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In this book, fundamental methods of nonlinear analysis are introduced, discussed and illustrated in straightforward examples. Every method considered is motivated and explained in its general form, but presented in an abstract framework as comprehensively as possible. Applications and generalizations are shown. In particular, a large number of methods is applied to boundary value problems for partial differential equations. The text is structured in two levels: a self-contained basic level and an advanced level - organized in appendices - for the more experienced reader. It thus serves as both a textbook for graduate-level courses and a reference book for mathematicians, engineers and applied scientists.