Using Schauder fixed-point theorem, existence of a fixed-point can be proven. I am wondering how to prove the uniqueness of a fixed point under some additional conditions. For example, consider the operator $R$ on a suitable convex function space (e.g. $C(0,T;\mathbb{H})$)
$$(Ru)(t)=K(t)u_0+\int_0^tK(t-s)F(u(s))ds$$
where
- $F(\cdot)$ is a locally-Lipschitz nonlinear operator on $\mathbb{H}$, a Hilbert space.
- $K(t)$ is a uniformly continuous semi-group of compact linear operators on $\mathbb{H}$ (i.e. $\|K(t)\|$ is continuous for $t>0$ and $K(t)$ is compact for every $t$).
- $u_0\in \mathbb{H}$ is fixed.
Now, Schauder fixed-point theorem guarantees the existence of a fixed-point for $R$; but how is it going to be unique?
Considering uniqueness, the semigroup $\{K(t)\}_{t\geq 0}$ doesn't have to be compact and it is enough that $\mathbb{H}$ is a Banach space.
Assume there are two fixed-points $u_1$ and $u_2\in C([0,T],\mathbb{H})$ of the operator $R$. There is the Lipschitz constant $L>0$ and $\rho>0$ such that $$ u,v \in B(u_0,\rho) \Rightarrow \|F(u)-F(v)\|\leq L\|u-v\|. $$ Since $u_1(0)=u_0=u_2(0)$ and $u_1$, $u_2$ are (uniformly) continuous, there is $\delta>0$ such that $$ u_1(s),\; u_2(s)\in B(u_0,\rho) \quad\text{for every }s\in[0,\delta]. $$ Moreover, we have a standard estimate $$ \|K(t)\|\leq M e^{\omega t},\quad\text{for every }t\geq 0 $$ for some $M\geq 1$ and $\omega\in \mathbb{R}$. Therefore, for $t\in[0,\delta]$, $$ \begin{aligned} \|u_1(t)-u_2(t)\|&=\left\|\int_0^t K(t-s)[F(u_1(s))-F(u_2(s))] ds\right\|\\ &\leq M\int_0^t e^{\omega (t-s)}\|F(u_1(s))-F(u_2(s))\|ds\\ &\leq M\int_0^te^{\omega (t-s)}L\|u_1(s)-u_2(s)\|ds. \end{aligned} $$ We use Gronwall lemma and get $\|u_1(t)-u_2(t)\|=0$ for $t\in[0,\delta]$.
Now, we argue as above inductively. We can also cover the compact set $u_1([0,T])$ by finite number of open neighbourhoods where $F$ is Lipschitz. Then we take the greatest Lipschitz constant and proceed as above.