Existence and stability of a q-Caputo fractional jerk differential equation having anti-periodic boundary conditions

Abstract

In this work, we analyze a q-fractional jerk problem having anti-periodic boundary conditions. The focus is on investigating whether a unique solution exists and remains stable under specific conditions. To prove the uniqueness of the solution, we employ a Banach fixed point theorem and a mathematical tool for establishing the presence of distinct fixed points. To demonstrate the availability of a solution, we utilize Leray–Schauder’s alternative, a method commonly employed in mathematical analysis. Furthermore, we examine and introduce different kinds of stability concepts for the given problem. In conclusion, we present several examples to illustrate and validate the outcomes of our study.

1 Introduction

Recently, a lot of researchers have shown a great interest in the field of q-calculus ($\mathcal{QC}$) and problems involving fractional q-differential equations (q-). The roots of $\mathcal{QC}$ can be traced back to 1908 with the work of Jackson in [1]. Additionally, q- were developed to characterize the variety of physical processes that emerged, such as discrete stochastic processes, discrete dynamical systems, quantum dynamics, and so on [2]. As the theory of $\mathcal{QC}$ progressed, some associated ideas have been presented and examined, including q-integral transform theory, q-Mittag-Leffler functions, q-gamma, q-beta functions, q-Laplace transform, and so forth (for more details, see [3–9]). These concepts find applications in understanding and solving problems related to $\mathcal{QC}$. The reader may refer to [10–17] for more details on $\mathcal{QC}$.

In 1978, Schot [18] introduced the concept of “jerk” $\mathcal{J}$, which is essentially the rate at which acceleration changes. It involves the third derivative of quantity represented by u. The idea of $\mathcal{J}$ has proven in several scientific fields, including acoustics, electrical circuits, mechanics, and dynamical processes. It also helps us to understand how acceleration is changing over time, providing valuable insights into the behavior of systems in various applications [19–25]. In three dimensions, a dynamic system can be represented as

$$ \upsilon '( \chi )=a, \qquad a'( \chi ) = e, \qquad e'( \chi )= f( \upsilon ,a,e), $$

and can be well written in the form of $\upsilon ''' = f(\upsilon , \upsilon ', \upsilon '')$. The is third order autonomous that has found applications in various scientific fields, such as signal processing, secure communication, electrical engineering, control systems, bio-mechanics, and economic systems [17, 22]. Marcelo and Silva [26] employed the algebraic techniques in 2020 to ascertain the exact structure for a polynomial $\mathcal{J}$ function, hence guaranteeing the nonchaotic behavior of the subsequent :

$$ \upsilon ''' = \mathcal{J} \bigl( \upsilon , \upsilon ', \upsilon '' \bigr). $$

They also provided the proof for nonchaotic behavior. It can also be useful to investigate the different kinds of ordinary and their nonchaotic behavior. The authors in [27] addressed an initial value problem of nonlinear 3rd order :

$$ \textstyle\begin{cases} \upsilon ''' + f ( \upsilon , \upsilon ', \upsilon '' )=0, \\ \upsilon (0)=0, \qquad \upsilon '(0)=\mathscr{B},\qquad \upsilon ''(0)=0. \end{cases} $$

By employing analytical methodologies, the authors were able to enhance the method known as the global error minimization method GEMM to generate estimations using analytical techniques. Their developed approaches were known to be more successful and efficient than previously known current methods when compared to known solutions and accurate numerical ones. The authors in [28] utilized the modified harmonic balance technique for the subsequent nonlinear :

$$ \mathbb{D}^{3}\upsilon (\chi ) + \xi \bigl( \upsilon (\chi ), \mathbb{D}^{1}\upsilon (\chi ),{}^{\mathscr{C} }\mathbb{D}^{2} \upsilon (\chi ) \bigr)=0, $$

under conditions $\upsilon (0)=0$, $\mathbb{D}^{1} \upsilon (0) = \mathscr{B}$, and $\mathbb{D}^{2} \upsilon (0)=0$. Sousa et al., by employing fixed point approach, studied stability of the modified impulsive fractional s

$$ \textstyle\begin{cases} {}^{\mathscr{H} }\mathbb{D}^{\alpha ,\beta , \psi}_{0^{+}} \upsilon (\chi ) = \xi ( \chi , \upsilon (\chi ) ), & \chi \in (s_{i}, t_{i+1} ), i=0,1, \dots , m, \\ \upsilon (\chi )= \uptau _{i} (\chi , \upsilon (t_{i}^{+}) ),& \chi \in (t_{i}, s_{i}], i=1,2,\dots , m, \end{cases} $$

where ${}^{\mathscr{H} }\mathbb{D}^{\alpha ,\beta , \psi}_{0^{+}} (\cdot )$ is the ψ-Hilfer fractional derivative with $\alpha \in (0,1]$, $\beta \in [0, 1]$, and

are prefixed numbers, $\xi \in C( \Lambda \times \mathbb{R})$ and $\uptau _{i} \in C([t_{i}, s_{i}]\times \mathbb{R})$ for all $i=1,2,\dots , m$, which are noninstantaneous impulses, here with [29]. Wang et al. in [30] studied the various forms of Ulam stability ($\mathscr{US}$) and existence, uniqueness ($\mathfrak{EU}$) for the following nonlinear implicit fractional integro-differential equations involving Caputo derivative ($\mathscr{CD}$) of fractional order:

where $\nu , \zeta >0$, $1<\alpha \leqslant 2$, $0 \leqslant \beta \leqslant 2$ and continuous functions are represented as ξ, . The authors introduced the ψ-Hilfer pseudo-fractional operator, motivated by the ψ-Hilfer fractional derivative and the theory of pseudo-analysis, and investigated a new class of important and essential results for pseudo-fractional calculus in a semi-ring $([a, b], \oplus , \odot )$, and some particular cases were discussed (for more instances, see related research works [31–37]). Houas et al., by using Riemann–Liouville () and q-fractional $\mathscr{CD}$, examined the $\mathfrak{EU}$, Ulam–Hyers ($\mathscr{UH}$), and Ulam–Hyers–Rassias ($\mathscr{UHR}$) stability of the solution to q-fractional problem ($\mathbb{FJP}$) as follows:

where $\chi \in \Lambda$, $\{\alpha ,\omega ,\theta \}\in (0,1]$, $\beta \geq 1$, , ${}^{\mathscr{RL} }\mathbb{D}^{\alpha}_{q }$, ${}^{\mathscr{C} }\mathbb{D}^{\mu}_{q }$, $\mu \in \{ \omega , \theta \}$ are the q -fractional and $\mathscr{CD}s$ respectively [38]. The q -$\mathbb{FI}$ is $\mathcal{I}_{q }^{\beta}$ having type and is given an appropriate function [38].

Influenced by the aforementioned works, we present the following q-Caputo fractional with anti-periodic boundary conditions ($\mathbb{ABC}$s):

(1)

where $0<\{\alpha , \omega ,\theta \}\leqslant 1$, $\beta \in (0,1]$, , q-fractional $\mathscr{CD}$ is ${}^{\mathscr{C} }\mathbb{D}^{\mu}_{q}$, $\mu \in \{\alpha , \omega ,\theta , \beta \}$ of order μ on Λ, are appropriate functions and $\nu , \zeta >0$.

We list the important points of this manuscript:

• 1: We implement Caputo q-fractional having $\mathbb{ABC}$s for the first time in the literature.

• 2: In this manuscript, we established the $\mathfrak{EU}$ and $\mathscr{US}$ results for the suggested Problem (1).

• 3: Different from previous papers that used nonlinear implicit fractional integrodifferential equations in [30] and and q-fractional $\mathscr{CD}$ [38], we get better results by employing q-fractional having $\mathbb{ABC}$s.

• 4: We also show the graphical representation of having $\mathbb{ABC}$s.

This research article is organized in the following manner: Sect. 2 clarifies some basic ideas in $\mathcal{QC}$ and provides related lemmas. In Sect. 3, we establish the $\mathfrak{EU}$ of solution for the proposed system (1) by employing the Leray-Schauder alternative and the Banach fixed point theorem. Various types of $\mathscr{US}$ have been discussed in Sect. 4. In Sect. 5 an example is also presented at the end to verify our results. Finally, conclusion is also provided in Sect. 6.

2 Basic concepts

The following Banach space $(\mathcal{F}, \|\cdot \|_{ \mathcal{F}})$ is needed to analyze the q-:

supplied with the norm

$$ \begin{aligned} \Vert \upsilon \Vert _{\mathcal{F}} &= \Vert \upsilon \Vert + \bigl\Vert {}^{ \mathscr{C} } \mathbb{D}^{\theta}_{q } \upsilon \bigr\Vert + \bigl\Vert {}^{\mathscr{C} } \mathbb{D}^{\omega}_{ q } \bigl({}^{ \mathscr{C} } \mathbb{D}^{\theta}_{q } \upsilon \bigr) \bigr\Vert \\ & = \sup_{\chi \in \Lambda} \bigl\vert \upsilon ( \chi ) \bigr\vert + \sup_{\chi \in \Lambda} \bigl\vert {}^{\mathscr{C} }\mathbb{D}^{\theta}_{q } \upsilon (\chi ) \bigr\vert + \sup_{ \chi \in \Lambda} \bigl\vert \bigl( {}^{ \mathscr{C} } \mathbb{D}^{\omega}_{q } \bigl( {}^{\mathscr{C} }\mathbbm{ }\mathbb{D}^{\theta}_{q } \upsilon \bigr) \bigr) (\chi ) \bigr\vert . \end{aligned} $$

The fractional $\mathcal{QC}$ is examined on $\mathfrak{T}_{ \chi _{0}} = \{0\} {\cup } \{ \chi : \chi =\chi _{0} q ^{ \mathcal{N}} \}$ for $\mathcal{N} \in N$, and $0 < q < 1$ in [39]. We shall denote $\mathcal{T}_{\chi _{0}}$ by $\mathcal{T}$. Let . Define $\lceil \mu \rceil _{q }=\frac {1-q ^{\mu}}{1-q }$ in [40].

Definition 2.1

([39])

The $(\chi -s)^{ \mathcal{N}}_{q}$ is a q-factorial function. The expression $\mathcal{N}\in N_{0}$ is given by

(2)

and $(\chi -s)^{ (0)}_{ q}=1$, where $N_{0} : = \{0, 1,2, \dots \}$. Also, for , we obtain

$$ (\chi -s)_{q }^{(\mu )}=\chi ^{ \mu} \prod_{l=0}^{\infty} \frac {\chi - s q ^{l}}{\chi -sq ^{\mu +l}}. $$

(3)

Algorithm 1 is useful in this regard [41]. The q-gamma function is defined by $\Gamma _{q }(\mu ) =(1-q )_{q }^{(\mu -1)} / (1- q )^{\mu -1}$, where and satisfies $\Gamma _{q }(\mu +1)={\lceil \mu \rceil}_{q }\Gamma _{ q }(\mu )$ s.t. ${\lceil \mu \rceil}_{q }=(1-q ^{\mu})(1-q )^{-1}$ [39]. Algorithm 2, written using MATLAB commands, calculates q-gamma well [41].

Definition 2.2

([42])

The q-derivative of a function is expressed by

$$ \mathbb{D}_{q }\upsilon (\chi ) = \biggl( \frac {\mathrm{d}}{ \mathrm{d} \chi} \biggr)_{ q }\upsilon ( \chi ) = \frac {\upsilon (\chi ) - \upsilon (q \chi )}{(1-q \chi )}, \quad \forall \chi \in \mathcal{T} \setminus \{0\}, $$

(4)

and $\mathbb{D}_{q }\upsilon (0)=\lim_{\chi \to 0} \mathbb{D}_{ q }\upsilon (\chi )$. Also the higher q-derivative of function υ is defined by $\mathbb{D}^{n}_{q }\upsilon (\chi )= \mathbb{D}_{q } [ \mathbb{D}^{n-1}_{q }\upsilon (\chi ) ]$, $\forall n\geq 1$, here $\mathbb{D}^{0}_{\chi{q}}\upsilon (\chi )=\upsilon (\chi )$.

Definition 2.3

([42])

The q-integral of the function υ is expressed by

$$ \mathcal{I}_{q }\upsilon (\chi ) = \int _{0}^{\chi}\upsilon (s) \,\mathrm{d}_{q } s =\chi (1-q )\sum_{l=0}^{\infty} q ^{l}\upsilon \bigl(\chi q ^{l}\bigr), \quad 0 \leqslant \chi \leqslant b, $$

(5)

provided the series absolutely converges. If $\chi _{1}\in [0,r]$, then

$$ \int _{\chi _{1}}^{r} \upsilon (s) \,\mathrm{d}_{q }s= \mathcal{I}_{q } \upsilon (r) - \mathcal{I}_{q } \upsilon ( \chi _{1}) = (1-q ) \sum_{l=0}^{\infty} q ^{l} \bigl[ r - \upsilon \bigl( r q ^{l} \bigr) - \chi _{1}\upsilon \bigl(\chi _{1} q ^{l}\bigr) \bigr], $$

(6)

whenever the series exists (see Algorithm 3 and [41]). The operator $\mathcal{I}_{ q }^{n}$ is given as $\mathcal{I}^{0}_{ q }\upsilon ( \chi ) = \upsilon (\chi )$ and $\mathcal{I}^{n}_{ q }\upsilon ( \chi ) = \mathcal{I}_{q } [ \mathcal{I}^{n-1}_{q }\upsilon (\chi ) ]$ for $n\geq 1$ and $\upsilon \in \mathcal{C} ( [0,r])$.

It has been verified that $\mathbb{D}_{q } [\mathcal{I}_{q }\upsilon (\chi ) ] = \upsilon (\chi )$ and $\mathcal{I}_{q } [\mathbb{D}_{q }\upsilon (\chi ) ] = \upsilon (\chi )-\upsilon (0)$ whenever the function υ is continuous at $\chi =0$ in [42]. The fractional type q-integral of the function υ is given by

$$ \mathcal{I}_{q }^{\mu}{\upsilon}(\chi ) = \int _{0}^{\chi} \frac {(\chi -q s)^{\mu -1}}{\Gamma _{q }(\mu )} \upsilon (s) \,\mathrm{d}_{q }s,\quad \chi >0, \mu >0, $$

$\mathcal{I}_{q}^{0}{\upsilon}(\chi )=\upsilon (\chi )$ [43].

Definition 2.4

([43])

The operator ${}^{\mathscr{C} } \mathbb{D}^{\mu}_{q}$ is the fractional q-$\mathscr{CD}$ of order μ given by

$$ {}^{\mathscr{C} }\mathbb{D}_{q }^{\mu}\upsilon (\chi )= \mathcal{I}_{q }^{\lceil \mu \rceil -\mu} {} \mathbb{D}_{ q }^{\lceil \mu \rceil} \upsilon (\chi ), \quad \mu >0, $$

and ${}^{\mathscr{C} } \mathbb{D}_{q}^{0}\upsilon (\chi )= \upsilon (\chi )$ where $\lceil \mu \rceil$ is the smallest integer greater than μ.

Lemma 2.5

([28])

Let $\mu , \sigma \geq 0$ and υ be a function defined in Λ. Then (i) $\mathcal{I}_{q}^{\mu}[\mathcal{I}_{q}^{\sigma}\upsilon ( \chi )] = \mathcal{I}_{q}^{\mu +\sigma}\upsilon (\chi )$; (ii) ${}^{\mathscr{C} }\mathbb{D}_{q}^{\mu}[\mathcal{I}_{q}^{ \mu}\upsilon (\chi )]=\upsilon (\chi )$; (iii) ${}^{\mathscr{C} }\mathbb{D}_{q}^{\mu}[\mathcal{I}_{q}^{ \sigma}\upsilon (\chi )]=\mathcal{I}_{q}^{\sigma -\mu}\upsilon ( \chi )$.

Lemma 2.6

([43])

Let . Then the following equality

$$ \mathcal{I}_{q }^{\mu}{}^{\mathscr{C} }\mathbb{D}_{q }^{ \mu} \upsilon (\chi ) = \upsilon (\chi )- \sum_{k=0}^{n-1} \frac {\chi ^{k}}{ \Gamma _{q }(k+1)} {}^{\mathscr{C} } \mathcal{D}^{k}_{q } \upsilon (0) $$

is satisfied, and n is the smallest integer greater than or equal to μ. Equivalently, we can also write it as $n= \lceil \mu \rceil +1$, $n-1 <\mu \leqslant n$.

Lemma 2.7

([43])

(a) For and $\sigma >-1$, we obtain

$$ \mathcal{I}_{ q }^{\mu} \bigl[ \chi ^{(\sigma )} \bigr] = \frac {\Gamma _{q }(\sigma +1)}{\Gamma _{q }(\mu +\sigma +1)} \chi ^{(\mu +\sigma )}. $$

If $\sigma =0$, we obtain $\mathcal{I}_{q}^{\mu}[1]=\frac {1}{\Gamma _{q}(\mu +1)} \chi ^{(\mu )}$. (b) Similarly, for derivative, $\sigma >-1$, we get

$$ {}^{\mathscr{C} }\mathbb{D}_{q }^{\mu} \bigl[ \chi ^{(\sigma )} \bigr] = \frac {\Gamma _{q }(\sigma +1)}{ \Gamma _{q }( \sigma -\mu +1)} \chi ^{(\sigma -\mu )}. $$

If $\sigma =0$, we obtain ${}^{\mathscr{C} }\mathbb{D}_{q}^{\mu}[1]=0$.

We also point out formulas in [14], which will be used in our results.

$$ \begin{aligned} & \bigl[a (\chi -s) \bigr]^{(\alpha )} = a^{\alpha}(\chi -s)^{ \alpha}, \\ &_{\chi}\mathbb{D}_{q }(\chi -s)^{ \alpha} = \lceil \alpha \rceil _{q }( - s)^{(\alpha -1)}, \\ &_{s}\mathbb{D}_{q }(-s)^{\alpha}=-\lceil \alpha \rceil _{ q }(\chi -q s)^{(\alpha -1)}. \end{aligned} $$

Lemma 2.8

(Leray-Schauder alternative [44])

Let $\rho :\mathcal{F}\rightarrow \mathcal{F}$ be a completely continuous operator (i.e., a map restricted to any bounded set in $\mathcal{F}$ is compact). Let

$$ \Phi (\rho ) = \bigl\{ \upsilon \in \mathcal{F} : \upsilon = \pi \rho ( \upsilon ) \textit{ for some } 0< \pi < 1 \bigr\} . $$

(7)

Then the set $\Phi (\rho )$ is unbounded, or ρ has at least one fixed point.

Lemma 2.9

(Banach fixed point theorem [45])

Let $\mathcal{F}$ be a Banach space and mapping $\rho : \mathcal{F} \to \mathcal{F}$ be a contraction on $\mathcal{F}$. Hence ρ has a unique fixed point.

We now examine the $\mathscr{US}$ for the q- (1), as discussed in [46]. For $\overline{x}>0$ and , we get

$$ \bigl\vert {}^{\mathscr{C} }\mathbb{D}^{\alpha}_{q } \bigl( {}^{ \mathscr{C} } \mathbb{D}^{\omega}_{q } \bigl( {}^{\mathscr{C} } \mathbb{D}^{\theta}_{q } \upsilon (\chi ) \bigr) \bigr) - \Theta ^{*}_{\upsilon ,\omega ,\theta} (\chi ) \bigr\vert \leqslant \overline{x} $$

(8)

and

$$ \bigl\vert {}^{\mathscr{C} }\mathbb{D}^{\alpha}_{q } \bigl( {}^{ \mathscr{C} }\mathbb{D}^{\omega}_{q } \bigl( {}^{\mathscr{C} } \mathbb{D}^{\theta}_{q } \upsilon ( \chi ) \bigr) \bigr) - \Theta ^{*}_{ \upsilon ,\omega ,\theta} (\chi ) \bigr\vert \leqslant \overline{x} h(\chi ) $$

(9)

for $\chi \in \Lambda$, where

$$ \begin{aligned} \Theta ^{*}_{\upsilon ,\omega ,\theta}(\chi ) ={}& \xi \bigl( \chi , \upsilon (\chi ), {}^{\mathscr{C} } \mathbb{D}^{\theta}_{q } \upsilon (\chi ), \bigl( {}^{C} \mathbb{D}^{\omega}_{q } \bigl( {}^{C}\mathbb{D}^{\theta}_{q }\upsilon (\chi ) \bigr) \bigr) \bigr) \\ & {}+ \int _{0}^{\chi} \frac {(\chi -q s)^{ \nu -1}}{\Gamma _{q }(\zeta )}g \bigl(s, \upsilon (s), {}^{\mathscr{C} }\mathbb{D}_{q }^{ \theta}\upsilon (s), \bigl( {}^{\mathscr{C} }\mathbb{D}^{\omega}_{ q } \bigl( {}^{\mathscr{C} }\mathbb{D}^{\theta}_{q } \upsilon (s) \bigr) \bigr) \bigr) \,\mathrm{d}_{q }s. \end{aligned} $$

Definition 2.10

([46])

The q- (1) demonstrates the stability as:

• 1: In $\mathscr{UH}$ sense, if there is a positive real number such that there is a solution b of the q- (1) for each $\overline{x}>0$ and for each solution υ of inequality (8) having

  

• 2: In $\mathscr{UHR}$ sense, concerning , if there is a real number such that for each $\overline{x}>0$ and for each solution υ of inequality (9) there ∃ a solution υ̂ of q- (1) with

  

Remark 2.1

A function $\upsilon \in \mathcal{F}$ is considered a solution of inequality (8) iff ∃ another function (which relies on υ) s.t. $|\varrho (\chi )|\leqslant \overline{x}$ for every $\chi \in \Lambda$ and

$$ \bigl\Vert {}^{\mathscr{C} }\mathbb{D}^{\alpha}_{q } \bigl( {}^{ \mathscr{C} }\mathbb{D}^{\omega}_{q } \bigl( {}^{\mathscr{C} } \mathbb{D}^{\theta}_{q }\upsilon (\chi ) \bigr) \bigr) - \Theta ^{*}_{ \upsilon , \omega , \theta}(\chi ) \bigr\Vert \leqslant \bigl\Vert \varrho (\chi ) \bigr\Vert ,\quad \chi \in \Lambda . $$

3 Existence and uniqueness results

In this section, we investigate the 1$\mathfrak{EU}$ of solution of problem (1).

Lemma 3.1

Consider $\phi \in \mathcal{C}(\Lambda )$. Thus, the solution of problem

(10)

for $0<\max \{ \alpha ,\omega , \theta \} \leqslant 1$, is given as

(11)

where $\phi \in \mathcal{F}$ is given as

$$ \begin{aligned} \phi (\chi ) ={}& \xi \bigl( \chi , \upsilon (\chi ), {}^{\mathscr{C} } \mathbb{D}^{\theta}_{q }\upsilon (\chi ), \bigl( {}^{ \mathscr{C} }\mathbb{D}^{\omega}_{q } \bigl( {}^{\mathscr{C} } \mathbb{D}^{\theta}_{q }\upsilon ( \chi ) \bigr) \bigr) \bigr) \\ &{} + \int _{0}^{\chi} \frac {(\chi -q s)^{\nu -1}}{\Gamma _{q }(\zeta )} g \bigl( s, \upsilon (s), {}^{\mathscr{C} } \mathbb{D}_{q }^{ \theta}\upsilon (s), \bigl( {}^{\mathscr{C} } \mathbb{D}^{\omega}_{ q } \bigl( {}^{\mathscr{C} } \mathbb{D}^{\theta}_{q } \upsilon (s) \bigr) \bigr) \bigr) \,\mathrm{d}_{q }s, \end{aligned} $$

and .

Proof

Now, let us consider

$$ {}^{\mathscr{C} }\mathbb{D}^{\alpha}_{q } \bigl( {}^{ \mathscr{C} }\mathbb{D}^{\omega}_{q } \bigl( {}^{\mathscr{C} } \mathbb{D}^{\theta}_{ q } \upsilon (\chi ) \bigr) \bigr) = \phi (\chi ),\quad \chi \in \Lambda . $$

(12)

Applying the operator $\mathcal{I}_{q}^{\alpha}$ on both sides of (12) and employing Lemma 2.6 with $n=1$, we obtain

(13)

Now, using the operator $\mathcal{I}_{q}^{\omega}$, (1) of Lemma 2.5, (a) of Lemma 2.7, and applying the same procedure on both sides of (13), we get

(14)

It follows that

$$ \upsilon (\chi ) = \mathcal{I}_{q }^{\alpha + \omega +\theta} \phi (\chi ) + c_{0} \frac {\chi ^{\omega +\theta}}{\Gamma _{\chi{q}}(\omega +\theta +1)}+c_{1} \frac {\chi ^{\theta}}{\Gamma _{q }(\theta +1)} + c_{2}, $$

(15)

where , ($j=0,1,2$). Using boundary constraints

(16)

Now, using the L.H.S of (16) in (15), we obtain

$$ \begin{aligned} &\upsilon (\chi )|_{\chi =0} = \mathcal{I}_{q }^{\alpha + \omega +\theta}\phi (\chi )+c_{0} \frac {\chi ^{\omega +\theta}}{\Gamma _{q }(\omega +\theta +1)}+c_{1} \frac {\chi ^{\theta}}{\Gamma _{q }(\theta +1)}+c_{2}, \\ &\upsilon (\chi )|_{\chi =0} = c_{2}. \end{aligned} $$

Similarly, using the R.H.S of (16) in (15), we obtain

Thus (16) becomes

By the 2nd boundary condition,

$$ \bigl( {}^{\mathscr{C} }\mathbb{D}^{\omega}_{q } \bigl( {}^{ \mathscr{C} } \mathbb{D}^{\theta}_{q }\upsilon (\chi ) \bigr) \bigr)|_{\chi =\delta}=0. $$

(17)

Applying ${}^{\mathscr{C} }\mathbb{D}^{\theta}_{q}$, (3) of Lemma 2.5 and (b) of Lemma 2.7 on both sides of (15), we get

$$\begin{aligned} {}^{\mathscr{C} }\mathbb{D}^{\theta}_{q } \upsilon (\chi ) = \mathcal{I}_{q }^{\alpha + \omega}\phi (\chi ) + c_{0} \frac {\chi ^{\omega}}{ \Gamma _{q }(\omega +1)}+c_{1}. \end{aligned}$$

(18)

Now, applying ${}^{\mathscr{C} }\mathbb{D}^{\omega}_{q}$ and the same procedure on both sides of (18), we get

$$ {}^{\mathscr{C} }\mathbb{D}^{\omega}_{q } \bigl( {}^{ \mathscr{C} }\mathbb{D}^{\theta}_{q }\upsilon (\chi ) \bigr)= \mathcal{I}_{q }^{\alpha} \phi (\chi )+c_{0}. $$

(19)

So, Eq. (19) becomes ${}^{\mathscr{C} }\mathbb{D}^{\omega}_{q} ({}^{\mathscr{C} } \mathbb{D}^{\theta}_{q} \upsilon (\chi ))|_{\chi =\delta}= \mathcal{I}_{q}^{\alpha}\phi (\delta )+c_{0}$. By Eq. (17), we get $c_{0} = - \mathcal{I}_{q}^{\alpha} \phi (\delta )$. Using the 3rd boundary condition,

(20)

Now, using the L.H.S of (20) in (15), we get

$$ {}^{\mathscr{C} }\mathbb{D}_{q }^{\beta} \upsilon (\chi )= \mathcal{I}_{q }^{\alpha +\omega +\theta -\beta}\phi (\chi ) +c_{0} \frac {\chi ^{\omega +\theta -\beta}}{\Gamma _{q }(\omega +\theta -\beta +1)}+c_{1} \frac {\chi ^{\theta -\beta}}{\Gamma _{q }(\theta -\beta +1)}. $$

So, at ${}^{\mathscr{C} }\mathbb{D}_{q}^{ \beta} \upsilon ( \chi )|_{ \chi =0}=c_{1}$, since $\theta -\beta \leqslant 0$ by Eq. (2). Now, using the R.H.S of (20) in (15), we have

So, (20) becomes

Putting all values in (15), we obtain

and

 □

We define an operator $\rho :\mathcal{F}\rightarrow \mathcal{F}$ by applying Lemma 3.1 as follows:

The following assumptions will be used in our upcoming results:

$\mathrm{(H_{1})}$:

;

$\mathrm{(H_{2})}$:

are continuous;

$\mathrm{(H_{3})}$:

∃ constant $\overline{y}>0$ in such a way that ∀ $\chi \in {\Lambda}$ and , $m=\{1,2,3\}$, we get

$$ \bigl\vert {\xi}(\chi , \upsilon _{1}, \upsilon _{2}, \upsilon _{3} )-{\xi}( \chi ,\hat{ \upsilon}_{1}, \hat{ \upsilon}_{2},\hat{ \upsilon}_{3}) \bigr\vert \leqslant \sum _{m=1}^{3}\overline{y}_{m} \vert \upsilon _{m}- \hat{ \upsilon}_{m} \vert ; $$

$\mathrm{(H_{4})}$:

∃ constant $\overline{z}>0$ in such a way that ∀ $\chi \in {\Lambda}$ and , $v=\{1,2,3\}$, we have

$$ \bigl\vert {g}(\upsilon , \upsilon _{1}, \upsilon _{2}, \upsilon _{3})-{g}( \chi ,\hat{\upsilon}_{1}, \hat{ \upsilon}_{2},\hat{\upsilon}_{3}) \bigr\vert \leqslant \sum _{v=1}^{3}\overline{z}_{v} \vert \upsilon _{v}- \hat{\upsilon}_{v} \vert ; $$

$\mathrm{(H_{5})}$:

∃ real constants $\varphi _{m}\geq 0$ ($m=1,2,3$) and $\varphi _{0}>0$ in such a way that for any ($m=1,2,3$) we have

$$ \bigl\vert {\xi}(\chi , \upsilon _{1}, \upsilon _{2}, \upsilon _{3}) \bigr\vert \leqslant \varphi _{0}+\varphi _{1} \vert \upsilon _{1} \vert +\varphi _{2} \vert \upsilon _{2} \vert + \varphi _{3} \vert \upsilon _{3} \vert ; $$

$\mathrm{(H_{6})}$:

∃ real constants $\wp _{v}\geq 0$ $(v=1,2,3)$ and $\wp _{0}>0$ in such a way that for any ($v=1,2,3$) we have

$$ \bigl\vert {g}(\chi , \upsilon _{1}, \upsilon _{2}, \upsilon _{3}) \bigr\vert \leqslant \wp _{0}+\wp _{1} \vert \upsilon _{1} \vert +\wp _{2} \vert \upsilon _{2} \vert +\wp _{3} \vert \upsilon _{3} \vert ; $$

$\mathrm{(H_{7})}$:

∃ an increasing and $\vartheta _{h}>0$, then the following inequality

$$ \mathcal{I}_{q }^{ \alpha + \omega +\theta} h(\chi ) \leqslant \vartheta _{h} {h(\chi )},\quad \chi \in \Lambda , $$

is satisfied.

In the following sections, we will employ the fixed point theory to confirm $\mathfrak{EU}$ of solution of q-fractional $\mathcal{J}$ problem outlined in (1). For simplicity, the following notations will be used in our upcoming results:

(21)

Theorem 3.2

Suppose that assumptions $\mathrm{(H_{2})}$, $\mathrm{(H_{3}),}$ and $\mathrm{(H_{4})}$ hold. Thus, q- (1) has a unique solution if

$$ \Biggl[\sum_{m=1}^{3} \overline{y}_{m}+\sum_{v=1}^{3} \overline{z}_{v} \Biggr] \Biggl( \sum_{i=1}^{3} \varpi _{i} \Biggr)< 1, $$

(22)

where $\varpi _{i}$, $i=1,2,3$, are given by (21).

Proof

First, we demonstrate that $\rho{\mathcal{W}}_{\epsilon}\subset \mathcal{W}_{\epsilon}$, where $\mathcal{W}_{\epsilon}= \{ \upsilon \in \mathcal{F}:\|\upsilon \|_{\mathcal{F}}\leqslant \epsilon \}$ with

$$ \epsilon \geq \frac { ( \Pi + \psi ) \sum_{i=1}^{3} \varpi _{i}}{1- ( \sum_{m=1}^{3}\overline{y}_{m}+\sum_{v=1}^{3}\overline{z}_{v} ) \sum_{i=1}^{3}\varpi _{i}}, $$

s.t. $\Pi = \sup_{ \chi \in \Lambda} | \xi (\chi ,0,0,0)|$, $\psi = \sup_{\chi \in \Lambda}| g( \chi ,0,0,0)|$, and $\varpi _{i}$, $i=1,2,3$, are given by (21). Using $\mathrm{(H_{3})}$ and $\mathrm{(H_{4})}$, we get

$$\begin{aligned} \Theta ^{*}_{\upsilon ,\omega ,\theta}(\chi ) ={}& \biggl\vert \xi \bigl( \chi , \upsilon ( \chi ), {}^{\mathscr{C} } \mathbb{D}^{\theta}_{ q } \upsilon ( \chi ), \bigl( {}^{\mathscr{C} }\mathbb{D}^{ \omega}_{q } \bigl( {}^{\mathscr{C} }\mathbb{D}^{\theta}_{ q }\upsilon (\chi ) \bigr) \bigr) \bigr) \\ & {}+ \int _{0}^{\chi} \frac { ( \chi -q s)^{\nu -1}}{\Gamma _{q }(\zeta )} g \bigl(s, \upsilon (s), {}^{\mathscr{C} } \mathbb{D}_{q }^{ \theta}\upsilon (s), \bigl( {}^{\mathscr{C} }\mathbb{D}^{\omega}_{ q } \bigl( {}^{\mathscr{C} }\mathbb{D}^{\theta}_{q } \upsilon (s) \bigr) \bigr) \bigr) \,\mathrm{d}_{q }s \biggr\vert \\ \leqslant {}&\biggl\vert \xi \bigl( \chi ,\upsilon (\chi ), {}^{ \mathscr{C} } \mathbb{D}^{\theta}_{q } \upsilon (\chi ), \bigl( {}^{ \mathscr{C} }\mathbb{D}^{\omega}_{q } \bigl( {}^{\mathscr{C} } \mathbb{D}^{\theta}_{q } \upsilon (\chi ) \bigr) \bigr) \bigr) \\ &{} + \int _{0}^{\chi} \frac {(\chi -q s)^{\nu -1}}{\Gamma _{q }(\zeta )} g \bigl( s, \upsilon (s), {}^{\mathscr{C} }\mathbb{D}_{q }^{ \theta}\upsilon (s), \bigl( {}^{\mathscr{C} }\mathbb{D}^{\omega}_{ q } \bigl( {}^{\mathscr{C} } \mathbb{D}^{\theta}_{q } \upsilon (s) \bigr) \bigr) \bigr) \,\mathrm{d}_{q }s \\ &{} - \xi (\chi ,0,0,0) - \int _{0}^{\chi} \frac { ( \chi -q s)^{ \nu -1}}{ \Gamma _{q } ( \zeta )}g(s,0,0,0) \,\mathrm{d}_{q }s \biggr\vert \\ & {}+ \bigl\vert \xi ( \chi ,0,0,0) \bigr\vert + \biggl\vert \int _{0}^{\chi} \frac {(\chi -q s)^{\nu -1}}{\Gamma _{q }(\zeta )} g (s,0,0,0) \,\mathrm{d}_{q } s \biggr\vert \\ \leqslant{}& \sum_{m=1}^{3} \overline{y}_{m} \bigl( \Vert \upsilon \Vert + \bigl\Vert {}^{\mathscr{C} }\mathbb{D}^{\theta}_{q }\upsilon \bigr\Vert + \bigl\Vert {}^{\mathscr{C} }\mathbb{D}^{\omega}_{q } \bigl({}^{ \mathscr{C} }\mathbb{D}^{ \theta}_{q } \upsilon \bigr) \bigr\Vert \bigr) + \Pi \\ &{} + \sum_{v=1}^{3} \overline{z}_{v} \bigl( \Vert \upsilon \Vert + \bigl\Vert {}^{\mathscr{C} } \mathbb{D}^{ \theta}_{q }\upsilon \bigr\Vert + \bigl\Vert {}^{\mathscr{C} }\mathbb{D}^{\omega}_{q }\bigl({}^{ \mathscr{C} } \mathbb{D}^{\theta}_{q }\upsilon \bigr) \bigr\Vert \bigr) + \psi \\ \leqslant{}& \sum_{m=1}^{3} \overline{y}_{m} \Vert \upsilon \Vert _{ \mathcal{F}}+\Pi +\sum _{v=1}^{3}\overline{z}_{v} \Vert \upsilon \Vert _{ \mathcal{F}}+\psi \\ \leqslant{} &\sum_{m=1}^{3} \overline{y}_{m}\epsilon +\Pi +\sum_{v=1}^{3} \overline{z}_{v}\epsilon +\psi \\ ={}& \Biggl(\sum_{m=1}^{3} \overline{y}_{m}+\sum_{v=1}^{3} \overline{z}_{v} \Biggr)\epsilon +\Pi +\psi . \end{aligned}$$

(23)

Then we get

Now, using (23), we obtain

Also, we have

and

From the definition of $\|\cdot \|_{ \mathcal{F}}$, we have

$$ \begin{aligned} \bigl\Vert \rho (\upsilon ) \bigr\Vert _{ \mathcal{F}} ={}& \bigl\Vert \rho (\upsilon ) \bigr\Vert + \bigl\Vert {}^{\mathscr{C} }\mathbb{D}^{\theta}_{q }\rho ( \upsilon ) \bigr\Vert + \bigl\Vert {}^{\mathscr{C} }\mathbb{D}^{ \omega}_{ q } \bigl({}^{\mathscr{C} }\mathbb{D}^{\theta}_{q }\rho ( \upsilon ) \bigr) \bigr\Vert \\ \leqslant{}& ( \overline{y}+\overline{z})\epsilon \varpi _{1}+(\Pi + \psi )\varpi _{1} +(\overline{y}+\overline{z})\epsilon \varpi _{2} \\ & {}+ ( \Pi +\psi )\varpi _{2}+(\overline{y}+\overline{z}) \epsilon \varpi _{3}+(\Pi +\psi )\varpi _{3} \\ ={}& \Biggl[ \sum_{m=1}^{3} \overline{y}_{m}+\sum_{v=1}^{3} \overline{z}_{v} \Biggr]\sum_{i=1}^{3} \varpi _{i}\epsilon +(\Pi +\psi )\sum_{i=1}^{3} \varpi _{i}\leqslant \epsilon , \end{aligned} $$

which means that $\rho \mathcal{W}_{\epsilon}\subset \mathcal{W}_{\epsilon}$. We now demonstrate that the ρ is an operator for a contraction mapping. Now $\upsilon , \hat{\upsilon}\in \mathcal{W}_{\epsilon}$ and $\chi \in \Lambda$, we obtain

By $\mathrm{(H_{3})}$ and $\mathrm{(H_{4})}$, we obtain

Also, by using $\mathrm{(H_{3})}$ and $\mathrm{(H_{4})}$, we obtain

and

Thus, we get

$$ \begin{aligned} \bigl\Vert \rho (\upsilon )-\rho (\hat{\upsilon}) \bigr\Vert _{\mathcal{F}} ={}& \bigl\Vert \rho ( \upsilon )-\rho (\hat{ \upsilon}) \bigr\Vert + \bigl\Vert {}^{\mathscr{C} } \mathbb{D}^{\theta}_{q } \rho (\upsilon )-{}^{\mathscr{C} } \mathbb{D}^{\theta}_{q }\rho ( \hat{\upsilon}) \bigr\Vert \\ & {}+ \bigl\Vert \bigl( {}^{\mathscr{C} } \mathbb{D}^{ \omega}_{ q } \bigl( {}^{ \mathscr{C} } \mathbb{D}^{\theta}_{q } \rho (\upsilon ) \bigr) \bigr) - \bigl( {}^{\mathscr{C} } \mathbb{D}^{\omega}_{q } \bigl( {}^{\mathscr{C} }\mathbb{D}^{ \theta}_{q } \rho ( \hat{ \upsilon}) \bigr) \bigr) \bigr\Vert \\ \leqslant{}& \Biggl[ \sum_{m=1}^{3} \overline{y}_{m} + \sum_{v=1}^{3} \overline{z}_{v} \Biggr] \sum_{i=1}^{3} \varpi _{i} \Vert \upsilon - \hat{\upsilon} \Vert _{\mathcal{F}}. \end{aligned} $$

We observe that ρ is a contraction operator by using (22). We infer that ρ has a unique fixed point that is a solution of (1) as a result of Lemma 2.9. □

By applying Lemma 2.8, we explore certain conditions where q- (1) has at least one solution in Theorem 3.2.

Theorem 3.3

Assume that hypotheses $\mathrm{(H_{5})}$ and $\mathrm{(H_{6})}$ hold. If

$$ \Biggl[\sum_{m=1}^{3} \varphi _{m}+\sum_{v=1}^{3}\wp _{v} \Biggr] \Biggl( \sum_{i=1}^{3} \varpi _{i} \Biggr)< 1 $$

(24)

is satisfied, then the proposed problem described by (1) has at least one solution within the domain Λ.

Proof

Our initial goal is to investigate the complete continuity of an operator $\rho : \mathcal{F} \rightarrow \mathcal{F}$. Considering function’s continuity Θ, we can also conclude that the operator ρ is also continuous. Assume that $\kappa \subset \mathcal{F}$ is bounded. Then there exists a positive constant $\mathfrak{P}$ s.t. $| \Theta ^{*}_{\upsilon , \omega ,\theta}(s)|\leqslant \mathfrak{P}$ for each $\upsilon \in \kappa$. Then, for any $\upsilon \in \kappa$ and using (21), we can find that

$$ \begin{aligned} \bigl\Vert \rho (\upsilon ) \bigr\Vert _{ \mathcal{F}} = \bigl\Vert \rho (\upsilon ) \bigr\Vert + \bigl\Vert {}^{ \mathscr{C} }\mathbb{D}^{\theta}_{q }\rho (\upsilon ) \bigr\Vert + \bigl\Vert {}^{\mathscr{C} }\mathbb{D}^{\omega}_{q } \bigl( {}^{ \mathscr{C} }\mathbb{D}^{\theta}_{q }\rho (\upsilon ) \bigr) \bigr\Vert \leqslant \mathfrak{P} \sum_{i=1}^{3} \varpi _{i}. \end{aligned} $$

The inequalities indicate that an operator ρ remains uniformly bounded. Furthermore, we will verify that ρ is equicontinuous. For $\upsilon \in \Lambda$ and , we get

(25)

Also, we obtain

$$\begin{aligned} \bigl\vert {}^{\mathscr{C} }\mathbb{D}_{ q }^{\theta} \rho \upsilon ( \chi _{1})- {}^{\mathscr{C} } \mathbb{D}_{q }^{ \theta} \rho \upsilon (\chi _{2}) \bigr\vert & \leqslant \mathfrak{P} \biggl[ \frac { \vert \chi _{1}^{\alpha +\omega}-\chi _{2}^{\alpha +\omega} \vert }{\Gamma _{q }(\alpha +\omega +1)}+ \frac {\delta ^{\alpha}}{\Gamma _{q }(\alpha +1)} \biggl( \frac { \vert \chi _{2}^{\omega}-\chi _{1}^{\omega} \vert }{\Gamma _{q }(\omega +1)} \biggr) \biggr] \end{aligned}$$

(26)

and

$$\begin{aligned} \bigl\vert {}^{\mathscr{C} }\mathbb{D}_{ q }^{\omega} \bigl( {}^{ \mathscr{C} } \mathbb{D}_{q }^{\theta} \rho \upsilon (\chi _{1}) \bigr) - {}^{\mathscr{C} }\mathbb{D}_{q }^{\omega} \bigl( {}^{ \mathscr{C} }\mathbb{D}_{q }^{\theta}\rho \upsilon ( \chi _{2}) \bigr) \bigr\vert & \leqslant \mathfrak{P} \biggl[ \frac {\chi _{1}^{\alpha} - \chi _{2}^{\alpha}}{\Gamma _{q }(\alpha +1)} \biggr]. \end{aligned}$$

(27)

The right-hand sides of (25), (26), (27) tend to zero independently of υ as $\chi _{1}\rightarrow \chi _{2}$. Therefore, an operator $\rho : \mathcal{F} \to \mathcal{F}$ is completely continuous by Arzelà–Ascoli theorem. Finally, we show that a set $\Upsilon = \{ \upsilon \in \mathcal{F} : \upsilon = \varepsilon \rho (\upsilon ), 0 < \varepsilon <1 \}$ is bounded. Let $\upsilon \in \Upsilon$, thus $\upsilon = \varepsilon \rho ( \upsilon )$. For every $\chi \in \Lambda$, we have $\upsilon (\chi )=\varepsilon \rho \upsilon (\chi )$. Then

$$ \begin{aligned} \bigl\vert \upsilon (\chi ) \bigr\vert \leqslant{}& \varpi _{1} \bigl[ ( \varphi _{1}+\wp _{1}) \Vert \upsilon \Vert +(\varphi _{2}+\wp _{2}) \bigl\Vert {}^{\mathscr{C} } \mathbb{D}_{q }^{\theta}( \upsilon ) \bigr\Vert \\ & {}+ ( \varphi _{3}+\wp _{3}) \bigl\Vert {}^{\mathscr{C} } \mathbb{D}_{q }^{\omega} \bigl( {}^{\mathscr{C} } \mathbb{D}_{ q }^{\theta}(\upsilon ) \bigr) \bigr\Vert \bigr]+ \varpi _{1}( \varphi _{0}+\wp _{0}). \end{aligned} $$

We also have

$$ \begin{aligned} & \bigl\vert {}^{\mathscr{C} }\mathbb{D}_{q }^{\theta} \upsilon ( \chi ) \bigr\vert \leqslant \varpi _{2} \bigl[ (\varphi _{1}+\wp _{1}) \Vert \upsilon \Vert +(\varphi _{2}+\wp _{2}) \bigl\Vert {}^{\mathscr{C} } \mathbb{D}_{q }^{\theta}(\upsilon ) \bigr\Vert \\ & \hphantom{\bigl\vert {}^{\mathscr{C} }\mathbb{D}_{q }^{\theta} \upsilon ( \chi ) \bigr\vert \leqslant}{}+ (\varphi _{3} + \wp _{3}) \bigl\Vert {}^{\mathscr{C} } \mathbb{D}_{q }^{ \omega} \bigl( {}^{\mathscr{C} } \mathbb{D}_{ q }^{\theta}(\upsilon ) \bigr) \bigr\Vert \bigr] + \varpi _{2}( \varphi _{0}+\wp _{0}), \\ &\bigl\vert {}^{\mathscr{C} }\mathbb{D}_{q }^{\omega} \bigl({}^{ \mathscr{C} }\mathbb{D}_{q }^{\theta}\upsilon (\chi ) \bigr) \bigr\vert \leqslant \varpi _{3} \bigl[ (\varphi _{1}+\wp _{1}) \Vert \upsilon \Vert + ( \varphi _{2}+\wp _{2}) \bigl\Vert {}^{\mathscr{C} } \mathbb{D}_{ q }^{ \theta} (\upsilon ) \bigr\Vert \\ &\hphantom{\bigl\vert {}^{\mathscr{C} }\mathbb{D}_{q }^{\omega} \bigl({}^{ \mathscr{C} }\mathbb{D}_{q }^{\theta}\upsilon (\chi ) \bigr) \bigr\vert \leqslant}{} + ( \varphi _{3}+\wp _{3}) \bigl\Vert {}^{\mathscr{C} } \mathbb{D}_{q }^{ \omega} \bigl( {}^{\mathscr{C} } \mathbb{D}_{ q }^{ \theta} ( \upsilon ) \bigr) \bigr\Vert \bigr] + \varpi _{3}( \varphi _{0}+\wp _{0}), \end{aligned} $$

which implies that

$$ \Vert \upsilon \Vert _{\mathcal{F}}\leqslant \Biggl[ \sum _{m=1}^{3}\varphi _{m}+ \sum _{v=1}^{3}\wp _{v} \Biggr] \sum _{i=1}^{3}\varpi _{i} \Vert \upsilon \Vert _{\mathcal{F}}+\sum_{i=1}^{3} \varpi _{i}(\varphi _{0}+\wp _{0}). $$

Consequently,

$$\begin{aligned} \Vert \upsilon \Vert _{\mathcal{F}}\leqslant \frac { \sum_{i=1}^{3} \varpi _{i}( \varphi _{0} + \wp _{0})}{ 1 - [\sum_{m=1}^{3}\varphi _{m}+\sum_{v=1}^{3}\wp _{v} ]\sum_{i=1}^{3}\varpi _{i} }, \end{aligned}$$

(28)

where $\varpi _{i}, i=1,2,3$, are given by (21). From (28), we see that $\|\upsilon \|_{\mathcal{F}}\leqslant \infty$. As a result, ϒ is bounded. We deduce that an operator ρ has a fixed point, which is the solution of q- (1) as a result of Lemma 2.8. □

4 Stability results

We study the $\mathscr{UH}$ and $\mathscr{UHR}$ stability [46] of q- in this section.

Theorem 4.1

Assume that $\mathrm{(H_{2})}$–$\mathrm{(H_{4})}$ and (22) hold. Then the q- (1) is $\mathscr{UH}$ stable.

Proof

Consider $\hat{\upsilon}\in \mathcal{F}$ to be the only solution to the problem

(29)

for $\chi \in{ \Lambda}$, WHERE $0 < \alpha , \omega ,\theta \leqslant \Lambda$. So that inequality (8) can be solved with υ in $\mathcal{F}$. Utilizing Remark 2.1, we obtain

$$ \upsilon (\chi )=\mathcal{I}_{q }^{\alpha +\omega +\theta}\phi _{ \upsilon}(\chi )+c_{0} \frac {\chi ^{\omega +\theta}}{\Gamma _{q }(\omega +\theta +1)}+c_{1} \frac {\chi ^{\theta}}{\Gamma _{q }(\theta +1)}+c_{2}+ \mathcal{I}_{q }^{\alpha +\omega +\theta} \varrho (\chi ), $$

where , $j=\{0,1,2\}$, $\phi _{\upsilon} ( \chi ) = \Theta ^{*}_{\upsilon ,\omega ,\theta}( \chi )$, and $|\varrho (\chi )|\leqslant \overline{x}$, $\chi \in \Lambda$. Thanks to Lemma 3.1,

Also, we have

$$ \begin{aligned} \bigl\vert \upsilon (\chi )-\hat{\upsilon}(\chi ) \bigr\vert ={}& \biggl\vert \upsilon (\chi ) - \mathcal{I}_{q }^{\alpha + \omega + \theta} \phi _{\hat{\upsilon}}(\chi )+c_{0} \frac {\chi ^{\omega +\theta}}{\Gamma _{q }(\omega + \theta +1)} \\ & {} +c_{1} \frac {\chi ^{\theta}}{\Gamma _{q }(\theta +1)} +c_{2}+ \mathcal{I}_{q }^{\alpha +\omega +\theta}\varrho (\chi ) \biggr\vert \\ ={}& \bigl\vert \upsilon (\chi )-\rho \upsilon (\chi )+\rho \upsilon (\chi )- \rho \hat{\upsilon}(\chi ) \bigr\vert \\ \leqslant{}& \bigl\vert \upsilon (\chi )- \rho{\upsilon}( \chi ) \bigr\vert + \bigl\vert \rho \upsilon (\chi )-\rho \hat{\upsilon}(\chi ) \bigr\vert . \end{aligned} $$

$\mathrm{(H_{3})}$ and $\mathrm{(H_{4})}$ imply that

where Eq. (21) provides $\varpi _{i}$, $i=\{1,2,3\}$. Next

If we put

we obtain . As a result, the q- (1) is $\mathscr{UH}$ stable. □

Theorem 4.2

Suppose that $\mathrm{(H_{2})}$–$\mathrm{(H_{4})}$, $\mathrm{(H_{7}),}$ and (22) hold. Then q- (1) is $\mathscr{UHR}$ stable in relation to h.

Proof

We have

$$ \upsilon (\chi ) = \mathcal{I}_{ q }^{ \alpha +\omega + \theta} \phi _{\upsilon}( \chi ) + c_{0} \frac {\chi ^{\omega + \theta}}{\Gamma _{q }(\omega +\theta +1)}+c_{1} \frac {\chi ^{\theta}}{\Gamma _{q }(\theta +1)}+c_{2}+ \mathcal{I}_{q }^{\alpha +\omega + \theta} \varrho (\chi ), $$

where $\chi \in \Lambda$, , $j=0,1,2$, and $|\varrho (\chi )|\leqslant \overline{x} h(\chi )$, and inequality (9) can be solved by using $\upsilon \in \mathcal{F}$. Taking $\hat{\upsilon}\in \mathcal{F}$ as the singular solution of (29), by Lemma 3.1, we have

$$ \bigl\vert \upsilon (\chi )-\rho \upsilon (\chi ) \bigr\vert = \bigl\vert \mathcal{I}_{ q }^{\alpha +\omega +\theta}\varrho (\chi ) \bigr\vert \leqslant \overline{x} \mathcal{I}_{q }^{\alpha +\omega +\theta}\bigl[h(\chi )\bigr] \leqslant \overline{x}\vartheta _{h} h(\chi ). $$

Also, we have

$$\begin{aligned} \bigl\vert \upsilon (\chi )-\hat{\upsilon}(\chi ) \bigr\vert ={}& \biggl\vert \upsilon (\chi ) - \mathcal{I}_{q }^{\alpha +\omega + \theta} \phi _{ \hat{ \upsilon}}(\chi ) + c_{0} \frac { \chi ^{ \omega + \theta}}{ \Gamma _{q }( \omega +\theta +1)} \\ & {} + c_{1} \frac {\chi ^{\theta}}{ \Gamma _{ q }(\theta +1)} + c_{2}+ \mathcal{I}_{ q }^{ \alpha + \omega +\theta} \varrho (\chi ) \biggr\vert \\ ={}& \bigl\vert \upsilon (\chi )-\rho{\upsilon}(\chi )+\rho \upsilon (\chi )- \rho \hat{\upsilon}(\chi ) \bigr\vert \\ \leqslant{}& \bigl\vert \upsilon (\chi )- \rho{\upsilon}( \chi ) \bigr\vert + \bigl\vert \rho \upsilon (\chi )-\rho \hat{\upsilon}(\chi ) \bigr\vert . \end{aligned}$$

So, by $\mathrm{(H_{3})}$, $\mathrm{(H_{4})}$, and $\mathrm{(H_{7})}$, we obtain

$$ \Vert \upsilon -\hat{\upsilon} \Vert _{\mathcal{F}}\leqslant \overline{x} \vartheta _{h} h(\chi ) + \Biggl[\sum_{m=1}^{3} \overline{y}_{m}+\sum_{v=1}^{3} \overline{z}_{v} \Biggr]\sum_{i=1}^{3} \varpi _{i} \Vert \upsilon - \hat{\upsilon} \Vert _{\mathcal{F}}. $$

Then we get

$$ \Vert \upsilon -\hat{\upsilon} \Vert _{\mathcal{F}}\leqslant \frac {\vartheta _{h}}{ 1- [ \sum_{m=1}^{3}\overline{y}_{m}+\sum_{v=1}^{3}\overline{z}_{v} ] \sum_{i=1}^{3}\varpi _{i}} \overline{x} h(\chi ), \quad \chi \in \Lambda . $$

If we take

we can obtain considering $\chi \in \Lambda$. Consequently, the $\mathscr{UHR}$ stability is achieved by q- (1). □

5 Examples and illustrative results

In this section, we check the correctness of the results by showing several examples. In the first example, we test q-Caputo fractional with $\mathbb{ABC}$s (1) for the changes of q in the range of zero and one according to the proposed theorems.

Example 5.1

Let

$$ \textstyle\begin{cases} {}^{\mathscr{C} }\mathbb{D}^{ \frac{1}{3} }_{ q } ( {}^{ \mathscr{C} } \mathbb{D}^{\frac{4}{5}}_{ q } ( {}^{\mathscr{C} } \mathbb{D}^{ \frac{3}{4}}_{ q } \upsilon ( \chi ) ) ) \\ \quad = \frac { \sinh ( e^{\chi}+2 ) }{ 4 } + \frac { \sqrt{ 15} e^{ - \chi} \vert \upsilon ( \chi ) \vert }{ 41 ( \chi +3) ( \vert \upsilon (\chi ) \vert + 1)} & \\ \qquad {} + \frac { \cos ( {}^{\mathscr{C} } \mathbb{D}^{\frac{3}{4 } }_{q } \upsilon ( \chi ) ) }{ 333 \sqrt{ \ln ( \chi +12) }} + \frac { \arctan ( {}^{\mathscr{C} } \mathbb{D}^{\frac{4}{5} }_{q } ( {}^{\mathscr{C} } \mathbb{D}^{ \frac{3}{4}}_{ q } \upsilon ( \chi ) ) ) }{ 22 ( \chi +3)} & \\ \qquad {} + \int _{0}^{ \chi } \frac { ( \chi - q s )^{ \frac{3}{2} - 1} }{ \Gamma _{ q } ( \frac {3}{2} )} [ \frac { \sqrt{ e^{2s}}{ \vert \upsilon (s) \vert }}{ 29(s+3) ( \vert \upsilon (s) \vert + 5) } + \frac {\cos ( {}^{\mathscr{C} } \mathbb{D}^{\frac{3}{4} }_{ q } \upsilon (s) ) }{ 345 \sqrt{e^{ s+19}}} & \\ \qquad {} + \frac { \sin ( {}^{\mathscr{C} } \mathbb{D}^{ \frac{4}{5} }_{ q } ( {}^{ \mathscr{C} }\mathbb{D}^{ \frac{3}{4} }_{ q } \upsilon (s) ) ) }{ 137 \ln ( \sqrt{s+34} ) } ] \,\mathrm{d}_{q }s,& \chi \in [0,1], \\ \upsilon (0)| = - \upsilon (1), \qquad ( {}^{ \mathscr{C} } \mathbb{D}^{ \frac{4}{5}}_{ q } ( {}^{ \mathscr{C} } \mathbb{D}^{ \frac{3}{4}}_{q } \upsilon ( \frac {7}{11} ) ) )=0,& \\ {}^{\mathscr{C} }\mathbb{D}_{q }^{\frac{5}{9}} \upsilon (0) = - {}^{\mathscr{C} } \mathbb{D}_{ q }^{ \frac{5}{9}} \upsilon (1),& \end{cases} $$

(30)

where $q\in \{ \frac {1}{5},\frac {2}{5}, \frac {3}{5} \} \subseteq (0,1)$, $\alpha = \frac {1}{3} \in (0,1]$, $\omega =\frac {4}{5} \in (0,1]$, $\nu = \zeta = \frac {3}{2}$, $\theta = \frac {3}{4}\in (0,1]$, , $\beta = \frac {5}{9} \in (0,1]$, , and

$$ \begin{aligned} &\bigl\vert {}^{\mathscr{C} } \mathbb{D}^{\frac{1}{3} }_{q } \bigl( {}^{ \mathscr{C} } \mathbb{D}^{ \frac{4}{5} }_{ q } \bigl( {}^{\mathscr{C} } \mathbb{D}^{ \frac{3}{4} }_{ q } \upsilon ( \chi ) \bigr) \bigr) - \Theta ^{*}_{ \upsilon , \frac{4}{5}, \frac{3}{4}} (\chi ) \bigr\vert \leqslant \overline{x}, \\ &\bigl\vert {}^{\mathscr{C} } \mathbb{D}^{ \frac{1}{3}}_{q } \bigl( {}^{\mathscr{C} } \mathbb{D}^{ \frac{4}{5}}_{ q } \bigl( {}^{\mathscr{C} } \mathbb{D}^{ \frac{3}{4} }_{ q } \upsilon (\chi ) \bigr) \bigr) - \Theta ^{*}_{ \upsilon , \frac{4}{5}, \frac{3}{4}}( \chi ) \bigr\vert \leqslant \overline{x} h(\chi ), \end{aligned} $$

where $\overline{x}>0$, , and

$$\begin{aligned} \Theta ^{*}_{\upsilon , \frac{4}{5}, \frac{3}{4}}(\chi ) ={}& \frac { \sinh ( e^{\chi}+2 )}{ 4 } + \frac { \sqrt{15} e^{ - \chi} \vert \upsilon (\chi ) \vert }{ 41 ( \chi +3)( \vert \upsilon (\chi ) \vert +1)}+ \frac { \cos ( {}^{ \mathscr{C} }\mathbb{D}^{ \frac{3}{4}}_{q } \upsilon (\chi ) )}{ 333 { \sqrt{ \ln ( \chi +12)}} } \\ &{} + \frac { \arctan ( {}^{\mathscr{C} } \mathbb{D}^{ \frac{4}{5}}_{ q } ( {}^{\mathscr{C} } \mathbb{D}^{ \frac{3}{4}}_{q } \upsilon (\chi ) ) ) }{ 22 ( \chi +3)} + \int _{0}^{\chi} \frac {(\chi - q s)^{ \frac{3}{2}-1}}{ \Gamma _{q } ( \frac {3}{2} ) } \biggl[ \frac {\sqrt{ e^{2s}}{ \vert \upsilon (s) \vert }}{ 29 (s+3) ( \vert \upsilon (s) \vert + 5 ) } \\ & {}+ \frac {\cos ( {}^{\mathscr{C} }\mathbb{D}^{ \frac{3}{4}}_{q } \upsilon (s) ) }{ 345 \sqrt{ e^{ s +19}}} + \frac { \sin ( {}^{\mathscr{C} } \mathbb{D}^{ \frac{4}{5}}_{q } ( {}^{\mathscr{C} }\mathbb{D}^{ \frac{3}{4}}_{q } \upsilon (s) ) ) }{ 137 \ln ( \sqrt{s+34} ) } \biggr] \,\mathrm{d}_{q }s. \end{aligned}$$

(31)

For $\chi \in \Lambda$ and , $m=1,2,3$, we obtain

$$\begin{aligned}& \bigl\vert {\xi}(\chi , \upsilon _{1}, \upsilon _{2}, \upsilon _{3} ) - {\xi}( \chi , \hat{ \upsilon}_{1}, \hat{ \upsilon}_{2},\hat{ \upsilon}_{3}) \bigr\vert \\& \quad \leqslant \frac { \sqrt{15}}{123} \vert \upsilon _{1} -\hat{ \upsilon}_{1} \vert + \frac {1}{333{ \sqrt{\ln (12)}}} \vert \upsilon _{2}-\hat{ \upsilon}_{2} \vert + \frac {1}{66} \vert \upsilon _{3}-\hat{ \upsilon}_{3} \vert , \end{aligned}$$

and similarly for , $v=1,2,3$, we get

$$\begin{aligned}& \bigl\vert g(\chi , \upsilon _{1}, \upsilon _{2}, \upsilon _{3} )-g(\chi , \hat{ \upsilon}_{1}, \hat{ \upsilon}_{2},\hat{ \upsilon}_{3}) \bigr\vert \\& \quad \leqslant \frac {\sqrt{e^{2}}}{435} \vert \upsilon _{1}-\hat{ \upsilon}_{1} \vert + \frac {1}{345{\sqrt{e^{19}}}} \vert \upsilon _{2}-\hat{ \upsilon}_{2} \vert + \frac {1}{137 \ln \sqrt{35}} \vert \upsilon _{3} - \hat{ \upsilon}_{3} \vert . \end{aligned}$$

Therefore, conditions $\mathrm{(H_{3})}$ and $\mathrm{(H_{4})}$ are satisfied with

$$ \begin{aligned} &\overline{y}_{1} = \frac {\sqrt{15}}{123},\qquad \overline{y}_{2} = \frac {1}{333{\sqrt{ \ln (12)}}}, \qquad \overline{y}_{3} = \frac {1}{66}, \\ &\overline{z}_{1} = \frac { \sqrt{e^{2}}}{435},\qquad \overline{z}_{2} = \frac {1}{345{ \sqrt{e^{19}}}}, \qquad \overline{z}_{3} = \frac {1}{137 \ln ( \sqrt{35}) }. \end{aligned} $$

Furthermore, thanks to Eq. (21), we get

(32)

and

The data in Table 1 show the values of $\varpi _{i}$, $i=1,2,3$, for three different values q. Because the relations of q-calculators depend on the number of repetitions n, after several steps, their value is fixed. This mathematical performance can be clearly seen in Tables 1 and 2. The approach is similar to each group of curves in Figs. 1a, 1b, and 1c, aligning with each other and reaching a stable value that precisely determines the correctness of the argument. By (22), we get

$$[\sum _{m=1}^3{\bar{y}}_m+\sum _{v=1}^3{\bar{z}}_v]\sum _{i=1}^3{\varpi }_i\approx \left\{\begin{array}{cc}0.5513,& q=\frac{1}{5},\\ 0.5435,& q=\frac{2}{5},\\ 0.5373,& q=\frac{3}{5},\end{array}\right\}<1.$$

(33)

The numerical values of relation (33) are shown in Table 2. It can be seen that after stabilizing the data of each column, these results are less than one (see Fig. 2). Therefore, the given q- (30) is addressed in Theorem 3.2, asserting that it possesses a unique solution within the interval Λ. Additionally, Theorem 4.1 states that the same q- (30) is $\mathscr{UH}$ stable having

In general, as q approaches 1, we will achieve stability of the results with a higher number of iterations. For $h ( \chi ) = \chi ^{\frac{\ln (3)}{5}}$, we obtain

$$\begin{array}{rl}{\mathcal{I}}_q^{\frac{1}{3}+\frac{4}{5}+\frac{3}{4}}[h(\chi )]& ={\mathcal{I}}_q^{\frac{1}{3}+\frac{4}{5}+\frac{3}{4}}\left[{\chi }^{\frac{ln(3)}{5}}\right]\\ & \approx \left\{\begin{array}{cc}0.0834{\chi }^{\frac{ln(3)}{5}},& q=\frac{1}{5},\\ 0.1173{\chi }^{\frac{ln(3)}{5}},& q=\frac{2}{5},\\ 0.1066{\chi }^{\frac{ln(3)}{5}},& q=\frac{3}{5},\end{array}\right\}\\ & ={\vartheta }_{h}h(\chi ).\end{array}$$

Table 3 shows these results. In addition, the curves drawn in Figs. 3a and 3b confirm the existence of $\vartheta _{h}$ and Ineq. (34) variables. Therefore, condition $\mathrm{(H_{7})}$ is fulfilled with $h(\chi ) = \chi ^{\frac{\ln (3)}{5}}$ and $\vartheta _{h}=0.0834, 0.1173, 0.1066$ whenever $q= \frac {1}{5}, \frac {2}{5}, \frac {3}{5}$, respectively. Theorem 4.2 indicates that the q- is $\mathscr{UHR}$ (30) stable s.t.

$$\begin{array}{rl}{\parallel \upsilon -\hat{\upsilon }\parallel }_{\mathcal{F}}& ⩽\frac{{\vartheta }_h}{1-({\sum }_{m=1}^3{\bar{y}}_m+{\sum }_{v=1}^3{\bar{z}}_v){\sum }_{i=1}^3{\varpi }_i}\bar{x}h(\chi )\\ & \approx \left\{\begin{array}{cc}0.1864,& q=\frac{1}{5},\\ 0.2570,& q=\frac{2}{5},\\ 0.2303,& q=\frac{3}{5},\end{array}\right\}\times \bar{x}h(\chi ),\bar{x}>0,\chi \in \mathrm{\Lambda }.\end{array}$$

(34)

Figure 1

2D plot of $\varpi _{i}$, $i=1,2,3$ for q-Caputo fractional (30) in Example 5.1 for three cases of q

Figure 2

Representation of Eq. (33) q-Caputo fractional (30) in Example 5.1 for three cases of q

Figure 3

2D plot of $\vartheta _{h}$ and Ineq. (34) for q-Caputo fractional (30) in Example 5.1 for three cases of q

Table 1 Numerical results for Δ and $\varpi _{i}$, $i=1,2,3$ in Example 5.1 for three cases of q

Table 2 Numerical results for Eq. (33) in Example 5.1 for three cases of q

Table 3 Numerical results of $\vartheta _{h}$ in $I_{q}^{\alpha +\omega +\theta}h(\chi )\leqslant \vartheta _{h}{h( \chi )}$, in Example 5.1 for three cases of q and $\chi \in \Lambda$

The next example shows the proven facts for changes in the order of the derivative α.

Example 5.2

We consider the q-Caputo fractional with $\mathbb{ABC}$s (30) in Example 5.1

$$ \textstyle\begin{cases} {}^{\mathscr{C} }\mathbb{D}^{ \alpha}_{\frac{3}{5}} ( {}^{ \mathscr{C} } \mathbb{D}^{\frac{4}{5}}_{ \frac{3}{5}} ( {}^{\mathscr{C} } \mathbb{D}^{ \frac{3}{4}}_{ \frac{3}{5}} \upsilon ( \chi ) ) ) \\ \quad = \frac { \sinh ( e^{\chi}+2 ) }{ 4 } + \frac { \sqrt{ 15} e^{ - \chi} \vert \upsilon ( \chi ) \vert }{ 41 ( \chi +3) ( \vert \upsilon (\chi ) \vert + 1)} & \\ \qquad {} + \frac { \cos ( {}^{\mathscr{C} } \mathbb{D}^{\frac{3}{4 } }_{\frac{3}{5}} \upsilon ( \chi ) ) }{ 333 \sqrt{ \ln ( \chi +12) }} + \frac { \arctan ( {}^{\mathscr{C} } \mathbb{D}^{\frac{4}{5} }_{\frac{3}{5}} ( {}^{\mathscr{C} } \mathbb{D}^{ \frac{3}{4}}_{\frac{3}{5}} \upsilon ( \chi ) ) ) }{ 22 ( \chi +3)} & \\ \qquad {} + \int _{0}^{ \chi } \frac { ( \chi - \frac {3}{5} s )^{ \frac{8}{5} - 1} }{ \Gamma _{ \frac{3}{5}} ( \frac {8}{5} )} [ \frac { \sqrt{ e^{2s}}{ \vert \upsilon (s) \vert }}{ 29(s+3) ( \vert \upsilon (s) \vert + 5) } + \frac {\cos ( {}^{\mathscr{C} } \mathbb{D}^{\frac{3}{4} }_{\frac{3}{5}} \upsilon (s) ) }{ 345 \sqrt{e^{ s+19}}} & \\ \qquad {} + \frac { \sin ( {}^{\mathscr{C} } \mathbb{D}^{ \frac{4}{5} }_{ \frac{3}{5}} ( {}^{ \mathscr{C} }\mathbb{D}^{ \frac{3}{4} }_{ \frac{3}{5}} \upsilon (s) ) ) }{ 137 \ln ( \sqrt{s+34} ) } ] \,\mathrm{d}_{\frac{3}{5}}s,& \chi \in [0,1], \\ \upsilon (0)| = - \upsilon (1), \qquad ( {}^{ \mathscr{C} } \mathbb{D}^{ \frac{4}{5}}_{ \frac{3}{5}} ( {}^{\mathscr{C} } \mathbb{D}^{ \frac{3}{4}}_{ q } \upsilon ( \frac {7}{11} ) ) )=0,& \\ {}^{\mathscr{C} }\mathbb{D}_{ \frac{3}{5}}^{ \frac{5}{9}} \upsilon (0) = - {}^{\mathscr{C} } \mathbb{D}_{ \frac{3}{5} }^{ \frac{5}{9}} \upsilon (1),& \end{cases} $$

(35)

with the difference that $q=\frac {3}{5}$ is fixed and α chooses $\{ \frac {1}{8},\frac {1}{6}, \frac {1}{3} \} \subseteq (0,1)$, $\omega =\frac {4}{5}$, $\nu = \zeta = \frac {8}{5}$, $\theta = \frac {3}{4}$, $\delta = \frac {7}{11}$, $\beta = \frac {5}{9}$, , and

$$ \begin{aligned} &\bigl\vert {}^{\mathscr{C} } \mathbb{D}^{\alpha }_{\frac{3}{5}} \bigl( {}^{ \mathscr{C} } \mathbb{D}^{ \frac{4}{5} }_{ \frac{3}{5}} \bigl( {}^{\mathscr{C} } \mathbb{D}^{ \frac{3}{4} }_{ \frac{3}{5}} \upsilon ( \chi ) \bigr) \bigr) - \Theta ^{*}_{ \upsilon , \frac{4}{5}, \frac{3}{4}} ( \chi ) \bigr\vert \leqslant \overline{x}, \\ &\bigl\vert {}^{\mathscr{C} } \mathbb{D}^{ \alpha}_{ \frac{3}{5}} \bigl( {}^{\mathscr{C} } \mathbb{D}^{ \frac{4}{5}}_{ \frac{3}{5} } \bigl( {}^{\mathscr{C} } \mathbb{D}^{ \frac{3}{4} }_{ \frac{3}{5} } \upsilon (\chi ) \bigr) \bigr) - \Theta ^{*}_{ \upsilon , \frac{4}{5}, \frac{3}{4}}( \chi ) \bigr\vert \leqslant \overline{x} h(\chi ), \end{aligned} $$

where $\overline{x}>0$, , and $\Theta ^{*}_{\upsilon , \frac{4}{5}, \frac{3}{4}}(\chi )$ is defined by (31). It was found that conditions $\mathrm{(H_{3})}$ and $\mathrm{(H_{4})}$ are satisfied with $\overline{y}_{1} = \frac {\sqrt{15}}{123}$, $\overline{y}_{2} = \frac {1}{333{\sqrt{ \ln 12}}}$, $\overline{y}_{3} = \frac {1}{66}$, and $\overline{z}_{1} = \frac { \sqrt{e^{2}}}{435}$, $\overline{z}_{2} = \frac {1}{345{ \sqrt{e^{19}}}}$, $\overline{z}_{3} = \frac {1}{137 \ln ( \sqrt{35}) }$. Thanks to Eq. (21), by using these data, we obtain and

The data in Table 4 show the values of $\varpi _{i}$, $i=1,2,3$, for three different values of derivative order α. The approach is similar to each group of curves in Figs. 4a, 4b, and 4c, aligning with each other and reaching a stable value that precisely determines the correctness of the argument. By (22), we get

$$[\sum _{m=1}^3{\bar{y}}_m+\sum _{v=1}^3{\bar{z}}_v]\sum _{i=1}^3{\varpi }_i\approx \left\{\begin{array}{cc}0.568,& \alpha =\frac{1}{8},\\ 0.563,& \alpha =\frac{1}{6},\\ 0.537,& \alpha =\frac{1}{3},\end{array}\right\}<1.$$

(36)

The numerical values of relation (36) are shown in Table 5. It can be seen that after stabilizing the data of each column, these results are less than one (see Fig. 5). Therefore, the given q- (35) is addressed in Theorem 3.2, asserting that it possesses a unique solution within the interval Λ. Additionally, Theorem 4.1 states that the same q- (35) is $\mathscr{UH}$ stable having

For $h ( \chi ) = \chi ^{\frac{\ln (3)}{5}}$, we have

$$\begin{array}{rl}{\mathcal{I}}_q^{\frac{1}{3}+\frac{4}{5}+\frac{3}{4}}[h(\chi )]& ={\mathcal{I}}_q^{\frac{1}{3}+\frac{4}{5}+\frac{3}{4}}\left[{\chi }^{\frac{ln(3)}{5}}\right]\\ & \approx \left\{\begin{array}{cc}0.097{\chi }^{\frac{ln(3)}{5}},& \alpha =\frac{1}{8},\\ 0.099{\chi }^{\frac{ln(3)}{5}},& \alpha =\frac{1}{6},\\ 0.107{\chi }^{\frac{ln(3)}{5}},& \alpha =\frac{1}{3},\end{array}\right\}={\vartheta }_{h}h(\chi ).\end{array}$$

Table 5 shows these results. In addition, the curves drawn in Figs. 6a and 6b confirm the existence of $\vartheta _{h}$ and Ineq. (37) variables. Therefore, condition $\mathrm{(H_{7})}$ is fulfilled with $h(\chi ) = \chi ^{\frac{\ln (3)}{5}}$ and $\vartheta _{h}=0.097, 0.099, 0.107$ whenever $\alpha = \frac {1}{5}, \frac {2}{5}, \frac {3}{5}$, respectively. Theorem 4.2 indicates that the q- is $\mathscr{UHR}$ (35) stable s.t.

$$\begin{array}{rl}{\parallel \upsilon -\hat{\upsilon }\parallel }_{\mathcal{F}}& ⩽\frac{{\vartheta }_h}{1-({\sum }_{m=1}^3{\bar{y}}_m+{\sum }_{v=1}^3{\bar{z}}_v){\sum }_{i=1}^3{\varpi }_i}\bar{x}h(\chi )\\ & \approx \left\{\begin{array}{cc}0.224,& \alpha =\frac{1}{8},\\ 0.227,& \alpha =\frac{1}{6},\\ 0.230,& \alpha =\frac{1}{3},\end{array}\right\}\times \bar{x}h(\chi ),\bar{x}>0,\chi \in \mathrm{\Lambda }.\end{array}$$

(37)

Figure 4

2D plot of $\varpi _{i}$, $i=1,2,3$, for q-Caputo fractional (35) in Example 5.2 for three cases of derivative order α

Figure 5

Representation of Eq. (36) q-Caputo fractional (35) in Example 5.2 for three cases of α

Figure 6

2D plot of $\vartheta _{h}$ and Ineq. (37) for q-Caputo fractional (35) in Example 5.2 for three cases of derivative order α

Algorithm 1

MATLAB lines for calculation q-factorial function

Algorithm 2

MATLAB lines for q-gamma function

Algorithm 3

MATLAB lines to calculate q-integral

Table 4 Numerical results of $\varpi _{i}$, $i=1,2,3$, in Example 5.2 for three cases of derivative order α

Table 5 Numerical results of Eq. (36), $\vartheta _{h}$ and Ineq. (37) in Example 5.2 for three cases of derivative order α

6 Conclusion

We analyzed the q-, involving both $\mathbb{ABC}s$ and q-fractional $\mathscr{CD}s$. Our main focus was on establishing certain conditions that guaranteed the $\mathfrak{EU}$ of solution. For the validity of the suggested system, given in (1), we employed the Banach fixed point theorem and Leray-Schauder alternative. Additionally, we also explored the $\mathscr{US}$ outcomes and examined the resolution of our model (1) in specific circumstances. Our primary theoretical findings are demonstrated by means of a few examples.

Supplement