Multivalued fractals in b-metric spaces

Monica Boriceanu; Marius Bota; Adrian Petruşel

Open Mathematics (2010)

  • Volume: 8, Issue: 2, page 367-377
  • ISSN: 2391-5455

Abstract

top
Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems theory in several topics of applied sciences. It is known that examples of fractals and multivalued fractals are coming from fixed point theory for single-valued and multivalued operators, via the so-called fractal and multi-fractal operators. On the other hand, the most common setting for the study of fractals and multi-fractals is the case of operators on complete or compact metric spaces. The purpose of this paper is to extend the study of fractal operator theory for multivalued operators on complete b-metric spaces.

How to cite

top

Monica Boriceanu, Marius Bota, and Adrian Petruşel. "Multivalued fractals in b-metric spaces." Open Mathematics 8.2 (2010): 367-377. <http://eudml.org/doc/269747>.

@article{MonicaBoriceanu2010,
abstract = {Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems theory in several topics of applied sciences. It is known that examples of fractals and multivalued fractals are coming from fixed point theory for single-valued and multivalued operators, via the so-called fractal and multi-fractal operators. On the other hand, the most common setting for the study of fractals and multi-fractals is the case of operators on complete or compact metric spaces. The purpose of this paper is to extend the study of fractal operator theory for multivalued operators on complete b-metric spaces.},
author = {Monica Boriceanu, Marius Bota, Adrian Petruşel},
journal = {Open Mathematics},
keywords = {Multivalued operator; Fixed point; Strict fixed point; b-metric space; Self-similar set; Fractal operator; Multi-fractal operator; multivalued operator; fixed point; strict fixed point; -metric space; self-similar set; fractal operator; multi-fractal operator},
language = {eng},
number = {2},
pages = {367-377},
title = {Multivalued fractals in b-metric spaces},
url = {http://eudml.org/doc/269747},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Monica Boriceanu
AU - Marius Bota
AU - Adrian Petruşel
TI - Multivalued fractals in b-metric spaces
JO - Open Mathematics
PY - 2010
VL - 8
IS - 2
SP - 367
EP - 377
AB - Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems theory in several topics of applied sciences. It is known that examples of fractals and multivalued fractals are coming from fixed point theory for single-valued and multivalued operators, via the so-called fractal and multi-fractal operators. On the other hand, the most common setting for the study of fractals and multi-fractals is the case of operators on complete or compact metric spaces. The purpose of this paper is to extend the study of fractal operator theory for multivalued operators on complete b-metric spaces.
LA - eng
KW - Multivalued operator; Fixed point; Strict fixed point; b-metric space; Self-similar set; Fractal operator; Multi-fractal operator; multivalued operator; fixed point; strict fixed point; -metric space; self-similar set; fractal operator; multi-fractal operator
UR - http://eudml.org/doc/269747
ER -

References

top
  1. [1] Andres J., Fišer J., Metric and topological multivalued fractals, Int. J. Bifurc. Chaos Appl. Sci. Engn., 2004, 14, 1277–1289 http://dx.doi.org/10.1142/S021812740400979X Zbl1057.28003
  2. [2] Andres J., Fišer J., Gabor G., Leśniak K., Multivalued fractals, Chaos Solitons & Fractals, 2005, 24, 665–700 http://dx.doi.org/10.1016/j.chaos.2004.09.029 Zbl1077.28002
  3. [3] Bakhtin I.A., The contraction mapping principle in almost metric spaces, Funct. Anal, Gos. Ped. Inst. Unianowsk, 1989, 30, 26–37 Zbl0748.47048
  4. [4] Barnsley M.F, Fractals Everywhere, Academic Press, Boston, 1988 
  5. [5] Berinde V, Generalized contractions in quasimetric spaces, Seminar on Fixed Point Theory, 1993, 3–9 
  6. [6] Berinde V, Sequences of operators and fixed points in quasimetric spaces, Studia Univ. Babeş-Bolyai, Math., 1996, 16,23–27 Zbl1005.54501
  7. [7] Blumenthal L.M., Theory and Applications of Distance Geometry, Oxford Univ. Press, Oxford, 1953 
  8. [8] Boriceanu M., Petruşel A., Rus I.A., Fixed point theorems for some multivalued generalized contractions in b-metric spaces, Internat. J. Math. Statistics, 2010, 6, 65–76 
  9. [9] Bourbaki N., Topologie générale, Herman, Paris, 1974 
  10. [10] Browder FE., On the convergence of successive approximations for nonlinear functional equations, Indag. Math., 1968,30,27–35 Zbl0155.19401
  11. [11] Chifu C, Petruşel A., Multivalued fractals and generalized multivalued contractions, Chaos Solitons & Fractals, 2008,36,203–210 http://dx.doi.org/10.1016/j.chaos.2006.06.027 Zbl1131.28005
  12. [12] Covitz H., Nadler S.B. jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math., 1970, 8, 5–11 http://dx.doi.org/10.1007/BF02771543 Zbl0192.59802
  13. [13] Czerwik S., Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Univ. Modena, 1998, 46, 263–276 Zbl0920.47050
  14. [14] El Naschie M.S., Iterated function systems and the two-slit experiment of quantum mechanics, Chaos Solitons & Fractals, 1994, 4, 1965–1968 http://dx.doi.org/10.1016/0960-0779(94)90011-6 
  15. [15] Fréchet M., Les espaces abstraits, Gauthier-Villars, Paris, 1928 
  16. [16] Heinonen J., Lectures on Analysis on Metric Spaces, Springer Berlin, 2001 
  17. [17] Hu S., Papageorgiou N.S., Handbook of Multivalued Analysis, Vol. I, II, Kluwer Acad. Publ., Dordrecht, 1997, 1999 Zbl0887.47001
  18. [18] Hutchinson J.E., Fractals and self-similarity, Indiana Univ. Math. J., 1981, 30, 713–747 http://dx.doi.org/10.1512/iumj.1981.30.30055 Zbl0598.28011
  19. [19] Jachymski J., Matkowski J., Światkowski T., Nonlinear contractions on semimetric spaces, J. Appl. Anal., 1995, 1, 125–134 http://dx.doi.org/10.1515/JAA.1995.125 Zbl1295.54055
  20. [20] Kirk W.A., Sims B. (Eds.), Handbook of Metric Fixed Point Theory, Kluwer Acad. Publ., Dordrecht, 2001 Zbl0970.54001
  21. [21] Llorens-Fuster E., Petruşel A., Yao J.C., Iterated function systems and well-posedness, Chaos Solitons & Fractals, 2009, 41, 1561–1568 http://dx.doi.org/10.1016/j.chaos.2008.06.019 Zbl1198.52014
  22. [22] Meir A., Keeler E., A theorem on contraction mappings, J. Math. Anal. Appl., 1969, 28, 326–329 http://dx.doi.org/10.1016/0022-247X(69)90031-6 
  23. [23] Nadler S.B. Jr., Multivalued contraction mappings, Pacific J. Math., 1969, 30, 475–488 Zbl0187.45002
  24. [24] Păcurar (Berinde) M., Iterative methods for fixed point approximation, Ph.D. thesis, Babeş-Bolyai University Cluj-Napoca, Romania, 2009 
  25. [25] Păcurar (Berinde) M., A fixed point result for ϕ-contractions on b-metric spaces without the boundedness assumption, preprint 
  26. [26] Petruşel A., Rus I.A., Well-posedness of the fixed point problem for multivalued operators, Applied Analysis and Differential Equations (Cârjă O., Vrabie I.I. (Eds.) World Scientific 2007, 295–306 Zbl1169.47037
  27. [27] Petruşel A., Rus I.A., Yao J.C., Well-posedness in the generalized sense of the fixed point problems for multivalued operators, Taiwanese J. Math., 2007, 11, 903–914 Zbl1149.54022
  28. [28] Rhoades B.E., Some theorems on weakly contractive maps, Nonlinear Anal., 2001, 47, 2683–2693 http://dx.doi.org/10.1016/S0362-546X(01)00388-1 Zbl1042.47521
  29. [29] Rus I.A., Petruşel A., Sîntămărian A., Data dependence of the fixed points set of some multivalued weakly Picard operators, Nonlinear Anal., 2003, 52, 1947–1959 http://dx.doi.org/10.1016/S0362-546X(02)00288-2 
  30. [30] Rus I.A., Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001 
  31. [31] Rus I.A., Picard operators and applications, Sci. Math. Japon., 2003, 58, 191–219 
  32. [32] Rus I.A., Strict fixed point theory, Fixed Point Theory, 2003, 4, 177–183 Zbl1070.47519
  33. [33] Rus I.A., The theory of a metrical fixed point theorem: theoretical and applicative relevances, Fixed Point Theory, 2008, 9, 541–559 Zbl1172.54030
  34. [34] Singh S.L., Bhatnagar C., Mishra S.N., Stability of iterative procedures for multivalued maps in metric spaces, Demonstratio Math., 2005, 37, 905–916 Zbl1100.47056
  35. [35] Singh S.L., Prasad B., Kumar A., Fractals via iterated functions and multifunctions, Chaos Solitons & Fractals, 2009, 39, 1224–1231 http://dx.doi.org/10.1016/j.chaos.2007.06.014 Zbl1197.28004

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.