The aim of this paper is to investigate a discrete integral transform on the real line, which seems to be better adapted for some applications then the Hermite transform (see for example [6]). Another complete orthonormal
system (CON) of functions on the real line, which was introduced by Wiener is more appropriate for nonlinear differential equations of mathematical physics. The reasons are that there exist linearization formulas with respect to the argument as well as with respect to the index and that the functions tend to zero as |x| tends to infinity as quickly as |x|^-1.

Attractors and the associated basins of attraction for triple logistic maps are detected and visualized numerically (see [8]) as well as by the technique of critical manifolds due to C. Mira (see e.g., [7], [5]).

The necessary and sufficient conditions for separation by subadditive and sublinear functions are given.

In this paper we investigate a system of functional equations
N∘N = id
N∘f_k = f_p-1-k∘N   k=0,...,p-1
in finite and infinite interval, where f₀,...,f_p-1 are given real functions. Under suitable assumptions on f_i, we prove that the system has a unique solution and this solution is continuous and decreasing.

The paper deals with boundary value problems of the form
(0.1)      x(t) - x(0) - ∫₀^td[A(s)]x(s) = f(t) - f(0),  t∈[0,1],
(0.2)      Mx(0) + ∫₀¹K(τ)d[x(τ)] = r.
Their solutions are functions regulated on [0,1] and regular on (0,1) (i.e. 2x(t) = x(t-)+x(t+) for all t∈(0,1)). We assume that A and K have bounded variations on [0,1], f is regulated on [0,1] and all of them are regular on (0,1). We derive conditions for the existence and uniqueness of solutions to the given problem. Furthermore, the relationship between the dimensions of the spaces of solutions of the corresponding homogeneous problem and of its adjoint is established. Special attention is paid to the case when the additional condition (0.2) reduces to the periodic boundary condition x(0) = x(1). It is known (cf. [13]) that in the case that A and f are continuous from the right at t = 0 and from the left at t = 1, the equation (0.1) reduces to the distributional differential equation
(0.3)      x' - A'x = f'.
Related results concerning the case of solutions left-continuous on (0,1) were obtained in [18] and similar questions for periodic problems and for linear differential equations with distributional coefficients of the form (0.3) were recently treated by Z. Wyderka [21], cf. also [2], [3] or [10].

Assume that (X,+) are commutative groups, and C is a subset of X fulfilling the conditions C+C ⊆ C and C-C = X. A function f: X→Y is called C-additive function if it satisfies functional equation f(x+y) = f(x) + f{y), for all x,y∈X such that y-x∈C∪(-C)∪{0}. In [1, Theorem 8.4] has been proved that every C-additive function f: X→Y is additive. In the proof the comutativity has been essentially used. Here we present a simple proof of an analogous statement in the case of arbitrary groups.

The present paper studies the finite groups with the following property: the square of each element in the group is central.

The functional differential equation (g(x'(t)))' = (Fx)(t) is considered. Here g is an increasing homomorphism on ℝ, g(0) = 0 and F: C¹(J)→L₁(J) is a continuous operator satisfying a growth condition with respect to x. A class of nonlinear functional boundary conditions is considered and sufficient conditions for the existence at least one positive and one negative solutions of the boundary value problems are given. Results are proved by the homotopy theory, the Leray-Schauder degree and the Borsuk theorem.