Solvability of a forced autonomous Duffing's equation with periodic boundary conditions in the presence of damping

Gupta, Chaitan P.. "Solvability of a forced autonomous Duffing's equation with periodic boundary conditions in the presence of damping." Applications of Mathematics 38.3 (1993): 195-203. <http://eudml.org/doc/15746>.

@article{Gupta1993,
abstract = {Let $g$: $\mathbf \{R\}\rightarrow \mathbf \{R\}$ be a continuous function, $e$: $[0,1]\rightarrow \mathbf \{R\}$ a function in $L^2[0,1]$ and let $c \in \mathbf \{R\}$, $c\ne 0$ be given. It is proved that Duffing’s equation $u^\{\prime \prime \} + cu^\{\prime \} + g(u)=e(x)$, $0<x<1$, $u(0)=u(1)$, $u^\{\prime \}(0)=u^\{\prime \}(1)$ in the presence of the damping term has at least one solution provided there exists an $\mathbf \{R\} > 0$ such that $g(u)u\ge 0$ for $|u|\ge \mathbf \{R\}$ and $\int ^\{1\}_\{0\}e(x)dx=0$. It is further proved that if $g$ is strictly increasing on $\mathbf \{R\}$ 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 $e\in L^2[0,1]$.},
author = {Gupta, Chaitan P.},
journal = {Applications of Mathematics},
keywords = {Dufiing's equation; damping; forced autonomous Duffing’s equation; Leray-Schauder continuation theorem; Wirtinger type inequalities; uniqueness; boundary value problem; forced autonomous Duffing's equation; Leray-Schauder continuation theorem; Wirtinger type inequalities; uniqueness; boundary value problem},
language = {eng},
number = {3},
pages = {195-203},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Solvability of a forced autonomous Duffing's equation with periodic boundary conditions in the presence of damping},
url = {http://eudml.org/doc/15746},
volume = {38},
year = {1993},
}

TY - JOUR
AU - Gupta, Chaitan P.
TI - Solvability of a forced autonomous Duffing's equation with periodic boundary conditions in the presence of damping
JO - Applications of Mathematics
PY - 1993
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 38
IS - 3
SP - 195
EP - 203
AB - Let $g$: $\mathbf {R}\rightarrow \mathbf {R}$ be a continuous function, $e$: $[0,1]\rightarrow \mathbf {R}$ a function in $L^2[0,1]$ and let $c \in \mathbf {R}$, $c\ne 0$ be given. It is proved that Duffing’s equation $u^{\prime \prime } + cu^{\prime } + g(u)=e(x)$, $0<x<1$, $u(0)=u(1)$, $u^{\prime }(0)=u^{\prime }(1)$ in the presence of the damping term has at least one solution provided there exists an $\mathbf {R} > 0$ such that $g(u)u\ge 0$ for $|u|\ge \mathbf {R}$ and $\int ^{1}_{0}e(x)dx=0$. It is further proved that if $g$ is strictly increasing on $\mathbf {R}$ 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 $e\in L^2[0,1]$.
LA - eng
KW - Dufiing's equation; damping; forced autonomous Duffing’s equation; Leray-Schauder continuation theorem; Wirtinger type inequalities; uniqueness; boundary value problem; forced autonomous Duffing's equation; Leray-Schauder continuation theorem; Wirtinger type inequalities; uniqueness; boundary value problem
UR - http://eudml.org/doc/15746
ER -