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

By Mohamed A. Khamsi

Offers up to date Banach house results.
* positive factors an in depth bibliography for out of doors reading.
* offers distinct routines that elucidate extra introductory fabric.

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

Example text

Fix x € M and let xn = T n (x), n = 1,2, · · ·. We break the argument into two steps, each of which illustrates something more. 2. FURTHER EXTENSIONS Step 1. lim d(xn,xn+i) OF BANACH'S PRINCIPLE 47 =0. Proof. Since T is contractive the sequence {d(xn,xn+i)} is monotone decreasing and bounded below so lim d(xn,xn+i) = r > 0. Assume r / 0 . Then n—*oo by the contractive condition -^ -^- < a(d(xn,xn+i)), n = l,2,···. Letting n —♦ 00 we see that 1 < lim a(d(xn,xn+{)), and since a 6 S this in n—>oo turn implies r = 0.

We break the argument into two steps, each of which illustrates something more. 2. FURTHER EXTENSIONS Step 1. lim d(xn,xn+i) OF BANACH'S PRINCIPLE 47 =0. Proof. Since T is contractive the sequence {d(xn,xn+i)} is monotone decreasing and bounded below so lim d(xn,xn+i) = r > 0. Assume r / 0 . Then n—*oo by the contractive condition -^ -^- < a(d(xn,xn+i)), n = l,2,···. Letting n —♦ 00 we see that 1 < lim a(d(xn,xn+{)), and since a 6 S this in n—>oo turn implies r = 0. This contradiction establishes Step 1.

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. Then in each of the sets C„ there n=l exists a point zn such that d{x, zn) = d{y, Zn) = -d{x, y). 5. ) By compactness of M the sequence {zn} has a subsequence {zn„} which converges to a point z 6 M and since each of the sets Cn is closed, oo z G f) Cn- Since the metric d is continuous, 71 = 1 d(x,z) =d(y,z) = -d(x,y).

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