Related papers: Stability of a functional equation deriving from q…

We prove H\"{o}lder-logarithmic stability estimates for the problem of finding an integrable function $v$ on $\mathbb{R}^d$ with a super-exponential decay at infinity from its Fourier transform $\mathcal{F} v$ given on the ball $B_r$. These…

In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation $$\int_{S}f(\sigma(y)xt)d\mu(t)-\int_{S}f(xyt)d\mu(t) = 2f(x)f(y), \;x,y\in S,$$ where $S$ is a semigroup, $\sigma$ is…

The purpose of the present paper is to solve (under some assumption on the domain) the equation $$ g(x+y)-g(x)-g(y)=xf(y)+yf(x). $$ After determining the general solutions, we will investigate the so--called alien solutions. %More…

Let $S$ be a semigroup and $X$ a Banach space. The functional equation $\phi (xyz)+ \phi (x) + \phi (y) + \phi (z) = \phi (xy) + \phi (yz) + \phi (xz)$ is said to be stable for the pair $(X, S)$ if and only if $f: S\to X$ satisfying $\|…

In this paper we describe the solutions of the functional equation \begin{equation*} F\Big(\frac{x+y}2\Big)+f_1(x)+f_2(y)=G \big(g_1(x)+g_2(y)) \end{equation*} defined on an open subinterval of $ \mathbb{R} $. Improving previous results we…

The bivariate series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $\theta (q,.)$ is an entire function. We prove a property of stabilization of the coefficients of the…

We consider the stability of the orthogonal Jensen additive and quadratic equations in $F$-spaces, through applying and extending the approach to the proof of a 2010 result of W.Frchner and J.Sikorska, we presenting a new method to get the…

In the present paper by the Fourier transform we show that every linear differential equations of $n$-th order has a solution in $L^1(\Bbb{R})$ which is infinitely differentiable in $\Bbb{R} \setminus \{0\}$. Moreover the Hyers-Ulam…

We study the stability of reaction-diffusion equations in presence of noise. The relationship of stability of solutions between the stochastic ordinary different equations and the corresponding stochastic reaction-diffusion equation is…

We study the functional equation \[ \sum_{i=1}^mf_i(b_ix+c_iy)= \sum_{k=1}^nu_k(y)v_k(x) \] with $x,y\in\mathbb{R}^d$ and $b_i,c_i\in {GL}(d,\mathbb{R})$, both in the classical context of continuous complex-valued functions and in the…

In this paper, we investigate the validity of a quantitative version of stability for the critical Hardy-H\'enon equation \begin{equation*} H(u):=\div(|x|^{-2a}\nabla u)+|x|^{-pb}|u|^{p-2}u=0,\quad u\in D_a^{1,2}(\R^n), \end{equation*}…

In this paper, we study the convergence of the Euler-Maruyama numerical solutions for pantograph stochastic functional differential equations which was proposed in [11]. We also show that the numerical solutions have the properties of…

Our main result is that we describe the solutions $g,f:S\rightarrow\mathbb{C}$ of the functional equation \[g(x\sigma(y))=g(x)g(y)-f(x)f(y)+\alpha f(x\sigma(y)),\quad x,y\in S,\] where $S$ is a semigroup, $\alpha \in \mathbb{C}$ is a fixed…

The stability of the solution to the equation $(*)\dot{u} = F(t,u)+f(t)$, $t\ge 0$, $u(0)=u_0$ is studied. Here $F(t,u)$ is a nonlinear operator in a Banach space $\mathcal{X}$ for any fixed $t\ge 0$ and $F(t,0)=0$, $\forall t\ge 0$. We…

We study the solutions of the integral Kannappan's and Van Vleck's functional equations $$\int_{S}f(xyt)d\mu(t)+\int_{S}f(x\sigma(y)t)d\mu(t) = 2f(x)f(y), \;x,y\in S;$$ $$\int_{S}f( x\sigma(y)t)d\mu(t)-\int_{S}f(xyt)d\mu(t) = 2f(x)f(y),…

For two polynomials of degrees $n$ and $m$ ($n\geq m$) $$ f\left( s\right) =a_{0}+a_{1}s+\ldots+a_{n-1}s^{n-1}+a_{n}s^{n}$$ and $$g\left( s\right) =b_{0}+b_{1}s+\ldots+b_{m-1}s^{m-1}+b_{m}s^{m}$$ we define a set of polynomials $f\bullet g…

We study the regularity of stable solutions to the problem $$ \left\{ \begin{array}{rcll} (-\Delta)^s u &=& f(u) & \text{in} \quad B_1\,, u &\equiv&0 & \text{in} \quad \mathbb R^n\setminus B_1\,, \end{array} \right. $$ where $s\in(0,1)$.…

In this paper, we deal with a type exponential functional equation as follows $$f(xy)=f(x)^{g(y)},$$ where $f$ and $g$ are two real valued functions on a commutative semigroup. Our aim of this paper is to proved that the above functional…

In this paper we extend the interior regularity results for stable solutions in [Cabr\'{e}, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] to operators with variable coefficients. We show that stable solutions to the semilinear…

In this paper, we introduce and solve the following additive-additive $(s,t)$-functional inequality \begin{eqnarray}\label{0.1} && \|g\left(x+y\right) -g(x) -g(y)\| +\| h(x+y) + h(x-y) -2 h(x) \| && \le \left\|s\left( 2…