As has been shown in [15], the disconjugacy character implies the constant sign of the Green's function (g_{M}) related to problem (1 - 2).
In other words, the disconjugacy character is only a sufficient condition in order to ensure the constant sign of a Green's function related to problem (1 - 2).
The deduction of such a property holds from an iterative technique that consists of solving related linear problems on and using suitable maximum principles which are equivalent to the constant sign of the associated Green's function.
As a consequence we see that the interval of constant sign of the Green's function and the one of the disconjugacy for the linear operator are not the same in general.
It also means the difference in the propagation of singularities along (partial X): in the Dirichlet case, all singularities move in one direction (constant sign of (lambda '_{mathsf {D},k})), while in the Neumann case, some move in the opposite direction (variable sign of (lambda '_{{mathsf {N}},k})); this effect plays a role also in the case when (mu hlesssim 1).
It is very well known that the validity of the method of lower and upper solutions, coupled with the monotone iterative techniques [1, 2], is equivalent to the constant sign of the Green's function related to the linear part of the studied problem [3, 4].
The relaxing single-crossing or Spence-Mirrlees condition is the constant sign of the cross partial derivative with respect to decision and type: (CS+) ∀ (q,θ) in Q × Θ : V ̂ q θ > 0, (CS–) ∀ (q,θ) in Q × Θ : V ̂ q θ < 0. This implies that the existence of a curve q0 dividing the (θ,q) plane into two single-crossing regions, with V q θ > 0 below q0 and V q θ < 0 above.
Arguing as in Step 1, we will obtain the values of the real parameter M for which (hat{g}_{ -1)^{ -1M}(t,s)) is of constant sign on a neighborhood of (s=a) and (s=b) for every fixed (tin(a,b)).
Suppose that (G') satisfies the following: 1. (G'(r)) is of constant sign on ((R_{1},R_{2})) or 2.
