Methods of nonlinear analysis : applications to differential equations / Pavel Drabek, Jaroslav Milota

This interesting book is a valuable contribution to the literature. It should be useful for a graduate course. The authors give an introduction to abstract methods in nonlinear analysis and applications to partial differential equations. They include preliminary material on linear analysis and try to give methods in enough generality to make the material interesting without trying to obtain the most general results. In general, they have succeeded very well in this.
In more detail, the authors first introduce some linear space theory and differential calculus on Banach spaces. They then discuss implicit function theorems, degree theory (in finite- and infinite-dimensional spaces) and global bifurcation theory.
They introduce monotone operators and then discuss monotonicity (in the order sense). Next they discuss variational methods including the mountain pass theorem, the saddle point theorem and Lyusternik-Shnirelʹman theory.
In the final chapter the authors briefly discuss classical and weak solutions of nonlinear elliptic equations and consider the application to these of many of the earlier abstract results.
It would have been nice to discuss very briefly how some of the partial differential equations considered arise in applications.
The book is generally well written. There is however a serious technical difficulty in the construction of the finite-dimensional degree. The methods of Section 4.3 are needed to complete the construction of the degree in Section 5, contrary to what is claimed. The argument on pp. 273–274 is incomplete because the set of singular values of F may locally disconnect RN and hence the concept introduced on p. 273 is not proved to be well defined. (MathSciNet)