Some New Extensions of Multivalued Contractions in a b-metric Space and Its Applications

Abstract

The Hβ-Hausdorff–Pompeiu b-metric for β∈[0,1] is introduced as a new variant of the Hausdorff–Pompeiu b-metric H. Various types of multi-valued Hβ-contractions are introduced and fixed point theorems are proved for such contractions in a b-metric space. The multi-valued Nadler contraction, Czervik contraction, q-quasi contraction, Hardy Rogers contraction, weak quasi contraction and Ciric contraction existing in literature are all one or the other type of multi-valued Hβ-contraction but the converse is not necessarily true. Proper examples are given in support of our claim. As applications of our results, we have proved the existence of a unique multi-valued fractal of an iterated multifunction system defined on a b-metric space and an existence theorem of Filippov type for an integral inclusion problem by introducing a generalized norm on the space of selections of the multifunction.