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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 $x^{(iv)}(t)=f(t,x(t),x'(t),x''(t),x'''(t)),\quad 0<t<1,$ $k_1(\overline x)=0, \quad k_2(\overline x)=0, \quad l_1(\overline x)=0, \quad l_2(\overline x)=0,$ where $$\overline x= (x(0),x(1), x'(0),x'(1),x''(0),x''(1))$$ and $$f:[0,1] \times \mathbb{R}^4 \to \mathbb{R}$$, $$k_j:\mathbb{R}^6 \to \mathbb{R}$$ and $$l_j: \mathbb{R}^6 \to \mathbb{R}$$, $$j=1,2$$, 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 ordinary differential equations 34B27 Green’s functions for ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations
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