Finiteness of the set of solutions of some boundary-value problems for ordinary differential equations.

*(English)*Zbl 0678.34023Conditions are given under which certain nonlinear boundary value problems have at most finitely many solutions. A principal result is the following: Theorem. Consider the scalar second order differential equation (1) \(x''+f(t,x,x')=0\) with Dirichlet boundary conditions (2) \(x(0)=x(\pi)=0\), where f is continuous on [0,\(\pi\) ]\(\times {\mathbb{R}}\times {\mathbb{R}}\) and analytic on \({\mathbb{R}}^ 2\) for each \(t\in [0,\pi]\). Let \(\xi\) (t,y) be the unique local solution of the initial value problem (1) with initial condition (3) \(x(0)=0\), \(x'(0)=y\), and let \(\Omega_ f\) be the (possibly empty) set of \(y\in {\mathbb{R}}\) such that \(\xi\) (\(\cdot,y)\) has interval of existence which includes at least [0,\(\pi\) ]. Then if there exists a compact subset K contained in \(\Omega_ f\) such that every possible solution x of (1)-(2) satisfies \(x'(0)\in K\), then the problem (1)-(2) has at most finitely many solutions. Several corollaries are given which ensure that the hypotheses of the theorem hold. An analogous theorem for the problem (1) with Neumann conditions (4) \(x'(0)=x'(\pi)=0\) is also proved together with corresponding corollaries. In the case of periodic boundary conditions, a first-order result is obtainable in a similar way.

Reviewer: L.Grimm

##### MSC:

34B15 | Nonlinear boundary value problems for ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |