C. High school of Haumman El fetouaki, Had Soualem, 26402, Morocco
Abstract:
In this paper, we define and study the Volkenborn integral and the Shnirelman integral for cosine families of bounded linear operators on some non-Archimedean Banach spaces over $\mathbb{Q}_{p}$ and $\mathbb{C}_{p}$ respectively. We give some functional calculus for cosine families of infinitesimal generator $A$ such that $A$ is a nilpotent operator on some non-Archimedean Banach spaces over $\mathbb{C}_{p}.$ Many results are proved and examples are given to support our work.
Keywords and phrases:volkenborn integral, Shnirelman integral, cosine families of bounded linear operators, $p$-adic theory.
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