
05820479
j
2010f.00842
Kroopnick, Allan J.
Bounded solutions to $x^{\prime \prime}+q(t)b(x)=f(t)$.
Int. J. Math. Educ. Sci. Technol. 41, No. 6, 829836 (2010).
2010
Taylor \& Francis, Abingdon, Oxfordshire
EN
I75
bounded
boundedness
absolutely integrable
oscillatory
continuously differentiable
nonlinear differential equations
doi:10.1080/00207391003777863
Summary: This article discusses the conditions under which all solutions to $x^{\prime \prime}+q(t)b(x)=f(t)$ are bounded on $[0,\infty)$. These results are generalizations of the linear case. A short discussion of the properties of bounded oscillatory solutions for both the linear and nonlinear cases when $f(t)=0$, $xb(x)>0$ and $b^{\prime}(x)>0$ for $x\ne 0$ is also provided. Finally, we shall see that the previous arguments may be applied to the more general nonlinear differential equation $x^{\prime \prime}+c(t,x,x^{\prime})+q(t)b(x)=f(t)$ with appropriate conditions on $c(t,x,x^{\prime})$.