Project Name :
Boundary Value Problems for Ordinary Differential Equations with Singularities
Donor Organization :
Shota Rustaveli National Science Foundation
Budget:
150 000 GEL
Grant number :
GNSF/ST09_175_3-101
Research Direction :
5 Mathematics, Mechanics
Sub Direction :
5-101 Differential Equations
Keyword(s) :
Differential equation; boundary value problem
Description :
During the last four decades a theory of boundary value problems for ordinary differential equations has been widened noticeably, and covered equations with singularities in a time or phase variables. Nowadays within the scope of this theory there are considered differential equations (differential systems) for which the order of singularities of the right-hand sides or the order of singularities of their one-sided majorants in the time variables is less than the order of the equation (the order of equaitons involving in the system). As for differential equations and systems, which do not satisfy the above-mentioned restriction (i. e. differential equations and systems with strong singularities), for them a theory of boundary value problems is not constructed up to now. The goal of the proposed project is to fill this gap.
In the framework of the project, we elaborate a unified method for investigation of boundary value problems for ordinary differential equations and systems with singularities, which means:
• to generalize a principle of a priori estimates for differential equations and systems with strong singularities, i. e. to prove such general statements which reduce one or another boundary value problem for the above-mentioned equations and systems to the uniform estimate of solutions of an analogous problem for one-parameter families of differential equations and systems with singularities;
• to establish a priori estimates of solutions of two-point, multi-point, and nonlocal boundary value problems for differential inequalities and systems of inequalities with strong singularities in a time variable and with singularities in phase variable.
Using this method, for ordinary differential equations and systems, impulsive and generalized differential systems, and functional-differential equations and systems with strong singularities, we investigate two-point, multi-point, and nonlocal boundary value problems. In particular, optimal in a certain sense conditions will be established guaranteeing, respectively, solvability and unique solvability of those problems as well as stability of their solutions with respect to integrally small perturbations of the considered equation or system.
Duration :
22/01/2010 - 31/12/2012
Project Leader :
Ivan Kiguradze
Project manager :
Nino Partsvania
22/01/2010 - 31/12/2012
Project participant(s) :
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