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Ordinary differential equations with discontinuous nonlinearities. (English) Zbl 0790.34020
Two problems concerning solvability of nonlinear boundary value problems with discontinuous right-hand-side are considered. The first one concerns the scalar second order equation $$y''=f_ \varepsilon(y,t)$$, $$y(0)=0=y(1)$$, where $$f_ \varepsilon (y,t)=\overline g(t)+\varepsilon g(y)$$, $$y \in(y_ 1,y_ 2)$$, $$f_ \varepsilon(y,t)=0$$, $$y \notin (y_ 1,y_ 2)$$, $$y_ 1<y_ 2$$ being fixed real numbers, $$\overline g,g \in C^ 1(\mathbb{R})$$ and $$\varepsilon$$ is a small real parameter. This problem is solved via the implicit functions theorem.
In the second part of the paper the author deals with solvability of the problem $${\mathcal L} y=f(y,t)$$, where $${\mathcal L} y=y^{(n)}+p_ 1(t) y^{(n-1)}+\cdots+p_ n(t)$$ is an $$n$$-th order invertible differential operator, with certain linear boundary conditions at $$t=0,1$$. This problem is solved using a modification of the Clarke-Ekeland dual action principle.
Reviewer: O.Došlý (Brno)
MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations