The name is absent

∞

^ δ^t [C_y (0,f^t (y)) + D_x(f^t(y)) J f_x(y), which is the condition in part a. Hence, the condition in part c
t=1

implies part a and the invasion is a controlled invasion from y.

Let T be the first t such that a_t > 0 and suppose that T > 0. Then a_t = 0 for t = 0,...,T-1, while 0 <

a_τ ≤y_τ = f(xT_-1) = fT(y) and x_τ = y_τ - a_τ = fT(y) - aT. Since a_τ > 0 there exists an ɛ > 0 such that
f(y_τ-1) - f(yT_-1 - ^ε) < aT. Consider the alternative feasible sequence 0T.₁ = ε, aT = aT - [f(yT_-1) - f(yT_-1-ε)], and
a_t = a_t for all other t. Then x_τ-1 = y_τ-1 - ɛ and x_t = x_t for t ≠Τ-1. Since the sequence {x_t,a_t} is optimal,

∞

0 ≥ ∑ δ^t [C(a,,y,) + D(x,) - C(a,,yt) - D(X,)]
t=0

= δ^τ-1[ C(0,yT-ι) + D(yT-ι) - C(ε,yT-ι) - D(yT-ι-ε) ]

+ δT [C(aT,f(yT-ι)) + D(xT) - C(aT+f(yT-1-ε)-f(yT-1),f(yT-1-ε)) - D(xT) ]

Dividing by ε and letting ε → 0 implies

0 ≥ -C_a(0,fT^-1(y)) + Dx(fT^-1(y)) + δ[C_a(aT,fT(y)) + Cy(aT,fT(y))]fx(f^r-1(y))

Since this contradicts the condition in the proposition it must be that T = 0 and A(y) > 0 for all y. ■

Proof of Lemma 4. For purposes of exposition the proof is divided into a sequence of subsidiary lemmas.

Define the lower, right and left Dini derivatives of V at y by

V(y +ε) -V(y)                V(y) -V(y -ε)

D₊ V ( y ) = lιmιnf_e→₀ ——-------— and D_- V ( y ) = ——---—----.

εε

Lemma 4.1. -C_a(y₀-x₀,y₀) + Dx(x₀) + δ D~V(f(x₀)) ≤ 0.

Proof. Since x₀- is feasible from y₀, the principle of optimality implies C(y₀-x₀+ ,y₀) + D(x₀- ) +
δV(f(x₀-ε)) - C(y₀-x₀,y₀) - D(x₀) - δV(f(x₀)) ≥ 0. Dividing by ε and taking liminf_εt0 establishes the result.
■

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