A function $f(x)$ vanishes at $x \to \pm \infty$ and satisfies this equation
$\frac{df}{dx} + f(x) = \delta(x) - \int^{+\infty}_{-\infty} g(x-y)f(y) \, dy$
How to obtain the fourier transform, $F(k)$ of $f(x)$ in terms of the Fourier transform $G(k)$ of $g(x)$?
$$ is\hat{f}+\hat{f}=\frac{1}{\sqrt{2\pi}}+\sqrt{2\pi}\hat{f}(s)\hat{g}(s) \\ \hat{f}=\frac{1}{\sqrt{2\pi}}\frac{1}{1+is+\sqrt{2\pi}\hat{g}(s)} $$