The $k$th conjugate point function for an even order linear differential equation

Abstract:

For an even order, two term equation ${L_n}y = p(x)y,p(x) > 0$, x in $[0,\infty )$, the kth conjugate point function ${\eta _k}(a)$ is defined and is shown to be a strictly increasing continuous function with domain [0, b) or $[0,\infty )$. Extremal solutions are defined as nontrivial solutions with $n - 1 + k$ zeros on $[a,{\eta _k}(a)]$, and are shown to have exactly $n - 1 + k$ zeros, with even order zeros at a and ${\eta _k}(a)$ and exactly $k - 1$ odd order zeros in $(a,{\eta _k}(a))$, thus establishing that ${\eta _k}(a) < {\eta _{k + 1}}(a)$.