Nonresonance conditions for fourth order nonlinear boundary value problems (Q1338428)

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 \[ \begin{multlined} f(t,u,u',u'',u''')= f_ 2(t,u,u',u'',u''')u''+ f_ 1(t,u,u',u'',u''')u'+\\ f_ 0(t,u,u',u'',u''')u+ r(t,u,u',u'',u'''),\end{multlined} \] where \(r\) is a bounded function. We obtain existence conditions related to uniqueness conditions for the solution of the linear problem \(u^{(4)}= au+ bu''\), \(u(0)= u(2\pi)\), \(u'(0)= u'(2\pi)\), \(u''(0)= u''(2\pi)\), \(u'''(0)= u'''(2\pi)\).