On multivalued maps for φ-contractions involving orbits with application

Abstract

In [14], Proinov established the existence of fixed point theorems regarding as a generalization of the Banach contraction principle (BCP) of self mapping under an influence of gauge function (GF). In this paper, we develop some existence results on φ-contraction for multivalued maps via b-Bianchini-Grandolfi gauge function (B-GGF) in class of b-metric spaces and consequently assure the existence results in the module of simulation function as well α-admissible mapping. An extensive set of nontrivial example is given to justify our claim. At the end, we give an application to prove the existence behavior for the system of integral inclusion.