Existence for 2-Point Boundary Value Problems in Two Ion Electrodiffusion

On a Neumann boundary value problem for the Painlevé II equation in two-ion electro-diffusion

“…From Lemma 6, z ′′ (1) ≥ 0, z(1) ≤ 1. Adopting the same arguments in the proof of Lemma 4, it is seen that at x = 1, 10) from the definition of α * (3.4), and so L > l.…”

“…In [10], a two-point boundary value problem was investigated which arises in the study of two ions of the same valency migrating across a liquid junction such as a membrane. The relevant model equations are derived in [1,2] and , in a more general multi-ion context, in [7].…”

“…Elimination of the ionic concentrations in the two-ion problem leads to a nonlinear equation for the unknown y, proportional to the electric field E, namely [10] y ′′ (x) = f (x, y(x); y(0), y(1)), x ∈ (0, 1) ( and the liquid junction occupies the region 0 ≤ x ≤ 1. The quantities l > 0, λ > 0 and D with −1 < D < 1 are parameters as described in [10].…”

“…The quantities l > 0, λ > 0 and D with −1 < D < 1 are parameters as described in [10]. As noted in [10], if y is a solution of (1.1) for the parameters l, λ, D then −y is a solution for l, λ, −D.…”

“…Thompson in [10] used degree theory and the well-established method of upper and lower solutions (see e.g. [4]) to prove the existence of a positive solution to the boundary value problem (1.1) − (1.4) when D > 0 and subject to the constraint…”