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Stability of $ψ$-Hilfer Impulsive Fractional Differential Equations
J. Vanterler da C. Sousa, Kishor D. Kucche, E. Capelas de Oliveira
TL;DR
The paper studies existence, uniqueness, and generalized δ-Ulam-Hyers-Rassias stability for impulsive fractional differential equations with a ψ-Hilfer derivative. It applies Banach’s fixed-point theorem under stated hypotheses and obtains a unique solution associated with each admissible approximate solution.
Problem
The paper addresses stability, existence, and uniqueness for impulsive fractional differential equations involving the ψ-Hilfer fractional derivative.
Method
The authors apply a fixed-point approach based on Banach’s fixed-point theorem to the impulsive fractional differential equation.
Results
Under conditions (H1)-(H4), every solution satisfying the prescribed inequality yields a unique solution y0 of the impulsive fractional differential equation.
Takeaways & Limitations
The results establish generalized δ-Ulam-Hyers-Rassias stability for the considered ψ-Hilfer impulsive fractional equation.
Takeaways & Limitations
The formulation is bounded to continuous f and gi with prefixed impulse intervals, and the main theorem requires conditions (H1)-(H4).
Abstract
from arXiv · showhide
In this paper, we investigate the sufficient conditions for existence and uniqueness of solutions and δ-Ulam-Hyers-Rassias stability of an impulsive fractional differential equation involving $ψ$-Hilfer fractional derivative. Fixed point approach is used to obtain our main results.
1. Introduction
The paper situates impulsive fractional equations within models of abrupt state changes and extends stability analysis to a ψ-Hilfer formulation with not-instantaneous impulses. It uses a fixed-point approach to study generalized δ-Ulam-Hyers-Rassias stability.
- Impulsive differential equations model evolutionary processes that abruptly change state and apply across mechanics, ecology, medicine, biology, and electrical engineering.
- Prior work established existence, uniqueness, and stability results for differential and integro-differential equations with not-instantaneous impulses.
- Fractional impulsive equations broadened this literature to stability, existence, and uniqueness questions, including formulations in Banach spaces.
- The paper aims to extend stability research on impulsive fractional differential equations involving a ψ-Hilfer fractional derivative.
- The authors investigate generalized δ-Ulam-Hyers-Rassias stability for the impulsive fractional equation using a fixed-point approach.
2. Preliminaries
The preliminaries define the weighted and piecewise weighted function spaces, generalized metric framework, fractional operators, and generalized δ-Ulam-Hyers-Rassias stability used in the proof. Banach’s fixed-point theorem supplies the central convergence and uniqueness tool.
- A δ-norm satisfies zero separation, δ-homogeneity, and a δ-triangle inequality for 0 < δ ≤ 1.
- The weighted space C1−γ;ψ(J, R) consists of functions whose ψ-weighted form extends continuously to J and is a Banach space.
- The piecewise weighted space PC1−γ;ψ(J, R) imposes corresponding weighted continuity and endpoint-limit conditions on each interval and is also Banach.
- The ψ-Hilfer derivative is introduced for increasing ψ with nonzero derivative, together with the parameters α and β governing the formulation.
- The generalized metric space and operator framework support Banach fixed-point reasoning for convergence to a unique fixed point.
- Generalized δ-Ulam-Hyers-Rassias stability requires every approximate solution satisfying the prescribed inequality to correspond to an exact solution within a controlled bound.
3. δ−Ulam-Hyers-Rassias stability
The paper establishes δ-Ulam-Hyers-Rassias stability by representing the impulsive fractional problem as a fixed-point operator and applying Banach’s theorem under hypotheses (H1)–(H4). The operator is shown to be well defined and strictly contractive, yielding a unique continuous solution.
- Assumptions: The nonlinear term f satisfies a Lipschitz condition with constant Lf, while each impulse function gi satisfies a corresponding Lipschitz condition with constant Lgi.These are the principal regularity assumptions used to control differences between candidate solutions.
- Stability estimates: The function ϕ is assumed nondecreasing and bounded through a positive constant Cϕ, supporting the stability estimates used in the contraction argument.The stability control also uses the impulse discrepancy relation and bounds introduced in the proof.
- Main theorem: The main stability result assumes conditions (H1)–(H4) and concludes that an approximate solution satisfying Eq.(2.2) corresponds to a unique solution.The theorem is stated for the impulsive fractional differential equation under the paper’s generalized δ-Ulam-Hyers-Rassias framework.
- Fixed-point construction: The operator Ω is constructed from the integral formulation and is shown to be well defined on X before contractivity is verified across three time intervals.The proof treats the initial interval, impulse intervals, and the intervals following impulses separately.
- Fixed-point conclusion: Banach’s fixed-point theorem gives a continuous fixed point y0, and the proof establishes that y0 is the unique continuous solution represented by Eq.(3.4).The iterates Ω^n y0 converge to y0, and the metric-space argument shows the relevant distance from a reference element is finite.