A theory is developed for the fixed-h stability of integration schemes based on A(α)-stabie formulas when applied to nonlinear parabolic-like stiff equations. The theory is based on a general multiplier technique whose properties we fully develop. Assuming that a few simply checkable criteria which are related to monotonicity properties of the nonlinearity around the computed solution are satisfied, we obtain various error bounds and boundedness results for that solution. Some practical implications of the theory are also given. © 1981, Taylor & Francis Group, LLC. All rights reserved.