We introduce and study the topological concepts of entropy points, expansivity and shadowing property for dynamical systems on noncompact nonmetrizable spaces, which generalize the relevant concepts for metric spaces. We also obtain various properties on uniform entropy on noncompact nonmetrizable spaces. The main result is a theorem which yields a relation between topological shadowing property and positive uniform entropy.

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It is well known that differential equations with piecewise constant arguments is a class of functional differential equations, which has fascinated many scholars in recent years. These delay differential equations have been successfully applied to diverse models in real life, especially in biology, physics, economics, etc. In this work, we are interested in the existence and uniqueness of asymptotically almost periodic solution for certain differential equation with piecewise constant arguments. Due to the particularity of the equations, we cannot use the traditional method to convert it into the difference equation with exponential dichotomy. Through constructing Cauchy matrix of the investigated system to find the corresponding Green matrix of the difference equation, we need the concept of exponential dichotomy and the Banach contraction fixed point theorem of the corresponding system. Then we give some sufficient conditions to obtain the existence and uniqueness of asymptotically almost periodic solutions for these systems.

Carbonate reservoir is one of the important reservoir in the world. Because of the characteristics of carbonate reservoir, horizontal well and acid fracturing have become a key technology for efficiently developing carbonate reservoir. Establishing corresponding mathematical models and analyzing transient pressure behaviors of this type of well‐reservoir configuration can provide a better understanding of fluid flow patterns in formation as well as estimations of important parameters. A mathematical model for a oil‐water two‐phase flow multiple‐fractured horizontal well with multiple finite‐conductivity fractures in triple media carbonate reservoir by conceptualizing vugs as spherical shapes is presented in this article. A semi‐analytical solution is obtained in the Laplace domain by using source function theory, Laplace transformation, and superposition principle. Analysis of transient pressure responses indicates that nine characteristic flow periods of multiple‐fractured horizontal wells with multiple finite‐conductivity fractures in triple media carbonate reservoir can be identified. Parametric analysis shows that water saturation of matrix, vug and fracture system and acid fracture, fracture half‐length, fracture number, fracture spacing and acid fracture conductivity can significantly influence the transient pressure responses of multiple‐fractured horizontal wells with multiple finite‐conductivity fractures in triple media carbonate reservoir. The model presented in this article can be applied to obtain important parameters pertinent to reservoir or fracture by type curve matching, and it can also provide useful information for optimizing fracture parameters.

In this note, we try to generalize the classical Cauchy-Lipschitz-Picard theorem on the global existence and uniqueness for the Cauchy initial value problem of the ordinary differential equation with global Lipschitz condition, and we try to weaken the global Lipschitz condition. We can also get the global existence and uniqueness.