## Generalized solutions of boundary value problems for ordinary linear differential equations of second order in the Colombeau algebra.(English)Zbl 0837.34026

Let $$A_1$$ be the set of all smooth functions $$\varphi : \mathbb{R} \to \mathbb{R}$$ such that $$\int \varphi (x) dx = 1$$, $$\int x^k \varphi (x) dx = 0$$ $$(0 < k \leq q)$$. Let $$E$$ be the ring of functions $$R : A_1 \to \mathbb{R}$$ such that $$|R (\varphi (x/ \varepsilon)/ \varepsilon) |\leq c \varepsilon^{-N}$$ for appropriate $$N \in \mathbb{N}$$, $$c > 0$$, $$\eta > 0$$, and all $$\varphi \in A_q$$ $$(q \geq N)$$, $$0 < \varepsilon < \eta$$. Let $$T \subset E$$ be the ideal of functions $$R$$ satisfying $$|R (\varphi (x/ \varepsilon)/ \varepsilon |\leq c \varepsilon^{\alpha (q) - N}$$ for appropriate function $$\alpha : \mathbb{N} \to \mathbb{R}^+$$ such that $$\alpha (q) \to \infty$$ as $$q \to \infty$$. Denoting $$\overline \mathbb{R} = E/T$$, the Colombeau algebra $$G$$ includes (roughly speaking) all smooth functions $$\mathbb{R} \to \overline \mathbb{R}$$ and may be regarded for a multiplicative theory of generalized functions, see J. F. Colombeau [New generalized functions and multiplication of distributions, North- Holland, Amsterdam (1984; Zbl 0532.46019)]. The author states conditions for the coefficients $$p,q,r \in G$$ which ensure the existence and the uniqueness of solutions $$x \in G$$ of the boundary value problem $$x''(t) + p(t)x'(t) + q(t)x(t) = r(t)$$, $$x(a) = d_1$$, $$x(b) = d_2$$ $$(d_1, d_2 \in \overline \mathbb{R})$$.
Reviewer: J.Chrastina (Brno)

### MSC:

 34B05 Linear boundary value problems for ordinary differential equations 34B99 Boundary value problems for ordinary differential equations 46F99 Distributions, generalized functions, distribution spaces

### Keywords:

Colombeau algebra; existence; uniqueness; boundary value problem

Zbl 0532.46019