Application of p-regularity theory to the Duffing equation

• Main Authors: Beata Medak, Alexey A Tret’yakov
• Format: Article
• Language: English
• Published: SpringerOpen 2017-06-01
• Series: Boundary Value Problems
• Subjects: p-regularity; p-factor operator; Duffing equation; nonlinear boundary value problems
• Online Access: http://link.springer.com/article/10.1186/s13661-017-0815-8

Abstract The paper studies a solution existence problem of the nonlinear Duffing equation of the form F ( x , μ , β ) = x ″ + x + μ x 3 − β sin t = 0 , β > 0 , μ ≠ 0 , $$ F(x,\mu, \beta)=x''+x+\mu x^{3}-\beta\sin t = 0,\quad \beta > 0, \mu\neq0, $$ where F : C 2 [ 0 , 2 π ] × R × R → C [ 0 , 2 π ] $F: \mathcal{C}^{2}[0,2\pi]\times\mathbb{R}\times \mathbb{R}\rightarrow\mathcal{C}[0,2\pi]$ and x ( 0 ) = x ( 2 π ) = 0 $x(0)=x(2\pi)=0$ using the p-regularity theory.