On a class of abstract convex cone valued functional equations

We present an approach to solving a number of functional equations for functions with values in abstract convex cones. Such cones seem to be good generalizations of, e.g., families of nonempty compact and convex subsets or nonempty closed, bounded and convex subsets of a normed space. Moreover, we study some related stability problems

A singular behaviour of a set-valued approximate orthogonal additivity

We show that unlikely to the single-valued case, the set-valued orthogonally additive equation is unstable. After presenting an example showing this phenomenon, we provide some special cases where a set-valued approximately orthogonally additive function can be approximated by the one which satisfies the equation of orthogonal additivity exactly

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

In the first part of the paper we collect and prove several properties of the Hausdorff metric and the Hukuhara difference. They are applied in the next part where a tool for solving several equations for set-valued functions is described

Alienation and the Stability Problem

Starting from the inequality |f(x + y) − f(x) − f(y) + g(x + y) − g(x)g(y)| ε, x, y ∈ S, where f is a complex valued function defined on a monoid S, we deal with two problems: the stability problem and the problem of alienation of the approximate additivity condition from the condition of approximate exponentiality

Orthogonal stability of the Cauchy equation on balls

We deal with stability of some functional equations postulated for orthogonal vectors in a ball centered at the origin. The maps considered are defined on a finite dimensional inner product space and take their values in a real sequentially complete linear topological space. The main result establishes the stability of the corresponding conditional Cauchy functional equation and as a consequence we obtain some other stability results. Results which do not involve the orthogonality relation are considered in more general structures

Orthogonal stability of the Cauchy functional equation on balls in normed linear spaces

We study the stability of some functional equations postulated for orthogonal vectors in a ball centered at the origin. The maps considered are defined on a finite-dimensional normed linear space with Birkhoff-James orthogonality and take their values in a real sequentially complete linear topological space. The main results establish the stability of the corresponding conditional Cauchy functional equation on a half-ball and in uniformly convex spaces on a whole ball. The methods used in the first part of the paper are similar to those from [10]. Since, however, now in a general structure, some additional problems arise, we need several new tools

Generalized orthogonal stability of some functional equations

We deal with a conditional functional inequality , where is a given orthogonality relation, is a given nonnegative number, and is a given real number. Under suitable assumptions, we prove that any solution of the above inequality has to be uniformly close to an orthogonally additive mapping , that is, satisfying the condition . In the sequel, we deal with some other functional inequalities and we also present some applications and generalizations of the first result.</p

On an equation of Sophie Germain

We deal with the following functional equation (Formula Presented.) which is motivated by the well known Sophie Germain identity. Some connections as well as some differences between this equation and the quadratic functional equation (Formula Presented.) are exhibited. In particular, the solutions of the quadratic functional equation are expressed in the language of biadditive and symmetric functions, while the solutions of the Sophie Germain functional equation are of the form: the square of an additive function multiplied by some constant. Our main theorem is valid for functions taking values in a unique factorization domain. We present also an example which shows that our main result does not hold in each integral domain