Approximation theorems for fuzzy set multifunctions in Vietoris topology. Physical implications of regularity

Document Type: Research Paper

Authors

1 Faculty of Mathematics, Alexandru Ioan Cuza" University of Iasi Iasi, Romania

2 Department of Physics, Gheorghe Asachi Technical University of Iasi, Iasi, Romania

Abstract

n this paper, we consider continuity properties
(especially, regularity, also viewed as an approximation property) for $%
mathcal{P}_{0}(X)$-valued set multifunctions ($X$ being a linear,
topological space), in order to obtain Egoroff and Lusin type theorems for
set multifunctions in the Vietoris hypertopology. Some mathematical
applications are established and several physical implications of the
mathematical model of regularity are presented, which allows a
classification of the physical models.

Keywords


[1] M. Agop, O. Niculescu, A. Timofte, L. Bibire, A. S. Ghenadi, A. Nicuta, C. Nejneru and
G. V. Munceleanu, Non-di erentiable mechanical model and its implications, International
Journal of Theoretical Physics, 49(7) (2010).
[2] J. Andres and J. Fiser, Metric and topological multivalued fractals, Internat. J. Bifur. Chaos
Appl. Sci. Engrg., 14(4) (2004), 1277-1289.

[3] J. Andres and M. Rypka, Multivalued fractals and hyperfractals, Internat. J. Bifur. Chaos
Appl. Sci. Engrg., 22(1) (2012).
[4] G. Apreutesei, Families of subsets and the coincidence of hypertopologies, Annals of the
Alexandru Ioan Cuza University - Mathematics, XLIX (2003), 1-18.
[5] D. Averna, Lusin type theorems for multifunctions, Scorza Dragoni's property and
Caratheodory selections, Boll. U.M.I., (7)(8-A) (1994), 193-201.
[6] T. Banakh and N. Novosad, Micro and macro fractals generated by multi-valued dynamical
systems, arXiv: 1304.7529v1 [math.GN], 28 April, (2013).
[7] G. Beer, Topologies on closed and closed convex sets, Kluwer Academic Publishers, 1993.
[8] S. Brown, Memory and mathesis: For a topological approach to psychology, Theory, Culture
and Society, 29(4-5) (2012), 137-164.
[9] P. di Lorenzo and G. di Maio, The Hausdor metric in the Melody Space: A new approach
to Melodic Similarity, the 9th International Conference on Music Perception and Cognition,
Alma Mater Studiorum University of Bologna, August 22-26, (2006).
[10] N. Dinculeanu, Measure theory and real functions (in Romanian), Ed. Did. si Ped., Bucuresti,
1964.
[11] A. R. El-Nabulsi, Fractional derivatives generalization of Einstein's eld equations, Indian
Journal of Physics, 87 (2013), 195-200.
[12] A. R. El-Nabulsi, New astrophysical aspects from Yukawa fractional potential correction to
the gravitational potential in D dimensions, Indian Journal of Physics, 86 (2012), 763-768.
[13] M. S. El Naschie, O. E. Rosler, I. Prigogine, eds., Quantum Mechanics, Di usion and Chaotic
Fractals, Elsevier, Oxford, 1995.
[14] H. Fu and Z. Xing, Mixing properties of set-valued maps on hyperspaces via Furstenberg
families, Chaos, Solitons & Fractals, 45(4) (2012), 439-443.
[15] A. Gavrilut, Continuity properties and Alexandro theorem in Vietoris topology, Fuzzy Sets
and Systems, 194 (2012), 76-89.
[16] A. Gavrilut, Alexandro theorem in Hausdor topology for null-null-additive set multifunc-
tions, Annals of the Alexandru Ioan Cuza University - Mathematics., LIX(2) (2013), 237-251.
[17] J. L. Gomez-Rueda, A. Illanes and H. Mendez, Dynamic properties for the induced maps in
the symmetric products, Chaos, Solitons & Fractals, 45(9-10) (2012), 1180-1187.
[18] C. Guo and D. Zhang, On the set-valued fuzzy measures, Information Sciences, 160 (2004),
13-25.
[19] S. Hawking and R. Penrose, The nature of space time, Princeton, Princeton University Press,
1996.
[20] S. Hu and N. S. Papageorgiou, Handbook of multivalued analysis, vol. I, Kluwer Acad. Publ.,
Dordrecht, 1997.
[21] Q. Jiang and H. Suzuki, Fuzzy measures on metric spaces, Fuzzy Sets and Systems, 83 (1996),
99-106.
[22] J. Kawabe, Regularity and Lusin's theorem for Riesz space-valued fuzzy measures, Fuzzy Sets
and Systems, 158 (2007), 895-903.
[23] H. Kunze, D. La Torre, F. Mendivil and E. R. Vrscay, Fractal based methods in analysis,
Springer, 2012.
[24] K. Lewin, G. M. Heider and F. Heider, Principles of topological psychology, McGraw-Hill,
New York, 1936.
[25] J. Li, J. Li and M. Yasuda, Approximation of fuzzy neural networks by using Lusin's theorem,
(2007), 86-92.
[26] J. Li and M. Yasuda, Lusin's theorem on fuzzy measure spaces, Fuzzy Sets and Systems, 146
(2004), 121-133.
[27] R. Li, A note on stronger forms of sensitivity for dynamical systems, Chaos, Solitons &
Fractals, 45(6) (2012), 753-758.
[28] L. Liu, Y. Wang and G. Wei, Topological entropy of continuous functions on topological
spaces, Chaos, Solitons & Fractals, 39(1) (2009), 417-427.
[29] Y. Lu, C. L. Tan, W. Huang and L. Fan, An approach to word image mathching based on
weighted Hausdor distance, Document Analysis and Recognition, Proceedings, 2001.

30] X. Ma, B. Hou and G. Liao, Chaos in hyperspace system, Chaos, Solitons & Fractals, 40(2)
(2009), 653-660.
[31] L. Nottale, Fractal space-time and microphysics: towards theory of scale relativity, World
Scienti c, Singapore, 1993.
[32] L. Nottale, Scale relativity and fractal space-time, a new approach to unifying relativity and
quantum mechanics, Imperial College Press, London, 2011.
[33] E. Pap, Null-additive set functions, Kluwer Acad. Publishers, Dordrecht, 1995.
[34] R. Penrose, The road to reality: a complete guide to the laws of the universe, London:
Jonathan Cape, 2004.
[35] A. Precupanu, A. Croitoru and Ch. Godet-Thobie, Set-valued Integrals (in Romanian), Iasi,
in progress.
[36] A. Precupanu, T. Precupanu, M. Turinici, N. Apreutesei Dumitriu, C. Stamate, B. R. Satco,
C. Vaideanu, G. Apreutesei, D. Rusu, A. C. Gavrilut and M. Apetrii, Modern directions in
multivalued analysis and optimization theory, Venus Publishing House, Iasi, (in Romanian),
2006.
[37] A. Precupanu and A. Gavrilut, A set-valued Egoro type theorem, Fuzzy Sets and Systems,
175 (2011), 87-95.
[38] A. Precupanu and A. Gavrilut, A set-valued Lusin type theorem, Fuzzy Sets and Systems,
204 (2012), 106-116.
[39] T. Precupanu, Linear topological spaces and elements of convex analysis (in Romanian), Ed.
Acad. Romania, 1992.
[40] P. Sharma and A. Nagar, Topological dynamics on hyperspaces, Applied General Topology,
11(1) (2010), 1-19.
[41] J. Song and J. Li, Regularity of null-additive fuzzy measure on metric spaces, Int. J. Gen.
Systems, 32 (2003), 271-279.
[42] Y. Wang, G. Wei, W. H. Campbell and S. Bourquin, A framework of induced hyperspace
dynamical systems equipped with the hit-or-miss topology, Chaos, Solitons & Fractals, 41(4)
(2009), 1708-1717.
[43] K. R. Wicks, Fractals and Hyperspaces, Springer-Verlag Berlin Heidelberg, 1991.