Fixed point results for set-valued contractions by altering distances in complete metric spaces Original Research Article

Nadler’s contraction principle has led to fixed point theory of set-valued contraction in non-linear analysis. Inspired by the results of Nadler, the fixed point theory of set-valued contraction has been further developed in different directions by many authors, in particular, by Reich, Mizoguchi–Takahashi, Feng–Liu and many others. In the present paper, the concept of generalized contractions for set-valued maps in metric spaces is introduced and the existence of fixed point for such a contraction are guaranteed by certain conditions. Our first result extends and generalizes the Nadler, Feng–Liu and Klim–Wardowski theorems and the second result is different from the Reich and Mizoguchi–Takahashi results. As a consequence, we derive some results related to fixed point of set-valued maps satisfying certain conditions of integral type.