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

The authors consider semilinear boundary value problems $u''(t)+ \lambda_1 u(t)+ g(t, u'(t))= f(t),\quad t\in I,\tag{1}$
$(Bu)(t)= 0,\quad t\in \partial I,\tag{2}$ where $$I= [0, \pi]$$, $$B$$ 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 $$u''(t)+ \lambda u(t)= 0$$, $$t\in I$$, $$(Bu)(t)= 0$$, $$t\in \partial I$$. 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 $$u'(t)$$ 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 $$u(t)$$. The authors establish the solvability of the problem (1), (2).

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations

### Keywords:

semilinear boundary value problems; solvability
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