By Robert F. Brown

Here is a publication that would be a pleasure to the mathematician or graduate pupil of arithmetic – or perhaps the well-prepared undergraduate – who would favor, with not less than history and coaching, to appreciate the various appealing effects on the center of nonlinear research. in accordance with carefully-expounded principles from numerous branches of topology, and illustrated through a wealth of figures that attest to the geometric nature of the exposition, the ebook could be of great assist in supplying its readers with an realizing of the math of the nonlinear phenomena that represent our genuine world.

This e-book is perfect for self-study for mathematicians and scholars attracted to such components of geometric and algebraic topology, sensible research, differential equations, and utilized arithmetic. it's a sharply centred and hugely readable view of nonlinear research through a working towards topologist who has visible a transparent route to understanding.

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**Extra info for A Topological Introduction to Nonlinear Analysis**

**Example text**

Notice that the condition imposes no restriction on how large r can be. Many functions that arise in interesting analytic problems turn out to satisfy "growth conditions" which, as IIx II becomes large enough, prevent IIf(x) II from growing at the same rate, so that eventually IIx II ~ II f (x) II, which produces the Leray-Schauder condition in this special form . The result that replaces the requirement in Schauder 's theorem, of a convex set carried to itself, by the Leray-Schauder boundary condition is really just a corollary of the Schauder theorem.

Then K is convex since it the cartesian product of all the closed intervals {-~, ~] . An open subset of K in the metric topology on 12 is open in the product topol6g/on K. 5 to conclude that the "Hilbert cube" K has the fpp. In the sort of analytic problems we will be interested in, there is a map f that takes a normed linear space X into itself, and the solution to the analytic problem is a fixed point of f . However, the Schauder theorem turns out to be awkward to use in 4 . Schauder Fixed Point Theory 27 Figure S.

5. The Forced Pendulum 33 Now let's suppose that L was not just a linear function but actually an isomorphism with a (continuous) inverse L -I . Then we could rewrite the differential equation Ly = y " = f(t , y, l) as y = L -I f(t, y, y'). This really would be a fixed point problem because we could define a function S by setting S( y) = L -I f(t, y, y') and then a solution to the differential equation would be described by y = S(y), that is, a fixed point of S. I can't really get away with all that for two reasons.