Please read: Amini-Harandi, A. "Fixed point theory for set-valued quasi-contraction maps in metric spaces." Applied Mathematics Letters 24.11 (2011): 1791-1794.

Maybe it will be satisfactory answer, if not, please consider another type of contractibility (see some papers by  B Mohammadi, S Rezapour, N Shahzad, H Aydi, MF Bota, E Karapınar, S Mitrović or Chen, Chi-Ming. "Some new fixed point theorems for set-valued contractions in complete metric spaces." Fixed Point Theory and Applications 2011.1 (2011): 1-8)

• Dear Professor Mieczyslaw Cichon,

Thank you of your responses but I modified my answer as follows:

Let (X,d) be a complete metric spaces and let T be a multi-valued quasi contraction mapping. Amini Harandi solved the problem only for 0

Sorry - the question was unclear for me. In my opinion you don't formulate very precisely what is interesting for you.

Nevertheless, even for a single-valued case please check examples in Berinde, V. (2009). Some remarks on a fixed point theorem for Ciric-type almost contractions. Carpathian J. Math, 25(2), 157-162 (another version of quasi contraction) or in F Kiany, A Amini-Harandi, Fixed point theory for generalized Ćirić quasi-contraction maps in metric spaces, Fixed Point Theory and Applications, February 2013, 2013:26.

Noone did answer to the open problem yet. You can ask such reality from N. Shahzad or Amini Harandi or others. Whether you downloaded the pdf file? Ask from themselfs. They asserted that their work is only a partial answer to that.