The contraction mapping theorem says that a contraction map on a complete metric space has a unique fixed point. That is, if \( X \) is a complete metric space with metric \( d \), and for every pair of points \( x \) and \( y \) in \( X \) we have

\( d(f(x), f(u)) \leq \lambda d(x, y) \)

for some constant \( \lambda < 1 \), then there exists a unique point \( p \) in \( X \) such that \( f(p) = p \). The point \( p \) is called a fixed point because it does not move when \( f \) is applied.

This theorem is often used to prove that equations have solutions. It also suggests a numerical algorithm for computing solutions: take a guess at a fixed point, then repeatedly apply the function. Not only does the sequence of approximations converge, but we can also say more. It converges geometrically with the distance between successive approximations decreasing by a factor of \( \lambda \) or less each time.

While the contraction mapping theorem is very useful, its hypotheses are too strong for many applications. There are many variations on the theorem, weakening some of the hypotheses and possibly strengthening others. For example, one might replace the requirement that \( f \) be a contraction with some more general condition while placing more requirements on \( X \) such as convexity or compactness.

Vittorino Pata presents a set of variations on the contraction mapping theorem in his book *Fixed Point Theorems and Applications*, starting with the most basic version of the theorem and proceeding to more sophisticated variations. The conclusion of every theorem is the existence of a fixed point, but one may have to give up uniqueness in exchange for more general hypotheses. For example, Brouwer’s fixed point theorem requires very little of the map \( f \), only that it be continuous, but in exchange requires a great deal of the space \( X \), namely that it be finite-dimensional, convex, and compact. Brouwer’s theorem assures us that there is a fixed point, but it may not be unique, and the theorem does not provide a practical algorithm for finding a fixed point.

Pata’s book promises in the title not only a collection of fixed point theorems but also applications. And indeed he does devote about as many pages to applications as to theorems. This is in contrast to many books that mention theory and applications in the title but are overwhelmingly occupied with theory. The book has three parts: theorems, applications, and problems. The first two sections account for about 45% of the book each, and the remaining 10% is a list of 50 problems. Not all applications are in the applications section of the book. Some quick applications, such as a proof of the fundamental theorem of algebra, are sprinkled in with the theorems.

The largest category of applications is differential equations. In these applications, the space \( X \) is a set of potential solutions, a subset of a Banach space, and the mapping \( f \) represents the differential equation. It is usually too much to ask that \( f \) be a contraction and so one needs the weaker hypotheses discussed above. In addition, Pata gives applications to analysis, such as a proof of the implicit function theorem and a theorem on the existence of Haar measures. He concludes with an application to game theory.

*Fixed Point Theorems and Applications* follows the theme of fixed points across multiple areas of math. It could be used as a textbook for a somewhat unusual functional analysis course, or for a real analysis course that introduces functional analysis. It would also be good preparation for a course in nonlinear partial differential equations or control theory.