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
DOI: https://doi.org/10.1090/S0025-5718-1970-0281376-4
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 Math, and Math. Phys., no. 49, Cambridge Univ. Press, New York, 1958. MR 21 #3738. MR 0104991 (21:3738)
  • [3] L. Landweber, "An iteration formula for Fredholm integral equations of the first kind," Amer. J. Math., v. 73, 1951, pp. 615-624. MR 13, 247. MR 0043348 (13:247c)
  • [4] V. M. Fridman, "Method of successive approximations for a Fredholm integral equation of the first kind," Uspehi Mat. Nauk, v. 11, 1956, no. 1 (67), pp. 233-234. (Russian) MR 17, 861. MR 0076183 (17:861a)
  • [5] A. E. Taylor, Introduction to Functional Analysis, Wiley, New York, and Chapman &Hall, London, 1958. MR 20 #5411. MR 0098966 (20:5411)
  • [6] F. Riesz & B. Sz.-Nagy, Functional Analysis, Akad. Kiadó, Budapest, 1953; English transl., Ungar, New York, 1955. MR 15, 132; MR 17, 175. MR 0071727 (17:175i)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65.75

Retrieve articles in all journals with MSC: 65.75


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1970-0281376-4
Keywords: Fredholm integral equations, iteration procedures, Picard's criterion
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society