We consider a Cauchy problem for the inhomogeneous differential equation given in terms of an unbounded linear operator A and Caputo fractional derivative order α∈(0,2) time. The previously known representation mild solution to such does not have conventional variation-of-constants like form, with propagator derived from associated homogeneous problem. Instead, it relies on existence two propag...

This paper presents several characterizations of a local α-times integrated C-semigroup {T(t); 0 ≤ t < τ} by means of functional equation, subgenerator, and well-posedness of an associated abstract Cauchy problem.We also discuss properties concerning the nondegeneracy of T(·), the injectivity of C, the closability of subgenerators, the commutativity of T(·), and extension of solutions of the as...

We address a Cauchy problem for a nonlinear elliptic PDE arising in glaciology. After recasting the Cauchy problem as an ill-posed operator equation, we prove (for values of a certain parameter allowing Hilbert space techniques) differentiability properties of the associated operator. We also suggest iterative methods which can be applied to solve the operator problem.