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On the differential geometry of numerical schemes and weak solutions of functional equations. (English) Zbl 1454.39031

The aim of this long paper is to describe some differential geometric properties of the analysis of weak solutions of functional equations. To accomplish this task some differential geometric structures are exhibited, like diffeology and Frölicher spaces. Their main properties are presented together with their mutual relations. These introductory results are then used to investigate functional equations of the form \(F(u,q)=0\) where \(F\) is a smooth map on suitable spaces and to prove an implicit functions theorem. Various examples are presented in full detail.

MSC:

39B05 General theory of functional equations and inequalities
58A40 Differential spaces
58B10 Differentiability questions for infinite-dimensional manifolds
46T20 Continuous and differentiable maps in nonlinear functional analysis
47J25 Iterative procedures involving nonlinear operators
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