Multiple positive solutions to mixed boundary value problems for singular ordinary differential equations on the whole line

Abstract

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,
lim_t→−∞ x(t) = ∫_−∞^+∞ g(s, x(s), x'(s)) ds,
lim_t→+∞ 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 = +∞.