We consider a multivalued discrete system and study its properties and the existence of its solution.
Similar content being viewed by others
References
F. S. de Blasi and F. Iervolino, “Equazioni differentiali con soluzioni a valore compatto convesso,” Boll. Unione Mat. Ital., 2, No. 4-5, 491–501 (1969).
V. A. Plotnikov, A. V. Plotnikov, and A. N. Vityuk, Differential Equations with Multivalued Right-Hand Sides. Asymptotic Methods [in Russian], AstroPrint, Odessa (1999).
A. V. Plotnikov and N. V. Skripnik, Differential Equations with Clear and Fuzzy Multivalued Right-Hand Sides. Asymptotic Methods [in Russian], AstroPrint, Odessa (2009).
N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, and N. V. Skripnik, Impulsive Differential Equations with Multivalued and Discontinuous Right-Hand Sides [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2007).
N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, and N. V. Skripnik, Differential Equations with Impulse Effects. Multivalued Right-Hand Sides with Discontinuities, de Gruyter, Berlin (2011).
V. Lakshmikantham, B. T. Granna, and D. J. Vasundhara, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge (2006).
V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, London (2003).
A. A. Tolstonogov, Differential Inclusions in Banach Spaces [in Russian], Nauka, Novosibirsk (1986).
N. V. Plotnikova, “Approximation of a bundle of solutions of linear differential inclusions,” Nelin. Kolyv., 12, No. 3, 386–400 (2009); English translation: Nonlin. Oscillat., 12, No. 3, 375–390 (2009).
A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer AP, Dordrecht (1988).
F. S. de Blasi and F. Iervolino, “Euler method for differential equations with set-valued solutions,” Boll. Unione Mat. Ital., 4, No. 4, 941–949 (1971).
I. A. Chahma, “Set-valued discrete approximation of state-constrained differential inclusions,” Bayreuth. Math. Schr., 67, 3–162 (2003).
R. Baier and T. Donchev, “Discrete approximation of impulsive differential inclusions,” Numer. Funct. Anal. Optim., 31, No. 6, 653–678 (2010).
R. B. Gu and W. J. Guo, “On mixing properties in set-valued discrete system,” Chaos, Solitons Fractals, 28, No. 3, 747–754 (2006).
A. Khan and P. Kumar, “Chaotic properties on time varying map and its set-valued extension,” Adv. Pure Math., 3, 359–364 (2013).
A. V. Plotnikov and T. A. Komleva, “The averaging of fuzzy linear differential inclusions on finite interval,” Dyn. Contin., Discrete Impuls. Syst., Ser. B., Appl. Algorithms, 23, No. 1, 1–9 (2016).
A. V. Plotnikov and T. A. Komleva, “The partial averaging of fuzzy differential inclusions on finite interval,” Int. J. Different. Equat., 2014 (2014), Article ID 307941.
A. V. Plotnikov, “A procedure of complete averaging for fuzzy differential inclusions on a finite segment,” Ukr. Mat. Zh., 67, No. 3, 366–374 (2015); English translation: Ukr. Math. J., 67, No. 3, 421–430 (2015).
V. A. Plotnikov, L. I. Plotnikova, and A. T. Yarovoi, “Averaging method for discrete systems and its application to control problems,” Nelin. Kolyv., 7, No. 2, 241–254 (2004); English translation: Nonlin. Oscillat., 7, No. 2, 240–253 (2004).
H. Roman-Flores, “A note on transitivity in set-valued discrete systems,” Chaos, Solitons Fractals, 17, No. 1, 99–104 (2003).
H. Roman-Flores and Y. Chalco-Cano, “Robinson’s chaos in set-valued discrete systems,” Chaos, Solitons Fractals, 25, No. 1, 33–42 (2005).
Y. M. Shi and G. R. Chen, “Chaos of time-varying discrete dynamic systems,” J. Different. Equat. Appl., 15, No. 5, 429–449 (2009).
M. Hukuhara, “Integration des applications mesurables dont la valeur est un compact convexe,” Funkc. Ekvacioj., No. 10, 205–223 (1967).
A. Plotnikov and N. Skripnik, “Existence and uniqueness theorems for generalized set differential equations,” Int. J. Control Sci. Eng., 2, No. 1, 1–6 (2012).
E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis [in Russian], Fizmatlit, Moscow (2004).
A. V. Plotnikov and N. V. Skripnik, “Conditions for the existence of local solutions of set-valued differential equations with generalized derivative,” Ukr. Mat. Zh., 65, No. 10, 1350–1362 (2013); English translation: Ukr. Math. J., 65, No. 10, 1498–1513 (2014).
M. V. Balashov and E. S. Polovinkin, “M-strongly convex subsets and their generating sets,” Mat. Sb., 191, No. 1, 27–64 (2000).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 11, pp. 1519–1524, November, 2018.
Rights and permissions
About this article
Cite this article
Komleva, T.A., Plotnikova, L.I. & Plotnikov, A.V. A Multivalued Discrete System and Its Properties. Ukr Math J 70, 1750–1757 (2019). https://doi.org/10.1007/s11253-019-01612-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-019-01612-z