- Research
- Open access
- Published:
α-Confluent-hyper-geometric stability of ξ-Hilfer impulsive nonlinear fractional Volterra integro-differential equation
Boundary Value Problems volume 2023, Article number: 4 (2023)
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 [3–5, 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 pβ-normed space:
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:
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
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
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:
-
(I1)
\(\| \mathfrak{z}\|_{\alpha }=0\) if and only if, \(\mathfrak{z}=0\),
-
(I2)
\(\| k\mathfrak{z}\|_{\alpha}= |k|^{\alpha }\|\mathfrak{z}\|_{\alpha}\) for all \(k\in{\mathbb{K}}\) and \(\mathfrak{z}\in{\mathcal{X}}\),
-
(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
with norm
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:
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
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
and
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
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
and
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
-
(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 \(\mathfrak{O}^{*}= \lbrace \tilde{o}\in{\mathfrak{O}} : \tilde{d}( \Pi ^{\mathfrak{k}} \tilde{o}, \Pi ^{\mathfrak{k}} \tilde{o}) < \infty \rbrace \),
-
(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
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
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 \(\epsilon > 0 \),
where \(\mathcal{C}_{\Xi} \) is a positive constant.
Remark 1
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:
-
(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 \(\mathfrak{w}_{i} \in{\mathbb{R}}\), \(i=1, 2\).
-
(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
Let \(v: [0, \rho ] \rightarrow \mathbb{R}\) be a function belonging to W̃ such that for \(\epsilon \geq 0\),
for all \(\mathfrak{z}\in{(s_{j}, t_{j+1}]}\), \(j=0, 1, \ldots, m\), and
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
where
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
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:
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
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
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
Now applying the results of Cases 1–3, we obtain that
where \(\mathbb{K}'=\max_{i=1,2,3} \mathbb{K}_{i}\), and
Inequality (3.8) indicates that
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
for all \(\mathfrak{z}\in{[0, t_{1}]}\). Also, by the same argument, using (3.4), we have
for all \(\mathfrak{z}\in{(t_{j}, s_{j}]}\) and \(j= 1, 2, \ldots, m\). Furthermore,
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
for every \(\mathfrak{z}\in{[0, \rho ]}\). Using Theorem 2 and (3.9), we have
for all \(\mathfrak{z}\in{[0, \rho ]}\), where
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
and \(v({\frac{1}{3}}^{+})=1+ v({\frac{1}{3}}^{-})\). Also, consider
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
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
Example 2
Let \(\Omega = [0, 3]\) and consider the fractional system
We also have
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
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
Example 3
Let \(\Omega = [0, 3]\) and consider the fractional system
Also, we have
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
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 \(V2=\frac{1}{100} \sin{ (\Xi (1/2, 1/3, \mathfrak{z})-1)} \):
Availability of data and materials
There are no data that we needed for this paper.
References
Aderyani, S.R., Saadati, R., Feckan, M.: The Cadariu–Radu method for existence, uniqueness and Gauss hypergeometric stability of Ω-Hilfer fractional differential equations. Mathematics 2021, 9 (2021)
Diaz, J.B., Margolis, B.: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305–309 (1968)
Jung, S.M.: Hyers–Ulam stability of linear differential equations of first order (III). J. Math. Anal. Appl. 311, 139–146 (2005)
Jung, S.M.: Hyers–Ulam stability of linear differential equations of first order (II). Appl. Math. Lett. 19, 854–858 (2006)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Equations. Elsevier, Amsterdam (2006)
Lakshmikantham, V., Simeonov, P.S.: Theory of Impulsive Differential Equations, vol. 6. World Scientific, Singapore (1989)
Lin, Z., Wei, W., Wang, J.: Existence and stability results for impulsive integro-differential equations. Facta Univ., Ser. Math. Inform. 29(2), 119–130 (2014)
Liu, K., Wang, J., O’Regan, D.: Ulam–Hyers–Mittag-Leffler stability for ψ-Hilfer fractional-order delay differential equations. Adv. Differ. Equ. 2019(1), 50 1–12 (2019)
Mottaghi, F., Li, C., Abdeljawad, T., Saadati, R., Ghaemi, M.B.: Existence and Kummer stability for a system of nonlinear ϕ-Hilfer fractional differential equations with application. Fractal Fract. 5(4), 200 (2021)
Norouzi, F., N’Guerekata, G.M.: A study of ϕ-Hilfer fractional differential system with application in financial crisis. Chaos Solitons Fractals X 6, 100056 (2021)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon & Breach, Yverdon (1993). 44
Shah, K., Wang, J., Khalil, H., Khan, R.A.: Existence and numerical solutions of a coupled system of integral BVP for fractional differential equations. Adv. Differ. Equ. 2018, 149 (2018)
Sousa, J.V.C., De Oliveira, E.C.: On the ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)
Sousa, J.V.C., Kucche, K.D., De Oliveira, E.C.: Stability of ψ-Hilfer impulsive fractional differential equations. Appl. Math. Lett. 88, 73–80 (2019)
Sousa, J.V.D.C., Oliveira, E.C.: The Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the ϕ-Hilfer operator. J. Fixed Point Theory Appl. 20, 96 (2018)
Wang, G., Zhou, M., Sun, L.: Hyers–Ulam stability of linear differential equations of first order. Appl. Math. Lett. 21, 1024–1028 (2008)
Wang, J., Ibrahim, A.G., O’Regan, D.: Topological structure of the solution set for fractional non-instantaneous impulsive evolution inclusions. J. Fixed Point Theory Appl. 20, Article ID 59 (2018)
Wang, J., Li, X.: A uniform method to Ulam–Hyers stability for some linear fractional equations. Mediterr. J. Math. 13, 625–635 (2016)
Wang, J., Lin, Z., Zhou, Y.: On the stability of new impulsive ordinary differential equations. Topol. Methods Nonlinear Anal. 46(1), 303–314 (2015)
Wang, J., Zhang, Y.: Existence and stability of solutions to nonlinear impulsive differential equations in λ-normed spaces. Electron. J. Differ. Equ. 2014, 83, 1–10 (2014)
Wang, J., Zhang, Y.: A class of nonlinear differential equations with fractional integrable impulses. Commun. Nonlinear Sci. Numer. Simul. 19(2), 3001–3010 (2014)
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.
Author information
Authors and Affiliations
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
Ethics declarations
Competing interests
The authors declare no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-023-01694-6