the monotone iterative method and zeros of bessel functions for nonlinear singular derivative dependent bvp in the presence of upper and lower solutions

in this paper we consider a class of nonlinear singular boundary value problems

-(x^αy′(x))′+ x^αf(x, y(x), x^αy′(x)= 0, 0 < x < 1, y′(0) = y′(1) = 0,

for α ≥ 1. we assume that the source function f (x, y, x^αy′) is lipschitz in x^αy′ and onesided lipschitz in y. the initial approximations are an upper solution u₀(x) and a lower solution v₀(x) which can be ordered in one way, v₀(x) ≤ u₀(x), or the other, u₀(x) ≤ v₀(x). wepropose an iterative scheme and establish the existence of solutions bounded by v₀ and u₀, and allow ∂f /∂y to take both positive and negative values. the method is constructive in nature and can be used to generate solutions of the nonlinear singular boundary value problems.