A Topological Introduction to Nonlinear Analysis

From the book reviews: "The basic goal of this book is to explain, prove and apply a famous result in bifurcation theory called the Krasnoselski-Rabinowitz theorem. ... a large portion of this book should be reasonably understandable even to upper-level undergraduates with a good real analysis course under their belts; certainly a beginning graduate student should find this book quite comprehensible, very informative, and enjoyable as well. The author deserves both congratulations and thanks for making such nontrivial mathematics so readily accessible." (Mark Hunacek, MAA Reviews, February, 2015)

Robert F. Brown is a Professor of Mathematics at UCLA. His research area includes algebraic topology that is included within topological fixed point theory. Professor Brown's most recent research concerns the fixed point theory of fiber maps of fiberings with singularities.

Preface.- Part I Fixed Point Existence Theory.- The Topological Point of View.- Ascoli-Arzela Theory.- Brouwer Fixed Point Theory.- Schauder Fixed Point Theory.- The Forced Pendulum.- Equilibrium Heat Distribution.- Generalized Bernstain Theory.- Part II Degree Theory.- Brouwer Degree.- Properties of the Brouwer Degree.- Leray-Schauder Degree.- Properties of the Leray-Schauder Degree.- The Mawhin Operator.- The Pendulum Swings back.- Part III Fixed Point Index Theory.- A Retraction Theorum.- The Fixed Point Index.- The Tubulur Reactor.- Fixed Points in a Cone.- Eigenvalues and Eigenvectors.- Part IV Bifurcation Theory.- A Separation Theorem.- Compact Linear Operators.- The Degree Calculation.- The Krasnoselskii-Rabinowitz Theorem.- Nonlinear Strum Liouville Theory.- More Strum Liouville Theory.- Euler Buckling.- Part V Appendices.