[Paper Review] Mild and classical solutions for fractional evolution differential equation

J. Vanterler da C. Sousa, Thabet Abdeljawad+1 more|arXiv (Cornell University)|Aug 14, 2019

Nonlinear Differential Equations Analysis39 references4 citations

TL;DR

This paper establishes the existence and uniqueness of mild and classical solutions for a fractional evolution differential equation involving the Hilfer fractional derivative in a Banach space. Using the Banach contraction principle, Gronwall's inequality, and an $(\alpha,\beta)$-resolvent operator function, the authors prove that under suitable conditions, a unique solution exists and is continuously dependent on initial data.

ABSTRACT

Investigating the existence, uniqueness, stability, continuous dependence of data among other properties of solutions of fractional differential equations, has been the object of study by an important range of researchers in the scientific community, especially in fractional calculus. And over the years, these properties have been investigated more vehemently, as they enable more general and new results. In this paper, we investigate the existence and uniqueness of a class of mild and classical solutions of the fractional evolution differential equation in the Banach space $Ω$. To obtain such results, we use fundamental tools, namely: Banach contraction theorem, Gronwall inequality and the $β$-times integrated $β$-times integrated $α$-resolvent operator function of an $(α,β)$-resolvent operator function.

Motivation & Objective

• To investigate the existence and uniqueness of mild and classical solutions for a class of fractional evolution differential equations with Hilfer derivatives.
• To extend existing results on fractional differential equations by incorporating the more general Hilfer derivative, which unifies Riemann-Liouville and Caputo derivatives.
• To establish continuous dependence of solutions on initial data and system parameters using functional analytic tools.
• To introduce a new class of solutions defined via the $\beta$-times integrated $\alpha$-resolvent operator function in the context of Hilfer fractional evolution equations.
• To provide a rigorous framework for studying fractional evolution equations with nonlocal conditions and distributed delays.

Proposed method

• Employ the Banach contraction theorem to prove the existence and uniqueness of mild solutions in a complete metric space of continuous functions.
• Apply the generalized Gronwall inequality to establish continuity and stability properties of the solution in the $C_{1-\gamma}$-space.
• Utilize the $\beta$-times integrated $\alpha$-resolvent operator function associated with the $(\alpha,\beta)$-resolvent operator to define the mild solution.
• Use the Hilfer fractional derivative $^H\!\mathbb{D}_{t_0^+}^{\alpha,\beta}$ with $0 < \alpha \leq 1$ and $0 \leq \/beta \leq 1$, which generalizes both Riemann-Liouville and Caputo derivatives.
• Define the mild solution via an integral equation involving the Mittag-Leffler function and the kernel $K_\alpha(t-s)$, derived from the resolvent operator.
• Verify that the mild solution satisfies the original fractional differential equation and initial condition, thereby proving it is a classical solution.

Experimental results

Research questions

• RQ1 Under what conditions does a mild solution exist for the fractional evolution equation with Hilfer derivative in a Banach space?
• RQ2 How can the uniqueness of the mild solution be established under nonlocal initial conditions and distributed delays?
• RQ3 What is the relationship between mild and classical solutions in the context of Hilfer-type fractional evolution equations?
• RQ4 How does the $\beta$-times integrated $\alpha$-resolvent operator function contribute to the solution structure of the equation?
• RQ5 What role does the generalized Gronwall inequality play in proving the continuous dependence of solutions on initial data and system parameters?

Key findings

• The existence and uniqueness of mild solutions are established under the Banach contraction principle, assuming Lipschitz continuity of the nonlinear term and boundedness of the resolvent operator.
• The classical solution is shown to coincide with the mild solution, thereby validating the mild solution as a strong candidate for physical and engineering applications.
• The solution is continuously dependent on initial data, as demonstrated through the Gronwall inequality and the boundedness of the resolvent operator in the $C_{1-\gamma}$-space.
• The solution satisfies the initial condition $I_{t_0^+}^{1-\gamma}u(t_0^+) + \sum_{k=1}^p C_k I_{t_0^+}^{1-\gamma}u(t_k) = u_0$, ensuring consistency with nonlocal conditions.
• The continuity of the solution in time is proven via the estimate $\|u(t+h) - u(t)\|_{C_{1-\gamma}} \leq \widetilde{\delta}h \mathbb{E}_\alpha[MC(1+rk)\Gamma(\alpha)a^\alpha]$, where $\mathbb{E}_\alpha$ is the Mittag-Leffler function.
• The proof confirms that the mild solution $u(t)$ satisfies the original equation (1.1)–(1.2), thus verifying it as a classical solution under the given assumptions.