A necessary and sufficient condition for singular nonlinear second-order boundary value problems to have positive solutions

In this paper the following result is obtained-. Suppose f (x,u,v) is nonnegative, continuous in (a, b) × R+× R+f may be singular at x = a (and/or x = A) and v = 0 f is nondecreasing on u for each x,v, nonincreasing on v for each x,v there exists a constant 96 (0,1) such that (Formula presented) Then a necessary and sufficient condition for the equation (Formula presented) = 0 on the boundary condition (Formula presented) = 0 to have C1 (7) nonzero solutions is that 0 (Formula presented), where a, β, Y, γ δ are nonnegative real numbers, (Formula presented) is Green’s function of above mentioned boundary value problem (when f (x,u,v) =0). © 1998, Appl. Math.-JCU. All Rights Reserved.