# Article

 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 $u'' + cu' + g(u)=e(x)$, $0 0$ 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 .

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