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Fixed point property for nonexpansive mappings on large classes in Köthe-Toeplitz duals of certain difference sequence spaces
... We show that for any m ∈ N there exists a large class of closed, bounded and convex subsets of Köthe-Toeplitz dual for X ∞ (△ m ) with fixed point property for nonexpansive mappings. We note that case m = 1 has recently been done by Nezir and Cankurt [21]. As we stated, here we present the general case for any m ∈ N. Now, we consider the following class of closed, bounded and convex subsets. ...
In 1970, Ces?ro sequence spaces was introduced by Shiue. In 1981, K?zmaz defined difference sequence spaces for ??, c0 and c. Then, in 1983, Orhan introduced Ces?ro difference sequence spaces. Both works used difference operator and investigated K?the-Toeplitz duals for the new Banach spaces they introduced. Later, various authors generalized these new spaces, especially the one introduced by Orhan. In this study, first we discuss the fixed point property for these spaces. Then, we recall that Goebel and Kuczumow showed that there exists a very large class of closed, bounded, convex subsets in Banach space of absolutely summable scalar sequences, ?1 with fixed point property for nonexpansive mappings. So we consider a Goebel and Kuczumow analogue result for a K?the-Toeplitz dual of a generalized Ces?ro difference sequence space. We show that there exists a large class of closed, bounded and convex subsets of these spaces with fixed point property for nonexpansive mappings.
In this paper we introduce the generalized difference Cesàro sequence
spaces Co (Am), Oxo (Am), Cp (Am), Op (Am) and lp (Amm) for 1 < p < ∞. We study some topological properties of these spaces. We obtain some inclusion relations involving these sequence spaces. These notions generalize many notions on difference Cesàro sequence spaces.
In the present paper, we introduce a new difference sequence space rqB (u,p) by using the Riesz mean and the B- difference matrix. We show rqB (u,p) is a complete linear metric space and is linearly isomorphic to the space l(p). We have also computed its α-, ß- and γ-duals. Furthermore, we have constructed the basis of rqB (u,p)and characterize a matrix class rqB (u,p), l∞.
In this paper, we define the sequence spaces ) , c f " and 0 c be the linear spaces of bounded, convergent, and null sequences ) ( k x x with complex terms, respectively, normed by k k x x sup f
Let
p = {\left( {p_{k} } \right)}^{\infty }_{{k = 0}}
be a bounded sequence of positive reals, m ∈ ℕ and u be s sequence of nonzero terms. If
x = {\left( {x_{k} } \right)}^{\infty }_{{k = 0}}
is any sequence of complex numbers we write Δ(m)
x for the sequence of the m–th order differences of x and
\Delta ^{{{\left( m \right)}}}_{u} X = {\left\{ {x = {\left( x \right)}^{\infty }_{{k = 0}} :u\Delta ^{{{\left( m \right)}}} x \in X} \right\}}
for any set X of sequences. We determine the α–, β– and γ–duals of the sets
\Delta ^{{{\left( m \right)}}}_{u} X
for X = c
0(p), c(p), ℓ∞(p) and characterize some matrix transformations between these spaces Δ(m)
X.
It is proved that for any the Cesàro sequence space ces is (kNUC) for any natural number k and it has the uniform Opial property. Moreover, weakly convergent sequence coefficient of those spaces is also calculated. It is also proved that for the spaces ces have property (L) and weak uniform normal structure. The packing rate of those spaces is also calculated.
An introductory course in summability theory for students, researchers, physicists, and engineers In creating this book, the authors' intent was to provide graduate students, researchers, physicists, and engineers with a reasonable introduction to summability theory. Over the course of nine chapters, the authors cover all of the fundamental concepts and equations informing summability theory and its applications, as well as some of its lesser known aspects. Following a brief introduction to the history of summability theory, general matrix methods are introduced, and the Silverman-Toeplitz theorem on regular matrices is discussed. A variety of special summability methods, including the Nörlund method, the Weighted Mean method, the Abel method, and the (C, 1) - method are next examined. An entire chapter is devoted to a discussion of some elementary Tauberian theorems involving certain summability methods. Following this are chapters devoted to matrix transforms of summability and absolute summability domains of reversible and normal methods; the notion of a perfect matrix method; matrix transforms of summability and absolute summability domains of the Cesàro and Riesz methods; convergence and the boundedness of sequences with speed; and convergence, boundedness, and summability with speed. Discusses results on matrix transforms of several matrix methods The only English-language textbook describing the notions of convergence, boundedness, and summability with speed, as well as their applications in approximation theory Compares the approximation orders of Fourier expansions in Banach spaces by different matrix methods Matrix transforms of summability domains of regular perfect matrix methods are examined Each chapter contains several solved examples and end-of-chapter exercises, including hints for solutions An Introductory Course in Summability Theory is the ideal first text in summability theory for graduate students, especially those having a good grasp of real and complex analysis. It is also a valuable reference for mathematics researchers and for physicists and engineers who work with Fourier series, Fourier transforms, or analytic continuation.
This book discusses recent developments in and contemporary research on summability theory, including general summability methods, direct theorems on summability, absolute and strong summability, special methods of summability, functional analytic methods in summability, and related topics and applications. All contributing authors are eminent scientists, researchers and scholars in their respective fields, and hail from around the world. The book can be used as a textbook for graduate and senior undergraduate students, and as a valuable reference guide for researchers and practitioners in the fields of summability theory and functional analysis.
Summability theory is generally used in analysis and applied mathematics. It plays an important part in the engineering sciences, and various aspects of the theory have long since been studied by researchers all over the world.
In this note we give an example of a weakly compact convex subset of L1[0, 1] that fails to have the fixed point property for nonexpansive maps. This answers a long-standing question which was recently raised again by S. Reich [7].
In this paper define the spaces l ∞ (Δ), c(Δ), and c 0 (Δ), where for instance l ∞ (Δ) = {x=(x k ):sup k |x k -x k + l |< ∞} , and compute their duals (continuous dual, α-dual, β-dual and γ-dual). We also determine necessary and sufficient conditions for a matrix A to map l ∞ (Δ) or c(Δ) into l ∞ or c , and investigate related questions.
The aim of this chapter is to present criteria for the most important geometric properties related to the metric fixed point theory in some classes of Banach function lattices, mainly in Orlicz spaces and Cesaro sequence spaces. We also give some informations about respective results for Musielak-Orlicz spaces, Orlicz-Lorentz spaces and Calderón-Lozanovsky spaces.
We introduce the strongly (Vλ,A,Δ(vm)n,p)-summable sequences and give the relation between the spaces of strongly (Vλ,A,Δ(vm)n,p)-summable sequences and strongly (Vλ,A,Δ(vm)n,p)-summable sequences with respect to an Orlicz function when A=(aik)A=(aik) is an infinite matrix of complex numbers, Δ(vm)n is a generalized difference operator and p=(pi)p=(pi) is a sequence of positive real numbers. Also we give natural relationship between strongly (Vλ,A,Δ(vm)n,p)-convergence with respect to an Orlicz function and strongly Sλ(A,Δ(vm)n)-statistical convergence.
In this paper, we prove that every noncommutative L1L1-space associated to a finite von Neumann algebra can be renormed to satisfy the fixed point property for nonexpansive affine mappings. Particular examples are L1(R)L1(R), where RR is the hyperfinite II1II1 factor and the function spaces L1[0,1]L1[0,1] and L1(μ)L1(μ) for any σσ-finite measure space. This property does not hold for the usual ‖⋅‖1‖⋅‖1 norm.
The definition of the pα-, pβ- and pγ-duals of a sequence space was defined by Et [Internat. J. Math. Math. Sci. 24 (2000) 785–791]. In this paper we compute pα- and N-duals of the sequence spaces Δmv(X) for X=ℓ∞, c and c0, and compute β- and γ-duals of the sequence spaces Δmv(X) for X=ℓ∞, c and c0.
The James' distortion theorems and their relationship to fixed points and to the more restrictive l1 and c0 are investigated. James' distortion theorems state that Banach spaces which contain isomorphic copies of l1 contain almost isometric copies. Proofs are presented to confirm the theorems.
The main aim of this article is to generalize the notion of almost convergent, Cesµaro summable and lacunary summable spaces by using a generalized difierence operator deflned associating a sequence of non-zero scalars and characterize some matrix classes involving these se- quence spaces. In this article we also introduce the idea of difierence inflnite matrices. It is expected that these investigations will generalize several notions associated with thus constructed spaces as well as of matrix transformations.
Let for all k∈N, and let ⦀⋅⦀ be the equivalent norm of ℓ1 defined by We prove that (ℓ1,⦀⋅⦀) has the fixed point property for nonexpansive self-mappings.
In this paper, we show that the weak nearly uniform smooth Banach spaces have the fixed point property for nonexpansive mappings.
This book is devoted to some aspects of fixed point theory. To the author’s mind the results considered and described in detail in the book are always couched in at least a metric framework, usually in a Banach spaces setting, and the methods typically involve both the topological and the geometric structure of the space in conjunction with metric constraints on the behavior of the mappings. The book contains some short chapters: 1. Preliminaries; 2. Banach’s contractions principle; 3. Nonexpansive mappings: introduction; 4. The basic fixed point theorems for nonexpansive mappings; 5. Scaling the convexity of the unit ball; 6. The modulus of convexity and normal structure; 7. Normal structure and smoothness; 8. Conditions involving compactness; 9. Sequential approximation techniques for nonexpansive mappings; 10. Weak sequential approximations; 11. Properties of fixed point sets and minimal sets; 12. Special properties of Hilbert spaces; 13. Applications to accretivity; 14. Ultrafilter methods; 15. Set-valued mappings; 16. Uniformly Lipschitzian mappings; 17. Rotative mappings; 18. The theorems of Brouwer and Schauder; 19. Lipschitzian mappings; 20. Minimal displacement; 21. The retraction problem; it contains an extensive bibliography. Undoubtedly, the book should be of interested to graduate students seeking a field of interest, to mathematicians interested in learning about the subject and to specialists.
Fixed Points of Nonexpansive Maps on Closed, Bounded, Convex Sets in $ell^{1}$
- T M Everest
T. M. Everest, Fixed Points of Nonexpansive Maps on Closed, Bounded, Convex Sets in ℓ 1,
Ph.D thesis, University of Pittsburgh, Pittsburgh, 2013.
Fixed point properties of some sets in $ell^1$
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W. Kaczor and S. Prus, Fixed point properties of some sets in ℓ 1, Proceedings of the International Conference on Fixed Point Theory and Applications, (2004), 11 pp.
Geometric properties related to fixed point theory in some Banach function lattices, Handbook of Metric Fixed Point Theory
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Banach-Saks property and property (α) in Cesàro sequence spaces
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sequence spaces, Southeast Asian Bull. Math., 24 (2000), 201-210.
On some generalized sequence spaces
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Der fixpunktsatz in funktionalraumen
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