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A boundary value problem with a periodic nonlinearity. (English) Zbl 0609.34021
Using variational methods, the author proves the existence of at least one solution of the Picard problem $-u''-u+g(u)=h,$ $u(0)=u(\pi)=0$, assuming $g: {\bbfR}\to {\bbfR}$ continuous, periodic and mean value zero and $h\in L\sp 1(0,\pi)$ such that $\int\sp{\pi}\sb{0}h(t)\sin (t)dt=0.$ This is a semi-linear problem at resonance which doesn’t satisfy the Landesman-Lazer type of conditions: related problems were previously studied by {\it E. N. Dancer} [Ann. Mat. Pura Appl., IV. Ser. 131, 167- 185 (1982; Zbl 0519.34011)] and {\it J. Mawhin} and {\it M. Willem} [J. Differ. Equations 52, 264-287 (1984; Zbl 0557.34036)].
Reviewer: G.Caristi

34B15Nonlinear boundary value problems for ODE
Full Text: DOI
[1] Castro, A.: Periodic solutions of the forced pendulum equation. Differential equations, 149-160 (1980)
[2] DAncer E.N., On the use of asymptotics in nonlinear boundary value problems, preprint. · Zbl 0519.34011
[3] MAwhin J. & WIllem M., Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. diff. Eqns (to appear). · Zbl 0557.34036
[4] Rabinowitz, P.: Some minimax theorems and applications to nonlinear partial differential equations. Nonlinear analysis: A collection of papers in honor of erich H. Rothe, 161-177 (1978)
[5] DRabek P., Existence and multiplicity results for some weakly nonlinear elliptic problems at resonance, Cas. Pest. Mat. (to appear). · Zbl 0524.35045