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By the approximation method introduced in \cite{FYW}, the existence and uniqueness are proved for a class of distribution-dependent stochastic functional differential equations (DDSFDEs). Moreover, combining the Harnack and shift-Harnack…
A linear quaternionic equation in one quaternionic variable q is of the form $a_1 q b_1+a_2 q b_2+ ... +a_m q b_m = c$, where the $a_i, b_j, c$ are given quaternionic coefficients. If introducing basis elements $\bf i, j, k$ of pure…
We consider the equation $\Delta_x u+u_{yy}+f(u)=0,\ x=(x_1,\dots,x_N)\in\mathbb{R}^N,\ y\in \mathbb{R},$ where $N\geq 2$ and $f$ is a sufficiently smooth function satisfying $f(0)=0$, $f'(0)<0$, and some natural additional conditions. We…
We prove a H\"{o}lder-logarithmic stability estimate for the problem of finding a sufficiently regular compactly supported function $v$ on $\mathbb{R}^d$ from its Fourier transform $\mathcal{F} v$ given on $[-r,r]^d$. This estimate relies…
The aim of this note is to investigate the asymptotic stability behaviour of the Cauchy and Jensen functional equations. Our main results show that if these equations hold for large arguments with small error, then they are also valid…
In this paper we obtain a result on Hyers-Ulam stability of the linear functional equation in a single variable $f(\varphi(x)) = g(x) \cdot f(x)$ on a complete metric group.
In this paper, we provide the sufficient and necessary conditions for the symmetry of the following stable L\'evy-type operator $\mathcal{L}$ on $\mathbb{R}$: $$\mathcal{L}=a(x){\Delta^{\alpha/2}}+b(x)\frac{\d}{\d x},$$ where $a,b$ are the…
The purpose of this paper is to investigate the following invariance equation involving two $2$-variable generalized Bajraktarevi\'c means, i.e., we aim to solve the functional equation $$…
In this paper, a two-step regularization method is used to solve an ill-posed spherical pseudo-differential equation in the presence of noisy data. For the first step of regularization we approximate the data by means of a spherical…
For an arbitrary parameter $p\in [1,+\infty]$, we consider the problem of exponential stabilization in the spatial $L^{p}$-norm, and $W^{1,p}$-norm, respectively, for a class of anti-stable linear parabolic PDEs with space-time-varying…
Let D be a domain in the quaternionic space H. We prove a differential criterion that characterizes Fueter-regular quaternionic functions f:bD -> H of class C^1. We find differential operators T and N, with complex coefficients, such that a…
Consider the general scalar balance law $\partial_t u + \Div f(t, x,u) = F(t,x,u)$ in several space dimensions. The aim of this note is to estimate the dependence of its solutions from the flow $f$ and from the source $F$. To this aim, a…
We consider the (in)stability problem of the inviscid 2D Boussinesq equations near a combination of a shear flow $v=(y,0)$ and a stratified temperature $\theta=\alpha y$ with $\alpha>\frac{1}{4}$. We show that for any $\epsilon>0$ there…
In this article we present solutions of certain polynomial equations in periodic nested radicals. We also present a new way to solve the general tetranomial equation with new functions. As application of these new functions we solve the…
In 1922 Ritt described polynomial solutions of the functional equation P(f)=Q(g). In this paper we describe solutions of the equation above in the case when P,Q are polynomials while f,g are allowed to be arbitrary entire functions. In…
The functional equations $ f^2+g^2=1 $ and $ f^2+2\alpha fg+g^2=1 $ are respectively called Fermat-type binomial and trinomial equations. It is of interest to know about the existence and form of the solutions of general quadratic…
In this paper, some global existence and uniform asymptotic stability results for fractional functional differential equations are proved. It is worthy mentioning that when $\alpha=1$ the initial value problem (1.1) reduces to a classical…
In this paper, using the fixed point and direct methods, we prove the generalized Hyers-Ulam-Rassias stability of a Cauchy-Jensen additive functional equation in various normed spaces. The concept of Hyers-Ulam-Rassias stability originated…
Recently, Kulikov (\cite{Ku}) has shown that certain convex functionals on weighted Bergman spaces are maximized by reproducing kernels. We show a sharp quantitative stability of these estimates with the optimal norm and the exponent and an…
In this work characterizations of notions of output stability for uncertain time-varying systems described by retarded functional differential equations are provided. Particularly, characterizations by means of Lyapunov and Razumikhin…