We deal with functional equations of the form
f(x+y) = F(f(x),f(y))
(so called addition formulas) assuming that the given binary operation F is associative but its domain of definition is disconnected (admits "singularities"). The function
Flu,v) := (u+v)/(1+uv)
serves here as a good example; the corresponding equation characterizes the hyperbolic tangent. Our considerations may be viewed as counterparts of L. Losonczi's [4] and K. Domańska's [2] results on local solutions of the functional equation
f(F(x,y)) = f(x) + f(y)
with the same behaviour of the given associative operation F.
Our results exhibit a crucial role of 1 that turns out to be the critical value towards the range of the unknown function. What concerns the domain we admit fairly general structures (groupoids, groups, 2-divisible groups). In the case where the domain forms a group admitting subgroups of index 2 the family of solutions enlarges considerably.

We study the existence of positive solutions of the integral equation
x(t) = ∫₀¹k(t,s)f(s,x(s),x'{s),...,x^(n-1)(s))ds,  n≥2
in both C^n-1[0,1] and W^n-1,p[0,1] spaces, where p≥1. The Krasnosielskii fixed point theorem on cone is used.

Let (X,+,-,0,Σ,μ) be an abelian complete measurable group with μ(X)>0. Let f: X→ℂ be a function. We will show that if A(f)∈L_p⁺(X×X,ℂ) where
A{f)(x,y) = f{x+y) + f(x-y) - 2f(x)f(y),   x,y∈X,
then f∈L_p⁺(X,ℂ) or there exists exactly one function g: X→ℂ with
g{x+y) + g(x-y) - 2g(x)g(y),   x,y∈X
such that f is equal to g almost everywhere with respect to the measure μ.
L_p⁺ denotes the space of all functions for which the upper integral of ∥f∥^p is finite.

We examine the relation between the Itô and Stratonovich integrals in Hilbert spaces. A transition formula has origin in the correction term of the Wong-Zakai approximation theorem.

Report of Meeting. The Fourth Debrecen-Katowice Winter Seminar on Functional Equations and Inequalities February 4-7, 2004 Matráháza, Hangary.