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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)),0<t<1,
k 1 (x ¯)=0,k 2 (x ¯)=0,l 1 (x ¯)=0,l 2 (x ¯)=0,

where x ¯=(x(0),x(1),x ' (0),x ' (1),x '' (0),x '' (1)) and f:[0,1]× 4 , k j : 6 and l j : 6 , 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 x ¯. The 2mth-order problem is also studied under analogous arguments.

34B15Nonlinear boundary value problems for ODE
34B27Green functions
34B05Linear boundary value problems for ODE