IVP- Has at most one solution

Question

Suppose $f(t, x)$ is nonincreasing with respect to $x$ for all $t \geq 0$ and $x \in \mathbb{R}$. Prove that the IVP problem $$ \left\{ \begin{array}{l} x'(t)=f(t,x) \\ x(t_0)=x_0 \end{array} \right. $$ has at most one solution for $t \geq t_0$.

Answer 1

Suppose that $x_1$ and $x_2$ are two different solutions. Let $h(t)=(x_1(t)-x_2(t))^2$. Then $h(t)\ge0$ for $t\ge t_0$ and $h(t_0)=0$. Take derivatives and use the equation to get $$ h'(t)=2(x_1(t)-x_2(t))(f(t,x_1(t))-f(t,x_2(t))). $$ Use the fact that $f(t,x)$ is non-increasing in $x$ for each $t\ge t_0$ to deduce that $h'(t)\le0$ and that $h$ is non-increasing.