H β -Hausdorff Functions and Common Fixed Points of Multivalued Operators in a b -Metric Space and Their Applications

Abstract

$H^{\beta }$ -Hausdorff functions for $\beta \in \left[0,1\right]$ are introduced, and common fixed-point theorems for a pair of multivalued operators satisfying generalized contraction conditions are proven in a $b$ -metric space. Our results are proper extensions and new variants of many contraction conditions existing in literature. In order to demonstrate applications of our result, we have proven an existence theorem for a unique common multivalued fractal of a pair of iterated multifunction systems and also an existence theorem for a common solution of a pair of Volterra-type integral equations.