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Nonresonance conditions for fourth order nonlinear boundary value problems. (English) Zbl 0810.34017
Summary: This paper is devoted to the study of the problem $$u^{(4)}= f(t,u,u',u'',u''')$$, $$u(0)= u(2\pi)$$, $$u'(0)= u'(2\pi)$$, $$u''(0)= u''(2\pi)$$, $$u'''(0)= u'''(2\pi)$$. We assume that $$f$$ can be written in the form $\begin{split} f(t,u,u',u'',u''')= f_ 2(t,u,u',u'',u''')u''+ f_ 1(t,u,u',u'',u''')u'+\\ f_ 0(t,u,u',u'',u''')u+ r(t,u,u',u'',u'''),\end{split}$ where $$r$$ is a bounded function. We obtain existence conditions related to uniqueness conditions for the solution of the linear problem $$u^{(4)}= au+ bu''$$, $$u(0)= u(2\pi)$$, $$u'(0)= u'(2\pi)$$, $$u''(0)= u''(2\pi)$$, $$u'''(0)= u'''(2\pi)$$.

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