Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T03:09:04.710Z Has data issue: false hasContentIssue false
  • English
  • Français

On uniformly contractive systems and quadratic equations in Banach space

Published online by Cambridge University Press:  17 April 2009

David K. Ruch
David K. Ruch
Affiliation:
Department of MathematicsSam Houston State UniversityHuntsvilleTX 77341United States of America
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The solution of quadratic equations using the contraction mapping principle is considered. A uniqueness result extending that given by Argyros is proved. Uniformly contractive systems theory is used to find approximate solutions and convergence criteria are given. In particular, only pointwise convergence of approximating operators is required to guarantee convergence of the approximate solutions. A theorem and algorithm for a continuation method are presented, and illustrated on Chandrasekhar's equation.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 52 , Issue 3 , December 1995 , pp. 441 - 455
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Anselone, P.M. and Moore, R.H., ‘An extension of the Newton-Kantorovič method for solving nonlinear equations with an application to elasticity’, J. Math. Anal. Appl. 13 (1966), 476–501.CrossRefGoogle Scholar
[2]Argyros, I.K., ‘Quadratic equations and applications to Chandreshekhar's and related equations’, Bull. Austral. Math. Soc. 32 (1985), 275–292.CrossRefGoogle Scholar
[3]Argyros, I.K., ‘On the solution by series of some nonlinear equations’, Rev. Acad. Cienc. Zaragoza 42 (1987), 19–23.Google Scholar
[4]Argyros, I.K., ‘On the approximation of solutions of compact operator equations in Banach space’, Proyecciones 14 (1988), 29–46.CrossRefGoogle Scholar
[5]Chandrasekhar, S., Radiative transfer (Dover, New York, 1960).Google Scholar
[6]Kupsch, J., ‘Estimates of the unitary integral’, Comm. Math. Phys. 19 (1960), 65–82.CrossRefGoogle Scholar
[7]McFarland, J.E., ‘An iterative solution of the quadratic equation in Banach space’, Proc. Amer. Math. Soc. (1958), 824–830.CrossRefGoogle Scholar
[8]Porter, W., ‘Synthesis of polynomic systems’, Siam J. Math. Anal. 11 (1980), 308–315.CrossRefGoogle Scholar
[9]Ball, L.B., ‘Quadratic equations in Banach space’, Rend. Circ. Math. Palermo Suppl, 10 (1961), 314–332.Google Scholar
[10]Rall, L.B., ‘On the uniqueness of solutions of quadratic equations’, Siam Rev. 11 (1969), 386–388.CrossRefGoogle Scholar
[11]Ruch, D.K., ‘Characterizations of compact bilinear maps’, Linear and Multilinear Algebra, 25 (1989), 297–307.CrossRefGoogle Scholar
[12]Ruch, D.K. and Van Fleet, P.J., ‘On multipower equations: Some iterative solutions and applications’, (preprint, 1994).Google Scholar