A relaxed Picard iteration process for setvalued operators of the monotone type
Author:
J. C. Dunn
Journal:
Proc. Amer. Math. Soc. 73 (1979), 319327
MSC:
Primary 47H10; Secondary 65J05
MathSciNet review:
518512
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Abstract: The fixed points of setvalued operators, , satisfying a condition of the monotonicity type on convex subsets X of a Hilbert space are approximated by a relaxation process, , in which is a singlevalued branch of T and the relaxation parameter is made to depend in a certain way on the prior history of the process. If is bounded on bounded subsets of X, then converges to 0 like . If is also continuous at and if , then . If satisfies a condition of the Lipschitz type at , then for some .
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 J. C. Dunn, Iterative construction of fixed points for multivalued operators of the monotone type, J. Functional Analysis 27 (1978), 3850. MR 477162 (81f:47056)
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 R. E. Bruck, Jr., The iterative solution of the equation for a monotone operator T in Hilbert space, Bull. Amer. Math. Soc. 79 (1973), 12581261. MR 0328692 (48:7034)
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 B. E. Rhoades, Fixed point iterations using infinite matrices. III, in Proc. Conf. on Computing Fixed Points with Applications (Clemson Univ., June 1974), Academic Press, New York (to appear). MR 0519589 (58:24922)
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 R. E. Bruck, A strongly convergent iterative solution of for a maximal monotone operator U in Hilbert space, J. Math. Anal. Appl. 48 (1974), 114126. MR 0361941 (50:14383)
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 F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197228. MR 0217658 (36:747)
 [10]
 T. E. Williamson, Jr., Geometric estimation of for monotonetype operator T in Hilbert space, Proc. Amer. Math. Soc. (to appear). MR 728603 (85a:47067)
 [11]
 J. C. Dunn, Rates of convergence for conditional gradient algorithms near singular and nonsingular extremals, SIAM J. Control Optimization (to appear). MR 525021 (80d:49025)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197905185128
PII:
S 00029939(1979)05185128
Keywords:
Setvalued Hilbert space operators,
fixed points,
relaxed Picard iterations,
residualdependent step lengths,
continuity conditions and convergence rates
Article copyright:
© Copyright 1979
American Mathematical Society
