Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Normed convex processes


Author: Stephen M. Robinson
Journal: Trans. Amer. Math. Soc. 174 (1972), 127-140
MSC: Primary 46B99; Secondary 47A99
DOI: https://doi.org/10.1090/S0002-9947-1972-0313769-9
MathSciNet review: 0313769
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that several well-known results about continuous linear operators on Banach spaces can be generalized to the wider class of convex processes, as defined by Rockafellar. In particular, the open mapping theorem and the standard bound for the norm of the inverse of a perturbed linear operator can be extended to convex processes. In the last part of the paper, these theorems are exploited to prove results about the stability of solution sets of certain operator inequalities and equations in Banach spaces. These results yield quantitative bounds for the displacement of the solution sets under perturbations in the operators and/or in the right-hand sides. They generalize the standard results on stability of unique solutions of linear operator equations.


References [Enhancements On Off] (What's this?)

  • [1] A. Ben-Israel, On error bounds for generalized inverses, SIAM J. Numer. Anal. 3 (1966), 585-592. MR 35 #6344. MR 0215504 (35:6344)
  • [2] C. Berge, Topological spaces, Macmillan, New York, 1963.
  • [3] D. Gale, The theory of linear economic models, McGraw-Hill, New York, 1960. MR 22 #6599. MR 0115801 (22:6599)
  • [4] A. J. Hoffman, On approximate solutions of systems of linear inequalities, J. Res. Nat. Bur. Standards 49 (1952), 263-265. MR 14, 455. MR 0051275 (14:455b)
  • [5] L. V. Kantorovič and G. P. Akilov, Functional analysis in normed spaces, Fizmatgiz, Moscow, 1959; English transl., Internat. Series of Monographs in Pure and Appl. Math., vol. 46, Macmillan, New York, 1964. MR 22 #9837; MR 35 #4699. MR 0119071 (22:9837)
  • [6] J. L. Kelley, General topology, Van Nostrand, Princeton, N. J., 1955. MR 16, 1136. MR 0070144 (16:1136c)
  • [7] S. M. Robinson, Bounds for error in the solution set of a perturbed linear program, Linear Algebra Appl. (to appear). MR 0317760 (47:6307)
  • [8] -, Extension of Newtons method to nonlinear functions with values in a cone, Numer. Math. 19 (1972), 341-347. MR 0314259 (47:2811)
  • [9] R. T. Rockafellar, Monotone processes of convex and concave type, Mem. Amer. Math. Soc. No. 77 (1967). MR 37 #825. MR 0225231 (37:825)
  • [10] -, Convex analysis, Princeton Univ. Press, Princeton, N. J., 1970.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46B99, 47A99

Retrieve articles in all journals with MSC: 46B99, 47A99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0313769-9
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society