On a general variational framework for existence and uniqueness in Differential Equations

Starting from the classic contraction mapping principle, we establish a general, flexible, variational setting that turns out to be applicable to many situations of existence in Differential Equations. This unifying feature is quite appealing and motivated our analysis. We show its potentiality with some selected examples including initial-value, Cauchy problems for ODEs; non-linear, monotone PDEs; linear and non-linear hyperbolic problems; and steady Navier–Stokes systems. Though the paper has the structure of a survey, we would like to explore in the future how this perspective could help in advancing for some new situations in PDEs.