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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.

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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