ABSTRACT
Chapter 10 is different from the previous chapters in the sense that it deals with nonlinear functional analysis, perhaps the most important part of which is fixed point theory with its applications in the solution of nonlinear differential and integral equations that appear in physics, chemistry, biology and economics. It incĺudes many of the most important results in fixed point theory, starting with Banach’s classical contraction principle for which several different proofs are presented, as well as various corollaries and examples. Furthermore, it presents results by Edelstein and Rakotch, the concept of nonlinear contraction by Boyd and Wong, and theorems by Meir–Keeler, Kannan, Chatterje and Zamfirescu. It also deals with results related to the concepts of Ćirić’s generalized contractions and quasi–contractions, and establishes the theorems by Reich, Hardy–Rogers, Caristi, and Bollenbacher and Hicks. The chapter closes with studies of the Mann iteration and fixed point theorems by Mann, Reinermann, Franks and Marzec for real functions on compact intervals of the real line.
