Some Applications of New Complex Function Space Constructed by Different Weights and Exponents

Abstract

In this article, we develop and study a new complex function space formed by varying the weights and exponents under a definite function. We investigate the geometric and topological characteristics of mapping ideals created using -numbers and this complex function space. Also, the action of shift mappings on this complex function space has been discussed. Finally, we introduced an extension of Caristi’s fixed point theorem on it.

1. Introduction

Numerous researchers are attempting to extend the Banach fixed point theorem [1] in a realistic manner. Kannan [2] recognized a subclass of mappings that execute the same fixed point operations as contractions but are not continuous. Ghoncheh [3] pioneered the study of Kannan mappings in modular vector spaces. Lebesgue spaces with variable exponents, , include Nakano sequence spaces. Across the second half of the twentieth century, it was thought that these variable exponent spaces offered an adequate framework for the mathematical components of a variety of problems for which the traditional Lebesgue spaces were inadequate. Due to the importance of these areas and their consequences, they have developed a reputation as an effective instrument for resolving a wide variety of problems; presently, the study of spaces is a developing field of research, with implications reaching across a broad range of mathematical disciplines [4]. The investigation of variable exponent Lebesgue spaces was accelerated further by the mathematical description of non-Newtonian fluid hydrodynamics [5, 6]. Non-Newtonian fluids, also known as electrorheological fluids, have a wide range of applications in a number of fields ranging from military science to civil engineering to orthopedics and beyond. Mapping ideal theory has a diverse range of applications in Banach space geometry, fixed point theory, spectral theory, and other areas of mathematics, as well as other fields of knowledge (for further information, see [7–13]). Bakery and Mohamed [14] studied the notion of a pre-quasi norm on Nakano sequence space with a variable exponent in the range . They explored the conditions under which it generates pre-quasi Banach and closed space when endowed with a particular pre-quasi norm as well as the Fatou property of various pre-quasi norms on it. Additionally, they showed the existence of a fixed point for Kannan pre-quasi norm contraction mappings on it as well as on the pre-quasi Banach operator ideal formed from this sequence space’s -numbers. In [15], they investigated some fixed points results of Kannan non-expansive mappings on generalized Cesàro backward difference sequence space of non-absolute type.

We will mark the complex and non-negative integers as and , respectively. By , we denote the space of all complex functions with complex variable. Assuming that , Bakery and El Dewaik [16] defined the following function space:where

They studied several of the topological and geometric properties for and even a pre-quasi ideal construction based on the and -numbers. Upper bounds for -numbers of infinite series of the weighted -th power forward shift operator on were also introduced for some entire functions. Further, they evaluated Caristi’s fixed point theorem in . For extra information on formal power series spaces and their behaviors, see [17–20]. We denote the space of every, finite rank, approximable, and compact bounded linear mappings from a Banach space into a Banach space by , , , and , and if , we mark , , , and , respectively. The ideal of all, finite rank, approximable, and compact mappings are denoted by , , , and . We will indicate the sequence of -numbers, approximation numbers, and Kolmogorov numbers for any bounded linear mapping by , , and . The mapping ideals constructed by the sequence of -numbers, approximation numbers, and Kolmogorov numbers in sequence space are marked by , , and . For any Banach spaces and , we will use the following notations.

Notations 1. (see [16])The purpose of this study is straightforward, as follows. In Section 3, we introduce and investigate the complex function space under the definite function . In Section 4, the mapping ideals constructed by -numbers and are presented. We have studied their geometric and topological properties. Specifically, we explore in Section 5 the upper limits of -numbers for infinite series of the weighted -th power forward and backward shift mapping on and their applications to various entire functions. Finally, in Section 6, we present an extension of Caristi’s fixed point theorem in .

2. Definitions and Preliminaries

Let , , , and denote the spaces of each, bounded, -absolutely summable, and null sequences of real numbers, respectively.

Definition 1. (see [16]). The function space is called a special space of formal power series (or in short ssfps), if it shows the following settings:(1), for all , where .(2)If and , for all , one has .(3)Suppose , then , where and marks the integral part of .

Theorem 1 (see [16]). is a mapping ideal, when is a ssfps.

We denote the space of finite formal power series by , i.e., if , one has with . Also, indicates the zero function of .

Definition 2. (see [16]). A subspace of the ssfps is said to be a pre-quasi normed ssfps, if there is a function which verifies the next conditions:(i)For , we have and .(ii)Suppose and , then there are with .(iii)Let ; then, there are such that .

Recall that if the space is complete, then is called a pre-quasi Banach ssfps.

Definition 3. (see [16]). A subspace of the ssfps is called a pre-modular ssfps, if there is a function which verifies the next conditions:(i)For , we have and .(ii)Suppose and , then there are with .(iii)Let ; then, there are such that .(iv)Suppose , for every ; then, .(v)There are so that .(vi).(vii)One has with , where .

Theorem 2 (see [16]). Every pre-modular ssfps is a pre-quasi normed ssfps.

Definition 4. (see [21]). A function is called an -number, if the sequence , for any , satisfies the following setup:(a)If , then .(b), for every , , .(c)The inequality holds, if , , and ; suppose that and are any two Banach spaces.(d)For and , then .(e)Suppose ; then, , whenever ,(f)Assume that represents the unit map on the -dimensional Hilbert space ; then, or .

The following are some instances of -numbers:(i)The k-th approximation number, , is presented as(ii)The k-th Kolmogorov number, , is presented as

Lemma 1 (see [7]). Assume that and , and we have maps and with , for each .

Definition 5. (see [7]). A Banach space is named simple if contains one and only one non-trivial closed ideal.

Theorem 3 (see [7]). Suppose is a Banach space with , and we have

Definition 6. (see [7]). A class is said to be a mapping ideal if every component satisfies the next setups:(i).(ii) is linear space on .(iii)Assume , , and ; then, .

Definition 7. (see [10]). A function is called a pre-quasi norm on the ideal if it satisfies the following setups:(1)Suppose , , and .(2)There is with , for all and .(3)One has so that , for every .(4)We get so that if , , and , then , where and are normed spaces.

Theorem 4 (see [10]). Every quasi norm is a pre-quasi norm on the same ideal.

With finite non-zero coordinates, we denote the space of every sequence by .

Theorem 5 (see [22]). Suppose type . If is a mapping ideal, we have(1)type .(2)Assume type and type ; then, type .(3)If and type , then type .(4) is solid, i.e., if type and , for all and , then type .

By card , we denote the number of elements of .

Lemma 2 (see [23]). Suppose is a bounded family of . Hence,

We will apply the next inequality [24]. For all and , we havewhere and .

Definition 8. (see [16]). Assume is a pre-quasi normed ssfps. A mapping is called forward shift, if , for all , where and .

Definition 9. (see [16]). Suppose is a pre-quasi normed ssfps. A mapping is called backward shift, if , for all , where and .

Definition 10. (see [20]). If , then .

Definition 11. (see [20]). If , then .

3. Pre-Modular ssfps

This section contains the space’s definition under the function , where , for all . We offer enough setups on to become pre-modular ssfps, which implies that is a pre-quasi Banach ssfps.

Let , and we define the following function space:

Theorem 6. If , then

Proof.

Hereafter, we will denote the space of all monotonic decreasing and monotonic increasing sequences of positive reals by and , respectively.

Theorem 7. is a ssfps, if it verifies the next setups:(a1).(a2), or with so that .

Proof.  (1-i) Assume ; then, and . We have . Since is bounded, we get and then . (1-ii) Let and . We have . Since is bounded, we have Then, . Therefore, by using components (1-i) and (1-ii), is linear. Clearly, , for all , where and .(2)Let , for all and . Then, . Since , for all , then Hence, and . Therefore, .(3)Let , be an increasing sequence, and there exists such that and is increasing. Therefore, and . One hasThis implies that and . Hence, .

Theorem 8. Let conditions (a1) and (a2) be satisfied; then, the space is a pre-modular Banach ssfps.

Proof. (i)Evidently, for all , then and .(ii)We have , for all and , for such that for every .(iii)For some , we obtain for all .(iv)It is clear from the proof part (2) of Theorem 7.(v)From the proof part (3) of Theorem 7, we have that .(vi)It is apparent that .(vii)There is with such that , for each and , if .Therefore, the space is pre-modular ssfps. To show that is a pre-modular Banach ssfps, suppose is a Cauchy sequence in ; then, for all , there is such that, for all , we getFor and , we obtainHence, is a Cauchy sequence in , for fixed , so , for fixed . Therefore, , for each . Finally, to explain that , we haveSo, . This implies that is a pre-modular Banach ssfps.

Taking into consideration Theorem 2, we put forward the following theorem.

Theorem 9. Let conditions (a1) and (a2) be satisfied. Then, the space is a pre-quasi Banach ssfps.

Theorem 10. Let conditions (a1) and (a2) be satisfied. Then, the space is a pre-quasi closed ssfps.

Proof. Assume that the setups are verified. From Theorem 9, the space is a pre-quasi normed ssfps. To show that is a pre-quasi closed ssfps, assume and ; then, for every , there is such that for all , one hasHence, for and , we getSo, is a convergent sequence in , for fixed . Therefore, , for fixed . Finally to prove that , we considerso . This finishes the proof.

4. Pre-Quasi Ideal

In this section, the mapping ideals constructed by -numbers and are presented. We have studied their geometric and topological structures. We will use the notation for , that is, , , and .

In view of Theorems 1 and 7, we conclude the next theorem.

Theorem 11. Let conditions (a1) and (a2) be satisfied. Then, is a mapping ideal.

4.1. Ideal of Finite Rank Mappings

In this section, enough setups (not necessary) on so that is dense in are investigated. This explains the non-linearity of the type spaces (Rhoades open problem [25]).

Example 1. The sequence satisfies and , for some .

Theorem 12. , whenever conditions (a1) and (a2) are satisfied.

Proof. It is clear that , since the space is a mapping ideal. Currently, we substantiate that . On taking , , with . Hence, , and assume , so there is such that , for some . While is decreasing, we getHence, there exist , , andSince is bounded,Let be monotonically increasing such that there exists a constant for which . Then, we have for thatSince , then , where . Since is increasing, inequalities (2)–(5) giveSince which gives a counter example of the converse statement, this finishes the proof.

According to Theorem 12, if (a1) and (a2) are fulfilled, then every compact mapping is represented by finite rank mappings; however, the reverse is not necessarily true.

4.2. Closed and Banach

In this part, we have investigated the sufficient conditions on such that the pre-quasi mapping ideal is Banach and closed.

Theorem 13. If and are Banach spaces and conditions (a1) and (a2) are satisfied, then the function is a pre-quasi norm on .

Proof. Suppose the conditions are verified, so verifies the next setups:(1)Let , then we have , and it is clear that , if and only if, , for all , if and only if, .(2)We have with , for every and .(3)One has for . Therefore, and . Therefore, for , one can see that(4)We have ; suppose , , and . Therefore, . Then, for , one can see that

Theorem 14. If and are Banach spaces and conditions (a1) and (a2) are satisfied, then is a pre-quasi Banach mapping ideal.

Proof. Suppose the conditions are verified, then the function is a pre-quasi norm on . Let be a Cauchy sequence in . Therefore, and . Assume ; then, by using conditions (iv) and (vii) of Definition 3 and since , we getThus, is a Cauchy sequence in . While the space is a Banach space, there exists with and since , for each , using Theorem 13 and the continuity of at , we obtainThus, we have ; then, .

Theorem 15. If and are Banach spaces and conditions (a1) and (a2) are satisfied, then is a pre-quasi closed mapping ideal.

Proof. Suppose the conditions are verified; then, the function is a pre-quasi norm on . Assume , with and . Therefore, and . Suppose ; then, from conditions (iv) and (vii) of Definition 3 and since , we getand then is a convergent sequence in . While the space is a Banach space, there exists with and since , for each , using Theorem 13 and the continuity of at , one can see thatand we have ; then, .

We deduce the following characteristics of the type using Theorem 5.

Theorem 16. For type , the following holds:(1)We have type .(2)If type and type , then type .(3)For all and type , then type .(4)The type is solid.

4.3. Smallness

We give here some inclusion relations concerning the space for different and .

Theorem 17. If and are Banach spaces with , , , for all , and setups (a1) and (a2) are satisfied, it is true that

Proof. Suppose . Therefore, and . One can see thathence . Next, if we take with , then and . Clearly, . By choosing with , then and . This finishes the proof.

In this part, we investigate the setups for which is small.

Theorem 18. If and are Banach spaces with , assume that the conditions (a1), (a2), and are satisfied, and hence is small, where .

Proof. Let . Therefore, one gets so that , for every . According to Dvoretzky’s theorem [26] with , there are quotient spaces and subspaces of that mapped onto by isomorphisms and with and . Let be the identity mapping on , be the quotient mapping from onto , and be the natural embedding mapping from into . Let , for all , be the Bernstein numbers [27]; we have thenfor . We have so thatAs , then . This contradicts . Therefore, and . Hence, the space is small.

By the same manner, we can easily conclude the next theorem.

Theorem 19. If and are Banach spaces with , assume that conditions (a1), (a2), and are satisfied, and hence is small, where .

4.4. Simpleness

We introduce an answer of the next question; for which , is the space simple?

Theorem 20. If , verify and , for all , and the setups (a1), (a2) are satisfied, then

Proof. Suppose there is and . According to Lemma 1, we can find and with . For every , one hasThis contradicts Theorem 17.

Corollary 1. If , verify and , for all , and the setups (a1), (a2) are satisfied, then .

Proof. It is clear from .

Theorem 21. If setups (a1), (a2) are satisfied with , then the space is simple.

Proof. Assume and . From Lemma 1, one has so that . We have . Therefore, . This implies that there is one non-trivial closed ideal in .

4.5. Spectrum

In this part, we expound the sufficient conditions on such that equals .

Theorem 22. If and are Banach spaces with and suppose setups (a1), (a2) are satisfied and , then

Proof. Let , and hence , where with , and , for all . We have , with , so , with . Therefore, , so .
Secondly, assume . Therefore, . Hence, we haveSince , then . Assume exists, for every . Therefore, . So, exists and is bounded. From the pre-quasi mapping ideal of , we obtainThis contradicts . Therefore, , for every . This gives . This provides the proof.

5. Application of Shift Mappings on

Specifically, we explore the upper limits of -numbers for infinite series of the weighted r-th power forward and backward shift mapping on and their applications to various entire functions in this section, where , for all .

Theorem 23. Let conditions (a1) and (a2) be satisfied, , and ; then, with .

Proof. Assume that the conditions are satisfied. For , since is increasing and bounded from above with , for all , thenThis gives with . Since , then there is with , for all . Hence, , and one gets . This completes the proof.

Theorem 24. Let conditions (a1) and (a2) be satisfied, , and ; then, with .

Proof. Assume the conditions are satisfied. For , since is increasing and bounded from above with , for all , thenThis gives with . Since , then there is with , for all . Hence, , and one gets . This completes the proof.

By , we denote the open unit disc in .

Theorem 25. Let conditions (a1) and (a2) be satisfied with . If , then every function in is analytic on . Furthermore, the convergence in implies the uniform convergence on , where is compact.

Proof. Let , and . Then, , with and . Therefore, . This givesAs , one gets , with . Hence, , with . Assume , with . Suppose , where , and we havewhere is increasing and bounded with and , for all . Clearly, ; then, . So, .

Theorem 26. If is the forward shift mapping on , we havewhere .

Proof. Let card and , for all , for which with and . Therefore, and .
Let be a mapping on with rank defined bySince , this gives . Define a mapping by , and we haveThis implies that , where . Then, the identity mapping will be , and from the definition of -numbers, we haveSince for all card , the last inequality is verified, so one can see thatIn contrary, let card , where . Define the mapping as . From the definition of approximation numbers, we haveSince for all card , the last inequality holds and by using Lemma 2, one hasThis completes the proof.

Theorem 27. If is the backward shift mapping on , thenwhere .

Proof. Assume card and , for every , where with and . Therefore, and .
Suppose is a mapping on with rank evident byAs . This implies that . Define a mapping by , and one getsTherefore, , where . Hence, the identity mapping will be , and in view of the definition of -numbers, one hasSince for every card , the last inequality is confirmed, and one obtainsIn contrary, let card , where . Define the mapping as . From the definition of approximation numbers, one getsSince for all card , the last inequality holds, and by using Lemma 2, one hasThis finishes the proof.

Theorem 28. If conditions (a1) and (a2) are satisfied with , let be a shift mapping on the space and ; then,

Proof. For , we have . One hasSince satisfies the triangle inequality, we have

Theorem 29. If conditions (a1) and (a2) are satisfied with , let be a shift mapping on the space and ; then,

Proof. Suppose , and one has . We haveAs verifies the triangle inequality, one can see that

Theorem 30. If conditions (a1) and (a2) are satisfied with , let be a shift mapping on ; then, the -numbers of this mapping are given by

Proof. Let card , where . Define the mapping as . Since the triangle inequality holds by , we haveAs for all card , the last inequality is verified, and one hasThis completes the proof.

Theorem 31. If conditions (a1) and (a2) are satisfied with , let be a shift mapping on ; then, the -numbers of this mapping are given by

Proof. Let card , where . Define the mapping as . Since the triangle inequality holds by , one getsAs for all card , the last inequality is verified, and one hasThis completes the proof.

The following theorems are direct actions of Theorem 30 and Definition 10.

Theorem 32. If conditions (a1) and (a2) are satisfied with , let be a shift mapping on and . The upper estimation of the -numbers of is given by

Theorem 33. If conditions (a1) and (a2) are satisfied with , let be a shift mapping on and . The upper estimation of the -numbers of is given by

The following theorems are direct actions of Theorem 31 and Definition 11.

Theorem 34. If conditions (a1) and (a2) are satisfied with , then the mapping on holds the following inequality:

Theorem 35. If conditions (a1) and (a2) are satisfied with and the mapping is defined on , then the upper estimation of the -numbers of is given by

6. Caristi’s Generalization of Fixed Point Theorem

In modular spaces, the Ekeland variational principle [28] cannot be applied because the modular does not really prove the triangle inequality. In this part, we consider an extension of Caristi’s fixed point theorem in in light of Farkas [28].

Definition 12. (a)The pre-quasi normed ssfps on is called -convex, if , for each and .(b) is -convergent to , if and only if, . If the -limit exists, then it is unique.(c) is -Cauchy, when .(d) is -closed, if for all -converging to , and hence .(e) is -bounded, when .(f)The -ball of radius and center , for every , is defined as(g)A pre-quasi normed ssfps on satisfies the Fatou property, if for any sequence with and any ,

Consider the fact that the -closedness of the -balls is determined by the Fatou property.

Theorem 36. Suppose setups (a1) and (a2) are satisfied; then, , for all , holds the Fatou property.

Proof. Assume the setups are fulfilled and with . Since the space is a pre-quasi closed space, then . Then, for any , one can see that

Theorem 37. The function , for all , does not satisfy the Fatou property, if setups (a1) and (a2) are satisfied with .

Proof. Let the conditions be fulfilled and with . Since the space is a pre-quasi closed space, then . Then, for any , we haveHence, does not satisfy the Fatou property.

Example 2. The space of functions is a pre-quasi normed ssfps, not quasi normed ssfps, and not a normed ssfps, where , for all .

Example 3. The space of functions , with , is a pre-quasi normed ssfps, quasi normed ssfps, and not a normed ssfps, where , for each .

Example 4. The space of functions is a pre-quasi normed ssfps, a quasi normed ssfps, and a normed ssfps, where , for all .

Definition 13. The function is said to be lower semicontinuous at if , for which denotes ’s neighborhood system.

Definition 14. The function is said to be proper, when

Theorem 38. If and is a -closed subset of , with , for all , and is a proper, -lower semicontinuous function with , assume that , , and with . So, we have which -converges to few , under the following conditions:(i), for every .(ii).(iii)When , then .

Proof. Set . Since , then . As is -lower semicontinuous, satisfies the Fatou property, and is -closed, we have that is -closed. Select withNext setSimilar to , one has and -closed. Suppose that we have built and . After that, select withSupposeTherefore, we construct the sequences and by induction. For constant , assume . One can see thatwhich givesSince is decreasing with , for each , one hasfor each , which gives that is Cauchy. Since is Banach space, has limits and satisfies. As , one haswhich implies that is decreasing. After that, assume . So, we get for which , for each , i.e.,As , with , one can see thatAs in the previous inequality, one getsThis implies thatThis finishes the proof.

We discuss the concept of Caristi’s fixed point theorem in using Theorem 38.

Theorem 39. If and is a -closed subset of , under , with , let and with . is a mapping and there is a function which is a proper and -lower semicontinuous under and(1), for any .(2), for any .Hence, there is a fixed point of in .

Proof. As , we have that is proper, bounded from below, and -lower semicontinuous. If , one hasAs , there is with . From Theorem 38, there is which -converges to few , withfor all . Suppose that , and one haswhich givesFrom condition (6), one can see thatInequality (6) givesWe have a contradiction. Hence, . This completes the proof.

Data Availability

No data were used to support this study.

Ethical Approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

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

Acknowledgments

This study was funded by the University of Jeddah, Saudi Arabia, under grant no. UJ-20-084-DR. The authors, therefore, acknowledge with thanks the University’s technical and financial support.