Khawlah Hashim Hussain
© 2023 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
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This study intends to provide and prove a novel stability theorem for the non-linear Volterra-Fredholm integro-differential equation with Caputo fractional derivative using the weighted space method and fixed-point technique. Specifically, the study investigates the H-U-R stability and semi-U-H-R stability results. Eventually, the investigation discusses an example of the capability of this method.
Caputo fractional derivative, Volterra-Fredholm system, stability, Mittag-Leffler kernel
As a result of their frequent appearance in a wide range of engineering and scientific disciplines, systems of fractional differential and integral equations are currently the focus of active research [1]. A system of integral-differential equations must therefore have approximate solutions. Besides, fractional derivatives provide a powerful tool for many types of physical modeling, such as stochastic dynamical systems, electrodynamics of complex medium, plasma physics, signal processing, economics, and so on researches [2, 3].
Budak et al. [4] reported that the stability issue of differential equations solutions presented. One of the most essential topics in differential equation theory is Ulam-Hyers stability. Because of the broad scope of fractional calculus, many authors focused on the study of stability for fractional differential equations [5-8]. In the same regard, fractional integro-differential equations also drew the attention of several authors [9-16].
Chalishajar and Kumar [5] enhanced a new direction of research via studied the existence and uniqueness of the solutions as well as discussed two types of stability. In same regard, Khan et al. [7] used Perov's fixed point theorem and generalized metric space to derive some relaxed requirements for the uniqueness of positive solutions to the aforementioned problem. Dong et al. [9] investigated the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the random fractional integro-differential equation using the fixed point theorem.
The stability theory of fractional integro-differential equations is a significant branch of fractional calculus. The Ulam-type stability of an integro-differential equation implies that we can find the exact solution to the problem near an approximate solution. Several varieties of Ulam-type stability for nonlinear fractional integro-differential equations have been studied in recent decades [5, 7, 15-17].
Recently, Sevgin and Sevli [12] examined the U-H stability and the U-H-R stability in formulations of fixed-point techniques for the nonlinear Volterra equation:
$\Delta^{\prime}(v)=A(v, \Delta(v))+\int_0^v \Phi(v, \varsigma, \Delta(\varsigma)) d \varsigma$ (1)
Vu and Van Hoa [15] addressed the nonlinear IVP of the Volterra equations, and they used the successive approximation approach to explain the U-H and U-H-R stability of the following equations.
$\Delta^{\prime}(v)=A(v, \Delta(v))+\int_a^v \Phi(v, \varsigma, \Delta(\varsigma)) d \varsigma, v \in[a, b]$ (2)
$\Delta(a)=\Delta_0$ (3)
Sousa and De Oliveira [18, 19] introduced U-H stability for the Volterra -Hilfer fractional problem using the Banach fixed-point approach.
${ }^H D_{0+}^{\alpha, \beta ; \psi \psi} \Delta(v)=A(v, \Delta(v))+\int_0^v \Phi(v, \varsigma, \Delta(\varsigma)) d \varsigma$ (4)
where, ${ }^H D_{0+}^{\alpha, \beta ; \psi}$ is the ψ-Hilfer fractional derivative.
Herein, the current study is interested in the following Caputo fractional nonlinear Volterra-Fredholm integro-differential problem:
$\Theta^{\prime}(v)+{ }^c D_{0+}^\alpha \Theta(v)=g(v, \Theta(v))+\int_0^v \Upsilon(v, \varsigma, \Theta(\varsigma)) d \varsigma+\int_0^1 \Psi(v, \varsigma, \Theta(\varsigma)) d \varsigma, v \in[0,1]$ (5)
$\Theta(0)=\eta$ (6)
where, $\Theta \in C^1[0,1], 0<\alpha<1, \Upsilon, \Psi:[0,1] \times[0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ and $g:[0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ are continuous functions.
Motivated by the above studies, the current study will investigate another problem of stability theorem for the non-linear Volterra-Fredholm integro-differential equation with Caputo fractional derivative using the weighted space method and fixed-point technique.
Therefore, the aim of this work is to investigate the H-U, H-U-R, and semi-U-H-R stability for the system (5) under some new standards.
In this segment, the study introduces some useful preliminaries for fractional derivatives [20, 21]. Moreover, we recall concepts of stability for Eq. (5).
Let:
$\rho(\Theta, \omega)=\sup _{v \in[0,1]} \frac{|\Theta(v)-\omega(v)|}{\xi(v)}, \Theta, \omega \in C^1[0,1]$ (7)
The weighted metric, where function $\xi$ is a continuous non-decreasing defined as $\zeta:[0,1] \rightarrow(0,+\infty)$ then there is $\xi \in[0,1)$, satisfies:
$\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) \varsigma(\varsigma) d \varsigma \leq \xi \zeta(v)$ (8)
Obviously, (C1[0,1], ρ) is a complete metric space.
Definition 2.1 [20, 21] Let $f:(0,+\infty) \rightarrow \mathbb{R}$ be integrable function, the R-L fractional integral is given by:
$I_{0+}^\alpha f(\Lambda)=\frac{1}{\Gamma(\alpha)} \int_0^{\Lambda}(\Lambda-v)^{\alpha-1} f(v) d v, \Lambda>0,0<\alpha<1$ (9)
Definition 2.2 [20, 21] The left Caputo fractional derivative of differentiable function f(υ) is given by:
${ }^c D_{0+}^\alpha f(\Lambda)=I_{0+}^{1-\alpha} f^{\prime}(\Lambda)=\frac{1}{\Gamma(1-\alpha)} \int_0^{\Lambda}(\Lambda-v)^{-\alpha} f^{\prime}(v) d v, 0<\alpha<1$ (10)
Definition 2.3 [20, 21] The function of Mittag-Leffler is given by:
$E_{\alpha, \beta}(z)=\sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k \alpha+\beta)}, \beta, \alpha, z \in \mathbb{C}, \operatorname{Re}(\beta)>0, \operatorname{Re}(\alpha)>0$ (11)
The Laplace transform of the Mittag-Leffler and Caputo derivative given by:
$\mathcal{L}\left\{v^{\beta-1} E_{\alpha, \beta}\left( \pm a v^\alpha\right)\right\}(\varsigma)=\frac{\varsigma^{\alpha-\beta}}{\left(\varsigma^\alpha \mp a\right)}, \operatorname{Re}(\varsigma)>|a|^{\frac{1}{\alpha}}, \operatorname{Re}(\beta)>0, \operatorname{Re}(\alpha)>0, \beta, \alpha \in \mathbb{C}$ (12)
$\mathcal{L}\left\{v^{\alpha k+\beta-1} E_{\alpha, \beta}^{(k)}\left( \pm a v^\alpha\right)\right\}(\varsigma)=\frac{k ! ^{\alpha-\beta}}{\left(\varsigma^\alpha \mp a\right)^{k+1}}, \operatorname{Re}(\varsigma)>|a|^{\frac{1}{\alpha}}, \alpha, \beta \in \mathbb{C}$, (13)
where:
$E_{\alpha, \beta}^{(k)}(y)=\frac{d^k}{d y^k} E_{\alpha, \beta}(y)=\sum_{j=0}^{\infty} \frac{(j+k) ! y^j}{j ! \Gamma(\alpha j+\alpha k+\beta)}, k=0,1,2, \ldots$, (14)
and
$\left.\mathcal{L}^c D_{0+}^\alpha f(v)\right\}(\varsigma)=\varsigma^\alpha \tilde{f}(\varsigma)-\varsigma^{\alpha-1} f(0), 0<\alpha<1$, (15)
respectively [20, 21].
Definition 2.4 [20, 21] If Λ(v) is a given differential function, satisfying:
$\left|\Lambda^{\prime}(v)+{ }^c D_{0+}^\alpha \Lambda(v)-g(v, \Lambda(v))-\int_0^v \Upsilon(v, \varsigma, \Lambda(\varsigma)) d \varsigma-\int_0^1 \Psi(v, \varsigma, \Lambda(\varsigma)) d \varsigma\right| \leq \theta, v \in[0,1], \theta>0$ (16)
There is C>0 and Θ(v) is a solution of the system in Eq. (5), where:
$|\Lambda(v)-\theta(v)| \leq C \theta, v \in[0,1]$ (17)
Then, the system in Eq. (5) has the U-H stability.
If Λ(v) satisfies (16) there is a solution Θ(v) of the system (5) and C>0, such that:
$|\Lambda(v)-\Theta(v)| \leq C \phi(v), v \in[0,1]$ (18)
where, the function φ is continuous nonnegative defined as ϕ:[0,1]→(0,+∞), then the system (1) has the semi-U-H-R stability.
If ϕ:[0,1]→(0,+∞) is a continuous and Λ(v) satisfying:
$\left|\Lambda^{\prime}(v)+{ }^c D_{0+}^\alpha \Lambda(v)-g(v, \Lambda(v))-\int_0^v \Upsilon(v, \varsigma, \Lambda(\varsigma)) d \varsigma-\int_0^1 \Psi(v, \varsigma, \Lambda(\varsigma))\right| \leq \varphi(v)$ (19)
There is C>0 and Θ(v) is a solution of the system (5), where:
$|\Lambda(v)-\Theta(v)|<-C \varphi(v), v \in[0,1]$ (20)
Then, the system in Eq. (1) has the U-H-R stability.
The study will investigate in this segment the stabilities of U-H-R, semi-U-H-R and U-H for the system (1) in C1[0,1].
3.1 U-H-R stability for the system in Eq. (5)
Here, the study will investigate the equivalent integral equation of the system (5) and study the U-H-R stability for the system (5) in (C1[0,1], ρ).
Lemma 3.1 Assume that$f:[0,1] \rightarrow \mathbb{R}$ is a continuous function, and $0<\alpha<1, \Theta(v) \in C^1[0,1]$, the unique solution of the following equation.
$\Theta^{\prime}(v)+{ }^c D_{0+}^\alpha \Theta(v)=f(v), \Theta(0)=\eta$ (21)
is given by:
$\Theta(v)=\eta+\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) f(\varsigma) d \varsigma$ (22)
Proof: The Laplace transforms of both $\Theta^{\prime}(v)$ and ${ }^c D_{0+}^\alpha \Theta(v)$ exist for $\Theta(v) \in C^1[0,1]$, applying the Laplace transform on two sides of Eq. (22). Then,
$s \widetilde{\Theta}(\varsigma)-\eta+\varsigma^\alpha \widetilde{\Theta}(\varsigma)-\varsigma^{\alpha-1} \Theta(0)=f(\varsigma)$ (23)
$\widetilde{\Theta}(\varsigma)=\frac{1}{\varsigma} \eta+\frac{1}{\varsigma^\alpha+\varsigma} \tilde{f}(\varsigma)$ (24)
It can take the inverse Laplace transform on the both sides of Eq. (23), then get:
$\Theta(v)=\eta+\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) f(\varsigma) d \varsigma$ (25)
Then, Θ(v) satisfies Eq. (21) $\Leftrightarrow \Theta(v)$ satisfies Eq. (23). As a result, Eq. (23) is the equivalent integral equation of Eq. (21).
Theorem 3.2 Assume that a function ζ is continuous non-decreasing defined as ζ: [0,1] → (0,∞), and there exists ξ ∈ [0, 1), satisfying:
$\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) \zeta(\varsigma) d \varsigma \leq \xi \zeta(v)$ (26)
The following hypotheses are introduced:
[D1] Assume that a continuous function g defined as g: $[0,1] \times \mathbb{R} \rightarrow \mathbb{R}$, such that:
$\left|g\left(v, h_1\right)-g\left(v, h_2\right)\right| \leq \epsilon_1\left|h_1-h_2\right|, v \in[0,1], h_1, h_2 \in \mathbb{R}$ (27)
with є1>0.
[D2] Suppose that the kernels $\Upsilon, \Psi:[0,1] \times[0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ are continuous functions satisfying:
$\left|\Upsilon\left(v, \varsigma, h_1\right)-\Upsilon\left(v, \varsigma, h_2\right)\right| \leq \epsilon_2^k\left|h_1-h_2\right|, v, \varsigma \in[0,1], h_1, h_2 \in \mathbb{R}$ (28)
$\left|\Psi\left(v, \varsigma, h_1\right)-\Psi\left(v, \varsigma, h_2\right)\right| \leq \epsilon_2^h\left|h_1-h_2\right|$ (29)
with $\epsilon_2^k, \epsilon_2^h>0$. If $\Lambda \in C^1[0,1]$ satisfies:
$\left|\Lambda^{\prime}(v)+{ }^c D_{0+}^\alpha \Lambda(v)-g(v, \Lambda(v))-\int_0^v \Upsilon(v, \varsigma, \Lambda(\varsigma)) d \varsigma-\int_0^1 \Psi(v, \varsigma, \Lambda(\varsigma)) d \varsigma\right| \leq \zeta(v), v \in[0,1]$ (30)
and if:
$\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi<1$ (31)
Then we have Θ(v) is a solution of the system (5) satisfies:
$|\Lambda(v)-\Theta(v)| \leq \frac{\xi \zeta(v)}{1-\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi}, v \in[0,1]$ (32)
Proof: Applying Lemma 3.1, the equivalent equation of Eq. (5) is given by:
$\Theta(v)=\eta+\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right)\left[g(\varsigma, \Theta(\varsigma))+\int_0^{\varsigma} \Upsilon(\varsigma, \tau, \Theta(\tau)) d \tau+\int_0^1 \Psi(s, \tau, \Theta(\tau)) d \tau\right] d \varsigma$ (33)
Define the operator $\Omega: C^1[0,1] \rightarrow C^1[0,1]$ by:
$\begin{gathered}(\Omega \omega)(v)=\eta+\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) g(\varsigma, \omega(\varsigma)) d \varsigma \\ +\int_0^v \int_0^{\varsigma} E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) \Upsilon(\varsigma, \tau, \omega(\tau)) d \tau d \varsigma \\ +\int_0^v \int_0^1 E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) \Psi(\varsigma, \tau, \omega(\tau)) d \tau d \varsigma, v \in[0,1], \omega \in C^1[0,1]\end{gathered}$ (34)
From the hypotheses [D1] and [D2], so Ω is continuous.
Now, we will show that Ω is strictly contractive in (C1[0,1], ρ). By weighted metric ρ definition and Eqns. (26)-(28), for any $\omega, w \in C^1[0,1]$, it can obtain:
$\begin{aligned} \rho(\Omega \omega, \Omega w) \leq & \sup _{v \in[0,1]} \frac{\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right)[g(\varsigma, w(\varsigma))-g(s, w(\varsigma))] d \varsigma\right|}{\zeta(v)} \\ & +\sup _{v \in[0,1]} \frac{\left|\int_0^v \int_0^{\varsigma} E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right)[\gamma(\varsigma, \tau, \omega(\tau))-\Upsilon(\varsigma, \tau, w(\tau))] d \tau \mathrm{d} \varsigma\right|}{\varsigma(v)} \\ & +\sup _{v \in[0,1]} \frac{\left|\int_0^v \int_0^1 E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right)[\Psi(\varsigma, \tau, \omega(\tau))-\Psi(\varsigma, \tau, w(\tau))] d \tau \mathrm{d} \varsigma\right|}{\varsigma(v)} \\ & \leq \epsilon_1 \sup _{v \in[0,1]} \frac{\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right)\right| \omega(\varsigma)-w(\varsigma)|d \varsigma|}{\varsigma(v)} \\ & +\epsilon_2^k \sup _{v \in[0,1]} \frac{\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) \int_0^{\varsigma}\right| \omega(\tau)-w(\tau)|d \tau d \varsigma|}{\varsigma(v)} \\ & +\epsilon_2^h \sup _{v \in[0,1]} \frac{\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) \int_0^1\right| \omega(\tau)-w(\tau)|d \tau d \varsigma|}{\varsigma(v)} \\ & \leq c_1 \xi \rho(\omega, w)+c_2^k \xi \rho(\omega, w)+c_2^h \xi(\omega, w)=\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi \rho(\omega, w)\end{aligned}$ (35)
From the hypothesis $\left(\epsilon_1+\epsilon_2^k+\epsilon_h\right) \xi<1$, so $\Omega$ is strictly contractive.
Here, it can assume that $\Lambda(v) \in C^1[0,1]$ satisfies Eq (30). By Eqns. (30), (26) and Lemma 3.1, then:
$\begin{gathered}\mid \Lambda(v)-\eta-\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right)\left[g(s, \Lambda(\varsigma))+\int_0^{\varsigma} \Upsilon(\varsigma, \tau, \Lambda(\tau)) d \tau+\int_0^1 \Psi(\varsigma, \tau, \Lambda(\tau)) d \tau \right] \\ d \varsigma \leq\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) \zeta(\varsigma) d \varsigma\right| \leq \xi \zeta(v),\end{gathered}$ (36)
The procedures for proving Eq (36) are the same as for proving Lemma 3.1. From Ω definition and Eq. (36), it can conclude:
$|(\Omega \Lambda)(v)-\Lambda(v)| \leq \xi \zeta(v)$ (37)
Then, by $\rho$ definition, it get:
$\rho(\Omega \Lambda, \Lambda) \leq \xi<1<\infty$ (38)
Let $C^*[0,1]=\left\{y \in C^1[0,1]: \rho(\Omega \Lambda, y)<\infty\right\}$. By Banach fixed-point theorem, there is a unique solution $\Theta \in C^*[0,1]$ such that ΩΘ=Θ, that means Θ is a solution of the system (33). Therefore, Θ is the solution of the system (1). Then:
$\rho(\Lambda, \Theta) \leq \frac{1}{1-\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi} \rho(\Omega \Lambda, \Lambda) \leq \frac{\xi}{1-\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi}$ (39)
From ρ definition, the Eq. (33) holds. This completes the proof.
3.2 Semi-U-H-R and U-H stabilities for the system (5)
The study will investigate in this segment the stabilities of U-H and semi-U-H-R in C1[0,1] for the system in Eq. (5).
Theorem 3.3 Assume that a function ζ is continuous non-decreasing defined as ζ: [0, 1] → (0, ∞) and there is ξ ∈ [0, 1) satisfying:
$\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) \zeta(\varsigma) d \varsigma \leq \xi \zeta(v)$ (40)
Let $\epsilon_1, \epsilon_2^{\mathrm{k}}, \epsilon_2^{\mathrm{h}}>0$ for $\left(\epsilon_1+\epsilon_2^{\mathrm{k}}+\epsilon_2^{\mathrm{h}}\right) \xi<1$. Assume that $\mathrm{h}:[0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ and $\Upsilon, \Psi:[0,1] \times[0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ are continuous functions satisfying:
$\left\{\begin{array}{c}\left|g\left(v, h_1\right)-g\left(v, h_2\right)\right| \leq \epsilon_1\left|h_1-h_2\right|, v \in[0,1], h_1, h_2 \in \mathbb{R} \\ \left|\Upsilon\left(v, \varsigma, h_1\right)-\Upsilon\left(v, \varsigma, h_2\right)\right| \leq \epsilon_2^k\left|h_1-h_2\right|, v, \varsigma \in[0,1], h_1, h_2 \in \mathbb{R} \\ \left|\Psi\left(v, \varsigma, h_1\right)-\Psi\left(v, \varsigma, h_2\right)\right| \leq \epsilon_2^h\left|h_1-h_2\right|, v, \varsigma \in[0,1], h_1, h_2 \in \mathbb{R}\end{array}\right.$ (41)
If $\Lambda \in C^1[0,1]$ satisfies:
$\left|\Lambda^{\prime}(v)+{ }^c D_{0+}^\alpha \Lambda(v)-g(v, \Lambda(v))-\int_0^v \Upsilon(v, \varsigma, \Lambda(\varsigma)) d \varsigma-\int_0^1 \Psi(v, \varsigma, \Lambda(\varsigma)) d \varsigma\right| \leq \theta, v \in[0,1]$ (42)
with θ>0, thus there is Θ(v) solution of the system (5) satisfies:
$|\Lambda(v)-\Theta(v)| \leq \frac{\theta \zeta(v)}{\left[1-\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi\right] \zeta(0)}\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) d \varsigma\right|, v \in[0,1]$ (43)
This means that under above conditions, the system (5) has the semi-U-H-R stability.
Proof. Consider $\Omega: C^1[0,1] \rightarrow C^1[0,1]$, defined by:
$(\Omega \omega)(v)=\eta+\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right)\left[g(s, \omega(\varsigma))+\int_0^{\varsigma} \Upsilon(\varsigma, \tau, \omega(\tau)) d \tau+\int_0^1 \Psi(\varsigma, \tau, \omega(\tau)) d \tau\right] d \varsigma$ (44)
where, $v \in[0,1], \omega \in C^1[0,1]$.
For any $\omega, w \in C^1[0,1]$, then it has:
$\rho(\Omega \omega, \Omega \omega) \leq\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi \rho(\omega, w)$ (45)
From $\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi<1$, then $\Omega$ is strictly contractive in $\left(C^1[0,1], \rho\right)$.
Next, suppose that $\Lambda(v) \in C^1[0,1]$ satisfies Eq. (42). By Eq. (42) and Lemma 3.1 it can get:
$\begin{gathered}\left|\Lambda(v)-\eta-\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right)\left[g(s, \Lambda(\varsigma))+\int_0^{\varsigma} \Upsilon(\varsigma, \tau, \Lambda(\tau)) d \tau+\int_0^1 \Psi(\varsigma, \tau, \Lambda(\tau)) d \tau\right] d \varsigma\right| \\ \leq \theta\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) d \varsigma\right|, v \in[0,1]\end{gathered}$ (46)
From the continuity of Mittag-Leffler function, we have $\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) d \varsigma\right|$ is a continuous nonnegative function. From the Eqns. (44) and (46), then:
$|(\Omega \Lambda)(v)-\Lambda(v)| \leq \theta\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) d \varsigma\right|$ (47)
Since ζ is a continuous function, it can get:
$\rho(\Omega \Lambda, \Lambda)=\sup _{v \in[0,1]} \frac{|(\Omega \Lambda)(v)-\Lambda(v)|}{\zeta(v)} \leq \sup _{v \in[0,1]} \frac{\theta\left|\int_0^v E_{1-\alpha, 1} \,\,\,\, \left(-(v-\varsigma)^{1-\alpha} \,\,\,\right) d \varsigma\right|}{\zeta(0)}<\infty$ (48)
Let $C^*[0,1]=\left\{y \in C^1[0,1]: \rho(\Omega \Lambda, y)<\infty\right\}$.
Applying the Banach fixed-point theorem, thus there is a solution $\Theta \in C^*[0,1]$ such that ΩΘ=Θ. That means Θ(v) is a unique solution of the system (5).
From the Banach fixed-point theorem and Eq. (48), then:
$\rho(\Lambda, \Theta) \leq \frac{1}{1-\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi} \rho(\Omega \Lambda, \Lambda) \leq \frac{\theta\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) d \zeta\right|}{\left[1-\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi\right] \zeta(0)}$ (49)
Thus, by the definition of ρ, then:
$|\Lambda(v)-\Theta(v)| \leq \frac{\theta}{\left[1-\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi\right] \zeta(0)} \zeta(v)\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) d \varsigma\right|$ (50)
where, $\zeta(v)\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) d \varsigma\right|$ is a continuous non-negative function. This completes the proof.
Remark 3.4 For any $v \in[0,1], \int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) d \varsigma$ is real number convergent series. Then, there exists N>0, such that:
$\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) d \varsigma\right|<N$ (51)
Theorem 3.5 Assume that $\epsilon_1, \epsilon_2^k, \epsilon_2^h, \xi$ are constants for which $\epsilon_1>0, \epsilon_2^k>0, \epsilon_2^h>0,0 \leq \xi<1,\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi<1$. Assume that g, Υ, and Ψ are continuous functions, such that:
$\left\{\begin{array}{c}\left|g\left(v, h_1\right)-g\left(v, h_2\right)\right| \leq \epsilon_1\left|h_1-h_2\right|, v \in[0,1], h_1, h_2 \in \mathbb{R} \\ \left|\Upsilon\left(v, \varsigma, h_1\right)-\Upsilon\left(v, \varsigma, h_2\right)\right| \leq \epsilon_2^k\left|h_1-h_2\right|, v, \varsigma \in[0,1], h_1, h_2 \in \mathbb{R} \\ \left|\Psi\left(v, \varsigma, h_1\right)-\Psi\left(v, \varsigma, h_2\right)\right| \leq \epsilon_2^k\left|h_1-h_2\right|, v, \varsigma \in[0,1], h_1, h_2 \in \mathbb{R}\end{array}\right.$
Let $\zeta:[\mathbf{0}, \mathbf{1}] \rightarrow(\mathbf{0}, \infty)$ be a continuous non-decreasing function, and satisfies:
$\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) \zeta(\varsigma) d \varsigma \leq \xi \zeta(v)$ (52)
If $\Lambda \in C^1[0,1]$ satisfies (4), with θ>0, then there is a solution Θ(v) of the system (5) such that:
$|\Lambda(v)-\Theta(v)| \leq \frac{N \zeta(1)}{\left[1-\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi\right] \zeta(0)} \theta, v \in[0,1]$ (53)
Proof. Since ζ is a continuous non-decreasing function,
$\zeta(v) \leq \zeta(1), v \in[0,1]$ (54)
By theorem 3.3, Eqns. (43) and (51), then it can obtain:
$|\Lambda(v)-\Theta(v)| \leq \frac{\theta}{\left[1-\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi\right] \zeta(0)} \zeta(v)\left|\int_0^v E_{1-\alpha, 1}\left(-(v-\varsigma)^{1-\alpha}\right) d \varsigma\right| \leq \frac{N \zeta(1)}{\left[1-\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi\right] \zeta(0)} \theta$ (55)
Theorem 3.5 shows that the system (5) has the U-H stability.
3.3 Illustrative example
Example 1. Let’s assume a fractional Volterra-Fredholm system as follows
$\Theta^{\prime}(v)+{ }^c D_{0+}^{\frac{1}{2}} \Theta(v)=\frac{1}{100}[v \cos \Theta(v)+\Theta(v) \sin v]+\frac{1}{50} \int_0^v \sin \Theta(\varsigma) d \varsigma+\frac{1}{50} \int_0^1 \cos \Theta(\varsigma) d \varsigma$ (56)
$\Theta(0)=0$ (57)
By comparison with the system (5), it can get:
$\begin{gathered}\alpha=\frac{1}{2}, g(v, \Theta(v))=\frac{1}{100}[v \cos \Theta(v)+\Theta(v) \sin v], \Upsilon(v, \varsigma, \Theta(\varsigma))=\frac{1}{50} \sin \Theta(\varsigma) \\ \Psi(v, \varsigma, \Theta(\varsigma))=\frac{1}{50} \sin \Theta(\varsigma), \Psi(v, \varsigma, \Theta(\varsigma))=\frac{1}{50} \cos \Theta(\varsigma) .\end{gathered}$ (58)
Then:
$\left\{\begin{array}{c}\left|g\left(v, h_1\right)-g\left(v, h_2\right)\right| \leq \frac{1}{50}\left|h_1-h_2\right|, h_1, h_2 \in \mathbb{R}, v \in[0,1] \\ \left|Y\left(v, \varsigma, h_1\right)-\Upsilon\left(v, \varsigma, h_2\right)\right| \leq \frac{1}{50}\left|h_1-h_2\right|, h_1, h_2 \in \mathbb{R}, v, \varsigma \in[0,1] \\ \left|\Psi\left(v, \varsigma, h_1\right)-\Psi\left(v, \varsigma, h_2\right)\right| \leq \frac{1}{50}\left|h_1-h_2\right|, h_1, h_2 \in \mathbb{R}, v, \varsigma \in[0,1]\end{array}\right.$ (59)
Let $\zeta(v)=e^v$ , it can obtain:
$\int_0^v E_{\frac{1}{2}, 1}\left(-(v-\varsigma)^{\frac{1}{2}}\right) e^{\varsigma} d \varsigma<e^v-1<\frac{3}{4} e^v, v \in[0,1]$ (60)
Here, it has $\epsilon_1=\epsilon_2^k=\epsilon_2^h=\frac{1}{50}, \xi=\frac{3}{4}$, and $\left(\epsilon_1+\epsilon_2^k+\epsilon_2^h\right) \xi=0.045<1$.
It can see that all the conditions in Theorems 3.2 and 3.5 are satisfied. Then, the system (56) is U-H stability, U-H-R stability and semi-U-H-R stability.
The objective of this study was to provide and demonstrate a novel stability theorem for the nonlinear Volterra-Fredholm integro-differential equation with Caputo fractional derivative utilising the weighted space method and fixed-point technique. The study specifically examines the H-U-R stability and semi-U-H-R stability results.
Besides, a class of nonlinear fractional Volterra-Fredholm integro-differential equations with initial conditions is discussed. By means of the Banach fixed-point techniques and weighted space, stability of the fractional nonlinear Volterra–Fredholm system has been tested. An illustrative example that demonstrates the applicability of the results has been included.
Discussing U-H-Mittag-Leffler stability [22] and finite-time stability [23] for the -Hilfer fractional Volterra-Fredholm integro-differential equations with time-varying delay terms would be a delightful extension of the current results. This will be the focus of future research.
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