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Zbl 1239.39010
Zhou, Zhan; Yu, Jianshe; Chen, Yuming
(Yu, Jian-She; Chen, Yu-Ming)
Homoclinic solutions in periodic difference equations with saturable nonlinearity.
(English)
[J] Sci. China, Math. 54, No. 1, 83-93 (2011). ISSN 1674-7283; ISSN 1869-1862/e

This paper is mainly concerned with the homoclinic solutions of the following difference equation $$L u_n-\omega u_n=\frac{\sigma\chi_n u^3_n}{1+c_nu^2_n},$$ where $$Lu_n=a_nu_{n+1}+a_{n-1}u_{n-1}+b_nu_n,$$ and $\{a_n\}$, $\{b_n\}$, $\{c_n\}$ and $\{\chi_n\}$ are positive real valued $T$-periodic sequences. By using the linking theorem in combination with periodic approximations, the authors establish sufficient conditions on the nonexistence and on the existence of homoclinic solutions for the above difference equation. Their results solve an open problem proposed by {\it A. Pankov} [Nonlinearity 19, No. 1, 27--40 (2006; Zbl 1220.35163)]. This paper is interesting and a good contribution in this area.
[Hui-Sheng Ding (Jiangxi)]
MSC 2000:
*39A23
39A20 Generalized difference equations

Keywords: homoclinic solution; periodic difference equation; linking theorem; periodic approximation; rational difference equation

Citations: Zbl 1220.35163

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