The paper investigates the nonlinear function spaces introduced by Giaquinta, Modica and Souček. It is shown that a function from ${Cart}^p(\Omega ,{𝐑}^m)$ is approximated by ${𝒞}^1$ functions strongly in ${𝒜}^q(\Omega ,{𝐑}^m)$ whenever $q<p$. An example is shown of a function which is in ${cart}^p(\Omega ,{𝐑}^2)$ but not in ${cart}^p(\Omega ,{𝐑}^2)$.

The Asymptotic Numerical Method (ANM) is a family of algorithms, based on computation of truncated vectorial series, for path following problems [2]. In this paper, we present and discuss some techniques to define local parameterization [4, 6, 7] in the ANM. We give some numerical comparisons of pseudo arc-length parameterization and local parameterization on non-linear elastic shells problems