In recent years, fractional differential equations are widely used in the many academic disciplines--viscoelastic mechanics, Fractal theory and so on. Furthermore, fractional differential equations can be used to describe some abnormal phenomenon. For instance, fractional convection-diffusion equation can be used to describe the fluid of abnormal infiltration phenomenon in the medium. In this p...

This paper is concerned with the existence and uniqueness of invariant measures for infinite-dimensional stochastic delay lattice systems defined on entire integer set. For Lipschitz drift diffusion terms, we prove by showing tightness a family probability distributions solutions in space continuous functions from finite interval to an space, based idea uniform tail...

We study definably complete locally o-minimal expansions of ordered groups in this paper. A definable continuous function defined on a closed, bounded and set behave like compact set. demonstrate uniform continuity Arzela-Ascoli-type theorem. propose notion special submanifolds with tubular neighborhoods show that any is decomposed into finitely many neighborhoods.

In this chapter, we begin to discuss the ways in which R differs from C[0,1]. In particular, we compare the characterization of compact subsets of R by HeineBorel with the characterization of compact subsets of C[0,1] by Arzela-Ascoli. We find that subsets of C[0,1] must satisfy more conditions than subsets of R if they are to be compact. Before we can begin to investigate this, we have a few p...

This work is devoted to the metrization of probabilistic spaces. More precisely, given such a space (G,D,?) and provided that triangle function ? continuous, we exhibit an explicit canonical metric ?D on G associated topology homeomorphic so-called strong topology. As applications, make advantage this present some fixed point theorems structures prove version Arzela-Ascoli theorem.

A system arising in the modeling of oil{recovery processes is analyzed. It consists of a hyperbolic conservation law governing the saturation and a ellip-tic equation for the pressure. By an operator splitting approach, an approximate solution is constructed. For this approximation appropriate a{priori bounds are derived. Applying the Arzela{Ascoli theorem, local existence and uniqueness of a c...

This paper is devoted to the study of periodic boundary value problems for nonlinear third order di¤erential equations subjected to impulsive e¤ects. We provide su¢ cient conditions on the nonlinearity and the impulse functions that guarantee the existence of at least one solution. Our approach is based on a priori estimates, the method of upper and lower solutions combined with an iterative te...

This paper deals with the asymptotic behavior of solutions to a class non-autonomous fractional stochastic $ p $-Laplacian equations delay driven by nonlinear colored noise on entire space \mathbb{R}^n $. We firstly considered existence continuous random dynamical system for as well uniform estimates respect time delay. then showed pullback asymptotical compactness and uniqueness tempered attra...

1. Compact operators on Banach spaces 2. Appendix: total boundedness and Arzela-Ascoli [Fredholm 1900/1903] treated compact operators as limiting cases of finite-rank operators. [1] [Riesz 1917] defined and made direct use of the compactness condition, more apt for Banach spaces. See [Riesz-Nagy 1952] for extensive discussion in the Hilbert-space situation, and many references to original paper...