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On semilinear problems with nonlinearities depending only on derivatives. (English) Zbl 0852.34018

The authors consider semilinear boundary value problems

${u}^{\text{'}\text{'}}\left(t\right)+{\lambda }_{1}u\left(t\right)+g\left(t,{u}^{\text{'}}\left(t\right)\right)=f\left(t\right),\phantom{\rule{1.em}{0ex}}t\in I,\phantom{\rule{2.em}{0ex}}\left(1\right)$
$\left(Bu\right)\left(t\right)=0,\phantom{\rule{1.em}{0ex}}t\in \partial I,\phantom{\rule{2.em}{0ex}}\left(2\right)$

where $I=\left[0,\pi \right]$, $B$ denotes either the Dirichlet or the Neumann or the periodic boundary conditions, respectively, and ${\lambda }_{1}$ is the first eigenvalue of the corresponding linear problem ${u}^{\text{'}\text{'}}\left(t\right)+\lambda u\left(t\right)=0$, $t\in I$, $\left(Bu\right)\left(t\right)=0$, $t\in \partial I$. The nonlinear function $g$ is supposed to be bounded and, in some cases, satisfies additional differentiability assumptions and asymptotic conditions. The authors emphasize the dependence of $g$ on the derivative of the solution ${u}^{\text{'}}\left(t\right)$ in order to show the qualitative difference of this case and the Landesman-Lazer-type problem in which the nonlinearity $g$ depends only on the solution $u\left(t\right)$. The authors establish the solvability of the problem (1), (2).

##### MSC:
 34B15 Nonlinear boundary value problems for ODE 34C25 Periodic solutions of ODE
##### Keywords:
semilinear boundary value problems; solvability