Common fixed points for pointwise Lipschitzian semigroups in modular function spaces

Buthinah A. Bin Dehaish; Mohamed A. Khamsi; Wojciech M. Kozlowski

doi:10.1186/1687-1812-2013-214

Common fixed points for pointwise Lipschitzian semigroups in modular function spaces

Buthinah A. Bin Dehaish^*, Mohamed A. Khamsi, Wojciech M. Kozlowski

^*Corresponding author for this work

King Fahd University of Petroleum & Minerals

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

Let C be a ρ-bounded, ρ-closed, convex subset of a modular function space L_ρ. We investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups of nonlinear mappings T _t : C → C, i.e. a family such that T₀(f) = f, T _s+t (f) = Ts o T_t(f) and ρ(T(f) - T(g)) ≤ α_t (f)α(f - g), where lim sup _t→∞αt (f) ≤ 1 for every f ∈ C. In particular, we prove that if L_ρ is uniformly convex, then the common fixed point is nonempty ρ-closed and convex.

Original language	English
Article number	214
Journal	Fixed Point Theory and Algorithms for Sciences and Engineering
Volume	2013
DOIs	https://doi.org/10.1186/1687-1812-2013-214
State	Published - Aug 2013

Bibliographical note

Funding Information:
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (247-006-D1433). The authors, therefore, acknowledge with thanks technical and financial support of DSR.

Keywords

Fixed point
Modular function space
Nonexpansive mapping
Orlicz space
Pointwise Lipschitzian mapping
Pointwise nonexpansive mapping
Semigroup
Uniform convexity

ASJC Scopus subject areas

Geometry and Topology
Applied Mathematics

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10.1186/1687-1812-2013-214

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title = "Common fixed points for pointwise Lipschitzian semigroups in modular function spaces",

abstract = "Let C be a ρ-bounded, ρ-closed, convex subset of a modular function space Lρ. We investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups of nonlinear mappings T t : C → C, i.e. a family such that T0(f) = f, T s+t (f) = Ts o Tt(f) and ρ(T(f) - T(g)) ≤ αt (f)α(f - g), where lim sup t→∞αt (f) ≤ 1 for every f ∈ C. In particular, we prove that if Lρ is uniformly convex, then the common fixed point is nonempty ρ-closed and convex.",

keywords = "Fixed point, Modular function space, Nonexpansive mapping, Orlicz space, Pointwise Lipschitzian mapping, Pointwise nonexpansive mapping, Semigroup, Uniform convexity",

author = "\{Bin Dehaish\}, \{Buthinah A.\} and Khamsi, \{Mohamed A.\} and Kozlowski, \{Wojciech M.\}",

note = "Funding Information: This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (247-006-D1433). The authors, therefore, acknowledge with thanks technical and financial support of DSR. ",

year = "2013",

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doi = "10.1186/1687-1812-2013-214",

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T1 - Common fixed points for pointwise Lipschitzian semigroups in modular function spaces

AU - Bin Dehaish, Buthinah A.

AU - Khamsi, Mohamed A.

AU - Kozlowski, Wojciech M.

N1 - Funding Information: This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (247-006-D1433). The authors, therefore, acknowledge with thanks technical and financial support of DSR.

PY - 2013/8

Y1 - 2013/8

N2 - Let C be a ρ-bounded, ρ-closed, convex subset of a modular function space Lρ. We investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups of nonlinear mappings T t : C → C, i.e. a family such that T0(f) = f, T s+t (f) = Ts o Tt(f) and ρ(T(f) - T(g)) ≤ αt (f)α(f - g), where lim sup t→∞αt (f) ≤ 1 for every f ∈ C. In particular, we prove that if Lρ is uniformly convex, then the common fixed point is nonempty ρ-closed and convex.

AB - Let C be a ρ-bounded, ρ-closed, convex subset of a modular function space Lρ. We investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups of nonlinear mappings T t : C → C, i.e. a family such that T0(f) = f, T s+t (f) = Ts o Tt(f) and ρ(T(f) - T(g)) ≤ αt (f)α(f - g), where lim sup t→∞αt (f) ≤ 1 for every f ∈ C. In particular, we prove that if Lρ is uniformly convex, then the common fixed point is nonempty ρ-closed and convex.

KW - Fixed point

KW - Modular function space

KW - Nonexpansive mapping

KW - Orlicz space

KW - Pointwise Lipschitzian mapping

KW - Pointwise nonexpansive mapping

KW - Semigroup

KW - Uniform convexity

UR - https://www.scopus.com/pages/publications/84902471725

U2 - 10.1186/1687-1812-2013-214

DO - 10.1186/1687-1812-2013-214

M3 - Article

AN - SCOPUS:84902471725

SN - 1687-1820

VL - 2013

JO - Fixed Point Theory and Algorithms for Sciences and Engineering

JF - Fixed Point Theory and Algorithms for Sciences and Engineering

M1 - 214

ER -

Common fixed points for pointwise Lipschitzian semigroups in modular function spaces

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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