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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References:

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