Green’s Functions in the Theory of Ordinary Differential Equations by Alberto Cabada

Author:Alberto Cabada
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

Remark 1.8.17.

For the case μ = −1, we have that if the equation Tx −λ x = y ≻ 0 has a positive solution x ≻ 0 then λ < r(T).

Now, we will study the eigenvalue equation

(1.8.8)

If operator T n [M] is invertible in X U , the previous differential equation is equivalent to the integral one

(1.8.9)

Denoting g M as the Green’s function related to operator T n [M], we have that

Having in mind the case in which Green’s function g M  ≤ 0 on J × J and vanishes at t = a or t = b, we assume the following condition:

(N g )Suppose that there is a continuous function ϕ(t) > 0 for all t ∈ (a, b) and , such that for a.e. s ∈ J, satisfying

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