Bounded solutions of linear and weakly nonlinear differential equations in a Banach space with an unbounded operator in the linear part
The paper presents necessary and sufficient conditions for the existence, on the whole real line, of a weak bounded solution of the following weakly nonlinear differential equation in a Banach space \(\mathbb{B}\) \[ \frac{dx}{dt}=A(t)x(t)+f(t), \;t\in J\subset \mathbb{R}, \] where \(f\) is a Banach-valued continuous and bounded operator and the operator \(A(t)\) is a closed operator with dense domain in \(\mathbb{B}\), where the domain is independent of \(t\). The approach is based on the assumption that the homogeneous equation \[ \frac{dx}{dt}=A(t)x(t), \;t\in J\subset \mathbb{R} \] admits an exponential dichotomy on the half-lines.
scientific article; zbMATH DE number 6105658
scientific article