Functional identities involving inverses on Banach algebras

The purpose of this paper is to characterize several classes of functional identities involving inverses with related mappings from a unital Banach algebra $\mathcal{A}$ over the complex field into a unital $\mathcal{A}$-bimodule $\mathcal{M}$. Let $N$ be a fixed invertible element in $\mathcal{A}$, $M$ be a fixed element in $\mathcal{M}$, and $n$ be a positive integer. We investigate the forms of additive mappings $f$, $g$ from $\mathcal{A}$ into $\mathcal{M}$ satisfying one of the following identities: \begin{equation*} \begin{aligned} &f(A)A- Ag(A) = 0\ &f(A)+ g(B)\star A= M\ &f(A)+A^{n}g(A^{-1})=0\ &f(A)+A^{n}g(B)=M \end{aligned} \qquad \begin{aligned} &$\text{for each invertible element}$~A\in$\mathcal{A}$; \ &$\text{whenever}$~ A,B\in$\mathcal{A}$~$\text{with}$~AB=N;\ &$\text{for each invertible element}$~A\in$\mathcal{A}$; \ &$\text{whenever}$~ A,B\in$\mathcal{A}$~$\text{with}$~AB=N, \end{aligned} \end{equation*} where $\star$ is either the Jordan product $A\star B=AB+BA$ or the Lie product $A\star B=AB-BA$.