Vector valued Banach limits and generalizations applied to the inhomogeneous Cauchy equation (Q1731441)

Given a mapping $\phi: V^2\to W$ we consider an inhomogeneous Cauchy equation \[ f(x+y)-f(x)-f(y)=\phi(x,y),\qquad x,y\in V \tag{1} \] with an unknown function $f: V\to W$. This equation has been studied for more than a half of a century. Although it is known that solvability of (1) is equivalent to some conditions imposed on $\phi$, finding explicit formulas for solutions constitutes another problem. \par In the paper the authors present a construction of a solution of (1) for a vector space $V$ over $\mathbb{Q}$ and with a given Hamel basis. Later, developing the concept of a vector-valued Banach limit (as introduced by \textit{R. Armario} et al. [Funct. Anal. Appl. 47, No. 4, 315--318 (2013; Zbl 1326.40005); translation from Funkts. Anal. Prilozh. 47, No. 4, 82--86 (2013)]), they construct a solution $f$ of (1) for $V$ being a rational vector space and $W$ being a normed space admitting such a limit $L$. The solution $f$ is then given in the form \[ f(x)=-L\left(\left(\phi(nx,x)\right)_{n\in\mathbb{N}}\right). \] The paper extends author's earlier results in [\textit{W. Prager} and \textit{J. Schwaiger}, Grazer Math. Ber. 363, 171--178 (2015; Zbl 1367.39006)], where the classical notion of a Banach limit was used to solve (1) in the case $f: \mathbb{R}\to\mathbb{R}$.