Well-Posedness of Variable-Order Fractional Abstract Cauchy Problems Under the Scarpi Framework
- 1School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China
- 2Key Laboratory of Data Science and Intelligence Education, Ministry of Education, Hainan Normal University, Haikou 571158, China
Abstract
We study the variable-order fractional abstract Cauchy problem SD0α(t)u(t)=Au(t)+f(t), u(0)=u0, in a Banach space X, where A is sectorial and α(t):[0,∞)→(0,1) is a Scarpi transition function. Using the Sonine-pair structure of the Scarpi kernels, we convert the problem into a Volterra integral equation and, adapting the resolvent-family framework of the constant-order theory to this variable-order setting, introduce a variable-order resolvent family through Laplace transform techniques. In contrast to the constant-order case, the variable-order resolvent family is strongly continuous and exponentially bounded, but no analyticity in t is claimed or used. For exponentially bounded transitions α(t)=α2+(α1−α2)e−ct, the factorization sq(s)=sα1·s(α2−α1)c/(s+c) links the variable-order resolvent to the constant-order α1-resolvent. We prove the existence of this resolvent family, establish the existence of mild solutions by direct construction via the variable-order resolvent family and Laplace transform methods, prove the uniqueness in the class of strong solutions of the Volterra integral equation via a duality argument, and derive continuous dependence on the data. To the best of our knowledge, this provides the first complete well-posedness theory for inhomogeneous Scarpi-type variable-order abstract Cauchy problems in general Banach spaces. Two examples (A=−λI and A=Δ) illustrate the theory; the latter yields well-posed variable-order subdiffusion equations, of which a numerical illustration is provided at the end of the paper.
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BibTeX
@article{lin2026wellposedness,
title = {{Well-Posedness of Variable-Order Fractional Abstract Cauchy Problems Under the Scarpi Framework}},
author = {Shiyou Lin and Junying Chen and Aijie Yang},
journal = {Fractal and Fractional},
year = {2026},
month = oct,
doi = {10.3390/fractalfract10110759},
url = {https://www.mdpi.com/2504-3110/10/11/759},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Well-Posedness of Variable-Order Fractional Abstract Cauchy Problems Under the Scarpi Framework
AU - Shiyou Lin
AU - Junying Chen
AU - Aijie Yang
PY - 2026
DA - 2026/10/29
JO - Fractal and Fractional
DO - 10.3390/fractalfract10110759
UR - https://www.mdpi.com/2504-3110/10/11/759
AB - We study the variable-order fractional abstract Cauchy problem SD0α(t)u(t)=Au(t)+f(t), u(0)=u0, in a Banach space X, where A is sectorial and α(t):[0,∞)→(0,1) is a Scarpi transition function. Using the Sonine-pair structure of the Scarpi kernels, we convert the problem into a Volterra integral equation and, adapting the resolvent-family framework of the constant-order theory to this variable-order setting, introduce a variable-order resolvent family through Laplace transform techniques. In contrast to the constant-order case, the variable-order resolvent family is strongly continuous and exponentially bounded, but no analyticity in t is claimed or used. For exponentially bounded transitions α(t)=α2+(α1−α2)e−ct, the factorization sq(s)=sα1·s(α2−α1)c/(s+c) links the variable-order resolvent to the constant-order α1-resolvent. We prove the existence of this resolvent family, establish the existence of mild solutions by direct construction via the variable-order resolvent family and Laplace transform methods, prove the uniqueness in the class of strong solutions of the Volterra integral equation via a duality argument, and derive continuous dependence on the data. To the best of our knowledge, this provides the first complete well-posedness theory for inhomogeneous Scarpi-type variable-order abstract Cauchy problems in general Banach spaces. Two examples (A=−λI and A=Δ) illustrate the theory; the latter yields well-posed variable-order subdiffusion equations, of which a numerical illustration is provided at the end of the paper.
SN - 2504-3110
ER -