Kannan Contraction Operator on the Domain of -Cesàro Matrix in and Related Prequasi Ideal with an Application of Nonlinear Difference Equations

Abstract

In this article, the sequence space has been built by the domain of -Cesàro matrix in Nakano sequence space , where and are sequences of positive reals with , and , with . Some topological and geometric behavior of , the multiplication maps acting on , and the eigenvalues distribution of operator ideal constructed by and -numbers have been examined. The existence of a fixed point of Kannan prequasi norm contraction mapping on this sequence space and on its prequasi operator ideal are investigated. Moreover, we indicate our results by some explanative examples and actions to the existence of solutions of nonlinear difference equations.

1. Introduction

As a remark of constant Lebesgue spaces, variable exponent Lebesgue spaces go again many years, and in successive centuries, variable Lebesgue and Sobolev spaces have been regularly studied. Next, many variable exponent real function spaces and complex function spaces have presented, for instance, Morrey spaces, Herz-Morrey spaces, Herz spaces, Hardy spaces, Besov spaces, Trieble-Lizorkin spaces, Fock spaces, Bessel potential spaces, and Bergman spaces with variable exponents. For three centuries, variable exponent function spaces have been extensively applied in approximation theory, image processing, and differential equations, and many variable exponent real function spaces and complex function spaces have shown. Thus far, the theory of variable exponent function spaces has pensively built upon on the boundedness of the Hardy-Littlewood maximal operator. This confines its technique in differential equations, optimization, and approximation. The spaces of all, bounded, -absolutely summable and convergent to zero sequences of complex numbers will be denoted by , , , and . . We denote the space of all, finite rank, approximable, and compact bounded linear maps from a Banach space into a Banach space by , , , and , and if , we indicate , , , and , respectively. The ideal of all, finite rank, approximable, and compact maps are indicated by , , , and . We label as 1 lies at the th coordinate, with

Definition 1 [1]. A function is called an -number, if the sequence for any , satisfies the following setup: (a), with (b) with and , (c), for all , , and , where and are any two Banach spaces(d)If and , then (e)Suppose , then , for all (f) or , where indicates the unit operator on the -dimensional Hilbert space We give some examples of -numbers as follows: (1)The -th Kolmogorov number, , where(2)The -th approximation number, , where

Notations 2 [2].

Some of ideals in the class of Banach spaces or Hilbert spaces are generated by scalar sequence spaces. For example, the ideal of compact maps is constructed by the space and , for . Pietsch [3] investigated the quasi-ideals for . He examined that the ideals of nuclear maps and of Hilbert Schmidt maps between Hilbert spaces are formed by and , respectively. He discussed that are dense in , and the algebra , where , generated simple Banach space. Pietsch [4] established that with , is small. Makarov and Faried [5] proved that for every infinite dimensional Banach spaces , , and , hence . Yaying et al. [6] constructed the sequence space, , whose its -Cesàro matrix in , with and . They studied the quasi Banach ideal of type , with and . They introduced its Schauder basis, , , and duals and determined certain matrix classes related to this sequence space. On sequence spaces, Baarir and Kara examined the compact maps on some Euler -difference sequence spaces [7], some difference sequence spaces of weighted means [8], the Riesz -difference sequence space [9], the -difference sequence space derived by weighted mean [10], and the th order difference sequence space of generalized weighted mean [11]. Mursaleen and Noman [12, 13] established the compact operators on some difference sequence spaces. Alotaibi et al. [14, 15] examined the compact operators on some Fibonacci difference sequence spaces and on a new sequence space related to spaces. The multiplication maps on Cesàro sequence spaces equipped with the Luxemburg norm investigated by Komal et al. [16]. lkhan et al. [17], investigated the multiplication maps on Cesàro second order function spaces. Recently, many authors in the literature have investigated some nonabsolute type sequence spaces and brought current exquisite papers, for examples, Mursaleen and Noman [18] introduced the sequence space and of nonabsolute type and proved that the spaces and are linearly isomorphic for , is a -normed space and a -space in the cases for and and constructed the basis of the space for . In [19], they examined the , , and duals of and of nonabsolute type, with . They detailed some related matrix classes and developed the properties of some other classes by means of a given basic lemma. On Cesàro summable sequences, Mursaleen and Baar [20] introduced some spaces of double sequences whose Cesàro transforms are bounded, convergent in the Pringsheim’s sense, null in the Pringsheim’s sense, both convergent in the Pringsheim’s sense and bounded, regularly convergent, and absolutely -summable, respectively, and investigated some topological properties of those sequence spaces. The Banach fixed point theorem [21] gave many mathematicians the way to examine many generalizations for the contraction maps defined on the space or on the space itself. Kannan [22] investigated an example of a class of operators with the identical fixed point actions as contractions though that fails to be continuous. Ghoncheh [23] was the only one who examined Kannan operators in modular vector spaces. He showed that the existence of a fixed point of Kannan mapping in complete modular spaces that have Fatou property. Bakery and Mohamed [24] explored the concept of the prequasi norm on Nakano sequence space such that its variable exponent in . They explained the sufficient conditions on its equipped with the definite prequasi norm to generate prequasi Banach and closed space and examined the Fatou property of different prequasi norms on it. More, they showed the existence of a fixed point of Kannan prequasi norm contraction maps on it and on the prequasi Banach operator ideal constructed by -numbers which belong to this sequence space. The next inequality will used in the sequel [25]: Assume and , for all , and , then

The aim of this paper is arranged as follows: In Section 3, we introduce the definition and some inclusion relations of the sequence space under the function . In Section 4, we investigate the enough setup on with definite function to form premodular private sequence space (), which gives that is a prequasi normed . In Section 5, we examine a multiplication map on and introduce the necessity and sufficient conditions on this sequence space such that the multiplication map is bounded, approximable, invertible, Fredholm, and closed range. In Section 6, firstly, we introduce the sufficient settings (not necessary) on, such thatis dense in. This investigates a negative answer of Rhoades [26] open problem about the linearity of -type spaces. Secondly, we introduce the conditions on so that the components of prequasi ideal are complete and closed. Thirdly, we investigate the sufficient settings on such that is strictly included for different weights and powers. We investigate the conditions for which the prequasi ideal is minimum. Fourthly, we explore the setting for which the Banach prequasi ideal is simple. Fifthly, we examine the sufficient setting on such that the class which sequence of eigenvalues in equals . In Section 7, the existence of a fixed point of Kannan prequasi norm contraction mapping on this sequence space and on its prequasi operator ideal generated by , and -numbers are presented. Additionally, in Section 8, we illustrate our results by some examples and applications to the existence of solutions of nonlinear difference equations. Finally, we introduce our conclusion in Section 9.

2. Definitions and Preliminaries

Lemma 3 [3]. Assume and , then there exist operators and such that , for all .

Definition 4 [3]. A Banach space is called simple if the algebra contains one and only one nontrivial closed ideal.

Theorem 5 [3]. If is a Banach space with , hence

Definition 6 [27]. An operator is called Fredholm if , , and is closed, where indicates the complement of .

Definition 7 [28]. A class is called an operator ideal if each element verifies the following conditions: (i), if indicates Banach space of one dimension(ii) is a linear space on (iii)Assume , , and , then , where and are normed spaces

Definition 8 [2]. A map is called a prequasi norm on the operator ideal , if it satisfies the following conditions: (1)For every , and (2)One has such that , with and (3)One has such that , for all (4)One has such that if , , and hence

Theorem 9 [2]. Every quasi norm on the ideal is a prequasi norm on the same ideal.

Definition 10 [29]. The linear space of sequences is called a private sequence space , if it verifies the next setup: (1), with (2) is solid, i.e., for , , and , with , then (3), where indicates the integral part of , if

Theorem 11 [29]. Suppose the linear sequence space be a , then be an operator ideal.

Definition 12 [29]. A subspace of the is called a premodular , if there exists a map verifies the next setup: (i)For all , , and , with is the zero vector of (ii)Assume and , one has with (iii) includes for some , with (iv)If , , one has (v)The inequality, holds, for (vi), where indicates the space of all sequences with finite none zero coordinates(vii)There are such that , with

Definition 13 [29]. The is called a prequasi normed , if verifies the conditions (i)–(iii) of Definition 12. When is complete equipped with , then is called a prequasi Banach .

Theorem 14 [29]. Every premodular is a prequasi normed .

Theorem 15 [29]. The function is a prequasi norm on , where , for all , whenever is a premodular .

Definition 16 [24]. A prequasi norm on satisfies the Fatou property, if for each sequence with and all then

Definition 17 [24]. A prequasi norm on the ideal , where , satisfies the Fatou property if for each sequence with and all , then

Definition 18. for all .
An element is called a fixed point of , if

Definition 19 [24]. An operator is named a Kannan -contraction, if there exists , so as to for all .

Definition 20 [24]. If is a prequasi normed (), and The operator is called -sequentially continuous at , if and only if, when then .

Definition 21 [24]. For the prequasi norm on the ideal , where , , and The operator is called -sequentially continuous at , if and only if, when , then .

Definition 22 [29]. If and is a prequasi normed . The operator is called a multiplication operator on , when , with . The multiplication operator is called created by , if .

Theorem 23 [30]. Assume -type If is an operator ideal, then the next setups are confirmed: (1) -type (2)Suppose -type and -type , then -type (3)Assume and -type , then -type (4)The sequence space is solid, i.e., if -type and , for all and , then -type

3. Main Results

3.1. The Sequence Space

The definition and some inclusion relations of the sequence space under the function in this section are presented.

Definition 24. Suppose , where be the space of all sequences of positive reals. The sequence space with the function is evident by:

Theorem 25. Assume , one has

Proof. Assume , one has

Remark 26. (1)Assume , , for all , , and , the sequence space examined by Yaying et al. [6](2)If , , for all and , hence , defined and studied by Ng and Lee [31]

Theorem 27. Pick up and , then be nonabsolute type.

Proof. Let , then . One has Hence, the sequence space is nonabsolute type.
Note that, we call the sequence space as -generalized Cesàro sequence space of nonabsolute type since it is generated by the domain of -Cesàro matrix in , where the -Cesàro matrix, , is defined as: The -Cesàro matrix may be shown clearly as:

Definition 28. Pick up . The -generalized Cesàro sequence space of absolute type is defined as:

Theorem 29. Assume with , then .

Proof. Suppose , as Therefore, For , we take ; we have and . For , we take ; we have and .

4. Premodular Private Sequence Space

In this section, we investigate the sufficient conditions on with definite function to form premodular , which gives that is a prequasi normed .

Here and after, the space of all monotonic decreasing and monotonic increasing sequences of positive reals will be indicated by and , respectively.

Theorem 30. is a , if the following conditions are satisfied:
(f1) with
(f2) with or , and there exists such that

Proof. (1-i)Let . We havetherefore, . (1-ii)Assume , and since , one hasTherefore, . According to conditions (1-i) and (1-ii), one has is a linear space.
Since and , we have Hence, , for all . (1)Assume , with and . We getthen . (2)Suppose , with and there exists with , one hasthen .

From Theorem 11, one has the following theorem.

Theorem 31. If the conditions (f1) and (f2) are confirmed, then is an operator ideal.

Theorem 32. is a premodular , if the conditions (f1) and (f2) are verified.

Proof. (i)Clearly, and (ii)One has with , for each and (iii)The inequality holds, with (iv)Obviously, from the proof part (31) of Theorem 30(v)Clearly, the proof part (32) of Theorem 30 that (vi)Definitely, (vii)There is with , for each and , if

Theorem 33. If the conditions (f1) and (f2) are verified, then is a prequasi Banach .

Proof. From Theorem 32, the space is a premodular . From Theorem 14, the space is a prequasi normed . To prove that is a prequasi Banach , suppose be a Cauchy sequence in , then for every , there exists such that for every , we have

Therefore, for and , one has Therefore, is a Cauchy sequence in ; for constant , this implies , for constant . Then, , for every . Finally, to prove that , we obtain hence . This gives that is a prequasi Banach .

According to Theorem 23, we explain the following properties of the -type .

Theorem 34. Assume -type The following settings are satisfied: (1)We have -type (2)If -type and -type , then -type (3)For all and -type , then -type (4)The -type is solid

5. Multiplication Operators on

In this section, we examine the multiplication map on the prequasi normed , , and introduce the necessity and sufficient conditions on such that the multiplication operator is bounded, invertible, approximable, Fredholm, and closed range.

Theorem 35. If , the conditions (f1) and (f2) are confirmed, then , if and only if,

Proof. Assume . Therefore, there exists such that , for all . Suppose , we have Hence, .
On the other hand, suppose and . Then, for every , there exist such that . One has Therefore, . Hence, .

Theorem 36. If and is a prequasi normed . Then, , for all and with , if and only if, is an isometry.

Proof. Assume the sufficient condition is satisfied. We have for all . Hence, is an isometry.
Assume the necessity condition is verified and for some . One has Also, if , clearly , which is a contradiction for the two cases. Hence, , for each .
The space of all sets with finite number of elements will be indicated by .

Theorem 37. If , the conditions (f1) and (f2) are verified. Then, , if and only if, .

Proof. Assume , hence . Let . Hence, one has so that the set . Suppose . Therefore, is an infinite set in . As for all . Hence, , which cannot have a convergent subsequence under . Then, . Which gives , this implies a contradiction. Therefore, . On the contrary, suppose . Hence, for each , we have . Therefore, for all , one has . Then, . Let , for every , where Clearly, as , for every . From with , one has

Therefore, . This implies is a limit of finite rank maps. Hence, .

Theorem 38. If , the conditions (f1) and (f2) are confirmed. Then, , if and only if, .

Proof. Clearly, as .

Corollary 39. If the conditions (f1) and (f2) are verified, then .

Proof. Since is created the multiplication map on . This implies and .

Theorem 40. Pick up be a prequasi Banach and . Then, there exist and so that , with , if and only if, is closed.

Proof. Let the sufficient setup be verified. Therefore, there exists such that , for each . To prove that is closed. Suppose is a limit point of . One has , for all such that . Clearly, the sequence is a Cauchy sequence. Since with , we have where Therefore, is a Cauchy sequence in . Since is complete. Then, there exists such that . As , one gets . But . Hence, . So . Then, is closed. Next, let the necessity condition be verified. Therefore, there exists such that , with . When , then for , we obtain this implies a contradiction. Then, , one has , with . This shows the theorem.

Theorem 41. If and is a prequasi Banach . Then, there exist and such that , for all , if and only if, is invertible.

Proof. Let the sufficient condition be confirmed. If with . From Theorem 35, the operators and are bounded linear. One has . Then, . After, assume be invertible. Hence, . Therefore, is closed. Hence, from Theorem 40, there exists such that , for all . One gets , if , with ; this implies which is an inconsistency, since is trivial. Hence, , for all . As . By using Theorem 35, there exists such that , for all . Then, one obtains , with .

Theorem 42. If is a prequasi Banach and . Then, is Fredholm operator, if and only if, (i) is a finite and (ii) , with .

Proof. Assume the sufficient setup be confirmed. Suppose be an infinite, then , for all . As s are linearly independent, we have that ; this gives an inconsistency. Therefore, must be finite. The setup (ii) follows from Theorem 40. After, assume the setups (i) and (ii) be satisfied. By using Theorem 40, the setup (ii) gives that is closed. The condition (i) implies that and . Then, is Fredholm.

6. Prequasi Ideal Properties

In this section, firstly, we give the sufficient conditions (not necessary) onso thatis dense in. This explains a negative answer of Rhoades [26] open problem about the linearity of -type spaces. Secondly, for which setup on are complete and closed? Thirdly, we investigate the sufficient conditions on so that is strictly contained for different weights and powers. We introduce the conditions such that is minimum. Fourthly, we give the setup such that the Banach prequasi ideal is simple. Fifthly, we explore the sufficient setup on so that the space of all bounded linear operators which sequence of eigenvalues in equals .

6.1. Finite Rank Prequasi Ideal

Theorem 43. , if the conditions and are confirmed. But the converse is not necessarily true.

Proof. To prove that . Since for all and is a linear space. Assume , we have . To prove that . We have . Suppose , one has . Since , assume , then there exists with , for some , where As is decreasing, one has Then, there exists such that rank and as , one gets Hence, we have From inequalities (4)–(34), we have Conversely, we have a counter example as where and , but is not satisfied. This gives the proof.

6.2. Banach and Closed Prequasi Ideal

Theorem 44. Suppose the conditions and be verified, then be a prequasi Banach ideal, where .

Proof. Since is a premodular , then from Theorem 15, is a prequasi norm on . Let be a Cauchy sequence in . Since , we have then is a Cauchy sequence in . As is a Banach space, hence there exists with As , for all . Then, from Definition 12 conditions (ii), (iii), and (v), we obtain Then, , hence .

Theorem 45. If and are normed spaces, the conditions and are confirmed; then, is a prequasi closed ideal, where .

Proof. Since is a premodular , from Theorem 15, is a prequasi norm on . Let , for all and . Since , one gets then is a convergent sequence in . As , for all . From Definition 12 conditions (ii), (iii), and (v), we have We get , so .

6.3. Minimum Prequasi Ideal

Theorem 46. If and are Banach spaces with and the conditions and are verified with and , for every , then

Proof. Assume , then . We have then . Next, if we take with , one obtains so that Hence, and . Obviously, . After, if we choose so that . One gets so that . This completes the proof.

Theorem 47. If and are Banach spaces with and the conditions and are verified, then is minimum.

Proof. Let the sufficient conditions be verified. Hence, , where is a prequasi Banach ideal. Assume , then there exists with , for all . From Dvoretzky’s theorem [32], for all , we have quotient spaces and subspaces of which can be operated onto by isomorphisms and with and . If is the identity map on , is the quotient map from to , and is the natural embedding map from to . Assume be the Bernstein numbers [33] hence for . One gets Therefore, for some , we have Hence, there is an inconsistency, when . Hence, and both cannot be infinite dimensional when . This finishes the proof.

Theorem 48. If and are Banach spaces with and the conditions and are verified, then is minimum.

6.4. Simple Banach Prequasi Ideal

Theorem 49. If and are Banach spaces with and the conditions and are verified with and , for all , then

Proof. Suppose and . From Lemma 3, one has and with . Then, for all , one has This contradicts Theorem 46. Hence, , which completes the proof.

Corollary 50. If and are Banach spaces with and the conditions and are verified with and , for all , then

Proof. Obviously, since .

Theorem 51. If and are Banach spaces with and the conditions and are confirmed, then is simple.

Proof. Suppose the closed ideal contains an operator . By using Lemma 3, there are with . This implies that . Hence, . Therefore, is simple Banach space.

6.5. Eigenvalues of -Type Operators

Notation 52.

Theorem 53. If and are Banach spaces with and the conditions and are verified, then

Proof. Assume , then and , for every . One gets , for each , then , for all . Hence, , so .
Secondly, assume . Hence, . Therefore, one can see Hence, Suppose exists, for all . Then, exists and bounded, for all . Hence, exists and bounded. Since is a prequasi operator ideal, one obtains Therefore, there is a contradiction, as . Then, , for all . This implies . This confirms the proof.

7. Kannan Contraction Operator

Theorem 54. The function confirms the Fatou property, for every , if the conditions and are verified.

Proof. Let with Since the space is a prequasi closed space, hence . Therefore, for each , one has

Theorem 55. The function does not satisfy the Fatou property, for every , if the conditions and are verified.

Proof. Let with Since the space is a prequasi closed space, hence . Therefore, for each , one gets Hence, does not satisfy the Fatou property.
Now, we give the sufficient conditions on equipped with definite prequasi norm such that there exists a unique fixed point of Kannan contraction operator.

Theorem 56. If the conditions and are verified, and is Kannan -contraction operator, where , for all , hence has a unique fixed point.

Proof. Assume , hence . As is a Kannan -contraction operator, one has Hence, for all with , one can see Therefore, is a Cauchy sequence in . As the space is prequasi Banach space. Hence, there is such that . To prove that . Since has the Fatou property, one has so . Therefore, is a fixed point of . To show that the fixed point is unique. Let we have two different fixed points of . Hence, we have Therefore,

Corollary 57. If the conditions and are verified, and is Kannan -contraction operator, where , for every , then has one and only one fixed point with

Proof. From Theorem 56, there exists a unique fixed point of . Hence, we have

Theorem 58. If the conditions and are verified, and , where , for every . The point is the only fixed point of , if the following conditions are confirmed: (a) is Kannan -contraction operator(b) is -sequentially continuous at (c)One has so that the sequence of iterates has a subsequence converging to

Proof. Let the sufficient conditions be verified. Assume be not a fixed point of , hence . From the conditions (b) and (c), we have As the operator is Kannan -contraction, one gets As , we have a contradiction. Therefore, is a fixed point of . To prove that the fixed point is unique. Let we have two different fixed points of . Then, one obtains Hence,

Example 59. Assume , where , with and As for each with , one gets For every with , one can see For all with and , one has

Hence, the operator is Kannan -contraction. As verifies the Fatou property. From Theorem 56, the operator has a unique fixed point

Assume with where with . As the prequasi norm is continuous, one has

Therefore, is not -sequentially continuous at . Hence, the operator is not continuous at .

Assume , for all .

As for each with , one obtains

If with , one has

For all with and , one gets

Hence, the operator is Kannan -contraction, and

Obviously, is -sequentially continuous at , and has a subsequence converging to . From Theorem 58, the vector is the only fixed point of .

Example 60. Suppose , with , for every and As for every with , one has For every with , then for every one gets

For each with and , one can see

Hence, the operator is Kannan -contraction. Clearly, is -sequentially continuous at , and there exists with so that the sequence of iterates has a subsequence converging to . From Theorem 58, the operator has one fixed point . Recall that is not continuous at .

Suppose , for every . As for every with , one has

For every with , then for every , one can see

For every with and , one has

Hence, the operator is Kannan -contraction. As verifies the Fatou property. From Theorem 56, the operator has a unique fixed point .

We examine the existence of a fixed point of Kannan contraction operator in the prequasi Banach operator ideal constructed by and -numbers.

Theorem 61. The prequasi norm does not satisfy the Fatou property, for all , if the conditions and are confirmed.

Proof. Assume the setup be satisfied and with Since the space is a prequasi closed ideal. Therefore, . Hence, for every , we have Then, does not satisfy the Fatou property.

Theorem 62. Suppose the conditions and be confirmed and , where , for all . The point is the unique fixed point of , if the following conditions are satisfied: (a) is Kannan -contraction mapping(b) is -sequentially continuous at a point (c)We have such that the sequence of iterates has a subsequence converging to

Proof. Assume the sufficient conditions be confirmed. Let be not a fixed point of , hence . From the setups (b) and (c), we get Since is Kannan -contraction operator, one has As , we have a contradiction. Therefore, is a fixed point of . To show that the fixed point is unique. Let we have two different fixed points of . Hence, one can see Hence,

Example 63. Let , where , for all and

As for every with , one has

For every with , one gets

For every with and , one has

Hence, the operator is Kannan -contraction and

Obviously, is -sequentially continuous at the zero operator , and has a subsequence converging to . From Theorem 62, the zero operator is the only fixed point of . Let be so that , where with . As the prequasi norm is continuous, one obtains

Therefore, is not -sequentially continuous at . Hence, the operator is not continuous at .

8. Application to the Existence of Solutions of Nonlinear Difference Equations

Summable equations as (87) are examined by Salimi et al. [34], Agarwal et al. [35], and Hussain et al. [36]. In this section, we search for a solution to (87) in , where the conditions and are verified and , for every . Consider the summable equations and suppose be defined by

Theorem 64. The summable equation (87) has a solution in , if , , , and , suppose there is a number such that and for every , one has

Proof. Assume the setup be verified. Let the mapping defined by equation (88). One has From Theorem 56, one gets a solution of equation (87) in

Example 65. Pick up the sequence space , where , for all . Consider the nonlinear difference equations: with and assume defined by

Obviously, there exists a number so that and for every , one obtains

From Theorem 64, the nonlinear difference equations (91) has a solution in .

9. Conclusion

In this article, we present some topological and geometric properties of , of the multiplication maps acting on , of the class , and of the class . We explain the existence of a fixed point of Kannan contraction map acting on these spaces. Some several numerical experiments are introduced to illustrate our results. More, some successful applications to the existence of solutions of nonlinear difference equations are discussed. This article has a number of advantages for researchers such as studying the fixed points of any contraction maps on this prequasi normed sequence space which is a generalization of the quasi normed sequence spaces, a new general space of solutions for many difference equations, examining the eigenvalue problem in this new settings and note that the closed operator ideals are certain to play an important function in the principle of Banach lattices.

Data Availability

No data were used.

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 competing interests.

Authors’ Contributions

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