Existence and Uniqueness Theorems for Generalized Set Differential Equations

In this paper the concept of generalized differentiability for set-valued mappings proposed by A.V. Plotnikov, N.V. Skripnik is used. The generalized set-valued differential equations with generalized derivative are considered and the existence and uniqueness theorems are proved.

F.S. de Blasi and F. Iervolino begun studying of set-valued differential equations (SDEs) in semilinear metric spaces [12,[19][20][21]. Now it developed in the theory of SDEs as an independent discipline. The properties of solutions, the impulsive SDEs, control systems and asymptotic methods for SDEs were considered [5,6,[9][10][11][16][17][18][19][20][21][22][23][24]. On the other hand, SDEs are useful in other areas of mathematics. For example, SDEs are used as an auxiliary tool to prove the existence results for differential inclusions. Also, one can employ SDEs in the investigation of fuzzy differential equations. Moreover, SDEs are a natural generalization of usual ordinary differential equations in finite (or infinite) dimensional Banach spaces [19].
In [9] a new concept of a derivative of a set-valued mapping that generalizes the concept of Hukuhara derivative was entered and a new type of a set-valued differential equation such that the diameter of its solution can whether increase or decrease (for example, to be periodic) was considered. In the ideological sense this definition of the derivative is close to the definitions proposed in [5,6,8].
In this paper the generalized set-valued differential equations with generalized derivative are considered and the existence and uniqueness theorems are proved.

The Generalized Derivative
∈ − ∆ + ∆ consider the following Hukuhara differences if these differences exist.
The differences (1) and (2)[ (3) and (4)] are called the right[left] differences. From the definition of the Hukuhara difference it follows that both one-sided differences exist only in the case when ( ) ∈ − ∆ + ∆ there exists only one of the one-sided differences, then using the properties of the Hukuhara difference, we get that the mapping Consider four types of limits corresponding to one of the difference types: ( ) So it is possible to say that in the point 0 t not more than two limits can exist (as we assumed that there exist only two of four Hukuhara differences).
Considering all above we have that there can exist only the following combinations of limits: a) (5) and (7); b) (6) and (8); c) (6) and (7); d) (5) and (8). Definition 2 [9]. If the corresponding two limits exist and are equal we will say that the mapping ( ) X ⋅ is differentiable in the generalized sense in the point 0 t and denote the generalized derivative by 0 ( ) DX t . Let us say that the set-valued mapping : is differentiable in the generalized sense on the interval I if it is differentiable in the generalized sense at every point of this interval.
Remark 1. Properties of the generalized derivative have been considered in [9]. Definition 3 [9]. The set-valued mapping if there exist a measurable set-valued mapping ( ) G t and a system of intervals 1 [ , ]

Generalized Differential Equations with the Generalized Derivative
First consider a differential equation with the generalized derivative that is similar to a differential equation with the Hukuhara derivative, i.e.
( , ) ( ) is said to be solution of differential equation (9) if it is absolutely continuous and satisfies (9)  t T . Remark 2. Unlike the case of differential equations with Hukuhara derivative, if a differential equation with the generalized derivative (9) has a solution then there exists an infinite number of solutions irrespective of the conditions on the right-hand side of the equation. Example 1. Consider the following differential equation with the generalized derivative (10) It is easy to check that the following set-valued mappings are the solutions of equation (10): Also it is possible to construct other solutions, thus only 1 ( ) X ⋅ will be the solution of the corresponding differential equation with the Hukuhara derivative and 1 ( ) X ⋅ and 2 ( ) X ⋅ are solutions of the differential equation with the generalized derivative (in the sense of [8]).
n ≥ be a space of all nonempty strictly convex closed sets of n R and all element of n R [27]. The following theorem of existence of the solution of equation (11)  [ , ] t t t a ∈ + . Then equation (11) is the ordinary differential equation with Hukuhara derivative (14) Therefore, using [17] we get that the equation (11) According to Definition 5 consider the following integral equation and prove the existence of solution on the some interval 0 0 [ , ] t t d