Quasi-nonexpansivity and two classical methods for solving nonlinear equations

Abstract:

Let F: ${{\mathbf {R}}_n} \to {{\mathbf {R}}_n}$ be a vector-valued function and let $J(x)$ denote the corresponding Jacobi matrix. The main result states that the functions $x - {J^{ - 1}}(x) \cdot F(x)$ and $x - \lambda {J^T}(x)\cdot F(x)$, where $\lambda$ is a certain positive number, are quasi-nonexpansive. This property is used for establishing the convergence of the Newton and the gradient methods in a finite-dimensional space.