| Title:
|
Solvability of a forced autonomous Duffing's equation with periodic boundary conditions in the presence of damping (English) |
| Author:
|
Gupta, Chaitan P. |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
38 |
| Issue:
|
3 |
| Year:
|
1993 |
| Pages:
|
195-203 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $g$: $\bold R\rightarrow \bold R$ be a continuous function, $e$: $[0,1]\rightarrow \bold R$ a function in $L^2[0,1]$ and let $c \in \bold R$, $c\neq 0$ be given. It is proved that Duffing's equation $u'' + cu' + g(u)=e(x)$, $0<x<1$, $u(0)=u(1)$, $u'(0)=u'(1)$ in the presence of the damping term has at least one solution provided there exists an $\bold R > 0$ such that $g(u)u\geq 0$ for $|u|\geq \bold R$ and $\int^{1}_{0}e(x)dx=0$. It is further proved that if $g$ is strictly increasing on $\bold R$ with $\lim_{u\rightarrow -\infty} g(u)=-\infty$, $\lim_{u\rightarrow \infty} g(u)=\infty$ and it Lipschitz continuous with Lipschitz constant $\alpha<4\pi^2+c^2$, then Duffing's equation given above has exactly one solution for every $e\in L^2[0,1]$. (English) |
| Keyword:
|
Dufiing's equation |
| Keyword:
|
damping |
| Keyword:
|
forced autonomous Duffing’s equation |
| Keyword:
|
Leray-Schauder continuation theorem |
| Keyword:
|
Wirtinger type inequalities |
| Keyword:
|
uniqueness |
| Keyword:
|
boundary value problem |
| MSC:
|
34B15 |
| MSC:
|
34C25 |
| MSC:
|
47H15 |
| MSC:
|
47J05 |
| MSC:
|
47N20 |
| idZBL:
|
Zbl 0785.34024 |
| idMR:
|
MR1218025 |
| DOI:
|
10.21136/AM.1993.104546 |
| . |
| Date available:
|
2008-05-20T18:45:30Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104546 |
| . |
| Reference:
|
[1] Gupta C. P.: On Functional Equations of Fred hoi in and Hammerstein type with Application to Existence of Periodic Solutions of Certain Ordinary Differential Equations.Journal of Integral Equations 3 (1981), 21-41. MR 0604314 |
| Reference:
|
[2] Gupta C. P., Mawhin J.: Asymptotic Conditions at the First two Eigenvalues for the Periodic Solutions of Lienard Differential Equations and an Inequality of E. Schmidt.Z. Anal. Anwendnngen 3 (1984), 33-42. MR 0739844, 10.4171/ZAA/88a |
| Reference:
|
[3] Gupta C. P., Nieto J. J., Sanchez L.: Periodic Solutions of Some Lienard and Duffing Equations.Jour. Math. Anal. & Appl. 140 (1989), 67-82. Zbl 0689.34032, MR 0997843, 10.1016/0022-247X(89)90094-2 |
| Reference:
|
[4] Loud W. S.: Periodic Solutions of $x" + cx' + g(x) = \epsilon f(t)$.Mem. Amer. Math. Soc., Providence, RI, 1959. MR 0107058 |
| Reference:
|
[5] Mawhin J.: Compacitè, Monotonie et Convexitè dans l'etude de problèmes aux limites semilinèaires.Sem. Anal. Moderne Université de Sherbrooke 19 (1981). |
| Reference:
|
[6] Mawhin J.: Landesman-Lazer type Problems for Non-linear Equations.Confer. Sem. Mat. Univ. Bari 147 (1977). MR 0477923 |
| Reference:
|
[7] Mawhin J.: Topological Degree Methods in Nonlinear Boundary Value Problems.CBMS Regional Conf. Math. Ser. Math, vol. 40, American Math. Society, Providence, RI, 1979. Zbl 0414.34025, MR 0525202 |
| Reference:
|
[8] Nieto J. J , and Rao V. S. H.: Periodic Solutions for Scalar Lienard Equations.Acta Math. Hung. 57 (1991), 15-27. MR 1128836, 10.1007/BF01903798 |
| . |
