• differential equation
• analisi

In this thesis we consider the differential equation of order two
$$u''+e(u)u'+g(u)=f(t),$$
where $e,g,f:\R \rightarrow \R$ are continuous functions and $g(0)=0$. Let $b$, $c$ and $M$ be three positive real numbers such that for every values of their variables
$e(u)\geq c$, $g'(u)\geq b$, $|f(t)|\leq M$.
We are interested in finding bounds on $u(t)$ and $u'(t)$.

In the first chapter we study the linear and non linear differential equation of order one
$$ cu'+bu=f(t)$$ and $$cu'+g(u)=f(t),$$
where $b$ and $c$ are positive real numbers, $f:\R \rightarrow \R$ is a bounded continuous function and $g:\R \rightarrow \R$ is a continuous function such that $g'(s)\geq bs$ for all real numbers $s$.
We find that both the solution and its derivative are bounded from above in the following way
$$\limsup_{t \to +\infty} |u(t)| \leq \frac{1}{b} \limsup_{t \to +\infty} |f(t)|,$$
$$\limsup_{t \to +\infty} |u'(t)| \leq \frac{2}{c} \limsup_{t \to +\infty} |f(t)|.$$
We then show that these estimates are optimal, in fact we exhibit a real continuous function $f$ such that these bounds are attained.

In the second chapter we report the estimations from above on the solution to differential equation of order two $u''+e(u)u'+g(u)=f(t)$, that we find in literature.
First we show the results obtained in the paper "Boundedness and Stability for the Damped and Forced Single Well Duffing Equation" by C. Fitouri and A. Halaux, that use an analytical approach and consider the simplified differential equation of order two
$$u''+cu'+g(u)=f(t).$$
In this paper the authors analyze the weakly damped case corresponding to the condition $c\leq 2\sqrt{b}$ and the strongly damped case $c\geq 2\sqrt{b}$, obtaining different estimates in the two cases.
We then report the estimation from above on the derivative $u'$ obtained in the paper "Boundedness and Convergence of Solutions of $x''+cx'+g(x)=e(t).$" by W. Loud. Here the author studies the general differential equation of order two, with a geometrical approach in the phase space.

If we compare the two papers, we notice that the analytical approach provides a better estimate for $u$, but it does not improve the result for $u'$ in the strong damped case obtained by Loud. Thus, at this point, the best estimates considering both papers are
$$\limsup_{t \to +\infty} |u(t)| \leq \frac{2}{c\sqrt{b}} \limsup_{t \to +\infty} |f(t)|,$$
$$\limsup_{t \to +\infty} |u'(t)| \leq \frac{4}{c} \limsup_{t \to +\infty} |f(t)|.$$

In the third chapter we study the linear differential equation of order two
$$u''+cu'+bu=f(t),$$
where $b$ and $c$ are positive real numbers and $f:\R \rightarrow \R$ is a bounded continuous function.
We find estimations from above and from below on $u$ and $u'$. We adapt the idea of chapter one and we prove that those estimations are optimal. In fact we show real bounded continuous functions $f$ such that $u$ and $u'$ satisfy
$$\limsup_{t \to +\infty}|u(t)| = A \limsup_{t \to +\infty} |f(t)| \leq \frac{1}{b} \limsup_{t \to +\infty} |f(t)|$$
$$\limsup_{t \to +\infty} |u'(t)| = B \limsup_{t \to +\infty} |f(t)| \leq \frac{2}{c} \limsup_{t \to +\infty} |f(t)|,$$
for suitable constants $A$ and $B$ depending on $b$ and $c$.
Moreover, under the condition $c>2\sqrt{b}$, we give an example of a real function $f$ such that these equalities are both obtained.

Our work supports the conjecture that the estimate on the derivative of the solution to $u''+e(u)u'+g(u)=f(t),$ should be
$$\limsup_{t \to +\infty} |u'(t)| \leq \frac{2}{c} \limsup_{t \to +\infty} |f(t)|,$$
and this is our main result. In fact, when studying the linear case, that estimate is obtained.