Inspired by the papers [2, 10] we will study, on 2-divisible groups that need not be abelian, the alienation problem between Drygas’ and the exponential Cauchy functional equations, which is expressed by the equation
f(x + y) + g(x + y) + g(x - y) = f(x)f(y) + 2g(x) + g(y) + g(-y).
We also consider an analogous problem for Drygas’ and the additive Cauchy functional equations as well as for Drygas’ and the logarithmic Cauchy functional equations. Interesting consequences of these results are presented.

In this work it was proved Matkowski’s fixed point theorem. The consequences of this theorem are also presented.

For a continuous and positive function w(λ), λ>0 and μ a positive measure on [0,∞) we consider the following D-logarithmic integral transform
DLog(w,μ)(T) :=∫₀^∞w(λ)ln(\frac{λ+T}{λ})dμ(λ),
where the integral is assumed to exist for T a positive operator on a complex Hilbert space H.

We show among others that, if A, B>0 with BA+AB ≥ 0, then
DLog(w,μ)(A) +DLog(w,μ)(B) ≥ DLog(w,μ)(A+B).
In particular we have
\frac{1}{6}π²+dilog(A+B) ≥ dilog(A) + dilog(B),
where the dilogarithmic function dilog : [0,∞)→ℝ is defined by
dilog(t) :=∫₁^t\frac{ln s}{1-s}ds, t ≥ 0.
Some examples for integral transform DLog(·,·) related to the operator monotone functions are also provided.

In this note, we establish some general results for two fundamental recursive sequences that are the basis of many well-known recursive sequences, as the Fibonacci sequence, Lucas sequence, Pell sequence, Pell-Lucas sequence, etc. We establish some general limit formulas, where the product of the first n terms of these sequences appears. Furthermore, we prove some general limits that connect these sequences to the number e (≈2:71828...).

We introduce the notion of an (α,β,γ) triple system, which generalizes the familiar generalized Jordan triple system related to a construction of simple Lie algebras. We then discuss its realization by considering some bilinear algebras and vice versa. Next, as a new concept, we study triality relations (a triality group and a triality derivation) associated with these triple systems; the relations are a generalization of the automorphisms and derivations of the triple systems. Also, we provide examples of several involutive triple systems with a tripotent element.

We consider the MHD system in a bounded domain Ω⊂ℝ^N, N = 2, 3, with Dirichlet boundary conditions. Using Dan Henry’s semigroup approach and Giga–Miyakawa estimates we construct global in time, unique solutions to fractional approximations of the MHD system in the base space (L²(Ω))^N × (L²(Ω))^N. Solutions to MHD system are obtained next as a limits of that fractional approximations.

In this paper our considerations are focused on some Markov chain associated with certain piecewise-deterministic Markov process with a statedependent jump intensity for which the exponential ergodicity was obtained in [4]. Using the results from [3] we show that the law of iterated logarithm holds for such a model.

We establish necessary and sufficient conditions allowing separation of pair of real functions by an m-convex and by an m-affine function. Some examples and a geometric interpretation of m-convexity of a function is exhibited, as well as a Jensen’s inequality for this kind of function.

In this paper we study a class of impulsive systems of stochastic differential equations with infinite Brownian motions. Sufficient conditions for the existence and uniqueness of solutions are established by mean of some fixed point theorems in vector Banach spaces. An example is provided to illustrate the theory.

Ring properties of amalgamated products are investigated. We offer new, elementary arguments which extend results from [5] and [12] to noncommutative setting and also give new properties of amalgamated rings.

In this paper we introduce some new types of contractive mappings by combining Caristi contraction, Ciric-quasi contraction and weak contraction in the framework of a metric space. We prove some fixed point theorems for such type of mappings over complete metric spaces with the help of '-diminishing property. Some examples are given in strengthening the hypothesis of our established theorems.

The convergence of sequences and non-unique fixed points are established in M-orbitally complete cone metric spaces over the strongly minihedral cone, and scalar weighted cone assuming the cone to be strongly minihedral. Appropriate examples and applications validate the established theory. Further, we provide one more answer to the question of the existence of the contractive condition in Cone metric spaces so that the fixed point is at the point of discontinuity of a map. Also, we provide a negative answer to a natural question of whether the contractive conditions in the obtained results can be replaced by its metric versions.