This paper considers the following boundary value problem: y⁽ⁿ⁾(t) = λP(t, y), t (0, 1), y^(j)(t_i)=0, J = 0, …, n_i−1, I = 1, …, r, where r 2,n_i 1 for i = 1,… r,∑_i=1^r n_i = n and 0 = t₁ < t₂ < … < t_r = 1 We shall determine those positive values of λ for which the boundary value problem has a solution that is “positive” in some sense. Specifically, criteria are developed for λ to constitute an interval, bounded as well as unbounded, more so explicit intervals of λ are presented. We also include examples to illustrate the importance of the results obtained.