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Grant 201/06/0254     1.1.2006 - 31.12.2008
Grantor: Grant Agency of Czech Republic (GACR)

Functional differential equations in Banach spaces

The project was aimed at the study of topics related to various kinds of boundary value problems for functional differential equations in a Banach space X. A particular attention was paid to the cases where X is the space of continuous functions on a bounded interval or X has finite dimension. For systems of ordinary and functional differential equations, we established efficient conditions guaranteeing the existence and uniqueness of a solution of the initial-value, periodic, and general non-local boundary value problem, as well as the validity of various theorems on differential inequalities, both in the linear and non-linear cases. In the case of higher-order scalar functional differential equations, efficient solvability conditions were established for the periodic boundary value problem, which are also new for ordinary differential equations. Properties of linear hyperbolic partial differential equations with discontinuous right-hand side, including the Fredholm property of the Darboux and Cauchy problems and the continuous dependence on initial conditions and parameters, were also studied. For certain classes of boundary value problems, we studied possibilities of application of the successive approximation method for the proof of the existence, approximate computation of a solution, and error estimation of the approximation. Some of the topics studied in the project are now investigated to such an extent that one can speak of a kind of completeness of the corresponding part of the theory, and the related groups of results can be published in the form of monographs.

 Main investigator:

Hakl Robert