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Upper and lower solution methods for fully nonlinear boundary value problems. (English) Zbl 1019.34015

The authors prove the existence of at least one solution to the fully nonlinear boundary problem

${x}^{\left(iv\right)}\left(t\right)=f\left(t,x\left(t\right),{x}^{\text{'}}\left(t\right),{x}^{\text{'}\text{'}}\left(t\right),{x}^{\text{'}\text{'}\text{'}}\left(t\right)\right),\phantom{\rule{1.em}{0ex}}0
${k}_{1}\left(\overline{x}\right)=0,\phantom{\rule{1.em}{0ex}}{k}_{2}\left(\overline{x}\right)=0,\phantom{\rule{1.em}{0ex}}{l}_{1}\left(\overline{x}\right)=0,\phantom{\rule{1.em}{0ex}}{l}_{2}\left(\overline{x}\right)=0,$

where $\overline{x}=\left(x\left(0\right),x\left(1\right),{x}^{\text{'}}\left(0\right),{x}^{\text{'}}\left(1\right),{x}^{\text{'}\text{'}}\left(0\right),{x}^{\text{'}\text{'}}\left(1\right)\right)$ and $f:\left[0,1\right]×{ℝ}^{4}\to ℝ$, ${k}_{j}:{ℝ}^{6}\to ℝ$ and ${l}_{j}:{ℝ}^{6}\to ℝ$, $j=1,2$, are continuous functions that satisfy some monotonicity properties.

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.

Similar results can be obtained for different choices of $\overline{x}$. The $2m$th-order problem is also studied under analogous arguments.

MSC:
 34B15 Nonlinear boundary value problems for ODE 34B27 Green functions 34B05 Linear boundary value problems for ODE