Abstract
This chapter focuses on a solution operator with infinite delays. For a retarded functional differential equation for which the solution at time t depends only on the past history over the interval [t - r, t], the solution operator takes the initial data on an interval [σ,-r, σ] into the restriction of the solution to [t - r, t] and is extremely simple. If the complete continuity is exploited to obtain results on the existence of periodic solutions for nonlinear equations, it must be supposed that the period ω is ≥ r. If the solution at time t of a retarded functional differential equation depends on the past history over (-∞, t], then the operator never becomes completely continuous so that spectral properties are not immediate and crude methods for obtaining periodic solutions will not work.
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