The characteristic of noncompact convexity and random fixed point theorem for set-valued operators

Poom Kumam; Somyot Plubtieng

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 1, page 269-279
  • ISSN: 0011-4642

Abstract

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Let ( Ω , Σ ) be a measurable space, X a Banach space whose characteristic of noncompact convexity is less than 1, C a bounded closed convex subset of X , K C ( C ) the family of all compact convex subsets of C . We prove that a set-valued nonexpansive mapping T C → K C ( C ) has a fixed point. Furthermore, if X is separable then we also prove that a set-valued nonexpansive operator T Ω × C → K C ( C ) has a random fixed point.

How to cite

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Kumam, Poom, and Plubtieng, Somyot. "The characteristic of noncompact convexity and random fixed point theorem for set-valued operators." Czechoslovak Mathematical Journal 57.1 (2007): 269-279. <http://eudml.org/doc/31129>.

@article{Kumam2007,
abstract = {Let $(\Omega ,\Sigma )$ be a measurable space, $X$ a Banach space whose characteristic of noncompact convexity is less than 1, $C$ a bounded closed convex subset of $X$, $KC(C)$ the family of all compact convex subsets of $C.$ We prove that a set-valued nonexpansive mapping $T\: C\rightarrow KC(C)$ has a fixed point. Furthermore, if $X$ is separable then we also prove that a set-valued nonexpansive operator $T\: \Omega \times C\rightarrow KC(C)$ has a random fixed point.},
author = {Kumam, Poom, Plubtieng, Somyot},
journal = {Czechoslovak Mathematical Journal},
keywords = {random fixed point; set-valued random operator; measure of noncompactness; random fixed point; set-valued random operator; measure of noncompactness},
language = {eng},
number = {1},
pages = {269-279},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The characteristic of noncompact convexity and random fixed point theorem for set-valued operators},
url = {http://eudml.org/doc/31129},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Kumam, Poom
AU - Plubtieng, Somyot
TI - The characteristic of noncompact convexity and random fixed point theorem for set-valued operators
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 1
SP - 269
EP - 279
AB - Let $(\Omega ,\Sigma )$ be a measurable space, $X$ a Banach space whose characteristic of noncompact convexity is less than 1, $C$ a bounded closed convex subset of $X$, $KC(C)$ the family of all compact convex subsets of $C.$ We prove that a set-valued nonexpansive mapping $T\: C\rightarrow KC(C)$ has a fixed point. Furthermore, if $X$ is separable then we also prove that a set-valued nonexpansive operator $T\: \Omega \times C\rightarrow KC(C)$ has a random fixed point.
LA - eng
KW - random fixed point; set-valued random operator; measure of noncompactness; random fixed point; set-valued random operator; measure of noncompactness
UR - http://eudml.org/doc/31129
ER -

References

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  1. Set-valued Analysis, Birkhäuser, Boston, 1990. (1990) MR1048347
  2. Measures of Noncompactness in Metric Fixed Point Theory; Advances and Applications Topics in Metric Fixed Point Theory, Birkhauser-Verlag, Basel 99, 1997. (1997) MR1483889
  3. Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1974. (1974) MR0787404
  4. 10.1155/S1085337503203080, Abstr. Appl. Anal. 6 (2003), 375–386. (2003) MR1982809DOI10.1155/S1085337503203080
  5. 10.1016/j.jmaa.2003.10.019, J. Math. Anal. Appl. 291 (2004), 100–108. (2004) MR2034060DOI10.1016/j.jmaa.2003.10.019
  6. 10.1090/S0002-9939-96-03062-6, Proc. Amer. Math. Soc. 124 (1996), 838–838. (1996) MR1301487DOI10.1090/S0002-9939-96-03062-6
  7. Topics in metric fixed point theorem, Cambridge University Press, Cambridge, 1990. (1990) MR1074005
  8. 10.2140/pjm.1977.68.85, Pacific J.  Math. 68 (1977), 85–90. (1977) Zbl0335.54036MR0451228DOI10.2140/pjm.1977.68.85
  9. Nonexpansive mappings in product spaces, set-valued mappings, and k-uniform rotundity, Proceedings of the Symposium Pure Mathematics, Vol. 45, part  2, American Mathematical Society, Providence, 1986, pp. 51–64. (1986) Zbl0594.47048MR0843594
  10. 10.1016/S0362-546X(01)00751-9, Nonlinear Anal. 50 (2002), 265–274. (2002) MR1904945DOI10.1016/S0362-546X(01)00751-9
  11. 10.1016/j.na.2004.01.003, Nonlinear Anal. 57 (2004), 23–34. (2004) MR2055985DOI10.1016/j.na.2004.01.003
  12. 10.1006/jmaa.1999.6454, J. Math. Anal. Appl. 237 (1999), 83–92. (1999) MR1708163DOI10.1006/jmaa.1999.6454
  13. Some random fixed point theorems, Fixed Point Theory and Applications, K.-K. Tan (ed.), World Scientific, Singapore, 1992, pp. 334–345. (1992) MR1190049
  14. 10.1137/0315056, SIAM J. Control Optim. 15 (1977), 859–903. (1977) Zbl0407.28006MR0486391DOI10.1137/0315056
  15. 10.1090/S0002-9939-1990-1021908-6, Proc. Amer. Math. Soc. 110 (1990), 395–400. (1990) MR1021908DOI10.1090/S0002-9939-1990-1021908-6
  16. Metric fixed point for multivalued mappings, Dissertationes Math. (Rozprawy Mat.) 389 (2000), 39. (2000) MR1799531
  17. 10.1090/S0002-9939-1993-1123670-8, Proc. Amer. Math. Soc. 117 (1993), 1089–1092. (1993) MR1123670DOI10.1090/S0002-9939-1993-1123670-8
  18. 10.1016/0362-546X(95)00013-L, Nonlinear Anal. 26 (1996), 1301–1311. (1996) MR1376105DOI10.1016/0362-546X(95)00013-L
  19. 10.1016/S0362-546X(99)00227-8, Nonlinear Anal. 43 (2001), 693–706. (2001) Zbl0988.47034MR1808203DOI10.1016/S0362-546X(99)00227-8
  20. 10.1007/BF01111112, Math. Z. 125 (1972), 17–31. (1972) Zbl0216.17302MR0306989DOI10.1007/BF01111112
  21. 10.1016/0362-546X(94)00268-M, Nonlinear Anal. 26 (1996), 1097–1102. (1996) MR1375652DOI10.1016/0362-546X(94)00268-M

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