The solution by iteration of nonlinear equations in Hilbert spaces

Author:
Şt. Măruşter

Journal:
Proc. Amer. Math. Soc. **63** (1977), 69-73

MSC:
Primary 47H10

MathSciNet review:
0636944

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The weak and strong convergence of the iterates generated by to a fixed point of the mapping are investigated, where *C* is a closed convex subset of a real Hilbert space. The basic assumptions are that *T* has at least one fixed point in *C*, and that is demiclosed at 0 and satisfies a certain condition of monotony. Some applications are given.

**[1]**Shmuel Agmon,*The relaxation method for linear inequalities*, Canadian J. Math.**6**(1954), 382–392. MR**0062786****[2]**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**, 10.1090/S0002-9904-1966-11544-6**[3]**J. B. Diaz and F. T. Metcalf,*On the set of subsequential limit points of successive approximations.*, Trans. Amer. Math. Soc.**135**(1969), 459–485. MR**0234327**, 10.1090/S0002-9947-1969-0234327-0**[4]**W. G. Dotson Jr.,*On the Mann iterative process*, Trans. Amer. Math. Soc.**149**(1970), 65–73. MR**0257828**, 10.1090/S0002-9947-1970-0257828-6**[5]**I. I. Eremin,*Generalization of the relaxation method of Motzkin and Agmon*, Uspehi Mat. Nauk**20**(1965), no. 2 (122), 183–187 (Russian). MR**0202473****[6]**T. S. Motzkin and I. J. Schoenberg,*The relaxation method for linear inequalities*, Canadian J. Math.**6**(1954), 393–404. MR**0062787****[7]**Curtis L. Outlaw,*Mean value iteration of nonexpansive mappings in a Banach space*, Pacific J. Math.**30**(1969), 747–750. MR**0247542****[8]**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**0326510****[9]**Helmut Schaefer,*Über die Methode sukzessiver Approximationen*, Jber. Deutsch. Math. Verein.**59**(1957), no. Abt. 1, 131–140 (German). MR**0084116****[10]**H. F. Senter and W. G. Dotson Jr.,*Approximating fixed points of nonexpansive mappings*, Proc. Amer. Math. Soc.**44**(1974), 375–380. MR**0346608**, 10.1090/S0002-9939-1974-0346608-8

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47H10

Retrieve articles in all journals with MSC: 47H10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0636944-2

Keywords:
Iteration,
fixed points,
demiclosed mappings,
monotone mappings

Article copyright:
© Copyright 1977
American Mathematical Society