Upper and lower solutions method for regular singular differential equations with quasi-derivative boundary conditions

In this paper we consider the class of nonlinear singular differential equations of the type - p ( x ) y ′ ( x ) ′ + q ( x ) f x , y ( x ) , p ( x ) y ′ ( x ) = 0 , 0 < x < 1 , subject to the boundary conditions lim x → 0 p ( x ) y ′ ( x ) = 0 , lim x → 1 p ( x ) y ′ ( x ) = 0 . Conditions on p ( x ) and q ( x ) are imposed so that x = 0 is a regular singular point. An approximation scheme which is iterative in nature is proposed to generate two monotonic sequences. To start the iteration we use upper and lower solutions which can be ordered in one way ( v 0 ⩽ u 0 ) or the other ( u 0 ⩽ v 0 ) . We prove some new existence results and determine the region of multiple solutions.