A sufficient and necessary condition for the convergence of the sequence of successive approximations to a unique fixed point

Suzuki, Tomonari

doi:10.1090/S0002-9939-08-09390-8

A sufficient and necessary condition for the convergence of the sequence of successive approximations to a unique fixed point
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Proc. Amer. Math. Soc. 136 (2008), 4089-4093 Request permission

Abstract:

If $(X, d)$ is a complete metric space and $T : X \to X$ is a contraction mapping, then the conclusion of the Banach-Caccioppoli contraction principle is that the sequence of successive approximations of $T$ starting from any point of the space converges to a unique fixed point. In this paper, we obtain a sufficient and necessary condition of the above conclusion in terms of the so-called strong Leader mappings.

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