Substitution method for generalized linear differential equations. (English) Zbl 0743.34023

Fraňková, Dana

Math. Bohem. 116, No. 4, 337-359 (1991).

The paper deals with the problem: how to find to a generalized linear differential equation (1) \(dx=d[A(t)]x+df\), where \(A,f\in BV_ n^{loc}(J)\), such an ordinary differential equation that both equations have the same solutions. This can be done by a substitution method described in the paper which is based on the notion of a logarithmic prolongation of \(A(t)\) along an increasing function \(v(t)\). This enables to obtain the properties of the solutions of (1) similar to those of the linear ordinary differential equations. As an example, there are found sufficient conditions for variational stability of the zero solution of the generalized linear differential equations.

Reviewer: M.Švec (Bratislava)

MSC:

34A30	Linear ordinary differential equations and systems
34D05	Asymptotic properties of solutions to ordinary differential equations
34A99	General theory for ordinary differential equations
34D99	Stability theory for ordinary differential equations

Keywords:

generalized linear differential equation; substitution method; variational stability