Maximum principles for ordinary differential inequalities of fourth and sixth order (Q913058)

The authors extend the maximum principle for second order differential inequalities to certain classes of differential inequalities of fourth and sixth order. For example, let \(L_ iu=u''+f_ i(x)u'+h_ i(x)u,\) \(i=1,2\), where \(f_ 1\), \(h_ 1\) are continuous on (a,b) and \(f_ 2,h_ 2\) are of class \(C^ 2(a,b)\). Let \(h_ i(x)\leq 0\), \(i=1,2\). If \(u\in C^ 4(a,b)\cap C^ 1[a,b]\) satisfies the differential inequalities \(L_ 1L_ 2u\geq 0\), \(u'(a)\geq 0\), \(u'(b)\leq 0\) and if u assumes zero as a minimum at a point in (a,b), then \(u(x)=0\) on [a,b].