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α-Confluent-hyper-geometric stability of ξ-Hilfer impulsive nonlinear fractional Volterra integro-differential equation

Abstract

The purpose of this work is to investigate the necessary conditions for the existence and uniqueness of solutions, and to introduce a new idea of α-confluent-hyper-geometric stability of an impulsive fractional differential equation with ξ-Hilfer fractional derivative. We use the Diaz–Margolis fixed point theorem to achieve this and illustrate the result with an example.

1 Introduction

Over time, fractional differential equations (FDEs) are crucial and interesting research areas because of their role in engineering, economics, substances sciences, physics, and biology. Scientists have implemented numerous mathematical methods through various research-orientated components of fractional differential structures [35, 8, 16, 17]. Impulsive fractional differential equations (IFDEs) are one of the attractive branches of FDEs for researchers due to their wide application in important positions, mainly in describing dynamics of populations subject to abrupt changes and other phenomena such as diseases and so forth. Moreover, these are mathematically attracting since the fractional derivative is nonlocal in contrast to the classical derivative, which leads to some obstacles in studying IFDEs. This challenging position motivates the researcher to work on this kind of problem, in particular, to study the existence, uniqueness, and stability, sometimes with noninstantaneous or instantaneous impulses; for instance, in [6, 10, 12, 14, 15, 19, 20] the reader can find some of these results.

Two main types of IFDEs are instantaneous and noninstantaneous. Wang and Zhang [6] proved the existence and uniqueness results for the following differential equation with no instantaneous impulses in a -normed space:

$$ \textstyle\begin{cases} \Upsilon '(\mathfrak{z})= \mathcal{H}(\mathfrak{z}, \Upsilon ( \mathfrak{z})), & \mathfrak{z}\in{(s_{j}, t_{j+1}]}, j=0, 1,\ldots, m, \\ \Upsilon (\mathfrak{z})= Q_{j}(\mathfrak{z}, \Upsilon (\mathfrak{z})) , & \mathfrak{z}\in{(t_{j}, s_{j}]}, j=1,2, \ldots, m. \end{cases} $$

The authors in [20] proved the existence and uniqueness of solutions for the ordinary nonlinear differential equation with no instantaneous impulses and introduced a new kind of stability, which is a generalization of the β-Ulam–Hyers–Rassias stability [18]. Also, the results were extended in [7] for impulsive integrodifferential equations with no instantaneous impulses.

Many researchers consider the mentioned area developed by fractional calculus; see [8, 11] and references therein. Concerning the stability result, some interesting extensions of Ulam–Hyers–Rassias were recently proved for different classes of IFDEs, for instance, Bielecki–Ulam-type stability [21], δ-Ulam–Hyers–Rassias stability [14], etc. These results are the motivation of this paper.

Let \(\lambda \in{(0, 1]}\), \(\varrho \in{[0, 1),}\) and let \({}^{\mathcal{H}}{\mathbb{D}}^{\lambda , \varrho ; \xi}_{0^{+}}( \cdot ) \) be a ξ-Hilfer fractional derivative (HFD) of order λ and type ϱ, and let \(I^{1- \mathfrak{X} ; \xi}_{0} \) be the ξ-Riemann–Liouville fractional integral (RLFI) of order \(1-\mathfrak{X} \) (\(\mathfrak{X}= \lambda + \varrho (1- \lambda )\)) with respect to the mapping ξ. In this study, we focus on the following ξ-HFD:

$$ \textstyle\begin{cases} {}^{\mathcal{H}}{\mathbb{D}}^{\lambda , \varrho ; \xi}_{0^{+}} v( \mathfrak{z}) ={\mathcal{K}}(v(\mathfrak{z}))+ \int _{0}^{ \mathfrak{z}} {\mathcal{J}}(\mathfrak{z}, s, v(s)) \,ds, & \mathfrak{z} \in{(s_{j}, t_{j+1}]}, j=0, 1,\ldots, m, \\ v(\mathfrak{z})= {\mathcal{Q}}_{j}(\mathfrak{z}, v(\mathfrak{z}_{j}^{+})) , & \mathfrak{z}\in{(t_{j}, s_{j}]}, j=1,2, \ldots, m, \\ I^{1- \gamma ; \phi}_{0} v(0)={\mathfrak{v}}_{0}, & {\mathfrak{v}}_{0} \in{\mathbb{R}\mathbbm{,}} \end{cases} $$
(1.1)

where \(t_{0} = s_{0}=0\), \(t_{j} \leq s_{j}\) for \(j= 1,\ldots, m\), and \(s_{j} < t_{i+1}\) for \(i=0, m\) are prefixed numbers in \([0, \rho ]\). Also, \(\mathcal{J}: {[0, \rho ]}^{2} \times \mathbb{R} \rightarrow \mathbb{R}\) is a given mapping, and \(\mathcal{K}\) is a linear operator. Moreover, the noninstantaneous impulses \({\mathcal{Q}}_{j}: (t_{j}, s_{j}] \times \mathbb{R} \rightarrow \mathbb{R}\), \(j= 1, 2, \ldots, m\), all are continuous functions.

In the following, we prove the existence and uniqueness results using the Diaz–Margolis theorem, present a new concept of stability, α-confluent-hyper-geometric stability, and conclude that the solution of equation (1.1) is stable in the α-confluent-hyper-geometric sense.

2 Preliminaries

In this section, we recall some basic concepts and fixed point theorem (FPT), which we use to prove the main results. We start with the confluent hypergeometric (CHG) functions, which are solutions of a hypergeometric differential equation and have different standard forms like Tricomi, Kummer, Coulomb wave, and so on [5]. In this paper, we use the solution of \(\mathfrak{z}\frac {d^{2} \mathfrak{u}}{dz} +(\delta _{2}- \mathfrak{z}) \frac {d\mathfrak{u}}{d\mathfrak{z}} - \delta _{1} \mathfrak{u}(\mathfrak{z})=0\), where \(\mathfrak{z}, \delta _{1} \in{\mathbb{C}}\) and \(\delta \in{\mathbb{C}\setminus {\mathbb{Z}}^{-}_{0}}\), which is the CHG function

$$\begin{aligned} \Xi (\delta _{1}, \delta _{2}; \mathfrak{z})={}_{1}{F}_{1}(\delta _{1}, \delta _{2}; \mathfrak{z})= \frac {\Gamma (\delta _{2})}{\Gamma (\delta _{1})} \sum _{k=0}^{ \infty} \frac {\Gamma (\delta _{1}+k)}{\Gamma (\delta _{2}+k)} \frac {\mathfrak{z}^{k}}{k!}. \end{aligned}$$
(2.1)

Series (2.1) is known as the CHG function of the first kind and was introduced in 1837; it converges for al \(\mathfrak{z}\) belonging to \(\mathbb{C}\). Clearly, for \(\delta _{1}=\delta _{2}=\delta \), \(\Xi (\delta , \delta ; \mathfrak{z})= \sum_{k=0}^{\infty} \frac {\mathfrak{z}^{k}}{k!}= e^{\mathfrak{z}}\) (see [1, 5] and references therein). We provide our stability result by applying series (2.1) on the real line \(\mathbb{R}\) as our control functions.

To define a new concept of stability, called the α-confluent-hyper-geometric stability, we consider the inequality

$$ \biggl\vert {}^{\mathcal{H}}{\mathbb{D}}^{\lambda , \varrho ; \xi}_{0^{+}} \mathcal{Z}(\mathfrak{z}) -{\mathcal{K}}\bigl(\mathcal{Z}(\mathfrak{z})\bigr) - \int _{0}^{\mathfrak{z}} {\mathcal{J}}\bigl(\mathfrak{z}, s, \mathcal{Z}(s)\bigr) \,ds \biggr\vert \leq \epsilon \Xi \bigl(\lambda , \varrho ; \bigl(\xi (\mathfrak{z})-\xi (0)\bigr)^{ \lambda}\bigr) $$

for \(\epsilon > 0\), where Ξ is the CHG function (see [5]).

Definition 1

Let \(\alpha \in{(0, 1],}\) and let \(\mathcal{Y}\) be a vector space over field \(\mathbb{F}\). A function \(\| \cdot \|_{\alpha }: \mathcal{Y} \rightarrow \mathbb{R}^{+} \cup \{0\}\) is called a α-norm [14] if it satisfies the following conditions:

  1. (I1)

    \(\| \mathfrak{z}\|_{\alpha }=0\) if and only if, \(\mathfrak{z}=0\),

  2. (I2)

    \(\| k\mathfrak{z}\|_{\alpha}= |k|^{\alpha }\|\mathfrak{z}\|_{\alpha}\) for all \(k\in{\mathbb{K}}\) and \(\mathfrak{z}\in{\mathcal{X}}\),

  3. (I3)

    \(\|\mathfrak{z}_{1} + \mathfrak{z}_{2}\|_{\alpha }\leq \|\mathfrak{z}_{1}\|_{ \alpha }+ \|\mathfrak{z}_{2}\|_{\alpha}\) for all \(\mathfrak{z}_{1}, \mathfrak{z}_{2} \in{\mathcal{X}}\).

Let \(\mathcal{M}([0, \rho ], \mathbb{R})\) be the space of continuous real-valued functions. Furthermore, the weighted space \(\mathcal{M}_{\mathfrak{X}, \xi} (0, \rho ] \) of continuous functions \(\xi : (0, \rho ] \rightarrow \mathbb{R}\) is defined as

$$\begin{aligned} \mathcal{M}_{1- \mathfrak{X}, \xi} [0, \rho ] = \bigl\lbrace \pi :(0, d] \rightarrow \mathbb{R}; \bigl(\xi (\mathfrak{z})- \xi (0)\bigr)^{1- \mathfrak{X}} \pi ( \mathfrak{z}) \in{\mathcal{M}[0, d]} \bigr\rbrace , \quad \text{where } 0 < \mathfrak{X} < 1, \end{aligned}$$

with norm

$$\begin{aligned} \Vert \pi \Vert _{\mathcal{M}_{1- \mathfrak{X}, \xi} [0, \rho ], \alpha } = \sup_{\mathfrak{z} \in{(0, \rho ]}} \bigl(\xi ( \mathfrak{z})- \xi (0)\bigr)^{1- \mathfrak{X}} \bigl\Vert \pi (\mathfrak{z}) \bigr\Vert _{\alpha}, \end{aligned}$$

where \(\xi : [0, \rho ] \longrightarrow \mathbb{R}\) is an arbitrary function, and \(\mathfrak{z} \in{[0, \rho ]}\).

The piecewise weighted space \(\mathcal{PM}_{\mathfrak{X}, \xi} (t_{\mathfrak{k}}, t_{\mathfrak{k}+1}]\) is defined as follows:

$$\begin{aligned} \mathcal{PM}_{1- \mathfrak{X}, \xi} [0, \rho ] =& \Bigl\{ \tilde{\pi}:(0, \rho ] \rightarrow \mathbb{R}; \bigl(\xi (\mathfrak{z})- \xi (t_{ \mathfrak{k}})- \xi (0) \bigr)^{1-\mathfrak{X}} \rho (\eta ) \in{ \mathcal{C}(t_{\mathfrak{k}}, t_{\mathfrak{k}+1}]}, \\ &\lim_{\mathfrak{z} \rightarrow t_{\mathfrak{k}}} \bigl(\xi (\mathfrak{z})- \xi (t_{\mathfrak{k}})- \xi (0) \bigr)^{1-\mathfrak{X}} \xi (\mathfrak{z}) < \infty \Bigr\} \end{aligned}$$

for all \(k=1,2, \ldots, m \), where \(0 < \mathfrak{X} < 1\). The space \(\mathcal{PM}_{\mathfrak{X}, \xi} (t_{\mathfrak{k}}, t_{\mathfrak{k}+1}]\) is equipped by the norm

$$\begin{aligned} \Vert \tilde{\pi} \Vert _{\mathcal{PM}_{1- \mathfrak{X}, \xi} [0, \rho ], \alpha } =\max_{k= 1,2, \ldots, m} \Bigl( \sup_{\eta \in{(t_{k}, t_{k+1}]}} \bigl(\xi (\mathfrak{z})- \xi (t_{\mathfrak{k}})- \xi (0) \bigr)^{1- \mathfrak{X}} \bigl\Vert \tilde{\pi}(\mathfrak{z}) \bigr\Vert _{\alpha } \Bigr), \end{aligned}$$

where \(\xi : [0, \rho ] \longrightarrow \mathbb{R}\) is an arbitrary function, \(\mathfrak{z} \in{[0, \rho ]}\), and there exist \(\tilde{\pi}(t_{\mathfrak{k}}^{-})\) and \(\tilde{\pi}(t_{\mathfrak{k}}^{+})\) for all \(\mathfrak{k}=1,2, \ldots, m \) with \(\tilde{\pi}(t_{\mathfrak{k}}^{-})= \tilde{\pi}(t_{\mathfrak{k}}^{+})\). The piecewise weighted space \(\mathcal{PM}_{1- \mathfrak{X}, \xi} [0, \rho ]\) and weighted space \(\mathcal{M}_{\mathfrak{X}, \xi} (0, \rho ]\) with the above defined norm are Banach spaces [14].

Definition 2

([9])

Let \((\mathfrak{s}, \rho )\) be an interval on real line (finite or infinite), let \(\lambda > 0\), let \(\xi (\mathfrak{z}): [\mathfrak{s}, \rho ] \rightarrow \mathbb{R}^{+}\) be a nondecreasing function on \((\mathfrak{s}, \rho ]\), and let \(\xi '(\mathfrak{z})\) be a continuous mapping on \((\mathfrak{s}, \rho )\). The left- and right-sided fractional integrals of a function ξ with respect to the function ξ on \([\mathfrak{s}, \rho ]\) are defined by

$$\begin{aligned} I^{\lambda ; \xi}_{\mathfrak{s}^{+}} w(x)= \frac {1}{\Gamma (\lambda )} \int _{\mathfrak{s}}^{x} \xi '( \mathfrak{z}) \bigl(\xi (x)- \xi (\mathfrak{z})\bigr)^{\lambda - 1} w( \mathfrak{z}) \,d \mathfrak{z} \end{aligned}$$

and

$$\begin{aligned} I^{\lambda ; \xi}_{\rho ^{-}} w(x)= \frac {1}{\Gamma (\lambda )} \int _{x}^{\rho }\xi '(\mathfrak{z}) \bigl(\xi (\mathfrak{z})- \xi (x)\bigr)^{ \lambda - 1} w(\mathfrak{z}) \,d \mathfrak{z}, \end{aligned}$$

respectively, where Γ is the gamma function.

It is worth mentioning here that for \(\lambda , \sigma > 0\), we have [13]

  • if \(\Delta (x)= (\xi (x)- \xi (\mathfrak{s}))^{\sigma - 1} \), then \(I^{\lambda ; \xi}_{\mathfrak{s}^{+}} \Delta (x)= C_{\lambda , \sigma}(\xi (x)- \xi (\mathfrak{s}))^{\lambda + \sigma - 1} \), and

  • if \(\Delta (x)= (\xi (\rho )- \xi (x))^{\sigma - 1} \), then \(I^{\lambda ; \xi}_{\rho ^{-}} \Delta (x)= C_{\lambda , \sigma} ( \phi (\rho )- \xi (x))^{\lambda + \sigma - 1} \),

where \(C_{\lambda , \sigma}= \frac {\Gamma (\sigma )}{\Gamma (\lambda + \sigma )}\). Also, the semigroup properties \(I^{\lambda ; \xi}_{\mathfrak{s}^{+}} I^{\varrho ; \xi}_{ \mathfrak{s}^{+}} \Lambda (x)= I^{\lambda + \varrho ; \xi}_{ \mathfrak{s}^{+}} \Lambda (x)\) and \(I^{\lambda ; \xi}_{\rho ^{-}} I^{\varrho ; \xi}_{\rho ^{-}} \Lambda (x)= I^{\lambda + \varrho ; \xi}_{\mathfrak{m}^{-}} \Lambda (x)\) are satisfied.

Definition 3

([9])

Let \((\mathfrak{s}, \rho )\subseteq \mathbb{R}\) be an interval (finite or infinite), \(\xi '(\mathfrak{z})\neq 0\) for all \(\mathfrak{z}\in{(\mathfrak{s}, \rho )}\), and \(\lambda > 0\), \(\mathfrak{k}\in{\mathbb{N}}\). The left-sided Riemann–Liouville derivative of a function χ with respect to ξ of order λ is defined by

$$ \begin{aligned} D^{\lambda , \xi}\chi (\mathfrak{z})&=\biggl( \frac {1}{\xi '(\mathfrak{z})} \frac {d}{dx}\biggr)^{\mathfrak{k}} I^{ \mathfrak{k}-\lambda , \xi} \chi (\eta ) \\ &=\frac {1}{\Gamma (\mathfrak{k}-\lambda )} \biggl( \frac {1}{\xi '(\mathfrak{z})}\frac {d}{dx} \biggr)^{\mathfrak{k}} \times \int _{\mathfrak{s}}^{\mathfrak{z}} \xi '(s) \bigl(\xi ( \mathfrak{z})- \xi (s)\bigr)^{ \mathfrak{k}-\lambda -1} \chi (s) \,ds. \end{aligned} $$

Definition 4

([9])

Let \(\mathfrak{k}\in{\mathbb{N}}\), \(\lambda \in{(\mathfrak{k}-1, \mathfrak{k})}\), \(\varpi =[\mathfrak{s}, \rho ]\) (\(-\infty \leq \mathfrak{s} < \rho \leq \infty \)), and let \(\hat{\pi}, \xi \in{\mathcal{C}^{n}([\mathfrak{s}, \rho ], \mathbb{R})} \) be two mappings with \(\xi (x) \) increasing for all \(x\in{\varpi}\). The left-sided and right-sided ξ-Hilfer fractional derivatives \({}^{\mathcal{H}}{\mathbb{D}}^{\lambda , \varrho ; \xi}_{0^{+}}( \cdot )\) of an arbitrary function π̂ of order λ and type \(\varrho \in{[0, 1)}\) are defined by

$$\begin{aligned} {}^{\mathcal{H}}{\mathbb{D}}^{\lambda , \varrho ; \xi}_{\mathfrak{s}^{+}} \hat{\pi}(x)= I^{\varrho (\mathfrak{k}-\lambda ); \phi}_{\mathfrak{s}^{+}} \biggl( \frac {1}{\xi '(x)} \frac {d}{dx} \biggr) ^{\mathfrak{k}} I^{(1- \varrho )(\mathfrak{k}-\lambda ); \xi}_{\mathfrak{s}^{+}} \hat{\pi}(x) \end{aligned}$$

and

$$\begin{aligned} {}^{\mathcal{H}}{\mathbb{D}}^{\lambda , \varrho ; \xi}_{\rho ^{-}} \hat{\pi}(x)= I^{\varrho (\mathfrak{k}-\lambda ); \xi}_{\rho ^{-}} \biggl( -\frac {1}{\xi '(x)} \frac {d}{dx} \biggr) ^{\mathfrak{k}} I^{(1- \varrho )(\mathfrak{k}-\lambda ); \xi}_{\rho ^{-}} \hat{\pi}(x), \end{aligned}$$

respectively.

Theorem 1

Let \(\varrho \in{[0,1)}\), \(\lambda > 0\), and \(\mathfrak{X}= \lambda + \varrho (1- \lambda )\). Then for \(\Lambda \in{\mathcal{C}^{1}[\mathfrak{s}, \rho ]}\), we have \({}^{\mathcal{H}}{\mathbb{D}}^{\lambda , \varrho ; \xi}_{\mathfrak{s}^{+}} I^{\lambda ; \xi}_{\mathfrak{s}^{+}} \Lambda (x)= \Lambda (x)\) and \({}^{\mathcal{H}}{\mathbb{D}}^{\lambda , \varrho ; \xi}_{\rho ^{-}} I^{ \lambda ; \xi}_{\rho ^{-}} \Lambda (x)= \Lambda (x)\). Also, we have

  • \(I^{\lambda ; \xi}_{\mathfrak{s}^{+}} {}^{\mathcal{H}}{\mathbb{D}}^{ \lambda , \varrho ; \xi}_{\mathfrak{s}^{+}} \Lambda (x) = \Lambda (x)- \frac {(\xi (x)-\xi (\mathfrak{s}))^{\gamma -1} I^{(1- \varrho )(1-\lambda ); \phi}_{\mathfrak{s}^{+}} \Lambda (\mathfrak{s})}{\Gamma (\gamma )} \),

  • \(I^{\lambda ; \xi}_{\rho ^{-}} {}^{\mathcal{H}}{\mathbb{D}}^{\lambda , \varrho ; \xi}_{\rho ^{-}} \Lambda (x) = \Lambda (x)- \frac {(\xi (\rho )-\xi (x))^{\gamma -1}I^{(1- \varrho )(1-\lambda ); \xi}_{\rho ^{-}} \Lambda (\rho )}{\Gamma (\gamma )} \).

Proof

See [13]. □

Now we recall an alternative FPT, which plays a crucial role in proving our main result and was proved by Diaz and Margolis [2] in 1967.

Theorem 2

([2])

Let \((\mathfrak{O}, \tilde{d}) \) be a generalized complete metric space, and let Π be a strictly contracting self-mapping with Lipschitz constant \(\kappa < 1\). Then either \(\tilde{d}(\Pi ^{\mathfrak{k} +1}\mathfrak{z}, \Pi ^{n}\mathfrak{z})=+ \infty \) for every \(\mathfrak{k} \in{\mathbb{N}}\), or

  1. (i)

    if there exists \(\mathfrak{k} \in{\mathbb{N}}\) such that \(\tilde{d}(\Pi ^{\mathfrak{k} +1}\mathfrak{z}, \Pi ^{n}\mathfrak{z})< \infty \) for some \(\mathfrak{z}\in{\mathfrak{O}}\), then the sequence \(\lbrace \Pi ^{\mathfrak{k}} \mathfrak{z}\rbrace \) converges to a unique fixed point \({\mathfrak{z}}^{*}\) of Π in the set \(\mathfrak{O}^{*}= \lbrace \tilde{o}\in{\mathfrak{O}} : \tilde{d}( \Pi ^{\mathfrak{k}} \tilde{o}, \Pi ^{\mathfrak{k}} \tilde{o}) < \infty \rbrace \),

  2. (ii)

    Furthermore, \(\tilde{d}(\mathfrak{z}, \mathfrak{z}^{*} ) \leq C_{\kappa} \tilde{d}( \mathfrak{z}, \Pi \mathfrak{z} )\) for all \(\mathfrak{z} \in{\mathfrak{O}}\), where \(C_{\kappa}=\frac {1}{1- \kappa}\), and \(\mathfrak{z}^{*}\) is defined in (i).

Definition 5

Let t \(\lambda \in{(0, 1]} \), \(\varrho \in{[0, 1)} \), and let \(\mathfrak{X}= \lambda + \varrho (1- \lambda )\) be nonnegative. The function \(v\in{\tilde{W}:= \mathcal{PM}_{1- \gamma ; \xi}([0, \rho ], \mathbb{R}) \bigcap_{j=0}^{m} \mathcal{M}^{1}((s_{j}, t_{j+1}], \mathbb{R})}\) is said to be a mild solution of IFDE (1.1) if

$$ v(\mathfrak{z}) = \textstyle\begin{cases} \frac{(\xi (\mathfrak{z})- \xi (0))^{\mathfrak{X}-1}}{\Gamma (\mathfrak{X})} \mathfrak{v}_{0} + I^{\lambda ; \xi}_{0^{+}} [ {\mathcal{K}}(v( \mathfrak{z}))+ \int _{0}^{\mathfrak{z}} {\mathcal{J}}(\mathfrak{z}, s, v(\mathfrak{z})) \,ds ], & \mathfrak{z}\in{[0, t_{1}]}, \\ {\mathcal{Q}}_{j}(\mathfrak{z}, v(\mathfrak{z}_{j}^{+})), & \mathfrak{z}\in{(t_{j}, s_{j}]}, \\ {\mathcal{Q}}_{j}(\mathfrak{z}, v(\mathfrak{z}_{j}^{+})) + I^{ \lambda ; \xi}_{0^{+}} [ {\mathcal{K}}(v(\mathfrak{z}))+ \int _{0}^{ \mathfrak{z}} {\mathcal{J}}(\mathfrak{z}, s, v(\mathfrak{z})) \,ds ], & \mathfrak{z}\in{(s_{j}, t_{j+1}]}, \end{cases} $$
(2.2)

for all \(j=1,2, \ldots, m\).

In the rest of the paper, we assume that for a continuously differentiable function \(v: [0, \rho ] \rightarrow \mathbb{R}\) and \(\epsilon \geq 0\), we have

$$ \textstyle\begin{cases} \vert {}^{\mathcal{H}}{\mathbb{D}}^{\lambda , \varrho ; \xi}_{0^{+}} v( \mathfrak{z}) - \mathcal{K}(v(\mathfrak{z}))- \int _{0}^{\mathfrak{z}} \mathcal{J}(\mathfrak{z}, s, v(s)) \,ds \vert \leq \epsilon \Xi , & \mathfrak{z}\in{(s_{j}, t_{j+1}]},j=0, 1, \ldots, m, \\ \vert v(\mathfrak{z})- \mathcal{Q}_{j}(\mathfrak{z}, v(\mathfrak{z}_{j}^{+})) \vert \leq \xi , & \mathfrak{z}\in{(t_{j}, s_{j}]},j= 1, \ldots, m, \end{cases} $$
(2.3)

for some positive ξ, where \(\alpha \in{(0,1]}\), and \(\Xi =\Xi (\lambda , \varrho ; (\xi (\eta )-\xi (0))^{\lambda})\).

Definition 6

Let \(\alpha \in{(0, 1]}\). Equation (1.1) has confluent-hyper-geometric stability (in short, CHG stability) with respect to \(\Xi (\lambda ,\varrho ;(\xi (\mathfrak{z})-\xi (0))^{\lambda})\) if for all \(v\in{(\mathcal{PC}[0, \rho ], \mathbb{R})}\) satisfying inequalities (2.3), there exists a solution \(w\in{(\mathcal{PM}[0, \rho ], \mathbb{R})}\) of equation (1.1) such that for all \(\epsilon > 0 \),

$$\begin{aligned} \bigl\vert w(\mathfrak{z})- v(\mathfrak{z}) \bigr\vert ^{\alpha }\leq C_{\Xi }\epsilon \Xi \bigl(\lambda , \varrho ; \bigl(\xi (\mathfrak{z})- \xi (0)\bigr)^{\lambda}\bigr) \quad \text{for all } \mathfrak{z} \in{[0, \rho ]}, \end{aligned}$$

where \(\mathcal{C}_{\Xi} \) is a positive constant.

Remark 1

([9, 14])

Let \(\mathfrak{w}\in{\mathcal{PM}([0, \rho ], \mathbb{R})}\) satisfy inequalities (2.3). Then we have the following integral inequalities:

  • For \(\mathfrak{z}\in{[0, t_{1}]}\),

    $$\begin{aligned} &\biggl\vert \mathfrak{w}(\mathfrak{z}) - \frac{(\xi (\mathfrak{z})- \xi (0))^{\gamma -1}}{\Gamma (\gamma )} \mathfrak{v}_{0} - I^{\lambda ; \xi}_{0^{+}} \biggl[ \int _{0}^{ \mathfrak{z}} \mathcal{J}\bigl(\mathfrak{z}, s, v(s) \bigr) \,ds \biggr] \\ &\quad {}-I^{\lambda ; \xi}_{0^{+}}\bigl(\mathcal{K}\bigl(v(\mathfrak{z})\bigr) \bigr) \biggr\vert \leq \epsilon \Xi \bigl(\lambda , \varrho ; \bigl({\xi ( \mathfrak{z})-\xi (0)}\bigr)^{ \lambda}\bigr); \end{aligned}$$
  • For \(\mathfrak{z}\in{(t_{j}, s_{j}]}\), \(j=0,1, \ldots, m\),

    $$\begin{aligned} \bigl\vert v(\mathfrak{z})- \mathcal{Q}_{j} \bigl(\mathfrak{z}, v \bigl(\mathfrak{z}_{j}^{+}\bigr)\bigr) \bigr\vert \leq \xi \epsilon \Phi \bigl(\lambda , \varrho ; \bigl({\xi (\mathfrak{z})-\xi (0)} \bigr)^{ \lambda}\bigr), \end{aligned}$$

    where ξ is defined in (2.3);

  • For \((s_{j}, t_{j+1}]\), and \(j=1, 2, \ldots, m\),

    $$\begin{aligned} &\biggl\vert \mathfrak{w}(\mathfrak{z}) - \mathcal{Q}_{j} \bigl( \mathfrak{z}, v\bigl( \mathfrak{z}_{j}^{+}\bigr)\bigr)- \frac{(\xi (\mathfrak{z})- \xi (0))^{\gamma -1}}{\Gamma (\gamma )} \mathfrak{v}_{0} - I^{\lambda ; \xi}_{0^{+}} \biggl[ \int _{0}^{ \mathfrak{z}} \mathcal{J}\bigl(\mathfrak{z}, s, v(s) \bigr) \,ds \biggr] \\ &\quad {}-I^{\lambda ; \xi}_{0^{+}}\bigl(\mathcal{K}\bigl(v(\mathfrak{z})\bigr) \bigr) \biggr\vert \leq (1+ \xi ) \epsilon \Xi \bigl(\lambda , \varrho ; \bigl({ \xi (\mathfrak{z})-\xi (0)}\bigr)^{ \lambda}\bigr), \end{aligned}$$

    where ξ is defined in (2.3).

3 Main results and stability analysis

To demonstrate our main point, we start by adopting the following hypotheses:

  1. (K1)

    The function \(\mathcal{J}: [0, \rho ]^{2} \times \mathbb{R} \rightarrow \mathbb{R}\) is continuous and is a Lipschitz function with respect to the third argument, i.e., for some \(\mathcal{L}_{\mathcal{J}} > 0\),

    $$\begin{aligned} \bigl\vert \mathcal{J}(\mathfrak{z}, s, \mathfrak{w}_{1})- \mathcal{J}( \mathfrak{z}, s, \mathfrak{w}_{2}) \bigr\vert \leq \mathcal{L}_{\mathcal{J}} \vert \mathfrak{w}_{1}- \mathfrak{w}_{2} \vert \end{aligned}$$
    (3.1)

    for all \(s, \mathfrak{z}\in{[0, \rho ]}\) and \(\mathfrak{w}_{i} \in{\mathbb{R}}\), \(i=1, 2\).

  2. (K2)

    \(\mathcal{Q}_{j}\in{\mathcal{M}_{1-\mathfrak{X}, \xi}([t_{i}, s_{i}] \times \mathbb{R}, \mathbb{R})}\), and there exist \(\mathcal{L}_{\mathcal{Q}_{j}} > 0\), \(j= 1,2 ,\ldots,m\), such that

    $$\begin{aligned} \bigl\vert \mathcal{Q}_{j}(\mathfrak{z}, \mathfrak{w}_{1})- \mathcal{Q}_{j}( \mathfrak{z}, \mathfrak{w}_{2}) \bigr\vert \leq \mathcal{L}_{\mathcal{Q}_{j}} \vert \mathfrak{w}_{1} - \mathfrak{w}_{2} \vert \end{aligned}$$
    (3.2)

    for all \(\mathfrak{z}\in{[t_{j}, s_{j}]}\) and \(\mathfrak{w}_{i} \in{\mathbb{R}}\), \(i=1, 2\).

Theorem 3

Assume that both conditions \((K1)\) and \((K2)\) are fulfilled. Also, assume that

$$\begin{aligned} M_{\mathcal{LQ}}:=\max_{j=1,2,\ldots, m} \biggl( \mathcal{L}_{{ \mathcal{Q}}_{j}} ^{\alpha }+ \biggl[ \frac {(\rho \mathcal{L}_{\mathcal{J}} + \Vert \mathcal{K} \Vert )\Gamma (\mathfrak{X}) (\xi (\rho )- \xi (0))^{\lambda}}{\Gamma (\lambda +1)} \biggr]^{\alpha } \biggr) < 1. \end{aligned}$$
(3.3)

Let \(v: [0, \rho ] \rightarrow \mathbb{R}\) be a function belonging to such that for \(\epsilon \geq 0\),

$$ \biggl\vert {}^{\mathcal{H}}{\mathbb{D}}^{\lambda , \varrho ; \xi}_{0^{+}} v( \mathfrak{z}) - \mathcal{K}\bigl(v(\mathfrak{z})\bigr)- \int _{0}^{\mathfrak{z}} \mathcal{J}\bigl(\mathfrak{z}, s, v(s)\bigr) \,ds \biggr\vert \leq \epsilon \Xi \bigl(\lambda , \varrho ; \bigl( \xi (\mathfrak{z})-\xi (0)\bigr)^{\lambda}\bigr) $$

for all \(\mathfrak{z}\in{(s_{j}, t_{j+1}]}\), \(j=0, 1, \ldots, m\), and

$$\begin{aligned} \bigl\vert v(\mathfrak{z})- \mathcal{Q}_{j}\bigl( \mathfrak{z}, v\bigl(\mathfrak{z}_{j}^{+}\bigr)\bigr) \bigr\vert \leq \xi \end{aligned}$$
(3.4)

for some positive ξ and for all \(\mathfrak{z}\in{(t_{j}, s_{j}]}\), \(j=1, \ldots, m\). Then there exists a unique continuous function \(\mathfrak{v}_{1}: [0, \rho ] \rightarrow \mathbb{R}\) satisfying equation (1.1) such that

$$ \bigl\vert v(\mathfrak{z}) - \mathfrak{v}_{1}( \mathfrak{z}) \bigr\vert \leq \frac { \epsilon \Xi (\lambda , \varrho ; (\xi (\mathfrak{z})-\xi (0))^{\lambda})}{1- \varpi}, $$
(3.5)

where

$$\begin{aligned} \varpi = \max_{j=1,2,\ldots, m} \biggl( \mathcal{L}_{{\mathcal{Q}}_{j}} ^{ \alpha }+ \biggl[ \frac {(\rho \mathcal{L}_{\mathcal{J}} + \Vert \mathcal{K} \Vert )\Gamma (\mathfrak{X}) (\xi (\rho )- \xi (0))^{\lambda}}{\Gamma (\lambda +1)} \biggr]^{\alpha } \biggr) \end{aligned}$$

for all \(\mathfrak{z}\in{[0, \rho ]}\).

Proof

Let \(\tilde{\mathfrak{D}}:=\mathcal{PM}_{1- \gamma , \xi} [0, \rho ] \) be endowed with the generalized metric

$$\begin{aligned} \tilde{\mathfrak{d}}(\zeta _{1}, \zeta _{2})&= \inf \bigl\lbrace \mathfrak{C}\in{[0, \infty ]}: \\ & \quad \bigl\vert \zeta _{1}( \mathfrak{z})- \zeta _{2}( \mathfrak{z}) \bigr\vert ^{\alpha } \leq \mathfrak{C} \epsilon \Xi \bigl(\lambda , \varrho ; \bigl(\xi (\mathfrak{z})- \xi (0)\bigr)^{\lambda}\bigr)\text{ for all } \mathfrak{z} \in{[0, \rho ]} \bigr\rbrace \end{aligned}$$
(3.6)

for \(\zeta _{1}, \zeta _{2} \in{\tilde{\mathfrak{D}}}\), where \(\alpha \in{(0,1]}\). It is not difficult to see that \((\tilde{\mathfrak{D}}, \tilde{\mathfrak{d}})\) is a complete generalized metric space [1]. Now we define the operator \(\Psi : \mathcal{PM}_{1- \mathfrak{X}, \xi} [0, \rho ] \rightarrow \mathcal{PM}_{1- \mathfrak{X}, \xi} [0, \rho ]\) as follows:

$$ (\Psi v) (\mathfrak{z}) = \textstyle\begin{cases} \frac{(\xi (\mathfrak{z})- \xi (0))^{\mathfrak{X}-1}}{\Gamma (\mathfrak{X})} \mathfrak{v}_{0} + I^{\lambda ; \xi}_{0^{+}} [ {\mathcal{K}}(v( \mathfrak{z}))+ \int _{0}^{\mathfrak{z}} {\mathcal{J}}(\mathfrak{z}, s, w(s)) \,ds ], & \eta \in{[0, t_{1}]}, \\ {\mathcal{Q}}_{j}(\mathfrak{z}, v(\mathfrak{z}_{j}^{+})), & \mathfrak{z}\in{(t_{j}, s_{j}]}, \\ {\mathcal{Q}}_{j}(s_{j}, v(\mathfrak{z}_{j}^{+})) + I^{\lambda ; \xi}_{0^{+}} [ {\mathcal{K}}(v(\mathfrak{z}))+ \int _{0}^{\mathfrak{z}} { \mathcal{J}}(\mathfrak{z}, s, v(s)) \,ds ], & \mathfrak{z}\in{(s_{j}, t_{j+1}]}, \end{cases} $$
(3.7)

for all \(j=1,2, \ldots, m\), where \(\lambda \in{(0, 1]}\), \(\varrho \in{[0, 1),}\) and \(\mathfrak{X}= \lambda + \varrho (1- \lambda )\). To prove that equation (1.1) has a unique solution, it suffices to show that (3.7) has a fixed point. To this aim, we use the Diaz–Margolis FPT. First, we will prove that \((\Psi v)(\mathfrak{z})\) is a strict contraction in three cases.

Case 1: For \(\mathfrak{z} \in{[0, t_{1}]}\) and arbitrary \(\mathfrak{v}_{i}\in{\tilde{\mathfrak{D}}}\), \(i= 1, 2\), and choose a constant \(\mathbb{K}_{1}\) such that \(\tilde{\mathfrak{d}}(\mathfrak{v}_{1}, \mathfrak{v}_{2}) \leq \mathbb{K}_{1}\), i.e., \(|\mathfrak{v}_{1}(\mathfrak{z})- \mathfrak{v}_{2}(\mathfrak{z})|^{ \alpha}\leq \mathbb{K}_{1} \epsilon \Xi (\lambda , \varrho ; (\xi ( \mathfrak{z})-\xi (0))^{\lambda})\). So, applying hypotheses \((K1)\) and \((K2)\), we have

$$\begin{aligned} & \bigl\vert (\Psi \mathfrak{v}_{1}) (\mathfrak{z}) - (\Psi \mathfrak{v}_{2}) ( \mathfrak{z}) \bigr\vert ^{\alpha} \\ &\quad \leq \biggl[ \frac {1}{\Gamma (\lambda )} \int _{0}^{\mathfrak{z}} \frac {\xi '(s)}{(\xi (\mathfrak{z})- \xi (s))^{\lambda}} \biggl[ \int _{0}^{s} \bigl\vert \mathcal{J}\bigl( \mathfrak{z}, s, \mathfrak{v}_{1}( \mathfrak{z})\bigr) - \mathcal{J} \bigl(\mathfrak{z}, s, \mathfrak{v}_{2}( \mathfrak{z})\bigr) \bigr\vert \,d\mathfrak{z} \biggr] \,ds \\ &\qquad {}+\frac { \Vert \mathcal{K} \Vert }{\Gamma (\lambda )} \int _{0}^{\mathfrak{z}} \frac {\xi '(s)}{(\xi (\mathfrak{z})- \xi (s))^{\lambda}} \bigl\vert \mathfrak{v}_{1}(s)- \mathfrak{v}_{2}(s) \bigr\vert \,ds \biggr]^{\alpha } \\ &\quad \leq \biggl[ \frac {(\rho \mathcal{L}_{\mathcal{J}} + \Vert \mathcal{K} \Vert )\mathbb{K}_{1} \epsilon}{\Gamma (\lambda )} \int _{0}^{\mathfrak{z}} \frac {\xi '(s)}{(\xi (\mathfrak{z})- \xi (s))^{\lambda}} \Xi \bigl( \lambda , \varrho ; \bigl(\xi (\mathfrak{z})-\xi (0)\bigr)^{\lambda}\bigr) \biggr]^{ \alpha } \\ &\quad \leq \biggl[ \frac {(\rho \mathcal{L}_{\mathcal{J}} + \Vert \mathcal{K} \Vert )\Gamma (\mathfrak{X}) (\xi (\rho )- \xi (0))^{\lambda}}{\Gamma (\lambda +1)} \biggr]^{\alpha } \mathbb{K}_{1} \epsilon \Xi \bigl(\lambda , \varrho ; \bigl(\xi ( \mathfrak{z})-\xi (0)\bigr)^{\lambda}\bigr) \end{aligned}$$

for all \(\mathfrak{v}_{1}, \mathfrak{v}_{2}\in{(\tilde{\mathfrak{D}}, \tilde{\mathfrak{d}})}\) and \(\eta \in{[0, t_{1}]}\).

Case 2: For \(\mathfrak{z}\in{(t_{j}, s_{j}],}\) choose a constant \(\mathbb{K}_{2}\) such that \(\tilde{\mathfrak{d}}(\mathfrak{v}_{1}, \mathfrak{v}_{2}) \leq \mathbb{K}_{2}\), i.e., \(|\mathfrak{v}_{1}(\mathfrak{z})- \mathfrak{v}_{2}(\mathfrak{z})|^{ \alpha }\leq \mathbb{K}_{2} \epsilon \Xi (\lambda , \varrho ; (\xi ( \mathfrak{z})-\xi (0))^{\lambda})\). Then by hypothesis \((K3)\) and equation (3.7) we have

$$\begin{aligned} \bigl\vert (\Xi \mathfrak{v}_{1}) (\mathfrak{z}) - (\Xi \mathfrak{v}_{2}) ( \mathfrak{z}) \bigr\vert ^{\alpha } \leq& \bigl\vert {\mathcal{Q}}_{j}\bigl(\mathfrak{z}, \mathfrak{v}_{1} \bigl(\mathfrak{z}_{j}^{+}\bigr)\bigr) - { \mathcal{Q}}_{j}\bigl( \mathfrak{z}, \mathfrak{v}_{2}\bigl( \mathfrak{z}_{j}^{+}\bigr)\bigr) \bigr\vert ^{\alpha } \\ \leq& \mathcal{L}_{{\mathcal{Q}}_{j}} ^{\alpha }\mathbb{K}_{2} \epsilon \Xi \bigl(\lambda , \varrho ; \bigl(\xi (\mathfrak{z})-\xi (0) \bigr)^{ \lambda}\bigr) \end{aligned}$$

for all \(\mathfrak{v}_{1}, \mathfrak{v}_{2}\in{(\tilde{\mathfrak{D}}, \tilde{\mathfrak{d}})}\) and \(\mathfrak{z}\in{(t_{j}, s_{j}]}\).

Case 3: For \(\mathfrak{z}\in{(s_{j}, t_{j+1}],}\) choose a constant \(\mathbb{K}_{3}\) such that \(\tilde{\mathfrak{d}}(\mathfrak{v}_{1}, \mathfrak{v}_{2}) \leq \mathbb{K}_{3}\), i.e., \(|\mathfrak{v}_{1}(\mathfrak{z})- \mathfrak{v}_{2}(\mathfrak{z})|^{ \alpha }\leq \mathbb{K}_{3} \epsilon \Xi (\lambda , \varrho ; (\xi ( \mathfrak{z})-\xi (0))^{\lambda})\), and then by hypotheses \((K1)\) and \((K3)\) and equation (3.7) we have

$$\begin{aligned} & \bigl\vert (\Psi \mathfrak{v}_{1}) (\mathfrak{z}) - (\Psi \mathfrak{v}_{2}) ( \mathfrak{z}) \bigr\vert ^{\alpha} \\ &\quad \leq \biggl[ {\mathcal{Q}}_{j}\bigl(s_{j}, \mathfrak{v}_{1} \bigl(\mathfrak{z}_{j}^{+}\bigr) \bigr) - {\mathcal{Q}}_{j}\bigl(s_{j}, \mathfrak{v}_{2} \bigl(\mathfrak{z}_{j}^{+}\bigr)\bigr) \\ &\qquad {}+ \frac {1}{\Gamma (\lambda )} \int _{0}^{\mathfrak{z}} \frac {\xi '(s)}{(\xi (\mathfrak{z})- \xi (s))^{\lambda}} \biggl[ \int _{0}^{s} \bigl\vert \mathcal{J}\bigl( \mathfrak{z}, s, \mathfrak{v}_{1}( \mathfrak{z})\bigr) - \mathcal{J} \bigl(\mathfrak{z}, s, \mathfrak{v}_{2}( \mathfrak{z})\bigr) \bigr\vert \,d\mathfrak{z} \biggr] \,ds \\ &\qquad {}+\frac { \Vert \mathcal{K} \Vert }{\Gamma (\lambda )} \int _{0}^{\mathfrak{z}} \frac {\xi '(s)}{(\xi (\mathfrak{z})- \xi (s))^{\lambda}} \bigl\vert \mathfrak{v}_{1}(s)- \mathfrak{v}_{2}(s) \bigr\vert \,ds \biggr]^{\alpha } \\ &\quad \leq \biggl[ \mathcal{L}_{\mathcal{Q}_{j}} \vert \mathfrak{v}_{1} - \mathfrak{v}_{2} \vert \\ &\qquad {}+ \frac {(\rho \mathcal{L}_{\mathcal{J}} + \Vert \mathcal{K} \Vert )\mathbb{K}_{3} \epsilon}{\Gamma (\lambda )} \int _{0}^{\mathfrak{z}} \frac {\xi '(s)}{(\xi (\mathfrak{z})- \xi (s))^{\lambda}} \Xi \bigl( \lambda , \varrho ; \bigl(\xi (\mathfrak{z})-\xi (0)\bigr)^{\lambda}\bigr) \biggr]^{ \alpha } \\ &\quad \leq \biggl[ \mathcal{L}_{\mathcal{Q}_{j}} + \frac {(\rho \mathcal{L}_{\mathcal{J}} + \Vert \mathcal{K} \Vert )\Gamma (\mathfrak{X}) (\xi (\rho )- \xi (0))^{\lambda}}{\Gamma (\lambda +1)} \biggr]^{\alpha }\mathbb{K}_{3} \epsilon \Xi \bigl(\lambda , \varrho ; \bigl(\xi ( \mathfrak{z})-\xi (0)\bigr)^{\lambda}\bigr). \end{aligned}$$

Now applying the results of Cases 1–3, we obtain that

$$\begin{aligned} \bigl\vert (\Psi \mathfrak{v}_{1}) (\mathfrak{z}) - (\Psi \mathfrak{v}_{2}) ( \mathfrak{z}) \bigr\vert ^{\alpha } \leq \varpi \mathbb{K}' \epsilon \Xi \bigl( \lambda , \varrho ; \bigl(\xi (\mathfrak{z})-\xi (0)\bigr)^{\lambda}\bigr), \end{aligned}$$
(3.8)

where \(\mathbb{K}'=\max_{i=1,2,3} \mathbb{K}_{i}\), and

$$\begin{aligned} \varpi = \max_{j=1,2,\ldots, m} \biggl( \mathcal{L}_{{\mathcal{Q}}_{j}} ^{ \alpha }+ \biggl[ \frac {(\rho \mathcal{L}_{\mathcal{J}} + \Vert \mathcal{K} \Vert )\Gamma (\mathfrak{X}) (\xi (\rho )- \xi (0))^{\lambda}}{\Gamma (\lambda +1)} \biggr]^{\alpha } \biggr). \end{aligned}$$

Inequality (3.8) indicates that

$$\begin{aligned} \begin{aligned}[b] &\tilde{\mathfrak{d}}\bigl((\Psi \mathfrak{v}_{1}), ( \Psi \mathfrak{v}_{2})\bigr) \\ &\quad \leq \max_{j=1,2,\ldots, m} \biggl( \mathcal{L}_{{\mathcal{Q}}_{j}} ^{ \alpha }+ \biggl[ \frac {( \rho \mathcal{L}_{\mathcal{J}} + \Vert \mathcal{K} \Vert )\Gamma (\mathfrak{X}) (\xi (\rho )- \xi (0))^{\lambda}}{\Gamma (\lambda +1)} \biggr]^{\alpha } \biggr) \tilde{\mathfrak{d}}(\mathfrak{v}_{1}, \mathfrak{v}_{2}). \end{aligned} \end{aligned}$$
(3.9)

Now, (3.3) and (3.9) show that Ψ is a strictly contractive operator on \([0, \rho ]\). Let \(\mathfrak{v}'\in{(\tilde{\mathfrak{D}}, \tilde{\mathfrak{d}})}\), so that \(\Psi \mathfrak{v}'\in{(\tilde{\mathfrak{D}}, \tilde{\mathfrak{d}}),}\) From the piecewise continuity property, using Remark 1, we have

$$\begin{aligned} & \bigl\vert \bigl(\Psi \mathfrak{v}'\bigr) (\mathfrak{z}) - \mathfrak{v}'(\mathfrak{z}) \bigr\vert ^{ \alpha } \\ &\quad \leq \biggl\vert \mathfrak{v}'(\mathfrak{z}) - \frac{(\xi (\mathfrak{z})- \xi (0))^{\mathfrak{X}-1}}{\Gamma (\mathfrak{X})} \mathfrak{w}_{0} \\ &\qquad {}- I^{\lambda ; \xi}_{0^{+}} \biggl[ \int _{0}^{s} \mathcal{J}\bigl(s, \mathfrak{z}, \mathfrak{v}'(\mathfrak{z})\bigr) \,d\mathfrak{z} - \mathcal{K}\bigl( \mathfrak{v}'(\mathfrak{z})\bigr) \biggr] \biggr\vert ^{\alpha} \\ &\quad \leq \bigl[\epsilon \Xi (\lambda , \varrho ; \bigl(\xi ( \mathfrak{z})-\xi (0)\bigr)\bigr]^{ \alpha } \end{aligned}$$

for all \(\mathfrak{z}\in{[0, t_{1}]}\). Also, by the same argument, using (3.4), we have

$$\begin{aligned} \bigl\vert \bigl(\Psi \mathfrak{v}'\bigr) (\mathfrak{z}) - \mathfrak{v}'(\mathfrak{z}) \bigr\vert ^{ \alpha }\leq \bigl\vert {\mathcal{Q}}_{j}\bigl(\mathfrak{z}, \mathfrak{v}' \bigl( \mathfrak{z}_{j}^{+}\bigr)\bigr) - \mathfrak{v}'(\mathfrak{z}) \bigr\vert ^{\alpha }\leq \bigl[ \xi \epsilon \Xi (\lambda , \varrho ; \bigl(\xi (\mathfrak{z})-\xi (0)\bigr) \bigr]^{ \alpha } \end{aligned}$$

for all \(\mathfrak{z}\in{(t_{j}, s_{j}]}\) and \(j= 1, 2, \ldots, m\). Furthermore,

$$\begin{aligned} \bigl\vert \bigl(\Psi \mathfrak{v}'\bigr) (\mathfrak{z}) - \mathfrak{v}'(\mathfrak{z}) \bigr\vert ^{ \alpha }&\leq \biggl\vert {\mathcal{Q}}_{j}\bigl(s_{j}, \mathfrak{v}' \bigl(\mathfrak{z}_{j}^{+}\bigr)\bigr) + I^{\lambda ; \phi}_{0^{+}} \biggl[ {\mathcal{K}}\bigl( \mathfrak{v}'( \mathfrak{z})\bigr)+ \int _{0}^{\mathfrak{z}}{\mathcal{J}}\bigl(\eta , s, \mathfrak{v}(\mathfrak{z})\bigr) \,ds \biggr] - \mathfrak{v}'( \mathfrak{z}) \biggr\vert ^{ \alpha}\\ &\leq \bigl[\xi + \epsilon \Xi (\lambda , \varrho ; \bigl(\xi (\mathfrak{z})- \xi (0)\bigr)\bigr]^{\alpha} \end{aligned}$$

for all \(\mathfrak{z}\in{(s_{j}, t_{j+1}]}\) and \(j=1, 2, \ldots, m\). So, in summary, \(\tilde{\mathfrak{d}}(\Psi \mathfrak{v}, \mathfrak{v}) \leq \infty \), and \(\tilde{\mathfrak{d}}(\Psi ^{\mathfrak{k}} \mathfrak{v}', \Psi ^{ \mathfrak{k}+1} \mathfrak{v}') < +\infty \) for all \(\mathfrak{k}\in{\mathbb{N}}\). According to Theorem 2, there exists a unique continuous function \(\mathfrak{v}^{*}: [0, \rho ] \rightarrow \mathbb{R}\) such that \(\Psi \mathfrak{v}^{*} = \mathfrak{v}^{*}\); \(\mathfrak{v}^{*}\) satisfies equation (1.1) for all \(\mathfrak{z}\in{[0, \rho ],}\) and

$$\begin{aligned} \mathfrak{v}^{*}(\mathfrak{z})= \frac{(\xi (\mathfrak{z})- \xi (0))^{\mathfrak{X}-1}}{\Gamma (\mathfrak{X})} \mathfrak{v}^{*}_{0} + I^{\lambda ; \xi}_{0^{+}} \biggl[ \int _{0}^{ \mathfrak{z}} \mathcal{J}\bigl(\mathfrak{z}, s, \mathfrak{v}^{*}(s)\bigr) \,ds + \mathcal{K}\bigl(\mathfrak{v}^{*}( \mathfrak{z})\bigr) \biggr], \end{aligned}$$

for every \(\mathfrak{z}\in{[0, \rho ]}\). Using Theorem 2 and (3.9), we have

$$\begin{aligned} \tilde{\mathfrak{d}}\bigl(\mathfrak{v}^{*}, \mathfrak{v}\bigr) \leq \frac {1}{1-\varpi} \end{aligned}$$

for all \(\mathfrak{z}\in{[0, \rho ]}\), where

$$\begin{aligned} \varpi = \max_{j=1,2,\ldots, m} \biggl( \mathcal{L}_{{\mathcal{Q}}_{j}} ^{ \alpha }+ \biggl[ \frac {(\rho \mathcal{L}_{\mathcal{J}} + \Vert \mathcal{K} \Vert )\Gamma (\mathfrak{X}) (\xi (\rho )- \xi (0))^{\lambda}}{\Gamma (\lambda +1)} \biggr]^{\alpha } \biggr). \end{aligned}$$

which justifies inequality (3.5). □

4 Application

Finally, we illustrate the above results by the following example.

Example 1

Let \(\Omega = [0, 1]\). Let \(v(\mathfrak{z})\) be a function on Ω, and let \(F: \Omega ^{2} \rightarrow \mathbb{R}\) be a real-valued continuous function such that \(|F(\mathfrak{z}, \gamma )v(\mathfrak{z})| < \frac{\Gamma (1/3)}{3\Gamma (2/9)}(e-1)^{-\frac{2}{9}}\). Consider the fractional system

$$ \textstyle\begin{cases} {}^{\mathcal{H}}{\mathbb{D}}^{\frac{1}{3}, \frac{2}{3}; e^{ \mathfrak{z}}}_{0^{+}} v(\mathfrak{z}) =\int _{0}^{1} F(\mathfrak{z}, \gamma ) v(\mathfrak{z}) \,d\mathfrak{z} + \int _{0}^{1} \sin ( \frac {3}{5}\mathfrak{z} v(s)) \,ds, & \mathfrak{z}\in{(0, \frac{1}{3}] \cup (\frac{2}{3},1]}, \\ v(\mathfrak{z})= \frac {1}{5} \sin (\mathfrak{z} v({\frac{1}{3}}^{+})) , & \mathfrak{z}\in{(\frac{1}{3}, \frac{2}{3}]}, \end{cases} $$
(4.1)

and \(v({\frac{1}{3}}^{+})=1+ v({\frac{1}{3}}^{-})\). Also, consider

$$ \textstyle\begin{cases} \vert {}^{\mathcal{H}}{\mathbb{D}}^{\frac{1}{3}, \frac{2}{3}; e^{\eta}}_{0^{+}} v(\mathfrak{z}) -\int _{0}^{1} F(\mathfrak{z}, \gamma ) v( \mathfrak{z}) \,d\mathfrak{z} \\ \quad {}- \int _{0}^{1} \sin ( \frac {3}{5} \mathfrak{z} v(s)) \,ds \vert \leq \epsilon \Xi , & \mathfrak{z} \in{(s_{j}, t_{j+1}]},j=0, 1, \ldots, m, \\ |v(\mathfrak{z})- \frac {1}{5} \sin (\mathfrak{z} v({\frac{1}{3}}^{+}))| \leq 1, & \mathfrak{z}\in{(t_{j}, s_{j}]},j= 1, \ldots, m, \end{cases} $$
(4.2)

where \(\Xi =\Xi (\frac{1}{3}, \frac{2}{3}; (e^{\mathfrak{z}}-1)^{ \frac{1}{3}})\). For all \(\mathfrak{z}\in{\Omega ,}\) a continuous real-valued function v on Ω, and \(v_{1}, v_{2}\in{\mathcal{PM}([0, \rho ], \mathbb{R}),}\) we have

$$\begin{aligned} \biggl\vert \sin \biggl( \frac {2}{3}\mathfrak{z} v_{1}(s) \biggr)- \sin \biggl( \frac {2}{3} \mathfrak{z} v_{2}(s)\biggr) \biggr\vert \leq \frac {2}{3} \bigl\vert \bigl(v_{1}( \mathfrak{z})- v_{2}( \mathfrak{z})\bigr) \bigr\vert . \end{aligned}$$

Furthermore, by assumption the operator \(3 \int _{0}^{1} F(\mathfrak{z}, \gamma ) v(\mathfrak{z}) \,d \mathfrak{z}\) is bounded, and we have \(|\int _{0}^{1} F(\mathfrak{z}, \gamma ) v(\mathfrak{z}) \,d \mathfrak{z}| \leq \frac{\Gamma (4/3)}{\Gamma (2/9)(e - 1)^{2/9} }\) for all \(\mathfrak{z}, \gamma \in{\Omega}\) and the function v. So hypotheses \((K1)\) and \((K2)\) are satisfied for \(\mathcal{L}_{\mathcal{J}}= \frac{2}{3}\), and \(\mathcal{L}_{\mathcal{Q}}= \frac{1}{5}\). Therefore by Theorem 2 equation (4.1) has at least one solution \(v: [0, 1] \rightarrow \mathbb{R}\) such that

$$ v(\mathfrak{z}) = \textstyle\begin{cases} \frac{(e^{\mathfrak{z}}- 1)^{-2/9}}{\Gamma (7/9)}\mathfrak{v}_{0}\\ \quad {}+ I^{ \lambda ; \xi}_{0^{+}} [ \int _{0}^{1} F(\mathfrak{z}, \gamma ) v( \mathfrak{z}) \,d\mathfrak{z} + \int _{0}^{1} \sin ( \frac {3}{5} \mathfrak{z} v(s)) \,ds \,ds ], & \mathfrak{z}\in{(0, \frac{1}{3}] \cup (\frac{2}{3},1]}, \\ \frac {1}{5} \sin (\mathfrak{z} v({\frac{1}{3}}^{+})) , & \mathfrak{z}\in{(\frac{1}{3}, \frac{2}{3}]}. \end{cases} $$
(4.3)

Example 2

Let \(\Omega = [0, 3]\) and consider the fractional system

$$ \textstyle\begin{cases} {}^{\mathcal{H}}{\mathbb{D}}^{\frac{1}{2}, \frac{1}{3}; \mathfrak{z}}_{0^{+}} v(\mathfrak{z}) = \sin (0.1 v(\mathfrak{z})) + \int _{0}^{ \mathfrak{z}} \frac{1}{7} \cos (\mathfrak{z} v(s)) \,ds, & \mathfrak{z} \in{[1, 3]}, \\ v(\mathfrak{z})= \frac{1}{100} \sin{ (\Xi (1/2, 1/3, 2/3, \mathfrak{z})-1)} , & \mathfrak{z}\in{[0, 1)}. \end{cases} $$
(4.4)

We also have

$$ \textstyle\begin{cases} \vert {}^{\mathcal{H}}{\mathbb{D}}^{\frac{1}{2}, \frac{1}{3}; \mathfrak{z}}_{0^{+}} v(\mathfrak{z}) - \sin (0.1 v(\mathfrak{z})) - \int _{0}^{ \mathfrak{z}} \frac{1}{7} \cos (\mathfrak{z} v(s)) \,ds \vert \leq \epsilon \Xi , & \mathfrak{z} \in{(1, 3]}, \\ \vert v(\mathfrak{z})- \frac{1}{100} \sin{ (\Xi (1/2, 1/3, 2/3, \mathfrak{z})-1)} \vert \leq 1, & \mathfrak{z}\in{[0, 1)}, \end{cases} $$
(4.5)

where \(\Xi =\Xi (\frac{1}{2}, \frac{1}{3}; (\mathfrak{z}-1)^{\frac{1}{3}})\). For all \(\mathfrak{z}\in{\Omega ,}\) a continuous real-valued function v on Ω, and \(v_{1}, v_{2}\in{\mathcal{PM}([0, \rho ], \mathbb{R}),}\) we have

$$\begin{aligned} \bigl\vert \sin \bigl( 0.1 \eta v_{1}(s)\bigr)- \sin \bigl( 0.1 \mathfrak{z} v_{2}(s)\bigr) \bigr\vert \leq \frac {1}{10} \bigl\vert \bigl(v_{1}(\mathfrak{z})- v_{2}(\mathfrak{z})\bigr) \bigr\vert . \end{aligned}$$

The same calculation is correct for \(\int _{0}^{\mathfrak{z}} \cos (\mathfrak{z} v(s)) \,ds\). So hypotheses \((K1)\) and \((K2)\) are satisfied for \(\mathcal{L}_{\mathcal{J}} = \frac{3}{7}\) and \(\mathcal{L}_{\mathcal{Q}}= \frac{1}{100}\). Also, taking \(\alpha = 0.5\), we have \(M_{\mathcal{LQ}}\approx 0.9513 < 1\) (hypothesis (3.3)). Therefore by Theorem 2 equation (4.4) has at least one solution \(v: [0, 1] \rightarrow \mathbb{R}\) such that

$$ v(\mathfrak{z}) = \textstyle\begin{cases} \frac{(\mathfrak{z}- 1)^{-1/3}}{\Gamma (2/3)}\mathfrak{v}_{0} + I^{1/2; \mathfrak{z}}_{0^{+}} [ \sin (0.1 v(\mathfrak{z})) - \int _{0}^{ \mathfrak{z}} \frac{1}{7} \cos (\mathfrak{z} v(s)) \,ds ], & \mathfrak{z}\in{(1, 3]}, \\ \frac{1}{100} \sin{ (\Xi (1/2, 1/3, \mathfrak{z})-1)} , & \mathfrak{z}\in{[0, 1]}. \end{cases} $$

Example 3

Let \(\Omega = [0, 3]\) and consider the fractional system

$$ \textstyle\begin{cases} {}^{\mathcal{H}}{\mathbb{D}}^{\frac{1}{2}, \frac{1}{3}; \mathfrak{z}}_{0^{+}} v(\mathfrak{z}) = \frac {1}{3+\Xi (1/2, 1/3, {\mathfrak{z}}^{1/2})}, & \mathfrak{z}\in{(1, 3]}, \\ v(\mathfrak{z})= \frac{1}{100} \sin{ (\Xi (1/2, 1/3, \mathfrak{z})-1)} , & \mathfrak{z}\in{[0, 1]}, \end{cases} $$
(4.6)

Also, we have

$$ \textstyle\begin{cases} \vert {}^{\mathcal{H}}{\mathbb{D}}^{\frac{1}{2}, \frac{1}{3}; \mathfrak{z}}_{0^{+}} v(\mathfrak{z}) - \frac{1}{3+ \Xi (1/2, 1/3, {\mathfrak{z}}^{1/2})} \vert \leq \epsilon \Xi , & \mathfrak{z} \in{[1, 3]}, \\ \vert v(\mathfrak{z})- \frac{1}{100} \sin{ (\Xi (1/2, 1/3, \mathfrak{z})-1)} \vert \leq 1, & \mathfrak{z}\in{[0, 1)}, \end{cases} $$

where \(\Xi =\Xi (\frac{1}{2}, \frac{1}{3}; \mathfrak{z}^{\frac{1}{2}})\). Let \(\Omega = [0, 3]\) and \(\alpha = 1/3\). The inequalities \(|\frac{1}{3+ \Xi (1/2, 1/3, {\mathfrak{z}}^{1/2})}| \leq 0.386\) for \(\mathfrak{z} \in{[0, 2)}\) and \(\frac{1}{100} \sin{ (\Xi (1/2, 1/3, \mathfrak{z})-1)} \leq 0.01 \) for \(\mathfrak{z} \in{[2, 3]}\) mean that \(\mathcal{L}_{\mathcal{J}}= 0.4 \) and \(\mathcal{L}_{\mathcal{Q}_{1}} = 0.01\). So hypotheses \((K1)\) and \((K2)\) are satisfied. Also, taking \(\alpha = 0.5\), we have \(M_{\mathcal{LQ}}\approx 0.3515 < 1\) (hypothesis (3.3)). Therefore by Theorem 2 equation (4.6) has at least one solution \(v: [0, 1] \rightarrow \mathbb{R}\) such that

$$ v(\mathfrak{z}) = \textstyle\begin{cases} \frac{(\mathfrak{z}- 1)^{-1/3}}{\Gamma (2/3)}\mathfrak{v}_{0} + I^{1/2; \mathfrak{z}}_{0^{+}} [ \frac {1}{3+\Xi (1/2, 1/3, {\mathfrak{z}}^{1/2})} ], & \mathfrak{z}\in{[1, 3]}, \\ \frac{1}{100} \sin{ (\Xi (1/2, 1/3, \mathfrak{z})-1)} , & \mathfrak{z}\in{[0, 1)}. \end{cases} $$

In the Fig. 1, we show the solution of equation (4.6), where \(V1= \frac{(\mathfrak{z}- 1)^{-1/3}}{\Gamma (2/3)}\mathfrak{v}_{0} + I^{1/2; \mathfrak{z}}_{0^{+}} [ \frac {1}{3+\Xi (1/2, 1/3, {\mathfrak{z}}^{1/2})} ]\), and \(V2=\frac{1}{100} \sin{ (\Xi (1/2, 1/3, \mathfrak{z})-1)} \):

Figure 1
figure 1

The solution of equation (4.6)

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Acknowledgements

The authors are thankful to the area editor for giving valuable comments and suggestions.

Funding

This research does not receive specific funding. The corresponding author is a full-time member of the School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran.

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Contributions

M.B.G., methodology. F.M., writing–original draft preparation. R.S., supervision and project administration. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Reza Saadati.

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Ghaemi, M.B., Mottaghi, F. & Saadati, R. α-Confluent-hyper-geometric stability of ξ-Hilfer impulsive nonlinear fractional Volterra integro-differential equation. Bound Value Probl 2023, 4 (2023). https://doi.org/10.1186/s13661-023-01694-6

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  • DOI: https://doi.org/10.1186/s13661-023-01694-6

MSC

  • 34N05
  • 26A33
  • 26E70
  • 34A08

Keywords

  • ξ-Hilfer fractional derivative
  • Impulsive fractional differential equation
  • α-Confluent-hyper-geometric stability
  • Diaz–Margolis fixed point theorem