Existence of multiple positive solutions of singular nonlinear boundary value problems

Abstract

a b s t r a c tFor a given positive integer N, we provide conditions on the nonlinear function f which guarantee that the boundary value problemhas N positive solutions. The nonlinear function f is allowed to be singular at y = 0 and t = 0 but is required to satisfy an integrability condition reminiscent of a condition first used by S. Taliaferro in 1979 and growth conditions as y increases similar to those assumed in the nonsingular case by Henderson and Thompson in 2000. Our main results depend on shooting methods.

A Review on a Class of Second Order Nonlinear Singular BVPs

“…where y ∈ R n and f is a continuous function from [0, T] × R n to R n . Based on shooting method given positive integer N, Baxley et al [181] derived conditions on the nonlinear function f which guarantee that the boundary value problem y = f (t, y), 0 < t < 1, y(0) = 0, y (r) = 0 has N positive solutions. The nonlinear function is allowed to be singular at y = 0 and t = 0 but is required to satisfy an integrability condition reminiscent of a condition first used by Taliaferro [130] and growth conditions are similar to those assumed in the nonsingular case by Henderson et al [182].…”