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Modular Contractions and Their Application

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Models, Algorithms, and Technologies for Network Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 32))

Abstract

The notion of a metric modular on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces and Orlicz spaces, were recently introduced and studied by the author [Chistyakov: Dokl. Math. 73(1):32–35, 2006 and Nonlinear Anal. 72(1):1–30, 2010]. In this chapter we present yet one more application of the metric modulars theory to the existence of fixed points of modular contractive maps in modular metric spaces. These are related to contracting generalized average velocities rather than metric distances, and the successive approximations of fixed points converge to the fixed points in the modular sense, which is weaker than the metric convergence. We prove the existence of solutions to a Carathéodory-type differential equation with the right-hand side from the Orlicz space.

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Acknowledgements

The individual research project No. 10-01-0071 “Metric modulars and their topological, geometric and econometric properties with applications” was supported by the Program “Scientific Foundation of the National Research University Higher School of Economics”. The work on the project has been carried out at Laboratory of Algorithms and Technologies for Networks Analysis, National Research University Higher School of Economics, and also partly supported by Ministry of Education and Science of Russian Federation, Grant No. 11.G34.31.0057. The author is grateful to Boris I. Goldengorin and Panos M. Pardalos for stimulating discussions on the results of this chapter.

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Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science and Laboratory of Algorithms and Technologies for Networks Analysis, National Research University Higher School of Economics, Bol’shaya Pechërskaya Street 25/12, Nizhny Novgorod, Russian Federation, 603155, Russia

    Vyacheslav V. Chistyakov

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  1. Vyacheslav V. Chistyakov
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Correspondence to Vyacheslav V. Chistyakov .

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Editors and Affiliations

  1. Higher School of Economics, Dept. of Applied Math. & Informatics, National Research University, B. Pecherskaya 25/12, Nizhny Novgorod, 603155, Russia

    Boris Goldengorin

  2. Higher School of Economics, Department of Applied Mathematics, National Research University, Nizhny, Novgorod, 603155, Russia

    Valery A. Kalyagin

  3. , Department of Industrial and Systems Eng, University of Florida, Weil Hall 401, Gainesville, 32611, Florida, USA

    Panos M. Pardalos

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Chistyakov, V.V. (2013). Modular Contractions and Their Application. In: Goldengorin, B., Kalyagin, V., Pardalos, P. (eds) Models, Algorithms, and Technologies for Network Analysis. Springer Proceedings in Mathematics & Statistics, vol 32. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5574-5_4

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