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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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