Nonstandard Methods in Fixed Point Theory pp 49-117 | Cite as

# Chapter 3

Chapter

## Abstract

Perhaps the most frequently cited fixed point theorem in analysis is the “Banach contraction mapping principle,” which states that if ( when

*M, d*) is a complete metric space and*T*is a contraction mapping from*M*to itself (there exists 0 <*k*< 1 such that*d*(*Tx*,*Ty*) ≤*kd*(*x*,*y*) for all*x*,*y*),then*T*has a unique fixed point in M. Moreover, for each*x*∈*M*the Picard iterates (*T*^{ n }(*x*)) converge to the fixed point. This theorem has its origins in Euler and Cauchy’s work [47] on the existence and uniqueness of a solution to the differential equation$$\{ \begin{array}{*{20}{c}} {dy/dx = f\left( {x,y} \right)} \\ {y\left( {{x_0}} \right) = {y_0}} \end{array}$$

*f*is a continuously differentiable function. In 1877, Lipschitz [144] simplified Cauchy’s proof using what we now know as the “Lipschitz condition.” (We should note that, interestingly, the method of Cauchy-Picard in fact was used before Cauchy.)## Keywords

Banach Space Convex Subset Nonexpansive Mapping Normal Structure Banach Lattice
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag New York Inc. 1990