Research On The Theory Of Differential Equations With Discontinuous Right-hand Sides And Related Problems

In recent years, differential equations with discontinuous right-hand sides and related problems, which are derived from realistic engineering, biological and physical backgrounds, have attracted a lot of researchers’ interests. In addition, the vector fields determined by the differential equations with discontinuous righthand sides are not smooth or global Lipschitz, many classical theories on differential equations are no longer applicable, leading to the development of theories and research methods is far from perfection. Therefore, the in-depth discussion from mathematical viewpoints for studying the existing problems of differential equations with discontinuous right-hand sides not only has important theoretical significance, but also has great practical significance. In this Ph.D. thesis, by using mathematical theories and methods such as the theory of set-valued mapping and differential inclusion、nonsmooth analysis and inequalities techniques,especially the theory of functional differential equations with discontinuous righthand sides and nonsmooth critical point theory, developing and improving the relevant theory of functional differential equations with discontinuous right-hand sides, we qualitatively study the dynamic behaviors of some neural networks with discontinuous activations、nonsmooth Lasota-Wazewska model and Nicholson’s blowflies model, which mainly include the existence、dissipativity and(asymptotical、exponential、finite-time) stability of equilibrium, existence and stability of(almost) periodic solutions, we also study a class of Kirchhoff type differential inclusion problem with nonsmooth potential function in unbounded domain and a class of parameter dependent p(x)-Kirchhoff type differential inclusion problem in bounded domain, by using the nonsmooth variational principle, we obtain some new existence and multiplicity results on the considered problems, respectively.These results not only conducive to the further development of mathematics, but also provide a reliable theoretical basis and effective key technologies and methods for science and engineering applications. The dissertation is divided into five chapters. The main contents are as follows:In Chapter 1, a review on the history、recent development and up-to-date progress of problems to be studied is presented. The research work of this thesis is briefly addressed as well, the motivations and significance of this work are also described.In Chapter 2, we introduce some basic preliminaries used in this thesis, which are mainly related to the content of set valued analysis、differential inclusion, functional differential equation, nonsmooth analysis and nonsmooth variational principle. In particular, we develop and extend a class of LaSalle invariance principle for studying the dissipativity problem for discontinuous differential equations.In Chapter 3, by using the fixed point theorems in set-valued analysis、the theory of topological degree, and combining with the nonsmooth analysis techniques、the generalized Lyapunov function(functional) methods and the inequality techniques, we separately study two class of neural network models with discontinuous right-hand sides and a class of memristor-based neural networks and obtain some new results on the existence、stability and dissipativity of equilibria and periodic solutions for the corresponding models. Furthermore, some recent results in the literatures are complemented and improved.In Chapter 4, we propose two classes of biological mathematical models,Lasota-Wazewska model and Nicholson’s blowflies model with discontinuous harvesting and time-varying delays, respectively, we also give a reasonable explanation for the discontinuous harvesting, by employing the nonsmooth analysis and a newly developed method, we obtain completely new criteria on the existence and stability of(almost) periodic solutions of the two investigated models, the established exponential stability of(almost) periodic systems contains the existence criteria of(almost) periodic solutions, we obtain a universal method for the existence and exponential stability of the considered two discontinuous biological mathematical models.In Chapter 5, we first study a class of Kirchhoff type differential inclusion problem in unbounded domain, overcome the difficulty of the techniques and theories for the lack of Sobolev embedding compactness and the differentiability of the nonlinear terms, by using the nonsmooth version mountain pass lemma, and combing the variational methods, we establish the new results on the existence of solutions of the considered problems. Moreover, by employing the nonsmooth version three critical point theorem, combing the the theories of Lebesgue and Sobolev variable exponent spaces, we study the existence and multiplicity of solutions for a class of p(x)-Kirchhoff type differential inclusion problem in bounded domain.