Computations of the critical groups and the nontrivial solutions for resonant type asymptotically linear Hamiltonian systems

2002 ◽  
Vol 49 (4) ◽  
pp. 481-499 ◽  
Author(s):  
Wenming Zou
Keyword(s):  
Hamiltonian Systems ◽  
Critical Groups ◽  
Nontrivial Solutions ◽  
Asymptotically Linear ◽  
Linear Hamiltonian Systems

Periodic Solutions of Asymptotically Linear Hamiltonian Systems without Twist Conditions

2010 ◽  
Vol 65 (5) ◽  
pp. 445-452
Author(s):  
Rong Cheng ◽  
Dongfeng Zhang
Keyword(s):  
Dynamical System ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Galerkin Approximation ◽  
Point Theory ◽  
Dynamical System Theory ◽  
Hamiltonian Form ◽  
Asymptotically Linear ◽  
Linear Hamiltonian Systems ◽  
Existence Of Periodic Solutions

In dynamical system theory, especially in many fields of applications from mechanics, Hamiltonian systems play an important role, since many related equations in mechanics can be written in an Hamiltonian form. In this paper, we study the existence of periodic solutions for a class of Hamiltonian systems. By applying the Galerkin approximation method together with a result of critical point theory, we establish the existence of periodic solutions of asymptotically linear Hamiltonian systems without twist conditions. Twist conditions play crucial roles in the study of periodic solutions for asymptotically linear Hamiltonian systems. The lack of twist conditions brings some difficulty to the study. To the authors’ knowledge, very little is known about the case, where twist conditions do not hold.

scholarly journals Multiple Results to Some Biharmonic Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xingdong Tang ◽  
Jihui Zhang
Keyword(s):  
Elliptic Problem ◽  
Degree Theory ◽  
Critical Groups ◽  
Biharmonic Operator ◽  
Nonlinear Elliptic Problem ◽  
Topological Degree Theory ◽  
Nontrivial Solutions ◽  
Asymptotically Linear ◽  
Nonlinear Elliptic ◽  
Biharmonic Problems

We study a nonlinear elliptic problem defined in a bounded domain involving biharmonic operator together with an asymptotically linear term. We establish at least three nontrivial solutions using the topological degree theory and the critical groups.

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