On the existence and nonexistence of positive solutions for nonlinear Sturm-Liouville boundary value problems

In this paper the existence and nonexistence results of positive solutions are obtained for Sturm-Liouville boundary value problem −(p(x)u′)′+q(x)u=f(x,u),x∈(0,1),au(0)−bp(0)u′(0)=0,cu(1)+dp(1)u′(1)=0, where p∈C1[0,1], q∈C[0,1], p(x)>0, q(x)⩾0 for x∈[0,1], f∈C([0,1]×R+), a,b,c,d⩾0 are constants and satisfy (a+b)(c+d)>0. The discussion is based on the positivity estimation for the Green's function of associated linear boundary value problem and the fixed point index theory in cones.