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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 $$ (\phi(u'))' = f(t,u,u') $$ submitted to various boundary conditions. In the equation, $\phi : \mathbb R \to \left]-a,a\right[$ is a homeomorphism such that $\phi(0) = 0$. An important motivation is the case of the curvature operator, where $\phi(s) = s/\sqrt{1 + s^2}$. The problems are reduced to fixed point problems in suitable function space, to which Leray–Schauder theory is applied.