Annual Lecture dedicated to the memory of Professor Andrzej Lasota.

The aim of this paper is to prove a general fixed point result by altering distances for two occasionally weakly compatible (owc) pairs of hybrid mappings and to reduce the study of fixed points of the pairs of mappings satisfying a contractive condition of integral type at the study of fixed points in metric spaces by altering distances satisfying a new type of implicit relations generalizing the result recently obtained by H.K. Pathak and Naseer Shahzad (see Bull. Belg. Math. Soc. Simon Stevin, 16 (2009), 1-12) which is of Gregus type.

Fejér-type inequalities as well as some refinement and a discrete version of the Hermite–Hadamard inequalities for strongly convex functions are presented.

In this paper, we prove a generalization of Titchmarsh’s theorem for the Bessel transform in the space L_p,α(ℝ⁺) for functions satisfying the (ψ,p)-Bessel Lipschitz condition.

Inspired by a problem posed by J. Matkowski in [10] we investigate the equation
f(p(x,y)(xf(y)+y) + (1−p(x,y))(yf(x)+x))) = f(x)f(y),  x,y∈ℝ,
where functions f: ℝ→ℝ, p: ℝ²→ℝ are assumed to be continuous.

An A-m-isometry is a bounded linear operator T on a Hilbert space ℍ satisfying an identity of the form Σ_k=0^m(-1)^m-k\binom{m}{k}T*^kAT^k = 0, where A is a positive (semi-definite) operator. In this paper, we show that the results for the supercyclicity and the hypercyclicity of m-isometries described in [6, 8] remain true for A-m-isometries.

Under appropriate conditions on abelian topological groups G and H, an orthogonality ⟂⊂G² and a σ-algebra ???? of subsets of G we prove that if at least one of the functions f,g,h: G→H satisfying
f(x+y) − g(x) − h(y) ∈ K  for x,y∈G such that x⟂y,
where K is a discrete subgroup of H, is continuous at a point or ????-measurable, then there exist: a continuous additive function A: G→H, a continuous biadditive and symmetric function B: G×G→H and constants a,b∈H such that
f(x) − B(x,x) − A(x) − a ∈ K,
g(x) − B(x,x) − A(x) − b ∈ K,
h(x) − B(x,x) − A(x) − a + b ∈ K
for x∈G and
B(x,y) = 0 for x,y∈G such that x⟂y.

Report of Meeting. The Twelfth Debrecen-Katowice Winter Seminar on Functional Equations and Inequalities, Hajdúszoboszló (Hungary), January 25-28, 2012.