拟巴拿赫空间中广义混合型二、四次函数方程的稳定性

Ulam stability of an additive-quadratic functional equation in F-space and quasi-Banach spaces

By adopting the direct method and fixed point method, we prove that the Hyers-Ulam stability of the following additive-quadratic functional equation \begin{equation} f(x+y, z+w)+f(x-y, z-w)-2 f(x, z)-2 f(x, w)=0 \end{equation} in $β$-homogeneous $F$-spaces and quasi-Banach spaces. There are some differences that we consider the target space with the $β$-homogeneous norm and quasi-norm. Overcoming the $β$-homogeneous norm and quasi-norm bottlenecks, we get some new results.

Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems

We employ positivity of Riesz functionals to establish representing measures (or approximate representing measures) for truncated multivariate moment sequences. For a truncated moment sequence $y$, we show that $y$ lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set $K \subseteq \re^n$ if and only if the associated Riesz functional $L_y$ is $K$-positive. For a determining set $K$, we prove that if $L_y$ is strictly $K$-positive, then $y$ admits a representing measure supported in $K$. As a consequence, we are able to solve the …

On the generalized quadratic mappings in quasi-Banach modules over a $C^*$--algebra

Let $n>2$ be a positive integer. In this paper, we obtain the general solution of the following functional equation n \sum_{1 \le i<j \le n} Q(x_i-x_j)=\sum_{i=1}^{n}Q(\sum_{j =1}^n x_j -n x_i) which is derived from the centroid of the $n$ distinct vectors $x_1, ..., x_n$ in an inner product space. Furthermore, we prove that a mapping $f$ between quasi-Banach modules over a $C^*$-algebra satisfying approximately the equation can be approximated by a quadratic mapping $Q$ satisfying exactly the equation such that $|f(x)-Q(x)|$ is bounded.

On existence theorems of a functional differential equations in partially ordered Banach algebras

In this paper we are concerned with existence results for a coupled system of quadratic functional differential equations. This system is reduced to a fixed point problem for a block operator matrix with nonlinear inputs. To prove the existence we are established some fixed point theorem of Dhage's type for the block matrix operator acting in partially ordered Banach algebras.

可加二次函数方程的模糊稳定性

Fuzzy stability of additive--quadratic functional equations

本文研究模糊巴拿赫空间中函数方程$$f (2x +y)+f (2x -y)= f (x + y)+ f (x -y)+2f (2x)-2f (x)$$的广义Hyers- Ulam稳定性.

Hanson-Wright inequality in Banach spaces

We discuss two-sided bounds for moments and tails of quadratic forms in Gaussian random variables with values in Banach spaces. We state a natural conjecture and show that it holds up to additional logarithmic factors. Moreover in a certain class of Banach spaces (including $L_r$-spaces) these logarithmic factors may be eliminated. As a corollary we derive upper bounds for tails and moments of quadratic forms in subgaussian random variables, which extend the Hanson-Wright inequality.

Maps completely preserving the quadratic operators

Let $\mathcal{A}$ and $\mathcal{B}$ be standard operator algebras on Banach spaces $\mathcal{X}$ and $\mathcal{Y}$, respectively. In this paper, we show that every bijection completely preserving quadratic operators from $\mathcal{A}$ onto $\mathcal{B}$ is either an isomorphism or (in the complex case) a conjugate isomorphism.

Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaces

Our main result gives an improved bound on the filling areas of closed curves in Banach spaces which are not closed geodesics. As applications we show rigidity of Pu's classical systolic inequality and investigate the isoperimetric constants of normed spaces. The latter has further applications concerning the regularity of minimal surfaces in Finsler manifolds.