Some Fixed Point Results in Premodular Special Space of Sequences and Their Associated Pre-Quasi-Operator Ideal

Bakery, Awad A.; Mohamed, Elsayed A. E.; Mohamed, O. M. Kalthum S. K.

doi:https://doi.org/10.1155/2021/4995428

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Fixed-Point Techniques and Applications to Real World Problems

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Volume 2021 | Article ID 4995428 | https://doi.org/10.1155/2021/4995428

Some Fixed Point Results in Premodular Special Space of Sequences and Their Associated Pre-Quasi-Operator Ideal

Awad A. Bakery,^1,2Elsayed A. E. Mohamed,^1,3and O. M. Kalthum S. K. Mohamed^1,4

Academic Editor: Anita Tomar

Received24 Aug 2021

Accepted03 Dec 2021

Published23 Dec 2021

Abstract

A weighted Nakano sequence space and the -numbers it contains are the subject of this article, which explains the concept of the pre-quasi-norm and its operator ideal. We show that both Kannan contraction and nonexpansive mappings acting on these spaces have a fixed point. A slew of numerical experiments back up our findings. The presence of summable equations’ solutions is shown to be useful in a number of ways. Weight and power of the weighted Nakano sequence space are used to define the parameters for this technique, resulting in customizable solutions.

1. Introduction

The spaces of all, bounded, -absolutely summable, and null sequences of real numbers will be denoted throughout the article by , , , and , respectively, where is the set of nonnegative integers. , while 1 lies in the ^th place, with .

Definition 1 (see [1]). A function , where is the space of all bounded linear operators from a Banach space into a Banach space and if , we write , which transforms every mapping to is said to be -number, if it is satisfying the following conditions: (a), for every (b), for every and , (c)Ideal property: , for every , , and , where and are any two Banach spaces(d)If and , we have (e)Rank property: if , then , for all (f)Norming property: or

The th approximation number, , is defined as

Notations 2 (see [2]). , where . Also, .

Fixed point theory, Banach space geometry, normal series theory, ideal transformations, and approximation theory are all examples of ideal operator theorems and summability. The concept of a pre-quasi-operator ideal is introduced and studied by Faried and Bakery [2]. Bakery and Abou Elmatty investigated some topological and geometric structures of in [3]. They proved that the space is a small pre-quasi-operator ideal and gave a strictly inclusion relation for different weights and powers. Several mathematicians were able to investigate many extensions for contraction mappings defined on the space or on the space itself thanks to the Banach fixed point theorem [4]. Kannan [5] investigated an example of a class of operators that perform the same fixed point actions as contractions but are not continuous. Ghoncheh [6] demonstrated the existence of a Kannan mapping fixed point in complete modular spaces with Fatou property (also see [7–10]). Bakery and Mohamed [11] examined the sufficient requirements on , variable exponent in under definite pre-quasi-norm so that there is a fixed point of Kannan pre-quasi-norm contraction mappings on this space. For the construction of pre-quasi-Banach and closed spaces, we use a weighted Nakano sequence space, , with various pre-quasi-norms in this study. Weighted Nakano sequence space’s pre-quasi-normal structural features, including the fixed point idea of Kannan pre-quasi-norm contraction and the Kannan pre-quasi-norm nonexpansive mapping in weighted Nakano sequence space, are improved. The existence of a fixed point for the Kannan pre-quasi-norm contraction mapping has been demonstrated using weighted Nakano sequence space and -numbers. Our talk concluded with various instances of how the information gathered could be put to good use in resolving a problem.

2. Preliminaries and Definitions

We indicate the space of all mappings , by .

Definition 3 (see [12]). If is a vector space and , a mapping is said to be modular: (a)If , with (b) holds, for each and (c)The inequality verifies, for every and .

Definition 4 (see [2]). If the following conditions hold: (1) is solid. This means if , , and , for every , then (2), where denotes the integral part of , when Then, is said to be a special space of sequences (sss).

Definition 5 (see [2]). If we have with the following: (i)if , (ii)suppose and , then there is for which (iii)the inequality, , for each , verifies for some (iv)if and , then (v)the inequality, , satisfies, for some (vi)assume is the space of finite sequences, one has (vii)we have so that for every Then, is said to be a premodular sss.

Example 1. Since for all , we have Hence, is a premodular (not a modular) on .

Definition 6 (see [11]). Assume is a sss. The function is called a pre-quasi-norm on , if it satisfies the conditions (i), (ii), and (iii) of Definition 5.

Theorem 7 (see [11]. Suppose is a premodular sss; then, is a pre-quasi-normed sss.

Theorem 8 (see [11]). Quasinormed sss is contained in pre-quasi-normed sss.

Definition 9 (see [13]). (a)The pre-quasi-norm 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, if (d) is -closed, if for every -converging to , then (e) is -bounded, if (f)The -ball of radius and center , for every , is defined as(g)A pre-quasi-norm on satisfies the Fatou property, when for every sequence with and every then Recall that the Fatou property implies the -closed of the -balls.

Definition 10 (see [14]). Let be the class of each bounded linear operators between any two Banach spaces. A subclass of is known as an operator ideal, if all element fulfills the following conditions: (i), where indicates Banach space of one dimension(ii)The space is linear over (iii)If , , and , then , where and are normed spaces

Pre-quasi-operator ideals are more general than quasioperator ideals.

Definition 11 (see [2]). A mapping is called a pre-quasi-norm when (a)let , , and (b)we have with , when and (c)we have so that , for all (d)we have such that , and then

Theorem 12 (see [11]). Suppose is a premodular sss, then is a pre-quasi-norm on .

Theorem 13 (see [2]). Quasi-normed ideal is contained in pre-quasi-normed ideal.

Lemma 14 (see [15]). If and for all , then

Lemma 15 (see [16]). Let and with ; then,

Lemma 16 (see [17]). Suppose and , for all ; then, where .

3. Main Results

3.1. The Sequence Space

Assume and , where denotes the set of positive reals. In [3], the weighted Nakano sequence space was defined as while

Theorem 17. If , then

Proof. (1)If , with , then defined and considered in [18, 19](2)If , with , then examined by many authors [16, 20, 21]

Theorem 18. The space is a Banach space, where .

Proof. Since we have the following: (i), for each and , if and only if, (ii)suppose , without loss of generality, let then(iii)assume , then there are and be such that Let ; since is nondecreasing and convex, one hasAs the ’s are nonnegative, one can see Then, the space is a normed space. Next, let be a Cauchy sequence in . Therefore, for every , we have such that for all , we obtain So, for and , one can see Hence, is a Cauchy sequence in , for fixed . This implies , for fixed . Hence, , for every . Since , therefore, . This implies that is a Banach space.

4. Pre-Quasi-Normed Sequence Space

To create pre-quasi-Banach and closed sequence space, we study the conditions on , where , for each . The Fatou property of has been investigated for various .

Theorem 19. (a1)Let be an increase.(a2)Either is a monotonic decrease or monotonic increase so that there is , where .Then, is a premodular sss.

Proof. (i)Evidently, and .(1-i) and (iii). Let . As , one gets Hence, .
(1-ii) and (ii). Let and . Since , one has where . Therefore, . From conditions (1-i) and (1-ii), one has which is linear. And for all , as (2) and (iv). Let , for all and . Since , for all , then we get .
(3) and (v). Suppose and is increasing. There is so that and is increasing; one can see Then, .
(vi) Obviously,
(vii) We have , for or , for such that

Theorem 20. Let the conditions (a1) and (a2) of Theorem 19 be satisfied, then be a pre-quasi-Banach sss.

Proof. From Theorems 19 and 7, we have which is a pre-quasi-normed sss. Suppose is a Cauchy sequence in . Therefore, for all , we have such that for all , we get Hence, for and , one obtains This implies is a Cauchy sequence in , for fixed . This explains , with fixed . Therefore, , for all . Also, one has ; hence, .

Theorem 21. The space is a pre-quasi-closed sss, whenever the conditions (a1) and (a2) of Theorem 19 are satisfied.

Proof. Let and ; hence, for all , one has such that for every , we obtain This gives Therefore, is a convergent sequence in , for constant . Hence, , with constant . Also, one gets Hence, .

Theorem 22. The function , for every , has the Fatou property, if the conditions (a1) and (a2) of Theorem 19conditions (a1) and (a2) of Theorem 19 are satisfied.

Proof. Assume and As is a pre-quasi-closed space, this implies . Hence, for all , we have

Theorem 23. The function does not hold the Fatou property, if the setups (a1) and (a2) of Theorem 19 are satisfied with .

Proof. Assume and As is a pre-quasi-closed space, this implies . Hence, for all , we have

Example 2. The function is a pre-quasi-norm and not quasi-norm, for all .

Example 3. The function is a pre-quasi-norm, quasi-norm, and not a norm on , for .

Example 4. The function is a norm on .

5. Kannan Contraction’s Fixed Points

Here, -Lipschitzian mapping acting on as Kannan -Lipschitzian mapping has been defined. We investigate the adequate requirements for a fixed point of Kannan contraction mapping on equipped with various pre-quasi-norms.

Definition 24. A mapping is said to be a Kannan -Lipschitzian, if there exists , such that for every . (1)Let ; then, the operator is called Kannan -contraction(2)For , then the operator is said to be Kannan -non-expansiveA vector is said to be a fixed point of , if

Theorem 25. Assume the conditions (a1) and (a2) of Theorem 19 are satisfied, and is Kannan -contraction mapping, where , for all ; hence, has a unique fixed point.

Proof. Let the setups be satisfied. Assume ; hence, . Since is a Kannan -contraction mapping, we obtain Therefore, for with , one has Hence, is a Cauchy sequence in, since is pre-quasi-Banach space. We have with, to show that . As verifies the Fatou property, we get Hence, . So is a fixed point of . To show the uniqueness of , let us have two different fixed points of . So, one has This implies

Example 5. Assume , where, for every and As for each with , one has For all with , one has For all with and , we get

Hence, is Kannan -contraction and holds one element , so that , by Theorem 25.

Corollary 26. Let conditions (a1) and (a2) of Theorem 19 be satisfied, and is Kannan -contraction mapping, where , for all ; then, has a unique fixed point with .

Proof. In view of Theorem 25, we have a unique fixed point of . Therefore, one gets

Definition 27. Assume is a pre-quasi-normed sss, and The operator is said to be -sequentially continuous at , if and only if, when , then .

Example 6. Suppose , where , for every and

is clearly both -sequentially continuous and discontinuous at .

Example 7. Assume is defined as in Example 5. Suppose is such that , where with .
As the pre-quasi-norm is continuous, we have

Therefore, is not -sequentially continuous at .

Theorem 28. If the conditions (a1) and (a2) of Theorem 19 are satisfied with and , where , for all , (1)suppose is Kannan -contraction mapping(2)assume is -sequentially continuous at a point (3)we have such that has a subsequence converging to ; then, is the only fixed point of

Proof. Suppose the settings are verified. Assume is not a fixed point of , then . From parts (54) and (55), one gets Since is Kannan -contraction, we obtain We get a contradiction when. To show the uniqueness of , suppose we have two different fixed points of . Therefore, one obtains Hence,

Example 8. Assume is defined as in Example 5. Let , for every .
As for each with , one has For all with , one has For all with and , we obtain

So, the mapping is Kannan -contraction and

Obviously, is -sequentially continuous at and contains a subsequence converging to . From Theorem 28, the vector is the unique fixed point of .

6. Kannan Nonexpansive Fixed Points

The uniform convexity (UUC 2) defined in [22] of the space has been investigated, where , for all . The property () and the -normal structure property of this space have been discussed. Finally, we present the sufficient conditions on this space such that the Kannan pre-quasi-norm nonexpansive mapping on it has a fixed point.

Definition 29 (see [23, 24]). (1)[25] Suppose and . LetIf , we put If , we put The function holds the uniform convexity (UC), if for all and , one has Note that for every , then , for very small . (2)[22] The function holds (UUC1), if for every and , we have with(3)[22] Let and . SupposeWhen , we put If , we place The function supports (UC 2), if for every and , one has Note that for all , , with very small . (4)[22] verifies (UUC 2); when and , one has with(5)[25] The function is strictly convex, (SC); when with and one obtains We will need the following comment here and later: for all and If , we set

Theorem 30. If the conditions (a1) and (a2) of Theorem 19 are satisfied with , then the pre-quasi-norm on is (UUC2).

Proof. Assume the settings are satisfied, and . Let so that From the definition of , we have which implies Consequent, put and Let ; we get By using the conditions, we get or Assume first Using Lemma 14, one gets This explains Since by adding inequalities 2 and 3, and from inequality 1, we have This gives Next, suppose Set As and the power function is convex, so Since we get For any , we have By Lemma 15, one gets Hence, This investigates Since by adding inequalities 5 and 6, one has Since by adding inequalities 7 and 8, and from inequality 1, we obtain This implies It is clear that By using inequalities 4 and 9 and Definition 29, if we put therefore, we have ; this implies is (UUC2).

Definition 31. Space is said to satisfy the property (R), if for all decreasing sequence of -closed and -convex nonempty subsets of with for some then one has

Theorem 32. If conditions (a1) and (a2) of Theorem 19 are satisfied with , then (1)Assume is a nonempty -closed and -convex subset of Let be withHence, one has a unique so that (2)The property (R) holds on

Proof. Suppose the conditions are satisfied. To show (51), assume as is -closed. So, one has . Hence, for every , one has with . Assume is not -Cauchy. Therefore, we get a subsequence and with , for every Furthermore, we have , for all As and for all , we obtain So for any . If we let , we get which is a contradiction. So, is -Cauchy. Since is -complete, then -converges to some . For every , one has the sequence converges to . As is -closed and -convex, one gets Surely converges to , which implies . By setting and using Theorem 22, as satisfies the Fatou property, we obtain Hence, . As is (UUC2), then is (SC); this implies is only one. To show (2), let , for some Since is increasing. Set . If . Else , for all . From (51), we get one point with , for every . A consistent proof will show that converges to some . Since are -convex, decreasing, and -closed, we get .

Definition 33. satisfies the -normal structure property, if for all nonempty -bounded, -convex, and -closed subset of not decreased to one point, we have with

Theorem 34. If the conditions (a1) and (a2) of Theorem 19 are satisfied with , then has the -normal structure property.

Proof. Suppose the setups are satisfied. Theorem 30 implies that is (UUC2). Let be a -bounded, -convex, and -closed subset of not decreased to one point. Hence, . Put . Suppose with Hence, For every , we have and Since is -convex, one obtains . Hence, for every Hence,

Lemma 35. Let be a nonempty P-bounded, P-convex, and 𝑃-closed subset of , where verifies the (R) property and the -quasi-normal property, and be a Kannan -nonexpansive mapping. For , suppose . Take Then, is a nonempty, -convex, and -closed subset of and

Proof. As , which implies , is a -closed and -convex subset of , as the -balls are -convex and -closed. Assume When , we get Else, suppose Take From the definition of , so Hence, which implies . Assume Hence, there is with . Then, Since is randomly positive, one has ; then, we have . This implies . As , we get ; this indicates is -invariant, consequent to prove that . As for every , let Then, . The definition of implies . So, . Therefore, we obtain , for every ; this gives

Theorem 36. Let be a nonempty, -convex, -closed, and -bounded subset of , where holds the -quasi-normal property and the () property, and be a Kannan -non-expansive mapping. Then, has a fixed point.

Proof. Take and , for every . In view of the definition of , we get , for all Let as defined in Lemma 35. Clearly, is a decreasing sequence of nonempty -bounded, -closed, and -convex subsets of . One has , from the property (). Let ; we have , for every . Suppose ; one has , which implies . Then, . One gets . Otherwise, ; this implies that has no a fixed point. Assume is as defined in Lemma 35. Since has not a fixed point and is -invariant, then . From the -quasi-normal property, we have with for every . According to Lemma 35, one gets . From definition of , hence, . Obviously, this explains which contradicts the definition of . Therefore, , which means that has a fixed point in .

We have the next corollary according to Theorems 32, 34, and 36.

Corollary 37. Pick up the conditions (a1) and (a2) of Theorem 19 to be satisfied with . Assume is a nonempty, -convex, -closed, and -bounded subset of , and is a Kannan -nonexpansive operator. Hence, has a fixed point.

Example 9. Suppose , where where and , for every . According to Example 5, the operator is Kannan -contraction. Hence, it is a Kannan -nonexpansive mapping. Evidently, is a nonempty, -convex, -closed, and -bounded subset of . In view of Corollary 37, the operator has a fixed point .

7. Kannan Contraction’s Fixed Points on Pre-Quasi-Ideal

In this part, we suppose and are Banach spaces. The Kannan contraction’s fixed points on , where , has been examined.

Theorem 38 (see [3]). If the conditions (a1) and (a2) of Theorem 19 are fulfilled, then is a pre-quasi-Banach operator ideal.

Theorem 39. If the conditions (a1) and (a2) of Theorem 19 are satisfied, then is a pre-quasi-closed operator ideal.

Proof. The class is a pre-quasi-operator ideal and follows from Theorems 19 and 12. Let , for every and . Hence, there is and since , one obtains Hence, is convergent in . This implies and while , for every and is a premodular sss. Hence, there exists with We have , then .

Definition 40. The function on satisfies the Fatou property, if for any sequence with and any , then

Theorem 41. Suppose the conditions (a1) and (a2) of Theorem 19 are satisfied, then the function , for all , does not hold the Fatou property.

Proof. Assume with Since the space is a pre-quasi-closed ideal, then . Hence, for any , we have

Definition 42. A mapping is said to be a Kannan -Lipschitzian, if we have , such that for every . (1)The mapping is said to be Kannan -contraction, whenever (2)The mapping is said to be Kannan -non-expansive, whenever

Definition 43. Suppose and is said to be -sequentially continuous at , if and only if, if , then .

Example 10. Assume , where
, for every and Evidently, is -sequentially continuous at the zero operator .
Let with , where with . As is continuous, one has So is not -sequentially continuous at .

Theorem 44. Assume the conditions (a1) and (a2) of Theorem 19 are satisfied and , where , for every . (i)Let is Kannan -contraction mapping(ii) is -sequentially continuous at a point (iii)We have such that the sequence has a subsequence converging to Then, is the only fixed point of .

Proof. Assume is not a fixed point of ; we have . By parts (ii) and (iii), one can see As is Kannan -contraction mapping, we get

This gives a contradiction as . Hence, is a fixed point of . For the uniqueness of the fixed point , assume we have two different fixed points of . Therefore, one gets This implies

Example 11. Assume , where , for every and If with , one has For each with , we get For each with and , one can see So, the mapping is Kannan -contraction and From Theorem 44, is the unique fixed point of , since is -sequentially continuous and converging to .

8. The Presence of Solutions to Summable Equations

The solution to (104), which is studied by some authors (see [26–28]), in , where is an increase and , for all , has been examined.

Consider the summable equations and suppose is defined by

Theorem 45. Assume and for all , there exists with Then, the summable equation (104) has a solution in

Proof. Suppose is defined by (105). We have

Hence, there is a unique solution of equation (104) in according to Theorem 25.

Example 12. Suppose we have , where , for all . Consider the summable equations where , and let be defined by Obviously,

By Theorem 45, the summable equation (108) has a unique solution in .

Example 13. Assume we have , where , for all . Consider the summable equations where , and let be defined by It is easy to see that

By Theorem 45, the summable equation (111) has one solution in .

Example 14. If we have , where , for all . Consider the summable equations such that and , and assume , where is defined by

According to Theorem 45, the summable equation (114) has a solution in .

9. Conclusion

There is a pre-quasi-normed space theorem that is more general than quasi-normed space. In weighted Nakano sequence space with the well-known pre-quasi-norm, we investigate the appropriate conditions for the generation of pre-quasi-Banach and closed spaces. Pre-quasi-normal structural properties of weighted Nakano sequence space, including the fixed point idea of Kannan pre-quasi-norm contraction and Kannan pre-quasi-norm nonexpansive mapping in weighted Nakano sequence space, are improved. Using weighted Nakano sequence space and -numbers, the presence of a fixed point for Kannan pre-quasi-norm contraction mapping has been proved. Toward the end of our discussion, we provided several examples of how the collected data could be used to solve an issue. The weight and power of the weighted Nakano sequence space can be used to define a wide range of circumstances under which existence findings can be found. Banach lattices are introduced in this article, and a new space of solutions for many difference equations is introduced, the spectrum of any bounded linear operator between any two Banach spaces with -numbers in this sequence space.

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 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, Saudi Arabia, under grant No. UJ-20-110-DR. The authors, therefore, acknowledge with thanks the University’s technical and financial support.

References

A. Pietsch, Eigenvalues and -Numbers, Cambridge University Press, New York, NY, USA, 1986.
N. Faried and A. A. Bakery, “Small operator ideals formed by numbers on generalized Cesàro and Orlicz sequence spaces,” Journal of Inequalities and Applications, vol. 1, 2018.
View at: Publisher Site | Google Scholar
A. A. Bakery and A. R. Abou Elmatty, “Pre-quasi simple Banach operator ideal generated by numbers,” Journal of Function Spaces, vol. 2020, Article ID 9164781, 11 pages, 2020.
View at: Publisher Site | Google Scholar
S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.
View at: Publisher Site | Google Scholar
R. Kannan, “Some results on fixed points- II,” The American Mathematical Monthly, vol. 76, no. 4, pp. 405–408, 1969.
View at: Publisher Site | Google Scholar
S. J. H. Ghoncheh, “Some fixed point theorems for Kannan mapping in the modular spaces,” Ciência e Natura, vol. 37, pp. 462–466, 2015.
View at: Publisher Site | Google Scholar
H. H. Alsulami, A. Roldan, E. Karapinar, and S. Radenovic, “Some inevitable remarks on “Tripled fixed point theorems for mixed monotone kannan type contractive mappings”,” Journal of Applied Mathematics, vol. 2014, Article ID 392301, 7 pages, 2014.
View at: Publisher Site | Google Scholar
E. Karapinar, “Best proximity points of Kannan type cyclic weak ϕ-contractions in ordered metric spaces,” Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 20, no. 3, pp. 51–64, 2012.
View at: Publisher Site | Google Scholar
U. Aksoy, E. Karapinar, and I. M. Erhan, “Fixed point theorems in complete modular metric spaces and an application to anti-periodic boundary value problems,” Univerzitet u Nišu, vol. 31, no. 17, pp. 5475–5488, 2017.
View at: Google Scholar
U. Aksoy, E. Karapinar, I. M. Erhan, and V. Rakocevic, “Meir-Keeler type contractions on modular metric spaces,” Univerzitet u Nišu, vol. 32, no. 10, pp. 3697–3707, 2018.
View at: Google Scholar
A. A. Bakery and O. S. K. Mohamed, “Kannan Prequasi Contraction Maps on Nakano Sequence Spaces,” Journal of Function Spaces, vol. 2020, Article ID 8871563, 10 pages, 2020.
View at: Publisher Site | Google Scholar
H. Nakano, “Modulared sequence spaces,” Proceedings of the Japan Academy, vol. 27, pp. 508–512, 1951.
View at: Google Scholar
A. A. Bakery and O. S. K. Mohamed, “Kannan nonexpansive maps on generalized Cesàro backward difference sequence space of non-absolute type with applications to summable equations,” Journal of Inequalities and Applications, vol. 2021, no. 1, 2021.
View at: Publisher Site | Google Scholar
A. Pietsch, Operator Ideals, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1980.
J. A. Clarkson, “Uniformly convex spaces,” Transactions of the American Mathematical Society, vol. 40, no. 3, pp. 396–414, 1936.
View at: Publisher Site | Google Scholar
K. Sundaresan, “Uniform convexity of Banach spaces ,” Studia Mathematica, vol. 39, no. 3, pp. 227–231, 1971.
View at: Publisher Site | Google Scholar
B. Altay and F. Başar, “Generalization of the sequence space ℓ(p) derived by weighted mean,” Journal of Mathematical Analysis and Applications, vol. 330, no. 1, pp. 174–185, 2007.
View at: Publisher Site | Google Scholar
S. T. Chen, “Geometry of Orlicz spaces,” Dissertations Math, vol. 356, pp. 1–204, 1996.
View at: Google Scholar
Y. A. Cui and H. Hudzik, “On the Banach-Saks and weak Banach-Saks properties of some Banach sequence spaces,” Acta Scientiarum Mathematicarum, vol. 65, pp. 179–187, 1996.
View at: Google Scholar
V. Klee, “Summability in spaces,” Studia Mathematica, vol. 25, pp. 277–280, 1964.
View at: Google Scholar
W. Orlicz, “Über konjugierte exponentenfolgen,” Studia Mathematica, vol. 3, no. 1, pp. 200–211, 1931.
View at: Publisher Site | Google Scholar
M. A. Khamsi and W. M. Kozlowski, Fixed Point Theory in Modular Function Spaces, Birkhauser, New York, 2015.
L. Diening, P. Harjulehto, P. Hästö, and M. Ruzicka, “Lebesgue and Sobolev spaces with variable exponents,” Tech. Rep., Springer, Berlin, 2011.
View at: Google Scholar
J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin, 1983.
View at: Publisher Site
H. Nakano, Topology of Linear Topological Spaces, Maruzen Co. Ltd., Tokyo, 1951.
P. Salimi, A. Latif, and N. Hussain, “Modified α-ψ-contractive mappings with applications,” Fixed Point Theory and Applications, vol. 2013, no. 1, 2013.
View at: Publisher Site | Google Scholar
R. P. Agarwal, N. Hussain, and M.-A. Taoudi, “Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations,” Abstract and Applied Analysis, vol. 2012, Article ID 245872, 15 pages, 2012.
View at: Publisher Site | Google Scholar
N. Hussain, A. R. Khan, and R. P. Agarwal, “Krasnosel’skii and Ky Fan type fixed point theorems in ordered Banach spaces,” J. Nonlinear Convex Anal., vol. 11, no. 3, pp. 475–489, 2010.
View at: Google Scholar

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Copyright © 2021 Awad A. Bakery et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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