We characterize all multi-dimensional real self-similar Gaussian Markov processes. Three types of covariance matrix functions occur: white-noise type functions, covariances that can be expressed by continuous matrix semigroups, and covariances based on non-continuous solutions of Cauchy's functional equation. Characterizing the latter requires us to develop some results on the representation theory of non-continuous matrix semigroups, which are presented in a companion paper. In dimension one, besides white noise, the self-similar Gaussian Markov processes reduce to a two-parameter family of time-changed Brownian motions. This observation simplifies several proofs of non-Markovianity of concrete processes found in the literature.

Polynomial Cauchy functional equations: A report

In this article, with simple and short proofs without applying fixed point theorems, some hyperstability results corresponding to the functional equations of Cauchy and Jensen are presented in 2-normed spaces. We also obtain some results on hyperstability for the general linear functional equation f(ax+by)=Af(x)+Bf(y)+C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f(ax+by)=Af(x)+Bf(y)+C$\\end{document}.

In this short note, we introduce probabilistic Cauchy functional equations, specifically, functional equations of the following form: where X 1 and X 2 represent two independent identically distributed real-valued random variables governed by a distribution µ having appropriate support on the real line. The symbol d = denotes equality in distribution. When µ follows an exponential distribution, we provide sufficient (regularity) conditions on the function f to ensure that the unique measurable solution to the above equation is solely linear. Furthermore, we present some partial results in the general case, establishing a connection to integrated Cauchy functional equations.

Functional equation plays a very important and interesting role in the area of mathematics, which involves simple algebraic manipulations and through which one can arrive an interesting solution. The theory of functional equations is also used in the development of other areas such as analysis, algebra, Geometry etc., the new methods and techniques are applied in solving problem in Information theory, Finance, Geometry, wireless sensor networks etc., In recent decades, the study of various types of stability of a functional equation such as HUS (Hyers-Ulam stability), HURS (Hyers-Ulam-Rassias stability) and generalized HUS of different types of functional equation and also for mixed type were discussed by many authors in various space. The problem of the stability of different functional equations has been widely studied by many authors, and more interesting results have been proved in the classical case (Archimedean). In recent years, the analogues results of the stability problem of these functional equations were investigated in non-Archimedean space. The aim of this study is to investigate the HUS of a mixed type of general Quadratic-Quartic Cauchy functional equation in non-Archimedean normed space. In this current article, we prove the generalized HUS for the following Quadratic-Quartic Cauchy functional equation over non-Archimedean Normed space.<img src=image/13428783_01.gif>

In this paper we give some hyperstability and stability results for the Cauchy and Jensen functional equations on restricted domains. We provide a simple and short proof for Brzdȩk’s result concerning a hyperstability result for the Cauchy equation.

The non-existence of a Hamel-basis and the general solution of Cauchy's functional equation for nonnegative numbers

Two entirely different Cauchy functional equations have been investigated in the literature with a couple of unknown operators [Formula: see text] and [Formula: see text] mapping a unitary ring [Formula: see text] into a domain of integrity [Formula: see text] satisfying the elimination law and keeping the operations of addition about operator [Formula: see text] and multiplication about operator [Formula: see text], respectively. A total of four studying syzygies achieved by multiplying these two fundamental Cauchy functional equations side by side are investigated. The novelty of the literature is to find out sufficient conditions upon [Formula: see text] deriving the model to be produced a strongly alienation phenomenon by using some new computation techniques.

In this paper, we consider a fixed point problem related to some contraction mappings and introduce new classes of Picard operators for such mappings in the framework of F-metric space, yielding some interesting and novel results. As application of the obtained results, we investigate the Hyers-Ulam stability of a fixed point problem, a Cauchy functional equation, and an integral equation. Also, we present the well-posedness of the fixed point problem and integral equation. Some illustrative examples are also provided to support the new findings.

I deal with an alienation problem for the system of two fundamental Cauchy functional equations with an unknown function f mapping a ring X into an integral domain Y and preserving binary operations of addition and multiplication, respectively. The resulting syzygies obtained by adding (resp. multiplying) these two equations side by side are discussed. The first of these two syzygies was first examined by Jean Dhombres in 1988 who proved that under some additional conditions concering the domain and range rings it forces f to be a ring homomorphism (alienation phenomenon). The novelty of the present paper is to look for sufficient conditions upon f solving the other syzygy to be alien.

On Nonlinear Random Approximation of 3-variable Cauchy Functional Equation

In this paper, we investigated the asymptotic stability behaviour of the Pexider–Cauchy functional equation in non-Archimedean spaces. We also showed that, under some conditions, if ∥f(x+y)−g(x)−h(y)∥⩽ε, then f,g and h can be approximated by additive mapping in non-Archimedean normed spaces. Finally, we deal with a functional inequality and its asymptotic behaviour.

Generalized Integrated Cauchy Functional Equation with Applications to Probability Models

Matrix homomorphism equations on a class of monoids and non Abelian groupoids

The presented work summarizes the relationships between stability results and separation theorems. We prove the equivalence between different types of theorems on separation by an additive map and different types of stability results concerning the stability of the Cauchy functional equation.

On martingale transformations of multidimensional Brownian Motion

Abstract 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 ) + 2 g ( x ) + g ( y ) + g ( - y ) . f\left( {x + y} \right) + g\left( {x + y} \right)g\left( {x - y} \right) = f\left( x \right)f\left( y \right) + 2g\left( x \right) + g\left( y \right) + g\left( { - y} \right). 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 paper, we generalize the recent hyperstability results obtained by Brzd{\k{e}}k and concerning the Cauchy functional equation \[f(x_1+x_2)=f(x_1)+f(x_2).\] The obtained results are in (2,\(\gamma\))-Banach spaces. The main tool used in the analysis is some fixed point theorem.

We present some hyperstability results for the well-known additive Cauchy functional equation f(x+y)=f(x)+f(y) in n-normed spaces, which correspond to several analogous outcomes proved for some other spaces. The main tool is a recent fixed-point theorem.

First we investigate the Hyers–Ulam stability of the Cauchy functional equation for mappings from bounded (unbounded) intervals into Banach spaces. Then we study the Hyers–Ulam stability of the functional equation f(xy)=xg(y)+h(x)y for mappings from bounded (unbounded) intervals into multi-normed spaces.