Project Name :
Singular boundary value problems for ordinary differential equations and for partial differential equations of hyperbolic type
Donor Organization :
CRDF-GRDF (Georgian-U. S. Bilateral Grants Program)
Budget:
27 720 USD
Grant number :
3318
Research Direction :
5 Mathematics, Mechanics
Sub Direction :
5-101 Differential Equations
Keyword(s) :
Singular boundary value problem
Description :
A theory of singular boundary value problems for ordinary differential equations and hyperbolic type partial differential equations with two variables is constructed. Namely, unimprovable in a certain sense sufficient conditions are obtained guaranteeing respectively:
(i) Fredholmness, unique solvability and well-posedness of two-point and multi-point boundary value problems and boundary value problems with integral conditions for higher order strongly singular linear ordinary differential equations and singular systems of generalized linear ordinary differential equations;
(ii) stability and asymptotic stability of systems of generalized (in particular, impulsive) linear ordinary differential equations, and well-posedness and weakly well-posedness of the Cauchy problem in an infinite interval for these systems;
(iii) solvability, unique solvability and well-posedness of two-point and multi-point weighted problems for higher order singular nonlinear differential equations and systems, and for second order singular functional differential equations;
(iv) existence of extremal solutions of two-point and the Kneser problems for second order nonlinear singular ordinary differential equations;
(v) solvability and unique solvability of nonlocal boundary value problems for two-dimensional singular differential systems;
(vi) solvability, unique solvability and well-posedness of the Dirichlet and Vallée-Poussin type initial-boundary and boundary value problems for higher order hyperbolic singular partial differential equations;
(vii) solvability and unique solvability of initial-boundary value problems in a rectangle and in an infinite strip for third order nonlinear hyperbolic equations;
(viii) existence and uniqueness of bounded and periodic solutions of nonlinear functional differential equations and systems of hyperbolic type nonlinear differential equations.
To construct the above-mentioned theory, we had to elaborate a new method which is based mainly on an a priori estimate of solutions of one-sided singular differential inequalities under the various boundary conditions, and on the general statement (so-called principle of an a priori boundedness) on solvability of an operator equation in the Banach space. This method can be successfully applied to the investigation of singular boundary value problems for ordinary and hyperbolic type partial functional differential equations.
Duration :
01/03/2003 - 31/08/2004
Project Leader :
Ivan Kiguradze
Project manager :
Project participant(s) :
Nino Partsvania
01/03/2003 - 31/08/2004
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