On the solvability of the matrix equation $(1+a{e}^{-\frac{\Vert X\Vert }{b}})X=Y$

Abstract

The treated matrix equation $(1+a{e}^{-\frac{\Vert X\Vert }{b}})X=Y$ in this short note has its origin in a modelling approach to describe the nonlinear time-dependent mechanical behaviour of rubber. We classify the solvability of $(1+a{e}^{-\frac{\Vert X\Vert }{b}})X=Y$ in general normed spaces $(E,\Vert \cdot \Vert )$ w.r.t. the parameters $a,b\in \mathbb{R}$, $b\ne 0$, and give an algorithm to numerically compute its solutions in $E={\mathbb{R}}^{m\times n}$, $m,n\in \mathbb{N}$, $m,n\ge 2$, equipped with the Frobenius norm.