Part of the Texts in Applied Mathematics book series (TAM, volume 39)
Nonlinear Equations and Their Solution by Iteration
Nonlinear functional analysis is the study of operators lacking the property of linearity. In this chapter, we consider nonlinear operator equations and their numerical solution. We begin the consideration of operator equations that take the form
Here, V is a Banach space, K is a subset of V, and T : K → V. The solutions of this equation are called fixed points of the operator T, as they are left unchanged by T. The most important method for analyzing the solvability theory for such equations is the Banach fixed-point theorem. We present the Banach fixed-point theorem and then discuss its application to the study of various iterative methods in numerical analysis.
$$ u = T(u),\;u \in K $$
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© Springer-Verlag New York, Inc. 2001