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Some geometric properties of generalized modular sequence space derived by the generalized de la Vallée-Poussin mean

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Abstract

In this paper, we define a generalized modular sequence space by using the generalized de la Vallée-Poussin mean with a generalized Riesz transformation. Moreover, we investigate the property (β) and the uniform Opial property which is equipped with the Luxemburg norm. Finally, we show that this space has the fixed point property.

1 Introduction

A number of mathematicians are studying the geometric properties of Banach spaces, because such properties were identified as an important characteristic of the Banach spaces. For example, if Banach spaces have some geometric properties such as the uniformly rotund, P-convexity, Q-convexity, Banach-Saks property, then they are reflexive spaces. The investigations of the metric geometry of Banach spaces date back to 1913, when Radon [1] introduced the Kadec-Klee property (sometimes called the Radon-Riesz property, or property (H)), and later, when Riesz [2, 3] showed that the classical L p -spaces, 1<p<, have the Kadec-Klee property. Although the space L 1 [0,1] (with Lebesgue measure) fails to have the Kadec-Klee property. In 1936, Clarkson [4] introduced the notion of the uniform convexity property (UC) or the uniform rotund property (UR) of Banach spaces, and it was shown that L p with 1<p< are examples of such space. In 1967, Opial [5] introduced a new property which was called the Opial property and proved that the sequence spaces l p (1<p<) have this property but L p [0,π] (p2, 1<p<) do not have it. In 1980, Huff [6] introduced the nearly uniform convexity for Banach spaces and he also proved that every nearly uniformly convex Banach space is reflexive and it has the uniformly Kadec-Klee property (UKK). In 1987, Rolewicz [7] defined the drop property and property (β) and the characterization of property (β), which is proved in [8]. In 1991, Kutzarova [8] defined and studied k-nearly uniformly Banach spaces. In 1992, Prus [9] introduced the notion of the uniform Opial property. There are many papers about the geometrical properties of sequence spaces. In 2003, Suantai [10, 11] defined the generalized Cesàro sequence space with a bounded sequence p=( p k ) of positive real numbers. In 2010, Şimşek et al. [12] introduced a new modular sequence space which is more general than the Cesàro sequence space defined by Shiue [13] and the generalized Cesàro sequence space defined by Suantai. In 2013, Mongkolkeha and Kumam [14] defined the generalized Cesàro sequence space ces ( p ) (q) for a bounded sequence p=( p k ) with p k 1 for all kN and q=( q k ) of positive real numbers. Recently, Şimşek et al. [15, 16] defined it by the modular sequence space with de la Vallée-Poussin’s mean and studied some geometric properties in these spaces. Some examples of the geometry of sequence spaces and their generalizations have been extensively studied in [1720].

The main purpose of this paper is to investigate the property (β) and the uniform Opial property equipped with the Luxemburg norm of the new modular sequence space, which is defined by using the generalized de la Vallée-Poussin mean with generalized Riesz transformation. Furthermore, we show that this space has the fixed point property.

2 Preliminaries and notations

Let l 0 be the space of all real sequences. For 1p<, the Cesàro sequence space ( ces p , for short) of Shue is defined by

ces p = { x l 0 : k = 1 ( 1 k i = 0 k | x ( i ) | ) p < }

equipped with the norm

x= ( k = 1 ( 1 k i = 1 k | x ( i ) | ) p ) 1 p .
(2.1)

The generalized Cesàro sequence space ces(p) for p=( p k ) a bounded sequence of positive real numbers with p k 1 for all kN of Suantai [10, 11] is defined by

ces ( p ) = { x l 0 : ϱ ( λ x ) <  for some  λ > 0 } ,

where

ϱ(x)= k = 1 ( 1 k i = 1 k | x ( i ) | ) p k

equipped with the Luxemburg norm

x=inf { ε > 0 : ϱ ( x ε ) 1 } .

In the case when p k =p, 1p< for all kN, the generalized Cesàro sequence space ces ( p ) is nothing but the Cesàro sequence space ces p and the Luxemburg norm is expressed by (2.1).

Let =( λ k ) be a nondecreasing sequence of positive real numbers tending to infinity and let λ 1 =1 and λ k + 1 λ k +1. The generalized de la Vallée-Poussin means of a sequence x=( x k ) is defined as follows:

t k (x)= 1 λ k j I k x j where  I k =[k λ k +1,k] for k1.

The modular sequence space V ϱ (λ;p) of Şimşek et al. [15, 16] is defined by de la Vallée-Poussin’s mean, namely

V ϱ (λ;p)= { x l 0 : ϱ ( τ x ) <  for some  τ > 0 } ,

where

ϱ(x)= k = 1 ( 1 λ k i I k | x ( i ) | ) p k

equipped with the Luxemburg norm

x=inf { τ > 0 : ϱ ( x τ ) 1 } .

Let q=( q k ) be a sequence of positive real numbers and Q k = i = 1 k q i . Then the Riesz transformation of x=( x k ) is defined as

t k = 1 Q k i = 1 k q i x i .
(2.2)

In 2012, Mursaleen et al. [21] has modified the definition of weighted statistical convergence due to Karakaya and Chishti [22], they showed that the definition must be as follows: A sequence x=( x k ) is weighted statistically convergent (or S N ¯ -convergent) to L if, for every ε>0,

lim k 1 Q k | { i Q k : q i | x i L | ε } | =0,

where Q k = i = 1 k q i as k. In the same year, Mongkolkeha and Kumam [14] defined the generalized Cesàro sequence space ces ( p ) (q) for a bounded sequence p=( p k ) with p k 1 for all kN and q=( q k ) of positive real numbers by

ces ( p ) (q)= { x l 0 : ϱ ( λ x ) <  for some  λ > 0 } ,

where

ϱ(x)= k = 1 ( 1 Q k i = 1 k q i | x ( i ) | ) p k

and Q k = i = 1 k q i with Q k = i = 1 k q i as k. Thus, we see that p k =p, 1p< for all kN; then ces ( p ) (q) reduces to ces p (q) defined by Khan [23]. Recently, Belen and Mohiuddine [24] generalized the concept of weighted statistical convergence due to Mursaleen et al. for a nondecreasing sequence ( λ k ) of positive real numbers tending to infinity and let λ k =1 and λ k + 1 λ k +1. That is, let a sequence q=( q k ) of nonnegative numbers be such that q 0 >0 and Q λ k = i I k q i as k and

σ k := 1 Q λ k i I k q i ,
(2.3)

where I k =[k λ k +1,k].

A sequence x=( x k ) is weighted λ-statistically convergent (or S N ¯ λ -convergent) to L if for every ε>0,

lim n 1 Q λ k | { i Q λ k : q i | x i L | ε } | =0.

Now, we define the new generalized modular sequence space for p=( p k ) a bounded sequence of positive real numbers with p k 1 for all kN and ( q k ) is a sequence of positive real numbers such that q 0 >0. Let Q λ k = i I k q i as k by

V ϱ (λ;p,q)= { x l 0 : ϱ ( τ x ) <  for some  τ > 0 } ,
(2.4)

where

ϱ(x)= k = 1 ( 1 Q λ k i I k q i | x ( i ) | ) p k

equipped with the Luxemburg norm

x=inf { τ > 0 : ϱ ( x τ ) 1 } ,

when Q λ k = i I k q i and I k =[k λ k +1,k] for k1.

By applying the reasoning of Remark 2.4 in [24], if we take λ k =k for all k1, then the weighted generalized modular sequence space V ϱ (λ;p,q) becomes the space ces ( p ) (q). If we take q k =1 for all k1, then the weighted generalized modular sequence space V ϱ (λ;p,q) becomes the space V ϱ (λ;p). Also, if we take λ k =k and q k =1 for all k1, then the weighted generalized modular sequence space V ϱ (λ;p,q) becomes the space ces ( p ) .

Let (X,) be a real Banach space and let B(X) (resp., S(X)) be a closed unit ball (resp., the unit sphere) of X. A point xS(X) is an H-point of B(X) if for any sequence ( x n ) in X such that x n 1 as n, the weak convergence of ( x n ) to x implies that x n x0 as n. If every point in S(X) is an H-point of B(X), then X is said to have the property (H). A Banach space X has the property (β) if for each ε>0 there exists δ>0 such that 1<x<1+δ implies α(conv(B(X){x})B(X))<ε, where α(A) denotes the Kuratowski measure noncompactness of a subset A of X defined as the infimum of all ε>0 such that A can be covered by a finite union of sets of diameter less than ε. The following characterization of the property (β) is very useful (see [25]): A Banach space X has the property (β) if and only if for each ε>0 there exists δ>0 such that for each element xB(X) and for each sequence ( x n ) in B(X) with sep( x n )ε there is an index k for which x + x k 2 <1δ where sep( x n )=inf{ x n x m :nm}>ε. A Banach space X is nearly uniformly convex (NUC) if for each ε>0 and every sequence ( x n ) in B(X) with sep( x n )ε, there exists δ(0,1) such that conv( x n )(1δ)B(X). A Banach space X is said to have the Opial property (see [5]) if every sequence ( x n ) weakly convergent to x 0 satisfies

lim n inf x n x 0 lim n inf x n x,

for every xX. Opial proved in [5] that the sequence spaces l p (1<p<) have this property but L p [0,π] (p2, 1<p<) do not have it. A Banach space X is said to have the uniform Opial property (see [9]), if for each ε>0 there exists τ>0 such that for any weakly null sequence ( x n ) in S(X) and xX with x>ε the following holds:

1+τ lim n inf x n x.

For example, the spaces in [19, 20] have the uniform Opial property.

The ball-measure of noncompactness was defined in [26, 27] by

β(A)=inf{ε>0:A can be covered by finitely many balls of diameterε}.

A Banach space X is said to have property (L) if lim ε 1 Δ(ε)=1, where Δ(ε)=inf{1inf[x:xA]:A is closed convex subset of B(X) with β(A)ε}. The function Δ is called the modulus of noncompact convexity (see [26]). It has been proved in [9] that property (L) is a useful tool in fixed point theory and that a Banach space X has property (L) if and only if it is reflexive and has the uniform Opial property.

Throughout this paper, we assume that lim k inf p k >1 and lim k sup p k < and for x l 0 , iN, we denote

In addition, we put M= sup k p k for all k1.

First, we start with a brief recollection of basic concepts and facts in modular space. For a real vector space X, a function ρ:X[0,] is called a modular if it satisfies the following conditions:

  1. (i)

    ρ(x)=0 if and only if x=0;

  2. (ii)

    ρ(αx)=ρ(x) for all scalar α with |α|=1;

  3. (iii)

    ρ(αx+βy)ρ(x)+ρ(y), for all x,yX and all α,β0 with α+β=1.

    The modular ρ is called convex if

  4. (iv)

    ρ(αx+βy)αρ(x)+βρ(y), for all x,yX and all α,β0 with α+β=1.

For modular ρ on X, the space

X ρ = { x X : ρ ( λ x ) 0  as  λ 0 + }

is called the modular space.

A sequence ( x n ) in X ρ is called modular convergent to x X ρ if there exists a λ>0 such that ρ(λ( x n x))0 as n.

A modular ρ is said to satisfy the Δ 2 -condition (ρ Δ 2 ) if for any ε>0 there exist constants K2 and a>0 such that

ρ(2u)Kρ(u)+ε

for all u X ρ with ρ(u)a.

If ρ satisfies the Δ 2 -condition for any a>0 with K2 dependent on a, we say that ρ has the strong Δ 2 -condition (ρ Δ 2 s ).

Lemma 2.1 [[28], Lemma 2.1]

If ρ Δ 2 s , then for any L>0 and ε>0, there exists δ=δ(L,ε)>0 such that

| ρ ( u + v ) ρ ( u ) | <ε,

whenever u,v X ρ with ρ(u)L, and ρ(v)δ.

Lemma 2.2 [[28], Lemma 2.3]

Convergences in norm and in modular sense are equivalent in X ρ if ρ Δ 2 .

Lemma 2.3 [[28], Lemma 2.4]

If ρ Δ 2 s , then for any ε>0 there exists δ=δ(ε)>0 such that x1+δ whenever ρ(x)1+ε.

3 Main results

In this section, we prove the property (β) and uniform Opial property in a generalized modular sequence space V ϱ (λ;p,q). Finally, we show that this space has the fixed point property. First we shall give some results which are very important for our consideration.

Proposition 3.1 The functional ϱ is a convex modular on V ϱ (λ;p,q).

Proof Let x,y V ϱ (λ;p,q). It is obvious that ϱ(x)=0 if and only if x=0 and ϱ(αx)=ϱ(x) for scalar α with |α|=1. Let α0, β0 with α+β=1. By the convexity of the function t | t | p k , for all kN, we have

ϱ ( α x + β y ) = k = 1 ( 1 Q λ k i I k | α q i x ( i ) + β q i y ( i ) | ) p k k = 1 ( α 1 Q λ k i I k q i | x ( i ) | + β 1 Q λ k i I k q i | y ( i ) | ) p k α k = 1 ( 1 Q λ k i I k q i | x ( i ) | ) p k + β k = 1 ( 1 Q λ k i I k q i | y ( i ) | ) p k = α ϱ ( x ) + β ϱ ( y ) .

 □

Proposition 3.2 For x V ϱ (λ;p,q), the modular ϱ on V ϱ (λ;p,q) satisfies the following properties:

  1. (i)

    if 0<a<1, then a M ϱ( x a )ϱ(x) and ϱ(ax)aϱ(x);

  2. (ii)

    if a>1, then ϱ(x) a M ϱ( x a );

  3. (iii)

    if a1, then ϱ(x)aϱ(x)ϱ(ax).

Proof (i) Let 0<a<1. Then we have

ϱ ( x ) = k = 1 ( 1 Q λ k i I k q i | x ( i ) | ) p k = k = 1 ( a Q λ k i I k q i | x ( i ) a | ) p k = k = 1 a p k ( 1 Q λ k i I k q i | x ( i ) a | ) p k k = 1 a M ( 1 Q λ k i I k q i | x ( i ) a | ) p k = a M k = 1 ( 1 Q λ k i I k q i | x ( i ) a | ) p k = a M ϱ ( x a ) .

By the convexity of modular ϱ, we have ϱ(ax)aϱ(x), so (i) is obtained.

(ii) Let a>1. Then

ϱ ( x ) = k = 1 ( 1 Q λ k i I k q i | x ( i ) | ) p k = k = 1 a p k ( 1 Q λ k i I k q i | x ( i ) a | ) p k a M k = 1 ( 1 Q λ k i I k q i | x ( i ) a | ) p k = a M ϱ ( x a ) .

Hence (ii) is satisfied. (iii) follows from the convexity of ϱ. □

Following the line of the proof in [10, 11, 17], we get the following results.

Proposition 3.3 For any x V ϱ (λ;p,q), we have

  1. (i)

    if x<1, then ϱ(x)x;

  2. (ii)

    if x>1, then ϱ(x)x;

  3. (iii)

    x=1 if and only if ϱ(x)=1;

  4. (iv)

    x<1 if and only if ϱ(x)<1;

  5. (v)

    x>1 if and only if ϱ(x)>1.

Proposition 3.4 For any x V ϱ (λ;p,q), we have

  1. (i)

    if 0<a<1 and x>a, then ϱ(x)> a M ;

  2. (ii)

    if a1 and x<a, then ϱ(x)< a M .

Proposition 3.5 Let ( x n ) be a sequence in V ϱ (λ;p,q).

  1. (i)

    If x n 1 as n, then ϱ( x n )1 as n.

  2. (ii)

    If ϱ( x n )0 as n, then x n 0 as n.

Lemma 3.6 For any x V ϱ (λ;p,q), there exist j 0 N and γ(0,1) such that ϱ( x j 2 ) 1 γ 2 ϱ( x j ) for all jN with j j 0 , where

and λ k corresponding to I k for k1.

Proof Let jN be fixed. So there exist k j N such that j I k j . Let α be a real number such that 1<α lim k inf p k , then there exists j 0 N such that α< p k j for all j j 0 . Choose γ(0,1) to be a real such that ( 1 2 ) α 1 γ 2 . Then for each x V ϱ (λ;p,q) and j j 0 , we have

ϱ ( x j 2 ) = k = k j ( 1 Q λ k i I k q i | x ( i ) 2 | ) p k = k = k j ( 1 2 ) p k ( 1 Q λ k i I k q i | x ( i ) | ) p k ( 1 2 ) α k = k j ( 1 Q λ k i I k q i | x ( i ) | ) p k 1 γ 2 ϱ ( x j ) .

 □

Lemma 3.7 For any x V ϱ (λ;p,q) and ε(0,1) there exists δ(0,1) such that ϱ(x)1ε implies x1δ.

Proof For a proof of this lemma, we apply and follow the line of the proof of Theorem 1.39(4) in [29]. Suppose that the lemma does not hold, then there exist ε>0 and x n V ϱ (λ;p,q) such that ϱ( x n )1ε and 1 2 x n 1. Let a n = 1 x n 1. Then a n 0 as n. Let L=sup{ϱ(2 x n ):nN}. By sup k p k <, i.e., ϱ Δ 2 s , there exists K2 such that

ϱ(2u)Kϱ(u)+1,
(3.1)

for every ul(p,θ) with ϱ(u)<1. By (3.1), we have ϱ(2 x n )Kϱ( x n )+1K+1 for all nN. Hence 0<L<. By Proposition 3.1 and Proposition 3.2(iii), we have

1 = ϱ ( x n x n ) = ϱ ( 2 a n x n + ( 1 a n ) x n ) a n ϱ ( 2 x n ) + ( 1 a n ) ϱ ( x n ) a n L + ( 1 ε ) 1 ε ,

which is a contradiction. □

Theorem 3.8 The space V ϱ (λ;p,q) is a Banach space with respect to the Luxemburg norm.

Proof Let ( x n )=( x n (i)) be a Cauchy sequence in V ϱ (λ;p,q) and ε(0,1). Thus there exists NN such that x n x m <ε for all n,mN. By Proposition 3.3(i), we have

ϱ( x n x m ) x n x m <εfor all n,mN.
(3.2)

That is,

k = 1 ( 1 Q λ k i I k q i | x n ( i ) x m ( i ) | ) p k <εfor all n,mN.
(3.3)

For fixed k, we see that

| q i x n ( i ) q i x m ( i ) | <εfor all n,mN.

Thus, let ( q i x n (i)) be a Cauchy sequence in for all iN. Since is complete, for each i1, there exists x(i)R such that q i x m (i) q i x(i) as m. Thus for fixed k and for each i I k , we have

q i | x n ( i ) x ( i ) | <εas m, for all nN.

This implies that

ϱ( x n x m )ϱ( x n x)as m.
(3.4)

That is,

k = 1 ( 1 Q λ k i I k q i | x n ( i ) x m ( i ) | ) p k k = 1 ( 1 Q λ k i I k q i | x n ( i ) x ( i ) | ) p k
(3.5)

as m. By (3.3), we have

ϱ( x n x) x n x<εfor all nN,

and hence x n x as n. So we have x n x V ϱ (λ;p,q). Since ( x n ) V ϱ (λ;p,q) and the linearity of the sequence space V ϱ (λ;p,q), we get x= x n ( x n x) V ϱ (λ;p,q). Therefore the sequence space V ϱ (λ;p,q) is a Banach space, with respect to the Luxemburg norm, and the proof is complete. □

Theorem 3.9 The space V ϱ (λ;p,q) has property (β).

Proof Let ε>0 and ( x n )B( V ϱ (λ;p,q)) with sep( x n )ε. For each jN, there exist k j N such that j I k j . Let

where λ k corresponds to I k for k1. This is so since for each iN, ( x n ( i ) ) n = 1 is bounded. By using the diagonal method, we see that for each jN we can find a subsequence ( x n l ) of ( x n ) such that ( x n l (i)) converges for each iN. Therefore, for any jN there exists an increasing sequence ( t j ) such that sep( ( x n l j ) l > t j )ε. Hence for each jN there exists a sequence of positive integers ( s j ) j = 1 with s 1 < s 2 < s 3 < such that x s j j ε 2 and, since ϱ Δ 2 s , by Lemma 2.2 we may assume that there exists η>0 such that ϱ( x s j j )η for all jN, that is,

k = k j ( 1 Q λ k i I k q i | x s j j ( i ) | ) p k η
(3.6)

for all jN. On the other hand, by Lemma 3.6, there exist j 0 N and γ(0,1) such that

ϱ ( u j 2 ) 1 γ 2 ϱ ( u j )
(3.7)

for all u V ϱ (λ;p,q) and j j 0 . From Lemma 3.7, there exists δ>0 such that for any y V ϱ (λ;p,q)

ϱ(y)1 γ η 4 y1δ.
(3.8)

Since again ϱ Δ 2 s , by Lemma 2.1, there exists δ 0 such that

| ϱ ( u + v ) ϱ ( u ) | < γ η 4 ,
(3.9)

whenever ϱ(u)1 and ϱ(v) δ 0 . Since xB( V ϱ (λ;p,q)), we have ϱ(x)1. Then there exists j j 0 such that ϱ( x j ) δ 0 . We put u= x s j j and v= x j ,

ϱ ( u 2 ) = k = k j ( 1 Q λ k i I k q i | x s j ( i ) 2 | ) p k < 1 and ϱ ( v 2 ) = k = k j ( 1 Q λ k i I k q i | x ( i ) 2 | ) p k < δ 0 .

From (3.7) and (3.9), we have

k = k k ( 1 Q λ k i I k q i | x ( i ) + x s j ( i ) 2 | ) p k = ϱ ( u + v 2 ) ϱ ( u 2 ) + γ η 4 1 γ 2 ( ϱ ( u ) ) + γ η 4 .
(3.10)

By (3.6), (3.9), (3.10), and convexity of the function f(t)= | t | p k , for all kN, we have

ϱ ( x + x s j 2 ) = k = 1 ( 1 Q λ k i I k q i | x ( i ) + x s j ( i ) 2 | ) p k = k = 1 k j 1 ( 1 Q λ k i I k q i | x ( i ) + x s j ( i ) 2 | ) p k + k = k j ( 1 Q λ k i I k q i | x ( i ) + x s j ( i ) 2 | ) p k 1 2 ( k = 1 k j 1 ( 1 Q λ k i I k q i | x ( i ) | ) p k + k = 1 k j 1 ( 1 Q λ k i I k q i | x s j ( i ) | ) p k ) + k = k j ( 1 Q λ k i I k q i | x s j ( i ) 2 | ) p k + γ η 4 1 2 ( k = 1 k j 1 ( 1 Q λ k i I k q i | x ( i ) | ) p k + k = 1 k j 1 ( 1 Q λ k i I k q i | x s j ( i ) | ) p j ) + 1 γ 2 k = k j ( 1 Q λ k i I k q i | x s j ( i ) | ) p k + γ η 4 = 1 2 k = 1 k j 1 ( 1 Q λ k i I k q i | x ( i ) | ) p k + 1 2 k = 1 k j 1 ( 1 Q λ k i I k q i | x s j ( i ) | ) p k + 1 γ 2 k = k j ( 1 Q λ k i I k q i | x s j ( i ) | ) p k + γ η 4 = 1 2 k = 1 k j 1 ( 1 Q λ n i I n q i | x ( i ) | ) p k + 1 2 k = 1 ( 1 Q λ k i I k q i | x s j ( i ) | ) p k γ 2 k = k j ( 1 Q λ k i I k q i | x s j ( i ) | ) p j + γ η 4 1 2 + 1 2 γ η 2 + γ η 4 = 1 γ η 4 .

So it follows from (3.8) that

x + x s j 2 1δ.

Therefore, the space V ϱ (λ;p,q) has property (β). □

By the facts that property (β) implies (NUC), and (NUC) implies property (UKK), property (H), and reflexivity (see [2931]). The following results are obtained directly from Theorem 3.9.

Corollary 3.10 The space V ϱ (λ;p) has property (β).

Corollary 3.11 The space V ϱ (λ;p,q) is nearly uniform convexity and reflexive.

Corollary 3.12 The space V ϱ (λ;p,q) has property (UKK) and property (H).

Corollary 3.13 The space V ϱ (λ;p) is nearly uniform convexity and reflexive.

Corollary 3.14 The space V ϱ (λ;p) has property (UKK) and (H).

Next, we will prove the uniform Opial property for the space V ϱ (λ;p,q).

Theorem 3.15 The space V ϱ (λ;p,q) has the uniform Opial property.

Proof Take any ε>0 and x V ϱ (λ;p,q) with xε. Let ( x n ) be a weakly null sequence in S( V ϱ (λ;p,q)). By ϱ Δ 2 s , hence by Lemma 2.2 there exists δ(0,1) independent of x such that ϱ(x)>δ. Also, by ϱ Δ 2 s and Lemma 2.1, one may assert that there exists δ 1 (0,δ) such that

| ϱ ( y + z ) ϱ ( y ) | < δ 4
(3.11)

whenever ϱ(y)1 and ϱ(z) δ 1 . Choose k 0 N such that

k = k 0 + 1 ( 1 Q λ k i I k q i | x ( i ) | ) p k < δ 1 4 .
(3.12)

So, we have

δ < k = 1 k 0 ( 1 Q λ k i I k q i | x ( i ) | ) p k + k = k 0 + 1 ( 1 Q λ k i I k q i | x ( i ) | ) p k k = 1 k 0 ( 1 Q λ k i I k q i | x ( i ) | ) p k + δ 1 4 ,
(3.13)

which implies that

k = 1 k 0 ( 1 Q λ k i I k q i | x ( i ) | ) p k > δ δ 1 4 > δ δ 4 = 3 δ 4 .
(3.14)

Since x n w 0, there exists n 0 N such that

3 δ 4 k = 1 k 0 ( 1 Q λ k i I k q i | x n ( i ) + x ( i ) | ) p k
(3.15)

for all n> n 0 , since weak convergence implies coordinatewise convergence. Again, by x n w 0, there exists n 1 N such that

x n | k o <1 ( 1 δ 4 ) 1 M
(3.16)

for all n> n 1 . Hence, by the triangle inequality of the norm, we get

x n | N k o > ( 1 δ 4 ) 1 M .
(3.17)

It follows from Proposition 3.3(ii) that

1 ϱ ( x n | N k o ( 1 δ 4 ) 1 M ) = k = k 0 + 1 ( 1 Q λ k i I k q i | x n ( i ) | ( 1 δ 4 ) 1 M ) p k ( 1 ( 1 δ 4 ) 1 M ) M k = k 0 + 1 ( 1 Q λ k i I k q i | x n ( i ) | ) p k
(3.18)

implies that

k = k 0 + 1 ( 1 Q λ k i I k q i | x n ( i ) | ) p k 1 δ 4
(3.19)

for all n> n 1 . By inequality (3.11), (3.12), (3.15), and (3.19), we have for any n> n 1

ϱ ( x n + x ) = k = 1 k 0 ( 1 Q λ k i I k q i | x n ( i ) + x ( i ) | ) p k + k = k 0 + 1 ( 1 Q λ k i I k q i | x n ( i ) + x ( i ) | ) p k > k = 1 k 0 ( 1 Q λ k i I k q i | x n ( i ) + x ( i ) | ) p k + k = k 0 + 1 ( 1 Q λ k i I k q i | x n ( i ) + x ( i ) | ) p k 3 δ 4 + k = k 0 + 1 ( 1 Q λ k i I k q i | x n ( i ) | ) p k δ 4 3 δ 4 + ( 1 δ 4 ) δ 4 1 + δ 4 .

Since ϱ Δ 2 s and by Lemma 2.3 there exists τ depending on δ only such that x n +x1+τ, which implies that lim n inf x n +x1+τ, hence the proof is complete. □

Corollary 3.16 The space V ϱ (λ;p) has the uniform Opial property.

Corollary 3.17 [[20], Theorem 2.6]

The space ces ( p ) has the uniform Opial property.

Corollary 3.18 [[19], Theorem 2]

For any 1<p<, the space ces p has the uniform Opial property.

Corollary 3.19 The space V ϱ (λ;p,q) has property (L) and the fixed point property.

Corollary 3.20 The space V ϱ (λ;p) has property (L) and the fixed point property.

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Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for the technical and financial support.

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Correspondence to Abdul Latif.

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Latif, A., Mongkolkeha, C. & Sintunavarat, W. Some geometric properties of generalized modular sequence space derived by the generalized de la Vallée-Poussin mean. J Inequal Appl 2014, 375 (2014) doi:10.1186/1029-242X-2014-375

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Keywords

  • generalized Cesàro sequence spaces
  • property (β)
  • uniform Opial property
  • Vallée-Poussin
  • generalized Riesz transformations
  • fixed point property