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

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: creation (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 house Ultrapowers (pages 243–271):

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Example text

Now choose n so that 1/n < 6. For each fc = 0,1, · · · ,n it is possible to choose a rational number Wk so that \f(k/n) — Wk\ < e/5. Now let g 6 P„ be the function for which g(k/n) = WkThen if x € [k/n, (fc + 1) / n ] , \f(x)-9(x)\ < \f(x)-f(k/n)\ + \f(k/n)-g(k/n)\ + < 2e/5 + \g(k/n)-g({k + l)/n)\ < 2e/5 + |/(fc/n)-/((fc + l ) / n ) | + 2 e / 5 < \g(k/n)-g(x)\ ε. Having established the above, it follows that given any / € C[0,1] there exists g € P such that d(f,g) < ε. This implies P = C[0,1]. What we have just observed is that there is a countable family of piecewise linear continuous functions which is dense in C[0,1].

THE CARISTI-EKELAND PRINCIPLE 57 In particular, if c*o < μ < a < β, then ά(χμ,Χα) < φ{χμ) - φ(χα) < ε. This proves that {χ α }<*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.

Proof. Assume (E) is false. Then Va: € M 3 g(x) 6 M such that x < g(x). It follows that d{x,g{x)) < ψ{χ) - (E) uses the Axiom of Choice, whereas the proof that (E) => (C) does not. In fact, these two theorems are equivalent only if one (*) 56 CHAPTER 3. METRIC CONTRACTION PRINCIPLES assumes (as we do) the Axiom of Choice.

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