Multiple positive solutions to mixed boundary value problems for singular ordinary differential equations on the whole line - Yuji, Yuji (Nonlinear Analysis: Modelling and Control, 2012)
This paper is concerned with the mixed boundary value problem of the second order singular ordinary differential equation [Φ(ρ(t)x’(t))]’ + f(t, x(t), x’(t)) = 0, t ∈ R, limt→−∞ x(t) = ∫−∞+∞ g(s, x(s), x’(s)) ds, limt→+∞ ρ(t)x’(t) = ∫−∞+∞h(s, x(s), x’ (s)) ds. Sufficient conditions to guarantee the existence of at least one positive solution are established. The emphasis is put on the one-dimensional p-Laplacian term [Φ(ρ(t)x’(t))]’ involved with the nonnegative function ρ satisfying ∫−∞+∞1/ρ(s) ds = +∞.