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Existence results on the one-dimensional Dirichlet problem suggested by the piecewise linear case. (English) Zbl 0595.34017
The problem at resonance (1) $u''+n\sp 2u+g(u)=f(x)$, (2) $u(0)=u(\pi)=0$ is solved under assumptions that $f\in L\sp 2(0,\pi)$, n is an integer number and $g\in C(R)$ satisfies three conditions. The most interesting of them is the growth condition which allows g to have a very large linear growth in the positive direction provided that it has sufficiently small growth in the negative one, and vice versa. The proof is based on a priori bounds for possible solutions of $u''+n\sp 2u+\lambda g(u)=\lambda f(x)$, $0<\lambda <1$.
Reviewer: W.Seda

34B15Nonlinear boundary value problems for ODE
34C15Nonlinear oscillations, coupled oscillators (ODE)
34C25Periodic solutions of ODE
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