A relaxed Picard iteration process for set-valued operators of the monotone type

Author:
J. C. Dunn

Journal:
Proc. Amer. Math. Soc. **73** (1979), 319-327

MSC:
Primary 47H10; Secondary 65J05

MathSciNet review:
518512

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Abstract: The fixed points of set-valued 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 single-valued 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 .

**[1]**J. C. Dunn,*Iterative construction of fixed points for multivalued operators of the monotone type*, J. Funct. Anal.**27**(1978), no. 1, 38–50. MR**477162**, 10.1016/0022-1236(78)90018-6**[2]**Ronald E. Bruck Jr.,*The iterative solution of the equation 𝑦∈𝑥+𝑇𝑥 for a monotone operator 𝑇 in Hilbert space*, Bull. Amer. Math. Soc.**79**(1973), 1258–1261. MR**0328692**, 10.1090/S0002-9904-1973-13404-4**[3]**B. E. Rhoades,*Fixed point iterations using infinite matrices. III*, Fixed points: algorithms and applications (Proc. First Internat. Conf., Clemson Univ., Clemson, S.C., 1974) Academic Press, New York, 1977, pp. 337–347. MR**0519589****[4]**Ronald E. Bruck Jr.,*A strongly convergent iterative solution of 0∈𝑈(𝑥) for a maximal monotone operator 𝑈 in Hilbert space*, J. Math. Anal. Appl.**48**(1974), 114–126. MR**0361941****[5]**E. H. Zarantonello,*Solving functional equations by contractive averaging*, U. S. Army Mathematics Research Center Technical Report #160, Wisconsin, June 1960.**[6]**Eduardo H. Zarantonello,*The closure of the numerical range contains the spectrum*, Bull. Amer. Math. Soc.**70**(1964), 781–787. MR**0173176**, 10.1090/S0002-9904-1964-11237-4**[7]**Eduardo H. Zarantonello,*The closure of the numerical range contains the spectrum*, Pacific J. Math.**22**(1967), 575–595. MR**0229079****[8]**A. B. Bakušinskiĭ and B. T. Poljak,*The solution of variational inequalities*, Dokl. Akad. Nauk SSSR**219**(1974), 1038–1041 (Russian). MR**0377619****[9]**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**0217658****[10]**T. E. Williamson,*A geometric approach to fixed points of non-self-mappings 𝑇:𝐷→𝑋*, Fixed points and nonexpansive mappings (Cincinnati, Ohio, 1982) Contemp. Math., vol. 18, Amer. Math. Soc., Providence, RI, 1983, pp. 247–253. MR**728603**, 10.1090/conm/018/728603**[11]**J. C. Dunn,*Rates of convergence for conditional gradient algorithms near singular and nonsingular extremals*, SIAM J. Control Optim.**17**(1979), no. 2, 187–211. MR**525021**, 10.1137/0317015

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1979-0518512-8

Keywords:
Set-valued Hilbert space operators,
fixed points,
relaxed Picard iterations,
residual-dependent step lengths,
continuity conditions and convergence rates

Article copyright:
© Copyright 1979
American Mathematical Society