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

In this note 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