Chapter 3

  • Asuman Güven Aksoy
  • Mohamed A. Khamsi
Part of the Universitext book series (UTX)


Perhaps the most frequently cited fixed point theorem in analysis is the “Banach contraction mapping principle,” which states that if (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 xM 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
$$\{ \begin{array}{*{20}{c}} {dy/dx = f\left( {x,y} \right)} \\ {y\left( {{x_0}} \right) = {y_0}} \end{array}$$
when 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.)


Banach Space Convex Subset Nonexpansive Mapping Normal Structure Banach Lattice 
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Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Asuman Güven Aksoy
    • 1
  • Mohamed A. Khamsi
    • 2
  1. 1.Department of MathematicsClaremont McKenna CollegeClaremontUSA
  2. 2.Department of Mathematical SciencesUniversity of Texas at El PasoEl PasoUSA

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