- Open Access
The fixed point property of Orlicz sequence spaces equipped with thep-Amemiya norm
© He et al.; licensee Springer. 2013
- Received: 12 August 2013
- Accepted: 20 November 2013
- Published: 13 December 2013
In this paper, the Opial modulus and the weakly convergent sequence coefficient ofOrlicz space endowed with the p-Amemiya norm are calculated, the criteria for the uniformOpial property as well as for weakly uniform normal structure ofare presented. It is shown that the Orlicz sequence space equipped with thep-Amemiya norm has the fixed point property if and only if it isreflexive.
MSC: 47H10, 46E30, 46B20.
- Orlicz sequence space
- Opial property
- weakly convergent sequence coefficient
- weakly uniform normal structure
- p-Amemiya norm
The aim of this paper is to present criteria for some important geometric propertiesrelated to the metric fixed point theory in Orlicz sequence spaces.
The Opial property originates from the fixed point theorem proved by Opial in. The uniform Opial property withrespect to the weak topology was defined in and Opial modulus was introduced in . It iswell known that the Opial property and normal structure of a Banach space Xplay an important role in metric fixed point theory for nonexpansive mappings, as wellas in the theory of differential and integral equations (see [1, 4–6]). The Opial property also plays an important role in the study ofweak convergence of iterates, random products of nonexpansive mappings and theasymptotic behavior of nonlinear semigroups [1, 7–9].Moreover, it can be introduced to the open unit ball of a complex Hilbert space,equipped with the hyperbolic metric, where it is useful in proving the existence offixed points of holomorphic self-mappings of X.
The coefficient was introduced by Bynum , who established their relations with normal structure andcalculated the value of . A reflexive Banach space X with has normalstructure and consequently it has the weakly fixed point property. This is probably oneof the Banach space constants which has been most widely studied, although withconsiderable confusion because there exist many equivalent definitions. Severaldifferent formulae for were found (see ), also see the work by Sims and Smyth .
The notion of p-Amemiya norm was introduced by Cui and Hudzik in , where they showed that the p-Amemiya normis equivalent to the Orlicz norm aswell as to the Luxemburg norm . They also illustrated thedescription of extreme points and strongly extreme points in Orlicz spaces equipped withthe p-Amemiya norm [12, 13]. In 2012, they presented the criteria for non-squareness,uniform non-squareness, and locally uniform non-squareness of these spaces. Chen and Cui (see [15, 16]) gave the criteria forcomplex extreme points and complex strict convexity in Orlicz function spaces equippedwith the p-Amemiya norm, and for complex mid-point locally uniform rotundityand complex rotundity of Orlicz sequence spaces equipped with the p-Amemiyanorm.
The rest of the paper is organized as follows. In the first section, some basic notions,terminology and original results are reviewed, which will be used throughout the paper.In Section 2, the Opial modulus of Orlicz space endowed with the p-Amemiya norm is calculated, and the criteria for the uniformOpial property of are presented. The weakly convergent sequence coefficient is calculated inSection 3. Finally, the necessary and sufficient condition for fixed pointproperties to exist in are given.
Let X be a Banach space. We denote by the unit ball ofX, by the unit sphere of X. Now we recall somenotions from fixed point theory.
A mapping defined on a subsetC of a Banach space X is said to be non-expansive if for all.
We say that a Banach space X has the fixed point property if for every weaklycompact convex subset and forevery nonexpansive , T has a fixedpoint of C.
It is known that does not have the fixed point property.
A map Φ is said to be an Orlicz function if , Φ is notidentically equal to zero, it is even and convex on the interval and left-continuous at .
And the convex modular by for any.
respectively. The Orlicz space equipped with the Luxemburg norm and the Orlicz norm isdenoted by and,respectively.
and define for all . Note that thefunctions andare convex. Moreover, the function isincreasing on for, but the function isincreasing on the interval only.
The Orlicz space equipped with the p-Amemiya norm will be denoted by.
It is obvious that for every and.
If the Orlicz function Φ vanishes only at zeroand, then the norm convergence and the modularconvergence are equivalent.
- 1.If, , then
If, then the p-Amemiya normis attained at every.
If, then the p-Amemiya normis attained at every.
Lemma 1.4 If the Orlicz function Φ vanishes only at zero,thenisorder continuous if and only if.
isreflexive if and only ifand.
Lemma 1.7 Let.hasa subspace isomorphic toif andonly if,where.
Proof Since the function isnondecreasing, exists.
then .This yields that the norm and theLuxemburg norm are equivalent. Since the p-Amemiya norm and the Luxemburg normare equivalent, has a subspace isomorphic to . □
Therefore, if , thespace has the Schur property, and it has the Opial property trivially because there is noweakly null sequence in . The case when is notinteresting if we consider the Opial modulus and weakly convergent sequence coefficient.For this reason we will assume that in thefollowing whenever the Opial modulus and the weakly convergent sequence coefficient areconsidered.
In this section we present some results on the Opial modulus. The obtained resultsextend the existing ones, which were presented by a number of papers studying thegeometry of Orlicz spaces endowed with the Luxemburg norm and the Orlicz norm,respectively. A formula for calculating the Opial modulus in Orlicz andMusielak-Orlicz spaces equipped with the Luxemburg or the Orlicz norm is found in[21–26].
Opial proved in  that () has thisproperty, but does not have it if,.
It is obvious that the uniform Opial property implies the Opial property.
It is easy to see that X has the uniform Opial property if and only if for any.
Theorem 2.1 If Φ is an Orlicz function,,,thendoesnot have the Opial property.
So .Therefore, does not have the Opial property. □
In the following we may assume that .
Proof Set .
which shows that the function is well defined.
Then ,,and as for any .
Since , there exists such that.
(as ). Then we have proved that weakly.
hence . By thearbitrariness of , wehave .
This is a contradiction.
this is also a contradiction. The two cases have shown that . □
Remark The main result presented in this paper generalizes the existing result tothe p-Amemiya norm. In the case that , the situationdegrades to the case of the classical Orlicz norm.
In the following we will consider the uniform Opial property for the Orlicz sequenceequipped with the p-Amemiya norm.
Let X be a Köthe sequence space with the semi-Fatouproperty and without order continuity of the norm.Then X does not have the uniform Opial property. Infact, we even have that for any, .
Theorem 2.4 Let,hasthe uniform Opial property if and only if.
But ,according to Lemma 1.2 and , this is a contradiction.
If , then does not have the order continuity property, by Lemma 2.3, the proof isfinished. □
In this section, our main aim is to calculate the weakly convergent sequence coefficientfor an Orlicz sequence space and further discuss the fixed point property of thisspace.
The weakly convergent sequence coefficient concerned with normal structure is animportant geometric parameter. It was introduced by Bynum  as follows.
For ,and . A formula for calculatingthe weakly convergent sequences of reflexive Orlicz and Musielak-Orlicz sequence spacesequipped with the Luxemburg or Amemiya norm is found in [21, 23], respectively.
Then ,so ,since ε is arbitrary, we have .
On the other hand, let in be an arbitrary asymptoticequidistant sequence such that weakly.
If , then , so .
- (2)If , set , then
hence we get again. Consequently . By thearbitrariness of in , it follows that. □
Theorem 3.2 Letand.hasweakly uniform normal structure if and only if.
This is a contradiction.
so . □
According to the above proof, we have the following.
Corollary 3.3 Let,and, then.
If , then .
- 2.If , then
Next, we discuss the fixed point property of .
Theorem 3.5and,,thencontainsan asymptotically isometric copy of.
for .It is obvious that P is linear.
which implies that contains an asymptotically isometric copy of . □
Theorem 3.6 Let,,thenhasthe fixed point property if and only if it is reflexive.
Proof Since a reflexive Banach space X with has the fixedpoint property, we only need to prove the necessity.
Suppose , then by Theorem 3.5,contains an asymptotically isometric copy of . Hencedoes not have the fixed point property.
Hence contains an asymptotically isometric copy of . ByTheorem 2 of , does not have the fixed point property. □
Theorem 3.7 If,,thendoesnot have the fixed point property.
Therefore, for any .
so we have , then P is an orderisometry of ontoa closed subspace of . □
This work was supported by the Provincial Education Department Fund (12531185) andpartly supported by the National Natural Science Foundation of China (61203191).
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