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Keywords:
generalized linear differential equation; substitution method; variational stability; logarithmic prolongation; ordinary linear differential equation with a substitution
generalized linear differential equation; substitution method; variational stability; logarithmic prolongation; ordinary linear differential equation with a substitution
Summary:
The generalized linear differential equation $dx=d[a(t)]x+df$ where $A,f\in BV^{loc}_n(J)$ and the matrices $I-\Delta^-\ A(t), I+\Delta^+\ A(t)$ are regular, can be transformed $\frac{dy}{ds}=B(s)y+g(s)$ using the notion of a logarithimc prolongation along an increasing function. This method enables to derive various results about generalized LDE from the well-known properties of ordinary LDE. As an example, the variational stability of the generalized LDE is investigated.
References:
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[S1] Schwabik Š.: Generalized differential equations. Fundamental results. Rozpravy ČSAV, Academia Praha 1985. MR 0823224 | Zbl 0594.34002
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