scholarly journals Some Topological and Geometric Properties of Some New Spaces ofλ-Convergent and Bounded Series

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Meltem Kaya ◽  
Hasan Furkan
Keyword(s):  
Fixed Point ◽  
Convexity Property ◽  
Point Property ◽  
Geometric Properties ◽  
Matrix Classes ◽  
Modulus Of Convexity ◽  
Opial Property ◽  
Weak Fixed Point Property ◽  
Inclusion Relations ◽  
Schauder Bases

The main purpose of this study is to introduce the spacescsλ,cs0λ, andbsλwhich areBK-spaces of nonabsolute type. We prove that these spaces are linearly isomorphic to the spacescs,cs0, andbs, respectively, and derive some inclusion relations. Additionally, Schauder bases of the spacescsλandcs0λhave been constructed and theα-,β-, andγ-duals of these spaces have been computed. Besides, we characterize some matrix classes from the spacescsλ,cs0λ, andbsλto the spaceslp,c, andc0, where1≤p≤∞. Finally, we examine some geometric properties of these spaces as Gurarǐ’s modulus of convexity, propertym∞, property(M), property WORTH, nonstrict Opial property, and weak fixed point property.

scholarly journals Some Geometric Properties of Lacunary Sequence Spaces Related to Fixed Point Property

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Chirasak Mongkolkeha ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Lacunary Sequence ◽  
Fixed Point Property ◽  
Sequence Spaces ◽  
Point Property ◽  
Geometric Properties ◽  
Opial Property ◽  
Uniform Opial Property

The main purpose of this paper is considering the lacunary sequence spaces defined by Karakaya (2007), by proving the property (β) and Uniform Opial property.

scholarly journals A note on properties that imply the fixed point property

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
S. Dhompongsa ◽  
A. Kaewkhao
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Property ◽  
Point Property ◽  
Geometric Properties ◽  
Weak Fixed Point Property

We give relationships between some Banach-space geometric properties that guarantee the weak fixed point property. The results extend some known results of Dalby and Xu.

GEOMETRIC AND FIXED POINT PROPERTIES IN PRODUCTS OF NORMED SPACES

2019 ◽  
Vol 99 (2) ◽  
pp. 262-273 ◽  
Author(s):  
M. VEENA SANGEETHA
Keyword(s):  
Fixed Point ◽  
Dimensional Subspace ◽  
Point Property ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Nonexpansive Maps ◽  
Uniform Rotundity ◽  
Fixed Point Properties ◽  
Weak Fixed Point Property ◽  
Monotone Norm

Given two (real) normed (linear) spaces $X$ and $Y$, let $X\otimes _{1}Y=(X\otimes Y,\Vert \cdot \Vert )$, where $\Vert (x,y)\Vert =\Vert x\Vert +\Vert y\Vert$. It is known that $X\otimes _{1}Y$ is $2$-UR if and only if both $X$ and $Y$ are UR (where we use UR as an abbreviation for uniformly rotund). We prove that if $X$ is $m$-dimensional and $Y$ is $k$-UR, then $X\otimes _{1}Y$ is $(m+k)$-UR. In the other direction, we observe that if $X\otimes _{1}Y$ is $k$-UR, then both $X$ and $Y$ are $(k-1)$-UR. Given a monotone norm $\Vert \cdot \Vert _{E}$ on $\mathbb{R}^{2}$, we let $X\otimes _{E}Y=(X\otimes Y,\Vert \cdot \Vert )$ where $\Vert (x,y)\Vert =\Vert (\Vert x\Vert _{X},\Vert y\Vert _{Y})\Vert _{E}$. It is known that if $X$ is uniformly rotund in every direction, $Y$ has the weak fixed point property for nonexpansive maps (WFPP) and $\Vert \cdot \Vert _{E}$ is strictly monotone, then $X\otimes _{E}Y$ has WFPP. Using the notion of $k$-uniform rotundity relative to every $k$-dimensional subspace we show that this result holds with a weaker condition on $X$.

scholarly journals Some properties of Generalized Fibonacci difference bounded and $p$-absolutely convergent sequences

2018 ◽  
Vol 36 (1) ◽  
pp. 37 ◽  
Author(s):  
Bipan Hazarika ◽  
Anupam Das
Keyword(s):  
Sequence Space ◽  
Geometric Properties ◽  
Matrix Classes ◽  
Band Matrix ◽  
Inclusion Relations ◽  
Alpha Beta ◽  
Convergent Sequences

The main objective of this paper is to introduced a new sequence space $l_{p}(\hat{F}(r,s)),$ $ 1\leq p \leq \infty$ by using the band matrix $\hat{F}(r,s).$ We also establish a few inclusion relations concerning this space and determine its $\alpha-,\beta-,\gamma-$duals. We also characterize some matrix classes on the space $l_{p}(\hat{F}(r,s))$ and examine some geometric properties of this space.

scholarly journals A class of spaces with weak normal structure

1994 ◽  
Vol 49 (3) ◽  
pp. 523-528 ◽  
Author(s):  
Brailey Sims
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Normal Structure ◽  
Fixed Point Property ◽  
Point Property ◽  
Weak Fixed Point Property

It has recently been shown that a Banach space enjoys the weak fixed point property if it is ε0-inquadrate for some ε0 < 2 and has WORTH; that is, if then, ║xn — x║ — ║xn + x║ → 0, for all x. We establish the stronger conclusion of weak normal structure under the substantially weaker assumption that the space has WORTH and is ‘ε0-inquadrate in every direction’ for some ε0 < 2.

scholarly journals On \(\ell_1\)-preduals distant by 1

2018 ◽  
Vol 72 (2) ◽  
pp. 41
Author(s):  
Łukasz Piasecki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Wide Class ◽  
Fixed Point Theory ◽  
General Setting ◽  
Point Theory ◽  
Fixed Point Property ◽  
Sufficient Condition ◽  
Point Property ◽  
Weak Fixed Point Property

For every predual \(X\) of \(\ell_1\) such that the standard basis in \(\ell_1\) is weak\(^*\) convergent, we give explicit models of all Banach spaces \(Y\) for which the Banach-Mazur distance \(d(X,Y)=1\). As a by-product of our considerations, we obtain some new results in metric fixed point theory. First, we show that the space \(\ell_1\), with a predual \(X\) as above, has the stable weak\(^*\) fixed point property if and only if it has almost stable weak\(^*\) fixed point property, i.e. the dual \(Y^*\) of every Banach space \(Y\) has the weak\(^*\) fixed point property (briefly, \(\sigma(Y^*,Y)\)-FPP) whenever \(d(X,Y)=1\). Then, we construct a predual \(X\) of \(\ell_1\) for which \(\ell_1\) lacks the stable \(\sigma(\ell_1,X)\)-FPP but it has almost stable \(\sigma(\ell_1,X)\)-FPP, which in turn is a strictly stronger property than the \(\sigma(\ell_1,X)\)-FPP. Finally, in the general setting of preduals of \(\ell_1\), we give a sufficient condition for almost stable weak\(^*\) fixed point property in \(\ell_1\) and we prove that for a wide class of spaces this condition is also necessary.

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