The following paper considers contractions in metric space:

K. Conrad, The contraction mapping theorem:

http://www.math.uconn.edu/~kconrad/blurbs/analysis/contraction.pdf

See, also, 

L. Peeler, Metric spaces and the contraction mapping principle:

http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Peeler.pdf

and

https://www.researchgate.net/publication/247818481_THE_CONTRACTION_MAPPING_THEOREM

and

downloads.hindawi.com/journals/ijmms/1982/632643.pdf

Conrad gives a proof of the contraction mapping theorem, Sect. 2, starting on page 2.

Article THE CONTRACTION MAPPING THEOREM

 Dear , you can see may paper quasi Banach space the quasi norm is a metric space may be called quasi metric . This hint may be help you? 

All the above answers are wrong and have not relation to my answer. Existence the fixed point for quasi-contraction when 1/2< a< 1 is still open and has not been solved yet. All of the papers relation to this problem are only light version of that problem. Read the following paper to get it right:

[1]. Rhoades, B. E. "Notes on the paper “Fixed point theory for set-valued quasi-contraction maps in metric spaces” by A. Amini-Harandi." Applied Mathematics Letters 25, no. 10 (2012): 1578.

[2] Amini-Harandi, A. "Fixed point theory for set-valued quasi-contraction maps in metric spaces." Applied Mathematics Letters 24, no. 11 (2011): 1791-1794.

Dear Professor Khojasteh. Could you give me the exact reference where Amini Harandi solved the problem for 0

Dear Professor Raul Fierro,

Not only Amini Harandi Solved the problem in a paper in Applied Mathematics Letter, but also Billy Rhoades introduced an easily way to solved in via Kannan type fixed point theorem. The references are as follows:

[1] Amini-Harandi, A. "Fixed point theory for set-valued quasi-contraction maps in metric spaces." Applied Mathematics Letters 24, no. 11 (2011): 1791-1794.

[2] Rhoades, B. E. "Notes on the paper “Fixed point theory for set-valued quasi-contraction maps in metric spaces” by A. Amini-Harandi." Applied Mathematics Letters 25, no. 10 (2012): 1578.