Abstract
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.
Similar content being viewed by others
1 Introduction and preliminaries
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[1]. The uniform Opial property withrespect to the weak topology was defined in [2]and Opial modulus was introduced in [3]. 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[8].
The coefficient was introduced by Bynum [10], 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 [4]), also see the work by Sims and Smyth [11].
The notion of p-Amemiya norm was introduced by Cui and Hudzik in [12], 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[14]. 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.
For any map ,define
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 .
For every Orlicz function Φ, we define its complementary function by theformula
And the convex modular by for any.
The Orlicz sequence space is defined as the set
The Luxemburg norm and the Orlicz norm are expressed as
and
respectively. The Orlicz space equipped with the Luxemburg norm and the Orlicz norm isdenoted by and,respectively.
For any and,define
and define for all . Note that thefunctions andare convex. Moreover, the function isincreasing on for, but the function isincreasing on the interval only.
Let . For any, define the p-Amemiya norm by theformula
The Orlicz space equipped with the p-Amemiya norm will be denoted by.
It is known that and. If and,
(see [12]).
Let bethe right-hand side derivative of Φ on and put . Define the function by
and the functions , by
It is obvious that for every and.
Set .
Definition 1.3[17]
We say that an Orlicz function Φ satisfies the -condition (, for short) if there exist constants andsuch that
For more details about Orlicz spaces, we refer to [13, 14, 18, 19].
Lemma 1.1[14]
Let.Thenforallwith.
Lemma 1.2[17]
If the Orlicz function Φ vanishes only at zeroand, then the norm convergence and the modularconvergence are equivalent.
Lemma 1.3[12]
For everyandeach, the following conditions hold.
-
1.
If, , then
-
2.
If, then the p-Amemiya normis attained at every.
-
3.
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.
Lemma 1.5 Assume, . Then,for anyand, thereexistssuchthat
Proof Let
Then , since. Without loss of generality, we assume and.Set .Since the modular convergence implies the norm convergence, so we can find suchthat implies . Thus, applying theconvexity of Φ and Lemma 1.1, we have
Replacing x, y by ,−y, respectively, in the above inequalities, we have
□
Lemma 1.6[20]
isreflexive if and only ifand.
Lemma 1.7 Let.hasa subspace isomorphic toif andonly if,where.
Proof Since the function isnondecreasing, exists.
If , by thecontinuity of , thereexists such that
Then .
For any ,we have , then (),therefore,
we have
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.
2 Opial modulus for Orlicz sequence spaces
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].
Definition 2.1[1]
We say that a Banach space X has the Opial property if for any weakly nullsequence in X and any there holds
Opial proved in [1] that () has thisproperty, but does not have it if,.
Definition 2.2[6]
We say that X has the uniform Opial property if for any thereexists such thatfor any with and any weakly null sequence in the unit sphere of X there holds
It is obvious that the uniform Opial property implies the Opial property.
Definition 2.3[2]
The Opial modulus of X is denoted by and itis defined for by the formula
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.
Proof Divide ℕ into a sequence of pairwise disjoint and infinite subsetsof ℕ such that as and define
Then the sequence is weakly convergent to zero. For any, we have
and for any ,
So .Therefore, does not have the Opial property. □
In the following we may assume that .
Theorem 2.2 Let Φ be an Orlicz functionsatisfying, . Then forany, we have
Proof Set .
(1) For fixed , consider the function
We have F is continuous on . Since
we have for any , thus
Moreover, , then thereexists unique such that
which shows that the function is well defined.
(2) Now, we will show that for any . For any , there exist with , and such that . Put
Then ,,and as for any .
Since , there exists such that.
For fixed ,due to ,we have
So, for any and which is small enough, we get
Then .
Take any ,there exists satisfying . Since
as well as the Young inequality, we have
(as ). Then we have proved that weakly.
Since x and havedisjoint supports, we get
hence . By thearbitrariness of , wehave .
(3) Assume that for some . Then there exists such that , so there exist and in such that , weakly and
For fixed ,there exists such that
for large enough. Set
And define
then (for all ), ,.For n large enough, we have
Case 1. is infinite. For any , we take. Then
This is a contradiction.
Case 2. is infinite. For any ,
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.
If , then
In the following we will consider the uniform Opial property for the Orlicz sequenceequipped with the p-Amemiya norm.
Lemma 2.3[25]
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.
Proof If and does not have the uniform Opial property, then there exists some such that. Therefore, for anysequence such that ,there are two sequences and in with ,and such that
Since
then
Thus we obtain ,then ,which means that the sequence is bounded. Then
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. □
3 Weakly convergent sequence coefficient for Orlicz sequence spaces
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.
Let X denote a reflexive infinite dimensional Banach space, without Schurproperty automatically. For each sequence in X, we define the asymptoticdiameter and asymptotic radius respectively by
The weakly convergent sequence coefficient concerned with normal structure is animportant geometric parameter. It was introduced by Bynum [10] as follows.
is the supremum of the set of all numbersM with the property that for each weakly convergent sequence with asymptotic diameter A, thereis some y in the closed convex hull of the sequence such that
A sequence in X is said to be asymptoticequidistant if
In this paper, we use the following equivalent definition of . This definition was introduced in [27], where it was proved that
It is obvious that (see[10]). A Banach space X is saidto have weakly uniform normal structure provided . See[11] for further information about thiscoefficient.
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.
Theorem 3.1 If, ,then
Proof Let . For any , thereexists such that
Then there exist andsuch that .Define
where have pairwise disjoint supports. Then() for all and all.According to the same method as in the proof of Theorem 2.2, we have weakly,
Then ,so ,since ε is arbitrary, we have .
On the other hand, let in be an arbitrary asymptoticequidistant sequence such that weakly.
Since , then by Lemma 1.5 for,there exists such that
Let and pick such that and choosesuch that . By as for ,we can find with such that . And so on,by induction, we find the sequence and of natural numbers with ,satisfying
Take .
-
(1)
If , then , so .
-
(2)
If , set , then
Therefore .
(3) If , 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.
Proof Necessity. If , by Theorem 3.1,
Assume , then for anythere exist and satisfying.By and , there exists such that. Hence,
This is a contradiction.
Sufficiency. If not, , then for any thereexists such that
We can find such that
Take satisfying . Then. Set
Then and is an asymptoticequidistant sequence. And because ,we have weakly. Finally,,then
which implies that
so . □
According to the above proof, we have the following.
Corollary 3.3 Let,and, then.
Remark In the case that , the situationdegrades to the case of classical Orlicz norm.
-
1.
If , then .
-
2.
If , then
Corollary 3.4
Next, we discuss the fixed point property of .
Theorem 3.5and,,thencontainsan asymptotically isometric copy of.
Proof If , then there exist the sequence and such that
Set
Define by
for .It is obvious that P is linear.
For any ,since , then there exists such that for all,hence
which implies .Moreover, for any ,we have
then .
On the other hand, for any , there exists such that
where (), then
and
Therefore,
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.
Suppose , then there exists such that for any , and for everysequence decreasing to 0, thereexist such that
Set ,then there exists such that.Hence, for any ,we have
Hence contains an asymptotically isometric copy of . ByTheorem 2 of [28], does not have the fixed point property. □
Theorem 3.7 If,,thendoesnot have the fixed point property.
Proof If , forany , define , then
Therefore, for any .
Moreover, we can prove that .Hence, define by
for all . Thenthe operator P is obviously linear, since
so we have , then P is an orderisometry of ontoa closed subspace of . □
References
Opial Z: Weak convergence of the sequence of successive approximations of nonexpansivemappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0
Prus S: Banach spaces with the uniform Opial property. Nonlinear Anal. 1992, 18: 697–704. 10.1016/0362-546X(92)90165-B
Lin PK, Tan KK, Xu HK: Demiclosedness principle and asymptotic behavior for asymptotically nonexpansivemappings. Nonlinear Anal. 1995, 24: 929–946. 10.1016/0362-546X(94)00128-5
Ayerbe JM, Benavides TD, Acedo GL: Measures of Noncompactness in Metric Fixed Point Theory. Birkhäuser, Basel; 1997.
Kirk WA, Sims B (Eds): Handbook of Metric Fixed Point Theory. Kluwer Academic, Dordrecht; 2001.
Krasnoselskii MA, Rutickii YB: Convex Functions and Orlicz Spaces. Noordhoff, Groningen; 1961.
Goebel K, Kirk WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.
Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings. Dekker, New York; 1984.
Gornicki J: Some remarks on almost convergence of the Picard iterates for nonexpansivemappings in Banach spaces which satisfy the Opial condition. Comment. Math. 1988, 29: 59–68.
Bynum WL: Normal structure coefficients for Banach spaces. Pac. J. Math. 1980, 86: 427–436. 10.2140/pjm.1980.86.427
Sims B, Smyth A: On some Banach space properties sufficient for weak normal structure and theirpermanence properties. Trans. Am. Math. Soc. 1999, 351(2):497–513. 10.1090/S0002-9947-99-01862-0
Cui YA, Duan LF, Hudzik H, Wisla M: Basic theory of p -Amemiya norm in Orlicz spaces ( ): extremepoints and rotundity in Orlicz spaces endowed with these norms. Nonlinear Anal. 2008, 69: 1797–1816.
Cui YA, Hudzik H, Li JJ, Wisla M: Strongly extreme points in Orlicz spaces equipped with the p -Amemiyanorm. Nonlinear Anal. 2009, 71: 6343–6364. 10.1016/j.na.2009.06.085
Cui YA, Hudzik H, Wisla M, Wlazlak K: Non-squareness properties of Orlicz spaces equipped with the p -Amemiyanorm. Nonlinear Anal. 2012, 75: 3973–3993. 10.1016/j.na.2012.02.014
Chen LL, Cui YA: Complex extreme points and complex rotundity in Orlicz function spaces equippedwith the p -Amemiya norm. Nonlinear Anal. 2010, 73: 1389–1393. 10.1016/j.na.2010.04.071
Chen LL, Cui YA: Complex rotundity of Orlicz sequence spaces equipped with the p -Amemiyanorm. J. Math. Anal. Appl. 2011, 378: 151–158. 10.1016/j.jmaa.2011.01.039
Chen ST Dissertationes Mathematicae 356. In Geometry of Orlicz Spaces. Istitute of Mathematics, Warszawa; 1996.
Maligranda L Seminars in Math. 5. In Orlicz Spaces and Interpolation. Universidade Estadual de Campinas, Campinas; 1989.
Musielak J Lecture Notes in Math. 1034. In Orlicz Spaces and Modular Spaces. Springer, Berlin; 1983.
Li XY, Cui YA: The dual space of Orlicz space equipped with p -Amemiya norm. J. Harbin Univ. Sci. Technol. 2011, 16(1):110–112.
Cui YA: Weakly convergent sequence coefficient in Köthe sequence spaces. Proc. Am. Math. Soc. 1998, 126: 195–201. 10.1090/S0002-9939-98-03483-2
Cui YA, Hudzik H: On the uniform Opial property in some modular sequence spaces. Funct. Approx. Comment. Math. 1998, 26: 93–102.
Cui YA, Hudzik H, Zhu HW: Maluta’s coefficient of Musielak-Orlicz sequence spaces equipped with Orlicznorm. Proc. Am. Math. Soc. 1998, 126: 115–121. 10.1090/S0002-9939-98-03839-8
Cui YA, Hudzik H: Maluta’s coefficient and Opial’s properties in Musielak-Orliczsequence spaces equipped with the Luxemburg norm. Nonlinear Anal. 1999, 35: 475–485. 10.1016/S0362-546X(97)00695-0
Cui YA, Hudzik H, Yu FF: On Opial properties and Opial modulus for Orlicz sequence spaces. Nonlinear Anal. 2003, 55: 335–350. 10.1016/S0362-546X(03)00213-X
Yao H, Wang T: Maluta’s coefficient of Musielak-Orlicz sequence spaces. Acta Math. Sin. Engl. Ser. 2005, 21: 699–704. 10.1007/s10114-004-0429-9
Zhang GL: Weakly convergent sequence coefficient of product space. Proc. Am. Math. Soc. 1992, 117(3):637–643.
Dowling PN, Lennai CJ, Turett B: Reflexivity and fixed-point property for nonexpansive maps. J. Math. Anal. Appl. 1996, 200: 653–662. 10.1006/jmaa.1996.0229
Acknowledgements
This work was supported by the Provincial Education Department Fund (12531185) andpartly supported by the National Natural Science Foundation of China (61203191).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authorsread and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
He, X., Cui, Y. & Hudzik, H. The fixed point property of Orlicz sequence spaces equipped with thep-Amemiya norm. Fixed Point Theory Appl 2013, 340 (2013). https://doi.org/10.1186/1687-1812-2013-340
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-340