#### Abstract

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.

#### 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

Lemma 21. (i) If and for every , one has (ii) Assume and for all with , one obtains

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