Random products of contractions in Banach spaces

Authors:
J. Dye, M. A. Khamsi and S. Reich

Journal:
Trans. Amer. Math. Soc. **325** (1991), 87-99

MSC:
Primary 47A05; Secondary 65J10

DOI:
https://doi.org/10.1090/S0002-9947-1991-0989572-5

MathSciNet review:
989572

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the random product of a finite number of contractions converges weakly in all smooth reflexive Banach spaces. If one of the contractions is compact, then the convergence is uniform.

**[1]**I. Amemiya and T. Ando,*Convergence of random products of contractions in Hilbert space*, Acta Sci. Math. (Szeged)**26**(1965), 239-244. MR**0187116 (32:4570)****[2]**F. E. Browder,*On some approximation methods for solutions of the Dirichlet problem for linear elliptic equations of arbitrary order*, J. Math. Mech.**7**(1958), 69-80. MR**0092070 (19:1057a)****[3]**R. E. Bruck,*Properties of fixed-point sets of nonexpansive mappings in Banach spaces*, Trans. Amer. Math. Soc.**179**(1973), 251-262. MR**0324491 (48:2843)****[4]**-,*Random products of contractions in metric and Banach spaces*, J. Math. Anal. Appl.**88**(1982), 319-332. MR**667060 (84a:47075)****[5]**-,*Asymptotic behavior of nonexpansive mappings*, Contemp. Math.**18**(1983), 1-47. MR**728592 (85d:47057)****[6]**R. E. Bruck and S. Reich,*Nonexpansive projections and resolvents of accretive operators in Banach spaces*, Houston J. Math.**3**(1977), 459-470. MR**0470761 (57:10507)****[7]**J. Dye,*Convergence of random products of compact contractions in Hilbert space*, Integral Equations Operator Theory**12**(1989), 12-22. MR**973044 (90c:47013)****[8]**-,*A generalization of a theorem of Amemiya and Ando on the convergence of random products of contractions in Hilbert space*, Integral Equations Operator Theory**12**(1989), 155-162. MR**986593 (90b:47067)****[9]**C. Franchetti and W. Light,*The alternating algorithm in uniformly convex spaces*, J. London Math. Soc.**29**(1984), 545-555. MR**754940 (85h:41064)****[10]**I. Halperin,*The product of projection operators*, Acta Sci. Math. (Szeged)**23**(1962), 96-99. MR**0141978 (25:5373)****[11]**C. Hamaker and D. C. Solmon,*The angles between the null spaces of*-*rays*, J. Math. Anal. Appl.**62**(1978), 1-23. MR**0463859 (57:3798)****[12]**U. Krengel,*Ergodic theorems*, De Gruyter, Berlin, 1985. MR**797411 (87i:28001)****[13]**T C. Lim,*Asymptotic centers and nonexpansive mappings in conjugate Banach spaces*, Pacific J. Math.**90**(1980), 135-143. MR**599326 (82h:47052)****[14]**P. L. Lions,*On the Schwarz alternating method*. I, Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, Pa., 1988, pp. 1-42. MR**972510 (90a:65248)****[15]**S. Reich,*Product formulas, nonlinear semigroups and accretive operators*, J. Funct. Anal.**36**(1980), 147-168. MR**569251 (81k:47076)****[16]**-,*Nonlinear semigroups, accretive operators and applications*, Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York, 1982, pp. 831-838. MR**728043 (85c:47060)****[17]**-,*A limit theorem for projections*, Linear and Multilinear Algebra**13**(1983), 281-290. MR**700890 (84i:47070)****[18]**K. T. Smith, D. C. Solmon, and S. L. Wagner,*Practical and mathematical aspects of reconstructing objects from radiographs*, Bull. Amer. Math. Soc.**83**(1977), 1227-1270. MR**0490032 (58:9394a)****[19]**J. E. Spingarn,*A projection method for least-squares solutions to over-determined systems of linear inequalities*, Linear Algebra Appl.**86**(1987), 211-236. MR**870941 (88k:65058)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
47A05,
65J10

Retrieve articles in all journals with MSC: 47A05, 65J10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1991-0989572-5

Keywords:
Contraction,
random product,
weak convergence

Article copyright:
© Copyright 1991
American Mathematical Society