Small solutions of operator equations

Integrodifference equations in patchy landscapes

“…follows from a theorem of Krein and Rutman (Krein and Rutman, 1950), which is stated as Theorem 2.5 and proved in Krasnosel'skii (1964a).…”

“…The desired results are now obtained from Theorems 2.10, 2.11, and 2.13 of Krasnosel'skii (1964a) which are summarized as the following theorem.…”

“…For existence, it is required that Q l is continuous, Q l (0) exists with dominant eigenvalue larger than unity and its asymptotic derivative Q l (∞) be the zero operator where the asymptotic derivative of Q l is defined as follows (see page 241 in Krasnosel'skii (1964a)). Since the asymptotic derivative is given by the zero operator, λ ∞ = 0, the above theorem applies.…”

“…Compactness of Q l follows from Theorem 7.1 on page 233 of Krasnosel'skii (1964a) which is stated below for convenience of the reader. To apply this theorem, we note the Beverton-Holt function (1.2.5) is bounded above by r 0 r 0 −1 .…”

“…To verify that Q l is Fréchet differentiable at zero we apply Theorem 7.3 on page 238 from Krasnosel'skii (1964a). First, we note that g(x, y, 0) = k(x, y)f (0, y) = 0 and ∂g(x, y, 0) ∂N = k(x, y)r 0 (y) := R(x, y).…”