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Zbl 0852.34018
Cañada, A.; Drábek, P.
On semilinear problems with nonlinearities depending only on derivatives. (English)
[J] SIAM J. Math. Anal. 27, No.2, 543-557 (1996). ISSN 0036-1410; ISSN 1095-7154/e

The authors consider semilinear boundary value problems $$u''(t)+ \lambda_1 u(t)+ g(t, u'(t))= f(t),\quad t\in I,\tag1$$ $$(Bu)(t)= 0,\quad t\in \partial I,\tag2$$ 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).
[A.I.Kolosov (Khar'kov)]

MSC 2000:: *34B15 Nonlinear boundary value problems of ODE
34C25 Periodic solutions of ODE

Keywords: semilinear boundary value problems; solvability

Cited in: Zbl 1033.34021

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