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Download e-book for iPad: An Introduction to Metric Spaces and Fixed Point Theory by Mohamed A. Khamsi

By Mohamed A. Khamsi

ISBN-10: 0471418250

ISBN-13: 9780471418252

ISBN-10: 1118033078

ISBN-13: 9781118033074

Content material:
Chapter 1 creation (pages 1–11):
Chapter 2 Metric areas (pages 13–40):
Chapter three Metric Contraction rules (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 mounted element concept (pages 197–241):
Chapter nine Banach area Ultrapowers (pages 243–271):

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

Example text

Also < ψ{χα) - ψ{χα+ι). < φ(χμ) - φ(χα). If μ < a, then by assumption ά(χμ,χα) Hence ά(χμ,χα+ι) < < ά(χμ,χα)+ά(χα,χα+ι) φ(χμ) - φ(χα) + φ(χα) - φ(χα+1) This establishes (l). Case 2. β is a limit ordinal. First note that (1) implies {φ(χα)}α<β is a nonincreasing net of real numbers. Since ψ is bounded below there exists r G R such that lim ψ(χα) — inf ψ(χα) = r- a—0- a<0 Let ε > 0. Then there exists c*o < β such that if c*o < a < β then φ{χα)

The following is an example of a seemingly more general formulation of Caristi's theorem which actually reduces to a simple application of the original result. 17 Let M be a complete metric space and suppose φ : M —> R is lower semicontinuous and bounded below. Suppose ξ : R —» R is continuous, strictly increasing, concave downward, and vanishes at 0, and suppose g : M —> M satisfies: ξ{ά(χ, g(x))) < φ{χ) -

Since ψ is bounded below there exists r G R such that lim ψ(χα) — inf ψ(χα) = r- a—0- a<0 Let ε > 0. Then there exists c*o < β such that if c*o < a < β then φ{χα) 0- = < = ν(^μ) - r ψ{χμ) - ψ{χ) ψ{χμ) - ψ(χβ)- Thus (1) holds for β, and (2) holds vacuously when β is a limit ordinal.

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


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