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Nonlinear convolution operators. Ordinary differential equations. (Nelinejnye operatory v svertkakh. Obyknovennye differentsial’nye uravneniya.) (Russian) Zbl 0693.46049

Sverdlovsk: Izdatel’stvo Ural’skogo Universiteta. 148 p. R. 0.80 (1989).
The theory of generalized solutions of nonlinear equations, translation invariant with respect to the unknown function of one real variable, is consequently studied.
As a foundation for the construction of a theory of generalized solutions serves the affine structure of the line, permitting to consider translation invariant operators and to represent them in the form of a convolution with functionals. This is a detailed repetition of the situation, which appears in the application of the theory of distributions to the foundation of the operational calculus of Heaviside.
In Theorem 1.1 the fundamental canonical isomorphism between the set of translation invariant operators and a set of functionals is established. An interesting theory of convolution equations is constructed for the space of analytic functionals, i.e. the analytic mappings of the Schwartz space \({\mathcal D}({\mathbb{R}})\) into \({\mathbb{C}}\). The convolution of analytic functionals inherits the fundamental properties of convolution of distributions. In particular, the associativity and the existence of the convolution of functionals, all of which (except, possibly, one) have compact support is shown, and moreover, the analytic functionals with support in the right semiaxis form a convolution algebra.
Theorem 2.4 guarantees the existence of a fundamental solution of a nonlinear differential equation in the case, where the highest derivative belongs to the linear part, which is a considerable improvement of the corresponding classical results.
The method is analogical to that one, which realizes the correspondence between distributions and classical functions, it is possible to construct a classical meaning to a generalized solution. Moreover, the classical meaning of a generalized solution is a classical solution of a classical equation, and conversely.
One must notice the applied aspect of the generalized discrete approximation of the right side of a differential equation. As is shown in numerous examples, this method allows to obtain information of differential equations to iteration schemes, which is very economical from the numerical point of view and nevertheless is approximating the solution globally.

MSC:

46F99 Distributions, generalized functions, distribution spaces
47E05 General theory of ordinary differential operators
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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