Positive solutions of two-point boundary value problems for second-order differential equations with the nonlinearity dependent on the derivative

The existence of positive solutions is established for two-point boundary value problems for second-order differential equations with the nonlinearity dependent on the derivative: {(Lφ)(x)=h(x)f(φ(x),φ′(x)),0≤x≤1,R1(φ)≡α1φ(0)+β1φ′(0)=0,R2(φ)≡α2φ(1)+β2φ′(1)=0, where (Lφ)(x)=−(p(x)φ′(x))′+q(x)φ(x)(Lφ)(x)=−(p(x)φ′(x))′+q(x)φ(x). Conditions are given in terms of the relative behaviors of the quotient f(u,v)|u|+|v| for |u|+|v||u|+|v| near 0 and +∞+∞.