The evolution operator approach to functional-differential equations with delay

Abstract

The nonlinear nonautonomous functional differential equation x ˙ ( t ) ∈ B ( t ) x ( t ) + F ( t , x t ) , t ⩾ s , x s = φ \dot x(t) \in B(t)\,x(t) + F(t,{x_t}),\;t \geqslant s,\,{x_s} = \varphi , is considered. The representation of the solution to this equation via the associated evolution operator is extended from the single-valued case to the general case of multivalued operators B ( t ) B(t) .