Periodic Solutions for a Class of Forced Liénard-type Equations

Periodic Solutions for a Class of Forced Liénard-type Equations By applying the topological degree theory, we establish some sufficient conditions for the existence on T-periodic solutions for the Liénard-type equation $$ {x}\ifmmode{''}\else$''$\fi + {\sum\limits_{i = 1}^n {h_{i} {\left( x \right)}{\left| {{x}\ifmmode{'}\else$'$\fi} \right|}^{{2\alpha _{i} }} + f_{1} {\left( x \right)}{\left| {{x}\ifmmode{'}\else$'$\fi} \right|}^{2} + f_{2} {\left( x \right)}{x}\ifmmode{'}\else$'$\fi + g{\left( {t,x} \right)} = p{\left( t \right)}.} } $$ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Periodic Solutions for a Class of Forced Liénard-type Equations

Periodic Solutions for a Class of Forced Liénard-type Equations

Acta Mathematicae Applicatae Sinica, English Series Vol. 21, No. 1 (2005) 81–92 Periodic Solutions for a Class of Forced Lie ´nard-type Equations 1 2 Bing-wen Liu , Li-hong Huang Department of Mathematics, Hunan University of Arts and Science, Changde, Hunan 415000, China (E-mail: liubw007@yahoo.com.cn) College of Mathematics and Econometrics, Hunan University, Changsha 410082, China Abstract By applying the topological degree theory, we establish some sufficient conditions for the existence on T -periodic solutions for the Lie ´nard-type equation 2α  2 x + h (x)|x | + f (x)|x | + f (x)x + g(t, x)= p(t). i 1 2 i=1 Our results extend and improve some known results in the literature. Keywords lie ´nard-type equation; periodic solutions; topological degree 2000 MR Subject Classification 34C25 1 Introduction In this paper, we study the existence of T -periodic solutions of the forced Lie ´nard-type equation 2α  2 x + h (x)|x | + f (x)|x | + f (x)x + g(t, x)= p(t), (1.1) i 1 2 i=1 where h (i =1, 2,··· ,n),f ,f ,p : R → R and g : R × R → R are continuous functions, p is i 1 2 T -periodic, g is T -periodic in the first variable, α ∈ R(i =1, 2,··· ,n)and α > 0. i i Clearly, when h (x) ≡ 0(i =1, 2,··· ,n), f (x) ≡ 0, f (x)= f (x)and g(t, x)= g(x), i 1 2 Equation (1.1) is reduced to the well-known Lie ´nard equation x + f (x)x + g(x)= p(t). (1.2) Therefore, Equation (1.1) may be referred to as a Lie ´nard-type equation. The relevance of (1.1) and (1.2) to physics, mechanics and engineering can be found in [2,12]. Hence, (1.1) and (1.2) have been the objects of intensive analysis by numerous authors. In particular, the problem of the existence of periodic solutions of (1.2) under various conditions on f and g has been extensively discussed in the literature, some of this results can be found in [1–14]...
Loading next page...
 
/lp/springer_journal/periodic-solutions-for-a-class-of-forced-li-nard-type-equations-MD2B3BDNWJ
Publisher
Springer-Verlag
Copyright
Copyright © 2005 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-005-0218-y
Publisher site
See Article on Publisher Site

Abstract

By applying the topological degree theory, we establish some sufficient conditions for the existence on T-periodic solutions for the Liénard-type equation $$ {x}\ifmmode{''}\else$''$\fi + {\sum\limits_{i = 1}^n {h_{i} {\left( x \right)}{\left| {{x}\ifmmode{'}\else$'$\fi} \right|}^{{2\alpha _{i} }} + f_{1} {\left( x \right)}{\left| {{x}\ifmmode{'}\else$'$\fi} \right|}^{2} + f_{2} {\left( x \right)}{x}\ifmmode{'}\else$'$\fi + g{\left( {t,x} \right)} = p{\left( t \right)}.} } $$

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jan 1, 2005

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches

$49/month

Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.

$588

$360/year

billed annually
Start Free Trial

14-day Free Trial