Some Operations on Mixed Monotone Operator in Banach Spaces

This paper discusses some operations on mixed monotone operator in Banach space, especially addition an multiplication operations. We will prove the sum and product of two mixed monotone operators. The proof using some relevant definitions. The result is the sum o of them is a mixed monotone operator and the product is  too if both  satisfy some conditions


Introduction
There are many classes of real functions. One of them is a monotone function. On the other side, we have known a term called mixed monotone operator, that is a function defined on the Cartesian Product of two subsets of Real Banach Space [1][2][3][4][5][6][7][8][9]13]. It has been well-known that under operation of addition, the sum of two monotone functions is monotone function too, but not for case of multiplication operation [10]. Inspired by both of result above, we interested on knowing how about addition and multiplication of two mixed monotone operator defined in real Banach space. Hence, we will prove it here. The objective of this research is to prove the sum and product of two mixed monotone operators and to find some conditions in order to the product is mixed monotone operator.

Research Method
To prove the sum and product of two mixed monotone operators done in three steps. Firstly by defining some partial ordering in subset of Banach space. Then to find some conditions of mixed monotone operators. The last showing that two relevant operations satisfy some definition.

Result and Discussion 1. Prelimineries
In this section we will give some definitions and theories of monotone function. Definition 1.1: Let be a nonvoid set and is relation on set . Relation is called partial ordering on if satisfy the following property:  Let is a Real Banach space, which is partially ordered by a cone , i.e., ≤ iff − ∈ . Definition 1.8: Let ⊂ . An operator : × → is said to be mixed monotone if ( , ) is nondecreasing in and nonincreasing in .

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[13]

Main Result
In this part we assume E is real Banach space which is partially ordered by a cone P, put ̃= {( , ) ∈ × | ≥ 0, ≤ 0}. It is clear that ̃ is a cone in × .

Proof:
To prove that ≼ is partial ordering in × firstly we show that the relation ≤ on × is reflexsive. For the condition ( ) dan ( ), we leave it.

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Let Let ⊂ and operator , : × → be mixed monotone operators. The sum of two mixed monoton operaton in the following Theorem 2.2. Theorem 2.2: Let Let ⊂ and operator , : × → . If and are mixed monotone operators then + is a mixed monotone operator.
The result of the product of two mixed monotone operators given in the following Theorem 2.3. The last, we show that is nonincreasing in .  Hence, in according to Definition 1.8, we conclude that is a mixed monotone operator.

Conclusion
In this paper we have the conlusion that the sum of two mixed monotone operator is mixed monotone operator, while the product is mixed monotone operator if both are positive operator.