Existence of basic solutions of first order linear homogeneous set-valued differential equations

A. V. Plotnikov; T. A. Komleva; N. V. Skripnik

doi:10.30970/ms.61.1.61-78

A. V. Plotnikov Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine
T. A. Komleva Odessa State Academy of Civil Engineering and Architecture
N. V. Skripnik Odessa I.I. Mechnikov National University, Odessa, Ukraine

DOI: https://doi.org/10.30970/ms.61.1.61-78

Keywords: set-valued mapping, set-valued differential equation, Cauchy problem, Hukuhara derivative

Abstract

The paper presents various derivatives of set-valued mappings,
their main properties and how they are related to each other.
Next, we consider Cauchy problems with linear homogeneous
set-valued differential equations with different types of
derivatives (Hukuhara derivative, PS-derivative and
BG-derivative). It is known that such initial value problems with
PS-derivative and BG-derivative have infinitely many solutions.
Two of these solutions are called basic. These are solutions such
that the diameter function of the solution section is a
monotonically increasing (the first basic solution) or monotonically
decreasing (the second basic solution) function. However, the second
basic solution does not always exist. We provide
conditions for the existence of basic solutions of such initial
value problems. It is shown that their existence depends on the
type of derivative, the matrix of coefficients on the right-hand
and the type of the initial set. Model examples are considered.

Author Biographies

A. V. Plotnikov, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine

Odessa State Academy of Civil Engineering and Architecture,

Odessa, Ukraine

T. A. Komleva, Odessa State Academy of Civil Engineering and Architecture

Odessa State Academy of Civil Engineering and Architecture

N. V. Skripnik, Odessa I.I. Mechnikov National University, Odessa, Ukraine

Odessa I.I. Mechnikov National University,
Odessa, Ukraine

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