Remark on methods of estimating the number of solutions of nonlinear boundary value problems for ordinary differential equations
The author extends the results by Petrov about the number of solutions of the Dirichlet problem (1) \(x'' = g(t,x) + f(t,x,x')\), \(x(0) = 0\), \(x (\pi) = 0\), to the case of a jumping nonlinearity \(g\) and \(f\) sublinear in \(x\), \(x'\). Assuming that \(\xi\) is a solution of (1), he proves that the number of solutions to (1) is not less than \(|m_+ - k |+ |k - m_- |\), where \(k(m_+\) or \(m_-)\) is the number of zeros in \((0, \pi)\) of a solution (which does not vanish in \(\pi)\) of the initial value problem \(-y'' = g_x (t, \xi) y + f_x (t, \xi, \xi') y + f_{x'} (t, \xi, \xi') y'\), \(y(0) = 0\), \(y'(0) = 1\), \((- z'' = g_+ (t) z^+ - g_- (t) z^-\), \(z(0) = 0\), \(z'(0) = 1\) or \(z'(0) = - 1)\).
scientific article; zbMATH DE number 898894