Interval – Valued Differential Equations with Generalized Derivative

In this paper the concept of generalized differentiab ility (proposed in [17]) for interval-valued mappings is used. The interval-valued differential equations with generalized derivative are considered and the existence theorem is proved.


Introduction
Lately the development of calculus in metric spaces became an object of attention of many researchers [1,11-14,16,18 and ref. herein ]. Earlier F.S. de Blasi and F. Iervolino begun studying of set-valued differential equations (SDEs) in semilinear metric spaces [5][6][7][8]. No w it transformed into the theory of SDEs as an independent discipline. The properties of solutions, the impulse SDEs, control systems and asymptotic methods for SDEs were considered. On the other hand, SDEs are useful in other areas of mathematics. For examp le, 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 generalizat ion of usual ordinary d ifferential equations in fin ite (or infinite) d imensional Banach spaces.
However all these equations have a natural lack -the diameter of a set-valued solution is a non-decreasing function. Possibly it is connected with the fact that these differential equations were entered by analogy with the single-valued theory.
But in the theory of ordinary d ifferential equations a solution in any mo ment of time is a point (so a solution does not possess the property of "thickness"). Therefore, the similar introduction of the differential equations for a set-valued case is not absolutely adequate.
In [17] 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 examp le, to be periodic) was considered. In the ideological sense this definition of the derivative is close to the definitions proposed in [2][3][4]15].
In this paper the interval-valued differential equations with generalized derivative are considered and the existence theorem is proved.

The Generalized Derivative
Let be a space of all nonempty closed intervals with Hausdorff metric Definiti on 1 [10]. Let A set such that is called a Hu kuhara difference of the sets X and Y and is denoted by . Let The Hukuhara d ifference of the sets and e xists iff and is equal to Let be an interval-valued mapping; be a -neighbourhood of a point For any consider the following Hukuhara differences if these differences exist. (1) The differences (1) and (2) (2) and (4); c) (2) and (3); d) (1) and (4). Consider four types of limits corresponding to one of the difference types: So it is possible to say that in the point not more than two limits can exist (as we assumed that there exist only two of four Hukuhara differences).

Definiti on 2.
If the corresponding two limits exist and are equal we will say that the mapping is differentiable in the generalized sense in the point and denote the generalized derivative by Let us say that the interval-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.
The interval-valued mapping is called absolutely continuous on the interval I if there exist a measurable interval-valued mapping and a system of intervals such that for all or

Differential Equations with the Generalized Derivative
Consider the differential equation with the generalized derivative (9) where are interval valued mappings; is a continuous function, function

Case Let
Then the interval-valued mapping satisfies the integral equation i.e. the functions satisfy the system of integral equations Therefore the functions satisfy the system of differential equations Using the Caratheodory theorem [9] we have that there exists a solution of this system defined on the interval

Case
. Then the interval-valued mapping satisfies the integral equation i.e. the functions satisfy the system of integral equations Therefore the functions satisfy the system of differential equations Using the Caratheodory theorem we have that there exists a solution of this system defined on the interval . Then there exists a solution of differential equation (9) provided that The fact that follows fro m the condition c) of the theorem: because on intervals where doesn't decrease.
3) Then we have So the solution of differential equation (9) exists on .

Conclusions
In this paper the concept of generalized differentiability (proposed in [17]) for interval-valued mappings is used. The interval-valued differential equations with generalized derivative are considered and the existence theorem is proved.