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Modular Lipschitzian and contractive maps
Ch. 1. P. 1–15.
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.
In book
Vol. 130: Springer Proceedings in Mathematics & Statistics. , Switzerland: Springer, 2015.
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
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
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
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
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
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
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
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
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
Chistyakov V., Nonlinear Analysis 2010 Vol. 72 No. 1 P. 15–30
The notion of a modular is introduced as follows. A (metric) modular on a set X is a function w:(0,∞)×X×X→[0,∞] satisfying, for all x,y,z∈X, the following three properties: x=y if and only if w(λ,x,y)=0 for all λ>0; w(λ,x,y)=w(λ,y,x) for all λ>0; w(λ+μ,x,y)≤w(λ,x,z)+w(μ,y,z) for all λ,μ>0. We show that, given x0∈X, the set Xw={x∈X:limλ→∞w(λ,x,x0)=0} is a ...
Added: January 25, 2013