On Ulam Stability of the Inhomogeneous Version of the General Linear Functional Equation

The Ulam stability concerns the following issue: how much an approximate solution to an equation differs from an exact solution to the equation. We prove a general Ulam stability result for the functional equation sigma(m)(i=1) A(i)f (sigma(n)(j=1)a(ij)x(j ))= D(x(1), . . . , x(n)),in the class of functions f mapping a module X, over a commutative ring K, into a Banach space Y, where m and n are fixed positive integers, a(ij) is an element of K for every i is an element of {1,. .., m} and j is an element of {1,. .., n}, A(1),..., A(m) are scalars, and the function D : X-n -> Y is fixed. In this way we generalize an earlier result of A. Bahyrycz and J. Olko. We also show some interesting consequences of this outcome, including conditions sufficient for the existence of solutions to the equation. Particular cases of the equation that we investigate are for instance the functional equations of Cauchy, Jensen, Jordan-von Neumann, Drygas, Fre & acute;chet, Popoviciu, Wright and many others.