On the solvability of the matrix equation $(1+ae^{-frac{|X|}{b}})X=Y$

The treated matrix equation $(1+ae^{-frac{|X|}{b}})X=Y$ 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 $(1+ae^{-frac{|X|}{b}})X=Y$ in general normed spaces $(E,|cdot|)$ w.r.t. the parameters $a,binmathbb{R}$, $b eq 0$, and give an algorithm to numerically compute its solutions in $E=mathbb{R}^{mtimes n}$, $m,ninmathbb{N}$, $m,ngeq 2$, equipped with the Frobenius norm.