ABSTRACT
Let C be a ring of generalized numbers, ε a sheaf of algebras on a topological space X and, for each open set Ω of X, P (Ω) a family of seminorms on ε (Ω). Then, we can construct a sheaf https://www.w3.org/1998/Math/MathML"> A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203745458/b4be5e7d-e785-4326-bbbf-b05e79859038/content/eq1230.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of (C, ε, P-algebras which generalize many previous approaches in the theory of generalized functions [2] [3] [4] [8] [15] [1]. The sheaf structure of https://www.w3.org/1998/Math/MathML"> A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203745458/b4be5e7d-e785-4326-bbbf-b05e79859038/content/eq1231.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> permits to define localization and microlocalization processes [9] [16] [17]. Then we develop the idea to adapt C, ε, P as parameters to each problem to solve [10] [13] [14] [11] [12]. Restriction and derivation can be defined and (with product) are good tools to pose and solve many non linear (and linear) problems with irregular data which have no solution in the distributional sense [14] [18]. We give here another application. We will show that some perturbation problems https://www.w3.org/1998/Math/MathML"> { φ ( ε ) d X d t = f ε ( t , X ) X ( 0 ) = ψ ( ε ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203745458/b4be5e7d-e785-4326-bbbf-b05e79859038/content/eq1232.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
