The development of the unified version mixing maps between arbitrary sets

The concepts of nonlinear mixed summable families and maps for the spaces that only non-void sets are developed. Several characterizations of the corresponding concepts are achieved and the proof for a general Pietsch Domination-type theorem is established. Furthermore, this work has presented plenty of composition and inclusion results between different classes of mappings in the abstract settings. Finally, a generalized notation of mixing maps and their characteristics are extended to a more general setting.


Notations and Preliminaries
In the beginning, some notations are introduced that will be used throughout this article. Let be an index set. The symbols , , and represent the sets of natural numbers, real numbers, and positive real numbers, respectively. The letter denotes the field of real or complex numbers. Let , , , and be non-void sets and be a non-void family of mappings from into . Let , and be Banach spaces and the closed unit ball of a Banach space is denoted by . The dual space of is denoted by . The letters and stand for pointed metric spaces. Suppose be a map from into , then can be defined to be Lipschitz if there is a nonnegative constant such that ( ) ( ) for all , in , where is the Lipschitz constant of ( ( )). In addition, let the space be the Lipschitz dual of that is the Banach space of real-valued maps defined on send the
Iraqi Journal of Science, 2022, Vol. 63, No. 3, pp: 1285-1298 1286 space point 0 to 0 with the Lipschitz norm ( ).Let and be compact Hausdorff topological spaces. The symbols ( ), ( ) and ( ) stand for the set of all Borel probability measures defined on , and , respectively. The value of at the element is denoted by ⟨ ⟩.

Introduction
The usual mathematical problems include nonlinear operators, occasionally influential on arbitrary sets with few (or none) algebraic structures, hence the extension of linear mechanisms to the nonlinear setting, besides its essential mathematical interest, is an important duty for potential applications. The full general version of maps (with no structure of the spaces included) would certainly be interesting for potential applications. Botelho et al. [1] defined the concept of R-S-abstract -summing map as follows. Let . A mapping is said to be R-S-abstract -summing if there is a constant such that for all , and , with . Several authors have investigated a special case version of the class of R-S-abstractsumming maps starting with the seminal papers [4], [5], [6] and [7] (linear version) and (Lipschitz version) and further explored applications in the nonlinear case can be found in [8] and [9]. This paper consists of 7 sections. In Section 3, inequality (2) is modified to construct the concept of H-Q-abstract -summing map which is quite useful to prove the main results under certain assumptions in the forthcoming sections. In Section 4, the nonlinear version concept of M-mixed ( )-summable family is defined in which the spaces are just arbitrary sets and establish an important characterization for this notion under certain hypotheses in abstract settings. In Section 5, the concept of H-M-(( ) )-mixing maps between arbitrary sets is constructed and several characterizations are proved. Afterwards, various compositions and inclusion results between different classes of mappings in abstract setting and a quite general of Pietsch [10] Domination-type Theorem are proved. Section 6 presented the proof of how Proposition 11 and Proposition 13 can be appealed in order to get some of the familiar characterizations that have appeared in the different generalizations of the concept of ( )mixing operators. It is obvious to see that for suitable choices of , , , , , , , , and , for a mapping to belong to one of such classes of mixing maps is equivalent to be H-M-(( ) )-mixing map and the corresponding characterizations that hold for this class is nothing but Proposition 11 and Proposition 13. Fnally, in Section 7, a notion of mixing maps is generalized and characterization for this notion to a more general setting is showed.

Properties of H-Q-abstract -summing maps
Let and be non-void families of mappings from into and into , respectively, and let be arbitrary maps satisfy the following conditions: for all nonzero in , in , in , in and . The infimum of such constants is denoted by ( ). Let ( ) be the class of all H-Q-abstract -summing maps from into . The next proposition has a similar implications as the nonlinear general Pietsch [10] Domination-type Theorem [Theorem 3.1], therefore it is omitted.  [4]. Define the following maps. ( With these choices one can obtain ( ( )) with ( ) and satisfy Inequality (4). The concept of M-mixed ( )-summable family can be constructed as follows.
where the infimum is taken over all nonzero families ( ) ( ). The next result will be used in the forthcoming section. Proof. Suppose that the family (( )) satisfies (6). Define a number as follows.
( ) Then is finite. Put and . Then . Now consider the compact, convex of ( ). Note that the equation and for This proves the necessity of the above condition. Conversely, suppose that a family (( )) is -mixed ( )-summable. Take any family ( ) ( ) such that ∑ | | | ( )| . Applying Hölder inequality, hence whenever ( ). This proves the sufficiency of the above condition.

Properties of H-M-(( ) )-mixing maps
Throughout this section, assume that be a vector space over the field and let ( ) be a finite family of semi-norms on . The topology induced by a finite family of semi-norms on is denoted by -topology on . If . Then ( ). For , , , and , from Proposition 9, it can be obtained that: In order to show the converse, (8) can be explained as for every discrete probability measure on and , , , and . Since ( ( ) ( ))-dense the set of all finitely supported probability measures on , then (9) in the set of all probability measures on , it follows that (9) satisfies all probability measures on and , , , . Taking the supremum over ( ) on the left side of (9) and using Proposition 9, it can be found that The following multiplication formula represents the main-point of the theory of H-M-(( ) )-mixing maps and it is somewhat inspired by analogous result in the linear theory. Proposition 12 Let . If the maps , , and satisfy conditions II and VI, respectively, then Proof. Suppose that ( ) and (( ) ) ( ). Given in , in , , in , and , then It can be noticed from By applying Hölder inequality and conditions (II) and (VI) one can obtain The following characterization is a quite general of unified Pietsch [10] Taking the supremum over on on the left side of (10) and from Proposition 9, then Conversely, suppose that is --(( ) )-mixing map. From Proposition 12 and using condition VII, be --abstract -summing map with ( ) (( ) ) ( ). Hence, by using Proposition 5, there exists a probability measure on such that   ((s, q), p , ..., p ) for all nonzero , ̃, ̃, ̃ and . The infimum of such constants is denoted by H ,..., H -M (( ) ) ( ). Let us denote by

Conclusions and discussion
This work is concerned with the development of the unified version of mixing maps between arbitrary sets. The innovative general approach has been avoided the multiplication and the appearance of apparently different proofs of Pietsch Domination-type theorems. Based on the good results are achieved in the present proofs, it has encouraged the forthcoming work to focus on developing new nonlinear prototypes of mixing operators with new general settings.

Disclosure and conflict of interest
The authors declare that they have no conflicts of interest.