert \rho \Vert = \sup_{t\in J} \vert \rho (t) \vert \). For any \(u \in B_{r}\), we have

$$\begin{aligned} \bigl\vert (\mathcal{K} u) (t) \bigr\vert \leq{}& \sup_{t\in J } \bigl\{ I_{q}^{ \beta } \bigl\vert g_{u}(t) \bigr\vert +I_{q}^{\alpha +\beta } \bigl\vert f_{u}(t) \bigr\vert \\ &{}+ \bigl\vert \mu _{1}(t) \bigr\vert \bigl[ I_{q}^{\beta +\sigma _{1}} \bigl\vert g_{u}( \eta _{1}) \bigr\vert +I_{q}^{\alpha +\beta +\sigma _{1}} \bigl\vert f_{u}( \eta _{1}) \bigr\vert \bigr] + \bigl\vert \mu _{2}(t) \bigr\vert \bigl[ \lambda _{2} \bigl(I_{q}^{ \beta +\sigma _{2}} \bigl\vert g_{u}(\eta _{2}) \bigr\vert \\ &{}+I_{q}^{\alpha +\beta +\sigma _{2}} \bigl\vert f_{u}(\eta _{2}) \bigr\vert \bigr) + b_{2}\bigl(I_{q}^{ \beta -\gamma } \bigl\vert g_{u}(T) \bigr\vert +I_{q}^{\alpha +\beta -\gamma } \bigl\vert f_{u}(T) \bigr\vert \bigr) +a_{2} \bigl(I_{q}^{\beta } \bigl\vert g_{u}(T) \bigr\vert \\ &{} +I_{q}^{\alpha +\beta } \bigl\vert f_{u}(T) \bigr\vert \bigr) \bigr] \bigr\} \leq \Theta _{2} \Vert \rho \Vert +\Theta _{1} \Vert \sigma \Vert < r, \end{aligned}$$

which implies that \(\mathcal{K} B_{r} \subset B_{r}\).

Step 2: \(\mathcal{K}_{2}\) is compact

In view of Step 1, we observe that the operator \(\mathcal{K}_{2}\) is uniformly bounded; indeed for any \(u \in B_{r}\):

$$\begin{aligned} \bigl\vert (\mathcal{K}_{2} u) (t) \bigr\vert \leq{}& I_{q}^{\beta } \bigl\vert g_{u}(t) \bigr\vert + \bigl\vert \mu _{1}(t) \bigr\vert \bigl(I_{q}^{\beta +\sigma _{1}} \bigl\vert g_{u}(\eta _{1}) \bigr\vert \bigr) \\ &{}+ \bigl\vert \mu _{2}(t) \bigr\vert \bigl[\lambda _{2} \bigl(I_{q}^{\beta + \sigma _{2}} \bigl\vert g_{u}(\eta _{2}) \bigr\vert \bigr) +b_{2} \bigl(I_{q}^{ \beta -\gamma } \bigl\vert g_{u}(T) \bigr\vert \bigr) +a_{2} \bigl(I_{q}^{\beta } \bigl\vert g_{u}(T) \bigr\vert \bigr) \bigr] \\ \leq {}&\Vert \rho \Vert \biggl[ \frac{T^{\beta }}{\Gamma _{q}(\beta +1)}+ \bigl\vert \mu _{1}(T) \bigr\vert \frac{\eta _{1}^{\beta +\sigma _{1}}}{\Gamma _{q}(\beta +\sigma _{1}+1)} \\ &{}+ \bigl\vert \mu _{2}(T) \bigr\vert \biggl( \lambda _{2} \frac{\eta _{2}^{\beta +\sigma _{2}}}{\Gamma _{q}(\beta +\sigma _{2}+1)} +b_{2}\frac{T^{\beta -\gamma }}{\Gamma _{q}(\beta -\gamma +1)} +a_{2} \frac{T^{\beta }}{\Gamma _{q}(\beta +1)} \biggr) \biggr] \\ \leq{}& \Theta _{2} \Vert \rho \Vert . \end{aligned}$$

Now, take \(t_{1}, t_{2} \in J\) by assuming \(t_{1}< t_{2}\) and \(u \in B_{r}\). Hence we have

$$\begin{aligned} & \bigl\vert \mathcal{K}_{2} u(t_{2})- \mathcal{K}_{2} u(t_{1}) \bigr\vert \\ &\quad \leq I_{q}^{ \beta } \bigl\vert g_{u}(t_{2}) - g_{u}(t_{1}) \bigr\vert + \bigl\vert \mu _{1}(t_{2}) - \mu _{1}(t_{1}) \bigr\vert \bigl(I_{q}^{\beta +\sigma _{1}} \bigl\vert g_{u}( \eta _{1}) \bigr\vert \bigr) \\ &\qquad {}+ \bigl\vert \mu _{2}(t_{2}) - \mu _{2}(t_{1}) \bigr\vert \bigl[\lambda _{2} \bigl(I_{q}^{\beta +\sigma _{2}} \bigl\vert g_{u}(\eta _{2}) \bigr\vert \bigr) +b_{2} \bigl(I_{q}^{\beta -\gamma } \bigl\vert g_{u}(T) \bigr\vert \bigr) +a_{2} \bigl(I_{q}^{ \beta } \bigl\vert g_{u}(T) \bigr\vert \bigr) \bigr] \\ &\quad \leq \frac{ \Vert \rho \Vert }{\Gamma _{q}(\beta +1)} \bigl[t_{2}^{\beta }-t_{1}^{ \beta }+2 (t_{2}-t_{1} )^{\beta } \bigr]+ \bigl\vert \mu _{1}(t_{2})- \mu _{1}(t_{1}) \bigr\vert \bigl(I_{q}^{\beta +\sigma _{1}} \Vert \rho \Vert \bigr) \\ &\qquad {}+ \bigl\vert \mu _{2}(t_{2})-\mu _{2}(t_{1}) \bigr\vert \bigl(\lambda _{2} \bigl(I_{q}^{ \beta +\sigma _{2}} \Vert \rho \Vert \bigr) +b_{2} \bigl(I_{q}^{\beta - \gamma } \Vert \rho \Vert \bigr) +a_{2} \bigl(I_{q}^{\beta } \Vert \rho \Vert \bigr) \bigr). \end{aligned}$$
(16)

The right-hand side of (16) tends to zero (not depending upon u) as \(t_{2} \rightarrow t_{1} \). This shows that \(\mathcal{K}_{2}\) is equicontinuous. From the above reasons, it is clear that \(\mathcal{K}_{2}\) is relatively compact on \(B_{r}\). Application of the Arzelà–Ascoli theorem proves the compactness of \(\mathcal{K}_{2}\) on \(B_{r}\).

Step 3: \(\mathcal{K}_{1}\) is Q-contractive.

From \((\mathbf{B}_{1} )\) and \((\mathbf{B}_{2} )\) and for each \(u,v \in B_{r}\), we have

$$\begin{aligned} \bigl\vert \mathcal{K}_{1} u(t)-\mathcal{K}_{1} v(t) \bigr\vert \leq{}& \sup_{t \in J} \bigl\{ I_{q}^{\alpha +\beta } \vert f_{u}-f_{v} \vert (t)+ \bigl\vert \mu _{1}(t) \bigr\vert \bigl(I_{q}^{\alpha +\beta +\sigma _{1}} \vert f_{u}-f_{v} \vert (\eta _{1})\bigr) \\ &{}+ \bigl\vert \mu _{2}(t) \bigr\vert \bigl[\lambda _{2} \bigl(I_{q}^{\alpha +\beta + \sigma _{2}} \vert f_{u}-f_{v} \vert (\eta _{2})\bigr) +b_{2}\bigl(I_{q}^{\alpha +\beta - \gamma } \vert f_{u}-f_{v} \vert (T)\bigr) \\ &{}+a_{2} \bigl(I_{q}^{\alpha +\beta } \vert f_{u}-f_{v} \vert (T)\bigr) \bigr] \bigr\} \\ \leq{}& L\Theta _{1} \Vert u-v \Vert . \end{aligned}$$

So, \(\Vert \mathcal{K}_{1} u-\mathcal{K}_{1} v\Vert \leq L\Theta _{1} \Vert u-v\Vert \). Thus \(\mathcal{K}_{1}\) is Q-contractive because of \(Q:=L\Theta _{1}<1\).

Step 4: \(\mathcal{K}\) is condensing.

As \(\mathcal{K}_{1}\) and \(\mathcal{K}_{2}\) are continuous Q–contraction and compact, respectively, thus by Lemma 10, \(\mathcal{K}: B_{r} \rightarrow B_{r}\) with \(\mathcal{K}=\mathcal{K}_{1}+\mathcal{K}_{2}\) is a condensing map on \(B_{r}\). From the above arguments, by Theorem 11, we conclude that the map \(\mathcal{K}\) has a fixed point, which leads to the existence of at least one solution for the sequential four-point q-CFBVP (3) in X. □

3.2 The second existence criterion

We now use another fixed point result due to Krasnoselskii–Zabreiko to demonstrate the following existence criterion for the sequential four-point q-CFBVP (3).

Theorem 13

([44])

Consider a completely continuous map \(\mathcal{K}\) on a Banach space X. If a bounded linear map \(\mathcal{L}\) exists on X so that 1 is not an eigenvalue of it and

$$ \lim_{ \Vert u \Vert \rightarrow \infty } \frac{ \Vert \mathcal{K} (u)-\mathcal{L} (u) \Vert }{ \Vert u \Vert }=0, $$

then \(\mathcal{K}\) has a fixed point in X.

Theorem 14

Consider the following assertions:

  1. (H1)

    \(f: J \times \mathbb{R} \rightarrow \mathbb{R}\) is continuous and for some \(t \in J\), \(f(t, 0) \neq 0\) and

    $$ \lim_{\|u\| \rightarrow \infty } \frac{f(t, u)}{u}=\lambda (t). $$
    (17)
  2. (H2)

    The function \(g: J \times \mathbb{R} \rightarrow \mathbb{R}\) is continuous and \(\exists A\in \mathbb{R}_{+} \) so that

    $$ \bigl\vert g \bigl(t, u(t)\bigr) \bigr\vert \leq A \bigl\vert u(t) \bigr\vert . $$

Then there exists at least one solution for the sequential four-point q-CFBVP (3) on J such that

$$ \lambda _{\max }:=\max_{t \in J} \bigl\vert \lambda (t) \bigr\vert < \frac{1-A\Theta _{2}}{\Theta _{1}}, $$
(18)

where \(\Theta _{1}\) and \(\Theta _{2}\) are, respectively, given by (11) and (12).

Proof

Consider a sequence \(\{u_{n} \} \subset B_{r} \) which converges to u. We know that f and g are continuous, so, by letting \(n\to \infty \), we get

$$ \vert f_{u_{n}} - f_{u} \vert (t) \to 0, \qquad \vert g_{u_{n}} - g_{u} \vert (t) \to 0. $$

Thus, for \(t \in J\), we write

$$\begin{aligned} \bigl\vert (\mathcal{K} u_{n}) (t)-(\mathcal{K} u) (t) \bigr\vert \leq{}& I_{q}^{ \beta } \vert g_{u_{n}}-g_{u} \vert (t)+I_{q}^{\alpha +\beta } \vert f_{u_{n}}-f_{u} \vert (t) \\ &{}+\mu _{1}(t) \bigl(I_{q}^{\beta +\sigma _{1}} \vert g_{u_{n}}-g_{u} \vert (\eta _{1})+I_{q}^{\alpha +\beta +\sigma _{1}} \vert f_{u_{n}}-f_{u} \vert (\eta _{1}) \bigr) \\ &{}+\mu _{2}(t) \bigl(\lambda _{2} \bigl(I_{q}^{\beta +\sigma _{2}} \vert g_{u_{n}}-g_{u} \vert (\eta _{2})+I_{q}^{\alpha +\beta + \sigma _{1}} \vert f_{u_{n}}-f_{u} \vert (\eta _{2}) \bigr) \\ & {}+b_{2} \bigl(I_{q}^{\beta -\gamma } \vert g_{u_{n}}-g_{u} \vert (T)+I_{q}^{\alpha +\beta -\gamma } \vert f_{u_{n}}-f_{u} \vert (T) \bigr) \\ & {}+a_{2} \bigl(I_{q}^{\beta } \vert g_{u_{n}}-g_{u} \vert (T)+I_{q}^{ \alpha +\beta } \vert f_{u_{n}}-f_{u} \vert (T) \bigr) \bigr) \to 0. \end{aligned}$$
(19)

Therefore the right-hand side of (19) tends to zero. Therefore, the continuity of \(\mathcal{K}\) is established. Now, for \(r>0\), we set \(N=\{u \in C(J,\mathbb{R});\|u\| \leq r\}\) and \(\|f^{*}\|=\sup_{(t,u)\in J\times N} \vert f_{u}(t) \vert \). Thus,

$$\begin{aligned} \bigl\vert (\mathcal{K} u) (t) \bigr\vert \leq{}& I_{q}^{\beta } \bigl\vert g_{u}(t) \bigr\vert +I_{q}^{\alpha +\beta } \bigl\vert f_{u}(t) \bigr\vert \\ &{}+ \bigl\vert \mu _{1}(t) \bigr\vert \bigl[ I_{q}^{\beta +\sigma _{1}} \bigl\vert g_{u}( \eta _{1}) \bigr\vert +I_{q}^{\alpha +\beta +\sigma _{1}} \bigl\vert f_{u}( \eta _{1}) \bigr\vert \bigr] \\ &{}+ \bigl\vert \mu _{2}(t) \bigr\vert \bigl[ \lambda _{2}\bigl(I_{q}^{\beta +\sigma _{2}} \bigl\vert g_{u}(\eta _{2}) \bigr\vert +I_{q}^{\alpha +\beta +\sigma _{2}} \bigl\vert f_{u}(\eta _{2}) \bigr\vert \bigr) \\ &{}+ b_{2}\bigl(I_{q}^{\beta -\gamma } \bigl\vert g_{u}(T) \bigr\vert +I_{q}^{\alpha + \beta -\gamma } \bigl\vert f_{u}(T) \bigr\vert \bigr) +a_{2}\bigl(I_{q}^{\beta } \bigl\vert g_{u}(T) \bigr\vert +I_{q}^{\alpha +\beta } \bigl\vert f_{u}(T) \bigr\vert \bigr) \bigr] \\ \leq{}& \bigl\{ I_{q}^{\alpha +\beta } 1(t) + \bigl\vert \mu _{1}(t) \bigr\vert \bigl[ I_{q}^{ \alpha +\beta +\sigma _{1}} 1(\eta _{1}) \bigr] + \bigl\vert \mu _{2}(t) \bigr\vert \bigl[ \lambda _{2}I_{q}^{\alpha +\beta +\sigma _{2}} 1(\eta _{2}) \\ &{}+ b_{2}I_{q}^{\alpha +\beta -\gamma } 1(T) +a_{2}I_{q}^{\alpha +\beta } 1(T) \bigr] \bigr\} \bigl\Vert f^{*} \bigr\Vert \\ &{}+ \bigl\{ I_{q}^{\beta } 1(t) + \bigl\vert \mu _{1}(t) \bigr\vert \bigl[ I_{q}^{ \beta +\sigma _{1}} 1(\eta _{1}) \bigr] + \bigl\vert \mu _{2}(t) \bigr\vert \bigl[ \lambda _{2}I_{q}^{\beta +\sigma _{2}} 1(\eta _{2}) + b_{2}I_{q}^{ \beta -\gamma } 1(T) \\ &{}+a_{2}I_{q}^{\beta }1(T) \bigr] \bigr\} Ar \leq \Theta _{1} \bigl\Vert f^{*} \bigr\Vert + \Theta _{2} Ar, \end{aligned}$$

which yields \(\|\mathcal{K} u\| \leq \Theta _{1}\|f^{*}\|+A\Theta _{2} r\). This shows the uniformly boundedness of \(\mathcal{K}\). We now claim that \(\mathcal{K}\) is equicontinuous.

Let \(t_{1}, t_{2} \in J\) via \(t_{1}< t_{2}\). Then, by setting \(\|f^{*}\|=\sup_{(t,u)\in J\times N} \vert f_{u}(t) \vert \), we obtain

$$\begin{aligned} & \bigl\vert \mathcal{K} u(t_{2})-\mathcal{K} u(t_{1}) \bigr\vert \\ &\quad \leq I_{q}^{ \beta } \bigl\vert g_{u}(t_{2}) - g_{u}(t_{1}) \bigr\vert +I_{q}^{\alpha +\beta } \bigl\vert f_{u}(t_{2}) - f_{u}(t_{1}) \bigr\vert \\ &\qquad {}+ \bigl\vert \mu _{1}(t_{2}) - \mu _{1}(t_{1}) \bigr\vert \bigl[ I_{q}^{\beta + \sigma _{1}} \bigl\vert g_{u}(\eta _{1}) \bigr\vert +I_{q}^{\alpha +\beta + \sigma _{1}} \bigl\vert f_{u}(\eta _{1}) \bigr\vert \bigr] \\ &\qquad {}+ \bigl\vert \mu _{2}(t_{2}) - \mu _{2}(t_{1}) \bigr\vert \bigl[ \lambda _{2} \bigl(I_{q}^{ \beta +\sigma _{2}} \bigl\vert g_{u}(\eta _{2}) \bigr\vert +I_{q}^{\alpha + \beta +\sigma _{2}} \bigl\vert f_{u}(\eta _{2}) \bigr\vert \bigr) \\ &\qquad {}+ b_{2}\bigl(I_{q}^{\beta -\gamma } \bigl\vert g_{u}(T) \bigr\vert +I_{q}^{\alpha + \beta -\gamma } \bigl\vert f_{u}(T) \bigr\vert \bigr) +a_{2}\bigl(I_{q}^{\beta } \bigl\vert g_{u}(T) \bigr\vert +I_{q}^{\alpha +\beta } \bigl\vert f_{u}(T) \bigr\vert \bigr) \bigr] \\ &\quad \leq \frac{Ar }{\Gamma _{q}(\beta +1)} \bigl[t_{2}^{\beta }-t_{1}^{ \beta }+2(t_{2}-t_{1})^{\beta } \bigr] \\ &\qquad {}+\frac{ \Vert f^{*} \Vert }{\Gamma _{q}(\alpha +\beta +1)} \bigl[t_{2}^{\alpha + \beta }-t_{1}^{\alpha +\beta }+2(t_{2}-t_{1})^{\alpha +\beta } \bigr] \\ &\qquad {}+ \bigl\vert \mu _{1}(t_{2}) - \mu _{1}(t_{1}) \bigr\vert \bigl[ I_{q}^{\beta + \sigma _{1}} 1(\eta _{1})Ar +I_{q}^{\alpha +\beta +\sigma _{1}} 1( \eta _{1}) \bigl\Vert f^{*} \bigr\Vert \bigr] \\ &\qquad {}+ \bigl\vert \mu _{2}(t_{2}) - \mu _{2}(t_{1}) \bigr\vert \bigl[ \lambda _{2} \bigl(I_{q}^{ \beta +\sigma _{2}}1(\eta _{2})Ar +I_{q}^{\alpha +\beta +\sigma _{2}} 1( \eta _{2}) \bigl\Vert f^{*} \bigr\Vert \bigr) \\ &\qquad {}+ b_{2}\bigl(I_{q}^{\beta -\gamma } 1(T)Ar +I_{q}^{\alpha +\beta -\gamma } 1(T) \bigl\Vert f^{*} \bigr\Vert \bigr) +a_{2}\bigl(I_{q}^{\beta }1(T)Ar +I_{q}^{\alpha +\beta } 1(T) \bigl\Vert f^{*} \bigr\Vert \bigr) \bigr]. \end{aligned}$$

It is clear that \(\vert \mathcal{K} u (t_{2} )-\mathcal{K}u (t_{1} ) \vert \rightarrow 0\) as \(t_{2} \rightarrow t_{1}\) independent of u. In consequence, from the above arguments, \(\mathcal{K}\) is relatively compact on N. Application of the Arzelà–Ascoli theorem proves the compactness of \(\mathcal{K}\) on N.

Now, by considering the sequential four-point q-CFBVP (3) to be linear by taking \(f_{u}(t)=f(t, u(t))=\lambda (t) u(t)\), the operator \(\mathcal{L}\), by Lemma 7, is formulated by

$$\begin{aligned} \mathcal{L}u(t) ={}& I_{q}^{\beta }g_{u}(t)+I_{q}^{\alpha +\beta } \lambda (t)u(t) \\ &{}+\mu _{1}(t) \bigl[I_{q}^{\beta +\sigma _{1}}g_{u}( \eta _{1})+I_{q}^{ \alpha +\beta +\sigma _{1}}\lambda (\eta _{1})u(\eta _{1}) \bigr] \\ &{}+\mu _{2}(t) \bigl[\lambda _{2} \bigl(I_{q}^{\beta +\sigma _{2}}g_{u}( \eta _{2})+I_{q}^{\alpha +\beta +\sigma _{2}} \lambda (\eta _{2})u( \eta _{2}) \bigr) \\ &{}-b_{2} \bigl(I_{q}^{\beta -\gamma }g_{u}(T)+I_{q}^{\alpha +\beta - \gamma } \lambda (T)u(T) \bigr) \\ &{}-a_{2} \bigl(I_{q}^{\beta }g_{u}(T)+I_{q}^{\alpha +\beta } \lambda (T)u(T) \bigr) \bigr]. \end{aligned}$$

Our next claim is that 1 is not an eigenvalue of \(\mathcal{L}\). If it is so, by (18), we estimate

$$\begin{aligned} \Vert u \Vert ={}& \sup_{t\in J} \bigl\vert (\mathcal{L}u) (t) \bigr\vert \\ \leq{}& \sup_{t\in J} \bigl\{ I_{q}^{\beta } \bigl\vert g_{u}(t) \bigr\vert +I_{q}^{ \alpha +\beta } \bigl\vert \lambda (t) \bigr\vert \bigl\vert u(t) \bigr\vert \\ &{}+ \bigl\vert \mu _{1}(t) \bigr\vert \bigl[ I_{q}^{\beta +\sigma _{1}} \bigl\vert g_{u}( \eta _{1}) \bigr\vert +I_{q}^{\alpha +\beta +\sigma _{1}} \bigl\vert \lambda ( \eta _{1}) \bigr\vert \bigl\vert u(\eta _{1}) \bigr\vert \bigr] \\ &{}+ \bigl\vert \mu _{2}(t) \bigr\vert \bigl[ \lambda _{2}\bigl(I_{q}^{\beta +\sigma _{2}} \bigl\vert g_{u}(\eta _{2}) \bigr\vert +I_{q}^{\alpha +\beta +\sigma _{2}} \bigl\vert \lambda (\eta _{2}) \bigr\vert \bigl\vert u(\eta _{2}) \bigr\vert \bigr) \\ &{}+ b_{2}\bigl(I_{q}^{\beta -\gamma } \bigl\vert g_{u}(T) \bigr\vert +I_{q}^{\alpha + \beta -\gamma } \bigl\vert \lambda (T) \bigr\vert \bigl\vert u(T) \bigr\vert \bigr) \\ &{}+a_{2}\bigl(I_{q}^{\beta } \bigl\vert g_{u}(T) \bigr\vert +I_{q}^{\alpha +\beta } \bigl\vert \lambda (T) \bigr\vert \bigl\vert u(T) \bigr\vert \bigr) \bigr] \bigr\} \\ \leq{}& (\lambda _{\max } \Theta _{1}+A\Theta _{2} ) \Vert u \Vert < \Vert u \Vert , \end{aligned}$$

which is not possible. Hence we established our claim.

Finally, we show that \(\|\mathcal{K}(u)-\mathcal{L}(u)\|/\|u\|\) vanishes as \(\|u\| \rightarrow \infty \). For \(t \in J\), one may write

$$\begin{aligned} & \bigl\vert (\mathcal{K} u) (t)-(\mathcal{L} u) (t) \bigr\vert \\ &\quad \leq I_{q}^{ \alpha +\beta } \bigl\vert f_{u}(t) - \lambda (t)u(t) \bigr\vert \\ &\qquad {}+ \bigl\vert \mu _{1}(t) \bigr\vert \bigl[ I_{q}^{\alpha +\beta +\sigma _{1}} \bigl\vert f_{u}(\eta _{1}) - \lambda (\eta _{1}) u(\eta _{1}) \bigr\vert \bigr] \\ &\qquad {}+ \bigl\vert \mu _{2}(t) \bigr\vert \bigl[ \lambda _{2}I_{q}^{\alpha +\beta + \sigma _{2}} \bigl\vert f_{u}( \eta _{2}) - \lambda (\eta _{2}) u(\eta _{2}) \bigr\vert \\ &\qquad {}+ b_{2} I_{q}^{\alpha +\beta -\gamma } \bigl\vert f_{u}(T) - \lambda (T) u(T) \bigr\vert +a_{2}I_{q}^{\alpha +\beta } \bigl\vert f_{u}(T) - \lambda (T) u(T) \bigr\vert \bigr] \\ &\quad \leq I_{q}^{\alpha +\beta } \biggl( \biggl\vert \frac{f_{u}(t)}{u(t)}-\lambda (t) \biggr\vert \bigl\vert u(t) \bigr\vert \biggr) \\ &\qquad {}+ \bigl\vert \mu _{1}(t) \bigr\vert \biggl[I_{q}^{\alpha +\beta +\sigma _{1}} \biggl( \biggl\vert \frac{f_{u}(\eta _{1})}{u(\eta _{1})}-\lambda (\eta _{1}) \biggr\vert \bigl\vert u( \eta _{1}) \bigr\vert \biggr) \biggr] \\ &\qquad {}+ \bigl\vert \mu _{2}(t) \bigr\vert \biggl[\lambda _{2} \biggl(I_{q}^{\alpha +\beta +\sigma _{2}} \biggl( \biggl\vert \frac{f_{u}(\eta _{2})}{u(\eta _{2})}-\lambda (\eta _{2}) \biggr\vert \bigl\vert u( \eta _{2}) \bigr\vert \biggr) \biggr) \\ &\qquad {}+b_{2} \biggl(I_{q}^{\alpha +\beta -\gamma } \biggl( \biggl\vert \frac{f_{u}(T)}{u(T)}-\lambda (T) \biggr\vert \bigl\vert u(T) \bigr\vert \biggr) \biggr) +a_{2} \biggl(I_{q}^{ \alpha +\beta } \biggl( \biggl\vert \frac{f_{u}(T)}{u(T)}-\lambda (T) \biggr\vert \bigl\vert u(T) \bigr\vert \biggr) \biggr) \biggr]. \end{aligned}$$

This means that

$$\begin{aligned} \frac{ \Vert \mathcal{K} u-\mathcal{L} u \Vert }{ \Vert u \Vert }\leq{}& I_{q}^{\alpha + \beta } \biggl( \biggl\vert \frac{f_{u}(t)}{u(t)}-\lambda (t) \biggr\vert \biggr) \\ &{}+ \bigl\vert \mu _{1}(t) \bigr\vert \biggl[I_{q}^{\alpha +\beta +\sigma _{1}} \biggl( \biggl\vert \frac{f_{u}(\eta _{1})}{u(\eta _{1})}-\lambda (\eta _{1}) \biggr\vert \biggr) \biggr] \\ &{}+ \bigl\vert \mu _{2}(t) \bigr\vert \biggl[\lambda _{2} \biggl(I_{q}^{\alpha +\beta +\sigma _{2}} \biggl( \biggl\vert \frac{f_{u}(\eta _{2})}{u(\eta _{2})}-\lambda (\eta _{2}) \biggr\vert \biggr) \biggr) \\ &{}+b_{2} \biggl(I_{q}^{\alpha +\beta -\gamma } \biggl( \biggl\vert \frac{f_{u}(T)}{u(T)}-\lambda (T) \biggr\vert \biggr) \biggr) +a_{2} \biggl(I_{q}^{ \alpha +\beta } \biggl( \biggl\vert \frac{f_{u}(T)}{u(T)}-\lambda (T) \biggr\vert \biggr) \biggr) \biggr]. \end{aligned}$$

By (17) and letting \(\|u\| \rightarrow \infty \), it is concluded that \(\vert \frac{f_{u}(\cdot )}{u}-\lambda (\cdot ) \vert \rightarrow 0\). Thus we obtain

$$ \lim_{ \Vert u \Vert \rightarrow \infty } \frac{ \Vert \mathcal{K} (u) -\mathcal{L} (u) \Vert }{ \Vert u \Vert }=0. $$

Consequently, by Theorem 13, the supposed sequential four-point q-CFBVP (3) admits a solution in X. The proof is ended. □

3.3 The third existence criterion

We now present our last existence criterion based on the O’Regan theorem [45].

Theorem 15

([45])

Consider a closed and convex set \(E \neq \emptyset \) belonging to a Banach space X containing an open set O. Define \(\mathcal{K}: \bar{O} \rightarrow E\) as \(\mathcal{K}=\mathcal{K}_{1}+\mathcal{K}_{2}\) subject to \(\mathcal{K}(\bar{O})\) being bounded. Moreover, \(\mathcal{K}_{1}: \bar{O} \rightarrow E\) is continuous and completely continuous, \(\mathcal{K}_{2}: \overline{O} \rightarrow E\) is nonlinear contraction (i.e, a nonnegative nondecreasing function \(\Upsilon :[0, \infty ) \rightarrow [0, \infty )\) exists which satisfies \(\Upsilon (t)< t\) for \(t>0\), and \(\Vert \mathcal{K}_{2} u-\mathcal{K}_{2} u' \Vert \leq \Upsilon ( \|u-u'\|)\), \(\forall u,u' \in O\).) Then either

  1. (C1)

    \(\mathcal{K}\) has a fixed point \(u \in \bar{O} \);

or

  1. (C2)

    there exist \(u \in \partial O\) and \(\mu \in (0,1)\) such that \(u=\mu \mathcal{K}(u)\).

Theorem 16

Let \(f, g\in C(J \times \mathbb{R}, \mathbb{R})\) and assume that:

  1. (D1)

    there exist a nonnegative mapping \(b \in C(J,[0, \infty ))\) and a nondecreasing function \(\mathbb{T}:[0, \infty ) \rightarrow (0, \infty )\) such that

    $$ \bigl\vert f(t, u) \bigr\vert \leq b(t) \mathbb{T}\bigl( \Vert u \Vert \bigr), \quad \forall (t, u) \in J \times \mathbb{R}; $$
  2. (D2)

    there exist a continuous function \(\phi _{1}:[0, \infty ) \rightarrow [0, \infty )\) and \(\kappa >0\) such that

    $$ \bigl\vert g(t, u)-g(t, v) \bigr\vert \leq \phi _{1} \bigl( \Vert u-v \Vert \bigr) \quad \textit{and} \quad \phi _{1} \bigl( \vert u \vert \bigr) \leq \kappa \vert u \vert , \quad \forall t \in J, u, v \in \mathbb{R}; $$
  3. (D3)

    there exists \(\varepsilon >0\) such that \(\sup_{\varepsilon \in (0, \infty )} [ \frac{\varepsilon }{\Theta _{1}b^{*} \mathbb{T}(\varepsilon )+l\Theta _{2}} ] >\frac{1}{1-\kappa \Theta _{2}}\), where \(l=\sup_{t \in J} |g(t, 0)| \) and \(\kappa \Theta _{2}<1\).

Then there exists a solution for the supposed sequential four-point q-CFBVP (3) on J.

Proof

We consider \(\mathcal{K}: X \rightarrow X\) defined by (13) as

$$ \mathcal{K} u(t)=\mathcal{K}_{1} u(t)+\mathcal{K}_{2} u(t),\quad t \in J, $$

where the operators \(\mathcal{K}_{1}\) and \(\mathcal{K}_{2}\) are, respectively, given in (14) and (15). By (D3), \(\exists \varepsilon >0\) so that

$$ \frac{\varepsilon }{\Theta _{1}b^{*} \mathbb{T}(\varepsilon )+l\Theta _{2}}> \frac{1}{1-k \Theta _{2}}, $$

and take \(B_{\varepsilon }=\{u \in X:\|u\|<\varepsilon \}\). We demonstrate the continuity and complete continuity of \(\mathcal{K}_{1}\). Before this, we prove the uniform boundedness of \(\mathcal{K}_{1}\). Taking any \(u \in \bar{B}_{\varepsilon }\), we have

$$\begin{aligned} \bigl\vert (\mathcal{K}_{1} u) (t) \bigr\vert \leq{}& I_{q}^{\alpha +\beta } \bigl\vert f_{u}(t) \bigr\vert + \bigl\vert \mu _{1}(t) \bigr\vert \bigl(I_{q}^{\alpha +\beta +\sigma _{1}} \bigl\vert f_{u}( \eta _{1}) \bigr\vert \bigr) \\ &{}+ \bigl\vert \mu _{2}(t) \bigr\vert \bigl[\lambda _{2} \bigl(I_{q}^{\alpha +\beta + \sigma _{2}} \bigl\vert f_{u}(\eta _{2}) \bigr\vert \bigr) +b_{2} \bigl(I_{q}^{\alpha +\beta -\gamma } \bigl\vert f_{u}(T) \bigr\vert \bigr)+a_{2}\bigl(I_{q}^{ \alpha +\beta } \bigl\vert f_{u}(T) \bigr\vert \bigr) \bigr] \\ \leq{}& \Theta _{1} b^{*} \mathbb{T}(\varepsilon ), \end{aligned}$$

in which \(b^{*} = \sup_{t\in J}\vert b(t) \vert \). Thus \(\mathcal{K}_{1}\) is uniformly bounded. Let \(t_{1}, t_{2} \in J\) such that \(t_{1}< t_{2}\). Then

$$\begin{aligned} & \bigl\vert (\mathcal{K}_{1} u) (t_{2})-( \mathcal{K}_{1} u) (t_{1}) \bigr\vert \\ &\quad \leq I_{q}^{\alpha +\beta } \bigl\vert f_{u}(t_{2}) - f_{u}(t_{1}) \bigr\vert + \bigl\vert \mu _{1}(t_{2}) - \mu _{1}(t_{1}) \bigr\vert \bigl(I_{q}^{\alpha +\beta +\sigma _{1}} \bigl\vert f_{u}( \eta _{1}) \bigr\vert \bigr) \\ &\qquad {}+ \bigl\vert \mu _{2}(t_{2}) - \mu _{2}(t_{1}) \bigr\vert \bigl[\lambda _{2} \bigl(I_{q}^{ \alpha +\beta +\sigma _{2}} \bigl\vert f_{u}(\eta _{2}) \bigr\vert \bigr) \\ &\qquad {}+b_{2}\bigl(I_{q}^{\alpha +\beta -\gamma } \bigl\vert f_{u}(T) \bigr\vert \bigr)+a_{2}\bigl(I_{q}^{\alpha + \beta } \bigl\vert f_{u}(T) \bigr\vert \bigr) \bigr] \\ &\quad \leq \frac{b^{*}\mathbb{T}(\varepsilon ) }{\Gamma _{q}(\alpha +\beta +1)} \bigl[t_{2}^{\alpha +\beta }-t_{1}^{\alpha +\beta }+2(t_{2}-t_{1})^{ \alpha +\beta } \bigr] \\ &\qquad {}+ \bigl\vert \mu _{1}(t_{2}) - \mu _{1}(t_{1}) \bigr\vert \bigl[ I_{q}^{\alpha + \beta +\sigma _{1}} 1(\eta _{1}) b^{*}\mathbb{T}(\varepsilon ) \bigr] \\ &\qquad {}+ \bigl\vert \mu _{2}(t_{2}) - \mu _{2}(t_{1}) \bigr\vert \bigl[ \lambda _{2}I_{q}^{ \alpha +\beta +\sigma _{2}} 1(\eta _{2})b^{*} \mathbb{T}(\varepsilon ) \\ &\qquad {}+ b_{2}I_{q}^{\alpha +\beta -\gamma } 1(T)b^{*} \mathbb{T}( \varepsilon ) +a_{2}I_{q}^{\alpha +\beta } 1(T)b^{*}\mathbb{T}( \varepsilon ) \bigr], \end{aligned}$$

which tends to zero as \(t_{2} \rightarrow t_{1}\) free of u. This gives the equicontinuity of \(\mathcal{K}_{1}\). Application of the Arzelà–Ascoli theorem proves the compactness of \(\mathcal{K}_{1}\) and consequently its complete continuity. Furthermore, the continuity of \(\mathcal{K}_{1}\) can be deduced from that of f by the hypothesis.

We now show that \(\mathcal{K}_{2}\) is a nonlinear contraction. By (D2) and for \(u, v \in B_{\varepsilon }\), we have

$$ \begin{aligned} & \bigl\vert (\mathcal{K}_{2} u) (t)-( \mathcal{K}_{2} v) (t) \bigr\vert \\ &\quad \leq I_{q}^{ \beta } \vert g_{u} - g_{v} \vert (t) + \bigl\vert \mu _{1}(t) \bigr\vert \bigl[ I_{q}^{ \beta +\sigma _{1}} \vert g_{u} - g_{v} \vert (\eta _{1}) \bigr] \\ &\qquad {}+ \bigl\vert \mu _{2}(t) \bigr\vert \bigl[ \lambda _{2}I_{q}^{\beta +\sigma _{2}} \vert g_{u} - g_{v} \vert (\eta _{2}) + b_{2}I_{q}^{\beta -\gamma } \vert g_{u} - g_{v} \vert (T) +a_{2}I_{q}^{\beta } \vert g_{u} - g_{v} \vert (T) \bigr] \\ &\quad \leq \Theta _{2} \phi _{1} \bigl( \bigl\Vert u(t) - v(t) \bigr\Vert \bigr) \\ &\quad \leq \Theta _{2} \kappa \bigl\Vert u(t) - v(t) \bigr\Vert . \end{aligned} $$

By setting \(\Upsilon (u)= \Theta _{2} \kappa u\), note that \(\Upsilon (0)=0\) and \(\Upsilon (u)= \Theta _{2} \kappa u< u\) for \(u>0\) since \(\kappa \Theta _{2} < 1\). Thus

$$ \Vert \mathcal{K}_{2} u-\mathcal{K}_{2} v \Vert \leq \Upsilon \bigl( \Vert u-v \Vert \bigr). $$

Hence \(\mathcal{K}_{2}\) is a nonlinear contraction. Now again, by (D2), for arbitrary \(u \in B_{\varepsilon }\), we estimate

$$ \bigl\vert g_{u}(t) \bigr\vert = \bigl\vert g(t, u) \bigr\vert \leq \bigl\vert g(t, u)-g(t, 0) \bigr\vert + \bigl\vert g(t, 0) \bigr\vert \leq \phi _{1}\bigl( \Vert u \Vert \bigr)+ \bigl\vert g(t, 0) \bigr\vert \leq \kappa \varepsilon + l . $$

where \(l=\sup_{t \in J}\vert g (t, 0)\vert \). Hence, we get

$$ \Vert \mathcal{K}_{2}u \Vert \leq \Theta _{2}(\kappa \varepsilon +l), $$

which confirms the boundedness of \(\mathcal{K}_{2}\). Thus, \(\mathcal{K} = \mathcal{K}_{1} + \mathcal{K}_{2}\) is bounded.

In the final step, we prove that the assumption \((\mathrm{C} 2)\) of Theorem 15 does not hold. To prove this, consider the existence of \(\mu \in (0,1)\) and \(u \in \partial B_{\varepsilon }\) such that \(u=\mu \mathcal{K} u\). So \(\|u\|=\varepsilon \) and

$$\begin{aligned} \bigl\vert u(t) \bigr\vert & = \mu \bigl\vert (\mathcal{K}u) (t) \bigr\vert \\ &= \mu \bigl\vert \mathcal{K}_{1} u(t)+\mathcal{K}_{2} u(t) \bigr\vert \\ &\leq \bigl\vert \mathcal{K}_{1} u(t) \bigr\vert + \bigl\vert \mathcal{K}_{2} u(t) \bigr\vert \\ &\leq \Theta _{1}b^{*} \mathbb{T}(\varepsilon )+ \Theta _{2}(\kappa \varepsilon +1) . \end{aligned}$$

Taking the supremum for all \(t \in J\) yields

$$ \Vert u \Vert \leq \Theta _{1}b^{*} \mathbb{T}( \varepsilon )+(\kappa \varepsilon +l) \Theta _{2}. $$

Hence, we get

$$ \frac{\varepsilon }{\Theta _{1}b^{*} \mathbb{T}(\varepsilon )+l \Theta _{2}} \leq \frac{1}{1- \kappa \Theta _{2}}, $$

which contradicts (D3). Thus \(\mathcal{K}_{1}\) and \(\mathcal{K}_{2}\) satisfy all the assertions of Theorem 15. Therefore, a fixed-point of \(\mathcal{K}\) in \(B_{\varepsilon }\) exists, which is the same solution of the sequential four-point q-CFBVP (3). The proof is finished. □

3.4 The uniqueness property

Finally, we investigate the uniqueness property for the solutions of the sequential four-point q-CFBVP (3) by referring to the Banach principle.

Theorem 17

Let

\((H_{4})\):

\(\exists a > 0\) satisfying

$$\big|g_{u}(t) - g_{v}(t) \big| \leq a\big\vert u(t)-v(t)\big\vert ,\quad \forall t \in J, u, v \in \mathbb{R}; $$
\((H_{5})\):

\(\exists \ell >0\) satisfying

$$\big|f_{u}(t) -f_{v}(t)\big| \leq \ell \big\vert u(t)-v(t)\big\vert , \quad \forall t \in J, u, v \in \mathbb{R}. $$

Then the sequential four-point q-CFBVP (3) has a unique solution on J if

$$ \ell \Theta _{1}+a \Theta _{2}< 1, $$
(20)

where \(\Theta _{1}\), \(\Theta _{2}\) are given in (11) and (12), respectively.

Proof

To prove the relevant result, define the ball \(B_{r}=\{u \in X:\|u\| \leq r\}\) for some \(r>0\) satisfying

$$ r \geq \frac{\Theta _{1} f_{0}^{*}+\Theta _{2} g_{0}^{*}}{1-\ell \Theta _{1} -a \Theta _{2}}, $$

where \(g_{0}^{*}=\sup_{t \in J}|g(t, 0)|\) and \(f_{0}^{*}=\sup_{t \in J}|f(t, 0)|\) and \(\Theta _{1}\) and \(\Theta _{2}\) are, respectively, given by (11) and (12). Now, we prove \(\mathcal{K} B_{r} \subset B_{r}\) in which the operator \(\mathcal{K}: X\rightarrow X\) is illustrated as (13). Similar to Step 1 in Theorem 12, for \(u \in B_{r}\), we get

$$\begin{aligned} \bigl\vert (\mathcal{K} u) (t) \bigr\vert \leq{}& I_{q}^{\beta } \bigl\vert g_{u}(t) \bigr\vert +I_{q}^{\alpha +\beta } \bigl\vert f_{u}(t) \bigr\vert \\ &{}+ \bigl\vert \mu _{1}(t) \bigr\vert \bigl[ I_{q}^{\beta +\sigma _{1}} \bigl\vert g_{u}( \eta _{1}) \bigr\vert \\ &{}+I_{q}^{\alpha +\beta +\sigma _{1}} \bigl\vert f_{u}(\eta _{1}) \bigr\vert \bigr] + \bigl\vert \mu _{2}(t) \bigr\vert \bigl[ \lambda _{2}\bigl(I_{q}^{\beta + \sigma _{2}} \bigl\vert g_{u}(\eta _{2}) \bigr\vert +I_{q}^{\alpha +\beta + \sigma _{2}} \bigl\vert f_{u}(\eta _{2}) \bigr\vert \bigr) \\ &{}+ b_{2}\bigl(I_{q}^{\beta -\gamma } \bigl\vert g_{u}(T) \bigr\vert +I_{q}^{\alpha + \beta -\gamma } \bigl\vert f_{u}(T) \bigr\vert \bigr) +a_{2}\bigl(I_{q}^{\beta } \bigl\vert g_{u}(T) \bigr\vert +I_{q}^{\alpha +\beta } \bigl\vert f_{u}(T) \bigr\vert \bigr) \bigr] \\ \leq{}& \bigl(\ell \Vert u \Vert +f_{0}^{*}\bigr) \sup_{t \in J} \bigl\{ I_{q}^{\alpha + \beta } 1(t) + \bigl\vert \mu _{1}(t) \bigr\vert \bigl[ I_{q}^{\alpha +\beta + \sigma _{1}} 1(\eta _{1}) \bigr] \\ &{}+ \bigl\vert \mu _{2}(t) \bigr\vert \bigl[ \lambda _{2}I_{q}^{\alpha +\beta + \sigma _{2}} 1(\eta _{2}) + b_{2}I_{q}^{\alpha +\beta -\gamma } 1(T) +a_{2}I_{q}^{ \alpha +\beta } 1(T) \bigr] \bigr\} \\ &{}+ \bigl(a \Vert u \Vert +g_{0}^{*}\bigr) \sup _{t \in J} \bigl\{ I_{q}^{\beta } 1(t) + \bigl\vert \mu _{1}(t) \bigr\vert \bigl[ I_{q}^{\beta +\sigma _{1}} 1(\eta _{1}) \bigr] + \bigl\vert \mu _{2}(t) \bigr\vert \bigl[ \lambda _{2}I_{q}^{\beta +\sigma _{2}} 1(\eta _{2}) \\ &{}+ b_{2}I_{q}^{\beta -\gamma } 1(T) +a_{2}I_{q}^{\beta } 1(T) \bigr] \bigr\} \\ \leq{}& \Theta _{1} \bigl(\ell r+f_{0}^{*} \bigr)+\Theta _{2} \bigl(a r+g_{0}^{*} \bigr)< r, \end{aligned}$$

which implies \(\|\mathcal{K}(u)\| \leq r\). Thus, \(\mathcal{K}\) maps \(B_{r}\) into itself. Next, we prove that \(\mathcal{K}\) is a contraction. For \(u, v \in X\), and applying (11) and (12), we have

$$\begin{aligned} \bigl\vert (\mathcal{K} u) (t)-(\mathcal{K} v) (t) \bigr\vert \leq{}& I_{q}^{\beta } \vert g_{u} - g_{v} \vert (t) +I_{q}^{\alpha +\beta } \vert f_{u} - g_{v} \vert (t) \\ &{}+ \bigl\vert \mu _{1}(t) \bigr\vert \bigl[ I_{q}^{\beta +\sigma _{1}} \vert g_{u} - g_{v} \vert (\eta _{1}) +I_{q}^{\alpha +\beta +\sigma _{1}} \vert f_{u} - f_{v} \vert (\eta _{1}) \bigr] \\ &{}+ \bigl\vert \mu _{2}(t) \bigr\vert \bigl[ \lambda _{2}\bigl(I_{q}^{\beta +\sigma _{2}} \vert g_{u} - g_{v} \vert (\eta _{2}) +I_{q}^{\alpha +\beta +\sigma _{2}} \vert f_{u} - f_{v} \vert (\eta _{2}) \bigr) \\ &{}+ b_{2}\bigl(I_{q}^{\beta -\gamma } \vert g_{u} - g_{v} \vert (T) +I_{q}^{ \alpha +\beta -\gamma } \vert f_{u} - f_{v} \vert (T) \bigr) \\ &{}+a_{2}\bigl(I_{q}^{\beta } \vert g_{u} - g_{v} \vert (T) +I_{q}^{\alpha + \beta } \vert f_{u} - f_{v} \vert (T) \bigr) \bigr] \\ \leq{}& (\ell \Theta _{1}+a\Theta _{2} ) \Vert u-v \Vert . \end{aligned}$$

Consequently, we get

$$ \bigl\Vert \mathcal{K} (u) (t)-\mathcal{K} (v) (t) \bigr\Vert \leq (\ell \Theta _{1}+a \Theta _{2} ) \Vert u-v \Vert . $$

Since \(\ell \Theta _{1}+a\Theta _{2} <1\), the above inequality proves that \(\mathcal{K}\) is a contraction. Thus application of the Banach principle shows that \(\mathcal{K}\) has a unique fixed point, corresponding to unique solution of the sequential four-point q-CFBVP (3) on J. This ends the proof. □

4 The criterion of Ulam–Hyers stability

Due to the importance of the notion of stability for possible solutions of different dynamical systems, in this section, we review two Ulam–Hyers and generalized Ulam–Hyers stabilities for solutions of the sequential four-point q-CFBVP (3). For more information, see [46–48].

Definition 18

([49])

The sequential four-point q-CFBVP (3) is Ulam–Hyers stable if \(\exists c^{*} \in \mathbb{R}_{+} \) such that \(\forall \varepsilon >0\) and \(\forall u^{*}(t)\in C(J,\mathbb{R}) \) as a solution function satisfying

$$ \bigl\vert { \mathcal{D}}_{q}^{\alpha } \bigl({ \mathcal{D}}_{q}^{\beta }u^{*}(t)-g\bigl(t, u^{*}(t)\bigr) \bigr) - f\bigl(t, u^{*}(t)\bigr) \bigr\vert < \varepsilon , $$
(21)

\(\exists u(t)\in C(J,\mathbb{R}) \) as the solution of the sequential four-point q-CFBVP (3) with

$$ \bigl\vert u^{*}(t)- u(t) \bigr\vert \leq \varepsilon c^{*}, \quad t \in J. $$

Definition 19

([49])

The sequential four-point q-CFBVP (3) is generalized Ulam–Hyers stable if \(\exists H \in C(\mathbb{R}^{+},\mathbb{R}^{+})\) with \(H(0)=0\) such that \(\forall \varepsilon >0\) and \(\forall u^{*}(t)\in C(J,\mathbb{R}) \) as a solution of

$$ \bigl\vert { \mathcal{D}}_{q}^{\alpha } \bigl({ \mathcal{D}}_{q}^{\beta }u^{*}(t)-g\bigl(t, u^{*}(t)\bigr) \bigr) - f\bigl(t, u^{*}(t)\bigr) \bigr\vert < \varepsilon , $$

\(\exists u(t)\in C(J,\mathbb{R}) \) as a solution of the sequential four-point q-CFBVP (3) with

$$ \bigl\vert u^{*}(t)-u(t) \bigr\vert \leq H(\varepsilon ),\quad t\in J. $$

Remark 1

([49])

It is evident that Def. 18 ⇒ Def. 19.

Remark 2

([49])

It is notable that \(u^{*}(t)\in C(J,\mathbb{R}) \) is a solution for (21) iff \(\exists G\in C(J,\mathbb{R})\) depending on \(u^{*}\) such that

  1. (1)

    \(\vert G(t)\vert < \varepsilon \), \(t\in J\).

  2. (2)

    \({ \mathcal{D}}_{q}^{\alpha } ({ \mathcal{D}}_{q}^{\beta }u^{*}(t)-g(t, u^{*}(t)) ) = f(t, u^{*}(t)) + G(t)\), \(t\in J\).

Now, we can discuss the above stabilities for solutions to the sequential four-point q-CFBVP (3).

Theorem 20

If \((H_{4})\) and \((H_{5})\) are fulfilled, then the sequential four-point q-CFBVP (3) is Ulam–Hyers stable on J and accordingly is generalized Ulam–Hyers stable whenever

$$ \ell \Theta _{1} + a\Theta _{2} < 1, $$

where \(\Theta _{1}\), \(\Theta _{2}\) are in the same forms given in (11) and (12), respectively.

Proof

For each \(\varepsilon >0 \) and each function \(u^{*}(t)\in C(J,\mathbb{R}) \) as a solution of the inequality

$$ \bigl\vert { \mathcal{D}}_{q}^{\alpha } \bigl({ \mathcal{D}}_{q}^{\beta }u(t)-g\bigl(t, u(t)\bigr) \bigr) - f \bigl(t, u(t)\bigr) \bigr\vert < \varepsilon , $$

a function \(G(t) \) exists which satisfies

$$ { \mathcal{D}}_{q}^{\alpha } \bigl({ \mathcal{D}}_{q}^{\beta }u(t)-g \bigl(t, u(t)\bigr) \bigr) = f\bigl(t, u(t)\bigr) + G(t) $$

with \(\vert G(t) \vert \leq \varepsilon \). It gives

$$\begin{aligned} u^{*}(t)={}& I_{q}^{\beta }g_{u^{*}}(t)+I_{q}^{\alpha +\beta }f_{u^{*}}(t) + I_{q}^{\alpha +\beta }G(t) \\ &{}+\mu _{1}(t) \bigl[I_{q}^{\beta +\sigma _{1}}g_{u^{*}}( \eta _{1})+I_{q}^{ \alpha +\beta +\sigma _{1}}f_{u^{*}}(\eta _{1}) \bigr] \\ &{}+\mu _{2}(t) \bigl[\lambda _{2} \bigl(I_{q}^{\beta +\sigma _{2}}g_{u^{*}}( \eta _{2})+I_{q}^{\alpha +\beta +\sigma _{2}}f_{u^{*}}( \eta _{2}) \bigr) \\ &{}-b_{2} \bigl(I_{q}^{\beta -\gamma }g_{u^{*}}(T)+I_{q}^{\alpha +\beta - \gamma }f_{u^{*}}(T) \bigr) -a_{2} \bigl(I_{q}^{\beta }g_{u^{*}}(T)+I_{q}^{ \alpha +\beta }f_{u^{*}}(T) \bigr) \bigr]. \end{aligned}$$

On the other side, let a unique function \(u(t)\in C(J,\mathbb{R}) \) be the solution of (3). Then \(u(t) \) is written by

$$\begin{aligned} u(t)={}& I_{q}^{\beta }g_{u}(t)+I_{q}^{\alpha +\beta }f_{u}(t) \\ &{}+\mu _{1}(t) \bigl[I_{q}^{\beta +\sigma _{1}}g_{u}( \eta _{1})+I_{q}^{ \alpha +\beta +\sigma _{1}}f_{u}(\eta _{1}) \bigr] \\ &{}+\mu _{2}(t) \bigl[\lambda _{2} \bigl(I_{q}^{\beta +\sigma _{2}}g_{u}( \eta _{2})+I_{q}^{\alpha +\beta +\sigma _{2}}f_{u}( \eta _{2}) \bigr) \\ &{}-b_{2} \bigl(I_{q}^{\beta -\gamma }g_{u}(T)+I_{q}^{\alpha +\beta - \gamma }f_{u}(T) \bigr) -a_{2} \bigl(I_{q}^{\beta }g_{u}(T)+I_{q}^{ \alpha +\beta }f_{u}(T) \bigr) \bigr]. \end{aligned}$$

We estimate

$$\begin{aligned} \bigl\vert u^{*}(t)- u(t) \bigr\vert \leq{}& I_{q}^{\alpha +\beta } \bigl\vert G(t) \bigr\vert + I_{q}^{\beta } \vert g_{u^{*}} - g_{u} \vert (t) +I_{q}^{ \alpha +\beta } \vert f_{u^{*}} - g_{u} \vert (t) \\ &{}+ \bigl\vert \mu _{1}(t) \bigr\vert \bigl[ I_{q}^{\beta +\sigma _{1}} \vert g_{u^{*}} - g_{u} \vert (\eta _{1}) +I_{q}^{\alpha +\beta +\sigma _{1}} \vert f_{u^{*}} - f_{u} \vert (\eta _{1}) \bigr] \\ &{}+ \bigl\vert \mu _{2}(t) \bigr\vert \bigl[ \lambda _{2}\bigl(I_{q}^{\beta +\sigma _{2}} \vert g_{u^{*}} - g_{u} \vert (\eta _{2}) +I_{q}^{\alpha +\beta + \sigma _{2}} \vert f_{u^{*}} - f_{u} \vert (\eta _{2}) \bigr) \\ &{}+ b_{2}\bigl(I_{q}^{\beta -\gamma } \vert g_{u^{*}} - g_{u} \vert (T) +I_{q}^{ \alpha +\beta -\gamma } \vert f_{u^{*}} - f_{u} \vert (T) \bigr) \\ &{}+a_{2}\bigl(I_{q}^{\beta } \vert g_{u^{*}} - g_{u} \vert (T) +I_{q}^{ \alpha +\beta } \vert f_{u^{*}} - f_{u} \vert (T) \bigr) \bigr] \\ \leq{}& \frac{\varepsilon T^{\alpha + \beta }}{\Gamma _{q}(\alpha + \beta + 1)} + (\ell \Theta _{1}+a\Theta _{2} ) \bigl\Vert u^{*}-u \bigr\Vert . \end{aligned}$$

Hence

$$ \bigl\Vert u^{*}-u \bigr\Vert \leq \frac{\varepsilon T^{\alpha + \beta }}{\Gamma _{q}(\alpha + \beta + 1)} + (\ell \Theta _{1}+a\Theta _{2} ) \bigl\Vert u^{*}-u \bigr\Vert , $$

where \(\Theta _{1}\), \(\Theta _{2}\) are the same constants as represented in (11) and (12), respectively. In consequence,

$$\begin{aligned} \bigl\Vert u^{*}-u \bigr\Vert &\leq \frac{\varepsilon T^{\alpha + \beta }}{\Gamma _{q}(\alpha + \beta + 1) [1 - (\ell \Theta _{1}+a\Theta _{2}) ] }. \end{aligned}$$

By assuming \(c^{*}= \frac{ T^{\alpha + \beta }}{\Gamma _{q}(\alpha + \beta + 1) [1 - (\ell \Theta _{1}+a\Theta _{2}) ] } \), the Ulam–Hyers stability for q-system (3) is satisfied. Also, for

$$ H(\varepsilon )= \frac{\varepsilon T^{\alpha + \beta }}{\Gamma _{q}(\alpha + \beta + 1) [1 - (\ell \Theta _{1}+a\Theta _{2}) ] } $$

with \(H(0)=0 \), the condition of the generalized Ulam–Hyers stability is fulfilled for solutions of the q-system (3). This completes the proof. □

5 Two examples

Here, we aim to present some examples to examine the obtained results.

Example 1

Let us consider the sequential four-point q-CFBVP with the following data:

$$ \textstyle\begin{cases} D^{1/3}_{1/4} [D^{2/3}_{1/4} u(t)-g(t, u(t)) ]=f(t, u(t)), \quad t \in J = [0,1], \\ u(0)+2{ \mathcal{D}}_{1/4}^{1/2} u(0)=2/5 I_{1/4}^{3/4}u(1/2), \\ 2 u(1)+ { \mathcal{D}}_{1/4}^{1/2} u(1)= 3/7 I_{1/4}^{1/4}u(3/4), \end{cases} $$
(22)

where \(\alpha =1 / 3\), \(\beta =2 / 3\), \(q=1/4\), \(T=1\), \(\gamma =1 / 2\), \(a_{1}=b_{2}=1\), \(a_{2}=b_{1}=2\), \(\sigma _{1}=3 / 4\), \(\sigma _{2}=1/4\), \(\lambda _{1}=2/5\), \(\lambda _{2}=3/7\), \(\eta _{1}=1 / 2\), \(\eta _{2}=3/4\) and \(g(t, u)\), \(f(t, u)\) are defined by

$$ f(t,u)=\frac{t u}{56(1+t)^{5}} \biggl(\frac{ \vert u \vert +2}{ \vert u \vert +1} \biggr)+ \frac{1}{2} \quad \text{and}\quad g(t,u)=\frac{3 t^{2}}{6} \biggl( \frac{ \vert u \vert }{3( \vert u \vert +1)} \biggr). $$

The continuity of f is obvious and we reach \(f(t, 0)=\frac{1}{2} \) (see Fig. 1). Now, we divide \(f(t, u)\) by u and we get

$$ \frac{f(t, u)}{u}=\frac{t}{56(1+t)^{5}} \biggl(1+\frac{1}{ \vert u \vert +1} \biggr)+ \frac{1}{2 u}. $$

Hence

$$ \lim_{\|u\| \rightarrow \infty } \frac{f(t, u)}{u}= \frac{t}{56(1+t)^{5}}. $$

Setting \(\lambda (t)=\frac{t}{56(1+t)^{5}}\), we get \(\lambda _{\max }=0.0179\). On the other side,

$$ \bigl\vert g(t, u) \bigr\vert \leq \frac{1}{6} \vert u \vert . $$

Letting \(A=1/6\), we obtain \(\Theta _{1}=3.5597\) and \(\Theta _{2}=4.8600\). since \((1-A\Theta _{2} ) / \Theta _{1}=0.0548>\lambda _{\max }\), where \(\Theta _{1}\) and \(\Theta _{2}\) are, respectively, given by Eqs. (11) and (12). therefore, by Theorem 14, the sequential four-point q-CFBVP (22) has a solution on \([0, 1]\).

Figure 1
figure 1

Graphs of the functions \(f(t,u)\) and \(g(t,u)\)

Full size image

Example 2

By considering \(\alpha =1 / 3\), \(\beta =2 / 3\), \(q=1/4\), \(T=1\), \(\gamma =1 / 2\), \(a_{1}=b_{2}=1\), \(a_{2}=b_{1}=2\), \(\sigma _{1}=3 / 4\), \(\sigma _{2}=1/4\), \(\lambda _{1}=2/5\), \(\lambda _{2}=3/7\), \(\eta _{1}=1 / 2\), \(\eta _{2}=3/4\) the sequential four-point q-CFBVP is then given by

$$ \textstyle\begin{cases} D^{1/3}_{1/4} [D^{2/3}_{1/4} u(t)-g(t, u(t)) ]=f(t, u(t)),\quad t \in J= [0,1], \\ u(0)+2{ \mathcal{D}}_{1/4}^{1/2} u(0)=2/5 I_{1/4}^{3/4}u(1/2), \\ 2 u(1)+ { \mathcal{D}}_{1/4}^{1/2} u(1)= 3/7 I_{1/4}^{1/4}u(3/4), \end{cases} $$
(23)

where \(f(t, u)\) and \(g(t, u)\) are given by (see Fig. 2)

$$ f(t, u)=\frac{1}{3 \sqrt{900+t^{2}}} \bigl(\arctan u+e^{-t} \bigr)\quad \text{and}\quad g(t, u)=\frac{1}{100 (1+t^{2} )} \sin u+ \frac{\cos t}{25}. $$
Figure 2
figure 2

Graphs of the functions \(f(t,u)\) and \(g(t,u)\)

Full size image

By usual computations, we obtain \(\Theta _{1}=3.5597\) and \(\Theta _{2}=4.8600\). Taking \(a =1/100\) and \(\ell =1/90\), it is clear that \((H_{4} )\) and \((H_{5} )\) are verified. Moreover, \(\ell \Theta _{1}+ a \Theta _{2} \approx 0.8775<1\). Thus, Theorem 17 is fulfilled and hence based on it, one can find that a unique solution exists for the sequential four-point q-CFBVP (23) on \([0,1]\). On the other side, as \(\ell \Theta _{1} + a\Theta _{2} <1\) is valid, so, by Theorem 20, the given sequential four-point q-CFBVP (23) is Ulam–Hyers and also generalized Ulam–Hyers stable on J.

6 Conclusions

In the present research, we considered a new boundary problem in the context of the quantum fractional operators. In other words, we defined a sequential q-fractional system of q-difference equation in which boundary conditions are designed as a linear combination of an unknown function and its q-derivative corresponding to a multiple of q-integrals in four points. The main focus of this research is on the solution’s existence and its uniqueness with the help of some methods inspired by several pure concepts in functional analysis. We used three different fixed-point methods for this aim relying on the measure of non-compactness and condensing operators and compact operators. The existence of a unique solution is investigated based on the Banach criterion. The investigation of stability of the given q-CFBVP system in two formats based on Ulam–Hyers’ conditions is implemented. Lastly, two examples are provided to ensure the findings. It is evident that this structure is more general and has many special applied cases. By assuming \(g(t, u(t)) = - \mu \in \mathbb{R}\) and \(a_{1} = b_{1} = a_{2} = b_{2} = 1\) and \(\sigma _{1} = \sigma _{2} = 1\) and by letting \(q \to 1\), our proposed sequential four-point q-CFBVP (3) is transformed into a fractional Langevin equation with integral conditions

$$ \textstyle\begin{cases} { \mathcal{D}}_{q}^{\alpha } ({ \mathcal{D}}_{q}^{\beta }u(t) + \mu ) )=f(t, u(t)), \quad t\in \mathrm{J}:=[0,T], \\ u(0)+ { \mathcal{D}}^{\gamma }u(0)=\lambda _{1} \int _{0}^{ \eta _{1}} u(s) \,\mathrm{d}s, \quad \eta _{1} \in (0,T), \\ u(T)+ { \mathcal{D}}^{\gamma }u(T)= \lambda _{2} \int _{0}^{ \eta _{2}} u(s) \,\mathrm{d}s,\quad \eta _{2} \in (0,T), \end{cases} $$

which is considered as one of the most important equations in mathematical physics. Therefore, one can observe that the research study presented in the manuscript is not only new in the existing structure, but will also lead to other various quantum fractional problems as special cases. In future studies, we can generalize our boundary conditions to multi-point ones and investigate similar results in the context of newly-defined fractional \((p,q)\)-operators in both cases of difference equations and inclusions.