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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 \(u^{(n)}= f(t,u)\), \(t\in [a,b]\), \(u^{(i)}(a)= u^{(i)}(b)= \lambda_ i\in \mathbb{R}\), \(i= 0,1,\dots, n-1\) is solved by means of the monotone iterative method. The best estimates for the constant \(M\) in the statement \(u^{(n)}+ Mu\geq 0\), \(M>0\) \((M<0)\), \(u^{(i)}(a)= u^{(i)}(b)\), \(i= 0,1,\dots, n-1\) imply that \(u\geq 0\) in \([a,b]\) (\(u\leq 0\) in \([a,b]\)) are contained for \(n=2\), \(M>0\), \(n=3\), \(M\neq 0\), \(n= 4\), \(M<0\) and for \(n= 2k\geq 6\) the known estimate for \(M<0\) is improved.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
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