A sequential implicit function theorem for the chords iteration

Year, pages: 2013, 1085 - 1100

In this paper we study the local convergence of the method

$$ 0\in f(p,x_k)+A(x_k+1-x_k)+F(x_k+1), $$

in order to find the solution of the generalized equation

$$ find x\in X such that 0\in f(p,x)+F(x). $$

We first show that under the strong metric regularity of the linearization of the associated mapping and some additional assumptions regarding dependence on the parameter and the relation between the operator $A$ and the Jacobian $\nabla_xf(\bar{p},\bar{x})$, we prove linear convergence of the method which is uniform in the parameter $p$. Then we go a step further and obtain a sequential implicit function theorem describing the dependence of the set of sequences of iterates of the parameter.

Nedelcheva, D. 2013. A sequential implicit function theorem for the chords iteration. In Mathematica Slovaca, vol. 63, no.5, pp. 1085-1100. 0139-9918. DOI: https://doi.org/10.2478/s12175-013-0157-7

Nedelcheva, D. (2013). A sequential implicit function theorem for the chords iteration. Mathematica Slovaca, 63(5), 1085-1100. 0139-9918. DOI: https://doi.org/10.2478/s12175-013-0157-7