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The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems. (English) Zbl 0807.34023
The problem $u\sp{(n)}= f(t,u)$, $t\in [a,b]$, $u\sp{(i)}(a)= u\sp{(i)}(b)= \lambda\sb i\in \bbfR$, $i= 0,1,\dots, n-1$ is solved by means of the monotone iterative method. The best estimates for the constant $M$ in the statement $u\sp{(n)}+ Mu\ge 0$, $M>0$ $(M<0)$, $u\sp{(i)}(a)= u\sp{(i)}(b)$, $i= 0,1,\dots, n-1$ imply that $u\ge 0$ in $[a,b]$ ($u\le 0$ in $[a,b]$) are contained for $n=2$, $M>0$, $n=3$, $M\ne 0$, $n= 4$, $M<0$ and for $n= 2k\ge 6$ the known estimate for $M<0$ is improved.

34B15Nonlinear boundary value problems for ODE
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