scholarly journals Some Aspects of Geometric Constants in Modular Spaces

2021 ◽  
Vol 1 (2) ◽  
pp. 151-163
Author(s):  
Zhijian Yang ◽  
Qi Liu ◽  
Muhammad Sarfraz ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Strict Convexity ◽  
Uniform Convexity ◽  
Normed Spaces ◽  
James Constant ◽  
Modular Spaces ◽  
The Relationship ◽  
Geometric Constants

In this paper, we generalize the typical geometric constants of Banach spaces to modular spaces. We study the equivalence between the convexity of modular and normed spaces, and obtain the relationship between ρ-Neumann-Jordan constant and ρ-James constant. In particular, we extend the convexity and smoothness modular, and obtain the criterion theorems of the uniform convexity and strict convexity.

scholarly journals Some Properties Concerning the JL(X) and YJ(X) Which Related to Some Special Inscribed Triangles of Unit Ball

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1285
Author(s):  
Asif Ahmad ◽  
Yuankang Fu ◽  
Yongjin Li
Keyword(s):  
Unit Ball ◽  
Banach Spaces ◽  
Uniformly Convex ◽  
Geometric Properties ◽  
James Constant ◽  
Von Neumann ◽  
Geometric Constants

In this paper, we will make some further discussions on the JL(X) and YJ(X) which are symmetric and related to the side lengths of some special inscribed triangles of the unit ball, and also introduce two new geometric constants L1(X,▵), L2(X,▵) which related to the perimeters of some special inscribed triangles of the unit ball. Firstly, we discuss the relations among JL(X), YJ(X) and some geometric properties of Banach spaces, including uniformly non-square and uniformly convex. It is worth noting that we point out that uniform non-square spaces can be characterized by the side lengths of some special inscribed triangles of unit ball. Secondly, we establish some inequalities for JL(X), YJ(X) and some significant geometric constants, including the James constant J(X) and the von Neumann-Jordan constant CNJ(X). Finally, we introduce the two new geometric constants L1(X,▵), L2(X,▵), and calculate the bounds of L1(X,▵) and L2(X,▵) as well as the values of L1(X,▵) and L2(X,▵) for two Banach spaces.

scholarly journals Complex Convexity of Orlicz Modular Sequence Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Lili Chen ◽  
Deyun Chen ◽  
Yang Jiang
Keyword(s):  
Sequence Space ◽  
Extreme Points ◽  
Strict Convexity ◽  
Sequence Spaces ◽  
Uniform Convexity ◽  
Uniformly Convex ◽  
Strictly Convex ◽  
Modular Spaces

The concepts of complex extreme points, complex strongly extreme points, complex strict convexity, and complex midpoint locally uniform convexity in general modular spaces are introduced. Then we prove that, for any Orlicz modular sequence spacelΦ,ρ,lΦ,ρis complex midpoint locally uniformly convex. As a corollary,lΦ,ρis also complex strictly convex.

scholarly journals Measures of noncompactness in modular spaces and fixed point theorems for multivalued nonexpansive mappings

2021 ◽  
Vol 23 (3) ◽  
Author(s):  
T. Domínguez Benavides ◽  
P. Lorenzo Ramírez
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Geometric Property ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Uniform Convexity ◽  
Vector Spaces ◽  
Multivalued Mappings ◽  
Measures Of Noncompactness ◽  
Modular Spaces

AbstractThis paper is devoted to state some fixed point results for multivalued mappings in modular vector spaces. For this purpose, we study the uniform noncompact convexity, a geometric property for modular spaces which is similar to nearly uniform convexity in the Banach spaces setting. Using this property, we state several new fixed point theorems for multivalued nonexpansive mappings in modular spaces.

scholarly journals Hyers-Ulam stability of quadratic forms in 2-normed spaces

2019 ◽  
Vol 52 (1) ◽  
pp. 496-502
Author(s):  
Won-Gil Park ◽  
Jae-Hyeong Bae
Keyword(s):  
Banach Spaces ◽  
Quadratic Forms ◽  
Functional Equations ◽  
Ulam Stability ◽  
Normed Spaces

AbstractIn this paper, we obtain Hyers-Ulam stability of the functional equationsf (x + y, z + w) + f (x − y, z − w) = 2f (x, z) + 2f (y, w),f (x + y, z − w) + f (x − y, z + w) = 2f (x, z) + 2f (y, w)andf (x + y, z − w) + f (x − y, z + w) = 2f (x, z) − 2f (y, w)in 2-Banach spaces. The quadratic forms ax2 + bxy + cy2, ax2 + by2 and axy are solutions of the above functional equations, respectively.

scholarly journals On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 116
Author(s):  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Product Space ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
The Other ◽  
Inner Product ◽  
Equivalent Characterization ◽  
Geometric Constant ◽  
Geometric Constants

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete space.

scholarly journals HUR-approximation of an ELTA functional equation

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4311-4328
Author(s):  
A.R. Sharifi ◽  
Azadi Kenary ◽  
B. Yousefi ◽  
R. Soltani
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Linear Mapping ◽  
Stability Theorem ◽  
Normed Spaces ◽  
The Stability ◽  
Rassias Stability

The main goal of this paper is study of the Hyers-Ulam-Rassias stability (briefly HUR-approximation) of the following Euler-Lagrange type additive(briefly ELTA) functional equation ?nj=1f (1/2 ?1?i?n,i?j rixi- 1/2 rjxj) + ?ni=1 rif(xi)=nf (1/2 ?ni=1 rixi) where r1,..., rn ? R, ?ni=k rk?0, and ri,rj?0 for some 1? i < j ? n, in fuzzy normed spaces. The concept of HUR-approximation originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

scholarly journals The generalized projection methods in countably normed spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sarah Tawfeek ◽  
Nashat Faried ◽  
H. A. El-Sharkawy
Keyword(s):  
Banach Spaces ◽  
Projection Operator ◽  
Metric Projection ◽  
Projection Methods ◽  
Generalized Projection ◽  
Uniformly Convex ◽  
Normed Spaces ◽  
Generalized Projection Operator ◽  
Set Valued Mapping ◽  
Uniformly Smooth

AbstractLet E be a Banach space with dual space $E^{*}$ E ∗ , and let K be a nonempty, closed, and convex subset of E. We generalize the concept of generalized projection operator “$\Pi _{K}: E \rightarrow K$ Π K : E → K ” from uniformly convex uniformly smooth Banach spaces to uniformly convex uniformly smooth countably normed spaces and study its properties. We show the relation between J-orthogonality and generalized projection operator $\Pi _{K}$ Π K and give examples to clarify this relation. We introduce a comparison between the metric projection operator $P_{K}$ P K and the generalized projection operator $\Pi _{K}$ Π K in uniformly convex uniformly smooth complete countably normed spaces, and we give an example explaining how to evaluate the metric projection $P_{K}$ P K and the generalized projection $\Pi _{K}$ Π K in some cases of countably normed spaces, and this example illustrates that the generalized projection operator $\Pi _{K}$ Π K in general is a set-valued mapping. Also we generalize the generalized projection operator “$\pi _{K}: E^{*} \rightarrow K$ π K : E ∗ → K ” from reflexive Banach spaces to uniformly convex uniformly smooth countably normed spaces and study its properties in these spaces.

scholarly journals Additive ρ-functional inequalities in β-homogeneous normed spaces

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1651-1658
Author(s):  
Choonkil Park
Keyword(s):  
Banach Spaces ◽  
Complex Number ◽  
Functional Equations ◽  
Additive Functional ◽  
Functional Inequalities ◽  
Ulam Stability ◽  
Normed Spaces ◽  
Homogeneous Complex ◽  
Fixed Complex

In this paper, we solve the following additive ?-functional inequalities ||f (x + y) - f (x) - f (y)|| ? ???(2f (x+y/2) - f(x) + -f (y))??, (1) where ? is a fixed complex number with |?|<1, and ??2f(x+y/2)-f(x)- f(y)???||?(f(x+y)-f(x)-f(y))||, (2) where ? is a fixed complex number with |?|<1/2 , and prove the Hyers-Ulam stability of the additive ?-functional inequalities (1) and (2) in ?-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of additive ?-functional equations associated with the additive ?-functional inequalities (1) and (2) in ?-homogeneous complex Banach spaces.

scholarly journals Some fixed point theorems in n-normed spaces

2020 ◽  
Vol 25 (3) ◽  
pp. 1-15 ◽  
Author(s):  
Hanan Sabah Lazam ◽  
Salwa Salman Abed
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Normed Space ◽  
Fixed Point Theorems ◽  
Normed Spaces ◽  
Weak Contraction ◽  
Contraction Mappings ◽  
Basic Properties ◽  
Definition Of

In this article, we recall the definition of a real n-normed space and some basic properties. fixed point theorems for types of Kannan, Chatterge, Zamfirescu, -Weak contraction and  - (,)-Weak contraction mappings in  Banach spaces.

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