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The mean-value iteration for set-valued mappings


Author: Peter K. F. Kuhfittig
Journal: Proc. Amer. Math. Soc. 80 (1980), 401-405
MSC: Primary 47H10
DOI: https://doi.org/10.1090/S0002-9939-1980-0580993-X
MathSciNet review: 580993
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Abstract | References | Similar Articles | Additional Information

Abstract: In this note Krasnoselskii's iteration procedure

$\displaystyle {x_{n + 1}} = \tfrac{1}{2}(I + T){x_n}$

is extended to certain classes of set-valued mappings.

References [Enhancements On Off] (What's this?)

  • [1] F. E. Browder and W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 571-575. MR 0190745 (32:8155b)
  • [2] D. Downing and W. A. Kirk, Fixed point theorems for set-valued mappings in metric and Banach spaces, Math. Japon. 22 (1977), 99-112. MR 0473934 (57:13592)
  • [3] M. Edelstein, A remark on a theorem of M. A. Krasnoselskii, Amer. Math. Monthly 73 (1966), 509-510. MR 0194866 (33:3072)
  • [4] R. Kannan, Some results on fixed points. II, Amer. Math. Monthly 76 (1969), 405-408. MR 0257838 (41:2487)
  • [5] M. A. Krasnoselskii, Two observations about the method of successive approximations, Uspehi Mat. Nauk 10 (1955), 123-127. (Russian) MR 0068119 (16:833a)
  • [6] T. C. Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space, Bull. Amer. Math. Soc. 80 (1974), 1123-1126. MR 0394333 (52:15136)
  • [7] W. R. Mann, Mean value methods in iterations, Proc. Amer. Math. Soc. 4 (1953), 506-510. MR 0054846 (14:988f)
  • [8] E. Michael, Topologies of spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 151-182. MR 0042109 (13:54f)
  • [9] S. B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475-488. MR 0254828 (40:8035)
  • [10] W. V. Petryshyn, Construction of fixed points of demicompact mappings in Hilbert spaces, J. Math. Anal. Appl. 14 (1966), 274-284. MR 0194942 (33:3147)
  • [11] W. V. Petryshyn and T. E. Williamson, Strong and weak convergence of the sequence of successive approximations of quasi-nonexpansive mappings, J. Math. Anal. Appl. 43 (1973), 459-497. MR 0326510 (48:4854)
  • [12] H. Schaefer, Ueber die Methode sukzessiver Approximationen, Jber. Deutsch. Math.-Verein. 59 (1957), 131-140. MR 0084116 (18:811g)
  • [13] H. F. Senter and W. G. Dotson, Jr., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375-380. MR 0346608 (49:11333)
  • [14] C. Shiau, K.-K. Tan and C. S. Wong, Quasi-nonexpansive multi-valued maps and selections, Fund. Math. 87 (1975), 109-119. MR 0372692 (51:8899)
  • [15] -, A class of quasi-nonexpansive multi-valued maps, Canad. Math. Bull. 18 (1975), 709-714. MR 0407667 (53:11439)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0580993-X
Keywords: Fixed points, set-valued mappings, iteration
Article copyright: © Copyright 1980 American Mathematical Society

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