Abstract
C0‐semigroups of linear operators play a crucial role in the solvability of evolution equations in the classical context. This paper is concerned with a brief conceptualization of C0‐semigroups on (ultrametric) free Banach spaces . In contrast with the classical setting, the parameter of a given C0‐semigroup belongs to a clopen ball Ωr of the ground field . As an illustration, we will discuss the solvability of some homogeneous p‐adic differential equations.
Highlights
Let (K, +, ·, | · |) be a ultrametric-valued field and let Ωr be the closed ball of K centered at 0 with radius r > 0, that is, Ωr = {κ ∈ K : |κ| ≤ r}
We provide the reader with a brief conceptualization of ultrametric counterparts of C0-semigroups in connection with the formalism of linear operators on free Banach and non-Archimedean Hilbert spaces, recently developed in [2,3,4,5,6]
The present paper is mainly motivated by the solvability of p-adic differential and partial differential equations [9, 11,12,13, 18] as strong solutions to the Cauchy problem related to several classes of differential and partial differential equations in the classical setting which can be expressed through C0-semigroups, see, for example, [15, 16]
Summary
(The ring of p-adic integers Zp is the unit ball of Qp centered at zero, that is, the set of all x ∈ Qp such that |x| ≤ 1, where | · | is the p-adic valuation of Qp). It does converge for all q ∈ Zp such that |q| < r = p−1/(p−1), where Zp denotes the ring of p-adic integers. For more on these and related issues, we refer the reader to [1, 7, 8, 18].
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