[Paper Review] Existence, uniqueness, estimation and continuous dependence of the solutions of a nonlinear integral and an integrodifferential equations of fractional order

J. Vanterler da C. Sousa, E. Capelas de Oliveira|arXiv (Cornell University)|Jun 5, 2018

Fractional Differential Equations Solutions30 references10 citations

TL;DR

This paper establishes the existence, uniqueness, estimation, and continuous dependence of solutions for nonlinear fractional Volterra integral and integrodifferential equations using the $ψ$-Hilfer fractional derivative. By introducing a specialized norm and leveraging fractional integral inequalities, the authors prove solution stability and derive explicit bounds involving the Mittag-Leffler function, advancing analytical theory in fractional calculus for general nonlinear systems.

ABSTRACT

By means of two fractional order integral inequalities we investigate the existence and uniqueness of the solutions of the fractional nonlinear Volterra integral equation and a fractional nonlinear integrodifferential equation in Banach space $C_ξ$, using an adequate norm, $||\cdot||_{ξ,\infty}$. We estimate the solutions and investigate their continuous dependence.

Motivation & Objective

• To establish the existence and uniqueness of solutions for nonlinear fractional Volterra integral and integrodifferential equations in Banach space $C_{\xi}$.
• To derive explicit solution estimates using fractional integral inequalities and the Mittag-Leffler function.
• To investigate continuous dependence of solutions on initial data and parameters in the context of $ψ$-Hilfer fractional derivatives.
• To generalize analytical results for fractional integral equations by introducing a tailored norm and metric structure.
• To contribute to the theoretical foundation of fractional calculus, particularly for nonlinear systems involving generalized fractional derivatives.

Proposed method

• The authors define a new norm $||\cdot||_{\xi,\infty}$ and a metric on the Banach space $C_{\xi}$ to analyze solution behavior.
• They employ two key fractional integral inequalities (Lemmas 4 and 5) and a corollary involving the Mittag-Leffler function to bound solution differences.
• The analysis uses the $ψ$-Hilfer fractional derivative and $ψ$-Riemann-Liouville integral to model the nonlinear integral and integrodifferential equations.
• The solution estimates are derived via contraction-type arguments under Lipschitz and boundedness conditions on the nonlinear terms.
• Continuous dependence is proven by bounding solution differences in terms of parameter and initial data perturbations using the Mittag-Leffler function.
• The framework is applied to both the fractional integral equation (1.1) and the fractional integrodifferential equation (1.2) with initial condition.

Experimental results

Research questions

• RQ1 Under what conditions does a nonlinear fractional Volterra integral equation with $ψ$-Hilfer derivative admit a unique solution in $C_{\xi}$?
• RQ2 How can one estimate the solution of a nonlinear fractional integral equation using fractional integral inequalities?
• RQ3 How does the solution of a nonlinear fractional integrodifferential equation depend continuously on initial data and parameters?
• RQ4 What role does the Mittag-Leffler function play in quantifying solution stability and error bounds?
• RQ5 Can the proposed norm and metric structure ensure completeness and convergence in the solution space for fractional integrodifferential problems?

Key findings

• The solution of the fractional integral equation (1.1) exists and is unique in the Banach space $C_{\xi}$ under appropriate Lipschitz and boundedness conditions.
• An explicit solution estimate is derived: $||x||_{\xi,\infty} \leq \frac{M}{1 - \overline{N}}$, where $M$ is a bound on the nonlinear term and $\overline{N} < 1$ is a Lipschitz constant.
• For continuous dependence, the difference between solutions $z_1(t)$ and $z_2(t)$ is bounded by $|z_1(t) - z_2(t)| \leq \frac{Q|\mu - \mu_0|}{1 - \overline{N}} \mathbb{E}_{\alpha}\left[\frac{\overline{N}}{1 - \overline{N}} r(t,t)(\psi(t) - \psi(a))^\alpha\right]$.
• The continuous dependence result is extended to parameter-dependent systems, yielding a bound involving the Mittag-Leffler function: $|z_1(t) - z_2(t)| \leq Q|\mu - \mu_0| \mathbb{E}_{\alpha}\left\{\overline{p}(t)\Gamma(\alpha) \mathbb{E}_{\alpha}(\overline{r}(t,t)\Gamma(\alpha)(\psi(t) - \psi(a))^\alpha)(\psi(t) - \psi(a))^\alpha\right\}$.
• The results are valid for the general $ψ$-Hilfer fractional derivative, extending prior work to a broader class of fractional differential equations.
• The theoretical framework is built on a new norm and metric, ensuring completeness of the solution space and enabling rigorous analysis of solution stability.