Fixed Point Theorems

In this chapter, we address the question of solving an equation \( \Phi(\textit{x}) \) from the point of view of Analysis. The unknown G may belong to an infinite-dimensional vector space.We will see several situations in whichwe can guarantee the existence of a solution and also obtain the solution as the limit of a sequence defined iteratively. We will prove several fundamental results: the Banach fixed point theorem, the local inversion theorem, the Cauchy–Lipschitz theorem, Brouwer’s theorem and the domain invariance theorem. We will also address the study of Ekeland’s variational principle and Nash’s implicit function theorem, which play a key role in many research fields.