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On sign-changing solution for some three-point boundary value problems. (English) Zbl 1069.34019

The authors consider the second-order autonomous differential equation \[ y''(t)+f(y)=0 \eqno (1) \] with three-point boundary conditions \[ y(0)=0,\quad \alpha y(\eta)=y(1), \eqno (2) \] where \(\alpha, \eta \in (0,1)\), \(f\in C(\mathbb R)\) is strictly increasing and \(f(0)=0\).
They suppose that \(\lambda_1<\lambda_2<\dots <\lambda_n <\dots \) is the sequence of positive solutions of the equation \[ \sin \sqrt x = \alpha \sin \eta \sqrt x. \] Then, under the assumption that there exists \(n_0\in \mathbb N\) such that \[ \lambda_{2n_0} < \lim_{x\to 0} {f(x)\over x} < \lambda _{2n_0+1},\quad \lim_{| x| \to \infty} {f(x)\over x} < 2(1-\alpha \eta), \] the authors prove the existence of a sign-changing solution of problem (1), (2). Moreover, they get at least one positive and at least one negative solution of (1), (2). Their proofs are based on the fixed-point index method.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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