arXiv subjects
Publications and source records attributed to Hikmatullo Ismatov.
We characterize the pointwise notion of almost equicontinuity for families of real-valued measurable functions on subsets of $\mathbb R^n$ of finite measure. The characterization is given by means of an integral truncated translation condition. We also provide examples showing that the finite measure assumption and the truncation are essential.
We study a stationary first--order mean field game on the $d$--dimensional torus. The system couples a Hamilton--Jacobi equation for the value function with a transport equation for the density of players. Our goal is to give a detailed and friendly exposition of the monotone--operator argument that yields existence and uniqueness of solutions. We first present a general framework in a Hilbert space and prove existence of a strong solution by adding a simple coercive regularisation and applying Minty's method. Then we specialise to the explicit Hamiltonian \[ H(p,m)=|p|^2-m, \] check all assumptions, and show how the abstract theorem gives existence and uniqueness for this concrete mean field game. The exposition is written in a slow and elementary way so that a motivated undergraduate can follow each step.
We give a self-contained and introductory account of some basic functional analytic tools needed to understand maximal monotone operators in Hilbert spaces. We review domains of (possibly unbounded) operators, closed sets and closed operators, and provide concrete examples of bounded and unbounded operators in both finite and infinite dimensions. We then explain in detail a fundamental result of Br\'ezis: if $A$ is a maximal monotone linear operator, then its domain is dense, $A$ is closed, and $(I+\lambda A)^{-1}$ is a non-expansive mapping for every $\lambda>0$. The Banach fixed point theorem (contraction mapping principle) is stated and used as a key ingredient in the analysis. The presentation is aimed at beginning graduate students and readers seeing these notions for the first time.