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Let $A$ be a nonselfadjoint positive operator in a real Hilbert space. This paper deals with the stability of a class of iterative schemes used to solve the operator equation $Au=f$. A corresponding class of parabolic equations can also be solved by means of these iterative schemes. Several sufficient conditions of stability are obtained which are expressed in terms of known operators and can be used a priori. The results can be applied to problems with variable coefficients and initial-boundary value problems.
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