The concept of b-metric space was introduced by Czerwik in 1993 .Some examples of b- metric space and some fixed point theorems in b- metric spaces can be found in M. Boriceanu , M. Bota , etc.
The fixed point theory of multi-valued mapping has been developed in different ways by many author, by Reich (1983), Mizoguchi-Takahashi (1989), Feng-Liu (2006) and many other.