Banach-Saks Type and Gurari < Modulus of Convexity of Some Banach Sequence Spaces

and Applied Analysis 3 that is,C τ is the set of those σ whose support has cardinality at most τ. Let us define


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 .Several authors including Cui and Hudzik [26][27][28][29], Cui and Meng [30], Suantai [31], and Lee [32] investigated the geometric properties of Cesàro sequence space ces().Also Cesàro-Orlicz sequence spaces equipped with Luxemburg norm have been studied in [5,[33][34][35][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][42][43].
Let us recall that a sequence {V()} ∞ =1 in a Banach space  is called ℎ  of  (or  for short) if for each  ∈  there exists a unique sequence A sequence space  with a linear topology is called a - if each of the projection maps   :  → C defined by   () = () for  = (()) ∞ =1 ∈  is continuous for each natural .A  é ℎ  is a complete metric linear space and the metric is generated by an -norm and a Fréchet space which is a -space is called an -; that is, a -space  2 Abstract and Applied Analysis is called an -space if  is a complete linear metric space.In other words,  is an -space if  is a Fréchet space with continuous coordinate projections.All the sequence spaces mentioned above are  spaces except the space  00 .
We say that  ∈  is order continuous if, for any sequence A Köthe sequence space  is said to be order continuous that if all sequences in  are order continuous.It is easy to see that  ∈  is order continuous if and only if ‖(0, 0, . . ., 0, (+ 1), ( + 2), . ..)‖ → 0 as  → ∞.
A Köthe sequence space  is said to have the   if, for any real sequence  ∈ ℓ 0 and any {  } in  such that   ↑  coordinatewisely and sup  ‖  ‖ < ∞, we have the fact that  ∈  and ‖  ‖ → ‖‖.
A Banach space  is said to have the ℎ-  if every bounded sequence {  } in  admits a subsequence {  } such that the sequence {  ()} is convergent in  with respect to the norm, where A Banach space  is said to have the  ℎ-  whenever given any weakly null sequence {  } in  there exists its subsequence {  } such that the sequence {  ()} converges to zero strongly.Given any  ∈ (1, ∞), we say that a Banach space (, ‖ ⋅ ‖) has the ℎ-     if there exists a constant  > 0 such that every weakly null sequence {  } has a subsequence {  ℓ } such that (see [22]) The Banach-Saks property of type  ∈ (1, ∞) and the weak Banach-Saks property for Ces à ro sequence spaces have been considered in [28].These properties and stronger property (  ) for Musielak-Orlicz and Nakano sequence spaces have been studied in [17].
We say that a Banach space  has the     if every nonexpansive self-mapping defined on a nonempty weakly compact convex subset  of  has a fixed point in .
Gurariǐ and Sozonov [70] proved that a normed linear space (, ‖ ⋅ ‖) is an inner product space if and only if, for every ,  ∈ () Zanco and Zucchi [71] showed an example of a normed space  with   (2) ̸ =   (2).Now, we will define Köthe sequence spaces (, ) and ℓ  (, 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  of C, we denote by () the sequence {  ()} such that   () = 1 for  ∈ , and   () = 0 otherwise.Further, we define that is, C  is the set of those  whose support has cardinality at most .Let us define where Δ  =   −  −1 .
Given any  ∈ Φ, we define the following sequence space, introduced in [55]: Sargent [55] established the relationship of this space to the space ℓ  (1 ≤  ≤ ∞) and characterized some matrix transformations.In [49], matrix classes (, ()) have been characterized, where  is assumed to be any -space.
Recently in [52], some of the geometric properties of () have been investigated.In [61], Tripathy and Sen extended the space () to (, ) as follows: for  ∈ Φ and  > 0.
It has been proved in [61] that, for 1 ≤  < ∞, (, ) is a Banach space if it is endowed with the norm and that one has the following.
It is easy to see that (, ) is a Köthe sequence space, indeed a -space with respect to its natural norm (see [55]).Note that throughout the present paper we will study the space (, ) except the case   = , for which it is reduced to the space ℓ ∞ .Now we will introduce the space ℓ  (, V).
and let V = {V  } ∞ =0 be arbitrary real sequences with all coordinates   and V  different from zero and let, for any It is obvious that this is a linear space.It is known (see [48]) that the functional is a norm in ℓ  (, V) and that the couple (ℓ  (, V), ‖ ⋅ ‖ ℓ  (,V) ) is a Banach space.The space ℓ  (, V) is a generalization of three spaces.Namely, one has the following.

Banach-Saks Type of Sequence Space 𝑚(𝜙,𝑝)
In this section, we investigate some properties of the space (, ) such as the Fatou property, the Banach-Saks property of type , and the weak fixed point property.Let us start with the following lemma.
Let us denote  = sup  ‖  ‖ (Φ,) .Then, since the supremum is homogeneous, we have Moreover, by the assumptions that {  } is non-decreasing and convergent to  coordinatewisely and by the Beppo-Levi theorem, we have Abstract and Applied Analysis 5 By using the norm of the space (, ), we have ()  ())            (,) Therefore, This completes the proof of the theorem.

Banach-Saks Type and Gurari< Modulus of
Sequence Spaces ℓ  (,V) Theorem 6.The space ℓ  (, V) has the Banach-Saks property of the type .
Let us define the matrix  = (, V) = {  } by for all ,  ∈ N, where   depends only on  and V  depends only on .The matrix  is called generalized weighted mean or factorable matrix.By  = (V, ) = (ℎ  ), we denote the inverse of the matrix (, V) as follows: Theorem 7.For  ∈ ℓ  (, V), by (39), one has the fact that the Gurariǐ modulus of convexity for the normed space ℓ  (, V) satisfies the inequality for any 0 ≤  ≤ 2.
Proof.Let  ∈ ℓ  (, V).By using (39), we have Consequently, we get for  ≥ 1 the inequality which is the desired result.