Utility of extension of functional equations—when possible

Abstract

Jean-Claude Falmagne observed in 1981 [On a recurrent misuse of classical functional equation result. Journal of Mathematical Psychology, 23, 190–193] that, even under regularity assumptions, not all solutions of the functional equation k ( s + t ) = k ( s ) + k ( t ) , important in many fields, also in the theory of choice, are of the form k ( s ) = Cs . This is certainly so when the domain of the equation (the set of ( s , t ) for which the equation is satisfied) is finite. We mention an example showing that this can happen even on some infinite, open, connected sets (open regions). The more general equations k ( s + t ) = ℓ ( s ) + n ( t ) and k ( s + t ) = m ( s ) n ( t ) , called Pexider equations, have been completely solved on R 2 . In case they are assumed valid only on an open region, they have been extended to R 2 and solved that way (the latter if k is not constant). In this paper their common generalization k ( s + t ) = ℓ ( s ) + m ( s ) n ( t ) is extended from open region to R 2 and then completely solved if k is not constant on any interval. Both the general solution without further regularity conditions and under weak regularity condition are given. This equation has further interesting particular cases, such as k ( s + t ) = ℓ ( s ) + m ( s ) k ( t ) and k ( s + t ) = k ( s ) + m ( s ) n ( t ) , that arose in characterization of geometric and power means and in a problem on equivalence of certain utility representations, respectively, where the equation may not hold on the whole real plane, only on an open region. Thus this problem is now solved too.