We use the approach from Czudek and Szarek (see [1]) to prove the central limit theorem for a stationary Markov chain generated by an iterative function system for a family of increasing, injective functions on [0, 1] with "contractive" properties. We introduce a new approach to prove existence of an unique invariant measure using e-property (see [2]).

Generalized commutative quaternions generalize elliptic, parabolic and hyperbolic quaternions, bicomplex numbers, complex hyperbolic numbers and hyperbolic complex numbers. In this paper, we use the Mersenne numbers and polynomials in the theory of these quaternions. We introduce and study generalized commutative Mersenne quaternion polynomials and generalized commutative Mersenne–Lucas quaternion polynomials.

Given S a semigroup. We study two Pexider-type functional equations
f(xy) + g(xy) = f(x) + f(y) + g(x)g(y),   x,y ∈ S,
and
∫_Sf(xyt)dμ(t) + ∫_Sg(xyt)dμ(t) = f(x) + f(y) + g(x)g(y),   x,y ∈ S,
for unknown functions f and g mapping S into ℂ, where μ is a linear combination of Dirac measures (δ_{z_i}})_i∈I for some fixed elements (z_i)_i∈I contained in S such that ∫_Sdμ(t) = 1.
The main goal of this paper is to solve the above two functional equations and examine whether or not they are equivalent to the systems of equations
f(xy) = f(x) + f(y),  g(xy) = g(x)g(y),   x,y ∈ S,
and
∫_Sf(xyt)dμ(t) = f(x) + f(y),  ∫_Sg(xyt)dμ(t) = g(x)g(y),   x,y ∈ S,
respectively.

This article examines integral inequalities dealing with functions of the form “a function raised to the power of another function” under varying monotonicity and convexity assumptions. First, we assess the validity of a referenced theorem on the subject. Specifically, we present a counterexample and identify a gap in its proof. We then propose an alternative version of the theorem with more flexible convexity assumptions. In addition, we establish new lower and upper bounds for the same integral using refined Hermite–Hadamard integral inequalities. A complementary variant is also discussed. Thus, our results fill gaps in the literature and extend existing results on integral inequalities under classical assumptions.

In this paper, the bidimensional extensions of the Fibonacci numbers are explored, along with a detailed examination of their properties, characteristics, and some identities. We introduce and study the matrices with bidimensional Fibonacci numbers, focusing in particular on their recurrence relation, key properties, determinant, and various other identities. It is our purpose to study the matrix version of bidimensional Fibonacci numbers and provide new results and sometimes extensions of some results existing in the literature. We aim to introduce these matrices using the bidimensional Fibonacci numbers and to give the determinant of these matrices.

We study a recent conjecture proposed by Horst Alzer and Janusz Matkowski concerning a bilinearity property of the Cauchy exponential difference for real-to-real functions. The original conjecture was affirmatively resolved by Tomasz Małolepszy. We deal with generalizations for real or complex mappings acting on a linear space.

In this paper, by Schauder's fixed point theorem and the Banach contraction principle, we consider the existence, uniqueness, and stability of convex solutions of a nonhomogeneous iterative functional differential equation. Finally, some examples were considered by our results.

We determine the solutions of the conditional Drygas equation for functions f₁ and f₂ that satisfy (y²+y)f₁(x) = (x²+x)f₂(y) for all (x, y)∈ℝ² under the additional conditions y = x², or y = log(x), x > 0 or y = exp(x).

In this work we focus on a dynamical system with jumps, where the intensity of the jumps depends on the system’s state. By verifying the assumptions of the theorem from [4], we show that our model satisfies the central limit theorem.

In this paper we will try to answer what conditions must be met by the fuzzy Xor to be used for cryptographic purposes. We will also show that defining the fuzzy Xor using other fuzzy connectives is not suitable for this purpose.

Let S be a semigroup, ℍ be the skew field of quaternions, and ψ: S→S be an anti-endomorphism. We determine the general solution of the functional equation
g(xy) - g(xψ(y)) = 2g(x)g(y),    x,y ∈ S,
where g: S→ℂ is the unknown function. And when S = M is a monoid, we solve the functional equation
g(xy) + g(xψ(y)) = 2g(x)g(y),   x,y ∈ M,
where g: M→ℍ is the unknown function.