An Introduction to Metric Spaces and Fixed Point Theory by Mohamed A. Khamsi

By Mohamed A. Khamsi

Offers updated Banach area results.
* good points an intensive bibliography for outdoor reading.
* presents specified routines that elucidate extra introductory fabric.

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The fixed point is unique and sequences of iterates of the original mapping always converge to this fixed point. It seems that any theorem which should claim to be a genuine extension of the contraction principle should offer just as much, and indeed, many such extensions appear in the literature. The extensions we present in this section are easy to describe and include many of the others. They are in no way definitive, however. The first extension we take up, which is due to Geraghty [63], was inspired by an earlier theorem of Rakotch [134].

A family C of subsets of a set X is called an (abstract) convexity structure if (1) Both 0 and X are in C. (2) C is stable under intersections; that is, if {Da}aei subfamily of C then f) Da € C. is any nonempty (3) C is stable for nested unions; that is, if {Da}aej is any nonempty subfamily of C which is totally ordered by set inclusion, then (J Da £ C. «6/ 36 CHAPTER 2. METRIC SPACES Unfortunately, for our purposes, there is even a problem with this abstract formulation. In the study of metric fixed point theory closed balls turn out to play a fundamental role.

The proof is left as an exercise. 9 If {xn\ is a Cauchy sequence in a metric space M and if {xn} has a subsequence {xnk} which converges to x E M, then lim xn — x. 3. 27 COMPLETENESS The diameter of a nonempty bounded subset D of a metric space is the number diam (D) = sup{d(x, y) : x,y 6 D). Note that we have diam (D) = diam (£>). 10 (Cantor's Intersection Theorem) A metric space (M,d) is complete if and only if given any descending sequence {Dn} of nonempty bounded closed subsets of M, Um diam (D„) = 0 => n—»oo I ' n=l Γ\ΌηφΖ.

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