Functional equations on restricted domains

An alternative Jensen's functional equation on semigroups

“…In addition, we will extend the result to functions defined on groups and will eventually show that in the case of functions f from a 2-divisible group to a uniquely divisible abelian group, the alternative Jensen's functional equation (3) is equivalent to Jensen's functional equation (2).…”

Equivariant perturbation in Gomory and Johnson’s infinite group problem—III: foundations for the k-dimensional case with applications to $$k=2$$ k = 2

“…where u, v ∈ R (see, e.g., [1,10,[21][22][23]). This result is known as the Interval Lemma in the integer programming community [19].…”

“…It is convenient to break it into convex sets F , which we then study independently. (5) and (6) are referred to as conditional Cauchy and Pexider equations or as Cauchy and Pexider equations on restricted domains [14,22]. It is clear that the Pexider equation imposes no conditions on the function values of f , g, and h outside of the projections p 1 (F ), p 2 (F ), and p 3 (F ), respectively, where the projections are as defined in (4).…”

Characterization of special classes of solutions for some functional equations on orthogonal vectors

“…Also in this case some suitable examples point out expressive situations: namely (3) 1 is non-equivalent to (3), and then neither to (1) 1 nor to (1) as the following functionals show.…”

“…Functional equations restricted to orthogonal vectors have been dealt with by several authors; specially, additive, quadratic and exponential equations on this restricted domain have been studied in very general structures and a lot of important results are now known (see [3], [4] for many useful references).…”