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

By Mohamed A. Khamsi

Content material:
Chapter 1 advent (pages 1–11):
Chapter 2 Metric areas (pages 13–40):
Chapter three Metric Contraction ideas (pages 41–69):
Chapter four Hyperconvex areas (pages 71–99):
Chapter five “Normal” constructions in Metric areas (pages 101–124):
Chapter 6 Banach areas: advent (pages 125–170):
Chapter 7 non-stop Mappings in Banach areas (pages 171–196):
Chapter eight Metric fastened element concept (pages 197–241):
Chapter nine Banach house Ultrapowers (pages 243–271):

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Extra info for An Introduction to Metric Spaces and Fixed Point Theory

Sample text

Consequently the limit r = lim d(Tn(x),x0) n—*oo exists and r > 0. Also, since M is compact, the sequence {T n (x)} has a convergent subsequence {T n *(x)}, say lim T n *(x) = z € M. Since {T n (x)} is fc—»CO decreasing, r = φ , χ 0 ) = um d(Tn" (x), xQ) = lim d(T nfc+1 (:r),zo) = fc—»oo fc—*oo d{T(z),xQ). But if z φ XQ then d(T(z),xo) = d(T(z),T(xo)) < d(z,xo) · This proves that any convergent subsequence of {T n (x)} converges to xo, so it must be the case that lim T n (x) = x 0 . 2 Further extensions of Banach's Principle The strength of the Contraction Mapping Principle lies in the fact that the underlying space is quite general (complete metric) while the conclusion is very strong.

On the other hand, if M is any compact metric space and if A Ç M, then A does have a metric convex hull, although it need not be unique. This is a classical result proved by Karl Menger in 1931. It rests on the following lemma. 1 Let C\ 2 C*2 =? · ■ ■ be a descending sequence of nonempty closed metrically convex subsets of a compact metric space (M,d). nonempty and metrically convex. oo Then f] Cn is n=l Proof. The fact that the intersection is nonempty is immediate from comoo pactness. Suppose x,y E f] Cn with x φ y.

Khamsi and William A. Kirk Copyright © 2001 John Wiley & Sons, Inc. 1 Banach's Contraction Principle Banach's Contraction Mapping Principle is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theorem in all of analysis. This is because the contractive condition on the mapping is simple and easy to test, because it requires only a complete metric space for its setting, because it provides a constructive algorithm, and because it finds almost canonical applications in the theory of differential and integral equations.

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