Nonresonance problems in the limit circle case

Abstract

This chapter presents existence results for the “nonresonant” singular boundary value problem

$$ \left\{ \begin{gathered} \frac{1}{{p(t)}}(p(t)y'(t))' + \mu q(t)y(t) = f(t,y(y)){\text{ a}}{\text{.e}}{\text{. on [0,1]}} \hfill \\ {\text{li}}{{\text{m}}_{t \to 0}} + p(t)y'(t) = y(1) = 0 \hfill \\ \end{gathered} \right.\ $$

(1.1)

where λ _{m −1} ≤ µ ≤ λ _m on [0,1] (or a more general condition described in section 11.2) with λ _{m −1} < µ < λ _m on a subset of [0,1] of positive measure; here λ _m , m =0,1, ... is the (m + 1)^st eigenvalue (described in more detail later) of

$$\left\{ \begin{gathered}Lu = \lambda u{\text{ a}}{\text{.e}}{\text{. on [0,1]}} \hfill \\{\text{li}}{{\text{m}}_{t \to {0^ + }}}p(t)u'(t) = u(1) = 0 \hfill \\ \end{gathered} \right.]$$

(1.2)

where $Lu = - \frac{1}{{pq}}(pu')'$.