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On a generalization of Picard’s theorem on solvability of a Fredholm integral equation of the first kind. (English. Russian original) Zbl 0609.47021

Sov. Math., Dokl. 31, 143-146 (1985); translation from Dokl. Akad. Nauk SSSR 280, 781-784 (1985).
Let X and Y be normed spaces and let A:X\(\to Y\) be a bounded linear operator. Consider the equation \[ (1)\quad Ax=y. \] The sequence \((R_ n)\) of bounded operators \(Y\to X\) is called a linear regularizer (LR) for the equation (1) if \(\forall x\in X,\exists x_ 0\in X:R_ nAx\to x_ 0\) and \(x-x_ 0\in \ker A\) [V. V. Ivanov, V. V. Vasin, and V. P. Tanana, Theory of linear ill-posed problems and its applications (Russian) (1978; Zbl 0489.65035)]. An \(LR(R_ n)\) for (1) is said to be soluble if the convergence set of \((R_ n)\{y\in Y| \exists x\in X:R_ ny\to x\}\) coincides with the range of A. A well known theorem of Picard concerns the solvability of the integral equation \[ (2)\quad Kx=\int^{b}_{a}k(s,t)x(t)dt=y(s) \] where \(K:L_ 1[a,b]\to L_ 2[a,b]\) and k(s,t) is continuous, symmetric and closed. The theorem can be reformulated to assert that a certain sequence \((R_ n)\) of operators of the form \[ (3)\quad R_ ny=\sum^{n}_{k-1}g_ k(y)e_ k \] where \((e_ k)\) is some basis in X and \(\{g_ k\}\subset Y'\), is a soluble LR for (2).
The authors state without proof several theorems concerning the existence of soluble LR’s, the non-solvability of certain LR’s, necessary conditions on X and Y for the existence of a solvable LR for (1), and related properties. It is remarked that (1) has LR’s of the form (3) whenever A is injective and X is reflexive with a basis.
Reviewer: R.W.Gross

MSC:

47A50 Equations and inequalities involving linear operators, with vector unknowns
45B05 Fredholm integral equations
45L05 Theoretical approximation of solutions to integral equations

Citations:

Zbl 0489.65035
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