Lower and upper solutions for a fully nonlinear beam equation. (English) Zbl 1172.34013

Summary: The boundary value problem
\[ u^{(iv)}= f(t,u,u',u'',u'''), \;0<t<1, \]
\[ u(0)= u'(1)= u''(0)= u'''(1)=0, \]
is considered, where \(f:[0,1]\times\mathbb R^4\to\mathbb R\) is a continuous function satisfying a Nagumo-type condition.
We prove the existence of a solution lying between lower and upper solutions using an a priori estimation, lower and upper solutions method and degree theory. The same arguments can be used, with adequate modifications, for any two-point boundary value problem, including all derivatives until order three, with the second and the third derivatives given in different end-points.
An application to the extended Fisher-Kolmogorov problem is given.


34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI


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