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Zbl 1019.34015
Ehme, Jeffrey; Eloe, Paul W.; Henderson, Johnny
Upper and lower solution methods for fully nonlinear boundary value problems.
(English)
[J] J. Differ. Equations 180, No.1, 51-64 (2002). ISSN 0022-0396

The authors prove the existence of at least one solution to the fully nonlinear boundary problem $$x^{(iv)}(t)=f(t,x(t),x'(t),x''(t),x'''(t)),\quad 0<t<1,$$ $$k_1(\overline x)=0, \quad k_2(\overline x)=0, \quad l_1(\overline x)=0, \quad l_2(\overline x)=0,$$ where $\overline x= (x(0),x(1), x'(0),x'(1),x''(0),x''(1))$ and $f:[0,1] \times \bbfR^4 \to \bbfR$, $k_j:\bbfR^6 \to \bbfR$ and $l_j: \bbfR^6 \to \bbfR$, $j=1,2$, are continuous functions that satisfy some monotonicity properties. \par Such solution is given as the limit of a sequence of solutions to adequate truncated problems. The result follows from Schauder's fixed-point and Kamke's convergence theorem. \par Similar results can be obtained for different choices of $\overline x$. The $2m$th-order problem is also studied under analogous arguments.
MSC 2000:
*34B15 Nonlinear boundary value problems of ODE
34B27 Green functions
34B05 Linear boundary value problems of ODE

Keywords: nonlinear boundary value problems; Nagumo condition; upper and lower solution

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