Well-Posedness of Variable-Order Fractional Abstract Cauchy Problems Under the Scarpi Framework

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.