Boundary value problems for second-order differential equations on unbounded domains in a Banach space.
The author studies the following two-point boundary value problem for a second-order nonlinear differential equation in a Banach space \(X\) \[ \frac{d^2 x}{dt^2}=f \biggl(t, x(t), \frac{d x(t)}{dt}\biggr),\quad t\geq 0, \qquad x(t)=x_0, \quad \frac{d x(\infty)}{dt}=y_\infty, \] where \(x_0, y_\infty\in X\) are given vectors, and \(f: [0,\infty)\times X\times X\to X\) is a given continuous function. By virtue of the Sadovskii fixed-point theorem, the existence of solutions is investigated. Besides, the Lipschitz condition for \(f\) is not required.
scientific article; zbMATH DE number 1861367