Banach-Saks Type and Gurariǐ Modulus of Convexity of Some Banach Sequence Spaces

Abstract

Banach-Saks type is calculated for two types of Banach sequence spaces and Gurariǐ modulus of convexity is estimated from above for the spaces of one type among them.

1. Introduction

Recently, there has been a lot of interest in investigating geometric properties of sequence spaces besides topological and some other usual properties. In the literature, there are many papers concerning geometric properties of various Banach sequence spaces. For example, geometry of Orlicz spaces and of Musielak-Orlicz spaces has been studied in [1–25]. Several authors including Cui and Hudzik [26–29], Cui and Meng [30], Suantai [31], and Lee [32] investigated the geometric properties of Cesàro sequence space ces(𝑝). Also Cesàro-Orlicz sequence spaces equipped with Luxemburg norm have been studied in [5, 33–36]. Additionally, geometry of Orlicz-Lorentz sequence spaces and of generalized Orlicz-Lorentz sequence space were studied in [37, 38]. Furthermore, Mursaleen et al. [39] studied some geometric properties of normed Euler sequence space. Additionally, Hudzik and Narloch [40] have studied relationships between monotonicity and complex rotundity properties with some consequences. Besides, some geometrical properties of Calderon-Lozanovskii sequence spaces have been investigated in [41–43].

Quite recently, Karakaya [20] defined a new sequence space involving lacunary sequence space equipped with the Luxemburg norm and studied Kadec-Klee (𝐻) and rotundity (𝑅) properties of these spaces. Further information on topological and geometric properties of sequence spaces can be found in [39, 44–62].

Let 𝑋 be a real Banach space and 𝑆(𝑋) and 𝐵(𝑋) be the unit sphere and the unit ball of 𝑋, respectively. Let ℓ⁰, 𝑐₀, 𝑐, ℓ_∞, and ℓ₁ be the spaces of all real sequences, null, convergent, and bounded sequences and absolutely convergent series, respectively, and let 𝑐₀₀ be the space of those real sequences which have only a finite number of nonzero coordinates and ℓ_𝑝={𝑥=𝑥(𝑖)∶∑^∞_𝑖=1|𝑥(𝑖)|^𝑝<∞}.

Note that 𝑐₀, 𝑐, and ℓ_∞ are Banach spaces with the sup-norm ‖𝑥‖_∞=sup_𝑖|𝑥(𝑖)| and ℓ_𝑝(1≤𝑝<∞) are Banach spaces with the norm ‖𝑥‖_𝑝=(∑|𝑥(𝑖)|^𝑝)^1/𝑝, while 𝑐₀₀ is not a Banach space with respect to any norm.

Let us recall that a sequence {𝑣(𝑖)}^∞_𝑖=1 in a Banach space 𝑋 is called 𝑆𝑐ℎ𝑎𝑢𝑑𝑒𝑟𝑏𝑎𝑠𝑖𝑠 of 𝑋 (or 𝑏𝑎𝑠𝑖𝑠 for short) if for each 𝑥∈𝑋 there exists a unique sequence {𝜆(𝑖)}^∞_𝑖=1 of scalars such that 𝑥=∑^∞_𝑖=1𝜆(𝑖)𝑣(𝑖), that is, lim_𝑛→∞∑^𝑛_𝑖=1𝜆(𝑖)𝑣(𝑖)=𝑥.

A sequence space 𝑋 with a linear topology is called a 𝐾-𝑠𝑝𝑎𝑐𝑒 if each of the projection maps 𝑃_𝑗∶𝑋→ℂ defined by 𝑃_𝑗(𝑥)=𝑥(𝑖) for 𝑥=(𝑥(𝑖))^∞_𝑖=1∈𝑋 is continuous for each natural 𝑗. A 𝐹𝑟𝑒𝑐́ℎ𝑒𝑡𝑠𝑝𝑎𝑐𝑒 is a complete metric linear space and the metric is generated by an 𝐹-norm and a Fréchet space which is a 𝐾-space is called an 𝐹𝐾-𝑠𝑝𝑎𝑐𝑒; that is, a 𝐾-space 𝑋 is called an 𝐹𝐾-space if 𝑋 is a complete linear metric space. In other words, 𝑋 is an 𝐹𝐾-space if 𝑋 is a Fréchet space with continuous coordinate projections. All the sequence spaces mentioned above are 𝐹𝐾 spaces except the space 𝑐₀₀.

An 𝐹𝐾-space 𝑋 which contains the space 𝑐₀₀ is said to have the 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝐴𝐾 if, for every sequence {𝑥(𝑖)}∈𝑋, 𝑥=∑^∞_𝑖=1𝑥(𝑖)𝑒(𝑖), where 𝑒(𝑖)=(0,0,…,1(𝑖th-place),0,0,…). The spaces 𝑐₀ equipped with the sup-norm,ℓ_𝑝(1≤𝑝<∞) equipped with the norm ‖𝑥‖=(∑^∞_𝑖=1|𝑥(𝑖)|^𝑝)^1/𝑝 and ℓ⁰ endowed with the metric 𝑑(𝑥,𝑦)=∑^∞_𝑖=12^−𝑖(|𝑥(𝑖)−𝑦(𝑖)|/(1+|𝑥(𝑖)−𝑦(𝑖)|)) have property 𝐴𝐾, while 𝑐 and ℓ_∞ do not have property 𝐴𝐾. Banach spaces with continuous coordinate projections are called 𝐵𝐾-spaces.

A Banach space 𝑋 is said to be a 𝐾𝑜𝑡̈ℎ𝑒𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠𝑝𝑎𝑐𝑒 (see [28, 63]) if 𝑋 is a subspace of ℓ⁰ such that(i)if 𝑥∈ℓ⁰,𝑦∈𝑋, and |𝑥(𝑖)|≤|𝑦(𝑖)|, for all 𝑖∈ℕ, then 𝑥∈𝑋 and ‖𝑥‖≤‖𝑦‖;(ii)there exists an element 𝑥∈𝑋 such that 𝑥(𝑖)>0 for all 𝑖∈ℕ.

We say that 𝑥∈𝑋 is order continuous if, for any sequence (𝑥_𝑛) in 𝑋₊ (the positive cone in 𝑋) such that 𝑥_𝑛(𝑖)≤|𝑥(𝑖)| and 𝑥_𝑛(𝑖)→0(𝑛→∞) for each 𝑖∈ℕ(𝑛→∞), we have ‖𝑥_𝑛‖→0(𝑛→∞) holds.

A Köthe sequence space 𝑋 is said to be order continuous that if all sequences in 𝑋 are order continuous. It is easy to see that 𝑥∈𝑋 is order continuous if and only if ‖(0,0,…,0,𝑥(𝑛+1),𝑥(𝑛+2),…)‖→0 as 𝑛→∞.

A Köthe sequence space 𝑋 is said to have the 𝐹𝑎𝑡𝑜𝑢𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 if, for any real sequence 𝑥∈ℓ⁰ and any {𝑥_𝑛} in 𝑋 such that 𝑥_𝑛↑𝑥coordinatewisely and sup_𝑛‖𝑥_𝑛‖<∞, we have the fact that 𝑥∈𝑋 and ‖𝑥_𝑛‖→‖𝑥‖.

A Banach space 𝑋 is said to have the 𝐵𝑎𝑛𝑎𝑐ℎ-𝑆𝑎𝑘𝑠𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 if every bounded sequence {𝑥_𝑛} in 𝑋 admits a subsequence {𝑧_𝑛} such that the sequence {𝑡_𝑘(𝑧)} is convergent in 𝑋 with respect to the norm, where 𝑡_𝑘(𝑧)=1𝑘𝑧₁+𝑧₂+⋯+𝑧_𝑘,∀𝑘∈ℕ. (1) A Banach space 𝑋 is said to have the 𝑤𝑒𝑎𝑘𝐵𝑎𝑛𝑎𝑐ℎ-𝑆𝑎𝑘𝑠𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 whenever given any weakly null sequence {𝑥_𝑛} in 𝑋 there exists its subsequence {𝑧_𝑛} such that the sequence {𝑡_𝑘(𝑧)} converges to zero strongly.

Given any 𝑝∈(1,∞), we say that a Banach space (𝑋,‖⋅‖) has the 𝐵𝑎𝑛𝑎𝑐ℎ-𝑆𝑎𝑘𝑠𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑜𝑓𝑡𝑦𝑝𝑒𝑝 if there exists a constant 𝑐>0 such that every weakly null sequence {𝑥_𝑘} has a subsequence {𝑥_𝑘ℓ} such that (see [22]) ^𝑛_ℓ=1𝑥_𝑘ℓ≤𝑐𝑛^1/𝑝(∀𝑛∈ℕ). (2) The Banach-Saks property of type 𝑝∈(1,∞) and the weak Banach-Saks property for Cesàro sequence spaces have been considered in [28]. These properties and stronger property (𝑆_𝑝) for Musielak-Orlicz and Nakano sequence spaces have been studied in [17].

We say that a Banach space 𝑋 has the 𝑤𝑒𝑎𝑘𝑓𝑖𝑥𝑒𝑑𝑝𝑜𝑖𝑛𝑡𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 if every nonexpansive self-mapping defined on a nonempty weakly compact convex subset 𝐴 of 𝑋 has a fixed point in 𝐴.

In [64], Garcia-Falset introduced the following coefficient for a Banach space (𝑋,‖⋅‖): 𝑅(𝑋)=supliminf_𝑛→∞𝑥_𝑛(𝑋),𝑥−𝑥∶𝑥∈𝐵_𝑛⊂𝐵(𝑋),𝑥_𝑛⟶0weakly (3) and he proved (see [64, 65]) that a Banach space 𝑋 with 𝑅(𝑋)<2 has the weak fixed point property.

Clarkson 𝑚𝑜𝑑𝑢𝑙𝑢𝑠𝑜𝑓𝑐𝑜𝑛𝑣𝑒𝑥𝑖𝑡𝑦 of a normed space (𝑋,‖⋅‖) is defined (see Clarkson [66] and Day [67]) by the formula 𝛿_𝑋(𝜀)=inf1−𝑥+𝑦2;𝑥,𝑦∈𝑆(𝑋),𝑥−𝑦=𝜀 (4) for any 𝜀∈[0,2]. The inequality 𝛿_𝑋(𝜀)>0 for all 𝜀∈(0,2] characterizes the uniform convexity of 𝑋 and the equality 𝛿_𝑋(2)=1 characterizes strict convexity (=rotundity) of 𝑋.

The Gurariǐ modulus of convexity of a normed space 𝑋 is defined (see [68, 69]) by 𝛽_𝑋(𝜀)=inf1−inf_𝛼∈[0,1]()𝑦;(𝑋),𝛼𝑥+1−𝛼𝑥,𝑦∈𝑆𝑥−𝑦=𝜀 (5) for any 𝜀∈[0,2]. It is obvious that 𝛿_𝑋(𝜀)≤𝛽_𝑋(𝜀) for any Banach space 𝑋 and any 𝜀∈[0,2]. It is also known that 𝛽_𝑋(𝜀)≤2𝛿_𝑋(𝜀) for any 𝜀∈[0,2] and that 𝑋 is rotund if and only if 𝛽_𝑋(𝜀)=2 and as well as that 𝑋 is uniformly convex if and only if 𝛽_𝑋(𝜀)>0 for any 𝜀∈[0,2]. Gurariǐ [68] proved that if 𝑋=𝑐₀ is renormed by the norm ‖𝑥‖=‖𝑥‖_∞+_𝑛=0𝑃𝑥_𝑛2^𝑛21/2,∀𝑥_𝑛∈𝑐₀, (6) then 𝛽_𝑋(𝜀)=0 for any 𝜀∈[0,2) and 𝛽_𝑋(2)=1.

Gurariǐ and Sozonov [70] proved that a normed linear space (𝑋,‖⋅‖) is an inner product space if and only if, for every 𝑥,𝑦∈𝑆(𝑋)inf_𝛼∈[0,1]𝛼𝑥+(1−𝛼)𝑦=𝑥+𝑦2. (7) Zanco and Zucchi [71] showed an example of a normed space 𝑋 with 𝛿_𝑋(2)≠𝛽_𝑋(2).

Now, we will define Köthe sequence spaces 𝑚(𝜙,𝑝) and ℓ_𝑝(𝑢,𝑣) that will be considered in this paper.

Let 𝒞 denote the set whose elements are finite sets of distinct positive integers. Given any element 𝜎 of 𝒞, we denote by 𝑐(𝜎) the sequence {𝑐_𝑛(𝜎)} such that 𝑐_𝑛(𝜎)=1 for 𝑛∈𝜎, and 𝑐_𝑛(𝜎)=0 otherwise. Further, we define 𝒞_𝜏=𝜎∈𝒞∶^∞_𝑛=1𝑐_𝑛(𝜎)≤𝜏; (8) that is, 𝒞_𝜏 is the set of those 𝜎 whose support has cardinality at most 𝜏. Let us define Φ=𝜙=𝜙_𝑛^∞_𝑛=1∈ℓ⁰∶𝜙₁>0,Δ𝜙_𝑘𝜙≥0,Δ^𝑘𝑘≤0,∀𝑘∈ℕ, (9) where Δ𝜙_𝑛=𝜙_𝑛−𝜙_𝑛−1.

Given any 𝜙∈Φ, we define the following sequence space, introduced in [55]: 𝑚𝜙=𝑥=𝑥_𝑛^∞_𝑛=1∈ℓ⁰∶sup_𝜏≥1sup_{𝜎∈𝒞𝜏}1𝜙_{𝜏𝑛∈𝜎}𝑥_𝑛<∞. (10) Sargent [55] established the relationship of this space to the space ℓ_𝑝(1≤𝑝≤∞) and characterized some matrix transformations. In [49], matrix classes (𝑋,𝑚(𝜙)) have been characterized, where 𝑋 is assumed to be any 𝐹𝐾-space.

Recently in [52], some of the geometric properties of 𝑚(𝜙) have been investigated. In [61], Tripathy and Sen extended the space 𝑚(𝜙) to 𝑚(𝜙,𝑝) as follows: 𝑚𝑥𝜙,𝑝∶=𝑥=_𝑛^∞_𝑛=1∈ℓ⁰∶sup_𝜏≥1sup_{𝜎∈𝒞𝜏}1𝜙_{𝜏𝑛∈𝜎}𝑥_𝑛^𝑝<∞, (11) for 𝜙∈Φ and 𝑝>0.

It has been proved in [61] that, for 1≤𝑝<∞, 𝑚(𝜙,𝑝) is a Banach space if it is endowed with the norm ‖𝑥‖_{𝑚(𝜙,𝑝)}=sup_𝜏≥1sup_{𝜎∈𝒞𝜏}1𝜙_{𝜏𝑛∈𝜎}𝑥_𝑛^𝑝1/𝑝, (12) and that one has the following.(i)If 𝜙_𝑛=1, for all 𝑛∈ℕ, then 𝑚(𝜙,𝑝)=ℓ_𝑝. Moreover, ℓ_𝑝⊆𝑚(𝜙,𝑝)⊆ℓ_∞.(ii)If 𝑝=1, then 𝑚(𝜙,𝑝)=𝑚(𝜙). Also, for any 𝑝≥1,𝑚(𝜙)⊆𝑚(𝜙,𝑝).(iii)𝑚(𝜙,𝑝)⊆𝑚(𝜓,𝑝) if and only if sup_𝜏≥1(𝜙_𝜏/𝜓_𝜏)<∞.

It is easy to see that 𝑚(𝜙,𝑝) is a Köthe sequence space, indeed a 𝐵𝐾-space with respect to its natural norm (see [55]). Note that throughout the present paper we will study the space 𝑚(𝜙,𝑝) except the case 𝜙_𝑛=𝑛, for which it is reduced to the space ℓ^∞.

Now we will introduce the space ℓ_𝑝(𝑢,𝑣). Let 𝑢={𝑢_𝑛}^∞_𝑛=0 and let 𝑣={𝑣_𝑛}^∞_𝑛=0 be arbitrary real sequences with all coordinates 𝑢_𝑘 and 𝑣_𝑘 different from zero and let, for any 𝑝∈[1,∞), ℓ_𝑝()=𝑥𝑢,𝑣𝑥=_𝑘^∞_𝑘=1∶^∞_𝑛=1^𝑛_𝑘=1𝑢_𝑛𝑣_𝑘𝑥(𝑘)^𝑝<∞. (13) It is obvious that this is a linear space. It is known (see [48]) that the functional ‖𝑥‖_{ℓ𝑝(𝑢,𝑣)}=_𝑛^𝑛_𝑘=1𝑢_𝑛𝑣_𝑘𝑥(𝑘)^𝑝1/𝑝 (14) is a norm in ℓ_𝑝(𝑢,𝑣) and that the couple (ℓ_𝑝(𝑢,𝑣),‖⋅‖_{ℓ𝑝(𝑢,𝑣)}) is a Banach space. The space ℓ_𝑝(𝑢,𝑣) is a generalization of three spaces. Namely, one has the following.(i)If (𝑣_𝑘)=(1,1,1,…) and (𝑢_𝑛)=(1/𝑛), then ℓ_𝑝(𝑢,𝑣) is the Cesáro sequence space 𝑋_𝑝 of nonabsolute type (see [32]) and ‖𝑥‖_{ℓ𝑝(𝑢,𝑣)}=‖𝑥‖_𝑋𝑝.(ii)Let {𝑝_𝑛} be a real sequence with 𝑝₁>0 and 𝑝_𝑛≠0 for all 𝑛∈ℕ. If (𝑣_𝑘)={𝑝_𝑘} and (𝑢_𝑛)=(1/𝑄_𝑛), where 𝑄_𝑛=𝑝₁+𝑝₂+⋯+𝑝_𝑛 for any 𝑛∈ℕ. Then we obtain that ℓ_𝑝(𝑢,𝑣) is the Riesz sequence space of nonabsolute type denoted by 𝛾^𝑝(𝑣) and that ‖𝑥‖_{ℓ𝑝(𝑢,𝑣)}=‖𝑥‖_{𝛾𝑝(𝑣)} (see [44]).(iii)Let {𝑝_𝑛} be a real sequence with 𝑝₁>0 and 𝑝_𝑛≠0, for all 𝑛∈ℕ, 𝑛≥2. If (𝑣_𝑘)=(𝑝_{𝑛−𝑘+1}) and (𝑢_𝑛)=(1/𝑄_𝑛), where 𝑄_𝑛=𝑝_𝑛+𝑝_𝑛−1+⋯+𝑝₁ for any 𝑛∈ℕ. Then ℓ_𝑝(𝑢,𝑣) reduces to the Nörlund sequence space of nonabsolute type denoted by 𝑁_𝑝 and ‖𝑥‖_{ℓ𝑝(𝑢,𝑣)}=‖𝑥‖_𝑁𝑝(see [60]).

2. Banach-Saks Type of Sequence Space 𝑚(𝜙,𝑝)

In this section, we investigate some properties of the space 𝑚(𝜙,𝑝) such as the Fatou property, the Banach-Saks property of type 𝑝, and the weak fixed point property. Let us start with the following lemma.

Lemma 1. If a 𝐵𝐾-space 𝑋 containing 𝑐₀₀ has the 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝐴𝐾, then it is order continuous, that is, ‖(0,0,…,𝑥(𝑛),𝑥(𝑛+1),…)‖→0 as 𝑛→∞ for any 𝑥∈𝑋.

Proof. From the definition of property 𝐴𝐾, we have that every 𝑥={𝑥(𝑖)}∈𝑋 has the unique representation 𝑥=∑^∞_𝑖=1𝑥(𝑖)𝑒(𝑖),thatis,𝑥^[𝑛]=∑^𝑛_𝑖=1𝑥(𝑖)𝑒(𝑖)→𝑥 as 𝑛→∞. Hence ‖𝑥−𝑥^[𝑛]‖_𝑋→0 as 𝑛→∞, that is, ‖(0,0,…,𝑥(𝑛),𝑥(𝑛+1),…)‖→0 as 𝑛→∞, which means that 𝑋 is order continuous.

Corollary 2. The space 𝑚(𝜙,𝑝) is order continuous.

Proof. It is easy to see that 𝑚(𝜙,𝑝) contains 𝑐₀₀ and that every 𝑥={𝑥(𝑖)}∈𝑚(𝜙,𝑝) has the unique representation 𝑥=∑^∞_𝑖=1𝑥(𝑖)𝑒(𝑖),thatis,𝑥^[𝑛]=∑^𝑛_𝑖=1𝑥(𝑖)𝑒(𝑖)→𝑥 as 𝑛→∞, which means that 𝑚(𝜙,𝑝) has the property 𝐴𝐾. Hence, from the above lemma, 𝑚(𝜙,𝑝) is order continuous.

Theorem 3. The space 𝑚(𝜙,𝑝) has the Fatou property.

Proof. Let x be any real sequence from (𝑙⁰)₊ and {𝑥_𝑛} be any non-decreasing sequence of non-negative elements from 𝑚(Φ,𝑝) such that 𝑥_𝑛(𝑖)→𝑥(𝑖) as 𝑛→∞ coordinatewisely and sup_𝑛‖𝑥_𝑛‖_{𝑚(Φ,𝑝)}<∞.
Let us denote 𝑠=sup_𝑛‖𝑥_𝑛‖_{𝑚(Φ,𝑝)}. Then, since the supremum is homogeneous, we have 1𝑠sup_𝜏≥1sup_{𝜎∈𝒞𝜏}1Φ_{𝜏𝑖∈𝜎}|𝑥_𝑛(𝑖)|^𝑝1/𝑝=sup_𝜏≥1sup_{𝜎∈𝒞𝜏}1Φ_{𝜏𝑖∈𝜎}𝑥_𝑛(𝑖)𝑠^𝑝1/𝑝≤sup_𝜏≥1sup_{𝜎∈𝒞𝜏}1Φ_{𝜏𝑖∈𝜎}𝑥_𝑛(𝑖)𝑥_{𝑛𝑚(Φ,𝑝)}^𝑝1/𝑝=1𝑥_{𝑛𝑚(Φ,𝑝)}𝑥_{𝑛𝑚(Φ,𝑝)}=1. (15) Moreover, by the assumptions that {𝑥_𝑛} is non-decreasing and convergent to 𝑥 coordinatewisely and by the Beppo-Levi theorem, we have 1𝑠lim_𝑛→∞sup_𝜏≥1sup_{𝜎∈𝒞𝜏}1Φ_{𝜏𝑖∈𝜎}|𝑥_𝑛(𝑖)|^𝑝1/𝑝=sup_𝜏≥1sup_{𝜎∈𝒞𝜏}1Φ_{𝜏𝑖∈𝜎}𝑥(𝑖)𝑠^𝑝1/𝑝=𝑥𝑠_{𝑚(Φ,𝑝)}≤1, (16) whence ‖𝑥‖_{𝑚(Φ,𝑝)}≤𝑠=sup_𝑛𝑥_{𝑛𝑚(Φ,𝑝)}=lim_𝑛→∞𝑥_{𝑛𝑚(Φ,𝑝)}<∞. (17) Therefore, 𝑥∈𝑚(Φ,𝑝). On the other hand, since 0≤𝑥_𝑛≤𝑥 for any natural number 𝑛 and the sequence {𝑥_𝑛} is non-decreasing, we obtain that the sequence {‖𝑥_𝑛‖_{𝑚(Φ,𝑝)}} is bounded from above by ‖𝑥‖_{𝑚(Φ,𝑝)}. In consequence lim_𝑛→∞‖𝑥_𝑛‖_{𝑚(Φ,𝑝)}≤‖𝑥‖_{𝑚(Φ,𝑝)}, which together with the opposite inequality proved already, yields that ‖𝑥‖_{𝑚(Φ,𝑝)}=lim_𝑛‖𝑥_𝑛‖_{𝑚(Φ,𝑝)}.

Theorem 4. The space 𝑚(𝜙,𝑝) has the Banach-Saks property of the type 𝑝.

Proof. Let {𝜖_𝑛} be a sequence of positive numbers with ∑^∞_𝑛=1𝜖_𝑛≤1/2. Let {𝑥_𝑛} be a weakly null sequence in 𝐵(𝑚(𝜙,𝑝)). Let us set 𝑥₀=0 and 𝑧₁=𝑥_𝑛1=𝑥₁. Then, there exists 𝜏₁∈ℕ such that _𝑖∈𝑠1𝑧₁(𝑖)𝑒(𝑖)_{𝑚(𝜙,𝑝)}<𝜖₁, (18) where 𝑠₁ consist of the elements of 𝜎 which exceed 𝜏₁. Since 𝑥→_𝑛^𝜔0 implies that 𝑥_𝑛→0 coordinatewise, there is 𝑛₂∈ℕ such that ^𝜏1_𝑖=1𝑥_𝑛(𝑖)𝑒(𝑖)_{𝑚(𝜙,𝑝)}<𝜖₁, (19) for all 𝑛≥𝑛₂. Set 𝑧₂=𝑥_𝑛2. Then there exists 𝜏₂>𝜏₁ such that _𝑖∈𝑠2𝑧₂(𝑖)𝑒(𝑖)_{𝑚(𝜙,𝑝)}<𝜖₂, (20) where 𝑠₂ consist of all elements of 𝜎 which exceed 𝜏₂. Using again the fact that 𝑥_𝑛→0 coordinatewise, there exists 𝑛₃>𝑛₂ such that ^𝜏2_𝑖=1𝑥_𝑛(𝑖)𝑒(𝑖)_{𝑚(𝜙,𝑝)}<𝜖₂, (21) for all 𝑛≥𝑛₃.
Continuing this process, we can find two increasing sequences {𝜏_𝑗} and {𝑛_𝑗} such that ^𝜏𝑗_𝑖=1𝑥_𝑛(𝑖)𝑒(𝑖)_{𝑚(𝜙,𝑝)}<𝜖_𝑗,foreach𝑛≥𝑛_𝑗+1,_{𝑖∈𝑠𝑗}𝑧_𝑗(𝑖)𝑒(𝑖)_{𝑚(𝜙,𝑝)}<𝜖_𝑗, (22) where 𝑧_𝑗=𝑥_𝑛𝑗 and 𝑠_𝑗 consist of the elements of 𝜎 which exceed 𝜏_𝑗. Since 𝜖_𝑗−1+𝜖_𝑗<1, we have 1𝜙_{𝜏𝑛∈𝜎}𝑧_𝑗(𝑛)≤𝜖_𝑗−1+𝜖_𝑗<1, (23) for all 𝑗∈ℕ and 𝜏≥1. Hence ^𝑛_𝑗=1𝑧_{𝑗𝑚(𝜙,𝑝)}=^𝑛_𝑗=1^𝜏𝑗−1_𝑖=1𝑧_𝑗(𝑖)𝑒(𝑖)+^𝜏𝑗_{𝑖=𝜏𝑗−1+1}𝑧_𝑗(𝑖)𝑒(𝑖)+_{𝑖∈𝑠𝑗}𝑧_𝑗(𝑖)𝑒(𝑖)_{𝑚(𝜙,𝑝)}≤^𝑛_𝑗=1^𝜏𝑗−1_𝑖=1𝑧_𝑗(𝑖)𝑒(𝑖)_{𝑚(𝜙,𝑝)}+^𝑛_𝑗=1^𝜏𝑗_{𝑖=𝜏𝑗−1+1}𝑧_𝑗(𝑖)𝑒(𝑖)_{𝑚(𝜙,𝑝)}≤^𝑛_𝑗=1^𝜏𝑗_{𝑖=𝜏𝑗−1+1}𝑧_𝑗(𝑖)𝑒(𝑖)_{𝑚(𝜙,𝑝)}+2^𝑛_𝑗=1𝜖_𝑗. (24) By using the norm of the space 𝑚(𝜙,𝑝), we have ^𝑛_𝑗=1^𝜏𝑗_{𝑖=𝜏𝑗−1+1}𝑧_𝑗(𝑖)𝑒(𝑖)_{𝑚(𝜙,𝑝)}=^𝑛_𝑗=1sup_𝜏≥1sup_{𝑠𝑗−1∈𝒞𝜏}1𝜙_{𝜏𝑛∈𝑠𝑗−1}𝑧_𝑗(𝑛)^𝑝≤^𝑛_𝑗=1sup_𝜏≥1sup_{𝜎∈𝒞𝜏}1𝜙_{𝜏𝑛∈𝜎}𝑧_𝑗(𝑛)^𝑝≤𝑛. (25) Therefore, ^𝑛_𝑗=1𝑧_{𝑗𝑚(𝜙,𝑝)}≤𝑛^1/𝑝+1≤2𝑛^1/𝑝. (26) This completes the proof of the theorem.

Theorem 5. For 1<𝑝<∞, the space 𝑚(𝜙,𝑝) has the weak fixed point property, if 𝐾>2^1−𝑝, where 𝐾=sup_𝜏≥1𝜙_𝜏<∞.

Proof. If 𝜓_𝜏=1, for all 𝜏∈ℕ, it follows that 𝑚𝜙,𝑝⊆ℓ_𝑝iffsup_𝜏≥1𝜙_𝜏^1/𝑝‖𝑥‖<∞,_{𝑚(𝜙,𝑝)}=sup_𝜏≥11𝜙_𝜏^1/𝑝‖𝑥‖_ℓ𝑝. (27) Hence 𝑅𝑚𝜙,𝑝=sup_𝜏≥11𝜙_𝜏^1/𝑝𝑅ℓ_𝑝=2𝐾^1/𝑝<2,since𝑅ℓ_𝑝=2^1/𝑝, (28) where 𝑅(𝑋) stands for the Garcia-Falset coefficient of 𝑋. Therefore, 𝑚(𝜙,𝑝) has in this case the weak fixed point property.

3. Banach-Saks Type and Gurariǐ Modulus of Sequence Spaces ℓ_𝑝(𝑢,𝑣)

Theorem 6. The space ℓ_𝑝(𝑢,𝑣) has the Banach-Saks property of the type 𝑝.

Proof. Let (𝜀_𝑛) be a sequence of positive numbers for which ∑^∞_𝑛=1𝜀_𝑛≤(1/2). Let {𝑥_𝑛} be a weakly null sequence in 𝐵(ℓ_𝑝(𝑢,𝑣)). Set 𝑡₀=𝑥₀=0 and 𝑡₁=𝑥_𝑛1=𝑥₁. Then there exists 𝑟₁∈ℕ such that ^∞_𝑖=𝑟1+1𝑡₁(𝑖)𝑒(𝑖)_{ℓ𝑝(𝑢,𝑣)}<𝜀_1. (29) Since the fact that {𝑥_𝑛} is a weakly null sequence implies that 𝑥_𝑛→0, coordinatewise, there is an 𝑛₂∈ℕ such that ^𝑟1_𝑖=1𝑥_𝑛(𝑖)𝑒(𝑖)_{ℓ𝑝(𝑢,𝑣)}<𝜀₁, (30) for all 𝑛≥𝑛₂. Set 𝑡₂=𝑥_𝑛2. Then there exists an 𝑟₂>𝑟₁ such that ^∞_𝑖=𝑟2+1𝑡₂(𝑖)𝑒(𝑖)_{ℓ𝑝(𝑢,𝑣)}<𝜀_2. (31) By using the fact that 𝑥_𝑛→0 coordinatewise, there exists an 𝑛₃>𝑛₂ such that ^𝑟2_𝑖=1𝑥_𝑛(𝑖)𝑒(𝑖)_{ℓ𝑝(𝑢,𝑣)}<𝜀_2, (32) for all 𝑛≥𝑛₃. Continuing this process, we can find by induction two increasing subsequences (𝑟_𝑖) and (𝑛_𝑖) of natural numbers such that ^𝑟𝑗_𝑖=1𝑥_𝑛(𝑖)𝑒(𝑖)_{ℓ𝑝(𝑢,𝑣)}<𝜀_𝑗, (33) for all 𝑛≥𝑛_𝑗+1 and ^∞_{𝑖=𝑟𝑗+1}𝑡_𝑗(𝑖)𝑒(𝑖)_{ℓ𝑝(𝑢,𝑣)}<𝜀_𝑗, (34) where 𝑡_𝑗=𝑥_𝑛𝑗. Hence, ^𝑛_𝑗=0𝑡_{𝑗ℓ𝑝(𝑢,𝑣)}=^𝑛_𝑗=0^𝑟𝑗−1_𝑖=0𝑡_𝑗(𝑖)𝑒(𝑖)+^𝑟𝑗_{𝑖=𝑟𝑗−1+1}𝑡_𝑗(𝑖)𝑒(𝑖)+^∞_{𝑖=𝑟𝑗+1}𝑡_𝑗(𝑖)𝑒(𝑖)_{ℓ𝑝(𝑢,𝑣)}≤^𝑛_𝑗=0^𝑟𝑗_{𝑖=𝑟𝑗−1+1}𝑡_𝑗(𝑖)𝑒(𝑖)_{ℓ𝑝(𝑢,𝑣)}+^𝑛_𝑗=0^𝑟𝑗−1_𝑖=0𝑡_𝑗(𝑖)𝑒(𝑖)_{ℓ𝑝(𝑢,𝑣)}+^𝑛_𝑗=0^∞_{𝑖=𝑟𝑗+1}𝑡_𝑗(𝑖)𝑒(𝑖)_{ℓ𝑝(𝑢,𝑣)}≤^𝑛_𝑗=0^𝑟𝑗_{𝑖=𝑟𝑗−1+1}𝑡_𝑗(𝑖)𝑒(𝑖)_{ℓ𝑝(𝑢,𝑣)}+2^𝑛_𝑗=0𝜀_𝑗. (35) On the other hand, since ‖𝑥_𝑛‖=(∑^∞_𝑖=0|∑^𝑖_𝑘=0𝑢_𝑖𝑣_𝑘𝑥_𝑛𝑗(𝑘)|^𝑝)^1/𝑝, it can be easily seen that ‖𝑥_𝑛‖<1. Therefore, ‖𝑥_𝑛‖^𝑝<1 and ^𝑛_𝑗=0^𝑟𝑗_{𝑖=𝑟𝑗−1+1}𝑡_𝑗(𝑖)𝑒(𝑖)^𝑝_{ℓ𝑝(𝑢,𝑣)}=^𝑛𝑟_𝑗=0^𝑗_{𝑖=𝑟𝑗−1+1}^𝑖_𝑘=0𝑢_𝑖𝑣_𝑘𝑡_𝑗(𝑘)^𝑝≤^𝑛∞_𝑗=0𝑖=0^𝑖_𝑘=0𝑢_𝑖𝑣_𝑘𝑡_𝑗(𝑘)^𝑝≤(𝑛+1). (36) Hence we obtain ^𝑛_𝑗=0^𝑟𝑗_{𝑖=𝑟𝑗−1+1}𝑡_𝑗(𝑖)𝑒(𝑖)≤^𝑛_𝑗=01^1/𝑝=(𝑛+1)^1/𝑝. (37) By using the inequality 1≤(𝑛+1)^1/𝑝 for all 𝑛∈ℕ and 1≤𝑝<∞, we have ^𝑛_𝑗=0𝑡_{𝑗𝑙𝑝(𝑢,𝑣)}≤(𝑛+1)^1/𝑝+1≤2(𝑛+1)^1/𝑝. (38) Therefore, the space ℓ_𝑝(𝑢,𝑣) has the Banach-Saks type 𝑝, which completes the proof of the theorem.

Let us define the matrix 𝐺=𝐺(𝑢,𝑣)={𝑔_𝑛𝑘} by 𝑔_𝑛𝑘=𝑢_𝑛𝑣_𝑘;0≤𝑘≤𝑛,0;𝑘>𝑛 (39) for all 𝑘,𝑛∈ℕ, where 𝑢_𝑛 depends only on 𝑛 and 𝑣_𝑘 depends only on 𝑘. The matrix 𝐺 is called generalized weighted mean or factorable matrix. By 𝐻=𝐻(𝑣,𝑢)=(ℎ_𝑛𝑘), we denote the inverse of the matrix 𝐺(𝑢,𝑣) as follows: ℎ_𝑛𝑘=(−1)^𝑛−𝑘𝑣_𝑛𝑢_𝑘;𝑛−1≤𝑘≤𝑛,0;0≤𝑛<𝑘or𝑛>𝑘+1. (40)

Theorem 7. For 𝑥∈ℓ_𝑝(𝑢,𝑣), by (39), one has the fact that the Gurariǐ modulus of convexity for the normed space ℓ_𝑝(𝑢,𝑣) satisfies the inequality 𝛽_{ℓ𝑝(𝑢,𝑣)}(𝜀)𝜀≤1−1−2^𝑝1/𝑝 (41) for any 0≤𝜀≤2.

Proof. Let 𝑥∈ℓ_𝑝(𝑢,𝑣). By using (39), we have ‖𝑥‖_{ℓ𝑝(𝑢,𝑣)}=‖𝐺(𝑢,𝑣)𝑥‖_ℓ𝑝=_𝑛(𝐺(𝑢,𝑣)𝑥)_𝑛^𝑝1/𝑝. (42) Let 0≤𝜀≤2. Then using (40), let us consider the following sequences: 𝑥=𝑥_𝑛=𝐻𝜀1−2^𝑝1/𝑝𝜀,𝐻2,𝑡,0,0,…𝑡=_𝑛=𝐻𝜀1−2^𝑝1/𝑝−𝜀,𝐻2,0,0,…. (43) Since 𝑦_𝑛=(𝐺𝑥)_𝑛 and 𝑧_𝑛=(𝐺𝑡)_𝑛, we have 𝑦=𝑦_𝑛=𝜀1−2^𝑝1/𝑝,𝜀2,𝑧,0,0,…𝑧=_𝑛=𝜀1−2^𝑝1/𝑝,−𝜀2,0,0,…. (44) By using the sequences given above, we obtain the following equalities: ‖𝑥‖^𝑝_{ℓ𝑝(𝑢,𝑣)}=‖𝐺(𝑢,𝑣)𝑥‖^𝑝_ℓ𝑝=𝜀1−2^{𝑝1/𝑝𝑝}+𝜀2^𝑝𝜀=1−2^𝑝+𝜀2^𝑝‖𝑡‖=1;_{𝑝ℓ𝑝(𝑢,𝑣)}=‖𝐺(𝑢,𝑣)𝑡‖^𝑝_ℓ𝑝=𝜀1−2^{𝑝1/𝑝𝑝}+−𝜀2^𝑝𝜀=1−2^𝑝+𝜀2^𝑝‖‖=1;𝑥−𝑡_{ℓ𝑝(𝑢,𝑣)}=‖𝐺(𝑢,𝑣)𝑥−𝐺(𝑢,𝑣)𝑡‖_ℓ𝑝=𝜀1−2^𝑝1/𝑝−𝜀1−2^{𝑝1/𝑝𝑝}+𝜀2−−𝜀2^𝑝1/𝑝=𝜀. (45) To complete the upper estimate of the Gurariǐ modulus of convexity, it remains to calculate the infimum of ‖𝛼𝑥+(1−𝛼)𝑡‖_{ℓ𝑝(𝑢,𝑣)} for 0≤𝛼≤1. We have inf_0≤𝛼≤1‖𝛼𝑥+(1−𝛼)𝑡‖_{ℓ𝑝(𝑢,𝑣)}=inf_0≤𝛼≤1‖𝛼𝐺(𝑢,𝑣)𝑥+(1−𝛼)𝐺(𝑢,𝑣)𝑡‖_ℓ𝑝=inf_0≤𝛼≤1𝛼𝜀1−2^𝑝1/𝑝+()𝜀1−𝛼1−2^{𝑝1/𝑝𝑝}+𝛼𝜀2+()−𝜀1−𝛼2^𝑝1/𝑝=inf_0≤𝛼≤1𝜀1−2^𝑝+|2𝛼−1|^𝑝𝜀2^𝑝1/𝑝=𝜀1−2^𝑝1/𝑝. (46) Consequently, we get for 𝑝≥1 the inequality 𝛽_{𝑙𝑝(𝑢,𝑣)}(𝜀)𝜀≤1−1−2^𝑝1/𝑝, (47) which is the desired result.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.