Multiplicity results for periodic solutions to a class of second order delay differential equations
Zhiming Guo Xiaomin Zhang
Communications on Pure & Applied Analysis 2010, 9(6): 1529-1542 doi: 10.3934/cpaa.2010.9.1529
In this paper, we study the following second order delay differential equation

$x''(t)=-f(x(t), x(t-\tau)).$

When $f$ possesses a symmetric property and grows asymptotically linear both at zero and at infinity, some new results for the existence and multiplicity of periodic solutions are obtained by using the critical point theory and $S^1$ geometrical index theory.

keywords: Hamiltonian systems critical point theory Second order delay differential equations multiple periodic solutions $S^1$ geometrical index theory.
Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case
Zhiming Guo Zhi-Chun Yang Xingfu Zou
Communications on Pure & Applied Analysis 2012, 11(5): 1825-1838 doi: 10.3934/cpaa.2012.11.1825
This paper deals with a class of non-local second order differential equations subject to the homogeneous Dirichlet boundary condition. The main concern is positive steady state of the boundary value problem, especially when the equation does not enjoy the monotonicity. Nonexistence, existence and uniqueness of positive steady state for the problem are addressed. In particular, developed is a technique that combines the method of super-sub solutions and the estimation of integral kernels, which enables us to obtain some sufficient conditions for the existence and uniqueness of a positive steady state. Two examples are given to illustrate the obtained results.
keywords: Dirichlet boundary value problems uniqueness. Non-local second order differential equation positive solutions existence
Periodic solutions for discrete convex Hamiltonian systems via Clarke duality
Jianshe Yu Honghua Bin Zhiming Guo
Discrete & Continuous Dynamical Systems - A 2006, 15(3): 939-950 doi: 10.3934/dcds.2006.15.939
Based on the Legendre transform, the dual action functional which corresponds to discrete Hamiltonian systems is given. In this paper, the existence of periodic solution for discrete convex Hamiltonian systems with forcing terms is obtained by using the dual least action principle and the perturbation technique.
keywords: perturbation technique dual least action principle discrete Hamiltonian systems periodic solution convex. Clarke duality
Dynamical behavior of a new oncolytic virotherapy model based on gene variation
Zizi Wang Zhiming Guo Huaqin Peng
Discrete & Continuous Dynamical Systems - S 2017, 10(5): 1079-1093 doi: 10.3934/dcdss.2017058

Oncolytic virotherapy is an experimental treatment of cancer patients. This method is based on the administration of replication-competent viruses that selectively destroy tumor cells but remain healthy tissue unaffected. In order to obtain optimal dosage for complete tumor eradication, we derive and analyze a new oncolytic virotherapy model with a fixed time period $τ $ and non-local infection which is caused by the diffusion of the target cells in a continuous bounded domain, where $τ $ is assumed to be the duration that oncolytic viruses spend to destroy the target cells and to release new viruses since they enter into the target cells. This model is a delayed reaction diffusion system with nonlocal reaction term. By analyzing the global stability of tumor cell eradication equilibrium, we give different treatment strategies for cancer therapy according to the different genes mutations (oncogene and antioncogene).

keywords: Oncolytic virotherapy tumor growth modeling gene mutation reaction-diffusion equations with delay global stability
A nonlocal diffusion population model with age structure and Dirichlet boundary condition
Yueding Yuan Zhiming Guo Moxun Tang
Communications on Pure & Applied Analysis 2015, 14(5): 2095-2115 doi: 10.3934/cpaa.2015.14.2095
In this paper, we study the global dynamics of a population model with age structure. The model is given by a nonlocal reaction-diffusion equation carrying a maturation time delay, together with the homogeneous Dirichlet boundary condition. The non-locality arises from spatial movements of the immature individuals. We are mainly concerned with the case when the birth rate decays as the mature population size becomes large. The analysis is rather subtle and it is inadequate to apply the powerful theory of monotone dynamical systems. By using the method of super-sub solutions, combined with the careful analysis of the kernel function in the nonlocal term, we prove nonexistence, existence and uniqueness of the positive steady states of the model. By establishing an appropriate comparison principle and applying the theory of dissipative systems, we obtain some sufficient conditions for the global asymptotic stability of the trivial solution and the unique positive steady state.
keywords: Nonlocal and delay model existence and uniqueness global asymptotic stability. positive steady states Dirichlet boundary condition

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