Asymptotic behavior of solutions to a class of differential equations in infinite-dimensional spaces
The author considers the asymptotic behavior of solutions to a coupled system of two nonlinear evolution equations of the form \[ \dot{x} = A(t)x + f(t,x,y), \quad x(t_0) = x_0, \] \[ \dot{y} = B(t)x + g(t,x,y), \quad y(t_0) = y_0, \] for \((x,y) \in X \times X\), \(X\) a Banach space. The existence of \(\delta\)-bounded solutions is studied, i.e., \((x(t),y(t))\) satisfies \[ \||x(t)\||\leq \delta_1(t), \quad \||y(t)\||\leq \delta_2(t), \] with nonnegative scalar functions \(\delta_i\). Criteria for the existence of such solutions are given and applied to functional-differential equations and infinite-dimensional systems of ordinary differential equations.
scientific article; zbMATH DE number 1389528