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.
作者机构：
[Xiao, Qizhen; Liu, Hongliang] Univ South China, Sch Math & Phys, Hengyang 421001, Peoples R China.;[Huang, Yao; Wang, Xiao] Natl Univ Def Technol, Dept Math, Changsha 410073, Hunan, Peoples R China.
通讯机构：
[Liu, Hongliang] U;Univ South China, Sch Math & Phys, Hengyang 421001, Peoples R China.
关键词：
Fixed-time bipartite flocking;Multi-agent systems;Structurally balanced graph diameter of clusters
摘要：
This work focuses on the fixed-time bipartite flocking control problem for multi-agent systems. By employing the structurally balanced signed graph theory and a fixed-time stability theorem, we can prove that all individuals can be partitioned into two disjoint clusters within a fixed time and they move to opposite directions with the same velocity. We find that the settling time merely depends on the parameters and network connectivity. Moreover, the parameters, network connectivity and the initial states have an important influence on the diameter of each disjoint cluster. The results are novel, which are firstly illustrated by both theoretical analysis and numerical simulations. (C) 2019 Elsevier Ltd. All rights reserved.
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：1687-1839
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):1-14 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：
Ground state solution;Kirchhoff type equations;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.
