Differential calculus on the ring of generalized Colombeau numbers and results in ordinary differential equations

Abstract In this paper, we introduce a differential calculus for functions between totally disconnected spaces within the context of Colombeau’s full generalized numbers and functions. We then study the properties of the Colombeau full differential algebra, including the Embedding Theorem, the Open Mapping Theorem, and the Fundamental Theorem of Calculus. Furthermore, we investigate the existence and uniqueness of solutions to differential equations in the space of Colombeau’s full-tempered generalized functions.