Annual Lecture dedicated to the memory of Professor Andrzej Lasota.

In this paper we prove a theorem on strong maximum principles for infinite implicit systems of parabolic differential-functional inequalities together with nonlocal inequalities with functionals in (n + 1)-dimensional sets more general than the cylindrical domain. Results obtained are generalizations of those from [1]–[8] and [10].

The aim of this paper it to characterize n-Jordan homomorphisms and to investigate their connection with n-homomorphisms as well as homomorphisms. Furthermore, the following implication is also verified: if a function ϕ is additive and (for a fixed integer n≥2) the mapping x ↦ ϕ(xⁿ)-ϕ(x)ⁿ satisfies some mild regularity assumption, then the function ϕ is an n-homomorphism or it is a continuous additive function.

We study the existence of positive periodic solutions of the equations
y''(x) - P'(x)y(x) + μQ'(x)f(x,y(x)) = 0,
y''(x) + P'(x)y(x) = μQ'(x)f(x,y(x)),
where μ>0, P and Q are real nondecreasing functions, P' and Q' are 1-periodic distributions, f is a continuous function and 1-periodic in the first variable. The Krasnosielski fixed point theorem on cone is used.

In this paper we prove a general fixed point theorem by altering distance for two pairs of owc hybrid mappings generalizing the main result from [10] and we reduce the study of fixed point of pairs of mappings satisfying a contractive condition of integral type at the study of fixed points in metric spaces by altering distance satisfying an implicit relation.

Report of Meeting. The Fourteenth Debrecen-Katowice Winter Seminar, Hajdúszoboszló (Hungary), January 29 - February 1, 2014.