On the analytical solution of the Cauchy problem for a linear set-valued differential equation with a Hukuhara derivative

Tatyana A. Komleva; Andrej V. Plotnikov; Natalia V. Skripnik

doi:10.56754/0719-0646.2802.391

DOI:

https://doi.org/10.56754/0719-0646.2802.391

Abstract

The article considers the Cauchy problem for a linear set-valued differential equation with the Hukuhara derivative and derives an analytical formula for its solution.

Keywords

Differential equation , linear , set-valued mapping , Hukuhara derivative

Mathematics Subject Classification:

26E25 , 34A60 , 34A30 , 34A12

Tatyana A. Komleva Odessa State Academy of Civil Engineering and Architecture, Didrihsona str. 4, 65029 Odessa, Ukraine. https://orcid.org/0000-0001-5970-6241
Andrej V. Plotnikov Odessa State Academy of Civil Engineering and Architecture, Didrihsona str. 4, 65029 Odessa, Ukraine. https://orcid.org/0000-0002-7864-0732
Natalia V. Skripnik Odessa I.I. Mechnikov National University, Vsevoloda Zmiienka str. 2, 65082 Odessa, Ukraine. https://orcid.org/0000-0002-3333-7553

Pages: 391-408
Date Published: 2026-05-31
Vol. 28 No. 2 (2026)

S. N. Avvakumov and Y. N. Kiselëv, “Support functions of some special sets, constructive smoothing procedures, and geometric difference,” in Problemy dinamicheskogo upravleniya. Vyp. 1. Moscow: Moskovskii Gosudarstvennyı Universtitet im. M. V. Lomonosova, Fakul’tet Vychislitel’noi Matematiki i Kibernetiki, 2005, pp. 24–110.

V. G. Boltyanski and J. Jerónimo Castro, “Centrally symmetric convex sets,” J. Convex Anal., vol. 14, no. 2, pp. 345–351, 2007.

F. S. De Blasi and F. Iervolino, “Equazioni differenziali con soluzioni a valore compatto convesso,” Boll. Unione Mat. Ital., IV. Ser., vol. 2, pp. 491–501, 1969.

G. E. Forsythe and C. B. Moler, Computer Solution of Linear Algebraic Systems. Englewood Cliffs, N. J.: Prentice-Hall, 1967.

J. Gipple, “The volume of n-balls,” Undergrad. Math J., vol. 15, no. 1, pp. 237–248, 2014.

G. H. Golub and C. F. Van Loan, Matrix computations, 4th ed. Baltimore, MD: The Johns Hopkins University Press, 2013.

A. Halder, “Smallest ellipsoid containing p-sum of ellipsoids with application to reachability analysis,” IEEE Trans. Autom. Control, vol. 66, no. 6, pp. 2512–2525, 2021, doi: 10.1109/TAC.2020.3009036.

M. Hukuhara, “Intégration des applications mesurables dont la valeur est un compact convexe,” Funkc. Ekvacioj, Ser. Int., vol. 10, pp. 205–223, 1967.

T. A. Komleva, A. V. Plotnikov, L. I. Plotnikova, and N. V. Skripnik, “Conditions for the existence of basic solutions of linear multivalued differential equations,” Ukr. Math. J., vol. 73, no. 5, pp. 758–783, 2021, doi: 10.1007/s11253-021-01958-3.

T. Komleva, A. Plotnikov, and N. Skripnik, “On solutions of a linear set-valued differential equation with a conformable fractional derivative,” Filomat, vol. 39, no. 33, pp. 11751–11764, 2025, doi: 10.2298/FIL2533751K.

T. Komleva, A. Plotnikov, and N. Skripnik, “On the solution of the Cauchy problem for a linear in homogeneous differential equation with a Hukuhara derivative,” Neliniini Kolyvannya, vol. 28, no. 2, pp. 196–205, 2025, doi: 10.3842/nosc.v28i2.1493.

T. A. Komleva, A. V. Plotnikov, and N. V. Skripnik, “Some properties of solutions of a linear set-valued differential equation with conformable fractional derivative,” Cubo, vol. 26, no. 2, pp. 191–215, 2024, doi: 10.56754/0719-0646.2602.191.

T. A. Komleva, A. V. Plotnikov, and N. V. Skripnik, “Solution of the Cauchy problem for impulsive linear set-valued differential equations with a conformable fractional-fractal derivative,” Mem. Differ. Equ. Math. Phys., vol. 97, pp. 81–92, 2026.

T. A. Komleva, L. I. Plotnikova, N. V. Skripnik, and A. V. Plotnikov, “Some remarks on linear set-valued differential equations,” Stud. Univ. Babes, -Bolyai, Math., vol. 65, no. 3, pp. 411–427, 2020, doi: 10.24193/subbmath.2020.3.09.

V. Lakshmikantham, T. Gnana Bhaskar, and J. Vasundhara Devi, Theory of set differential equations in metric spaces. Cambridge: Cambridge Scientific Publishers, 2006.

V. Lakshmikantham and R. N. Mohapatra, Theory of fuzzy differential equations and inclusions, ser. Ser. Math. Anal. Appl. London: Taylor & Francis, 2003, vol. 6.

A. A. Martynyuk, G. T. Stamov, and I. M. Stamova, “Fractional-like Hukuhara derivatives in the theory of set-valued differential equations,” Chaos Solitons Fractals, vol. 131, 2020, Art. ID 109487, doi: 10.1016/j.chaos.2019.109487.

A.A.Martynyuk, Qualitative analysis of set-valued differential equations. Cham: Birkhäuser, 2019, doi: 10.1007/978-3-030-07644-3.

N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, and N. V. Skripnik, Differential Equations with Impulse Effects. Multivalued Right-hand Sides with Discontinuities, ser. De Gruyter Stud. Math. Berlin: de Gruyter, 2011, vol. 40, doi: 10.1515/9783110218176.

B. Piątek, “On the Riemann integral of set-valued functions,” Zeszyty Naukowe. Matematyka Stosowana/Politechnika Śląska, no. 2, pp. 5–18, 2012.

A. V. Plotnikov, T. A. Komleva, and N. V. Skripnik, “Existence of basic solutions of first order linear homogeneous set-valued differential equations,” Mat. Stud., vol. 61, no. 1, pp. 61–78, 2024, doi: 10.30970/ms.61.1.61-78.

A. V. Plotnikov and N. V. Skripnik, Differential Equations with Clear and Fuzzy Set-Valued Right-Hand Side. Asymptotic Methods. Odessa: AstroPrint, 2009.

A. V. Plotnikov, T. A. Komleva, and L. I. Plotnikova, “Averaging of a system of set-valued differential equations with the Hukuhara derivative,” Journal of Uncertain Systems, vol. 13, no. 1, pp. 3–13, 2019.

V. A. Plotnikov, A. V. Plotnikov, and A. N. Vityuk, Differential Equations with Set-Valued Right Part. Asymptotic Methods. Odessa: AstroPrint, 1999.

E. S. Polovinkin, Set-Valued Analysis and Differential Inclusions. Moscow: Fizmatlit, 2014.

A. Tolstonogov, Differential inclusions in a Banach space., ser. Math. Appl. Kluwer Academic Publishers, 2000, vol. 524.