By James Dugundji, Andrzej Granas

http://www.amazon.com/Fixed-Theory-Springer-Monographs-Mathematics/dp/0387001735

The concept of fastened issues is among the strongest instruments of recent arithmetic. This ebook includes a transparent, designated and well-organized presentation of the most important effects, including an interesting set of historic notes and an in depth bibliography describing extra advancements and applications.

From the reviews:

"Granas-Dugundji’s publication is an encyclopedic survey of the classical mounted aspect conception of constant maps (the paintings of Poincaré, Brouwer, Lefschetz-Hopf, Leray-Schauder) and all its a variety of smooth extensions. this can be definitely the main realized booklet ever prone to be released in this subject."

-Felix Browder, Rutgers University

"The idea of fastened issues is among the strongest instruments of contemporary arithmetic. not just is it used every day in natural and utilized arithmetic, however it additionally serves as a bridge among research and Topology, and offers a really fruitful zone of interplay among the 2. This e-book incorporates a transparent, specific and well-organized presentation of the key effects, including an enjoyable set of old notes and an intensive bibliography describing additional advancements and applications"

-Haim Brézis, Rutgers University

"This is the main accomplished, well-written and whole ebook on fastened element concept up to now. The e-book stories with reference to each point of fastened aspect idea … . The e-book consists of an in depth literature at the topic and plenty of examples. a number of the fascinating effects, given as routines, represent an extension of the speculation verified more often than not textual content. … i like to recommend this glorious quantity on mounted aspect thought to a person attracted to this center topic of nonlinear analysis." (A.G. Kartsatos, Mathematical stories, 2004d)

"Thousands of papers were released in fastened element thought and its functions … . The e-book of Granas and Dugundji is a brand new and significant contribution to this function. … the most textual content is self-contained, the required heritage fabric being amassed in an appendix … . i will assert this monograph represents a great occasion and i'm certain it is going to turn into a reference paintings within the box. … this publication could be super important to somebody fascinated with mounted aspect idea … ." (R. Precup, ZAA-Zeitschrift für research und ihre Anwendungen, Vol. 23 (2), 2004)

"The new version of Granas and Dugundji’s publication is, in my view, crucial and whole survey within the final years on mounted aspect conception … . the most textual content is self-contained … . New and engaging effects and purposes are available everywhere in the publication. the fashion is alert and delightful. The technical presentation of the e-book is phenomenal. … it is going to be a really priceless paintings for somebody … focused on mounted aspect thought specifically and nonlinear research in general." (Adrian Petrusel, Studia Universitatis Babes-Bolyai Mathematica, Vol. XLIX (1), 2004)

"This monograph provides a gently worked-out account of the main easy rules and functions of the idea of mounted issues. earlier, a therapy of a number of the mentioned themes has been unavailable in ebook shape. The presentation is self-contained and is available to a large spectrum of readers." (L’Enseignement Mathematique, Vol. forty nine (3-4), 2003)

"Really, this publication is an updated and thoroughly labored out unified account of (not merely) classical ends up in mounted aspect idea … . The presentation is self-contained and is out there to a wide spectrum of readers. the most textual content is complemented through a number of routines, special reviews, and a accomplished bibliography. … the wealthy bibliography contains nearly 50 pages. The publication turns out to be useful to all mathematicians who care for fastened issues. it's going to turn into an compulsory quantity for each severe mathematical library." (Peter Zabreiko, Zentralblatt MATH, Vol. 1025, 2003)

"This monograph presents a massive textual content … on classical themes in fastened element conception that lie at the borderline of topology and nonlinear practical research. … The ebook is easily comprehensible and calls for just a easy wisdom of topology and useful research … . it may be warmly steered to a large spectrum of readers – to graduate scholars, specialists and everyone who needs to turn into familiar with the fundamental components and deeper houses of this a part of useful analysis." (EMS publication, March, 2005)

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**Additional info for Fixed Point Theory (Springer Monographs in Mathematics)**

**Example text**

For any other sequence {Yn} satisfying 'P(Yn) ---+ 0 we get 'P(x n ) +'P(Yn) ---+ 0, so, from (**) as before, d(xn,Yn) ---+ 0 and therefore Yn ---+ U also. 5) Let (X, d) be complete and F : X ---+ X continuous. Assume that the function 'P(x) = d(x,Fx) has the pmperty (**) and infxEx d(x, Fx) = O. Then F has a unique fixed point. THEOREM. 5) ; for if d(Fx , Fy) :S ad( x, y) with constant a < 1, then the condition (**) is satisfied with 'P (x) = d(x, Fx) because (1 - a)d(x, y) :S d(x, y) - d(Fx , Fy) :S d(x, Fx) + d(y, Fy) and infxEx d(x , Fx) = 0, since we have seen that d(Fnx, Fn+lx) ---+ 0 for each x E X.

We now present some fixed point theorems based on the considerations of order; these results have proved of importance in algebra, the theory of automata, mathematical linguistics, linear functional analysis, approximation theory and the theory of critical points. 1. The Knaster-Tarski Theorem Let (P, ::S) be a partially ordered set and M cPa nonempty subset. Recall that an upper (respectively lower) bound for M is an element pEP with m ::S p (respectively m t p) for each m E M; the supremum of M, if it exists, is an upper bound for M which is a lower bound for the set of all upper bounds of M.

Prove: T has a fixed point. [Define by induction a sequence {Yn} in K(xo , r) such that Yn +l E T(Yn) and d(Yn+I' Yn) ::; a n (1 - a)r. Then, using the estimate d(Yn+p , Yn) ::; [1 + a + a 2 + ... + a P- I Ja n (1- a)r, show that {Yn} is a Cauchy sequence and hence converges to a c ertain Yo E K(xo,r) . 2) Let (X, d) be complete, U C X open , and {Ht : U -> CB(X)} an a-contractive family of maps which are fixed point free on the boundary au. p(t')). Establish the existence of a maximal element (to, xo) in Q .