General Solution of the n-D Pompeiu Functional Equation

The functional equations have been studied for decades. The functional equations can be applied in mathematical modeling for various problems on physics, engineering or even economics. In this paper, we study the pompeiu functional equation of the form f (x+y+xy) = f (x)+f (y)+f (x) f (y)  for all x, y ∈ X where X is a real Banach space. Next, we generalize the pompeiu functional equation to n -dimensional functional equation which is in the form  f( πn(xi + 1) = πn(f(xi) + 1)-1 for all xi ∈ X as i=1,2,3...,n. We then solve this equation for its general solution. We can prove that a general solution obtained from n -D functional equation is f (x) = M(x +1) - 1 for all x ∈ X, where M(x) is a multiplicative function. This solution is also a solution of the classical pompeiu functional equation.