Existence problems for homoclinic solutions
Let \(f: \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n\) be a continuous function. The author considers the problem \[ \dot x= f(t,x),\quad x(-\infty)= x(+\infty),\tag{P} \] with \(x(\pm\infty):= \lim_{t\to\pm\infty} x(t)\in \mathbb{R}^n\). The author reduces (P) to a fixed-point problem for a convenient operator defined in the functional space \[ C_l:= \{x: \mathbb{R}\to \mathbb{R}^n\mid \exists x(\pm\infty)\in \mathbb{R}^n\}, \] and then applies the continuation method to establish the existence results for problem (P).
scientific article; zbMATH DE number 1780519