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In the present paper, we extend the concept of contraction in a new manner by introducing D-contraction defined on a family F of bounded functions. We also introduce a new notion of a fixed function on a metric space. Some fixed function theorems along with illustrative examples and application are also given to verify the effectiveness of our results.

In the present article, the notion of αH-ψH-multivalued contractive type mappings is introduced and some fixed point results in complete metric spaces are studied. These theorems generalize Nadler’s (Multivalued contraction mappings, Pac. J. Math., 30, 475–488, 1969) and Suzuki-Kikkawa's (Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal., 69, 2942–2949, 2008) results that exist in the literature. The effectiveness of the obtained results has been verified with the help of some comparative examples. Moreover, a homotopy result has been presented as an application.