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Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation
J. Vanterler da C. Sousa, E. Capelas de Oliveira
TL;DR
The paper addresses stability of a nonlinear fractional Volterra integro-differential equation under Hyers-Ulam-Rassias and Hyers-Ulam notions. It applies a fixed-point method with the ψ-Hilfer fractional derivative and proves existence, uniqueness, and stability results under Lipschitz and contraction conditions.
Problem
The paper studies the stability of a nonlinear fractional Volterra integro-differential equation using generalized fractional operators.
Method
The authors use a fixed-point method on continuous functions equipped with a generalized metric, under Lipschitz hypotheses for f and k.
Results
There exists a unique continuous solution u0 satisfying the integral equation and the stated stability estimate under the theorem’s hypotheses.
Takeaways & Limitations
The fixed-point framework establishes Hyers-Ulam-Rassias and Hyers-Ulam stability for the fractional Volterra equation within the stated assumptions.
Takeaways & Limitations
The main theorem assumes a bounded interval and contraction condition 0 < TLf + T^2Lk < 1, together with Lipschitz conditions on f and k.
Abstract
from arXiv · showhide
Using the $ψ-$Hilfer fractional derivative, we present a study of the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the fractional Volterra integral-differential equation by means of fixed-point method.
1. Introduction
The paper studies Hyers-Ulam-Rassias and Hyers-Ulam stability for a nonlinear fractional Volterra integro-differential equation using generalized fractional operators. It develops these stability results for the ψ-Hilfer fractional derivative and ψ-Riemann-Liouville fractional integral framework.
- Fractional differential-equation models have applications including population dynamics and erythrocytes sedimentation rate.
- The study uses recent generalized fractional differentiation and integration operators to obtain generalized stability results and recover usual results as particular cases.
- The paper’s motivation is to study Hyers-Ulam-Rassias and Hyers-Ulam stability for a nonlinear fractional Volterra integro-differential equation.
- The equation is considered on I = [0, T], with continuous nonlinear terms f and k and parameters governing the ψ-Hilfer derivative and ψ-Riemann-Liouville fractional integral.
- The paper introduces preliminary definitions and develops the Hyers-Ulam-Rassias and Hyers-Ulam stability results in separate sections.
2. Preliminaries
The preliminaries define the ψ-Hilfer fractional derivative and the two stability notions, then state a fixed-point theorem used to analyze the fractional equation. Hyers-Ulam-Rassias stability allows a nonnegative control function, while Hyers-Ulam stability is its constant-control special case.
- The section introduces the ψ-Hilfer fractional derivative for integrable functions on an interval and an increasing ψ with nonzero derivative.
- Hyers-Ulam-Rassias stability requires every approximate solution controlled by Φ(t) to lie within CΦ(t) of an exact solution.
- Hyers-Ulam stability is defined when the control function Φ(t) in the preceding stability inequalities is constant.
- The stated theorem is presented as the key preliminary result for proving the paper’s Hyers-Ulam and Hyers-Ulam-Rassias stability claims.
- The fixed-point result states that iterates of a strictly contractive operator converge to a unique fixed point in the relevant metric subspace.
3. Mains Results
The main results establish existence, uniqueness, and Hyers-Ulam-type stability for the fractional Volterra integro-differential equation under Lipschitz and contraction conditions. A fixed-point operator on a complete generalized metric space provides the proof framework.
- Assumptions: The analysis introduces Lipschitz hypotheses for f and k, with contraction condition 0 < MLf + M2Lk < 1.The corresponding conditions are stated for the nonlinear term and kernel on I.
- Stability: The fixed-point estimates use the generalized distance and the bound d(u, Ωu) ≤ M to establish the Hyers-Ulam-Rassias and Hyers-Ulam conclusions.The weighted metric framework relates pointwise deviations to the control function Φ.
- Scope: The paper notes that analogous Hyers-Ulam-Rassias stability results can be considered on a limited closed interval using the same hypotheses.This observation connects the main theorem to the bounded-interval setting.
- Stability: For an approximate solution u with residual bounded by ε, the results provide a unique continuous exact solution u0 within the corresponding stability bound.Theorem 3 states this conclusion under the ψ-Hilfer assumptions and the contraction condition involving T, Lf, and Lk.
- Fixed-point framework: The proof defines a generalized metric on continuous functions and applies a fixed-point operator to the fractional integral equation.The function space X consists of real-valued continuous functions, and the metric is shown to be complete.
- Existence and uniqueness: There exists a continuous fixed point u0 satisfying the fractional integral equation, and u0 is unique.The iterates of the operator converge to u0, while the generalized contraction theorem yields uniqueness.