Abstract

Because of no Lipschitz condition for upper semi-continuous (usc for short) multifunctions and some other technical difficulties, only the second order polynomial-like iterative equation with multifunctions was discussed but the general case of order n remains open. In this paper we consider the general case for a special class of multifunctions, called unblended multifunctions. We investigate the set of all jumps for iterates of those multifunctions and consider the piecewise Lipschitz condition. Then we prove the existence of usc multi-valued solutions for a modified form of this equation, which gives the existence of usc multi-valued solutions for this equation of general order n in the inclusion sense.

Author information

Contact details are reproduced from the original publication and may be historical.

Bing Xu

Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China

Kazimierz Nikodem

Dept. of Mathematics, University of Bielsko-Biała, 43-309 Bielsko-Biała, Poland

Weinian Zhang

Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China

matzwn@126.com

Keywords

  • Iteration
  • functional equation
  • multifunction
  • upper semi-continuity
  • unblended

Mathematics Subject Classification

39B12, 37E05, 54C60

References

  1. J. P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Boston (1990).Google Scholar
  2. K. Baron, W. Jarczyk: Recent results on functional equations in a single variable, perspectives and open problems, Aequationes Math. 61 (2001) 1–48.https://doi.org/10.1007/s000100050159Google Scholar
  3. C. Choi, D. Nam: Interpolation for partly hidden diffusion processes, Stochastic Processes Appl. 113 (2004) 199–216.https://doi.org/10.1016/j.spa.2004.03.014Google Scholar
  4. E. Fontich: Transversal homoclinic points of a class of conservative diffeomorphisms, J. Differ. Equations 87 (1990) 1–27.https://doi.org/10.1016/0022-0396(90)90012-EGoogle Scholar
  5. S. J. Greenfield, R. D. Nussbaum: Dynamics of a quadratic map in two complex variables, J. Differ. Equations 169 (2001) 57–141.https://doi.org/10.1006/jdeq.2000.3895Google Scholar
  6. W. Jarczyk, T. Powierża: On the smallest set-valued iterative roots of bijections, Int. J. Bifurcation Chaos Appl. Sci. Eng. 13 (2003) 1889–1893.https://doi.org/10.1142/S0218127403007710Google Scholar
  7. K. H. Johansson, A. Rantzer, K. J. Åström: Fast switches in relay feedback systems, Automatica 35 (1999) 539–552.https://doi.org/10.1016/S0005-1098(98)00160-5Google Scholar
  8. E. Klein, A. C. Thompson: Theory of Correspondences, John Wiley & Sons, New York (1984).Google Scholar
  9. M. Kuczma, B. Choczewski, R. Ger: Iterative Functional Equations, Encycl. Math. Appl. 32, Cambridge University Press, Cambridge (1990).https://doi.org/10.1017/CBO9781139086639Google Scholar
  10. J. Lawry: A framework for linguistic modelling, Artif. Intell. 155 (2004) 1–39.https://doi.org/10.1016/j.artint.2003.10.001Google Scholar
  11. M. Malenica: On the solutions of the functional equation φ(x) + φ2 (x) = F (x), Math. Vesn. 6 (1982) 301–305.Google Scholar
  12. P. J. McCarthy: Ultrafunctions, projective function geometry, and polynomial functional equations, Proc. Lond. Math. Soc. 53 (1986) 321–339.https://doi.org/10.1112/plms/s3-53.2.321Google Scholar
  13. K. Nikodem: On Jensen’s functional equation for set-valued functions, Rad. Mat. 3 (1987) 23–33.https://doi.org/10.1007/BF01836150Google Scholar
  14. K. Nikodem, W. Zhang: On a multivalued iterative equation, Publ. Math. 64 (2004) 427–435.https://doi.org/10.5486/PMD.2004.2967Google Scholar
  15. J. Olko: Semigroups of set-valued functions, Publ. Math. 51 (1997) 81–96.https://doi.org/10.5486/PMD.1997.1777Google Scholar
  16. D. Popa, N. Vornicescu: Locally compact set-valued solutions for the general linear equation, Aequationes Math. 67 (2004) 205–215.https://doi.org/10.1007/s00010-003-2699-1Google Scholar
  17. T. Powierża: Set-valued iterative square roots of bijections, Bull. Pol. Acad. Sci., Math. 47 (1999) 377–383.Google Scholar
  18. T. Powierża: Strong set-valued iterative roots, Grazer Math. Ber. 344 (2001) 51–56.Google Scholar
  19. T. Powierża: Higher order set-valued iterative roots of bijections, Publ. Math. 61 (2002) 315–324.https://doi.org/10.5486/PMD.2002.2555Google Scholar
  20. H. Rådström: An embedding theorem for space of convex sets, Proc. Amer. Math. Soc. 3 (1952) 165–169.https://doi.org/10.1090/S0002-9939-1952-0045938-2Google Scholar
  21. A. Smajdor: Iterations of multivalued functions, Pr. Nauk. Uniw. Śląsk. Katowicach 759 (1985) 1–59.Google Scholar
  22. A. Smajdor: Increasing iteration semigroups of Jensen set-valued functions, Aequationes Math. 56 (1998) 131–142.https://doi.org/10.1007/s000100050049Google Scholar
  23. W. Smajdor: Local set-valued solutions of the Jensen and Pexider functional equations, Publ. Math. 43 (1993) 255–263.https://doi.org/10.5486/PMD.1993.1243Google Scholar
  24. R. M. Starr: General Equilibrium Theory, Cambridge University Press, Cambridge (1997).https://doi.org/10.1017/CBO9781139174749Google Scholar
  25. J. Tabor, M. Zoldak: Iterative equations in Banach spaces, J. Math. Anal. Appl. 299 (2004) 651–662.https://doi.org/10.1016/j.jmaa.2004.06.011Google Scholar
  26. Gy. Targonski: Topics in Iteration Theory, Vandenhoeck & Ruprecht, Göttingen (1981).Google Scholar
  27. R. Węgrzyk: Fixed-point theorems for multivalued functions and their application to functional equations, Diss. Math. 201 (1982) 1–28.Google Scholar
  28. D. Yang, W. Zhang: Characteristic solutions of polynomial-like iterative equations, Aequationes Math. 67 (2004) 80–105.https://doi.org/10.1007/s00010-003-2708-4Google Scholar
  29. W. Zhang: Solutions of equivariance for a polynomial-like iterative equation, Proc. R. Soc. Edinb., Sect. A, Math. 130 (2000) 1153–1163.https://doi.org/10.1017/S0308210500000615Google Scholar

Suggested citation

B. Xu, K. Nikodem, W. Zhang. “On a Multivalued Iterative Equation of Order n.” Journal of Convex Analysis 18 (2011), No. 3, 673–686. https://doi.org/10.68381/jca18042

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.