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On the Bolza problem. (English) Zbl 0923.34025

A sufficient condition is obtained for the existence of infinitely many solutions to the problem \[ \ddot x+\nabla V(x)= 0,\quad x(0)= x_0,\quad x(T)= x_1, \] where \(T>0\), \(x_0,x_1\in \mathbb{R}^n\) are given. It is assumed that \(V\in C^2(\mathbb{R}^n,\mathbb{R})\) is even and for some \(p>2\) satisfies the inequality \(0<pV(x)\leq (\nabla V(x),x)\) for all large \(| x|\).

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
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