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Approximation to fixed points of generalized nonexpansive mappings


Author: Chi Song Wong
Journal: Proc. Amer. Math. Soc. 54 (1976), 93-97
DOI: https://doi.org/10.1090/S0002-9939-1976-0390850-9
MathSciNet review: 0390850
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Abstract | References | Additional Information

Abstract: Let $ K$ be a convex subset of a uniformly convex Banach space. It is proved that if $ K$ is compact, then the fixed points of a continuous generalized nonexpansive self-mapping $ T$ on $ K$ can be approximated by the iterates of $ {T_t}$ with $ t \in (0,1),{T_t}(x) = (1 - t)x + tT(x),x \in K;{T_t}$ is asymptotically regular if $ T$ has a fixed point.


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  • [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), 566-570. MR 32 #8155a. MR 0190744 (32:8155a)
  • [2] M. Edelstein, A remark on a theorem of M. A. Krasnoselski, Amer. Math. Monthly 73 (1966), 509-510. MR 33 #3072. MR 0194866 (33:3072)
  • [3] K. Goebel, W. A. Kirk and Tawfik N. Shimi, A fixed point theorem in uniformly convex spaces, Bol. Un. Mat. Ital., 47 (1973), 65-75. MR 47#9367. MR 0320834 (47:9367)
  • [4] G. F. Hardy and T. D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull. 16 (1973), 201-206. MR 48 #2847. MR 0324495 (48:2847)
  • [5] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71-76. MR 41 #2486. MR 0257837 (41:2486)
  • [6] -, Some results on fixed points. II, Amer. Math. Monthly 76 (1969), 405-408. MR 41 #2487. MR 0257838 (41:2487)
  • [7] -, Some results on fixed points. III, Fund. Math. 70 (1971), 169-177. MR 44#879. MR 0283649 (44:879)
  • [8] -, Some results on fixed points. IV, Fund. Math. 74 (1972), 181-187. MR 45 #9310. MR 0300264 (45:9310)
  • [9] -, Fixed point theorems in reflexive Banach spaces, Proc. Amer. Math. Soc. 38 (1973), 111-118. MR 47 #2448. MR 0313896 (47:2448)
  • [10] W. V. Petryshyn and T. E. Williamson, Jr., Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl. 43 (1973), 459-497. MR 48 #4854. MR 0326510 (48:4854)
  • [11] Simeon Reich, Some remarks concerning contraction mappings, Canad. Math. Bull. 14 (1971), 121-124. MR 45 #1145. MR 0292057 (45:1145)
  • [12] -, Kannan's fixed point theorem, Bol. Un. Mat. Ital. (4) 4 (1971), 1-11. MR 46 #4293. MR 0305163 (46:4293)
  • [13] -, Fixed points of contractive functions, Bol. Un. Mat. Ital. (4) 4 (1972), 26-42. MR 46 #8206. MR 0309095 (46:8206)
  • [14] H. Schaefer, Über die Methode sukzessiver Approximationen, Jber. Deutsch Math. Verein. 59 (1957), 131-140. MR 18, 811. MR 0084116 (18:811g)
  • [15] Paolo Soardi, Su un problema di punto unito di S. Reich, Bol. Un. Mat. Ital. (4) 4 (1971), 841-845. MR 46#741. MR 0301583 (46:741)
  • [16] Chi Song Wong, Fixed point theorems for generalized nonexpansive mappings, J. Australian Math. Soc. 18 (1974), 265-276. MR 0358753 (50:11212)
  • [17] -, Common fixed points for two mappings, Pacific J. Math. 48 (1973), 299-312.
  • [18] -, A fixed point theorem for a class of mappings, Math. Ann. 204 (1973), 97-103. MR 0367964 (51:4206)
  • [19] -, Fixed points of certain self maps on an interval, Proc. Amer. Math. Soc. 42 (1974), 234-235. MR 0325869 (48:4215)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0390850-9
Article copyright: © Copyright 1976 American Mathematical Society

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