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A note on the solvability of singular boundary value problems. (English) Zbl 0853.34028
The author considers the singular boundary value problem $$\bigl( 1/q(t) \bigr) \bigl( p(t)u' \bigr)' = f \bigl( t,u,p (t)u' \bigr),\ t \in (0,1),$$ $$u(0) = u(1) = 0, \quad \text{when} \quad \int^1_0 \bigl( 1/p(t) \bigr) dt < \infty,$$ or $$u(1) = 0,\ u(t) \in C \bigl( [0,1] \bigr), \quad \text{when} \quad \int^1_0 \bigl( 1/p (t) \bigr) dt = \infty,$$ where $p(t)$, $q(t) > 0$ on $(0,1]$, $q \in C (0,1]$, $p \in C [0,1]$ or $p \in C (0,1]$ and $f \in C ([0,1] \times\bbfR^2)$ or $f \in C((0,1] \times\bbfR^2)$. Under the assumption that $f$ is bounded or $f$ fulfills the classical sign conditions in the second variable and the Nagumo-type conditions in the third variable, he proves the existence. The typical conditions imposed on $p$ and $q$ are $$\int^1_0 q(t) dt < \infty, \quad \int^1_0 \bigl( 1/p (t) \bigr) \int^1_0 q(s) dsdt < \infty. $$ The proof is based on the upper and lower solutions method and the Schauder fixed point theorem. The results complete the earlier ones by the author in [Nonlinear Anal., Theory Methods Appl. 21, 153-159 (1993; Zbl 0790.34027)].

34B15Nonlinear boundary value problems for ODE
Full Text: DOI
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