Let $H$ be a Hilbert space. Suppose I have a function $f\colon [0,1] \to H$ defined as the unique solution of $$f(x) = F(x, f(x))$$ where $F\colon \mathbb{R} \times H \to H$ is some given function which is continuous in the first argument and may be differentiable in some sense in the second argument.
I want to be able to say that $f$ is continuous, given some assumptions of $F$. What kind of theorems can I use for this?
It is sufficient that $F$ is a contraction in the second argument:
Suppose that $F$ is Lipschitz in the sense that $$\|F(x,h) - F(x,g)\| \le L_1 \, \| h - g\|,\qquad \|F(x,h) - F(y,h)\| \le L_2 \, |x - y|$$ for some constants $L_1, L_2$ and all $x,y \in [0,1]$, $f,g \in H$.
Then, $$ \| f(x) - f(y) \| \le \| F(x,f(x)) - F(x,f(y))\| + \|F(x,f(y))-F(y,f(y))\| \le L_1 \, \|f(x) - f(y)\| + L_2 \, |x - y|.$$ In case $L_1 < 1$ this implies $$ \|f(x) - f(y)\| \le \frac{L_2}{1-L_1} \, |x-y|.$$