A comparison of various definitions of contractive mappings

Abstract

A number of authors have defined contractive type mappings on a complete metric space <italic>X</italic> which are generalizations of the well-known Banach contraction, and which have the property that each such mapping has a unique fixed point. The fixed point can always be found by using Picard iteration, beginning with some initial choice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x 0 element-of upper X"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mo>∈</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{x_0} \in X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper we compare this multitude of definitions. <italic>X</italic> denotes a complete metric space with distance function <italic>d</italic>, and <italic>f</italic> a function mapping <italic>X</italic> into itself.