Uniformly Nonsquare in Orlicz Space Equipped with the Mazur-Orlicz -Norm

Abstract

The definition of uniformly nonsquareness in Banach spaces is extended to -normed spaces. Most of the results from this paper concern (uniformly) nonsquareness in the sense of James or in the sense of Schäffer in Orlicz spaces equipped with the Mazur-Orlicz -norm. It is well known that uniform nonsquareness in the sense of Schäffer and in the sense of James are equivalent in Banach spaces. In this paper, we found that uniform nonsquareness in the sense of James and in the sense of Schäffer are not equivalent for -normed spaces. Criteria for Orlicz spaces equipped with the Mazur-Orlicz -norm to be nonsquare and uniformly nonsquare in the sense of James or in the sense of Schäffer are given.

1. Introduction and Preliminaries

As well known, Orlicz space is a generalization of classical Lebesgue space. The theory of Orlicz space has important applications in control theory, fixed point theory, ergodic theory, probability theory, and theory of vector analytic function and has been intensively developed during the last decades. In 2018, Cui et al. discussed the monotonicity of Orlicz space that generated by the monotone continuous function equipped with Mazur-Orlicz -norm (see [1, 2]). In 2020, Bai et al. given criteria that Orlicz spaces that generated by the monotone function equipped with Mazur-Orlicz -norm have strictly monotonicity and upper locally uniform monotonicity, and they get the conclusion that for each and if and only if is convex function on . So, in order to studying geometric properties of Orlicz spaces equipped with the Mazur-Orlicz -norm, we need to assume that is convex see [3].

Inspired -convex spaces, in 1964, the definition of uniformly nonsquare in normed linear space was introduced by James (see [4]). In 1976, the concept of uniformly nonsquare in normed linear space was introduced by Schäffer (see [5]). A lot of results concerning with uniformly nonsquare in Banach space are known. Among the great number of papers concerning this topic, we list here a little [4–10]. Particularly, whether or not uniformly nonsquare Banach space has fixed point of nonexpansive mapping has been discussed as an open problem. Until 2005, Garca-Falset et al. solved the open problem and obtained that uniformly nonsquare Banach space has fixed point property (see [11]).

The aim of this paper is to give criteria that Orlicz spaces equipped with the Mazur Orlicz-norm are nonsquare and uniformly nonsquare in the sense of James or in the sense of Schäffer.

1.1. Introduction

Denoted by and the sets of natural and real numbers, respectively. Let . Given any real linear space , the functional is called an -norm if the following conditions are satisfied: (i) if and only if (ii) for all (iii) for all (iv) whenever and for any , in and and in

We say that is an -normed space if it is complete with respect to the -norm topology.

Definition 1. Let be a finite measure space and be the space of all (equivalence) classes of -measurable real-valued functions defined on . A function : is called an Orlicz function if for all , even, convex, and . Any Orlicz function determines a mapping defined by the formula called the modular. The order ideal for some in is called an Orlicz space.

The space is an -normed space with respect to the following lattice -norm, called the Mazur-Orlicz -norm [12]:

Definition 2. We say that satisfies -condition (, for short) there are constants and such that whenever .

Definition 3. Let be an Orlicz function, be the right derivative of , and be the right-inverse function of . Then, we call the complementary function of .

Definition 4. We say satisfies -condition (, for short) there exist and such that whenever . It is well known that if and only if .

Lemma 5. (see [13]).
if and only if there exist constants and such that whenever .

The characteristic of convexity of is defined by , where is the Clarkson modulus of convexity of . A Banach space is said to be uniformly nonsquare whenever . Since satisfies the definition of a finite tree, we can get that is superreflexive (see [4]). Now, we extend the definition of nonsquare and uniformly nonsquare to -normed space.

Definition 6. An -normed space is said to be nonsquare in the sense of James if for any it is verified that whenever .

Definition 7. An -normed space is said to be nonsquare in the sense of Schäffer if for any it is verified that whenever .

Definition 8. An -normed space is said to be uniformly nonsquare in the sense of James if for any there exists such that whenever .

Definition 9. An -normed space is said to be uniformly nonsquare in the sense of Schäffer if for any there exists such that whenever .

2. Main Results

Lemma 10. Suppose that . If there exists such that for any, then there exists which satisfy .

Proof. If there exists a sequence such that and , then thanks to , a contradiction.

Theorem 11. is nonsquare in the sense of Schäffer if and only if .

Proof. Necessity. Assume that . We show that there exists a strictly increasing sequence numbers such that for each , and We take and which satisfy . There exists a sequence of pairwise disjoint sets in , for some , such that , , . Let . Then, , and we can get the following On the other hand, for any , there exists such that whenever .
Combing (11) and (13), the following inequalities are found to be true: Similar to the selection method of sequence , the selection of sequence of pairwise disjoint sets in satisfies , . Supposing , we verity that and . According to the definition of -norm, we can obtain that . Furthermore, by , we deduce that which implies that . Namely contradiction with , the proof of necessity is completed.

Sufficiency. Assume on the contrary that is not nonsquare in the sense of Schäffer, then has elements satisfying and , for some . Using , we conclude that .

Due to the convexity of we have the following inequality

and then

Observing that the Orlicz function has the property , the following inequality can be obtained

Hence holds.

Since , the following formula can be obtained whenever .

From the above analysis,

Therefore, we obtain that

It means that , hence , a contradiction. The proof is complete.

Theorem 12. The following conditions are equivalent: (i) is uniformly nonsquare in the sense of James(ii) is nonsquare in the sense of James(iii)

Proof. It is obvious that . We only need to prove and .
. Assume that does not satisfy -condition. Since the idea is similar to the proof of Theorem 11, we can find, , , and . Construct such that and . Then, and and and for any . For convenience, denote and which shows that .
Otherwise, we let and ; then, , and . By the arbitrariness of , we obtain that and .
Then, by the definition of Mazur-Orlicz -norm, we conclude that It shows that is not a nonsquare in the sense of James. Namely, we have .
We prove the implication . Obviously, by the convexity of , we have For the sake of simplicity, we set . Next, we want to prove that the inequality holds.
Since , we have ,which implies that holds.
Without loss of generality, we may assume . Then, there exists which satisfies and as .
Since , for a fixed find such that for every , where . Namely, Since as , we can obtain . Then, there exists such that whenever .
Combining conditions (27) and (28) and , we derive a contradiction: where .
We get that . Hence, we can obtained that which shows that is uniformly nonsquare in the sense of James.

Theorem 13. is uniformly nonsquare in the sense of Schäffer if and only if and .

Proof. Since the necessity of is similar to the one given in Theorem 11, we only state the results without the proof. We only need to prove the necessity of -condition.
Suppose that -condition is invalid. There exists a strictly increasing sequence which satisfy and Without loss of generality, we may take with Let , it means that . Then, .
Divided into disjoint two subsets and with .
Suppose , we also have and .
Consequently, by the convexity of , we deduce that then the following inequality is found to be true To prove the reverse inequalities, notice that by (31) and (32), we can obtain the following inequality That is , it follows that Next, we will prove that . Otherwise, we assume that . Without loss of generality, we may suppose there exists , which satisfies the following inequality Take large enough for which whenever . By the convexity of and (37), this lead to a contradiction: By inequality (36), we have the following results It is easily to see that From (34) and (40), it follows that .
Using the method of the proof of , we can obtain that . This implies that is not uniformly nonsquare in the sense of Schäffer, that is .
Sufficiency. Assume for the contrary that is not uniformly nonsquare in the sense of Schäffer. We can find subsequences satisfying and and and for some .
Without loss of generality, we may take such that Since , there exists such that whenever .
Put The following equation can be obtained by integrating both sides of the above formula Since , there exists and such that whenever .
Hence, we have the following results Put Using , there exists such that whenever and .
Obviously, We can choose such that whenever .
Therefore, we have this shows that Define , then we derive that It follows that holds. Thanks to Lemma 10, there exists such that, which contradicts with the assumption and . So is uniformly nonsquare in the sense of Schäffer. The proof is complete.

Example 14. Put The derivative of the function is . Obviously, So, from the Lemma 1.3, we immediately have does not satisfy the -condition.
On the other hand, we can get that . Thus Then Using Lemma 1.3 in [6], we have satisfy the -condition, i.e., . Then, is uniformly nonsquare in the sense of James and is not uniformly nonsquare in the sense of Schäffer. The proof is complete.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for valuable comments and suggestions for improving this work. This work is supported by the National Natural Science Foundation of China under Grants 11871181 and the Natural Science Foundation of Heilongjiang Province under Grants A2018006.