**3. Discussion**

The results presented in this paper provide information about the sign and dependence on the extremes *a* and *b* of the Green function of the problem (4) and its derivatives when the two-point boundary conditions are admissible, property which encompasses many types of boundary conditions usually covered in the literature (for instance, conjugate or focal boundary conditions). By doing so, this paper extends (and to a small degree corrects, as discussed in the Remark 5) the results of Eloe and Ridenhour in [1], a fine piece of Green function theory that is considered a reference in the subject. The paper goes beyond to address the *p*-alternate and strongly admissible cases, for which results on the signs of higher derivatives on the interval are provided. Thus, whilst both [1] and the Section 2.1 yield sign results only for derivatives up to max(*<sup>α</sup>*1, *β*1)-th order, in the case of *p*-alternate they are supplied for derivatives up to *αA* + 1 (if *α*1 > 0) and *βB* + 1 (if *α*1 = 0) orders, and in the case of strongly admissible conditions, for derivatives up to (*n* − 1)-th order. As stated in the Introduction, this is relevant since the maximum value of the integer *μ* of the problem (6) which allows a cone-based approach is limited by the order of the highest derivative of *<sup>G</sup>*(*<sup>x</sup>*, *t*) with constant sign, so that finding results for higher derivatives of *<sup>G</sup>*(*<sup>x</sup>*, *t*) permits increasing the applicability of the cone theory to such problems.

One question that is left open is whether it is possible to find conditions on the sign of the coefficients of *L* which gran<sup>t</sup> a constant sign of every derivative of *<sup>G</sup>*(*<sup>x</sup>*, *t*) on (*a*, *b*) up to the (*n* − 1)-th order, for any strongly admissible boundary conditions. We hypothesize an affirmative response, but a proper proof is still pending.

To conclude, other areas that can benefit from an extension of these sign findings are those of boundary conditions mixing different derivatives or those with integral conditions. The determination of the sign of the Green function of fractional boundary value problems is also a topic that has raised interest recently, as part of more sophisticated mechanisms to find solutions of other related non-linear fractional boundary value problems (see for instance [23–26]). However, there is a lot to do in this area, since most of these cases require the explicit calculation of the associated Green function, and this calculation is only possible in the simplest ones. A more generic approach that provided signs without having to solve fractional differential equations, similar to that presented here, would, therefore, be very welcome.

**Author Contributions:** Conceptualization, P.A.B.; methodology, P.A.B. and L.J.; investigation, P.A.B.; validation, P.A.B. and L.J.; writing—original draft preparation, P.A.B.; writing—review and editing, L.J.; visualization, P.A.B. and L.J.; supervision, P.A.B.; project administration, L.J.; funding acquisition, L.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work has been supported by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) gran<sup>t</sup> MTM2017-89664-P.

**Conflicts of Interest:** The authors declare no conflict of interest.
