摘要：
This paper focuses on the problem of the finite-time flocking with uniform minimal distance for second-order multi-agent systems. The solutions of these issues can be viewed as the reasonable explanations of the bird flocks or fish schools. A new discontinuous protocol, which combines a singular communication function with a weighted sum of sign functions of the relative velocities among agents, is proposed to guarantee that the agents can attract and repel with each other. Since the communication weight is singular, the existence and uniqueness theorem cannot be applied directly. However, by imposing some suitable conditions on the initial states and using the skill of the proof by contradiction, the existence of the global smooth solution is obtained. Furthermore, employing a finite time stability theory and constructing a Lyapunov function ingeniously, a flocking with least distance for the multi-agent system is acquired within a finite-time. Moreover, the bound of settling time can be estimated by the parameters and the initial states and this relationship show that the more the number of particles, the larger the bound of convergence time. Finally, numerical simulations are provided to demonstrate the effectiveness of the theoretical results.
摘要：
This work focuses on the fixed-time consensus problem for multi-agent systems. By employing an improved fixed-time stability theorem and Lyapunov functions, it is proved that the proposed protocols can achieve fixed-time consensus when the interaction topology is strongly connected and detail-balanced. In particular, the settling time of the presented protocols is less than that of the existing ones, which is illustrated by both theoretical analysis and numerical simulations.
期刊：
JOURNAL OF APPLIED ANALYSIS AND COMPUTATION,2019年9(4):1393-1406 ISSN：2156-907X
通讯作者：
Xiao, Qizhen
作者机构：
[Xiao, Qizhen; Liu, Hongliang; Liu, Zhisu] Univ South China, Sch Math & Phys, Hengyang 421001, Peoples R China.;[Shi, Hongxia] Hunan First Normal Univ, Sch Math & Computat Sci, Changsha 410205, Hunan, Peoples R China.;[Chen, Haibo] Cent S Univ, Sch Math & Stat, Changsha 410083, Hunan, Peoples R China.
通讯机构：
[Xiao, Qizhen] U;Univ South China, Sch Math & Phys, Hengyang 421001, Peoples R China.
关键词：
Biharmonic equation;ground state and nodal solution;variational methods;Hessian matric;energy doubling
期刊：
Advances in Difference Equations,2018年2018(1):1-18 ISSN：1687-1847
通讯作者：
Liu, Hongliang
作者机构：
[Xiao, Qizhen; Liu, Hongliang; Ouyang, Zigen] Univ South China, Sch Math & Phys, Hengyang, Peoples R China.
通讯机构：
[Liu, Hongliang] U;Univ South China, Sch Math & Phys, Hengyang, Peoples R China.
关键词：
Biharmonic equation;Variational methods;High energy solutions;Concentration of solutions
摘要：
We consider the following nonlinear biharmonic equations:
$$ \Delta^{2} u-\Delta u+ V_{\lambda }(x)u=f(x,u),\quad \text{in } \mathbb{R}^{N}, $$
where
$V_{\lambda }(x)$
is allowed to be sign-changing and f is an indefinite function. Under some suitable assumptions, the existence of nontrivial solutions and the high energy solutions are obtained by using variational methods. Moreover, the phenomenon of concentration of solutions is explored. The results extend the main conclusions in recent literature.
作者机构：
[Xiao, Qizhen; Liu, Hongliang; Liu, Zhisu] Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
通讯机构：
[Liu, Zhisu] U;Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
关键词：
Fourth-order elliptic equation;Logarithmic nonlinearity;Ground state solution
摘要：
In this paper, we study the existence of ground state solutions of nonlinear elliptic equation with logarithmic nonlinearity by the Linking theorem and logarithmic Sobolev inequality. Our results are quite different from those in the case of polynomial nonlinearity. (C) 2017 Elsevier Ltd. All rights reserved.
期刊：
ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS,2016年2016(90) ISSN：1417-3875
通讯作者：
Chen, Haibo
作者机构：
[Xiao, Qizhen; Liu, Hongliang] Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.;[Chen, Haibo] Cent S Univ, Sch Math & Stat, Changsha 410083, Hunan, Peoples R China.
通讯机构：
[Chen, Haibo] C;Cent S Univ, Sch Math & Stat, Changsha 410083, Hunan, Peoples R China.
关键词：
Kirchhoff type equations;ground state solution;quasilinear;variational methods
摘要：
We study the ground state solutions of the following quasilinear Kirchhoff type equation -( 1 + b integral(R3) vertical bar del u vertical bar(2) dx) Delta u + V (x) u - [Delta(u(2))]u = vertical bar u vertical bar(10)u + mu vertical bar u vertical bar(p-1)u, x is an element of R, where b >= 0 and mu is a positive parameter. Under some suitable conditions on V (x), we obtain the existence of ground state solutions of the above equation with 1 < p < 11.
期刊：
Abstract and Applied Analysis,2014年2014(SI18):1-15 ISSN：1085-3375
通讯作者：
Ouyang, Zigen
作者机构：
[Ouyang, Zigen; Liu, Hongliang] Univ South China, Sch Math & Phys, Hengyang 421001, Peoples R China.
通讯机构：
[Ouyang, Zigen] U;Univ South China, Sch Math & Phys, Hengyang 421001, Peoples R China.
关键词：
A class of fractional order three-point boundary value system with resonance is investigated in this paper. Using some techniques of inequalities;a completely new method is incorporated. We transform the problem into an integral equation with a pair of undetermined parameters. The topological degree theory is applied to determine the particular value of the parameters so that the system has a solution. Published: 2014 First available in Project Euclid: 3 October 2014 zbMATH: 07022356 MathSciNet: MR3224310 Digital Object Identifier: 10.1155/2014/419514
摘要：
A class of fractional order three-point boundary value system with resonance is investigated in this paper. Using some techniques of inequalities, a completely new method is incorporated. We transform the problem into an integral equation with a pair of undetermined parameters. The topological degree theory is applied to determine the particular value of the parameters so that the system has a solution.
