Two-point boundary value problems for planar systems: a lower and upper solutions approach
The paper is devoted to ensure the existence of solutions of the two dimensional system \[ (P) \begin{cases}x^{\prime}=f(t, x, y), & y^{\prime}=g(t, x, y) \\ (x(a), y(a)) \in \ell_{S}, & (x(b), y(b)) \in \ell_{A},\end{cases} \] where \(f:[a, b] \times \mathbb{R}^{2} \rightarrow \mathbb{R}\) and \(g:[a, b] \times \mathbb{R}^{2} \rightarrow \mathbb{R}\) are continuous functions, and \(\ell_{S}\) and \(\ell_{A}\) are two lines in the plane, the ``starting line'' and the ``arrival line'', respectively. They define a general concept of lower solution \(\alpha\) and upper solution \(\beta\), and consider both the well-ordered case \(\alpha \leq \beta\) and the non-well-ordered case \(\alpha \not \leq \beta\). In the well-ordered case they consider different situations, in the first one they deal with the case when both lines \(\ell_{S}\) and \(\ell_{A}\) are not vertical, in the second one, only one of the two lines is allowed to be vertical, in the third one, the case when both lines are vertical is studied. Moreover, some applications to second order equations are given, in particular for an equation involving the mean curvature operator. In the non well-ordered, they consider nonlinearities that are bounded perturbations of linear ones, and homogeneous Dirichlet boundary conditions. Moreover, they assume that the associated linear system is at resonance and avoid the interaction with higher order eigenvalues. An application of the obtained results is given when a condition of Landesman-Lazer type is assumed. In the last section some possible extensions to higher order dimensional systems are suggested. The existence results are deduced by means of topological degree arguments.
scientific article; zbMATH DE number 7436220
scientific article