ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

9. Ordinary Differential Equations and Dynamical Systems

1991 MS Classification: 34Axx Anichini, Giuseppe, Dipartimento di Matematica - Università di Modena, Italy. Conti Giuseppe, Istituto di Matematica - Facoltà di Architettura Università di Firemze, Italy Boundary value problems for implicit ODE’s in a singular case An existence result for boundary value problems for a first order linear (system of) differential equations like x′ = A(t)x + g(t, x, x′), with a nonlinear perturbation depending on the derivative, is sought. The boundary condition we are dealing with will be in the form Mx(t_{1}) + Nx(t_{2}) = r ∈ R^{n} where M and N are constant n × n real matrices. If U(t, s) is the so called evolution matrix related to the homogeneous linear differential equation associated to the given one, then the assumption ker[M + NU(t_{2}, t_{1})] ≠ 0, together some other technical conditions, is considered in order to get the existence result. The main tools we are using are some properties of the generalized inverse of a matrix and a fixed point theorem for set-valued maps which is a generalizations of the Darbo’s theorem.