We consider a generalized set-valued differential equation with generalized derivative and prove the theorems on existence and uniqueness of its solution for the cases of interval-valued and set-valued mappings.
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M. Hukuhara, “Integration des applications mesurables dont la valeur est un compact convexe,” Funk. Ekvacioj., No. 10, 205–223 (1967).
T. F. Bridgland, “Trajectory integrals of set-valued functions,” Pacif. J. Math., 33, No. 1, 43–68 (1970).
Yu. N. Tyurin, “Mathematical formulation of a simplified model of production planning,” Ékonom. Mat. Met., 1, No. 3, 391–409 (1965).
H. T. Banks and M. Q. Jacobs, “A differential calculus for multifunctions,” J. Math. Anal. Appl., No. 29, 246–272 (1970).
H. Radstrom, “An embedding theorem for spaces of convex sets,” Proc. Amer. Math. Soc., No. 3, 165–169 (1952).
A.V. Plotnikov, “Differentiation of multivalued mappings. T -derivative,” Ukr. Mat. Zh., 52, No. 8, 1119–1126 (2000); English translation: Ukr. Math. J., 52, No. 8, 1282–1291 (2000).
V. A. Plotnikov, A.V. Plotnikov, and A. N. Vityuk, Differential Equations with Set-Valued Right-Hand Side. Asymptotic Methods [in Russian], AstroPrint, Odessa (1999).
A. I. Vityuk, “Fractional differentiation of set-valued mappings,” Dop. Nats. Akad. Nauk Ukr., No. 10, 75–78 (2003).
B. Bede and L. Stefanini, “Generalized Hukuhara differentiability of interval-valued functions and interval differential equations,” Working Paper Ser. Econom. Math. Statist. Univ. Urbino “Carlo Bo” (2008).
A.V. Plotnikov and N. V. Skripnik, “Set-valued differential equations with generalized derivative,” J. Adv. Res. Pure Math., 3, No. 1, 144–160 (2011).
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).
Y. Chalco-Cano, H. Romin-Flores, and M. D. Jiminez-Gamero, “Generalized derivative and π-derivative for set-valued functions,” Inf. Sci., 181, No. 1, 2177–2188 (2011).
A. Lasota and A. Strauss, “Asymptotic behavior for differential equations which cannot be locally linearized,” J. Different. Equat., 3, No. 10, 152–172 (1971).
M. Martelli and A. Vignoli, “On differentiability of multi-valued maps,” Boll. Unione Mat. Ital., 4, No. 10, 701–712 (1974).
A.V. Plotnikov and N. V. Skripnik, Differential Equations with Clear and Fuzzy Set-Valued Right-Hand Sides. Asymptotic Methods [in Russian], AstroPrint, Odessa (2009).
B. Bede and S. G. Gal, “Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations,” Fuzzy Sets Syst., No. 151, 581–599 (2005).
L. Stefanini and B. Bede, “Generalized Hukuhara differentiability of interval-valued functions and interval differential equations,” Nonlin. Anal., Theory, Methods, Appl., Ser. A, 71, No. 3–4, 1311–1328 (2009).
N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, and N. V. Skripnik, Impulsive Differential Equations with Set-Valued and Discontinuous Right-Hand Sides [in Russian], Institute of Mathematics of the Ukrainian National Academy of Sciences, Kiev (2007).
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).
A. J. Brandao Lopes Pinto, F. S. de Blasi, and F. Iervolino, “Uniqueness and existence theorems for differential equations with compact convex-valued solutions,” Boll. Unione Mat. Ital., No. 4, 534–538 (1970).
V. Lakshmikantham, T. G. Bhaskar, and D. J. Vasundhara, Theory of Set Differential Equations in Metric Spaces, Cambridge Sci. Publ., Cambridge (2006).
V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, London (2003).
T. A. Komleva, A.V. Plotnikov, and N. V. Skripnik, “Differential equations with set-valued solutions,” Ukr. Mat. Zh., 60, No. 10, 1326–1337 (2008); English translation: Ukr. Math. J., 60, No. 10, 1540–1556 (2008).
A.V. Plotnikov and A.V. Tumbrukaki, “Integrodifferential equations with multivalued solutions,” Ukr. Mat. Zh., 52, No. 3, 359–367 (2000); English translation: Ukr. Math. J., 52, No. 3, 413–423 (2000).
A.V. Plotnikov and T. A. Komleva, “Averaging of set integrodifferential equations,” Appl. Math., 1, No. 2, 99–105 (2011).
M. Piszczek, “On a multivalued second order differential problem with Hukuhara derivative,” Opusc. Math., 28, No. 2, 151–161 (2008).
A.V. Arsirii and A.V. Plotnikov, “Systems of control over set-valued trajectories with terminal quality criterion,” Ukr. Mat. Zh., 61, No. 8, 1142–1147 (2009); English translation: Ukr. Math. J., 61, No. 8, 1349–1356 (2009).
A.V. Plotnikov and A.V. Arsirii, “Piecewise-constant control set systems,” Amer. J. Comput. Appl. Math., 1, No. 2, 89–92 (2011).
V. A. Plotnikov and O. D. Kichmarenko, “Averaging of controlled equations with Hukuhara derivative,” Nelin. Kolyvannya, 9, No. 3, 376–385 (2006); English translation: Nonlin. Oscillations, 9, No. 3, 365–374 (2006).
F. S. de Blasi, V. Lakshmikantham, and T. G. Bhaskar, “An existence theorem for set differential inclusions in a semilinear metric space,” Control Cybernet., 36, No. 3, 571–582 (2007).
N. V. Plotnikova, “Systems of linear differential equations with π-derivative and linear differential inclusions,” Mat. Sb., 196, No. 11, 127–140 (2005).
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).
N. Skripnik, “Interval-valued differential equations with generalized derivative,” Appl. Math., 2, No. 4, 116–120 (2012).
N. V. Plotnikova, “Approximation of a bundle of solutions of linear differential inclusions,” Nelin. Kolyvannya, 9, No. 3, 386–400 (2006); English translation: Nonlin. Oscillations, 9, No. 3, 375–390 (2006).
A. A. Tolstonogov, Differential Inclusions in Banach Spaces [in Russian], Nauka, Novosibirsk (1986).
A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer, Dordrecht (1988).
E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis [in Russian], Fizmatlit, Moscow (2004).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 10, pp. 1350–1362, October, 2013.
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Plotnikov, A.V., Skripnik, N.V. Conditions for the Existence of Local Solutions of Set-Valued Differential Equations with Generalized Derivative. Ukr Math J 65, 1498–1513 (2014). https://doi.org/10.1007/s11253-014-0875-1
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DOI: https://doi.org/10.1007/s11253-014-0875-1