Modular Lipschitzian and contractive maps

Vyacheslav V. Chistyakov

doi:10.1007/978-3-319-18567-5_1

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Modular Lipschitzian and contractive maps

Ch. 1. P. 1–15.

Vyacheslav V. Chistyakov

In the context of metric modular spaces, introduced recently by the author, we define the notion of modular Lipschitzian maps between modular spaces, as an extension of the notion of Lipschitzian maps between metric spaces, and address a modular version of Banach’s Fixed Point Theorem for modular contractive maps. We show that the assumptions in our fixed point theorem are sharp and that it guarantees the existence of fixed points in cases when Banach’s Theorem is inapplicable.

Language: English

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Keywords: модулярная сходимость неподвижная точка modular convergence fixed point modular space модулярное пространство Lipschitzian map липшицево отображение

In book

Optimization, Control, and Applications in the Information Age: In Honor of Panos M. Pardalos's 60th Birthday

Vol. 130: Springer Proceedings in Mathematics & Statistics. , Switzerland: Springer, 2015.

Quasi-confined modes produced by the Lugiato-Lefever model with a localized pump and the pseudo-Raman term

Gromov E., Malomed B. A., Physics Letters A 2026 Vol. 567 Article 131219

We introduce an extended nonlinear Lugiato-Lefever equation (LLE) with the pseudo-stimulated-Raman-scattering (pseudo-SRS) cubic term, linear damping/gain, and spatial inhomogeneous (weekly or strongly localized) pump. The LLE is derived, in the extended adiabatic approximation, from the underlying Zakharov’s system (ZS), which includes a viscosity term acting on its low-frequency (LF) component and the pump supporting the ...

Added: November 28, 2025

Analysis of Caputo Sequential Fractional Differential Equations with Generalized Riemann–Liouville Boundary Conditions

Nallappan G., Murugesan M., Seralan V. et al., Fractal and Fractional 2024 Vol. 8 No. 8 Article 457

This paper delves into a novel category of nonlocal boundary value problems concerning nonlinear sequential fractional differential equations, coupled with a unique form of generalized Riemann–Liouville fractional differential integral boundary conditions. For single-valued maps, we employ a transformation technique to convert the provided system into an equivalent fixed-point problem, which we then address using standard ...

Added: October 15, 2025

On the equivalence of some statements on fixed points of contractions

Semenov P., Functional Analysis and Its Applications 2017 Vol. 51 No. 4 P. 318–321

It is shown that a series of recent (2012–2016) generalizations of the notion of contraction (F-contraction, weak F-contraction, etc.) in fact reduce to known notions of contraction (due to Browder, Boyd and Wong, Meir and Keeler, etc.). ...

Added: April 10, 2018

О свойствах модулярных пространств

В.В.Чистяков, В кн.: Труды Математического центра имени Н.И.ЛобачевскогоТ. 54: Теория функций, ее приложения и смежные вопросы.: Каз.: Издательство Казанского математического общества и Академии наук РТ, 2017. С. 394–398.

We present basic concepts of the theory of modular spaces on arbitrary sets, which extends simultaneously the theory of such spaces on linear sets and the theory of metric spaces. We study the relationship between (three) modular spaces and metrics on them in the convex and nonconvex cases. We define the modular notions of convergence, topology and completeness. We ...

Added: August 31, 2017

Lipschitz stability of operators in Banach spaces

Protasov V., Proceedings of the Steklov Institute of Mathematics 2013 Vol. 268 No. 1 P. 268–279

We consider approximations of an arbitrarymap F: X → Y between Banach spaces X and Y by an affine operator A: X → Y in the Lipschitz metric: the difference F — A has to be Lipschitz continuous with a small constant ɛ > 0. In the case Y = ℝ we show that if F can be affinely ɛ-approximated on any straight line in X, then it can be globally 2ɛ-approximated by an affine operator on X. The constant 2ɛ is sharp. ...

Added: February 23, 2016

A Simple Proof of the Formula for the Betti Numbers of the Quasihomogeneous Hilbert Schemes

Buryak A., Feigin B. L., Nakajima H., International Mathematics Research Notices 2015 No. 13

In a recent paper, the first two authors proved that the generating series of the Poincare polynomials of the quasihomogeneous Hilbert schemes of points in the plane has a simple decomposition in an infinite product. In this paper, we give a very short geometrical proof of that formula. ...

Added: October 10, 2015

Неподвижные точки модулярно сжимающих отображений

Chistyakov V., Труды Математического центра им. Н.И. Лобачевского 2013 Т. 46 С. 56–62

In the context of modular metric spaces we prove a generalization of the Banach fixed point theorem for modular contractive mappings. ...

Added: August 29, 2013

Modular contractions and their application

Vyacheslav V. Chistyakov, , in: Models, Algorithms, and Technologies for Network AnalysisIssue 32.: NY: Springer, 2013. P. 65–92.

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 ...

Added: August 29, 2013

A fixed point theorem for contractions in modular metric spaces

Chistyakov Vyacheslav V., / Series math "arxiv.org". 2011. No. 1112.5561v1.

The notion of a (metric) modular on an arbitrary set and the corresponding modular space, more general than a metric space, were introduced and studied recently by the author [V.V. Chistyakov, Metric modulars and their application, Dokl. Math. 73 (1) (2006) 32–35, and Modular metric spaces, I: Basic concepts, Nonlinear Anal. 72 (1) (2010) 1–14]. ...

Added: February 6, 2013