A new proper generalization of metric called as
-metric is introduced by Khojasteh et al. (Mathematical Problems in Engineering (2013) Article ID 504609). In this paper, first we prove the Caristi type fixed point theorem in an alternative and comparatively new way in the context of
-metric. We also investigate two
(family of nonempty closed and bounded subsets of a set X
). Furthermore, using the obtained
, we prove two new fixed point results for multi-functions which generalize the results of Nadler and Lim type in the context of such spaces. In order to illustrate the usability of our results, we equipped them with competent examples.
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