Existence and Uniqueness Theorems for Generalized Set Differential Equations

Abstract

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.

Some remarks on linear set-valued differential equations

“…Now, we consider linear differential equation (3.1) with PS-derivative and BGderivative. By [1,32,33,34,35], this set-valued differential equation (3.1) has at least one solution. Moreover, one of these solutions (the one whose diameter is a non-decreasing function) coincides with the solution of the corresponding differential equation (3.2).…”

One Control Problem for a Set-valued Object

“…A.V. Plotnikov and N.V. Skripnik took advantage of some approaches that were used in [4,14] and introduced a new definition of a derivative, and studied its properties [5][6][7][8][9].…”

“…Now, we consider linear set-valued differential equations 0 ( ) = ( ), T Now, we consider linear set-valued differential equation 1with PS-derivative and BG-derivative. By [5][6][7][8][9][10][11][12], setvalued differential equation 1with PS(BG)-derivative has at least one solution. Moreover, one of these solutions (the one whose diameter is a non-decreasing function) coincides with the solution of the corresponding Hukuhara differential equation and always exists.…”

Solution of the pollutant concentration optimization problem with restrictions on the intensity of sources

“…We emphasize that the existence of a weak solution to problems of mathematical physics are considered, for example, in [11][12][13], where the restrictions on the physical coefficients are similar to those given above.…”