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

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By Mohamed A. Khamsi

ISBN-10: 0471418250

ISBN-13: 9780471418252

Provides updated Banach area results.
* positive aspects an in depth bibliography for outdoor reading.
* offers distinct workouts that elucidate extra introductory fabric.

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Example text

TTien there exists x € M suc/i ί/ιαί g(x) = x. (**) 58 CHAPTER 3. METRIC CONTRACTION PRINCIPLES Proof. Introduce the partial order > in M as follows. For x,y € M say that y > χ <=> max{d(x, y),cd(f(x), f(y))} < φ(/(χ)) - ), and for α,β € / set β > a <=> Χβ > xa. ι is a nonincreasing net in R + so there exists r > 0 such that = r. \imip(f(xa)) a Let ε > 0. Then there exists ao 6 / such that a > ao implies r < tp(f(xa)) a > ao, m a x i d ^ ^ ^ ) ^ ^ / ^ ) , / ^ ) ) } <

METRIC CONTRACTION PRINCIPLES Proof. Introduce the partial order > in M as follows. For x,y € M say that y > χ <=> max{d(x, y),cd(f(x), f(y))} < φ(/(χ)) - ), and for α,β € / set β > a <=> Χβ > xa. ι is a nonincreasing net in R + so there exists r > 0 such that = r. \imip(f(xa)) a Let ε > 0. Then there exists ao 6 / such that a > ao implies r < tp(f(xa)) a > ao, m a x i d ^ ^ ^ ) ^ ^ / ^ ) , / ^ ) ) } <

Uniqueness of z follows from the contractive condition on T. , rj J. r > 0 => ^ ( r j ) ~* VK1"))· This extension of Banach's Principle is due to Browder [25]. 48 CHAPTER 3. 2) d(T(x),T(y))(d(x,y)). Then T has a unique fixed point z, and {Tn(x)} converges to z, for each x 6 M. Proof. This theorem is actually a special case of the previous theorem. First introduce the function φ : R —» [0,1) by setting ^>(0) = φ(0) and φ(ί) = Ά for t > 0. To see that φ is in the class S suppose φ{ίη) —» 1. Then {i„} must be bounded (otherwise, lim inf φ(ίη) = 0 ) .

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


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