Dense Short Solution Segments from Monotonic Delayed Arguments

Abstract

AbstractWe construct a delay functional d on an open subset of the space $$C^1_r=C^1([-r,0],\mathbb {R})$$ C r 1 = C 1 ( [ - r , 0 ] , R ) and find $$h\in (0,r)$$ h ∈ ( 0 , r ) so that the equation $$\begin{aligned} x'(t)=-x(t-d(x_t)) \end{aligned}$$ x ′ ( t ) = - x ( t - d ( x t ) ) defines a continuous semiflow of continuously differentiable solution operators on the solution manifold $$\begin{aligned} X=\{\phi \in C^1_r:\phi '(0)=-\phi (-d(\phi ))\}, \end{aligned}$$ X = { ϕ ∈ C r 1 : ϕ ′ ( 0 ) = - ϕ ( - d ( ϕ ) ) } , and along each solution the delayed argument $$t-d(x_t)$$ t - d ( x t ) is strictly increasing, and there exists a solution whose short segments$$\begin{aligned} x_{t,short}=x(t+\cdot )\in C^2_h,\quad t\ge 0, \end{aligned}$$ x t , s h o r t = x ( t + · ) ∈ C h 2 , t ≥ 0 , are dense in an infinite-dimensional subset of the space $$C^2_h$$ C h 2 . The result supplements earlier work on complicated motion caused by state-dependent delay with oscillatory delayed arguments.