Newton's Method and Secant Method for Set-Valued Mappings

For finding zeros or fixed points of set-valued maps, the fact that the space of convex, compact, nonempty sets of R is not a vector space presents a major disadvantage. Therefore, fixed point iterations or variants of Newton’s method, in which the derivative is applied only to a smooth single-valued part of the set-valued map, are often applied for calculations. We will embed the set-valued map with convex, compact images (i.e. by embedding its images) and shift the problem to the Banach space of directed sets. This Banach space extends the arithmetic operations of convex sets and allows to consider the Fréchet-derivative or divided differences of maps that have embedded convex images. For the transformed problem, Newton’s method and the secant method in Banach spaces are applied via directed sets. The results can be visualized as usually nonconvex sets in R.