Linear distributional differential equations of the second order. (English) Zbl 0819.34007

Tvrdý, Milan

Math. Bohem. 119, No. 4, 415-436 (1994).

The author is concerned with the linear second order equation (1) \((pu')' + q'u = f''\) with distributional coefficients. Supposing \(p,q \in BV_ R\), \(p(t) \neq 0\) on \([0,T]\), \(p^{-1} \in BV\) and \(f \in G_ R\), where \(G_ R\) stands for the set of all regular functions on \([0, T]\) and \(BV_ R\) denotes the space of the functions of bounded variation on \([0,T]\) and regular on \([0,T]\), he obtains the basic existence and uniqueness results and the solution properties for the homogeneous equation \((pu')' + q'u = 0\) and the equation \((1)\) with the solutions from the space \(G_ R\). The results generalize several known results due to F. V. Atkinson, J. Ligȩza, R. Pfaff, A. B. Mingarelli and to M. Pelant and the author. Finally, the example of a boundary value problem is given, in which the obtained results are utilized and a formula suitable for the numerical approximation of the solution is derived.

Reviewer: J.Kalas (Brno)

MSC:

34A37	Ordinary differential equations with impulses
46F99	Distributions, generalized functions, distribution spaces
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
65J99	Numerical analysis in abstract spaces
34B05	Linear boundary value problems for ordinary differential equations

Keywords:

regulated functions; Perron-Stieltjes integral; Kurzweil integral; generalized differential equation; linear second order equation; distributional coefficients; existence; uniqueness; boundary value problem; numerical approximation