Convexity and the difference property

We show that the class of all delta-convex selfmappings of R$\mathbb{R}$ (differences of two convex functions) enjoys the difference property in the sense of N.G. de Bruijn. The Q$\mathbb{Q}$-differentiability technique has been applied as a proof tool.     

Smale’s point estimate theory for Newton’s method on Lie groups

We introduce the notion of the (one-parameter subgroup) γ$\gamma$-condition for a map f$f$ from a Lie group to its Lie algebra and establish α$\alpha$-theory and γ$\gamma$-theory for Newton’s method for a map f$f$ satisfying this condition. Applications to analytic maps are provided, and Smale’s α$\alpha$-theory and γ$\gamma$-theory are extended and developed. Examples arising from initial value problems on Lie group are presented to illustrate applications of our results.