Well-posedness for time-fractional evolution equations by the Yosida approximation

Abstract

We consider an initial value problem for a time-fractional evolution equation in a Hilbert space X: ∂_t^α (u(t)−a) = Au(t) for $0<t<T$, where ∂_t^α denotes a fractional differential operator of Caputo type with order 0<α<1, u: (0,T) → X, a describes an initial value, and A, with domain 𝒟(A), is the generator of a contraction C₀ semigroup in X. We prove a fractional Hille-Yosida theorem characterizing the unique existence of a weak solution for every a∈ X. We also discuss the unique existence of a strong solution for a ∈ 𝒟(A), as well as the continuity of weak solutions. Our results are applicable, for example, to time-fractional transport equations and Boltzmann equations. The proofs are mainly based on constructing an approximating sequence of solutions using the Yosida approximation.

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BibTeX RIS

BibTeX

@misc{chorfi2026wellposedness,
  title = {{Well-posedness for time-fractional evolution equations by the Yosida approximation}},
  author = {S. E. Chorfi and F. Gölgeleyen and M. Yamamoto},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35001},
  eprint = {2609.35001},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35001},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Well-posedness for time-fractional evolution equations by the Yosida approximation
AU  - S. E. Chorfi
AU  - F. Gölgeleyen
AU  - M. Yamamoto
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.35001
UR  - https://arxiv.org/abs/2609.35001
AB  - We consider an initial value problem for a time-fractional evolution equation in a Hilbert space X: ∂_t^α (u(t)−a) = Au(t) for $0<t<T$, where ∂_t^α denotes a fractional differential operator of Caputo type with order 0<α<1, u: (0,T) → X, a describes an initial value, and A, with domain 𝒟(A), is the generator of a contraction C_0 semigroup in X. We prove a fractional Hille-Yosida theorem characterizing the unique existence of a weak solution for every a∈ X. We also discuss the unique existence of a strong solution for a ∈ 𝒟(A), as well as the continuity of weak solutions. Our results are applicable, for example, to time-fractional transport equations and Boltzmann equations. The proofs are mainly based on constructing an approximating sequence of solutions using the Yosida approximation.
ER  -

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