Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On iteration procedures for equations of the first kind, $ Ax=y$, and Picard's criterion for the existence of a solution

Authors: J. B. Diaz and F. T. Metcalf
Journal: Math. Comp. 24 (1970), 923-935
MSC: Primary 65.75
MathSciNet review: 0281376
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that the (not identically zero) linear operator $ A$, on a real Hilbert space $ H$ to itself, is compact, selfadjoint, and positive semidefinite; that $ y$ is a vector of $ H$ which is perpendicular to the null space of $ A$; and that $ \mu $ is a real number such that $ 0 < \mu < 2/\vert\vert A\vert\vert$. Then, the "iteration scheme" $ {x_{n = + 1}} = {x_n} + \mu (y - A{x_n}),n = 0,1,2, \cdot \cdot \cdot $, yields a strongly convergent sequence of vectors $ \{x_n\}_{n = 0}^\infty$ if and only if "Picard's criterion" for the existence of a solution of $ Ax = y$ holds (i.e., if and only if $ y$ is perpendicular to the null space of $ A$, and $ \sum\nolimits_{k = 1}^\infty {{{(y,{u_k})}^2}} /\lambda _k^2 < \infty $, where the $ {u_k}$ and the $ {\lambda _k}$ are the orthonormalized eigenvectors, and the corresponding eigenvalues, of $ A$, respectively). An analogous result holds when $ A$ is only required to be compact.

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

  • [1] E. Picard, "Sur un théorème générale relatif aux équations intégrales de première espèce et sur quelques problèmes de physique mathématique," Rend. Cire. Mat. Palermo, v. 29, 1910, pp. 79-97.
  • [2] F. Smithies, Integral equations, Cambridge Tracts in Mathematics and Mathematical Physics, no. 49, Cambridge University Press, New York, 1958. MR 0104991
  • [3] L. Landweber, An iteration formula for Fredholm integral equations of the first kind, Amer. J. Math. 73 (1951), 615–624. MR 0043348,
  • [4] V. M. Fridman, Method of successive approximations for a Fredholm integral equation of the 1st kind, Uspehi Mat. Nauk (N.S.) 11 (1956), no. 1(67), 233–234 (Russian). MR 0076183
  • [5] Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966
  • [6] Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65.75

Retrieve articles in all journals with MSC: 65.75

Additional Information

Keywords: Fredholm integral equations, iteration procedures, Picard's criterion
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society