Relative Uniform Convergence of Sequence of Functions Related to -Spaces Defined by Orlicz Functions

Abstract

The Orlicz function-defined sequence spaces of functions by relative uniform convergence of sequences related to -absolutely summable spaces are a new concept that is introduced in this article. We look at its various attributes, such as solidity, completeness, and symmetry. We also look at a few insertional connections involving these spaces.

1. Introduction

The spaces of all, convergent, null, -absolutely summable, and limited sequences are identified by the symbols , and . Sargent [1] established the space in 1960 after researching its many characteristics and determining its relationship to the sequence space . Following that, a large number of researchers looked into this sequence space from various angles. Tripathy and Sen [2] introduced the space which generalized the space ; they also examined other properties.

The term “Orlicz function” refers to a continuous, nondecreasing, convex function that has the following characteristics: , , for , and for .

If there is a constant such that for all values of , then an Orlicz function is considered to satisfy the -condition for all values of .

Numerous authors, such as Bhardwaj and Singh [3], have introduced the concept of lacunary sequences defined by the Orlicz function and satisfied some basic property; Bilgin [4] has studied on difference sequence defined by the Orlicz function; Gungor et al. [5] have introduced the -convergence sequences defined by the Orlicz function and have initiated their different property; Tripathy and Mahanta [6] have introduced the -space in the setting vector valued Sargent type sequences defined by the Orlicz function and established their different algebraic and topological property; and Parashar and Choudhary [7] have extended the -space introduced and investigated by Lindenstrauss and Tzafriri to the setting of Paranormed sequence spaces defined by the Orlicz function. This motivated others to study different types of new sequence spaces defined by the Orlicz function.

The modulus function, first presented by Nakano [8], is the function that results when replaces the convexity of the Orlicz function.

The concept of relative uniform convergence of a set of functions with respect to a scale function was initially put forward by Moore [9].

Later, Chittenden [10] proposed the notion of relative uniform convergence of a sequence of functions.

2. Definition and Background

2.1. Relative Uniform Convergence

Definition 1. For every tiny positive number , there is an integer such that for every , the inequality holds. This sequence of functions is represented by , defined on the compact domain .

Scale function describes the function of the previous equation.

Many others researchers like Demirci et al. [11], Demirci and Orhan [12], Sahin and Dirik [13], and Devi and Tripathy ([14–16]) have explored the idea further.

Presume that any subsets of natural number that limit the number of items to fall under the category of . stands for a sequence of real integers that does not decrease for any such that .

As established by Sargent [1], the sequence space is defined as follows:

The sequence space and the Orlicz function concepts were first introduced by Lindenstrauss and Tzafriri [17].

The -space with respect to the norm becomes an Orlicz sequence space, a Banach space. It is a little close to the -space, an Orlicz sequence space with the formula for .

The following sequence of functions are used in this article:

We introduced the sequence of function space in this article: where represents the scaling function on a compact region .

The space is normed by where the scale function is represented by .

Example 2. Let for all
Consider an Orlicz function defined by Let a sequence of function defined by The above sequence of functions is relative uniform convergence with respect to the scale function where Then, we get, for

3. Main Results

In this section, we formulate the hypothesis of the results of this article and establish them.

We state the following results without proof, which can be established using standard techniques.

Theorem 3. A normed linear space based on the norm established by the space is

Remark 4. The spaces and are normed spaces, normed by respectively, where is the scale function.

Theorem 5. The class of sequences , where , is a -normed space defined by where

Proof. First, we establish that is a normed space for . (1)Clearly, , if and only if the null operator, for all Hence, the null sequence of functions(2)We have, for and be any scalarLet Then, Therefore, As a result, where , is a normed space of type .
We now show that is a whole -normed space.
If is a Cauchy sequence in , then is a scale function for and exists over where
Let for each fixed
After that, there is a positive integer such that for a given , for every and
Taking varying of
Let then, we have from (20) for every and
Therefore, Since is complete, is convergence in for , with regard to the scaling function . Therefore, for any fixed is a Cauchy sequence in .
Let as
Now, we show that and as , for each, fixmeaning that for any related to the scale function
Hence, as is a linear space.
Also, Therefore, for , is the whole -normed space.

Theorem 6. The space if and only if

Proof. Suppose and
Then, Now, Hence,
Conversely, suppose that Next, suppose Then, there exists a natural sequence such that
Let .
Consequently, exists in such a way that Therefore,
Since, we arrive at a contradiction.
Hence, We make the following conclusion without providing any proof in light of the previous theorem.

Remark 7. Let , then

Proof. The previous remark’s proof is identical to that of Theorem 6.

Theorem 8. Suppose is an Orlicz function that satisfies the -condition.
Then, (1).(2)

Proof. (1)Let Consequently, there is such that, Let and with such that for
Now, where the summation is over and the summation is over
Since is continuous, then we have For we use the fact that Because is convex and has a nondecreasing nature, There exists such that, given that meets the requirement of , Hence, From (38) and (39), it follows that .
Hence,
Let
Therefore, and
Consequently, there are and such that Let then, Therefore,
Hence,

Theorem 9. is a solid space.

Proof. Let . Consider for all , where Then, Hence,
It follows that is solid.

We make the subsequent conclusion without providing any proof in the light of Theorem 9.

Remark 10. For the whole number , the space is solid.

Regarding the description of -space, we arrive at the following conclusion.

Remark 11. The space is symmetric.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was funded by the University of Jeddah, Jeddah, Saudi Arabia, under grant no. (UJ-23-DR-127). The authors, therefore, acknowledge with thanks the University of Jeddah for its technical and financial support.