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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)
34B15 Nonlinear boundary value problems for ordinary differential equations