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Let J be a connected component of the complement of I. Then Z : J → {0, 1, 2, . . } is a continuous function and J is connected so its image Z(J) must be a connected subset of the nonnegative integers. So it must be a single integer. 20) have ℜ(z) < 0 for (α , β ) belonging to the open region bounded below by curve C0 and bounded above by curve {(α , β ) : β = −α , α ≤ 1} which meet at (α , β ) = (1, −1). 1. At least one root satisfies ℜ(z) > 0 for (α , β ) belonging to the open complementary region on the right.

In this case, K(s) = (sI − A − e−sr B)−1 is a matrix-valued function, I being the identity matrix, and therefore K is too. 9) are matrix-valued as well with 1 replaced by the identity matrix I. Matrix-valued solution k is the fundamental matrix solution in this case. 1) of the form x(t) = eλ t v, v = 0 where λ is complex and v is a vector whose components are complex. It is useful to have the notation expλ for the continuous function defined on [−r, 0] by expλ (θ ) = eλ θ . Using it, we see that the state xt corresponding to x(t) is xt = eλ t (expλ )v because xt (θ ) = x(t + θ ) = eλ (t+θ ) v = eλ t expλ (θ )v For x(t) to be a solution, we must have x (t) = λ eλ t v = L(xt ) = eλ t L(expλ v) or λ v = L(expλ v) Writing v = ∑ j v j e j where {e j } j is the standard basis for Cn , then L(expλ v) = ∑ j v j L(expλ e j ).

Show that x(θ ) and β (θ ) are strictly decreasing in θ ∈ (0, π /2). Chapter 3 Existence of Solutions Abstract Existence and uniqueness of solutions of discrete–delay differential equations is established by the method of steps, appealing to classical ODE results. More general delay equations require a more general framework for existence and uniqueness. This includes some peculiar notation endemic to the subject and the identification of the appropriate state space for delay equations. Solutions either extend to the entire half–line or blowup in finite time.