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

The treated matrix equation $(1+a{e}^{-\frac{\mathrm{\parallel }X\mathrm{\parallel }}{b}})X=Y$ 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 $(1+a{e}^{-\frac{\mathrm{\parallel }X\mathrm{\parallel }}{b}})X=Y$ in general normed spaces $(E,\mathrm{\parallel }\cdot \mathrm{\parallel })$ w.r.t. the parameters $a,b\in \mathbb{R}$, $b\mathrm{\ne }0$, 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.