The classical result about continuity of midconvex functions, which are bounded from above in a neighbourhood of a single point, is extended on functions on lattice ordered 2-divisible Archimedian groups.

Let α ≥ 0, α ≠ 1 and f: [0,∞[→ℝ be such that f(x+y) = f(x)+f(y)+o(max{x^α,y^α}) as max{x,y}↓0. We show that f(x) = a(x)+o(x^α), where a is an additive function, and we use this result in studying a second order generalized derivative for real functions.

We present a proof of the following theorem. Let E_n ⊂ [0,1] be a sequence of measurable sets with measures μ(E_n) ≥ δ > 0. There is a subsequence whose intersection contains a Cantor set.

The purpose of this paper is to give a new proof of the Borel formula for the convolution product of integrable distributions.

The criteria for an entirely bounded solution of a quasi-linear differential system are developed via asymptotic boundary value problems. The same principle allows us to deduce at the same time the existence of periodic orbits, when assuming additionally periodicity in time variables of the related right-hand sides. For almost periodicity, the situation is unfortunately not so straightforward. Nevertheless, for the Lipschitzean uniformly almost periodic (in time variables) systems, we are able to show that every bounded solution becomes almost periodic as well.

The first part of the paper deals with classification of solutions to the equations
u" + σg(t,u⁽ⁱ⁾) = 0,    i = 0,1; σ² = 1, t≥0.
The second part is devoted to systems of the form
u"(t) = A(t)u⁽ⁱ⁾(t) - g(t,u(h(t)),u'(h(t))),  t∈[0,l]
u⁽ⁱ⁾(0) = u⁽ⁱ⁾(1) = 0, i = 0,1.

It is shown that from the fact that the unique periodic solution of homogeneous system of equations is the trivial one it follows the existence of periodic solutions of nonhomogeneous systems of equations in the Colombeau algebra.

We deal with a conditional functional equation
ϕ(x) = ϕ(y)  implies  f(x+y) = f(x)+f(y).
Under some assumptions imposed on function ϕ, the domains and ranges of ϕ and f we derive the additivity of f. As a consequence we obtain two Hyers-Ulam stability results related to the equation considered.