Boundedness and Continuity of the Riemann-Liouville Fractional Integral in Bochner-Lebesgue Spaces

Concepts de base

The Riemann-Liouville fractional integral of order α > 0 is a bounded linear operator from Lp(t0, t1; X) to various function spaces, including Hölder continuous spaces, Bochner-Sobolev spaces, and spaces of bounded mean oscillation.

Résumé

The manuscript extends previous work on the boundedness and continuity properties of the Riemann-Liouville (RL) fractional integral of order α > 0 in Bochner-Lebesgue spaces. The key highlights and insights are:

For p ∈ (1, ∞) and α ∈ (1/p, ∞), the RL fractional integral Jα
t0,t is a bounded linear operator from Lp(t0, t1; X) to C([t0, t1]; X).

For p ∈ (1, ∞) and α ∈ (n + 1/p, n + 1 + 1/p), n ∈ N, the RL fractional integral Jα
t0,t is a bounded linear operator from Lp(t0, t1; X) to the Hölder continuous space Hn,q([t0, t1]; X), where q = α - (n + 1/p).

For p ∈ (1, ∞) and α = n + 1/p, n ∈ N*, the RL fractional integral Jα
t0,t is a bounded linear operator from Lp(t0, t1; X) to the Banach space BKn,p
γ (t0, t1; X), which is a subspace of the Bochner-Sobolev space W n,p(t0, t1; X) with additional regularity conditions.

The case p = 1 and α ≥ 1 is addressed, showing that Jα
t0,t is a bounded linear operator from L1(t0, t1; X) to the RL fractional Bochner-Sobolev space W γ,1
RL(t0, t1; X), for γ ∈ (0, α].

For p = ∞ and α ∈ (0, 1), the RL fractional integral Jα
t0,t is a bounded linear operator from L∞(t0, t1; X) to the Hölder continuous space H0,α([t0, t1]; X).

The results provide a comprehensive understanding of the boundedness and continuity properties of the Riemann-Liouville fractional integral in various function spaces.

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Idées clés tirées de

The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces IV

by Paul... à arxiv.org 10-02-2024

https://arxiv.org/pdf/2410.00830.pdf

Questions plus approfondies

How can the results be extended to other types of fractional integrals, such as the Caputo or Hadamard fractional integrals?

The results concerning the boundedness and continuity of the Riemann-Liouville fractional integral can indeed be extended to other types of fractional integrals, such as the Caputo and Hadamard fractional integrals. The Caputo fractional integral, defined similarly to the Riemann-Liouville integral but with a focus on initial value problems, allows for the incorporation of initial conditions in a more natural way. This is particularly useful in applications where initial conditions are specified in the classical sense.
To extend the results, one can leverage the relationships between these fractional integrals. For instance, the Caputo fractional integral can be expressed in terms of the Riemann-Liouville integral, which means that the boundedness results established for the Riemann-Liouville integral can be adapted for the Caputo integral by considering the appropriate function spaces.
Similarly, the Hadamard fractional integral, which is defined using a different approach involving the limit of a sequence of integrals, can also be analyzed within the framework of Bochner-Lebesgue spaces. The continuity and boundedness properties can be shown to hold under similar conditions, particularly by utilizing the properties of the underlying function spaces and the continuity of the involved operators.

What are the implications of these boundedness and continuity properties for the numerical approximation and computational analysis of fractional differential equations?

The boundedness and continuity properties of the Riemann-Liouville fractional integral have significant implications for the numerical approximation and computational analysis of fractional differential equations (FDEs). These properties ensure that the fractional integral behaves well under various function spaces, which is crucial for the stability and convergence of numerical methods.
For instance, when employing numerical methods such as finite difference or finite element methods to solve FDEs, the boundedness of the fractional integral guarantees that the numerical solutions will not exhibit unbounded growth, thus ensuring stability. The continuity of the fractional integral implies that small changes in the input function will result in small changes in the output, which is essential for the convergence of iterative numerical schemes.
Moreover, these properties facilitate the development of efficient algorithms for approximating solutions to FDEs. By ensuring that the fractional integral maps functions from one space to another in a controlled manner, numerical analysts can design algorithms that exploit these mappings to achieve better accuracy and efficiency in computations.

Are there any connections between the regularity of the Riemann-Liouville fractional integral and the well-posedness or stability of fractional-order partial differential equations?

Yes, there are significant connections between the regularity of the Riemann-Liouville fractional integral and the well-posedness or stability of fractional-order partial differential equations (PDEs). The regularity of the fractional integral plays a crucial role in determining the smoothness of the solutions to fractional-order PDEs.
Well-posedness, which typically refers to the existence, uniqueness, and continuous dependence of solutions on initial and boundary data, is closely tied to the regularity of the operators involved. The boundedness and continuity results established for the Riemann-Liouville fractional integral imply that the solutions to fractional-order PDEs will inherit certain regularity properties from the initial data. This is particularly important in ensuring that solutions remain stable under perturbations.
Furthermore, the regularity of the fractional integral can influence the stability of the solutions. If the fractional integral is continuous and bounded, it can help ensure that the solutions to the associated fractional-order PDEs do not exhibit pathological behavior, such as blow-up or oscillations, which can compromise stability.
In summary, the interplay between the regularity of the Riemann-Liouville fractional integral and the well-posedness of fractional-order PDEs is fundamental in establishing the stability and reliability of solutions, making it a critical area of study in the analysis of fractional calculus.