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

Author

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  • Reny George

    (Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
    Department of Mathematics and Computer Science, St. Thomas College, Bhilai 490009, India)

  • Hemanth Kumar Pathak

    (SOS in Mathematics, Pt. Ravishankar Shukla University, Raipur 492010, India)

Abstract

The H β -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 H β -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 H β -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.

Suggested Citation

  • Reny George & Hemanth Kumar Pathak, 2020. "Some New Extensions of Multivalued Contractions in a b-metric Space and Its Applications," Mathematics, MDPI, vol. 9(1), pages 1-21, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2020:i:1:p:12-:d:466845
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    References listed on IDEAS

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    1. Chifu, Cristian & Petruşel, Adrian, 2008. "Multivalued fractals and generalized multivalued contractions," Chaos, Solitons & Fractals, Elsevier, vol. 36(2), pages 203-210.
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