Geodesic
On a~criterion for solvability of Fredholm equations
Sbornik. Mathematics, Tome 55 (1986) no. 1, pp. 113-119

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Fredholm equations of the first and second kinds are considered in an abstract Hilbert space. A unified solvability criterion is given for them in terms of nonnegativity of bilinear forms. For Fredholm equations of the second kind it is possible to pass from an arbitrary equation to an equation with nonnegative operator, some solution of which can be used to construct a solution of the original equation that is minimal in norm. Questions connected with restrictions on the norms of solutions are answered for these equations. The results are then carried over to Fredholm integral equations of both kinds and to linear algebraic systems. Bibliography: 4 titles. 
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     title = {On a~criterion for solvability of {Fredholm} equations},
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N. K. Al'bov. On a~criterion for solvability of Fredholm equations. Sbornik. Mathematics, Tome 55 (1986) no. 1, pp. 113-119. http://geodesic.mathdoc.fr/item/SM_1986_55_1_a6/