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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.
Highlights
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
To demonstrate the applications of our results, we prove 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
The H β -Hausdorff–Pompeiu b-metric is introduced as a new tool in metric fixed point theory and new variants of Nadler, Ciric, Hardy–Rogers contraction principles for multi-valued mappings are established in a b-metric space
Summary
Pompeiu in [1] initiated the study of distance between two sets and introduced the Pompeiu metric. Hausdorff [2] further studied this concept and thereby introduced the Hausdorff–Pompeiu metric H induced by the metric d of a metric space ( X, d), as follows: For any two subsets A and B of X, the function H given by H ( A, B) = max{supx∈ A d( x, B), supx∈ B d( x, A)} is a metric for the set of compact subsets of X. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims. = max{ β sup d( x, B) + (1 − β) sup d( x, A), β sup d( x, A) x∈ A.
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