In the present paper it is proved that every C-polynomial function f: X→Y is a polynomial function, provided C fulfils conditions (1), (2) and X and Y are divisible commutative groups.

We deal with the functional equation
ϕ(F(x)) = g(x,ϕ(x))
where functions F and g are given, defined in open subsets of Banach spaces and taking values in Banach spaces as well. We prove theorems on the existtence and uniqueness of solutions of the equation in classes of differentiable functions. As corollaries we get some results on the conjugacy of diffeomorphisms. Analogous results have been known in the finite dimensional case only.

The paper includes purely combinatorial proof of a theorem that implies the following theorem, stated by Hugo Steinhaus in [6, p.35]: consider a chessboard (rectangular, not necessarily square) with some "mined" squares on. Assume that the king, while moving in accordance with the chess rules, cannot go across the chessboard from the left edge to the right one without meeting a mined square. Then the rook can go across the chessboard from the upper edge to the lower one moving exclusively on mined squares. All proofs of the Steinhaus theorem already published (see [5] and remarks on some proofs in [7, p.211]) are incomplete, except the hexagonal variant proved by Gale in [1].
Steinhaus theorem was thought in [6, p.269] as the lemma in the proof of the Brouwer fixed point theorem for the square (cf. Šaškin [5] and Gale [1]). It can also be used as the lemma for the mountain climbing theorem of Homma [2] (see Mioduszewski [3]). In this paper the Steinhaus theorem is used in the proof of a discrete analogue of the Jordan curve theorem (see Stout [8], where different proof is stated; cf. also Rosenfeld [4]).

The existence of one-one and one-one onto choice functions on a family of subsets of a given nonempty set is studied.

The equation x"(t) = f(t,x(α(t)), x'(β(t))) for t∈[a,b], where the functions α, β deviated argument of type [a,b] → [a,b] is considered.
A sufficient condition for existence of the end b of the interval [a,b], such that there exists the solution x of the above equation on [a,b] fulfilling the boundary value conditions x(a) = A, x(b) = B and ‖x'(a)‖ = v > 0, where the constants a, v and vectors A, B are given, is proved.

We have found conditions for the nonlinearity f which are sufficient for the existence of at least two solutions to the Neumann problem for the equation u" + f(t,u,u') = s.

The paper establishes sufficient conditions for the existence of solutions of a one-parameter differential equation x" = f(t,x,x',λ) satisfying some of the following boundary conditions:
γ(x) = 0,  x'(a) = x'(b) = 0,
x'(a) = x'(b) = 0,  x(c)-x(d) = 0
and
x'(a) = x'(t₀) = x'(b) = 0.
Here γ is a functional. The application is given for a class of one-parameter functional boundary value problems.

We consider discret time dynamical systems with multiplicative perturbations. We give a sufficient condition for the asymptotic stability of Markov operators on measures generated by dynamical systems with multiplicative perturbations.