Two-point boundary value problems and extremal points for linear differential equations (Q1260814)

Let (E) \(L_ n y+py=0\) be a differential equation, where \(L_ 0 y=r_ 0(x)y\), \(L_ i y=r_ i(x){d\over d_ x} L_{i-1} y\), \(i=1,\dots,n\). Consider the boundary value problem consisting of equation (E), under certain conditions on \(r_ i\) and \(p\) on \([a,\infty)\), and boundary conditions (BC) \(L_{j_ 0}y(b)=L_{j_ 1} y(b)=\cdots =L_{j_{k- 1}} y(b)=0=L_{j_ k} y(c)=\cdots= L_{j_{n-1}} y(c)\), \(1\leq k\leq n- 1\), \(a\leq b<c<\infty\), \(0\leq j_ 0< j_ 1<\cdots< j_{k-1}\leq n-1\), \(0\leq j_ k<\cdots< j_{n-1}\leq n-1\). For \(a\) fixed \(b\) let \(A(b)\) be the set of points \(c\), \(a\leq b< c<\infty\) such that the problem (E), (BC) has a nontrivial solution, and let \(\Theta(b)=\inf A(b)\) if \(A\neq\emptyset\) and \(\Theta(b)=+\infty\) if \(A=\emptyset\). The point \(\Theta(b)\) is called the extremal point of \((b)\) for (BC). In this paper different classes of boundary conditions (BC) are investigated and a number of comparison theorems for extremal points of two different classes of (BC) is proved. The concept of the so-called \(m\)-admissibility of (BC) plays here an important role.