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Finiteness of the set of solutions of some boundary-value problems for ordinary differential equations. (English) Zbl 0678.34023
Conditions 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
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