Set-valued solutions of a two-variable functional equation with involutions

In this work, we give some characterizations or representations of set-valued solutions defined on a commutative monoid (M, +) with values in a Hausdorff topological vector space of the following two-variable functional equation with involutions: F(x + y, z + w) + F(x + sigma(y), z + tau(w)) = alpha F(x, z) + beta F(y, w), where alpha, beta are fixed nonnegative real numbers and sigma, tau : M -> M are involutions (i.e.,sigma(x + y) = sigma(x) + sigma(y) and sigma circle sigma(x) = x for all x, y is an element of M).