Project Name :
Boundary Value Problems in an Infinite Interval for Non-autonomous Ordinary Differential Equations
Donor Organization :
Georgian National Science Foundation
Budget:
77 537 GEL
Grant number :
GNSF/ST06/3-002
Research Direction :
5 Mathematics, Mechanics
Sub Direction :
5-101 Differential Equations
Keyword(s) :
Non-autonomous equation; boundary value problem
Description :
In contrast to the finite interval, a unified theory of boundary value problems in an infinite interval for non-autonomous differential equations did not exist for that time, and the aim of the project was to fill to some extent this gap. To achieve this goal, a new method is elaborated which, first of all, means the establishment of a priori estimates for solutions of differential equations in an infinite interval under various boundary conditions and the non-standard generalization of fixed point theorems.
On the basis of the above-mentioned method, boundary value problems in an infinite interval are investigated for non-autonomous ordinary differential equations and systems, generalized differential systems, impulsive differential systems, and functional-differential systems. In particular, optimal, in a sense, conditions are established guaranteeing the solvability, unique solvability and well-posedness of
(a) periodic type boundary value problems,
(b) problems on bounded and vanishing at infinity solutions, and
(c) problems with conditions at infinity, in particular, problems on transitional solutions.
The obtained results are applied to the qualitative theory of non-autonomous differential equations. Namely:
• the existence of a multi-parametric family of proper solutions (i.e. nontrivial solutions, defined in the neighborhood of infinity) of strongly nonlinear differential equations and systems are established, and their asymptotic (in particular, oscillatory) properties are studied;
• new theorems on zones of the Lyapunov stability and asymptotic stability are proved for two-dimensional linear differential systems with periodic coefficients.
Duration :
01/10/2006 - 30/09/2008
Project Leader :
Ivan Kiguradze
Project manager :
Project participant(s) :
Nino Partsvania
01/10/2006 - 30/09/2008
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