Justyna Sikorska

Norm derivatives and characterizations of inner product spaces

Norm Derivatives Characterizations of Inner Product Spaces Orthogonality Relations Norm Derivatives and Heights Perpendicular Bisectors in Real Normed Spaces Bisectrices in Real Normed Spaces Areas of Triangles in Normed Real Spaces.

Exponential functional equation on spheres

Abstract We study the exponential functional equation f ( x + y ) = f ( x ) f ( y ) on spheres in real normed linear spaces. Regardless of the solutions of this equation, which are already known, we investigate its stability and consider the pexiderized version of it.

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.

Hyperstability of the Fréchet Equation and a Characterization of Inner Product Spaces

We prove some stability and hyperstability results for the well-known Frechet equation stemming from one of the characterizations of the inner product spaces. As the main tool, we use a fixed point theorem for the function spaces. We finish the paper with some new inequalities characterizing the inner product spaces.

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.

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.

Sandwich Theorems for Orthogonally Additive Functions

Let p be an orthogonally subadditive mapping, q an orthogonally superadditive mapping such that p ≤ q or q ≤ p. We prove that under some additional assumptions there exists a unique orthogonally additive mapping f such that p ≤ f ≤ q or q ≤ f ≤ p, respectively.