Bounded and L²‐solutions to a second order nonlinear differential equation with a square integrable forcing term

Abstract

This paper presents two theorems concerning the nonlinear differential equationx″+c(t)f(x)x′+a(t,x)=e(t), wheree(t)is a continuous square-integrable function. The first theorem gives sufficient conditions when all the solutions of this equation are bounded while the second theorem discusses when all the solutions are inL2[0,∞).

Boundedness of Solutions for Second Order Differential Equations

“…Besides (Qarawani, 2012) and (Tunc ¸& Ayhan 2015) mentioned earlier, (Kroopnick 1999), (Kroopnick 2008), (Nápoles Valdes, 2013) and (Tiryaki & Zafer, 2013) also used this approach. Other related theorems include (Athanassov, 1987), (Bonheure & Fabry, 2009), ( Capietto, Dambrosio, & Liu 2009), (Cecchi, Došlá, & Marini 2007), (Cheng 2008), ( Jiang, Rao, & Shi 2013), (Kroopnick, 2010), (Tunc¸C., & Tunc¸E., 2007), ( Tunc ¸, 2010), (Tunc ¸, 2011) and (Wang, McKee, & Usyk, 2010).…”

A high-order discontinuous Galerkin approximation to ordinary differential equations with applications to elastodynamics

“…Clearly, we have | e | ≤ | e π | + | e h |. By exploiting properties (28)- (30) and by employing estimates (26) and (27) we can bound the term | e π | as…”