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]...
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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

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