On the existence and multiplicity of branches of nodal solutions for a class of parameter-dependent Sturm-Liouville problems via the shooting map (Q5933763)

The authors consider the first-order planar differential system NEWLINE\[NEWLINEz'= Z(t,z;s)\tag{1}NEWLINE\]NEWLINE in the open domain \(\Omega\subset \mathbb{R}^2\), where \(Z= Z(t,z;s)\): \([0,T]\times \Omega\times I\to \mathbb{R}^2\) is a continuous function depending on the real parameter \(s\in I\) (\(I\subset \mathbb{R}\) is an interval).NEWLINENEWLINENEWLINEThe solutions to (1) satisfying the boundary conditions of Sturm-Liouville type NEWLINE\[NEWLINEz(0)\in \mathbb{L}^s_0,\quad z(T)\in \mathbb{L}^s_T,\quad s\in I,\tag{2}NEWLINE\]NEWLINE where, for each \(s\in I\), \(\mathbb{L}^s_0\) and \(\mathbb{L}^s_T\) are two straight half-lines passing through the origin \(O= (o,o)\in \mathbb{R}^2\) and depending continuously on the parameter \(s\).NEWLINENEWLINENEWLINEThe authors prove the existence of connected branches of solutions of boundary value problem (1), (2).