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A min-max theorem and its applications to nonconservative systems. (English) Zbl 1027.34049
The authors give a nonvariational version of a min-max principle by A. Lazer, and use it to prove a unique existence result on the periodic boundary value problem for the nonconservative system $u''(t)+Au'(t)+\nabla G(u,t)=e(t),$ assuming that the following conditions hold: $B_1+\alpha(\|u\|)I\leq\nabla^2G(u,t)\leq B_2-\beta(\|u\|)I,\quad \int_1^\infty\min\{\alpha(s),\beta(s)\} ds=+\infty,$ where $$u\in\mathbb R^n$$, $$A$$, $$B_1$$ and $$B_2$$ are real symmetric $$n\times n$$-matrices, and the eigenvalues of $$B_1$$ and $$B_2$$ are $$N_i^2$$ and $$(N_i + 1)^2$$, $$i =1,\dots,n$$, respectively; here, $$N_i$$ are nonnegative integers, $$\alpha (s)$$ and $$\beta(s)$$ are two positive nonincreasing functions for $$s\in [0, \infty)$$, $$\nabla f$$ and $$\nabla^2 f$$ are the gradient and the Hessian of $$f$$, respectively. Examples show that the main results extend known results.

MSC:
 34C25 Periodic solutions to ordinary differential equations 49J35 Existence of solutions for minimax problems 34B15 Nonlinear boundary value problems for ordinary differential equations
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