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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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