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On the Directonal Derivative Sets and Differentials of the Set Valued Maps

In this paper directional derivative sets and differentials of a given set valued map are studied. Relations between the set valued map and compact subsets of the directional derivative sets of the given map are investigated. Upper and lower contingent cones of some plane sets are calculated.


Introduction
Set valued maps arise in the mathematical models of physics, economics and biology. They are important tools for investigation of the optimal control, optimization and game theory problems. In studying of these problems it is often required to deal with differential or directional derivative sets of the given set valued map. In general, the differential notions of the set valued maps are based on the various types of tangent and contingent cones. In this paper, the upper and lower differentials of the set valued map are defined via upper and lower Bouligand contingent cones, which are used in many problems of set valued and nonsmooth analysis. The definition of directional derivative set is different from the definition of differential, but they also are closely connected with the concept of contingent cones. In the presented paper, the connections between differentials (upper and lower) and directional derivative sets (upper and lower) of a set valued map are studied. Upper and lower contingent cones of the sets given on the plane are calculated. The paper is organized as follows: In Section 2 the upper and lower contingent cones of the sets given on the plane are calculated (Example 2.1 and Example 2.2). In Section 3 connections between directional upper (lower) derivative sets and upper (lower) differentials are given. It is shown that if the set valued map is not locally Lipschitz continuous, then lower derivative set in the direction p and the value of the lower differential at p does not coincide (Example 3.1).
For scalar variable set valued maps, it is proved that upper derivative set in the direction p and the value of the upper differential at p are equal (Theorem 3.1). In Section 4 the properties of the compact subsets of the directional derivative sets and differentials are investigated. The Hausdorff deviation of the cone generated by a compact subset of the lower directional derivative set from the given set valued map is estimated (Theorem 4.1).

Upper and Lower Contingent Cones
Let us give the definitions of upper and lover contingent cones.
Definition 2.1 Let X be a Banach space, K ⊂ X be a closed set and x ∈ X. The sets are called the upper and lower contingent cone of the set K at x ∈ X respectively, where d(y, K) = inf z∈K y − z , i.e. d(y, K) is the distance from the point y to the set K.
T U K (x) and T L K (x) are closed cones in the space X. It is obvious that u ∈ T U K (x) if and only if there exist sequences Similarly, u ∈ T L K (x) if and only if there exist δ * > 0 and s(·) : (0, δ * ] → X such that s(δ) → 0 as δ → 0 + and the inclusion x(δ) = x + δu + δs(δ) ∈ K is verified for every δ ∈ [0, δ * ]. Now, let us calculate contingent cones of the sets given on the plane.

Thus, we have
Example 2.2 Let the set Ω ⊂ R 2 be defined as . By virtue of (2.9) we have , Ω = 0, Note that similarly to the Example 2.1 it is possible to show that

Upper and Lower Directional Derivative Sets
In this section upper and lower differentials and directional derivative sets of the set valued maps are investigated. The graph of the set valued map F (·) : X Y is denoted by grF (·) and is defined as grF (·) = {(x, y) ∈ X × Y : y ∈ F (x)} , where X and Y are Banach spaces.
Definition 3.1 Let F (·) : X Y be a given set valued map. Assume that for every x ∈ X there exist L x ≥ 0 and r x > 0 such that for every y ∈ B(x, r x ) and z ∈ B(x, r x ) the inequality is satisfied. Then the set valued map F (·) is called locally Lipschitz continuous.
Here h (F (y), F (z)) denotes the Hausdorff distance between the sets F (y) and F (z), is called the upper differential of the set valued map F (·) at the point (x, y).
Here T U grF (·) (x, y) is upper contingent cone of the set grF (·) at the point (x, y).
is called the lower differential of the set valued map F (·) at the point (x, y).
Here T L grF (·) (x, y) is lower contingent cone of the set grF (·) at the point (x, y).

Now, let us formulate definitions of the upper and lower directional derivative sets of a given set valued map.
Definition 3.4 Let F (·) : X Y be a set valued map, (x, y) ∈ X × Y and p ∈ X. The is called upper derivative set of the set valued map F (·) at the point (x, y) in the direction p.
Definition 3.5 Let F (·) : X Y be a set valued map, (x, y) ∈ X × Y and p ∈ X. The set is called lower derivative set of the set valued map F (·) at the point (x, y) in the direction p.
It is obvious that for given set valued F (·) : X Y the inclusions are satisfied for every (x, y) ∈ grF (·) and p ∈ X.
The following propositions characterize lower and upper derivative sets and differentials of the set valued maps.
Proposition 3.1 Let F (·) : X Y be a set valued map, grF (·) be a closed set, (x, y) ∈ grF (·). Then for every p ∈ X the inclusions are verified.
If F (·) : X Y is a locally Lipschitz continuous set valued map, then for every p ∈ X the equalities hold.
Note that if F (·) is not a locally Lipschitz continuous set-valued map, then the equality (3.1) The map F (·) : R R defined by (3.1) is not locally Lipschitz continuous on R. Since for every δ > 0, then we conclude that lim Now let us show that 0 ∈ D L F (0, 0)|(1), which is equivalent to the inclusion (1, 0) ∈ T L grF (·) (0, 0).
Then for each p ∈ R \ {0} the equality holds.
Since v ∈ D U F (x, y)|(p) is arbitrarily chosen, we obtain validity of the inclusion (3.8).

Properties of the Compact Subsets of the Directional Derivative Sets
In this section, the relations between the compact subsets of the directional derivative sets and the given set valued map are studied.
The Hausdorff deviation of the set E from the set D is denoted by h * (E, D) and defined Theorem 4.1 Let F : X Y be a set valued map, grF (·) be a closed set, p ∈ X, G ⊂ Y be a compact set and (x, y) ∈ grF (·). Assume that the inclusion is satisfied. Then the equality (4.2) lim δ→0 1 δ h * (y + δG, F (x + δp)) = 0 is valid.
Proof. Let us assume the contrary, i.e., let the equality (4.2) do not be satisfied. Then there exist a sequence {δ i } ∞ i=1 and α * > 0 such that δ i → 0 + as i → +∞ and Let µ * < α * be an arbitrary number. It follows from (4.3) that there exists N 1 > 0 such that h * (y + δ i G, F (x + δ i p)) > µ * 2 δ i for every i > N 1 . Then for each i > N 1 there exists a g i ∈ G such that and hence Since G ⊂ Y is a compact set, g i ∈ G for every i = 1, 2, . . . , then without loss of generality we may assume that g i → g * as i → ∞ and g * ∈ G.