Solution to integral equation Show that the non-linear integral equation
$v(x)=\cos^2(x)+\int_0^x e^{-v^2(s)}ds, \ x\in [0,\infty)$
has a solution in $C^1([0,\infty))$.
In previous questions of this sort, we have been able to use the contraction mapping theorem, but are having difficulty using it here due to the infinite domain and haven't been able to find a contraction yet.

I would try to argue that you can solve over an interval $[0,a]$ by choosing $a$ small enough. Then continue by solving
$$
       v(x) = \cos^{2}x + \int_{0}^{a}e^{-v^{2}(s)}ds+\int_{a}^{x}e^{-v^{2}(s)}ds \\
       = \cos^{2}x+A+\int_{a}^{x}e^{-v^{2}(s)}ds.
$$
