We prove an extension theorem for Pexider equation on a restricted domain in a uniquely 2-divisible Abelian topological group which has a base of open neighbourhoods of 0 satisfying some 2-divisibility conditions.

Abstract. Inspired by the papers by Abbas, Aczél and by Chudziak and Tabor, we consider the problem of existence and uniqueness of extensions for the generalized Pexider equationwhere D is a nonempty open subset of a normed space. We show that the connectedness of D, assumed in the mentioned above papers, can be weakened.Mathematics Subject Classification. 39B52, 39B82.

Let (S, +) be a commutative semigroup, σ : S → S be an endomorphism with σ2 = id and let K be a field of characteristic different from 2. Inspired by the problem of strong alienation of the Jensen equation and the exponential Cauchy equation, we study the solutions f, g : S → K of the functional equation $$f(x + y) + f(x + \sigma (y)) + g(x + y) = 2f(x) + g(x)g(y)\;\;\;\;{\rm for}\;\;x,y \in S.$$ We also consider an analogous problem for the Jensen and the d’Alembert equations as well as for the d’Alembert and the exponential Cauchy equations.

No abstract

We prove that if the principle of equivalent utility under the cumulative prospect theory is positively homogeneous on a relatively small family of risks for every non-negative initial wealth level, then a value function is linear for gains and losses, but, in general, it needs not be linear.