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

AbstractIn 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