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Zbl 0609.34021
Ward, J.R.jun.
A boundary value problem with a periodic nonlinearity. (English)
[J] Nonlinear Anal., Theory Methods Appl. 10, 207-213 (1986). ISSN 0362-546X

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)].
[G.Caristi]

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

Keywords: nonlinear boundary value problems; variational methods; second order differential equations; Picard problem

Citations: Zbl 0519.34011; Zbl 0557.34036

Cited in: Zbl 0706.34019

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