Let I be an interval and M,N: I×I→I some means with the strict internality property. Suppose that ϕ: I→ℝ is a non-constant and continuous solution of the functional equation
ϕ(M(x,y)) + ϕ(N(x,y)) = ϕ(x) + ϕ(y).
Then ϕ is one-to-one; moreover for every lower semicontinuous function f: I→ℝ satisfying the inequality
f(M(x,y)) + f(N(x,y)) ≤ f(x) + f(y),
the function f◦ϕ^-1 is convex on ϕ(I). This is a generalization of an earlier result of Zs. Páles. An application to the a-Wright convex function is given.

We prove an extension of Lagrange's increment inequality without using Lagrange's mean value theorem and the Hahn-Banach theorems.

A short description of the classical Bochner integral is presented together with the McShane concept of integration based on Riemann type integral sums. The corresponding classes are compared and it will be shown that the situation is different for finite- and infinite- dimensional valued vector functions.

Uniqueness, exact multiplicity and stability of periodic solutions to the periodically forced pendulum equation are discussed. All of this can be considered as a further specification of contributions to the problem of Moser and especially Mawhin's conjecture.

Sufficient conditions for the existence of solutions to the boundary value problems with a Carathéodory right side for the second order ordinary differential systems are established by means of a continuous approximations.

The note deals with differential equations of the second order with Borel measures as coefficients. The problem of existence and uniqueness of solutions is discussed. The Ritz-Galerkin method is used for determining of approximate solutions.

In this article some equations of second order are considered, whose nonlinearity satisfies a global Lipschitz condition. It is shown that the equations with additional conditions admit unique global solutions in the Colombeau algebra ????(ℝ¹).

We consider the second order differential equation
x" = f(t,x,x'),
where f is a Carathéodory function. We prove the existence of at least one solution of the equation satisfying the nonlinear boundary conditions
g₁(x(a),x'(a)) = 0, g₂(x(b),x'(b)) = 0.
Our methods of proofs are based on the topological degree arguments for auxiliary operator equation.

Sufficient conditions for the existence of solutions of one-parameter functional boundary value problems of the type
x" = f(t,x,x_t,x',x'_t,λ),
(x₀,x'₀) ∈ {(ϕ,χ+c); c∈R}, α(x|_J) = A, β(x(T)-x|_J) = B
are given. Here f: J×R×C_r×R×C_r×R→R is continuous, ϕ,χ∈C_r, α,β are continuous increasing functionals, A,B∈R and x|_J is the restriction of x to J=[0,T]. Results are proved by the Leray-Schauder degree method.