经典分析与常微分方程

We explicitly construct a heat kernel as a Neumann series for certain function spaces, such as $L^{1}$, $L^{2}$, and Hilbert spaces, associated to a locally compact Hausdorff space $\mathfrak{X}$ with Borel $\sigma$-algebra $\mathcal{B}$,…

The aim of this note is to present the simple observation that a slight refinement of the Contraction Mapping Principle allows one to recover the precise convergence rate in the Picard-Lindel\"of Theorem.

A theorem of Hoischen states that given a positive continuous function $\varepsilon:\mathbb{R}\to\mathbb{R}$, an integer $n\geq 0$, and a closed discrete set $E\subseteq\mathbb{R}$, any $C^n$ function $f:\mathbb{R}\to\mathbb{R}$ can be…

We study the existence of positive eigenvalues with associated nonnegative mild eigenfunctions for a class of abstract initial value problems in Banach spaces with functional, possibly nonlocal, initial conditions. The framework includes…

Let $(x_1,y_1),\ldots,(x_n,y_n)$ be distinct non-constant and non-degenerate solutions of the classical Lotka-Volterra system \begin{equation}\notag \begin{split} x'&= axy + bx\\ y'&= cxy + dy, \end{split} \end{equation} where…

In this paper, we prove a complex version of the incidence estimate of Guth, Solomon and Wang for tubes obeying certain strong spacing conditions, and we use one of our new estimates to resolve a discretized variant of Falconer's distance…

It is widely believed that the energy functional $E_p:(\mathbb{S}^2)^n \rightarrow \mathbb{R}$ $$ E_p = \sum_{i,j=1 \atop i \neq j}^{n} \frac{1}{\|x_i-x_j\|^p}$$ has a number of critical points, $\nabla E(x) = 0$, that grows exponentially…

The error of approximation of the $2\pi$-periodic sawtooth function $(\pi-x)/2$, $0\leq x<2\pi$, by its $n$-th Fourier polynomial is shown to be bounded by arccot$((2n+1)\sin(x/2))$. Related asymptotically tight inequalities with explicit…

We prove existence results for Dirichlet boundary value problems for equations of the type \begin{align*} \left( \Phi(k(t) x'(t) ) \right)' = f(t, x(t) , x'(t) ) \qquad \text{for a.e. } t \in I:=[0,T] , \end{align*} where $\Phi : J \to…

The standard text book theory of ODEs lacks a general method to solve linear equations having variable coefficients, providing instead a collection of special techniques for particular classes of equations. The present article addresses…

We study a second-order differential equation involving a quasi-derivative, leading to a non-self-adjoint Sturm--Liouville-type problem with four coefficient functions. To analyze this equation, we develop a generalized Pr\"ufer…

In the literature concerning the Laguerre-type weight function $x^\lambda w_0(x), x\in[0,+\infty)$, the Jacobi-type weight function $(1-x)^{\alpha}(1+x)^{\beta}w_0(x),x\in[-1,1]$, and the shifted Jacobi-type weight function…

In this paper, we introduce a new generalized derivative, which we term the specular derivative. We establish the Quasi-Rolles' Theorem, the Quasi-Mean Value Theorem, and the Fundamental Theorem of Calculus in light of the specular…

In this paper we propose some continuation theorems for the periodic problem \begin{equation*} \begin{cases} \, x_{i}' = g_{i}(t,x_{i+1}), &i=1,\ldots,n-1, \\ \, x_{n}' = h(t,x_{1},\ldots,x_{n}), \\ \, x_{i}(0)=x_{i}(T), &i=1,\ldots,n,…

We study a wide class of linear inhomogeneous boundary-value problems for $r$th order ODE-systems depending on a parameter $\mu$ in a general metric space $\mathcal M$. The solutions belong to the Sobolev spaces $(W^{n+r}_p)^m$,…

Let $\mathbb{H}^{n}$ be the Heisenberg group and $Q = 2n+2$. For $1 < q < \infty$, $\gamma > 0$ and an exponent function $p(\cdot)$ on $\mathbb{H}^n$, which satisfy log-H\"older conditions, with $0 < p_{-} \leq p_{+} < \infty$, we introduce…

In this paper, we give a new approach to the Bochner-Riesz summability. As a result, we show that the Bochner-Riesz operator $\mathbf{S}^\delta, 0<\Re\delta<{1\over 2}$ is bounded on $\mathbf{L}^p(\mathbb{R}^n)$ for ${n-1\over 2n}\leq…

The Malmquist-Takenaka (MT) system is a complete orthonormal system in $H^2(\mathbf{T})$ generated by an arbitrary sequence of points $a_n$ in the unit disk with $\sum_n (1-|a_n|) = \infty$. The point $a_n$ is responsible for multiplying…

In this work we study the asymptotic behavior of a class of damped second-order gradient systems $$ \ddot{u}(t) + a\dot{u}(t) + \nabla W(u(t)) = 0, $$ under assumptions ensuring local convexity of the potential near equilibrium and…

In this paper, we investigate the representations of rational numbers via continued fraction, Egyptian fraction, and Engel fraction expansions. Given $m \in \mathbb{N}$, denote by $C_m, E_m, E_m^*$ the sets of rational numbers whose…