Integral Dissipative Set-valued Maps

There are different forms of dissipativity as considered in , , , [7] and .These dissipativity conditions have their relationships with monotonicity in the sense of Minty , if a map A is maximal dissipative then -A is maximal monotone.However, there exists some multifunctions that are integral dissipative,but not necessarily dissipative. Such multifunctions exist on the attainable trajectories of neutral functional differential inclusions. Integrals of set-valued maps (multivalued maps, multifunctions) were considered by Aumann in , the properties of these integrals and their convex hull were established. In [5] Kisielewicz considered some further properties of the space of Aumann integrable set-valued maps. He also considered integral dissipative set-valued maps. This dissipativity condition is applicable in the case where the dissipativity of a system is on the average over a (continuous) period of time. In this work, we shall consider some properties of these integral dissipative set-valued maps, and show that their convex hull is also integral dissipative. We proved that the space of these maps is a complete metric space with respect to Hausdorff topology.


INTRODUCTION
There are different forms of dissipativity as considered in [1] , [4] , [5] , [7] and [8] .These dissipativity conditions have their relationships with monotonicity in the sense of Minty [6] , if a map A is maximal dissipative then -A is maximal monotone.However, there exists some multifunctions that are integral dissipative,but not necessarily dissipative. Such multifunctions exist on the attainable trajectories of neutral functional differential inclusions. Integrals of set-valued maps (multivalued maps, multifunctions) were considered by Aumann in [2] , the properties of these integrals and their convex hull were established. In [5] Kisielewicz considered some further properties of the space of Aumann integrable set-valued maps. He also considered integral dissipative set-valued maps. This dissipativity condition is applicable in the case where the dissipativity of a system is on the average over a (continuous) period of time. In this work, we shall consider some properties of these integral dissipative set-valued maps, and show that their convex hull is also integral dissipative. We proved that the space of these maps is a complete metric space with respect to Hausdorff topology.

Set-Valued Maps
Let X, Y be sets , a map Y X F → : is said to be a setvalued map(multi-valued map, or multifunction) if for every . By a selection of a setvalued map F, we mean a single-valued map f, such that The basic properties of set-valued maps have been extensively considered in [1] and [5] .

Convex Hull Let
be a set-valued map, the intersection of all the convex sets in Y containing F is called the convex hull of F, it is denoted by co(F).

The Space (Comp(X), h)
Let Comp(X) be the family of all non-empty compact subsets of a metric space ) , , the Hausdorff distance, h, on Comp(X), is defined as: The family of all Aumann integrable maps from I to n ℜ , shall be denoted by ) , ( as follows: ( X cl , the space of all closed and bounded subsets of X. X is a complete metric space , and hence ( ) The following inequality is satisfied :

RESULTS AND DISCUSSION
The following results due to Aumann [2] shall be used to prove our main results.

Main Results
We first showed that if a set-valued map F is integral dissipative, so is its convex hull. We then showed that the space of all integral dissipative maps is complete.
dissipative, then its convex hull coF is also integral dissipative.

Theorem 2 Let
) ( n F be a Cauchy sequence of integral dissipative maps, which converges to F. Then F is also integral dissipative.

Proof
We have to show that F is Aumann integrable and integral dissipative. Let

CONCLUSION
The space of Aumann integrable maps is sequentially complete with respect to Hausdorff toplogy. A subspace of this space with the property of integral dissipativity is also complete. Furthermore, the convex hull of integral dissipative multivalued map is also integral dissipative. This can be used to establish a relaxation theorem for integral inclusions having such multifunctions.