Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Fixed point iterations using infinite matrices


Author: B. E. Rhoades
Journal: Trans. Amer. Math. Soc. 196 (1974), 161-176
MSC: Primary 47H10; Secondary 65J05
MathSciNet review: 0348565
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let E be a closed, bounded, convex subset of a Banach space $ X,f:E \to E$. Consider the iteration scheme defined by $ {\bar x_0} = {x_0} \in E,{\bar x_{n + 1}} = f({x_n}),{x_n} = \Sigma _{k = 0}^n{a_{nk}}{\bar x_k},\;n \geq 1$, where A is a regular weighted mean matrix. For particular spaces X and functions f we show that this iterative scheme converges to a fixed point of f.


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

  • [1] H. G. Barone, Limit points of sequences and their transforms by methods of summability, Duke Math. J. 5 (1939), 740-752. MR 1, 218. MR 0001323 (1:218a)
  • [2] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197-228. MR 36 #747. MR 0217658 (36:747)
  • [3] L. B. Ciric, Generalized contractions and fixed-point theorems, Publ. Inst. Math. (Beograd) 12 (1971), 19-26. MR 0309092 (46:8203)
  • [4] W. G. Dotson, Jr., On the Mann iterative process, Trans. Amer. Math. Soc. 149 (1970), 65-73. MR 41 #2477. MR 0257828 (41:2477)
  • [5] W. G. Dotson, Jr. and W. R. Mann, A generalized corollary of the Browder-Kirk fixed point theorem, Pacific J. Math. 26 (1968), 455-459. MR 38 #5080. MR 0236786 (38:5080)
  • [6] R. L. Franks and R. P. Marzec, A theorem on mean value iterations, Proc. Amer. Math. Soc. 30 (1971), 324-326. MR 43 #6375. MR 0280656 (43:6375)
  • [7] C. W. Groetsch, A note on segmenting Mann iterates, J. Math. Anal. Appl. 40 (1972), 369-372. MR 0341204 (49:5954)
  • [8] G. H. Hardy, Divergent series, Clarendon Press, Oxford, 1949. MR 11, 25. MR 0030620 (11:25a)
  • [9] B. P. Hillam, Fixed point iterations and infinite matrices, and subsequential limit points of fixed point sets, Ph.D. Dissertation, University of California, Riverside, Calif., June, 1973.
  • [10] G. G. Johnson, Fixed points by mean value iteration, Proc. Amer. Math. Soc. 34 (1972), 193-194. MR 45 #1006. MR 0291918 (45:1006)
  • [11] R. Kannan, Some results on fixed points. III, Fund. Math. 70 (1971), no. 2, 169-177. MR 44 #879. MR 0283649 (44:879)
  • [12] -, Construction of fixed points of a class of nonlinear mappings, J. Math. Anal. Appl. 41 (1973), 430-438. MR 0320837 (47:9370)
  • [13] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. MR 14, 988. MR 0054846 (14:988f)
  • [14] C. L. Outlaw, Mean value iteration of nonexpansive mappings in a Banach space, Pacific J. Math. 30 (1969), 747-750. MR 40 #807. MR 0247542 (40:807)
  • [15] C. L. Outlaw and C. W. Groetsch, Averaging iteration in a Banach space, Bull. Amer. Math. Soc. 75 (1969), 430-432. MR 39 #835. MR 0239478 (39:835)
  • [16] J. Reinermann, Über Toeplitzsche Iterationsverfahren und einige ihre Anwendungen in der konstruktiven Fixpunktheorie, Studia Math. 32 (1969), 209-227. [Zbl. 176, 123]. MR 46 #2508. MR 0303371 (46:2508)
  • [17] -, Über Fixpunkte kontrahierender Abbildungen und schwach konvergente Toeplitz-Verfahren, Arch. Math. (Basel) 20 (1969), 59-64 MR 43 #2583. MR 0276843 (43:2583)
  • [18] B. E. Rhoades, Fixed point iterations using infinite matrices, Preliminary report, Notices Amer. Math. Soc. 19 (1972), A-516. Abstract #72T-B142.
  • [19] B. E. Rhoades, A comparison of various definitions of contractive mappings (in preparation).
  • [20] J. Schauder, Der Fixpunktsatz in Functionräumen, Studia Math. 2 (1930), 171-180.
  • [21] T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math. 23 (1972), 292-298. MR 0310859 (46:9957)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47H10, 65J05

Retrieve articles in all journals with MSC: 47H10, 65J05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0348565-1
PII: S 0002-9947(1974)0348565-1
Keywords: Cesàro matrix, contraction, fixed point iterations, quasi-nonexpansive, regular matrix, strictly pseudocontractive, weighted mean matrix
Article copyright: © Copyright 1974 American Mathematical Society