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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
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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.


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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