A brief summary of scientific research and achievements of Kazimierz Nikodem, professor of mathematics.

We study the problem of solvability of the equation
ϕ(x) = ∫_Ωg(ω)ϕ(f(x,ω))P(dω) + F(x)
where P is a probability measure on a σ-algebra of subsets of Ω, assuming Hölder continuity of F on the range of f.

This article has two main parts. In the first part we show that some of the basic theory of generalized polynomials on commutative semigroups can be extended to all semigroups. In the second part we show that if a sub-semigroup S of a group G generates G in the sense that G = S·S⁻¹, then a generalized polynomial on S with values in an Abelian group H can be extended to a generalized polynomial on G into H. Finally there is a short discussion of the extendability of exponential functions and generalized exponential polynomials.

Let X be an Abelian group, Y be a commutative monoid, K ⊂Y be a submonoid and F : X → 2^Y \ {∅} be a set-valued map. Under some additional assumptions on ideals ℐ₁ in X and ℐ₂ in X², we prove that if F is ℐ_2-almost everywhere K-additive, then there exists a unique up to K K-additive set-valued map G : X → 2^Y \{∅} such that F = G ℐ₁-almost everywhere in X. Our considerations refers to the well known de Bruijn’s result [1].

New upper bounds for the weighted Chebyshev functional under various conditions, including those of Steffensen type, are given. The obtained results are used to establish some new bounds for the Jensen functional.

We consider sequences of Steklov type operators and an associated functional equation. For a suitable sequence, we establish asymptotic formulas.

Cancellation conditions play a central role in the representation theory of measurement for a weak order on a finite two-dimensional Cartesian product set X. A weak order has an additive representation if and only if it violates no cancellation conditions. Given X, a longstanding open problem is to determine the simplest set of cancellation conditions that is violated by every linear order that is not additively representable. Here, we report that the simplest set of cancellation conditions on a 5 by 5 product X is obtained.

In this paper, we consider homogeneous quasideviation means generated by real functions (defined on (0,∞)) which are concave around the point 1 and possess certain upper estimates near 0 and ∞. It turns out that their concave envelopes can be completely determined. Using this description, we establish sufficient conditions for the Hardy property of the homogeneous quasideviation mean and we also furnish an upper estimates for its Hardy constant.

Motivated by the Szostok problem on functions with monotonic differences (2005, 2007), we consider a-Wright convex functions as a generalization of Wright convex functions. An application of these results to obtain new proofs of known results as well as new results is presented.

We analyze in our paper questions of the theory of relativity. We approach this theory from the point of view of velocities and their composition. This is where the functional equations appear. Solving them leads to a world where velocities are bounded from above, the upper bound being exactly the “speed of light”.

We present an adaptive method of approximate integration of convex (as well as concave) functions based on a certain refinement of the celebrated Hermite–Hadamard inequality. Numerical experiments are performed and the role of harmonic numbers is shown.