Solution of an implicit Fourier transform equation

Question

How does one solve the following equation ($\hat{a}(k)$ denotes the Fourier transform of $a(x)$ and $q$ is real positive): $$\hat{a}(k)=f(k)\widehat{a^q}(k).$$ This equation appeared in some paper. Numerically, I tried solving the above equation iteratively. First, I assumed an initial solution for $a(x)$ and then performed fast Fourier transform (FFT) of $a^q$. Then, I took inverse transform of the resulting right hand side of the above equation to get a new $a(x)$. The procedure is repeated until convergence. I could not get a converged solution in MATLAB, where I basically tried the following iteratively: $$aHat=f(k)\times fft(a^q);$$ $$a=ifft(aHat);$$ Is the above procedure supposed to work? Is there any other way to solve the above equation? I appreciate your help.