Set-valued Orthogonal Additivity

Abstract

We study the set-valued Cauchy equation postulated for orthogonal vectors. We give its general solution as well as we look for selections of functions satisfying the equation.

On a Method of Solving Some Functional Equations for Set-Valued Functions

“…For the convenience of the reader, in the next lemma we recall from [14] some properties of this convergence.…”

“…In the case of setvalued functions the same methods cannot be used directly (cf., [14]). This is because for the Minkowski difference A − B = {a − b : a ∈ A, b ∈ B}, it is not, in general, true that (A + B) − B equals A.…”

“…In a three-dimensional Euclidean space, by means of (10) we obtain the formula for the distribution law of velocities in an ideal gas at a fixed temperature. There are also applications of (10) in actuarial mathematics in a premium calculation principle (cf., [7]): it was shown, namely, that the variance principle is the only covariance-additive premium principle (for further references concerning (10), see [14]). …”

“…Our goal was to drop the not natural completeness assumption in the target space. However, in order not to go into notational details we assume the domain to be an inner product space (for general settings of the domain see [14]). Proof We apply Theorem 3.1 with β = 2.…”

A Singular Behaviour of a Set-Valued Approximate Orthogonal Additivity

“…where F maps an orthogonality space X into the family of non-empty subsets of a topological space Y (for nonempty sets A and B, by A + B we understand We know the solutions of (2) (see Sikorska [17] The main aim of this paper is to study approximately orthogonally additive multi-valued functions. We will answer the question whether for an approximately orthogonally additive function there exists a function which satisfies the conditional Cauchy equation exactly and which is close (in the sense of the Hausdorff distance) to the given function.…”