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

By Mohamed A. Khamsi

Offers updated Banach area results.
* positive factors an in depth bibliography for out of doors reading.
* presents particular routines that elucidate extra introductory fabric.

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2. 10 (The space B[0,1]) This space consists of all bounded realvalued functions defined on [0,1] with the distance d(f,g) for f,g € B[0,1] taken as above: d{f,g) = SMp{\f(t)-g(t)\:t S [0,1]}. Clearly the space C[0,1] is a subspace (in fact, a closed subspace) of B[0,1]. Indeed here we can think of 'subspace' in both the metric and the algebraic sense. There are other ways of assigning a distance to the collection of all continuous functions defined on [0,1]. 11 Let X be the space consisting of all continuous real-valued functions defined on the closed unit interval [0,1], with the distance d(f,g) between two such functions f, g taken as d{f,g)= f Jo \f(t)-g(t)\dt.

8 Let M be a complete metric space and suppose f : M —► M satisfies d (f (x), f (y)) <Φ(ά (x, y)) for each x,y 6 M, where φ : R + —> [0, oo) is upper semi-continuous from the right and satisfies 0 < φ(ί) < t for t > 0. Then f has a unique fixed point, x, and {fn (x)} converges to x for each x € M. Proof. Fix x G M and let xn —Tn(x),n= into two steps. 1 , 2 , · · · . 2. FURTHER EXTENSIONS Step 1. lim d(xn,xn+i) n—»oo OF BANACH'S 49 PRINCIPLE = 0. Proof. Since T is contractive the sequence {d(xn,xn+\)} is monotone decreasing and bounded below so lim d{xn,xn+\) = r > 0.

S — M). Any metric space M is isometric with a dense subset of a complete metric space called the completion of M. ) One way to see this is to consider the space Mc of equivalence classes of all Cauchy sequences in M, where two Cauchy sequences {xn} and {zn} are said to be equivalent (written {xn} ~ {zn}) if hrn d(xn, zn) = 0. Let n—*oo [{*„}] = { { * n } Ç M : { z n } ~ { * » } } and for x* = [{x„}], y* = [{yn}}, set d*(x",y*) = lim n—*oo d{xn,yn). It is easily verified that (i) the above limit does indeed exist; (ii) the space {Mc,d") is complete; and (iii) M is isometric to the subspace of (Mc,d*) consisting of all equivalence classes of the form [{x}], x € M.

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