Periodic boundary-value problems and the Dancer-Fucik spectrum under conditions of resonance

We prove the existence of solutions to the nonlinear $2 pi$-periodic problem $$displaylines{ u''(x)+mu u^+(x)-u u^-(x)+g(x,u(x))=f(x),,quad xin (0,2pi),,cr u(0)-u(2pi) =0 ,, quad u'(0) - u'(2pi)=0, }$$ where the point $(mu,u)$ lies in the Dancer-Fucik spectrum, with $$ 0< frac...

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Bibliographic Details
Published in:Electronic journal of differential equations 2011-08, Vol.2011 (112), p.1-34
Main Authors: David A. Bliss, James Buerger, Adolfo J. Rumbos
Format: Article
Language:English
Subjects:
Dancer-Fucik spectrum
jumping nonlinearities
Resonance
saddle point theorem
Online Access:Get full text
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Summary:We prove the existence of solutions to the nonlinear $2 pi$-periodic problem $$displaylines{ u''(x)+mu u^+(x)-u u^-(x)+g(x,u(x))=f(x),,quad xin (0,2pi),,cr u(0)-u(2pi) =0 ,, quad u'(0) - u'(2pi)=0, }$$ where the point $(mu,u)$ lies in the Dancer-Fucik spectrum, with $$ 0< frac{4}{9}mu leqslant u
ISSN:1072-6691