Suppose we are given an IVP $x = f(x), x(0) = x_0 $, x ∈ R^n for which we know that the unique solution x(t, x_0) exists globally in time. Construct an example where x(t, x_0) is bounded but lim→+∞ x(t, x_0) does not exist. Show however that $$\int_{0}^{\infty} ||f(x(s,x_0))||ds$$ < ∞, then necessarily lim→+∞ x(t,x_0) exists.
I could not figure out such examples. Can anyone has some idea about this?
Hopefully your differential equation is $\dot x=f(x)$.
Then consider the equation of circular motion with $f(x)=Jx$, $J^2=-I$ or more concrete, $f((x_1,x_2))=(-x_2,x_2)$.
The finiteness condition tells you that the curve length of $x$ in the state space, $$ \int_0^\infty \|\dot x(t)\|\,dt $$ is finite, and thus this curve has an end point.
Another way to see this is that $x(t_n)$ in any time series $t_n\to\infty$ is a Cauchy sequence since $$ \|x(t_n)-x(t_m)\|\le \int_{\min(t_m,t_n)}^\infty\|\dot x(t)\|\,dt $$ and $$ \int_T^\infty\|\dot x(t)\|\,dt\xrightarrow[\ T\to\infty\ ]{}0 $$