POSITIVE SOLUTIONS OF SUPERLINEAR EIGENVALUE PROBLEMS VIA A MONOTONE ITERATIVE TECHNIQUE

The superlinear eigenvalue problems considered in this paper are of the form u′'(x) + λa(x) uv(x) = 0, 0 < x < 1, αu(0) - βu′(0) = 0, γu(1) + δu′(1) = 0 with v > 1. In this paper the existence of a positive solution (λ, u) of the problem is shown by constructing an iterative sequence {λk, uk} of solutions of a related linear eigenvalue problem. It is then shown that λk ↗ λ and uk ↘ u uniformly on [0, 1]. The assumptions are 1. 1. a(x) is positive and in C(0, 1) ∩ L1(0, 1), 2. 2. γβ + γα + αδ ≠ 0, 3. 3. 3.1. a. α, β, γ, δ ≥ 0 or 3.2. b. α > 0, β ≥ 0, δ < 0, 0 < γ < -αδ (α + β) or 3.3. c. β < 0, γ > 0, δ ≥ 0, 0 < α < -γβ (γ + δ). © 1979.