This paper surveys a number of recent results obtained by C. Bereanu and the author in existence results for second order differential equations of the form
(ϕ(u'))' = f(t,u,u')
submitted to various boundary conditions. In the equation, ϕ: ℝ → ≤ ]-a,a[ is a homeomorphism such that ϕ(0) = 0. An important motivation is the case of the curvature operator, where ϕ(s) = s/√(1+s²). The problems are reduced to fixed point problems in suitable function space, to which Leray-Schauder theory is applied.