Cabada, Alberto The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems. (English) Zbl 0807.34023 J. Math. Anal. Appl. 185, No. 2, 302-320 (1994). 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. Reviewer: W.Šeda (Bratislava) Cited in 137 Documents MSC: 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:lower and upper solution; maximum principles; monotone iterative method PDFBibTeX XMLCite \textit{A. Cabada}, J. Math. Anal. Appl. 185, No. 2, 302--320 (1994; Zbl 0807.34023) Full Text: DOI