| Shrnutí: | The <inline-formula><math display="inline"><semantics><msup><mi>H</mi><mi>β</mi></msup></semantics></math></inline-formula>-Hausdorff–Pompeiu b-metric for <inline-formula><math display="inline"><semantics><mrow><mi>β</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> is introduced as a new variant of the Hausdorff–Pompeiu b-metric <i>H</i>. Various types of multi-valued <inline-formula><math display="inline"><semantics><msup><mi>H</mi><mi>β</mi></msup></semantics></math></inline-formula>-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 <inline-formula><math display="inline"><semantics><msup><mi>H</mi><mi>β</mi></msup></semantics></math></inline-formula>-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.
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