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Iterative Solution of a Nonlinear Singularly Perturbed Equation in a Banach Space and its Application to Differential Equations

Valery Y. Glizer


The paper deals with an equation, considered on some linear subspace of a Banach space. This equation consists of linear unbounded and nonlinear bounded parts. The linear part depends on a small positive parameter. Replacing this parameter by zero yields the reduced equation. It is assumed that a solution of the latter (if it exists) does not belong to the linear subspace. This implies that the original equation is singularly perturbed. An iterative method for the approximate solution of this singularly perturbed equation, uniformly valid with respect to the small parameter, is developed. This method is applied to some differential equations in classical and generalized formulations.

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