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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
37/2009

Subharmonics and homoclinics for a class of Hamiltonian-like equations

Percy Makita

Abstract

We study the existence of periodic and homoclinic solutions for a class of non-autonomous second order advanced-delayed differential equations of the type $$ \ddot{u}(t)+f_{0}(t,u(t))=\sum_{i=1}^{N}[f_{i}(t,u(t+\tau_{i})-u(t)) -f_{i}(t-\tau_{i},u(t)-u(t-\tau_{i}))]. $$ We prove, under some growth conditions on the non-linearities, the existence of non-constant periodic solutions with period any given positive integer. Using very simple arguments, the existence of a non-trivial homoclinic solution is also established. This homoclinic is obtained as the limit of subharmonics. An application to the existence of periodic and homoclinic travelling waves in an infinite lattice of partciles with $N$-nearest-neighbour interaction and on-site potential is given.

Received:
Jul 20, 2009
Published:
Aug 3, 2009
MSC Codes:
34C25, 34C37, 37K60
Keywords:
Advanced-delayed differential equations, critical point theory, periodic and homoclinic solutions

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Preprint
2009 Repository Open Access
Percy D. Makita

Subharmonics and homoclinics for a class of Hamiltonian-like equations