2012 Annual Lecture dedicated to the memory of Professor Andrzej Lasota.

The paper describes briefly a history of filtering problems of Markov processes and then concentrates on ergodic properties of filtering process. A mistake in a famous Kunita paper on ergodicity of filtering processes is shown. Then the paper reviews various attempts trying to correct this mistake.

Using a Bessel translation operator, we obtain a generalization of Theorem 2.2 in [3] for the Bessel transform for functions satisfying the (ψ,δ,β)-Bessel Lipschitz condition in the space L_2,α(ℝ⁺).

The existence and uniqueness of a classical solution to a semilinear hyperbolic differential Darboux problem together with semilinear nonlocal conditions in a bounded domain are studied. The Banach fixed point theorem is applied.

The alienation phenomenon of ring homomorphisms may briefly be described as follows: under some reasonable assumptions, a map f between two rings satisfies the functional equation
(*)       f(x + y) + f(xy) = f(x) + f(y) + f(x)f(y)
if and only if f is both additive and multiplicative. Although this fact is surprising for itself it turns out that that kind of alienation has also deeper roots. Namely, observe that the right hand side of equation (*) is of the form Q(f(x), f(y)) with the map Q(u, v) = u+v+uv being a special rational associative operation. This gives rise to the following question: given an abstract rational associative operation Q does the equation
f(x + y) + f(xy) = Q(f(x), f(y))
force f to be a ring homomorphism (with the target ring being a field)?
Plainly, in general, that is not the case. Nevertheless, the 2-homogeneity of f happens to be a necessary and sufficient condition for that effect provided that the range of f is large enough.

Periodicity of an entire function is characterized by the behavior of coefficients of its Maclaurin expansion.

We study Carathéodory functions f: D→Y , where (T, ????) is a measurable space, X, Y are metric spaces and D ⊂ T×X. In the case when ???? is complete and Y is a separable Banach space, we give a characterization of such functions.

Having in mind the ideas of J. Moreau, T. Strömberg and Á. Száz, for any function f and g of one power set ????(X) to another ????(Y), we define an other function (f⚹g) of ????(X) to ????(Y ) such that
(f⚹g)(A) = ∩{f(U)∪g(V): A⊂U∪V⊂X}
for all A⊂X. Thus (f⚹g) is a generalized infimal convolution of f and g.
We show that if f and g preserve arbitrary unions, then (f⚹g) also preserves arbitrary unions. Moreover, if F and G are relations on X to Y such that
F(x) = f({x})  and  G(x) = g({x})
for all x∈X, then
(f⚹g)(A) = F⋂G[A]
for all A⊂X.

Report of Meeting. The Thirteenth Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities, Zakopane (Poland), January 30 – February 2, 2013.