Open Access

Some geometric properties of a new modular space defined by Zweier operator

Fixed Point Theory and Applications20132013:165

https://doi.org/10.1186/1687-1812-2013-165

Received: 14 December 2012

Accepted: 5 June 2013

Published: 25 June 2013

Abstract

In this paper, we define the modular space Z σ ( s , p ) by using the Zweier operator and a modular. Then, we consider it equipped with the Luxemburg norm and also examine the uniform Opial property and property β. Finally, we show that this space has the fixed point property.

MSC:40A05, 46A45, 46B20.

Keywords

Zweier operator Luxemburg norm modular space uniform Opial property property ( β )

Dedication

Dedicated to Professor Hari M Srivastava

1 Introduction

In literature, there are many papers about the geometrical properties of different sequence spaces such as [19]. Opial [10] introduced the Opial property and proved that the sequence spaces p ( 1 < p < ) have this property but L p [ 0 , 2 π ] ( p 2 , 1 < p < ) does not have it. Franchetti [11] showed that any infinite dimensional Banach space has an equivalent norm that satisfies the Opial property. Later, Prus [12] introduced and investigated the uniform Opial property for Banach spaces. The Opial property is important because Banach spaces with this property have the weak fixed point property.

2 Definition and preliminaries

Let ( X , ) be a real Banach space and let S ( X ) (resp. B ( X ) ) be the unit sphere (resp. the unit ball) of X. A Banach space X has the Opial property if for any weakly null sequence { x n } in X and any x in X { 0 } , the inequality lim n inf x < lim n inf x n + x holds. We say that X has the uniform Opial property if for any ε > 0 there exists r > 0 such that for any x X with x ε and any weakly null sequence { x n } in the unit sphere of X, the inequality 1 + r lim n inf x n + x holds.

For a bounded set A X , the ball-measure of noncompactness was defined by β ( A ) = inf { ε > 0 : A  can be covered by finitely many balls with diameter  ε } . The function Δ defined by Δ ( ε ) = inf { 1 inf ( x : x A ) : A  is closed convex subset of  B ( X )  with  β ( A ) ε } is called the modulus of noncompact convexity. A Banach space X is said to have property  ( L ) , if lim ε 1 Δ ( ε ) = 1 . This property is an important concept in the fixed point theory and a Banach space X possesses property ( L ) if and only if it is reflexive and has the uniform Opial property.

A Banach space X is said to satisfy the weak fixed point property if every nonempty weakly compact convex subset C and every nonexpansive mapping T : C C ( T x T y x y , x , y C ) have a fixed point, that is, there exists x C such that T ( x ) = x . Property ( L ) and the fixed point property were also studied by Goebel and Kirk [13], Toledano et al. [14], Benavides [15], Benavides and Phothi [16]. A Banach space X is said to have property ( H ) if every weakly convergent sequence on the unit sphere is convergent in norm. Clarkson [17] introduced the uniform convexity, and it is known that the uniform convexity implies the reflexivity of Banach spaces. Huff [18] introduced the concept of nearly uniform convexity of Banach spaces. A Banach space X is called uniformly convex (UC) if for each ε > 0 , there is δ > 0 such that for x , y S ( X ) , the inequality x y > ε implies that 1 2 ( x + y ) < 1 δ . For any x B ( X ) , the drop determined by x is the set D ( x , B ( X ) ) = conv ( { x } B ( X ) ) . A Banach space X has the drop property ( D ) if for every closed set C disjoint with B ( X ) , there exists an element x C such that D ( x , B ( X ) ) C = { x } . Rolewicz [19] showed that the Banach space X is reflexive if X has the drop property. Later, Montesinos [20] extended this result and proved that X has the drop property if and only if X is reflexive and has property ( H ) . A sequence { x n } is said to be ε-separated sequence for some ε > 0 if
sep ( x n ) = inf { x n x m : n m } > ε .

A Banach space X is called nearly uniformly convex (NUC) if for every ε > 0 , there exists δ ( 0 , 1 ) such that for every sequence ( x n ) B ( X ) with sep ( x n ) > ε , we have conv ( x n ) ( ( 1 δ ) B ( X ) ) . Huff [18] proved that every (NUC) Banach space is reflexive and has property  ( H ) . A Banach space X has property ( β ) if and only if for each ε > 0 , there exists δ > 0 such that for each element x B ( X ) and each sequence ( x n ) in B ( X ) with sep ( x n ) ε , there is an index k for which x + x k 2 < 1 δ .

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

     
  3. (iii)

    ρ ( α x + β y ) ρ ( x ) + ρ ( y ) for all x , y X and all α , β 0 with α + β = 1 .

     
The modular ρ is called convex if
  1. (iv)

    ρ ( α x + β y ) α ρ ( x ) + β ρ ( y ) for all x , y X and all α , β 0 with α + β = 1 .

     
For any modular ρ on X, the space
X ρ = { x X : ρ ( σ x ) <  for some  σ > 0 }
is called a modular space. In general, the modular is not subadditive and thus it does not behave as a norm or a distance. But we can associate the modular with an F-norm. A functional : X [ 0 , ] defines an F-norm if and only if
  1. (i)

    x = 0 x = 0 ,

     
  2. (ii)

    α x = x whenever | α | = 1 ,

     
  3. (iii)

    x + y x + y ,

     
  4. (iv)

    if α n α and x n x 0 , then α n x n α x 0 .

     
F-norm defines a distance on X by d ( x , y ) = x y . If the linear metric space ( X , d ) is complete, then it is called an F-space. The modular space X ρ can be equipped with the following F-norm:
x = inf { α > 0 : ρ ( x α ) α } .

If the modular ρ is convex, then the equality x = inf { α > 0 : ρ ( x α ) 1 } defines a norm which is called the Luxemburg norm.

A modular ρ is said to satisfy the δ 2 -condition if for any ε > 0 , there exist constants K 2 , a > 0 such that ρ ( 2 u ) K ρ ( u ) + ε for all u X ρ with ρ ( u ) a . If ρ provides the δ 2 -condition for any a > 0 with K 2 dependent on a, then ρ provides the strong δ 2 -condition (briefly ρ δ 2 s ).

Let us denote by 0 the space of all real sequences. The Cesàro sequence spaces
Ces p = { x 0 : n = 1 ( n 1 i = 1 n | x i | ) p < } , 1 p < ,
and
Ces = { x 0 : sup n n 1 i = 1 n | x i | < } ,
were introduced by Shiue [21]. Jagers [22] determined the Köthe duals of the sequence space Ces p ( 1 < p < ). It can be shown that the inclusion p Ces p is strict for 1 < p < although it does not hold for p = 1 . Also, Suantai [23] defined the generalized Cesàro sequence space by
c e s ( p ) = { x 0 : ρ ( λ x ) <  for some  λ > 0 } ,
where ρ ( x ) = n = 1 ( 1 n i = 1 n | x ( i ) | ) p n . If p = ( p n ) is bounded, then
ces ( p ) = { x = ( x k ) : n = 1 ( n 1 i = 1 n | x ( i ) | ) p n < } .
The sequence space C ( s , p ) was defined by Bilgin [24] as follows:
C ( s , p ) = { x = ( x k ) : r = 0 ( 2 r r k s | x k | ) p r < , s 0 }
for p = ( p r ) with inf p r > 0 , where r denotes a sum over the ranges 2 r k < 2 r + 1 . The special case of C ( s , p ) for s = 0 is the space
Ces ( p ) = { x = ( x k ) : r = 0 ( 2 r r | x k | ) p r < }
which was introduced by Lim [25]. Also, the inclusion Ces ( p ) C ( s , p ) holds. A paranorm on C ( s , p ) is given by
ρ ( x ) = ( r = 0 ( 2 r r k s | x k | ) p r ) 1 / M

for M = max ( 1 , H ) and H = sup p r < .

The Z-transform of a sequence x = ( x k ) is defined by ( Z x ) n = y n = α x n + ( 1 α ) x n 1 by using the Zweier operator
Z = ( z n k ) = { α , k = n , 1 α , k = n 1 , 0 , otherwise for  n , k N  and  α F { 0 } ,

where is the field of all complex or real numbers. The Zweier operator was studied by Şengönül and Kayaduman [26].

Now we introduce a new modular sequence space Z σ ( s , p ) by
Z σ ( s , p ) = { x 0 : σ ( t x ) < ,  for some  t > 0 } ,
where σ ( x ) = r = 0 ( 2 r r k s | α x k + ( 1 α ) x k 1 | ) p r < and s 0 . If we take α = 1 , then Z σ ( s , p ) = C ( s , p ) ; if α = 1 and s = 0 , then Z σ ( s , p ) = Ces ( p ) . It can be easily seen that σ : Z σ ( s , p ) [ 0 , ] is a modular on Z σ ( s , p ) . We define the Luxemburg norm on the sequence space Z σ ( s , p ) as follows:
x = inf { t > 0 : σ ( x t ) 1 } , x Z σ ( s , p ) .

It is easy to see that the space Z σ ( s , p ) is a Banach space with respect to the Luxemburg norm.

Throughout the paper, suppose that p = ( p r ) is bounded with p r > 1 for all r N and
for i N and x 0 . In addition, we will require the following inequalities:
| a k + b k | p k C ( | a k | p k + | b k | p k ) , | a k + b k | t k | a k | t k + | b k | t k ,

where t k = p k M 1 and C = max { 1 , 2 H 1 } with H = sup p k .

3 Main results

Since p is reflexive and convex, ( p ) -type spaces have many useful applications, and it is natural to consider a geometric structure of these spaces. From this point of view, we generalized the space C ( s , p ) by using the Zweier operator and then obtained the equality Z σ ( s , p ) = Ces ( p ) , that is, it was seen that the structure of the space Ces ( p ) was preserved. In this section, our goal is to investigate a geometric structure of the modular space Z σ ( s , p ) related to the fixed point theory. For this, we will examine property ( β ) and the uniform Opial property for Z σ ( s , p ) . Finally, we will give some fixed point results. To do this, we need some results which are important in our opinion.

Lemma 3.1 [2]

If σ δ 2 s , then for any L > 0 and ε > 0 , there exists δ > 0 such that
| σ ( u + v ) σ ( u ) | < ε ,

where u , v X σ with σ ( u ) L and σ ( v ) δ .

Lemma 3.2 [2]

If σ δ 2 s , convergence in norm and in modular are equivalent in X σ .

Lemma 3.3 [2]

If σ δ 2 s , then for any ε > 0 , there exists δ = δ ( ε ) > 0 such that x 1 + δ implies σ ( x ) 1 + ε .

Now we give the following two lemmas without proof.

Lemma 3.4 If x L < 1 for any x Z σ ( s , p ) , then σ ( x ) x L .

Lemma 3.5 For any x Z σ ( s , p ) , x L = 1 if and only if σ ( x ) = 1 .

Lemma 3.6 If lim inf p r > 1 , then for any x Z σ ( s , p ) , there exist k 0 N and μ ( 0 , 1 ) such that
σ ( x k 2 ) 1 μ 2 σ ( x k )

for all k N with k k 0 , where and 2 r k < 2 r + 1 .

Proof Let k N be fixed. Then there exists r k N such that k I r k . Let γ be a real number 1 < γ lim inf p r , and so there exists k 0 N such that γ < p r k for all k k 0 . Choose μ ( 0 , 1 ) such that ( 1 2 ) γ 1 μ 2 . Therefore, we have
σ ( x k 2 ) = r = 0 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) 2 | ) p r = r = 0 ( 1 2 ) p r ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r ( 1 2 ) γ r = 0 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r < 1 μ 2 σ ( x k )

for each x Z σ ( s , p ) and k k 0 . □

Lemma 3.7 If σ δ 2 s , then for any ε ( 0 , 1 ) , there exists δ ( 0 , 1 ) such that σ ( x ) 1 ε implies x 1 δ .

Proof Suppose that lemma does not hold. So, there exist ε > 0 and x n Z σ ( s , p ) such that σ ( x n ) < 1 ε and 1 2 x n 1 . Take s n = 1 x n 1 , and so s n 0 as n . Let P = sup { σ ( 2 x n ) : n N } . There exists D 2 such that
σ ( 2 u ) D σ ( u ) + 1
(3.2)
for every u Z σ ( s , p ) with σ ( u ) < 1 , since σ δ 2 s . We have
σ ( 2 x n ) D σ ( x n ) + 1 < D + 1
for all n N by (3.1). Therefore, 0 < P < and from Lemma 3.5 we have
1 = σ ( x n x n ) = σ ( 2 s n x n + ( 1 s n ) x n ) s n σ ( 2 x n ) + ( 1 s n ) σ ( x n ) s n P + ( 1 ε ) ( 1 ε ) .

This is a contradiction. So, the proof is complete. □

Theorem 3.8 The space Z σ ( s , p ) has property ( β ) .

Proof Let ε > 0 and ( x n ) B ( Z σ ( s , p ) ) with sep ( x n ) ε and x B ( Z σ ( s , p ) ) . For each l N , we can find r k N such that 2 r k l < 2 r k + 1 . Let
Since for each i N , ( x n ( i ) ) i = 1 is bounded, by using the diagonal method, we can find a subsequence ( x n j ) of ( x n ) such that ( x n j ( i ) ) converges for each i N with 1 i l . Therefore, there exists an increasing sequence of positive integers t l such that sep ( ( x n j l ) j t l ) ε . Thus, there exists a sequence of positive integers ( r l ) l = 1 with r 1 < r 2 < such that x r l l ε 2 for all l N . Since σ δ 2 s , there is η > 0 such that
σ ( x r l l ) η for all  l N
(3.4)
from Lemma 3.3. However, there exist k 0 N and μ ( 0 , 1 ) such that
σ ( v k 2 ) 1 μ 2 σ ( v k )
(3.5)
for all v Z σ ( s , p ) and k k 0 by Lemma 3.6. There exists δ > 0 such that
σ ( y ) 1 μ η 4 y 1 δ
(3.6)

for any y Z σ ( s , p ) by Lemma 3.7.

By Lemma 3.1, there exists δ 0 such that
| σ ( u + v ) σ ( u ) | < μ η 4 ,
(3.7)
where σ ( u ) 1 and σ ( v ) δ 0 . Hence, we get that σ ( x ) 1 since x B ( Z σ ( s , p ) ) . Then there exists k k 0 such that σ ( x k ) δ 0 . Let u = x r l l and v = x l . Then
σ ( u 2 ) < 1 and σ ( v 2 ) < δ 0 .
We obtain from (3.3) and (3.5) that
σ ( u + v 2 ) σ ( u 2 ) + μ η 4 1 μ 2 σ ( u ) + μ η 4 .
(3.8)
Choose s i = r l i . By the inequalities (3.2), (3.3), (3.6) and the convexity of the function f ( u ) = | u | p r , we have
σ ( x + x s k 2 ) = r = 0 ( 2 r r k s | α ( x ( k ) + x s i ( k ) ) + ( 1 α ) ( x ( k 1 ) + x s i ( k 1 ) ) 2 | ) p r = r = 0 r k 1 ( 2 r r k s | α ( x ( k ) + x s i ( k ) ) + ( 1 α ) ( x ( k 1 ) + x s i ( k 1 ) ) 2 | ) p r + r = r k ( 2 r r k s | α ( x ( k ) + x s i ( k ) ) + ( 1 α ) ( x ( k 1 ) + x s i ( k 1 ) ) 2 | ) p r 1 2 r = 0 r k 1 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r + 1 2 r = 0 r k 1 ( 2 r r k s | α x s i ( k ) + ( 1 α ) x s i ( k 1 ) | ) p r + r = r k ( 2 r r k s | α x s i ( k ) + ( 1 α ) x s i ( k 1 ) 2 | ) p r + μ η 4 1 2 r = 0 r k 1 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r + 1 2 r = 0 r k 1 ( 2 r r k s | α x s i ( k ) + ( 1 α ) x s i ( k 1 ) | ) p r + 1 μ 2 r = r k ( 2 r r k s | α x s i ( k ) + ( 1 α ) x s i ( k 1 ) 2 | ) p r + μ η 4 1 2 r = 0 r k 1 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r + 1 2 r = 0 ( 2 r r k s | α x s i ( k ) + ( 1 α ) x s i ( k 1 ) | ) p r μ 2 r = r k ( 2 r r k s | α x s i ( k ) + ( 1 α ) x s i ( k 1 ) 2 | ) p r + μ η 4 1 2 + 1 2 μ η 2 + μ η 4 = 1 μ η 4 .

So, the inequality (3.4) implies that x + x s k 2 1 δ . Consequently, the space Z σ ( s , p ) possesses property ( β ) . □

Since property ( β ) implies NUC, NUC implies property ( D ) and property ( D ) implies reflexivity, we can give the following result from Theorem 3.8.

Corollary 3.9 The space Z σ ( s , p ) is nearly uniform convex, reflexive and also it has property ( D ) .

Theorem 3.10 The space Z σ ( s , p ) has the uniform Opial property.

Proof Let ε > 0 and x Z σ ( s , p ) be such that x ε and ( x n ) be a weakly null sequence in S ( Z σ ( s , p ) ) . By σ δ 2 s , there exists ζ ( 0 , 1 ) independent of x such that σ ( x ) > ζ by Lemma 3.2. Also since σ δ 2 s , by Lemma 3.1, there is ζ 1 ( 0 , ζ ) such that
| σ ( y + z ) σ ( y ) | < ζ 4
(3.10)
whenever σ ( y ) 1 and σ ( z ) ζ 1 . Take r 0 N such that
r = r 0 + 1 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r < ζ 1 4 .
(3.11)
Hence, we have
ζ < r = 1 r 0 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r + r = r 0 + 1 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r r = 1 r 0 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r + ζ 1 4
(3.12)
and this implies that
r = 1 r 0 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r > ζ ζ 1 4 > ζ ζ 4 = 3 ζ 4 .
(3.13)
Since x n w 0 , by the inequality (3.10), there exists r 0 N such that
r = 1 r 0 ( 2 r r k s | α ( x n ( k ) + x ( k ) ) + ( 1 α ) ( x n ( k 1 ) + x ( k 1 ) ) | ) p r > 3 ζ 4 .
(3.14)
Again, by x n w 0 , there is r 1 > r 0 such that for all r > r 1
x n | r 0 < 1 ( 1 ζ 4 ) 1 / M ,
(3.15)
where p r M N for all r N . Therefore, we obtain that
x n | N r 0 > ( 1 ζ 4 ) 1 / M
(3.16)
by the triangle inequality of the norm. It follows from the definition of the Luxemburg norm that
1 σ ( x n | N r 0 ( 1 ζ 4 ) 1 / M ) = r = r 0 + 1 ( 2 r r k s | α x n ( k ) + ( 1 α ) x n ( k 1 ) | ( 1 ζ 4 ) 1 / M ) p r ( 1 ( 1 ζ 4 ) 1 / M ) M r = r 0 + 1 ( 2 r r k s | α x n ( k ) + ( 1 α ) x n ( k 1 ) | ) p r
(3.17)
and this implies that
r = r 0 + 1 ( 2 r r k s | α x n ( k ) + ( 1 α ) x n ( k 1 ) | ) p r 1 ζ 4 .
(3.18)
By (3.7), (3.8), (3.11), (3.15) and since x n w 0 x n 0 (coordinatewise), we have for any r > r 1 that
σ ( x n + x ) = r = 1 r 0 ( 2 r r k s | α ( x n ( k ) + x ( k ) ) + ( 1 α ) ( x n ( k 1 ) + x ( k 1 ) ) | ) p r + r = r 0 + 1 ( 2 r r k s | α ( x n ( k ) + x ( k ) ) + ( 1 α ) ( x n ( k 1 ) + x ( k 1 ) ) | ) p r r = r 0 + 1 ( 2 r r k s | α ( x n ( k ) + x ( k ) ) + ( 1 α ) ( x n ( k 1 ) + x ( k 1 ) ) | ) p r ζ 4 + 3 ζ 4 3 ζ 4 + ( 1 ζ 4 ) ζ 4 = 1 + ζ 4 .

Since σ δ 2 s , it follows from Lemma 3.3 that there is τ depending on ζ only such that x n + x 1 + τ . □

Corollary 3.11 The space Z σ ( s , p ) has property ( L ) and the fixed point property.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Firat University
(2)
Department of Statistics, Bitlis Eren University
(3)
Department of Mathematics, Muş Alparslan University

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© Et et al.; licensee Springer. 2013

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