Periodic solutions of the second-order differential equations with singularity

✍ Scribed by Zaihong Wang

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 231 KB
• Volume: 58
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2004.05.006

✦ Synopsis

In this paper, we deal with the existence of periodic solutions of the second-order di erential equations x + g(x) = p(t) with singularity near origin. By using the phase-plane analysis methods, we prove that the given equation has at least one periodic solution when g(x) exhibits semilinear condition for positive large x and singularity condition at origin and the primitive function G(x)(= x 1 g(s) ds) tends to positive inÿnity as x → 0 + . Moreover, we construct a counter-example to show that our result is no longer valid when G(x) tends to some ÿnite constant as x → 0 + .