scholarly journals Modular Uniform Convexity in Every Direction in Lp(·) and Its Applications

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 870
Author(s):  
Mostafa Bachar ◽  
Osvaldo Méndez
Keyword(s):  
Fixed Point ◽  
Lebesgue Space ◽  
Variable Exponent ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Uniform Convexity ◽  
Uniformly Convex ◽  
Made In

We prove that the Lebesgue space of variable exponent L p ( · ) ( Ω ) is modularly uniformly convex in every direction provided the exponent p is finite a.e. and different from 1 a.e. The notion of uniform convexity in every direction was first introduced by Garkavi for the case of a norm. The contribution made in this work lies in the discovery of a modular, uniform-convexity-like structure of L p ( · ) ( Ω ) , which holds even when the behavior of the exponent p ( · ) precludes uniform convexity of the Luxembourg norm. Specifically, we show that the modular ρ ( u ) = ∫ Ω | u ( x ) | d x possesses a uniform-convexity-like structure even if the variable exponent is not bounded away from 1 or ∞. Our result is new and we present an application to fixed point theory.

scholarly journals Modular Uniform Convexity of Lebesgue Spaces of Variable Integrability

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 708 ◽  
Author(s):  
Mostafa Bachar ◽  
Osvaldo Mendez ◽  
Messaoud Bounkhel
Keyword(s):  
Fixed Point ◽  
Lebesgue Space ◽  
Variable Exponent ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Uniform Convexity ◽  
Convexity Property ◽  
Lebesgue Spaces ◽  
Central Result ◽  
Variable Integrability

We analyze the modular geometry of the Lebesgue space with variable exponent, L p ( · ) . Our central result is that L p ( · ) possesses a modular uniform convexity property. Part of the novelty is that the property holds even in the case sup x ∈ Ω p ( x ) = ∞ . We present specific applications to fixed point theory.

scholarly journals Modular Uniform Convexity of Lebesgue Spaces of Variable Integrability

2018 ◽  
Author(s):  
Mostafa Bachar ◽  
Osvaldo Mendez ◽  
Messaoud Bounkhel
Keyword(s):  
Fixed Point ◽  
Lebesgue Space ◽  
Variable Exponent ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Uniform Convexity ◽  
Convexity Property ◽  
Lebesgue Spaces ◽  
Variable Exponent Lebesgue Space ◽  
Variable Integrability

We analyze the modular geometry of the variable exponent Lebesgue space Lp(.). We show that Lp(.) possesses a modular uniform convexity property. Part of the novelty is that the property holds even in the case supp(x) = ∞ . We present specific applications to fixed point theory. xÆΩ

scholarly journals Erratum: Abdou, A. A. N. and Khamsi, M.A. Fixed Points of Kannan Maps in the Variable Exponent Sequence Spaces ℓp(·). Mathematics 2020, 8, 76

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 578
Author(s):  
Afrah A. N. Abdou ◽  
Mohamed Amine Khamsi
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Variable Exponent ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Sequence Spaces ◽  
Vector Spaces ◽  
Nonexpansive Maps ◽  
Banach Contraction

Kannan maps have inspired a branch of metric fixed point theory devoted to the extension of the classical Banach contraction principle. The study of these maps in modular vector spaces was attempted timidly and was not successful. In this work, we look at this problem in the variable exponent sequence spaces lp(·). We prove the modular version of most of the known facts about these maps in metric and Banach spaces. In particular, our results for Kannan nonexpansive maps in the modular sense were never attempted before.

scholarly journals Fixed Points of Kannan Maps in the Variable Exponent Sequence Spaces ℓp(·)

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 76 ◽  
Author(s):  
Afrah Abdou ◽  
Mohamed Khamsi
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Variable Exponent ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Sequence Spaces ◽  
Vector Spaces ◽  
Nonexpansive Maps ◽  
Banach Contraction

Kannan maps have inspired a branch of metric fixed point theory devoted to the extension of the classical Banach contraction principle. The study of these maps in modular vector spaces was attempted timidly and was not successful. In this work, we look at this problem in the variable exponent sequence spaces ℓ p ( · ) . We prove the modular version of most of the known facts about these maps in metric and Banach spaces. In particular, our results for Kannan nonexpansive maps in the modular sense were never attempted before.

scholarly journals Approximating Fixed Points of Reich–Suzuki Type Nonexpansive Mappings in Hyperbolic Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Kifayat Ullah ◽  
Junaid Ahmad ◽  
Manuel De La Sen ◽  
Muhammad Naveed Khan
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Iterative Process ◽  
Fixed Point Theory ◽  
Nonexpansive Mappings ◽  
Point Theory ◽  
Hyperbolic Spaces ◽  
Uniformly Convex ◽  
Convergence Results ◽  
Hyperbolic Metric Space

In this work, we prove some strong and Δ convergence results for Reich-Suzuki type nonexpansive mappings through M iterative process. A uniformly convex hyperbolic metric space is used as underlying setting for our approach. We also provide an illustrate numerical example. Our results improve and extend some recently announced results of the metric fixed-point theory.

scholarly journals Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 187-196 ◽  
Author(s):  
Kifayat Ullah ◽  
Muhammad Arshad
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Fixed Points ◽  
Fixed Point Theory ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Point Theory ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Uniformly Convex Banach Spaces

In this paper we propose a new three-step iteration process, called M iteration process, for approximation of fixed points. Some weak and strong convergence theorems are proved for Suzuki generalized nonexpansive mappings in the setting of uniformly convex Banach spaces. Numerical example is given to show the efficiency of new iteration process. Our results are the extension, improvement and generalization of many known results in the literature of iterations in fixed point theory.

scholarly journals Concentration analysis in Banach spaces

2016 ◽  
Vol 18 (03) ◽  
pp. 1550038 ◽  
Author(s):  
Sergio Solimini ◽  
Cyril Tintarev
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Uniformly Convex ◽  
Opial Condition ◽  
Uniformly Convex Banach Spaces ◽  
Weak Star ◽  
Compactness Arguments

The concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach–Alaoglu weak-star compactness theorem. We prove existence of profile decompositions for general bounded sequences in uniformly convex Banach spaces equipped with a group of bijective isometries, thus generalizing analogous results previously obtained for Sobolev spaces and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are based on the notion of [Formula: see text]-convergence by Lim [Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179–182] instead of weak convergence, and the two modes coincide if and only if the norm satisfies the well-known Opial condition, in particular, in Hilbert spaces and [Formula: see text]-spaces, but not in [Formula: see text], [Formula: see text]. [Formula: see text]-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies the connection of [Formula: see text]-convergence with the Brezis–Lieb lemma and gives a version of the latter without an assumption of convergence a.e.

scholarly journals Iterative Construction of Fixed Points for Operators Endowed with Condition E in Metric Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Junaid Ahmad ◽  
Kifayat Ullah ◽  
Hüseyin Işik ◽  
Muhammad Arshad ◽  
Manuel de la Sen
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Iterative Process ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Uniformly Convex ◽  
Uniformly Convex Hyperbolic Space ◽  
Iterative Procedures ◽  
Iterative Construction

We consider the class of mappings endowed with the condition E in a nonlinear domain called 2-uniformly convex hyperbolic space. We provide some strong and Δ -convergence theorems for this class of mappings under the Agarwal iterative process. In order to support the main outcome, we procure an example of mappings endowed with the condition E and prove that its Agarwal iterative process is more effective than Mann and Ishikawa iterative processes. Simultaneously, our results hold in uniformly convex Banach, CAT(0), and some CAT( κ ) spaces. This approach essentially provides a new setting for researchers who are working on the iterative procedures in fixed point theory and applications.

scholarly journals On Milman’s moduli for Banach spaces

2001 ◽  
Vol 6 (2) ◽  
pp. 115-129 ◽  
Author(s):  
Elisabetta Maluta ◽  
Stanislaw Prus ◽  
Mariusz Szczepanik
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Uniform Convexity ◽  
Infinite Dimensional ◽  
Uniform Smoothness

We show that infinite dimensional geometric moduli introduced by Milman are strongly related to nearly uniform convexity and nearly uniform smoothness. An application of those moduli to fixed point theory is given.

Modeling the dynamics of hepatitis E via the Caputo–Fabrizio derivative

2019 ◽  
Vol 14 (3) ◽  
pp. 311 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Zakia Hammouch ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Numerical Scheme ◽  
Arbitrary Order ◽  
Fixed Point Theory ◽  
Hepatitis E ◽  
Point Theory ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Essential Properties

A virus that causes hepatitis E is known as (HEV) and regarded on of the reason for lever inflammation. In mathematical aspects a very low attention has been paid to HEV dynamics. Therefore, the present work explores the HEV dynamics in fractional derivative. The Caputo–Fabriizo derivative is used to study the dynamics of HEV. First, the essential properties of the model will be presented and then describe the HEV model with CF derivative. Application of fixed point theory is used to obtain the existence and uniqueness results associated to the model. By using Adams–Bashfirth numerical scheme the solution is obtained. Some numerical results and tables for arbitrary order derivative are presented.

Sign in / Sign up

Export Citation Format

Share Document