Given $f(x) = \text{sign}^+(x) = \chi_{(0,\infty)}(x)$, is it possible to a sequence $f_n$ such that $f_n$ is smooth, $f_n' \geq 0$ and $$f_n \to f?$$ In some sense of convergence?
Preferably $f_n''$ should be bounded but this is not necessary.
How to do these approximations? Does it have something to do with Yosida approximation?
There are already some examples on how to construct such an approximation in the comments above. So I'll just say something about the limitations:
It is easy to get pointwise convergence. However you will not get uniform convergence (uniformly converging $f_n$ would imply that the limit $f$ is continuous, which it isn't). Also you cannot get any uniform bounds on the derivative, since $\int_{-\epsilon}^\epsilon f_n'(x) dx = f_n(\epsilon) - f_n(-\epsilon) \rightarrow 1$ for any $\epsilon > 0$, which also implies that none of the higher derivatives are bounded.
The one additional thing you can get are integral bounds, so $$\|f-f_n\|_{L^p} = \sqrt[p]{ \int_{\mathbb{R}} |f(x)-f_n(x)|^p dx} \rightarrow 0$$ is possible for any $p\in [1,\infty)$, but again not uniformly for all $p$ (this would imply uniform convergence). However there are again no such bounds on the derivative, so $\|f'-f_n'\|_{L^p}$ does not converge to 0. (This would imply that $f$ has a weak derivative, which it hasn't. If you want more information about that, you will have to look into Sobolev spaces.)