Existence of Positive Solutions for Singular Second-orderm-Point Boundary Value Problems

Abstract

The singular second-order m-point boundary value problem

$$ \left\{ \begin{aligned} & - {\left( {L\varphi } \right)}{\left( x \right)} = h{\left( x \right)}f{\left( {\varphi {\left( x \right)}} \right)},{\kern 1pt} 0 < x < 1, \\ & \varphi {\left( 0 \right)} = 0,{\kern 1pt} \varphi {\left( 1 \right)} = {\sum\limits_{i = 1}^{m - 2} {a_{i} \varphi {\left( {\xi _{i} } \right)}} } \\ \end{aligned} \right. $$

, is considered under some conditions concerning the first eigenvalue of the relevant linear operators, where (Lϕ)(x) = (p(x)ϕ′(x))′ + q(x)ϕ(x) and ξ _i ∈ (0, 1) with 0 < ξ₁ < ξ₂ < · · · < ξ _m−2 < 1, a _i ∈ [0, ∞). h(x) is allowed to be singular at x = 0 and x = 1. The existence of positive solutions is obtained by means of fixed point index theory. Similar conclusions hold for some other m-point boundary value conditions.