# zbMATH — the first resource for mathematics

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
Full Text: