3

I am supervising an end of degree project related to Sturm-Liouville problems. In the paper Singular Sturm comparison theorems I saw the next strange integrals in Theorem 1:

Let $P(x)$, $p(x)$ be continuous functions on the open, finite or infinite interval $(a,b)$ (but not necessarily at its endpoints), and $P(x)\geq p(x)$, $P (x)\not\equiv p(x)$ on $(a, b)$. (i) Suppose that the differential equation $$u′′+p(x)u=0,\quad a<x<b,$$ has a solution $u$ which satisfies the boundary conditions $$\int_a\frac{dx}{u^2(x)}=\infty,\ \int^b\frac{dx}{u^2(x)}=\infty.$$

In the paper, there is no explanation about how they are defined. There is one reference in the paper to the book Ordinary Differential Equations by Philip Hartman. In this book, the notation is used extensively but I could neither found their definition.

My thoughts: they don't seem to be primitives; they neither seem to be improper integrals in an infinite interval.

Can someone help me, please?

Math wind
  • 51
  • This is a just a guess, but the notation suggests the following interpretation to me $$\int_a f(x)dx=\lim_{c\to a+}\int_c^d f(x)dx,$$ and similarly for the other. Does this make sense in context? – saulspatz Nov 27 '18 at 08:13
  • 2
    IMO, "near $x=a$" and "near $x=b$". As the integral diverges at $a$ and $b$, the other bound is unimportant. –  Nov 27 '18 at 10:06
  • @YvesDaoust, thank you very much for your answer. I think that this make sense. – Math wind Nov 28 '18 at 07:54

0 Answers0