General boundedness theorems to some second order nonlinear differential equation with integrable forcing term (Q1902935)

Summary: We present a boundedness theorem to the equation \(x'' + c(t,x,x') + a(t)b(x) = e(t)\) where \(e(t)\) is a continuous absolutely integrable function over the nonnegative real line. We then extend the result to the equation \(x'' + c(t,x,x') + a(t,x) = e(t)\). The first theorem provides the motivation for the second theorem. Also, an example illustrating the theory is then given.