Existence and Uniqueness Theorem for Set-Valued Volterra Integral Equations

Andrej V. Plotnikov^{1, 2,} and Natalia V. Skripnik²

¹Department of Applied Mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine

²Department of Optimal Control and Economic Cybernetics, Odessa National University named after I.I. Mechnikov, Odessa, Ukraine

Cite this paper

Andrej V. Plotnikov and Natalia V. Skripnik. Existence and Uniqueness Theorem for Set-Valued Volterra Integral Equations. American Journal of Applied Mathematics and Statistics. 2013; 1(3):41-45. doi: 10.12691/AJAMS-1-3-2

Abstract

The space of nonempty compact sets of is well-known to be a nonlinear space. This fact essentially complicates the research of set-valued differential and integral equations. In this article we consider set-valued Volterra integral equations and prove the existence and uniqueness theorem.

Keywords

set-valued integral equation, existence, uniqueness, set-valued differential equation

Copyright

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References

[1]	de Blasi, F.S., Iervolino, F., “Equazioni differentiali con soluzioni a valore compatto convesso.” Boll. Unione Mat. Ital. 2 (4-5). 491-501. 1969.
[2]	Drici, Z., Mcrae, F.A., Vasundhara Devi, J., “Set differential equations with causal operators.” Math. Probl. Eng. 2. 185-194. 2005.
[3]	Galanis, G.N., Gnana Bhaskar, T., Lakshmikantham, V., Palamides, P.K., “Set value functions in Frechet spaces: Continuity, Hukuhara differentiability and applications to set differential equations.” Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 61. 559-575. 2005.
[4]	Galanis, G.N., Tenali, G.B., Lakshmikantham, V., “Set differential equations in Frechet spaces.” J. Appl. Anal. 14. 103-113. 2008.
[5]	Lakshmikantham, V., “Set differential equations versus fuzzy differential equations.” Appl. Math. Comput. 164. 277-294. 2005.
[6]	Lakshmikantham, V., Granna Bhaskar, T., Vasundhara Devi, J., Theory of set differential equations in metric spaces. Cambridge Scientific Publishers, 2006.
[7]	Laksmikantham, V., Leela, S., Vatsala, A.S., “Interconnection between set and fuzzy differential equations.” Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods. 54. 351-360. 2003.
[8]	Lakshmikantham, V., Mohapatra, R.N., Theory of fuzzy differential equations and inclusions. London, Taylor & Francis, 2003.
[9]	Perestyuk, N.A., Plotnikov, V.A., Samoilenko, A.M., Skripnik, N.V., Differential equations with impulse effects: multivalued right-hand sides with discontinuities. de Gruyter Stud. Math.: 40, Berlin/Boston: Walter De Gruyter GmbH\& Co., 2011.
[10]	Piszczek, M., “On a multivalued second order differential problem with Hukuhara derivative.” Opuscula Math. 28 (2). 151-161. 2008.
[11]	Plotnikov, A.V., Skripnik, N.V., Differential equations with ''clear'' and fuzzy multivalued right-hand side. Asymptotics methods. Odessa, AstroPrint, 2009. (in Russian).
[12]	Plotnikov, V.A., Plotnikov, A.V., Vityuk, A.N., Differential equations with multivalued right-hand side. Asymptotic methods. Odessa, AstroPrint, 1999. (in Russian).
[13]	Plotnikova, N.V., “Approximation of a pencil of solutions of linear differential inclusions.” Nonlinear Oscil. (N. Y.). 9 (3). 375-390. 2006.
[14]	Tolstonogov, A., Differential inclusions in a Banach space. Dordrecht, Kluwer Academic Publishers, 2000.
[15]	Plotnikov, A.V., Tumbrukaki, A.V., “Integro-differential equations with multivalued solutions.” Ukr. Math. J. 52(3). 413-423. 2000.
[16]	Ahmad, B., Sivasundaram,, S., “-stability of impulsive hybrid setvalued differential equations with delay by perturbing lyapunov functions.” Commun. Appl. Anal. 12 (2). 137-146. 2008.
[17]	Arsirii, A.V., Plotnikov, A.V., “Systems of control over set-valued trajectories with terminal quality criterion.” Ukr. Math. J. 61 (8). 1349-1356. 2009.
[18]	Plotnikov, A.V., Arsirii, A.V., “Piecewise constant control set systems.” American Journal of Computational and Applied Mathematics. 1 (2). 89-92. 2011.
[19]	Plotnikov, V.A., Kichmarenko, O.D., “Averaging of controlled equations with the Hukuhara derivative.” Nonlinear Oscil. (N. Y.). 9(3). 365-374. 2006.
[20]	Plotnikov, A.V., Skripnik, N.V., “Existence and Uniqueness Theorem for Set Integral Equations.” J. Adv. Res. Dyn. Control Syst. 5 (2). 65-72. 2013.
[21]	Tise, I., “Set integral equations in metric spaces.” Math. Morav. 13 (1). 95-102. 2009.
[22]	Plotnikov, A.V., “Averaging differential embeddings with Hukuhara derivative.” Ukr. Math. J. 41 (1). 112-115. 1989.
[23]	Balachandran, K., Prakash, P., “On fuzzy Volterra integral equations with deviating arguments.” Journal of Applied Mathematics and Stochastic Analysis. 2004 (2). 169-176. 2004.
[24]	Friedman, M., Ming, M., Kandel, A., “Numerical solution of fuzzy differential and integral equations.” Fuzzy Sets and System. 106. 35-48. 1999.
[25]	Wang, H., Liu, Y., “Existence results for fuzzy integral equations of fractional order.” Int. Journal of Math. Analysis. 5 (17). 811-818. 2011.
[26]	Rådström, H., “An embedding theorem for spaces of convex sets.” Proc. Amer. Math. Soc. 3. 165-169. 1952.
[27]	Hukuhara, M., “Integration des applications mesurables dont la valeur est un compact convexe.” Funkcial. Ekvac. 10. 205-223. 1967.
[28]	Polovinkin, E.S., “Strongly convex analysis.” Sb. Math. 187. 259-286. 1996.
[29]	Balashov, M.V., Polovinkin, E.S., “M-strongly convex subsets and their generating sets.” Sb. Math. 191. 25-60. 2000.