Existence and extendability of solutions of the equation \(g(t,x)=0\) (Q2754802)

The authors prove a non-local version of the implicit function theorem. Under some mild assumptions a solution exists between such values of \(t\) in which \(g_x'(t,x)=0\) for some \(x\) (including \(x=\pm \infty\); this extension of a usual definition of a critical point is crucial for explaining well-known examples of non-extendability of local solutions). The problem of existence of piecewise-smooth and periodic solutions is also considered.