The nonlinear initial-value problemu″(t)+f(t,u(t))=0,u(t0)+bu′(t0)=c,t0≥0,b≤0,c≥0, is considered for positive solutions on [t0, ∞). Existence of positive solutions is proved without the hypothesis thatf(t, ω)≥0 (or ≤0), using the lattice fixed point theorem. A monotonicity condition inf(t, ω) is used to prove the uniqueness of the solution of the initial-value problem. Whenf(t, ω)≥0 (or ≤0), uniqueness is also obtained under a sublinearity condition onf(t, ω).