Why downsample exhibit wider bandwidth when doing discrete time fourier transform? Based on Nyquist sampling theorem, sampling would have a convolution in frequency domain, and reasonably the bandwidth of each convolution would be as that of the original signal.
Nevertheless, the discrete time Fourier transform of a signal ,which has been  downsampled , has the $M$ multiples bandwidth of each convolution.($M$ is the downsample factor). I could not have got the idea, what is the true relation between time/frequency domain with underlying discrete time Fourier transform? 

Sampling gets convolution in frequency domain:


Downsampling gets more wider bandwidth:
You don't get "more" bandwidth. That's simply a misunderstanding of your x-axis. Your signal after downsampling occupies a $M$ times larger part of the Nyquist rate, but that's because the Nyquist rate got reduced by a factor of $M$, not because the signal got wider!