Existence and nonexistence of solutions on opposing mixed convection problems in boundary layer theory

We introduce an integral equation to study the opposing mixed convection problems in boundary layer theory. This equation is of singularities and two integrands take negative values. By means of some special analytical techniques, we prove the existence and the nonexistence of positive solutions of this equation and utilize it to treat analytically the mixed convection parameter ε<−1ε<−1 and the temperature parameter λ>0λ>0 involved in the problems mentioned above. Previous results only treated the case λ=0λ=0 or ε≥−1ε≥−1.