Advances in Nonlinear Analysis and ApplicationsView this Special Issue
Kannan Nonexpansive Mappings on Nakano Sequence Space of Soft Reals with Some Applications
We developed the operators ideal in this article by extending -soft reals and a particular space of sequences with soft real numbers. The criteria necessary for the Nakano sequence space of soft real numbers given with the definite function to be prequasi Banach and closed are investigated. This space’s () and normal structural features are illustrated. Fixed points have been introduced for Kannan contraction and nonexpansive mapping. Finally, we investigate whether the Kannan contraction mapping has a fixed point in the prequasi operator ideal with which it is linked. By examining some real-world instances and their applications, it is demonstrated that there exist solutions to nonlinear difference equations.
The study of variable exponent Lebesgue spaces received additional impetus from the mathematical explanation of non-Newtonian fluids’ hydrodynamics (see [1, 2]). Electrorheological fluids have various applications in various fields, including military science, civil engineering, and orthopedics. Since the publication of the Banach fixed point theorem , there have been numerous developments in the field of mathematics. While contractions have fixed point actions, Kannan  illustrated a noncontinuous mapping. In Reference , a single attempt was made to explain Kannan operators in modular vector spaces, and this was the only one that worked. Mitrovi et al.  defined a cone -metric space over Banach algebra as a generalization of metric spaces, rectangular metric spaces, b-metric spaces, rectangular b-metric spaces, -generalized metric spaces, cone b-metric spaces over Banach algebra, and rectangular cone b-metric spaces over Banach algebra. They provided fixed point results for Banach and Kannan in cone -metric spaces over Banach algebra. Debnath et al.  showed the existence and uniqueness of common fixed points for pairs of self-maps of the Kannan, Reich, and Chatterjea types in a complete metric space. Younis et al.  used concepts from graph theory and fixed point theory to provide a fixed point result for Kannan-type mappings in the context of freshly published graphical b-metric spaces. They provided suitable examples of graphs that corroborated the existing theory. They demonstrated the anticipated results by applying them to several nonlinear issues encountered in engineering and research. Younis and Singh  discovered adequate conditions for the existence of solutions to certain classes of Hammerstein integral equations and fractional differential equations. They extended the concept of Kannan mappings in terms of F-contraction in the context of b-metric-like spaces and provided a series of novel and nontrivial instances, as well as computer simulations, to demonstrate the established results, therefore introducing the concept in a novel way. On the other hand, several unresolved issues are offered to enthusiastic readers. More information on Kannan’s fixed point theorems can be found here (see [10–15]). The mathematics underpinnings of fuzzy set theory, which were pioneered by Zadeh  in 1965 and have made significant progress, are well understood in fuzzy theory. The fuzzy theory has the potential to be applied to various real-world problems. The possibility theory, for example, has been developed by several researchers, including Dubois and Prade  and Nahmias . The contribution of probability theory, fuzzy set theory, and rough sets to the study of uncertainty is critical. Yet, these theories have some limitations as well as advantages. The theory of soft sets, developed by Molodtsov , was introduced as a new mathematical strategy for dealing with uncertainties to overcome these characteristics. Soft sets have been widely used in various disciplines and technologies. In particular, Maji et al. [20, 21] studied several operations on soft sets and applied their findings to decision-making problems in the literature. Several writers, including Chen , Pei and Miao , Zou and Xiao , and Kong et al. , have discovered significant characteristics of soft sets. Soft semirings, soft ideals, and idealistic soft semirings were all investigated by Feng et al. . Das and Samanta developed the ideas of a soft real number and a soft real set in  and discussed the characteristics of each concept. These principles served as the foundation for their investigation into the concept of “soft metrics” in “.” (See [29, 30] for a more in-depth examination.) Based on the idea of soft elements of soft metric spaces, Abbas et al.  developed the concept of soft contraction mapping, which they named “soft contraction mapping.” They focused on fixed points of soft contraction maps and obtained, among other things, a soft Banach contraction principle as a result of their efforts. In their paper, Abbas et al.  demonstrated that every complete soft metric induces an equivalent complete usual metric. They obtained in a direct way soft metric versions of various significant fixed point theorems for metric spaces, such as the Banach contraction principle, Kannan and Meir-Keeler fixed point theorems, and Caristi theorem, Kirk’s, among other things. In , Chen and Lin presented an extension of the Meir and Keeler fixed point theorem to soft metric spaces, which was previously published. Many researchers working on sequence spaces and summability theory were involved in introducing fuzzy sequence spaces and studying their many characteristics. When it comes to fuzzy numbers, Nuray and Sava  defined and explored the Nakano sequences of fuzzy numbers, equipped with a definite function. The following theories use operators’ ideals: fixed point theory, Banach space geometry, normal series theory, approximation theory, and ideal transformations. For additional evidence, see [35–37]. According to Faried and Bakery , prequasi operator ideals are broader than quasioperator ideals. This study is aimed at introducing a certain space of soft real number sequences, abbreviated (csss), under a pre-quasi-quasi function (csss). The structure of the ideal operators has been described using this space and -numbers. The conditions essential to generate prequasi Banach and closed (csss) supplied with the definite function are investigated. This space’s () and normal structure properties are illustrated. Fixed points have been introduced for Kannan contraction and nonexpansive mapping. Finally, we investigate whether the Kannan contraction mapping has a fixed point in the prequasi operator ideal with which it is linked. A few real-world examples and applications demonstrate the existence of solutions to nonlinear difference equations.
2. Definitions and Preliminaries
Assume that is the set of real numbers and is the set of nonnegative integers. We denote the collection of all nonempty bounded subsets of by and is the set of parameters.
Definition 1 (see ). A soft real set denoted by , or simply by , is a mapping . If is a single-valued mapping on taking values in , then is called a soft element of or a soft real number. If is a single-valued mapping on taking values in the set of nonnegative real numbers, then is called a nonnegative soft real number. We shall denote the set of nonnegative soft real numbers (corresponding to ) by . A constant soft real number is a soft real number such that for each , we have , where is some real number.
Definition 2 (see ). For two soft real numbers , , we say that (a) if , for all (b) if , for all (c) if , for all (d) if , for all
Note that the relation is a partial order on . The additive identity and multiplicative identity in are denoted by and , respectively.
The arithmetic operations on are defined as follows:
The absolute value of is defined by
Let , where for all . Assume is defined by
Note that (1) is a complete metric space(2) for all (3), for all
Definition 3. A sequence of soft real numbers is said to be (a)bounded if the set of soft real numbers is bounded; i.e., if a sequence is bounded, then there are two soft real numbers such that (b)convergent to a soft real number if, for every , there exists such that , for all
By and , we indicate the spaces of bounded and -absolutely summable sequences of reals. Assume is the classes of all sequence spaces of soft reals. If , where is the space of positive real sequences, we introduce Nakano sequences of soft reals such as  and marked it by where The space , where and , for all , is a Banach space. Suppose , one has
Lemma 4 (see ). If and , for all , one gets where .
3. Some Properties of
We have investigated in this section the certain space of sequences of soft real numbers under definite function to form prequasi (csss). We present sufficient conditions of under definite function to construct prequasi Banach and closed (csss). The Fatou property of different prequasi norms on has been explained. We have explored the uniform convexity (UUC2), the property (), and this space’s -normal structure property.
Definition 5. The linear space is called a certain space of sequences of soft reals (csss), when (1), where , for marks at the place(2) is solid, i.e., if , , and , for all , one has (3), where indicates the integral part of , assume
Definition 6. A subclass of is said to be a premodular (csss), if one has holds the following conditions: (i)Suppose , with , where (ii)We have , the inequality holds, for all and (iii)One has , the inequality satisfies, for all (iv)When , for all , we have (v)The inequality verifies, for some (vi)Assume is the space of finite sequences of soft real numbers, one has the closure of (vii)We have with where , for every
Definition 7. If is a (csss). The function is said to be a prequasi norm on , if it satisfies the following settings: (i)Suppose , with , where (ii)One has , the inequality verifies, for all and (iii)We have , the inequality satisfies, for all
Evidently, by the last two definitions, one has the following two theorems.
Theorem 8. Assume is a premodular (csss), then it is prequasi normed (csss).
Theorem 9. is a prequasi normed (csss), when it is quasinormed (csss).
Definition 10. (a)The function on is called -convex, when for all 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 -converges to , one has (e) is -bounded, assume (f)The -ball of radius and center , for all , is denoted by (g)A prequasi norm on verifies the Fatou property, if for all sequence with and every , we have
Recall that the Fatou property gives the -closedness of the -balls. We will indicate the space of all increasing sequences of reals by .
Theorem 11. , where , for every , is a premodular (csss), if with .
Proof. (i) Clearly, and .
(1-i) Assume . Then, Hence, .
(ii) We have with , for every .
(1-ii) Suppose and , one has Since . By parts (1-i) and (1-ii), we have is linear. And for every as
(iii) One has with , for every and .
(2) If , for every and . Then then .
(iv) Evidently, from (24).
(3) Assume , one has so . (v) From (25), there are .
(vi) Clearly the closure of .
(vii) One gets , for or , for with
Theorem 12. Assume with , one has which is a prequasi Banach (csss), where , for all .
Proof. From Theorems 11 and 8, the space is a prequasi normed (csss). If is a Cauchy sequence in , then for all , we have such that for every , we obtain Therefore, Since is a complete metric space, so is a Cauchy sequence in , for constant . Then, , for fixed . So , for all . As Then, .
Theorem 13. If with , we have a prequasi closed (csss), where , for all .
Proof. By Theorems 11 and 8, the space is a prequasi normed (csss). When and , one has for every , there is such that for every , one gets Therefore, Since is a complete metric space, so is a convergent sequence in , for constant . Then, , for fixed . As We have .
Theorem 14. The function verifies the Fatou property, when so that , for every .
Proof. Assume with As is a prequasi closed space, we have . For every , then
Theorem 15. The function does not satisfy the Fatou property, for every , if and , for every .
Proof. Assume with As is a prequasi closed space, we have . For all , one can see
Example 16. For , the function is a norm on .
Example 17. The function is a prequasi norm (not a norm) on .
Example 18. The function is a prequasi norm (not a quasinorm) on .
Example 19. The function is a prequasi norm, quasi norm, and not a norm on , for .
Definition 20. (1) If and . Mark For , let Suppose , we take (2) The function holds (UUC2) when for all and , one has such that (3) The function is strictly convex, (SC), when for every with and one gets
In the next part of this section, we will use the function as , for all .
Theorem 22. If so that , one has is (UUC2).
Proof. Suppose and . If with By using the definition of , one can see then Assume and . For all one has Therefore, or Let first In view of Lemma 21, part (i), one gets then Since by summing inequalities 2 and 3, and from inequality 1, one can see This implies After, assume Put Since and the power function is convex. Hence, As one has For all one obtains In view of Lemma 21, part (ii), one gets So then As by summing inequalities 5 and 6, we have As by summing inequalities 7 and 8, and from inequality 1, then So Evidently, From inequalities 4 and 9, and Definition 20, when we take Therefore, we have so is (UUC2).
Definition 23. The space verifies the property (), if and only if, for every decreasing sequence of -closed and -convex nonempty subsets of so that for some then
By denoting a nonempty -closed and -convex subset of .
Theorem 24. Suppose so that , we have (i)if such that One has a unique with (ii) satisfies the property ().
Proof. To prove (i), if as is -closed, we have . Then, for every , we have so that . Assume is not -Cauchy. There is a subsequence and so that for all Also, we obtain for every As for all , one has So for every . By choosing we have This is a contradiction. Hence, is -Cauchy. Since is -complete, one has -converges to some . For every , we have -converges to . As is -closed and -convex, we have