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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 213878, 14 pages
http://dx.doi.org/10.1155/2011/213878
Research Article

Some Topological and Geometrical Properties of a New Difference Sequence Space

1Department of Mathematics, Faculty of Arts and Science, Gaziosmanpaşa University, 60250 Tokat, Turkey
2Faculty of Education, Inönü University, 44280 Malatya, Turkey

Received 9 August 2010; Accepted 25 January 2011

Academic Editor: Narcisa C.Β Apreutesei

Copyright Β© 2011 Serkan Demiriz and Celal Çakan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce the new difference sequence space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) . Further, it is proved that the space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) is the BK-space including the space 𝑏 𝑣 𝑝 , which is the space of sequences of pbounded variation. We also show that the spaces π‘Ž π‘Ÿ 𝑝 ( Ξ” ) , and β„“ 𝑝 are linearly isomorphic for 1 ≀ 𝑝 < ∞ . Furthermore, the basis and the 𝛼 - , 𝛽 - and 𝛾 - duals of the space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) are determined. We devote the final section of the paper to examine some geometric properties of the space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) .

1. Preliminaries, Background, and Notation

By πœ” , we will denote the space of all real valued sequences. Any vector subspace of πœ” is called as a sequence space. We will write β„“ ∞ , 𝑐 , and 𝑐 0 for the spaces of all bounded, convergent and null sequences, respectively. Also, by 𝑏 𝑠 , 𝑐 𝑠 , β„“ 1 and β„“ 𝑝 ; we denote the spaces of all bounded, convergent, absolutely, and 𝑝 -absolutely convergent series, respectively, where 1 < 𝑝 < ∞ .

A sequence space πœ† with a linear topology is called a 𝐾 -space, provided each of the maps 𝑝 𝑖 ∢ πœ† β†’ β„‚ defined by 𝑝 𝑖 ( π‘₯ ) = π‘₯ 𝑖 is continuous for all 𝑖 ∈ β„• , where β„‚ denotes the complex field and β„• = { 0 , 1 , 2 , … } . A 𝐾 -space πœ† is called an FK-space, provided πœ† is a complete linear metric space. An 𝐹 𝐾 -space whose topology is normable is called a BK-space (see [1, pages 272-273]).

Let πœ† , πœ‡ be two sequence spaces and 𝐴 = ( π‘Ž 𝑛 π‘˜ ) an infinite matrix of real or complex numbers π‘Ž 𝑛 π‘˜ , where 𝑛 , π‘˜ ∈ β„• . Then, we say that 𝐴 defines a matrix mapping from πœ† into πœ‡ , and we denote it by writing 𝐴 ∢ πœ† β†’ πœ‡ ; if for every sequence π‘₯ = ( π‘₯ π‘˜ ) ∈ πœ† , the sequence 𝐴 π‘₯ = { ( 𝐴 π‘₯ ) 𝑛 } , the 𝐴 -transform of π‘₯ , is in πœ‡ , where ( 𝐴 π‘₯ ) 𝑛 =  π‘˜ π‘Ž 𝑛 π‘˜ π‘₯ π‘˜ ( 𝑛 ∈ β„• ) . ( 1 . 1 ) For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞ . By ( πœ† ∢ πœ‡ ) we denote the class of all matrices 𝐴 such that 𝐴 ∢ πœ† β†’ πœ‡ . Thus, 𝐴 ∈ ( πœ† ∢ πœ‡ ) if and only if the series on the right side of (1.1) converges for each 𝑛 ∈ β„• and every π‘₯ ∈ πœ† , and we have 𝐴 π‘₯ = { ( 𝐴 π‘₯ ) 𝑛 } 𝑛 ∈ β„• ∈ πœ‡ for all π‘₯ ∈ πœ† . A sequence π‘₯ is said to be 𝐴 -summable to 𝛼 if 𝐴 π‘₯ converges to 𝛼 which is called as the 𝐴 -limit of π‘₯ .

If a normed sequence space πœ† contains a sequence ( 𝑏 𝑛 ) with the property that for every π‘₯ ∈ πœ† , there is a unique sequence of scalars ( 𝛼 𝑛 ) such that l i m 𝑛 ⟢ ∞ β€– β€– ξ€· 𝛼 π‘₯ βˆ’ 0 𝑏 0 + 𝛼 1 𝑏 1 + β‹― + 𝛼 𝑛 𝑏 𝑛 ξ€Έ β€– β€– = 0 , ( 1 . 2 ) then ( 𝑏 𝑛 ) is called a Schauder basis (or briefly basis) for πœ† . The series βˆ‘ 𝛼 π‘˜ 𝑏 π‘˜ which has the sum π‘₯ is then called the expansion of π‘₯ with respect to ( 𝑏 𝑛 ) and written as βˆ‘ 𝛼 π‘₯ = π‘˜ 𝑏 π‘˜ .

For a sequence space πœ† , the matrix domain πœ† 𝐴 of an infinite matrix 𝐴 is defined by πœ† 𝐴 = ξ€½ ξ€· π‘₯ π‘₯ = π‘˜ ξ€Έ ξ€Ύ ∈ πœ” ∢ 𝐴 π‘₯ ∈ πœ† , ( 1 . 3 ) which is a sequence space. The new sequence space πœ† 𝐴 generated by the limitation matrix 𝐴 from the space πœ† either includes the space πœ† or is included by the space πœ† , in general; that is, the space πœ† 𝐴 is the expansion or the contraction of the original space πœ† .

We will define 𝐡 π‘Ÿ = ( 𝑏 π‘Ÿ 𝑛 π‘˜ ) by 𝑏 π‘Ÿ 𝑛 π‘˜ = ⎧ βŽͺ βŽͺ ⎨ βŽͺ βŽͺ ⎩ 1 βˆ’ π‘Ÿ π‘Ÿ 𝑛 + 1 π‘˜ π‘Ÿ , 0 ≀ π‘˜ ≀ 𝑛 βˆ’ 1 , 𝑛 + 1 𝑛 + 1 , π‘˜ = 𝑛 , 0 , π‘˜ > 𝑛 , ( 0 < π‘Ÿ < 1 ) , ( 1 . 4 ) for all 𝑛 , π‘˜ ∈ β„• and denote the collection of all finite subsets of β„• by β„± . We will also use the convention that any term with negative subscript is equal to naught.

The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has been recently employed by Wang [2], Ng and Lee [3], Malkowsky [4], and Altay et al. [5]. They introduced the sequence spaces ( β„“ 𝑝 ) 𝑁 π‘ž in [2], ( β„“ 𝑝 ) 𝐢 1 = 𝑋 𝑝 in [3], ( β„“ ∞ ) 𝑅 𝑑 = π‘Ÿ 𝑑 ∞ , 𝑐 𝑅 𝑑 = π‘Ÿ 𝑑 𝑐 and ( 𝑐 0 ) 𝑅 𝑑 = π‘Ÿ 𝑑 0 in [4] and ( β„“ 𝑝 ) 𝐸 π‘Ÿ = 𝑒 π‘Ÿ 𝑝 in [5]; where 𝑁 π‘ž , 𝐢 1 , 𝑅 𝑑 and 𝐸 π‘Ÿ denote the Nörlund, arithmetic, Riesz and Euler means, respectively, and 1 ≀ 𝑝 ≀ ∞ .

Recently, there has been a lot of interest in investigating geometric properties of sequence spaces besides topological and some other usual properties. In literature, there are many papers concerning the geometric properties of different sequence spaces. For example, in [6], Mursaleen et al. studied some geometric properties of normed Euler sequence space. Şimşek and Karakaya [7] investigated the geometric properties of sequence space β„“ 𝜌 ( 𝑒 , 𝑣 , 𝑝 ) equipped with Luxemburg norm. Further information on geometric properties of sequence space can be found in [8, 9].

The main purpose of the present paper is to introduce the difference sequence space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) together with matrix domain and is to derive some inclusion relations concerning with π‘Ž π‘Ÿ 𝑝 ( Ξ” ) . Also, we investigate some topological properties of this new space and furthermore characterize geometric properties concerning Banach-Saks type 𝑝 .

2. π‘Ž π‘Ÿ 𝑝 ( Ξ” ) Difference Sequence Space

In the present section, we introduce the difference sequence space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) and emphasize its some properties. Although the difference sequence space πœ† ( Ξ” ) corresponding to the space πœ† was defined by Kızmaz [10] as follows: πœ† ξ€½ ξ€· π‘₯ ( Ξ” ) = π‘₯ = π‘˜ ξ€Έ ξ€· π‘₯ ∈ πœ” ∢ π‘˜ βˆ’ π‘₯ π‘˜ + 1 ξ€Έ ξ€Ύ ∈ πœ† , ( 2 . 1 ) the difference sequence space corresponding to the space β„“ 𝑝 was not examined, where πœ† denotes the anyone of the spaces 𝑐 0 , 𝑐 or β„“ ∞ . So, Başar and Altay have recently studied the sequence space 𝑏 𝑣 𝑝 , the space of 𝑝 -bounded variation, in [11] defined by 𝑏 𝑣 𝑝 = ξ€½ ξ€· π‘₯ π‘₯ = π‘˜ ξ€Έ ξ€· π‘₯ ∈ πœ” ∢ π‘˜ βˆ’ π‘₯ π‘˜ βˆ’ 1 ξ€Έ ∈ β„“ 𝑝 ξ€Ύ , 1 ≀ 𝑝 < ∞ , ( 2 . 2 ) which fills up the gap in the existing literature. Recently, Aydín and Başar [12] studied the sequence spaces π‘Ž π‘Ÿ 0 and π‘Ž π‘Ÿ 𝑐 , defined by π‘Ž π‘Ÿ 0 = ξƒ― ξ€· π‘₯ π‘₯ = π‘˜ ξ€Έ ∈ πœ” ∢ l i m 𝑛 ⟢ ∞ 1 𝑛 + 1 𝑛  π‘˜ = 0 ξ€· 1 + π‘Ÿ π‘˜ ξ€Έ π‘₯ π‘˜ ξƒ° , π‘Ž = 0 π‘Ÿ 𝑐 = ξƒ― ξ€· π‘₯ π‘₯ = π‘˜ ξ€Έ ∈ πœ” ∢ l i m 𝑛 ⟢ ∞ 1 𝑛 + 1 𝑛  π‘˜ = 0 ξ€· 1 + π‘Ÿ π‘˜ ξ€Έ π‘₯ π‘˜ ξƒ° . e x i s t s ( 2 . 3 ) Aydín and Başar [13] introduced the difference sequence spaces π‘Ž π‘Ÿ 0 ( Ξ” ) and π‘Ž π‘Ÿ 𝑐 ( Ξ” ) , defined by π‘Ž π‘Ÿ 0 ξƒ― ξ€· π‘₯ ( Ξ” ) = π‘₯ = π‘˜ ξ€Έ ∈ πœ” ∢ l i m 𝑛 ⟢ ∞ 1 𝑛 + 1 𝑛  π‘˜ = 0 ξ€· 1 + π‘Ÿ π‘˜ π‘₯ ξ€Έ ξ€· π‘˜ βˆ’ π‘₯ π‘˜ βˆ’ 1 ξ€Έ ξƒ° , π‘Ž = 0 π‘Ÿ 𝑐 ξƒ― ξ€· π‘₯ ( Ξ” ) = π‘₯ = π‘˜ ξ€Έ ∈ πœ” ∢ l i m 𝑛 ⟢ ∞ 1 𝑛 + 1 𝑛  π‘˜ = 0 ξ€· 1 + π‘Ÿ π‘˜ π‘₯ ξ€Έ ξ€· π‘˜ βˆ’ π‘₯ π‘˜ βˆ’ 1 ξ€Έ ξƒ° . e x i s t s ( 2 . 4 ) Aydín [14] introduced π‘Ž π‘Ÿ 𝑝 sequence space, defined by π‘Ž π‘Ÿ 𝑝 = ⎧ βŽͺ ⎨ βŽͺ ⎩ ξ€· π‘₯ π‘₯ = π‘˜ ξ€Έ  ∈ πœ” ∢ 𝑛 | | | | | 1 𝑛 + 1 𝑛  π‘˜ = 0 ξ€· 1 + π‘Ÿ π‘˜ ξ€Έ π‘₯ π‘˜ | | | | | 𝑝 ⎫ βŽͺ ⎬ βŽͺ ⎭ < ∞ ; 1 ≀ 𝑝 < ∞ . ( 2 . 5 ) Define the matrix Ξ” = ( 𝛿 𝑛 π‘˜ ) by 𝛿 𝑛 π‘˜ = ξƒ― ( βˆ’ 1 ) 𝑛 βˆ’ π‘˜ , 𝑛 βˆ’ 1 ≀ π‘˜ ≀ 𝑛 , 0 , 0 ≀ π‘˜ < 𝑛 βˆ’ 1 o r π‘˜ > 𝑛 . ( 2 . 6 ) As was made by Başar and Altay in [11], we treat slightly more different than Kızmaz and the other authors following him and employ the technique obtaining a new sequence space by the matrix domain of a triangle limitation method. We will introduce the sequence space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) which is a natural continuation of Aydín and Başar [13], as follows: π‘Ž π‘Ÿ 𝑝 ⎧ βŽͺ ⎨ βŽͺ ⎩ ξ€· π‘₯ ( Ξ” ) = π‘₯ = π‘˜ ξ€Έ  ∈ πœ” ∢ 𝑛 | | | | | 1 𝑛 + 1 𝑛  π‘˜ = 0 ξ€· 1 + π‘Ÿ π‘˜ π‘₯ ξ€Έ ξ€· π‘˜ βˆ’ π‘₯ π‘˜ βˆ’ 1 ξ€Έ | | | | | 𝑝 ⎫ βŽͺ ⎬ βŽͺ ⎭ < ∞ ; 1 ≀ 𝑝 < ∞ . ( 2 . 7 ) With the notation of (1.3), we may redefine the space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) by π‘Ž π‘Ÿ 𝑝 ξ€· π‘Ž ( Ξ” ) = π‘Ÿ 𝑝 ξ€Έ Ξ” . ( 2 . 8 )

Define the sequence 𝑦 = { 𝑦 𝑛 ( π‘Ÿ ) } which will be frequently used as the 𝐡 π‘Ÿ -transform of a sequence π‘₯ = ( π‘₯ π‘˜ ) , that is, 𝑦 𝑛 ( π‘Ÿ ) = 𝑛 βˆ’ 1  π‘˜ = 0 ( 1 βˆ’ π‘Ÿ ) π‘Ÿ π‘˜ π‘₯ 1 + 𝑛 π‘˜ + 1 + π‘Ÿ 𝑛 π‘₯ 1 + 𝑛 𝑛 , 𝑛 ∈ β„• . ( 2 . 9 ) Now, we may begin with the following theorem which is essential in the text.

Theorem 2.1. The set π‘Ž π‘Ÿ 𝑝 ( Ξ” ) becomes the linear space with the coordinatewise addition and scalar multiplication which is the BK-space with the norm β€– π‘₯ β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) = β€– 𝑦 β€– β„“ 𝑝 , ( 2 . 1 0 ) where 1 ≀ 𝑝 < ∞ .

Proof. Since the proof is routine, we omit the details of the proof.

Theorem 2.2. The space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) is linearly isomorphic to the space β„“ 𝑝 ; that is, π‘Ž π‘Ÿ 𝑝 ( Ξ” ) β‰… β„“ 𝑝 , where 1 ≀ 𝑝 < ∞ .

Proof. It is enough to show the existence of a linear bijection between the spaces π‘Ž π‘Ÿ 𝑝 ( Ξ” ) and β„“ 𝑝 for 1 ≀ 𝑝 < ∞ . Consider the transformation 𝑇 defined, with the notation of (2.9), from π‘Ž π‘Ÿ 𝑝 ( Ξ” ) to β„“ 𝑝 by π‘₯ ↦ 𝑦 = 𝑇 π‘₯ . The linearity of 𝑇 is clear. Furthermore, it is trivial that π‘₯ = πœƒ whenever 𝑇 π‘₯ = πœƒ , and hence, 𝑇 is injective.
We assume that 𝑦 ∈ β„“ 𝑝 for 1 ≀ 𝑝 < ∞ and define the sequence π‘₯ = ( π‘₯ π‘˜ ) by π‘₯ 𝑛 ( π‘Ÿ ) = 𝑛  π‘˜ π‘˜ = 0  𝑗 = π‘˜ βˆ’ 1 ( βˆ’ 1 ) π‘˜ βˆ’ 𝑗 1 + 𝑗 1 + π‘Ÿ π‘˜ 𝑦 𝑗 ; π‘˜ ∈ β„• . ( 2 . 1 1 ) Then, since ( Ξ” π‘₯ ) 𝑛 = 𝑛  𝑗 = 𝑛 βˆ’ 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 1 + 𝑗 1 + π‘Ÿ 𝑛 𝑦 𝑗 ; 𝑛 ∈ β„• , ( 2 . 1 2 ) we get that  𝑛 | | | | | 1 1 + 𝑛 𝑛  π‘˜ = 0 ξ€· 1 + π‘Ÿ π‘˜ ξ€Έ π‘˜  𝑗 = π‘˜ βˆ’ 1 ( βˆ’ 1 ) π‘˜ βˆ’ 𝑗 1 + 𝑗 1 + π‘Ÿ π‘˜ 𝑦 𝑗 | | | | | 𝑝 =  𝑛 | | 𝑦 𝑛 | | 𝑝 < ∞ . ( 2 . 1 3 ) Thus, we have that π‘₯ ∈ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) . In addition, one can derive that β€– π‘₯ β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) = βŽ› ⎜ ⎜ ⎝  𝑛 | | | | | 1 1 + 𝑛 𝑛  π‘˜ = 0 ξ€· 1 + π‘Ÿ π‘˜ π‘₯ ξ€Έ ξ€· π‘˜ βˆ’ π‘₯ π‘˜ βˆ’ 1 ξ€Έ | | | | | 𝑝 ⎞ ⎟ ⎟ ⎠ 1 / 𝑝 = βŽ› ⎜ ⎜ ⎝  𝑛 | | | | | 1 1 + 𝑛 𝑛  π‘˜ = 0 ξ€· 1 + π‘Ÿ π‘˜ ξ€Έ π‘˜  𝑗 = π‘˜ βˆ’ 1 ( βˆ’ 1 ) π‘˜ βˆ’ 𝑗 1 + 𝑗 1 + π‘Ÿ π‘˜ 𝑦 𝑗 | | | | | 𝑝 ⎞ ⎟ ⎟ ⎠ 1 / 𝑝 = β€– 𝑦 β€– β„“ 𝑝 ( 2 . 1 4 ) which means that 𝑇 is surjective and is norm preserving. Hence, 𝑇 is a linear bijection.

We wish to exhibit some inclusion relations concerning with the space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) .

Theorem 2.3. The inclusion 𝑏 𝑣 𝑝 βŠ‚ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) strictly holds for 1 < 𝑝 < ∞ .

Proof. To prove the validity of the inclusion 𝑏 𝑣 𝑝 βŠ‚ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) for 1 < 𝑝 < ∞ , it suffices to show the existence of a number 𝐾 > 0 such that β€– π‘₯ β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) ≀ 𝐾 β€– π‘₯ β€– 𝑏 𝑣 𝑝 for every π‘₯ ∈ 𝑏 𝑣 𝑝 .
Let π‘₯ ∈ 𝑏 𝑣 𝑝 and 1 < 𝑝 < ∞ . Then, we obtain  𝑛 | | | | | 1 1 + 𝑛 𝑛  π‘˜ = 0 ξ€· 1 + π‘Ÿ π‘˜ π‘₯ ξ€Έ ξ€· π‘˜ βˆ’ π‘₯ π‘˜ βˆ’ 1 ξ€Έ | | | | | 𝑝 ≀  𝑛  2 𝑛  π‘˜ = 0 | | π‘₯ π‘˜ βˆ’ π‘₯ π‘˜ βˆ’ 1 | | ξƒ­ 1 + 𝑛 𝑝 < 2 𝑝 ξ‚΅ 𝑝 ξ‚Ά 𝑝 βˆ’ 1 𝑝  𝑛 | | π‘₯ 𝑛 βˆ’ π‘₯ 𝑛 βˆ’ 1 | | 𝑝 , ( 2 . 1 5 ) as expected, β€– π‘₯ β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) ≀ ξ‚΅ 2 𝑝 ξ‚Ά 𝑝 βˆ’ 1 β€– π‘₯ β€– 𝑏 𝑣 𝑝 , ( 2 . 1 6 ) for 1 < 𝑝 < ∞ .
Furthermore, let us consider the sequence π‘₯ = { π‘₯ 𝑛 ( π‘Ÿ ) } defined by π‘₯ 𝑛 ( π‘Ÿ ) = 𝑛  π‘˜ = 0 ( βˆ’ 1 ) 𝑛 1 + π‘Ÿ π‘˜ , 𝑛 ∈ β„• . ( 2 . 1 7 ) Then, the sequence π‘₯ is in π‘Ž π‘Ÿ 𝑝 ( Ξ” ) βˆ’ 𝑏 𝑣 𝑝 , as asserted.

Lemma 2.4 (see [11, Theorem  2.4]). The inclusion β„“ 𝑝 βŠ‚ 𝑏 𝑣 𝑝 strictly holds for 1 < 𝑝 < ∞ .

Combining Lemma 2.4 and Theorem 2.3, we get the following corollary.

Corollary 2.5. The inclusion β„“ 𝑝 βŠ‚ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) strictly holds for 1 < 𝑝 < ∞ .

3. The Basis for the Space π‘Ž π‘Ÿ 𝑝 ( Ξ” )

In the present section, we will give a sequence of the points of the space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) which forms a basis for the space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) , where 1 ≀ 𝑝 < ∞ .

Theorem 3.1. Define the matrix 𝐡 π‘Ÿ = { 𝑏 𝑛 ( π‘˜ ) ( π‘Ÿ ) } 𝑛 ∈ β„• of elements of the space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) for every fixed π‘˜ ∈ β„• by 𝑏 𝑛 ( π‘˜ ) ⎧ βŽͺ βŽͺ ⎨ βŽͺ βŽͺ ⎩ ( π‘Ÿ ) = 1 + π‘˜ 1 + π‘Ÿ π‘˜ βˆ’ 1 + π‘˜ 1 + π‘Ÿ π‘˜ + 1 , 0 ≀ π‘˜ ≀ 𝑛 βˆ’ 1 , 1 + 𝑛 1 + π‘Ÿ 𝑛 , π‘˜ = 𝑛 , 0 , π‘˜ > 𝑛 , ( 3 . 1 ) for every fixed π‘˜ ∈ β„• . Then, the sequence { 𝑏 ( π‘˜ ) ( π‘Ÿ ) } π‘˜ ∈ β„• is a basis for the space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) , and any π‘₯ ∈ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) has a unique representation of the form  π‘₯ = π‘˜ πœ† π‘˜ ( π‘Ÿ ) 𝑏 ( π‘˜ ) ( π‘Ÿ ) , ( 3 . 2 ) where πœ† π‘˜ ( π‘Ÿ ) = ( 𝐡 π‘Ÿ π‘₯ ) π‘˜ for all π‘˜ ∈ β„• and 1 ≀ 𝑝 < ∞ .

Proof. It is clear that { 𝑏 ( π‘˜ ) ( π‘Ÿ ) } βŠ‚ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) , since 𝐡 π‘Ÿ 𝑏 ( π‘˜ ) ( π‘Ÿ ) = 𝑒 ( π‘˜ ) ∈ β„“ 𝑝 , π‘˜ = 0 , 1 , 2 , … , ( 3 . 3 ) for 1 ≀ 𝑝 < ∞ ; here, 𝑒 ( π‘˜ ) is the sequence whose only nonzero term is 1 in the π‘˜ th place for each π‘˜ ∈ β„• . Let π‘₯ ∈ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) be given. For every nonnegative integer π‘š , we set π‘₯ [ π‘š ] = π‘š  π‘˜ = 0 πœ† π‘˜ ( π‘Ÿ ) 𝑏 ( π‘˜ ) ( π‘Ÿ ) . ( 3 . 4 ) Then, by applying 𝐡 π‘Ÿ to (3.4), we obtain with (3.3) that 𝐡 π‘Ÿ π‘₯ [ π‘š ] = π‘š  π‘˜ = 0 πœ† π‘˜ ( π‘Ÿ ) 𝐡 π‘Ÿ 𝑏 ( π‘˜ ) ( π‘Ÿ ) = π‘š  π‘˜ = 0 ξ€· 𝐡 π‘Ÿ π‘₯ ξ€Έ π‘˜ 𝑒 ( π‘˜ ) , ξ€½ 𝐡 π‘Ÿ ξ€· π‘₯ βˆ’ π‘₯ [ π‘š ] ξ€Έ ξ€Ύ 𝑖 = ξƒ― ξ€· 𝐡 0 , 0 ≀ 𝑖 ≀ π‘š , π‘Ÿ π‘₯ ξ€Έ 𝑖 , 𝑖 > π‘š , ( 3 . 5 ) where 𝑖 , π‘š ∈ β„• . For a given πœ€ > 0 , there is an integer π‘š 0 such that  ∞  𝑖 = π‘š | | ξ€· 𝐡 π‘Ÿ π‘₯ ξ€Έ 𝑖 | | 𝑝 ξƒ­ 1 / 𝑝 < πœ€ 2 , ( 3 . 6 ) for all π‘š β‰₯ π‘š 0 . Hence, β€– β€– π‘₯ βˆ’ π‘₯ [ π‘š ] β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) =  ∞  𝑖 = π‘š | | ξ€· 𝐡 π‘Ÿ π‘₯ ξ€Έ 𝑖 | | 𝑝 ξƒ­ 1 / 𝑝 ≀  ∞  𝑖 = π‘š 0 | | ξ€· 𝐡 π‘Ÿ π‘₯ ξ€Έ 𝑖 | | 𝑝 ξƒ­ 1 / 𝑝 < πœ€ 2 < πœ€ , ( 3 . 7 ) for all π‘š β‰₯ π‘š 0 , which proves that π‘₯ ∈ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) is represented as in (3.2).
Let us show the uniqueness of representation for π‘₯ ∈ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) given by (3.2). Assume, on the contrary, that there exists a representation βˆ‘ π‘₯ = π‘˜ πœ‡ π‘˜ ( π‘Ÿ ) 𝑏 ( π‘˜ ) ( π‘Ÿ ) . Since the linear transformation 𝑇 , from π‘Ž π‘Ÿ 𝑝 ( Ξ” ) to β„“ 𝑝 , used in Theorem 2.2 is continuous, at this stage, we have ξ€· 𝐡 π‘Ÿ π‘₯ ξ€Έ 𝑛 =  π‘˜ πœ‡ π‘˜ ( ξ€½ 𝐡 π‘Ÿ ) π‘Ÿ 𝑏 ( π‘˜ ) ( ξ€Ύ π‘Ÿ ) 𝑛 =  π‘˜ πœ‡ π‘˜ ( π‘Ÿ ) 𝑒 𝑛 ( π‘˜ ) = πœ‡ 𝑛 ( π‘Ÿ ) ; 𝑛 ∈ β„• , ( 3 . 8 ) which contradicts the fact that ( 𝐡 π‘Ÿ π‘₯ ) 𝑛 = πœ† 𝑛 ( π‘Ÿ ) for all 𝑛 ∈ β„• . Hence, the representation (3.2) of π‘₯ ∈ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) is unique. This step concludes the proof.

4. The 𝛼 -, 𝛽 -, and 𝛾 -Duals of the Space π‘Ž π‘Ÿ 𝑝 ( Ξ” )

In this section, we state and prove theorems determining the 𝛼 -, 𝛽 -, and 𝛾 -duals of the space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) . Since the case 𝑝 = 1 can be proved by the same analogy and can be found in the literature, we omit the proof of that case and consider only the case 1 < 𝑝 < ∞ in the proof of Theorems 4.44.6.

For the sequence spaces πœ† and πœ‡ , define the set 𝑆 ( πœ† , πœ‡ ) by 𝑆 ξ€½ ξ€· 𝑧 ( πœ† , πœ‡ ) = 𝑧 = π‘˜ ξ€Έ ξ€· π‘₯ ∈ πœ” ∢ π‘₯ 𝑧 = π‘˜ 𝑧 π‘˜ ξ€Έ ξ€Ύ ∈ πœ‡ , βˆ€ π‘₯ ∈ πœ† . ( 4 . 1 ) With the notation of (4.1), 𝛼 -, 𝛽 - and 𝛾 -duals of a sequence space πœ† , which are, respectively, denoted by πœ† 𝛼 , πœ† 𝛽 and πœ† 𝛾 , are defined by πœ† 𝛼 ξ€· = 𝑆 πœ† , β„“ 1 ξ€Έ , πœ† 𝛽 = 𝑆 ( πœ† , 𝑐 𝑠 ) , πœ† 𝛾 = 𝑆 ( πœ† , 𝑏 𝑠 ) . ( 4 . 2 ) It is well-known for the sequence spaces πœ† and πœ‡ that πœ† 𝛼 βŠ† πœ† 𝛽 βŠ† πœ† 𝛾 and πœ† πœ‚ βŠƒ πœ‡ πœ‚ whenever πœ† βŠ‚ πœ‡ , where πœ‚ ∈ { 𝛼 , 𝛽 , 𝛾 } .

We begin with to quoting the lemmas due to Stieglitz and Tietz [15], which are needed in the proof of the following theorems.

Lemma 4.1. 𝐴 ∈ ( β„“ 𝑝 ∢ β„“ 1 ) if and only if s u p 𝐾 ∈ β„±  π‘˜ | | | | |  𝑛 ∈ 𝐾 π‘Ž 𝑛 π‘˜ | | | | | π‘ž < ∞ ( 1 < 𝑝 ≀ ∞ ) . ( 4 . 3 )

Lemma 4.2. 𝐴 ∈ ( β„“ 𝑝 ∢ 𝑐 ) if and only if l i m 𝑛 ⟢ ∞ π‘Ž 𝑛 π‘˜ e x i s t s f o r e a c h π‘˜ ∈ β„• , ( 4 . 4 ) s u p 𝑛 ∈ β„•  π‘˜ | | π‘Ž 𝑛 π‘˜ | | π‘ž < ∞ 1 < 𝑝 < ∞ . ( 4 . 5 )

Lemma 4.3. 𝐴 ∈ ( β„“ 𝑝 ∢ β„“ ∞ ) if and only if (4.5) holds.

Theorem 4.4. Define the set π‘Ž π‘Ÿ π‘ž by π‘Ž r π‘ž = ⎧ βŽͺ ⎨ βŽͺ ⎩ ξ€· π‘Ž π‘Ž = π‘˜ ξ€Έ ∈ πœ” ∢ s u p 𝐾 ∈ β„±  π‘˜ | | | | |  𝑛 ∈ 𝐾 𝑐 π‘Ÿ 𝑛 π‘˜ | | | | | π‘ž ⎫ βŽͺ ⎬ βŽͺ ⎭ < ∞ , ( 4 . 6 ) where 𝐢 π‘Ÿ = ( 𝑐 π‘Ÿ 𝑛 π‘˜ ) is defined via the sequence π‘Ž = ( π‘Ž 𝑛 ) by 𝑐 π‘Ÿ 𝑛 π‘˜ = ⎧ βŽͺ βŽͺ ⎨ βŽͺ βŽͺ ⎩ ξ‚€ 1 + π‘˜ 1 + π‘Ÿ π‘˜ βˆ’ 1 + π‘˜ 1 + π‘Ÿ π‘˜ + 1  π‘Ž 𝑛 , 0 ≀ π‘˜ ≀ 𝑛 βˆ’ 1 , 1 + 𝑛 1 + π‘Ÿ 𝑛 π‘Ž 𝑛 , π‘˜ = 𝑛 , 0 , π‘˜ > 𝑛 . ( 4 . 7 ) for all 𝑛 , π‘˜ ∈ β„• . Then, { π‘Ž r 𝑝 ( Ξ” ) } 𝛼 = π‘Ž π‘Ÿ π‘ž , where 1 < 𝑝 < ∞ .

Proof. Bearing in mind the relation (2.9), we immediately derive that π‘Ž 𝑛 π‘₯ 𝑛 = 𝑛  π‘˜ π‘˜ = 0  𝑗 = π‘˜ βˆ’ 1 ( βˆ’ 1 ) π‘˜ βˆ’ 𝑗 1 + 𝑗 1 + π‘Ÿ π‘˜ π‘Ž 𝑛 𝑦 𝑗 = ( 𝐢 π‘Ÿ 𝑦 ) 𝑛 , 𝑛 ∈ β„• . ( 4 . 8 ) It follows from (4.8) that π‘Ž π‘₯ = ( π‘Ž 𝑛 π‘₯ 𝑛 ) ∈ β„“ 1 whenever π‘₯ ∈ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) if and only if 𝐢 π‘Ÿ 𝑦 ∈ β„“ 1 whenever 𝑦 ∈ β„“ 𝑝 . This means that π‘Ž = ( π‘Ž 𝑛 ) ∈ { π‘Ž π‘Ÿ 𝑝 ( Ξ” ) } 𝛼 if and only if 𝐢 π‘Ÿ ∈ ( β„“ 𝑝 ∢ β„“ 1 ) . Then, we derive by Lemma 4.1 with 𝐢 π‘Ÿ instead of 𝐴 that s u p 𝐾 ∈ β„±  π‘˜ | | | | |  𝑛 ∈ 𝐾 𝑐 π‘Ÿ 𝑛 π‘˜ | | | | | π‘ž < ∞ . ( 4 . 9 ) This yields the desired consequence that { π‘Ž π‘Ÿ 𝑝 ( Ξ” ) } 𝛼 = π‘Ž π‘Ÿ π‘ž .

Theorem 4.5. Define the sets π‘Ž π‘Ÿ 1 , π‘Ž π‘Ÿ 2 , and π‘Ž π‘Ÿ 3 by π‘Ž π‘Ÿ 1 = ξƒ― ξ€· π‘Ž π‘Ž = π‘˜ ξ€Έ ∈ πœ” ∢ s u p 𝑛 ∈ β„•  π‘˜ | | 𝑒 π‘Ÿ 𝑛 π‘˜ | | π‘ž ξƒ° , π‘Ž < ∞ π‘Ÿ 2 = ξƒ― ξ€· π‘Ž π‘Ž = π‘˜ ξ€Έ ∈ πœ” ∢ ∞  𝑗 = π‘˜ π‘Ž 𝑗 ξƒ° , π‘Ž e x i s t s f o r e a c h fi x e d π‘˜ ∈ β„• π‘Ÿ 3 =  ξ€· π‘Ž π‘Ž = π‘˜ ξ€Έ  ∈ πœ” ∢ 1 + 𝑛 1 + π‘Ÿ 𝑛 π‘Ž 𝑛   , ∈ 𝑐 𝑠 ( 4 . 1 0 ) where 𝐸 = ( 𝑒 π‘Ÿ 𝑛 π‘˜ ) is defined by 𝑒 π‘Ÿ 𝑛 π‘˜ = ⎧ βŽͺ βŽͺ βŽͺ ⎨ βŽͺ βŽͺ βŽͺ ⎩  π‘Ž ( π‘˜ + 1 ) π‘˜ 1 + π‘Ÿ π‘˜ + ξ‚€ 1 1 + π‘Ÿ π‘˜ βˆ’ 1 1 + π‘Ÿ π‘˜ + 1  𝑛  𝑗 = π‘˜ + 1 π‘Ž 𝑗 ξƒ­ , 0 ≀ π‘˜ ≀ 𝑛 βˆ’ 1 , 1 + 𝑛 1 + π‘Ÿ 𝑛 π‘Ž 𝑛 , π‘˜ = 𝑛 , 0 , π‘˜ > 𝑛 , ( 4 . 1 1 ) for all 𝑛 , π‘˜ ∈ β„• . Then, { π‘Ž π‘Ÿ 𝑝 ( Ξ” ) } 𝛽 = π‘Ž π‘Ÿ 1 ∩ π‘Ž π‘Ÿ 2 ∩ π‘Ž π‘Ÿ 3 , where 1 < 𝑝 < ∞ .

Proof. Consider the equation 𝑛  π‘˜ = 0 π‘Ž π‘˜ π‘₯ π‘˜ = 𝑛  π‘˜ = 0 ξƒ― π‘˜  𝑗 = 0  𝑗  𝑖 = 𝑗 βˆ’ 1 ( βˆ’ 1 ) 𝑖 βˆ’ 𝑗 1 + 𝑖 1 + π‘Ÿ 𝑗 𝑦 𝑗 π‘Ž ξƒ­ ξƒ° π‘˜ = 𝑛 βˆ’ 1  π‘˜ = 0  π‘Ž ( π‘˜ + 1 ) π‘˜ 1 + π‘Ÿ π‘˜ + ξ‚€ 1 1 + π‘Ÿ π‘˜ βˆ’ 1 1 + π‘Ÿ π‘˜ + 1  𝑛  𝑗 = π‘˜ + 1 π‘Ž 𝑗 ξƒ­ 𝑦 π‘˜ + 1 + 𝑛 1 + π‘Ÿ 𝑛 π‘Ž 𝑛 𝑦 𝑛 = ( 𝐸 𝑦 ) 𝑛 𝑛 ∈ β„• . ( 4 . 1 2 ) Thus, we deduce from Lemma 4.2 with (4.12) that π‘Ž π‘₯ = ( π‘Ž π‘˜ π‘₯ π‘˜ ) ∈ 𝑐 𝑠 whenever π‘₯ = ( π‘₯ π‘˜ ) ∈ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) if and only if 𝐸 𝑦 ∈ 𝑐 whenever 𝑦 = ( 𝑦 π‘˜ ) ∈ β„“ 𝑝 . That is to say that π‘Ž = ( π‘Ž π‘˜ ) ∈ { π‘Ž π‘Ÿ 𝑝 ( Ξ” ) } 𝛽 if and only if 𝐸 ∈ ( β„“ 𝑝 ∢ 𝑐 ) . Therefore, we derive from (4.4) and (4.5) that { π‘Ž π‘Ÿ 𝑝 ( Ξ” ) } 𝛽 = π‘Ž π‘Ÿ 1 ∩ π‘Ž π‘Ÿ 2 ∩ π‘Ž π‘Ÿ 3 .

Theorem 4.6. { π‘Ž π‘Ÿ 𝑝 ( Ξ” ) } 𝛾 = π‘Ž π‘Ÿ 1 , where 1 < 𝑝 < ∞ .

Proof. It is natural that the present theorem may be proved by the same technique used in the proof of Theorems 4.4 and 4.5, above. But, we prefer here the following classical way.
Let π‘Ž = ( π‘Ž π‘˜ ) ∈ π‘Ž π‘Ÿ 1 and π‘₯ = ( π‘₯ π‘˜ ) ∈ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) . Then, we obtain by applying Hölder’s inequality that | | | | | 𝑛  π‘˜ = 0 π‘Ž π‘˜ π‘₯ π‘˜ | | | | | = | | | | | 𝑛  π‘˜ = 0 ξƒ― π‘˜  𝑗 = 0  𝑗  𝑖 = 𝑗 βˆ’ 1 ( βˆ’ 1 ) 𝑖 βˆ’ 𝑗 1 + 𝑖 1 + π‘Ÿ 𝑗 𝑦 𝑗 π‘Ž ξƒ­ ξƒ° π‘˜ | | | | | = | | | | | 𝑛  π‘˜ = 0 𝑒 π‘Ÿ 𝑛 π‘˜ 𝑦 π‘˜ | | | | | ≀  𝑛  π‘˜ = 0 | | 𝑒 π‘Ÿ 𝑛 π‘˜ | | π‘ž ξƒͺ 1 / π‘ž  𝑛  π‘˜ = 0 | | 𝑦 π‘˜ | | 𝑝 ξƒͺ 1 / 𝑝 , ( 4 . 1 3 ) which gives us by taking supremum over 𝑛 ∈ β„• that s u p 𝑛 ∈ β„• | | | | | 𝑛  π‘˜ = 0 π‘Ž π‘˜ π‘₯ π‘˜ | | | | | ≀ s u p 𝑛 ∈ β„• ⎑ ⎒ ⎒ ⎣  𝑛  π‘˜ = 0 | | 𝑒 π‘Ÿ 𝑛 π‘˜ | | π‘ž ξƒͺ 1 / π‘ž  𝑛  π‘˜ = 0 | | 𝑦 π‘˜ | | 𝑝 ξƒͺ 1 / 𝑝 ⎀ βŽ₯ βŽ₯ ⎦ ≀ β€– 𝑦 β€– β„“ 𝑝 β‹…  s u p 𝑛 𝑛 ∈ β„•  π‘˜ = 0 | | 𝑒 π‘Ÿ 𝑛 π‘˜ | | π‘ž ξƒͺ 1 / π‘ž < ∞ . ( 4 . 1 4 ) This means that π‘Ž = ( π‘Ž π‘˜ ) ∈ { π‘Ž π‘Ÿ 𝑝 ( Ξ” ) } 𝛾 . Hence, π‘Ž π‘Ÿ 1 βŠ‚ ξ€½ π‘Ž π‘Ÿ 𝑝 ξ€Ύ ( Ξ” ) 𝛾 . ( 4 . 1 5 ) Conversely, let π‘Ž = ( π‘Ž π‘˜ ) ∈ { π‘Ž π‘Ÿ 𝑝 ( Ξ” ) } 𝛾 and π‘₯ = ( π‘₯ π‘˜ ) ∈ π‘Ž π‘Ÿ 𝑝 ( Ξ” ) . Then, one can easily see that ( βˆ‘ 𝑛 π‘˜ = 0 𝑒 π‘Ÿ 𝑛 π‘˜ 𝑦 π‘˜ ) 𝑛 ∈ β„• ∈ β„“ ∞ whenever ( π‘Ž π‘˜ π‘₯ π‘˜ ) ∈ 𝑏 𝑠 . This shows that the triangle matrix 𝐸 = ( 𝑒 π‘Ÿ 𝑛 π‘˜ ) , defined by (4.11), is in the class ( β„“ 𝑝 ∢ β„“ ∞ ) . Hence, the condition (4.5) holds with 𝑒 π‘Ÿ 𝑛 π‘˜ instead of π‘Ž 𝑛 π‘˜ which yields that π‘Ž = ( π‘Ž π‘˜ ) ∈ π‘Ž π‘Ÿ 1 . That is to say that ξ€½ π‘Ž π‘Ÿ 𝑝 ξ€Ύ ( Ξ” ) 𝛾 βŠ‚ π‘Ž π‘Ÿ 1 . ( 4 . 1 6 ) Therefore, by combining the inclusions (4.15) and (4.16), we deduce that the 𝛾 -dual of the space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) is the set π‘Ž π‘Ÿ 1 , and this step completes the proof.

5. Some Geometric Properties of the Space π‘Ž π‘Ÿ 𝑝 ( Ξ” )

In this section, we study some geometric properties of the space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) .

A Banach space 𝑋 is said to have the Banach-Saks property if every bounded sequence ( π‘₯ 𝑛 ) in 𝑋 admits a subsequence ( 𝑧 𝑛 ) such that the sequence { 𝑑 π‘˜ ( 𝑧 ) } is convergent in the norm in 𝑋 [16], where 𝑑 π‘˜ 1 ( 𝑧 ) = ξ€· 𝑧 π‘˜ + 1 0 + 𝑧 1 + β‹― + 𝑧 π‘˜ ξ€Έ ( π‘˜ ∈ β„• ) . ( 5 . 1 ) A Banach space 𝑋 is said to have the weak Banach-Saks property whenever given any weakly null sequence ( π‘₯ 𝑛 ) βŠ‚ 𝑋 and there exists a subsequence ( 𝑧 𝑛 ) of ( π‘₯ 𝑛 ) such that the sequence { 𝑑 π‘˜ ( 𝑧 ) } strongly convergent to zero.

In [17], García-Falset introduce the following coefficient: ξ‚» 𝑅 ( 𝑋 ) = s u p l i m i n f 𝑛 ⟢ ∞ β€– β€– π‘₯ 𝑛 β€– β€– ∢ ξ€· π‘₯ βˆ’ π‘₯ 𝑛 ξ€Έ βŠ‚ 𝐡 ( 𝑋 ) , π‘₯ 𝑛 πœ” ξ‚Ό βˆ’ β†’ 0 , π‘₯ ∈ 𝐡 ( 𝑋 ) , ( 5 . 2 ) where 𝐡 ( 𝑋 ) denotes the unit ball of 𝑋 .

Remark 5.1. A Banach space 𝑋 with 𝑅 ( 𝑋 ) < 2 has the weak fixed point property, [18].

Let 1 < 𝑝 < ∞ . A Banach space is said to have the Banach-Saks type 𝑝 or property ( B S ) 𝑝 , if every weakly null sequence ( π‘₯ π‘˜ ) has a subsequence ( π‘₯ π‘˜ 𝑙 ) such that for some 𝐢 > 0 , β€– β€– β€– β€– 𝑛  𝑙 = 0 π‘₯ π‘˜ 𝑙 β€– β€– β€– β€– < 𝐢 ( 𝑛 + 1 ) 1 / 𝑝 , ( 5 . 3 ) for all 𝑛 ∈ β„• (see [19]).

Now, we may give the following results related to the some geometric properties, mentioned above, of the space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) .

Theorem 5.2. The space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) has the Banach-Saks type 𝑝 .

Proof. Let ( πœ€ 𝑛 ) be a sequence of positive numbers for which βˆ‘ πœ€ 𝑛 ≀ 1 / 2 , and also let ( π‘₯ 𝑛 ) be a weakly null sequence in 𝐡 ( π‘Ž π‘Ÿ 𝑝 ( Ξ” ) ) . Set 𝑏 0 = π‘₯ 0 = 0 and 𝑏 1 = π‘₯ 𝑛 1 = π‘₯ 1 . Then, there exists π‘š 1 ∈ β„• such that β€– β€– β€– β€– ∞  𝑖 = π‘š 1 + 1 𝑏 1 ( 𝑖 ) 𝑒 ( 𝑖 ) β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) < πœ€ 1 . ( 5 . 4 ) Since ( π‘₯ 𝑛 ) is a weakly null sequence implies π‘₯ 𝑛 β†’ 0 coordinatewise, there is an 𝑛 2 ∈ β„• such that β€– β€– β€– β€– π‘š 1  𝑖 = 0 π‘₯ 𝑛 ( 𝑖 ) 𝑒 ( 𝑖 ) β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) < πœ€ 1 , ( 5 . 5 ) where 𝑛 β‰₯ 𝑛 2 . Set 𝑏 2 = π‘₯ 𝑛 2 . Then, there exists an π‘š 2 > π‘š 1 such that β€– β€– β€– β€– ∞  𝑖 = π‘š 2 + 1 𝑏 2 ( 𝑖 ) 𝑒 ( 𝑖 ) β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) < πœ€ 2 . ( 5 . 6 ) By using the fact that π‘₯ 𝑛 β†’ 0 coordinatewise, there exists an 𝑛 3 > 𝑛 2 such that β€– β€– β€– β€– π‘š 2  𝑖 = 0 π‘₯ 𝑛 ( 𝑖 ) 𝑒 ( 𝑖 ) β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) < πœ€ 2 , ( 5 . 7 ) where 𝑛 β‰₯ 𝑛 3 .
If we continue this process, we can find two increasing subsequences ( π‘š 𝑖 ) and ( 𝑛 𝑖 ) such that β€– β€– β€– β€– π‘š 𝑗  𝑖 = 0 π‘₯ 𝑛 ( 𝑖 ) 𝑒 ( 𝑖 ) β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) < πœ€ 𝑗 , ( 5 . 8 ) for each 𝑛 β‰₯ 𝑛 𝑗 + 1 and β€– β€– β€– β€– ∞  𝑖 = π‘š 𝑗 + 1 𝑏 𝑗 ( 𝑖 ) 𝑒 ( 𝑖 ) β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) < πœ€ 𝑗 , ( 5 . 9 ) where 𝑏 𝑗 = π‘₯ 𝑛 𝑗 . Hence, β€– β€– β€– β€– 𝑛  𝑗 = 0 𝑏 𝑗 β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) = β€– β€– β€– β€– 𝑛  𝑗 = 0 βŽ› ⎜ ⎜ ⎝ π‘š 𝑗 βˆ’ 1  𝑖 = 0 𝑏 𝑗 ( 𝑖 ) 𝑒 ( 𝑖 ) + π‘š 𝑗  𝑖 = π‘š 𝑗 βˆ’ 1 + 1 𝑏 𝑗 ( 𝑖 ) 𝑒 ( 𝑖 ) + ∞  𝑖 = π‘š 𝑗 + 1 𝑏 𝑗 ( 𝑖 ) 𝑒 ( 𝑖 ) ⎞ ⎟ ⎟ ⎠ β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) ≀ β€– β€– β€– β€– 𝑛  𝑗 = 0  π‘š 𝑗 βˆ’ 1  𝑖 = 0 𝑏 𝑗 ( 𝑖 ) 𝑒 ( 𝑖 ) ξƒͺ β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) + β€– β€– β€– β€– 𝑛  𝑗 = 0 βŽ› ⎜ ⎜ ⎝ π‘š 𝑗  𝑖 = π‘š 𝑗 βˆ’ 1 + 1 𝑏 𝑗 ( 𝑖 ) 𝑒 ( 𝑖 ) ⎞ ⎟ ⎟ ⎠ β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) + β€– β€– β€– β€– 𝑛  𝑗 = 0 βŽ› ⎜ ⎜ ⎝ ∞  𝑖 = π‘š 𝑗 + 1 𝑏 𝑗 ( 𝑖 ) 𝑒 ( 𝑖 ) ⎞ ⎟ ⎟ ⎠ β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) ≀ β€– β€– β€– β€– 𝑛  𝑗 = 0 βŽ› ⎜ ⎜ ⎝ π‘š 𝑗  𝑖 = π‘š 𝑗 βˆ’ 1 + 1 𝑏 𝑗 ( 𝑖 ) 𝑒 ( 𝑖 ) ⎞ ⎟ ⎟ ⎠ β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) + 2 𝑛  𝑗 = 0 πœ€ 𝑗 . ( 5 . 1 0 ) On the other hand, it can be seen that β€– π‘₯ 𝑛 β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) < 1 . Therefore, β€– π‘₯ 𝑛 β€– 𝑝 π‘Ž π‘Ÿ 𝑝 ( Ξ” ) < 1 . We have β€– β€– β€– β€– 𝑛  𝑗 = 0 βŽ› ⎜ ⎜ ⎝ π‘š 𝑗  𝑖 = π‘š 𝑗 βˆ’ 1 + 1 𝑏 𝑗 ( 𝑖 ) 𝑒 ( 𝑖 ) ⎞ ⎟ ⎟ ⎠ β€– β€– β€– β€– 𝑝 π‘Ž π‘Ÿ 𝑝 ( Ξ” ) = 𝑛  π‘š 𝑗 = 0 𝑗  𝑖 = π‘š 𝑗 βˆ’ 1 + 1 | | | | | 𝑖 βˆ’ 1  π‘˜ = 0 ( 1 βˆ’ π‘Ÿ ) π‘Ÿ π‘˜ π‘₯ 1 + 𝑖 𝑗 ( π‘˜ ) + 1 + π‘Ÿ 𝑖 π‘₯ 1 + 𝑖 𝑖 | | | | | ( π‘˜ ) 𝑝 ≀ 𝑛  ∞ 𝑗 = 0  𝑖 = 0 | | | | | i βˆ’ 1  π‘˜ = 0 ( 1 βˆ’ π‘Ÿ ) π‘Ÿ π‘˜ π‘₯ 1 + 𝑖 𝑗 ( π‘˜ ) + 1 + π‘Ÿ 𝑖 π‘₯ 1 + 𝑖 𝑖 | | | | | ( π‘˜ ) 𝑝 ≀ ( 𝑛 + 1 ) . ( 5 . 1 1 ) Hence, we obtain β€– β€– β€– β€– 𝑛  𝑗 = 0 βŽ› ⎜ ⎜ ⎝ π‘š 𝑗  𝑖 = π‘š 𝑗 βˆ’ 1 + 1 𝑏 𝑗 ( 𝑖 ) 𝑒 ( 𝑖 ) ⎞ ⎟ ⎟ ⎠ β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) ≀ ( 𝑛 + 1 ) 1 / 𝑝 . ( 5 . 1 2 ) By using the fact 1 ≀ ( 𝑛 + 1 ) 1 / 𝑝 for all 𝑛 ∈ β„• , we have β€– β€– β€– β€– 𝑛  𝑗 = 0 𝑏 𝑗 β€– β€– β€– β€– π‘Ž π‘Ÿ 𝑝 ( Ξ” ) ≀ ( 𝑛 + 1 ) 1 / 𝑝 + 1 ≀ 2 ( 𝑛 + 1 ) 1 / 𝑝 . ( 5 . 1 3 ) Hence, π‘Ž π‘Ÿ 𝑝 ( Ξ” ) has the Banach-Saks type 𝑝 . This completes the proof of the theorem.

Remark 5.3. Note that 𝑅 ( π‘Ž π‘Ÿ 𝑝 ( Ξ” ) ) = 𝑅 ( β„“ 𝑝 ) = 2 1 / 𝑝 , since π‘Ž π‘Ÿ 𝑝 ( Ξ” ) is linearly isomorphic to β„“ 𝑝 .

Hence, by the Remarks 5.1 and 5.3, we have the following.

Theorem 5.4. The space π‘Ž π‘Ÿ 𝑝 ( Ξ” ) has the weak fixed point property, where 1 < 𝑝 < ∞ .

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