Existence of positive solutions for boundary-value problems with singularities in phase variables

The singular boundary-value problem (g(x′))′ = μf(t,x,x′), x′(0) = 0, x(T) = b > 0 is considered. Here μ is the parameter and f(t,x,y), which satisfies local Carathéodory conditions on [0, T] × (ℝ\{b}) × (ℝ\{0}), may be singular at the values x = b and y = 0 of the phase variables x and y, respectively. Conditions guaranteeing the existence of a positive solution to the above problem for suitable positive values of μ are given. The proofs are based on regularization and sequential techniques and use the topological transversality theorem.