I am trying to show that the IVP
$$x'=\sqrt{x(t)}+1, t\in[0,1],\\x(0)=0, (t_0=0)$$
has a unique solution and show whether the initial value problem satisfies the assumptions of Picard’s Theorem, using the hint:
(Hint: Consider the function $F(x)=\int_0^x\frac{1}{\sqrt{z}+1},\mathrm{d}x$ show that $F$ is one-to-one (injective), and function $y=F^{-1}$ satisfies the initial value problem).
I am struggling show that $f(t,x)$ is Lipschitz and am struggling to find the Lipschitz constant $L$. So far all I have found is $$|f(t,u)-f(t,v)|=|\sqrt{u}-\sqrt{v}|\leq L|u-v|.$$ I know that once I have found $L$, I can use it to help show $f(t,x)$ satisfies Picard's Theorem
Any help would be really appreciated, thank you
Best Answer
The inequality $|\sqrt{u}-\sqrt{v}|\leq L|u-v|$ for $u,v \in [0,1]$ can not be true.
Reason: suppose that the inequality is true. Then we get , with $v=0:$
$$\sqrt{u} \le L u$$
for all $u \in [0,1].$ For $u>0$ this gives
$$\frac{1}{\sqrt{u}} \le L.$$
But this is not possible, since $\frac{1}{\sqrt{u}} \to \infty$ for $u \to 0+.$
To show that the the initial value problem has a unique solution use the hint !