Conditions for the existence of basic solutions of linear multivalued differential equations

  • T. A. Komleva Odessa State Academy of Civil Engineering and Architecture
  • A. B. Plotnikov Odessa State Academy of Civil Engineering and Architecture https://orcid.org/0000-0002-7864-0732
  • L. I. Plotnikova Odessa National Polytechnic University
  • N. V. Skripnik Odessa I.I. Mechnikov National University
Keywords: set-valued differential equation, Hukuhara derivative, conditions of existence

Abstract

UDC 517.9

In this paper, we discusses various definitions and properties of the derivative of a set-valued mapping.
Also, we consider a linear set-valued differential equation and investigate the problem of existence of solutions of this equation with Hukuhara derivative, PS-derivative and BG-derivative.
The obtained results are illustrated with model examples.

References

F. S. de Blasi, F. Iervolino, Equazioni differentiali con soluzioni a valore compatto convesso, (Italian), Boll. Unione Mat. Ital., 2, No. 4-5, 491 - 501 (1969)

V. Lakshmikantham, T. Granna Bhaskar, J. Vasundhara Devi, Theory of set differential equations in metric spaces, Cambridge Sci. Publ. (2006).

V. Lupulescu, D. O’Regan, A new derivative concept for set-valued and fuzzy-valued functions. Differential and integral calculus in quasilinear metric spaces, Fuzzy Sets and Syst., 404, 75 – 110 (2021). DOI: https://doi.org/10.1016/j.fss.2020.04.002

A. A. Martynyuk, Qualitative analysis of set-valued differential equations, Springer Nature Switzerland AG, Birkhäuser, Cham (2019), https://doi.org/10.1007/978-3-030-07644-3 DOI: https://doi.org/10.1007/978-3-030-07644-3

N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, N. V. Skripnik, Differential equations with impulse effects: multivalued right-hand sides with discontinuities, de Gruyter Stud. Math., 40, Walter De Gruyter GmbH& Co, Berlin; Boston (2011), https://doi.org/10.1515/9783110218176 DOI: https://doi.org/10.1515/9783110218176

A. V. Plotnikov, N. V. Skripnik, Дифференциальные уравнения с «четкой» и нечеткой многозначной правой частью. Асимптотические методы (Russian) [[Differenczial`ny`e uravneniya s “chetkoj” i nechetkoj mnogoznachnoj pravoj chast`yu. Asimptoticheskie metody]], AstroPrint, Odessa (2009)

V. A. Plotnikov, A. V. Plotnikov, A. N. Vityuk, Дифференциальные уравнения с многозначной правой частью. Асимптотические методы (Russian) Differenczial`ny`e uravneniya s mnogoznachnoj pravoj chast`yu.Asimptoticheskie metody]], AstroPrint, Odessa (1999).

A. Tolstonogov, Differential inclusions in a Banach space, Kluwer Acad. Publ., Dordrecht (2000), https://doi.org/10.1007/978-94-015-9490-5 DOI: https://doi.org/10.1007/978-94-015-9490-5

A. V. Plotnikov, T. A. Komleva, L. I. Plotnikova, Averaging of a system of set-valued differential equations with the Hukuhara derivative, J. Uncertain Syst., 13, 3 – 13 (2019).

M. Hukuhara, Integration des applications mesurables dont la valeur est un compact convexe, (French), Funkcial. Ekvac., 10, 205 - 223 (1967).

H. Minkowski, Zur Geometrie der Zahlen, Verhandlungen des III Internationalen Mathematiker-Kongresses in Heidelberg, Heidelberg, Berlin (1904), p. 164 – 173.

T. A. Komleva, L. I. Plotnikova, N. V. Skripnik, A. V. Plotnikov, Some remarks on linear set-valued differential equations, Stud. Univ. Babe¸s-Bolyai Math., 65, № 3, 415 – 431 (2020); https://doi.org/10.24193/subbmath.2020.3.09 DOI: https://doi.org/10.24193/subbmath.2020.3.09

A. V. Plotnikov, N. V. Skripnik, Set-valued differential equations with generalized derivative, J. Adv. Res. Pure Math., 3, № 1, 144 – 160 (2011); https://doi.org/10.5373/jarpm.475.062210 DOI: https://doi.org/10.5373/jarpm.475.062210

¸ S. E. Amrahov, A. Khastan, N. Gasilov, A. G. Fatullayev, Relationship between Bede – Gal differentiable set-valued functions and their associated support functions, Fuzzy Sets and Syst., 265, 57 – 72 (2016); https://doi.org/10.1016/j.fss.2015.12.002 DOI: https://doi.org/10.1016/j.fss.2015.12.002

M. T. Malinowski, Second type Hukuhara differentiable solutions to the delay set-valued differential equations, Appl. Math. and Comput., 218, 9427 – 9437 (2012); https://doi.org/10.1016/j.amc.2012.03.027. DOI: https://doi.org/10.1016/j.amc.2012.03.027

M. T. Malinowski, On set differential equations in Banach spaces — a second type Hukuhara differentiability approach, Appl. Math. and Comput., 219, 289 – 305 (2012); https://doi.org/10.1016/j.amc.2012.06.019. DOI: https://doi.org/10.1016/j.amc.2012.06.019

H. Vu, L. S. Dong, Initial value problem for second-order random fuzzy differential equations, Adv. Difference Equat., 2015, Article 373 (2015), 23 p.; DOI: https://doi.org/10.1186/s13662-015-0710-5 DOI: https://doi.org/10.1186/s13662-015-0710-5

H. Vu, N. Van Hoa, On impulsive fuzzy functional differential equations, Iran. J. Fuzzy Syst., 13, № 4, 79 – 94 (2016); https://doi.org/10.22111/IJFS.2016.2597

E. S. Polovinkin, Mnogoznachny`j analiz i differenczial`ny`e vklyucheniya, Fizmatlit, Moskva (2014).

T. F. Bridgland, Trajectory integrals of set valued functions, Pacif. J. Math., 33, № 1, 43 – 68 (1970). DOI: https://doi.org/10.2140/pjm.1970.33.43

H. T. Banks, M. Q. Jacobs, A differential calculus for multifunctions, J. Math. Anal. and Appl., 29, 246 – 272 (1970); https://doi.org/10.1016/0022-247X(70)90078-8 DOI: https://doi.org/10.1016/0022-247X(70)90078-8

Yu. N. Tyurin, Mathematical statement of the simplified model of industrial planning (in Russian), Econ. Math. Meth., 3, 391 – 409 (1965).

A. V. Plotnikov, Differenczirovanie mnogoznachny`kh otobrazhenij, $T $-proizvodnaya, Ukr. mat. zhurn., 52, № 8, 1119 – 1126 (2000).

Y. Chalco-Cano, H. Roman-Flores, M. D. Jimenez-Gamero, Generalized derivative and pi -derivative for set-valued functions, Inform. Sci., 181, № 11, 2177 – 2188 (2011); https://doi.org/10.1016/j.ins.2011.01.023. DOI: https://doi.org/10.1016/j.ins.2011.01.023

A. Lasota, A. Strauss, Asymptotic behavior for differential equations which cannot be locally linearized, J. Different. Equat., 10, 152 – 172 (1971); https://doi.org/10.1016/0022-0396(71)90103-3 DOI: https://doi.org/10.1016/0022-0396(71)90103-3

M. Martelli, A. Vignoli, On differentiability of multi-valued maps, Boll. Unione Mat. Ital., 10, 701 – 712 (1974).

N. V. Plotnikova, Systems of linear differential equations with $pi$ -derivative and linear differential inclusions, Sb. Math., 196, № 11, 1677 – 1691 (2005); https://doi.org/10.1070/SM2005v196n11ABEH003727. DOI: https://doi.org/10.1070/SM2005v196n11ABEH003726

N. V. Hoa, N. D. Phu, Fuzzy functional integro-differential equations under generalized H-differentiability, J. Intell. Fuzzy Syst., 26, 2073 – 2085 (2014); https://doi.org/10.3233/IFS-130883. DOI: https://doi.org/10.3233/IFS-130883

N. D. Phu, N. N. Hung, Minimum stability control problem and time-optimal control problem for fuzzy linear control systems, Fuzzy Sets and Syst., 371, 1 – 24 (2019); DOI: https://doi.org/10.1016/j.fss.2018.09.005. DOI: https://doi.org/10.1016/j.fss.2018.09.005

B. Bede, S. G. Gal, Almost periodic fuzzy-number-valued functions, Fuzzy Sets and Syst., 147, 385 – 403 (2004); https://doi.org/ 10.1016/j.fss.2003.08.004 DOI: https://doi.org/10.1016/j.fss.2003.08.004

B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equation, Fuzzy Sets and Syst., 151, 581 – 599 (2005); https://doi.org/ 10.1016/j.fss.2004.08.001 DOI: https://doi.org/10.1016/j.fss.2004.08.001

L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal., 71, 1311 – 1328 (2009); https://doi.org/10.1016/j.na.2008.12.005 DOI: https://doi.org/10.1016/j.na.2008.12.005

N. V. Plotnikova,Approksimacziya puchka reshenij linejny`kh differenczial`ny`kh vklyuchenij, Nelinijni kolivannya, 9, № 3, 386 – 400 (2006).

V. G. Boltyanski, J. Jer´onimo Castro, Centrally symmetric convex sets, J. Convex Anal., 14, № 2, 345 – 351 (2007).

A. V. Plotnikov, N. V. Skripnik, Existence and uniqueness theorems for generalized set differential equations, Int. J. Control Sci. and Eng., 2, № 1, 1 – 6 (2012); https://doi.org/10.5923/j.control.20120201.01. DOI: https://doi.org/10.5923/j.Control.20120201.01

A. V. Plotnikov, N. V. Skripnik, An existence and uniqueness theorem to the Cauchy problem for generalized set differential equations, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal., 20, № 4, 433 – 445 (2013).

A. V. Plotnikov, N. V. Skripnik, Mnogoznachny`e differenczial`ny`e uravneniya s obobshhennoj proizvodnoj, Ukr. mat. zhurn., 65, № 10, 1350 – 1362 (2013).

Published
24.05.2021
How to Cite
Komleva, T. A., A. B. Plotnikov, L. I. Plotnikova, and N. V. Skripnik. “Conditions for the Existence of Basic Solutions of Linear Multivalued Differential Equations”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 5, May 2021, pp. 651 -73, doi:10.37863/umzh.v73i5.6356.
Section
Research articles