Existence of Homoclinic Solutions for a Class of Damped Vibration Problems

• Huijuan Xu ^[1] ; Shan Jiang ^[1] ; Guanggang Liu ^[1]
  1. [1] Liaocheng University

     Liaocheng University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 2, 2022
• Idioma: inglés
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• Resumen

  • In this paper, we consider the existence of homoclinic solution for a class of damped vibration problem x¨(t)+(q(t)IN×N+B)x˙(t)+(12q(t)B−A(t))x(t)+Hx(t,x(t))=f(t).

    For every k∈N, we obtain the 2kT-periodic solution xk by a standard minimizing argument. By taking the limit of {xk}, we get a solution x0 of this problem. We prove that x0 satisfies x0→0 and x˙0→0 as t→±∞, and therefore x0 is a homoclinic solution of the problem.