**1. Introduction**

Let *J* be a compact interval in R and let us consider the real disfocal differential operator *L*: *Cn*(*J*) → *C*(*J*) defined by

$$Ly = y^{(n)}(\mathbf{x}) + a\_{n-1}(\mathbf{x})y^{(n-1)}(\mathbf{x}) + \dots + a\_0(\mathbf{x})y(\mathbf{x}), \quad \mathbf{x} \in I,\tag{1}$$

where *aj*(*x*) ∈ *<sup>C</sup>*(*J*), 0 ≤ *j* ≤ *n* − 1. Following Eloe and Ridenhour [1], let Ω*l* be the set whose members are collections of *l* different ordered integer indices *i* such that 0 ≤ *i* ≤ *n* − 1, let *k* ∈ N be such that 1 ≤ *k* ≤ *n* − 1, let *α* ∈ Ω*k* be the set {*<sup>α</sup>*1, ... , *<sup>α</sup>k*} and *β* ∈ Ω*<sup>n</sup>*−*<sup>k</sup>* be the set {*β*1, ... , *β<sup>n</sup>*−*<sup>k</sup>*}, both associated to the homogeneous boundary conditions

$$y^{(a\_i)}(a) = 0, \ i = 1, 2, \dots, k, \ a\_i \in a,\tag{2}$$

$$y^{(\beta\_i)}(b) = 0, \ i = 1, 2, \dots, n - k, \ \beta\_i \in \beta,\tag{3}$$

where [*a*, *b*] ⊂ *J*. Throughout this paper we will impose the condition that, for any integer m such that 1 ≤ *m* ≤ *n*, at least *m* terms of the sequence *α*1, ... , *αk*, *β*1, ... , *β<sup>n</sup>*−*<sup>k</sup>* are less than *m*. Due to their resemblance with the conditions defined by Butler and Erbe in [2], we will call them admissible boundary conditions (note that (2) and (3) are not exactly the same boundary conditions defined by Butler and Erbe since the latter applied to the so-called quasiderivatives of *y*(*x*) and not to derivatives). In particular, if for every integer *m* such that 1 ≤ *m* ≤ *p* + 1, exactly *m* terms of the sequence *α*1, ... , *αk*, *β*1, ... , *β<sup>n</sup>*−*<sup>k</sup>* are less than *m*, we will say that the boundary conditions are *p*-alternate. In the case *p* = *n* − 1 we will call the boundary conditions strongly admissible. The admissible conditions cover well known cases like conjugate boundary conditions (*<sup>α</sup>*1 = 0, *α*2 = 1, ... , *αk* = *k* − 1 and *β*1 = 0, *β*2 = 1, ... , *β<sup>n</sup>*−*<sup>k</sup>* = *n* − *k* − 1), focal boundary conditions (right focal with *α*1 = 0, *α*2 = 1, ... , *αk* = *k* − 1 and *β*1 = *k*, *β*2 = *k* + 1, ... , *β<sup>n</sup>*−*<sup>k</sup>* = *n* − 1 or left focal with *α*1 = *n* − *k*, *α*2 = *n* − *k* + 1, ... , *αk* = *n* − 1 and

*β*1 = 0, *β*2 = 1, ... , *β<sup>n</sup>*−*<sup>k</sup>* = *n* − *k* − 1) and many other. The focal boundary conditions are also strongly admissible (or (*n* − 1)-alternate).

The purpose of this paper will be to provide results on the sign of *<sup>G</sup>*(*<sup>x</sup>*, *t*), the Green function associated to the problem

$$\begin{aligned} Ly &= 0, \quad x \in (a, b), \\\\ y^{(a)}(a) &= 0, \; a\_i \in a; \quad y^{(\beta\_i)}(b) = 0, \; \beta\_i \in \beta\_i \end{aligned} \tag{4}$$

as well as some of its partial derivatives with regards to *x*, both in the interval (*a*, *b*) and at the extremes *a* and *b*. We will also analyze the dependence of the absolute value of *<sup>G</sup>*(*<sup>x</sup>*, *t*) and its derivatives with respect to the extremes *a* and *b*. In this sense, this paper represents an extension of the work by Eloe and Ridenhour [1] which in turn extended previous results from Peterson [3,4], Elias [5] and Peterson and Ridenhour [6]. Note that the disfocality of *L* on [*a*, *b*], according to Nehari [7], implies that *y*(*x*) ≡ 0 is the only solution of *Ly* = 0 satisfying *y*(*i*)(*xi*) = 0, *i* = 0, 1, 2, ... , *n* − 1, with *xi* ∈ [*a*, *b*], and also guarantees the existence of the Green function of (4).

It is well known (see for instance [8], Chapter 3) that problems of the type

$$\begin{aligned} \text{Ly} &= f, \quad \text{x} \in (a, b), \\\\ y^{(a\_i)}(a) &= 0, \; \mathfrak{a}\_i \in \mathfrak{a}; \quad y^{(\beta\_i)}(b) = 0, \; \beta\_i \in \beta\_i \end{aligned} \tag{5}$$

with *f* ∈ *<sup>C</sup>*[*<sup>a</sup>*, *b*] being an input function, have a solution given by *y*(*x*) = *ba <sup>G</sup>*(*<sup>x</sup>*, *<sup>t</sup>*)*f*(*t*)*dt*. Therefore, the knowledge of the sign of *<sup>G</sup>*(*<sup>x</sup>*, *t*) and its derivatives can provide information on the sign of the solution *y*(*x*) and these same derivatives, at least when *f* does not change sign on (*a*, *b*). This was already used by Eloe and Ridenhour in [1] to show that a clamped beam is stiffer that a simply supported beam. Likewise, the evolution of *<sup>G</sup>*(*<sup>x</sup>*, *t*) as *a* or *b* vary can also provide insights on the dependence of the value of *y*(*x*) on these extremes and can allow comparing the effect of a longer separation of the extremes when the same input function *f* is applied to a system modeled by (5).

The knowledge about the sign of *<sup>G</sup>*(*<sup>x</sup>*, *t*) is also useful to find information about the eigenvalues and eigenfunctions of the general problem

$$\begin{aligned} Ly &= \lambda \sum\_{l=0}^{\mu} c\_l(\mathbf{x}) y^{(l)}(\mathbf{x}), \ \mathbf{x} \in (a, b), \\ y^{(a\_i)}(a) &= 0, \ \mathbf{a}\_i \in \mathbf{a}; \ y^{(\beta\_i)}(b) = 0, \ \beta\_i \in \beta, \end{aligned} \tag{6}$$

with *μ* ≤ *n* − 1, *cl*(*x*) ∈ *C*(*J*) for 0 ≤ *l* ≤ *μ*. These problems are tackled by converting them in the equivalent integral problem

$$My(\mathbf{x}) = \frac{1}{\lambda}y(\mathbf{x}), \ x \in [a, b],\tag{7}$$

where *M* is the operator *M*: *<sup>C</sup><sup>μ</sup>*[*<sup>a</sup>*, *b*] → *Cn*[*<sup>a</sup>*, *b*] defined by

$$My(\mathbf{x}) = \int\_{a}^{b} \mathbf{G}(\mathbf{x}, t) \sum\_{l=0}^{\mu} c\_{l}(t) y^{(l)}(t) dt, \ \mathbf{x} \in [a, b]. \tag{8}$$

If the partial derivative of *<sup>G</sup>*(*<sup>x</sup>*, *t*) of the highest order whose sign is constant on (*a*, *b*) is not lower than *μ*, it is possible to define a cone *P* associated to that partial derivative such that *MP* ⊂ *P* and, with the help of the cone theory elaborated by Krein and Rutman [9] and Krasnosel'skii [10], prove that there exists a solution of (7) associated to the smallest eigenvalue *λ*. Moreover, it is possible to determine some properties of *λ* and even compare the values of *λ* for different boundary conditions. Refs. [11–17] are examples that follow this approach. In all these, therefore, the knowledge of the sign of the derivatives of *<sup>G</sup>*(*<sup>x</sup>*, *t*) is critical.

The non-linear version of (6), namely

$$Ly = \lambda f(y, \mathbf{x}), \ \mathbf{x} \in (a, b), \tag{9}$$

subject to different homogeneous, mixed or integral boundary conditions (see for instance [18,19]), is also addressed usually by converting it in the integral problem

$$\frac{1}{\lambda}y = \int\_{a}^{b} G(\mathbf{x}, t) f(y(t), t) dt, \ \mathbf{x} \in (a, b). \tag{10}$$

In most of these problems, the information about the sign of the Green function is relevant to apply other tools (fixed-point theorems, upper and lower solutions method, fixed-point index theory, etc.) to determine the existence of a solution. In some of them, the knowledge of the sign of the partial derivatives can help to achieve the same goal ([18,20,21]).

As for a physical applicability, problems of the type (5), (6) and (9) appear in many situations, like the study of the deflections of beams, both straight ones with non-homogeneous cross-sections in free vibration (which are subject to the fourth-order linear Euler-Bernoulli equation) and curved ones with different shapes. An account of these and other applications can be found in [22], Chapter IV.

Throughout the paper we will use the terms *<sup>G</sup>*(*<sup>α</sup>*, *β*, *x*, *t*) and *Ga*,*<sup>b</sup>*(*<sup>x</sup>*, *t*) (and further *Ga*,*<sup>b</sup>*(*<sup>α</sup>*, *β*, *x*, *t*)) when we want to highlight the dependence of the Green function of (4) on the boundary conditions (*<sup>α</sup>*, *β*) and the extremes *a*, *b*, respectively. That will be particularly useful when we manipulate Green functions subject to different boundary conditions or different extremes. We will denote by *<sup>H</sup>*(*<sup>x</sup>*, *t*) and *<sup>I</sup>*(*<sup>x</sup>*, *t*) the partial derivatives of *<sup>G</sup>*(*<sup>α</sup>*, *β*, *x*, *t*) with respect to the extreme *b* and *a*, respectively, that is

$$H(\mathbf{x},t) = \frac{\partial G(\mathbf{x},t)}{\partial b}, \quad I(\mathbf{x},t) = \frac{\partial G(\mathbf{x},t)}{\partial a}, \quad (\mathbf{x},t) \in [a,b] \times [a,b]. \tag{11}$$

We will say that *a*, *b* are interior to *A*, *B* if *A* ≤ *a* < *b* ≤ *B* and *A* < *a* or *b* < *B*. We will use the expression *card*{*D*} to denote the number of elements (or cardinal) of the set *D*.

Likewise, if we assume that *y* is a function with (*n* − 1)*th* derivative in [*a*, *b*], we will make use of the following nomenclature associated to (*<sup>α</sup>*, *β*):


To make these definitions clear, let us use some examples. Let us assume that *n* = 8, *k* = 4, *α* = {0, 1, 2, 5} and *β* = {3, 4, 5, <sup>7</sup>}. Then *αA* = 2 (since 3 ∈/ *α*), *βB* = 5 (since 6 ∈/ *β* but also 5 ∈ *α*), *<sup>K</sup>*(*<sup>α</sup>*, *β*) = 6 (since 6 ∈/ *α* ∪ *β* and 7 ∈ *β*), *<sup>S</sup>*(*α*) = 0 + 1 + 2 + 5 = 8 and *S*(*β*) = 3 + 4 + 5 + 7 = 19. Likewise, let us assume that *n* = 7, *k* = 2, *α* = {3, 5} and *β* = {0, 1, 2, 4, <sup>5</sup>}. Then *αA* = 3, *βB* = 2, *<sup>K</sup>*(*<sup>α</sup>*, *β*) = 6, *<sup>S</sup>*(*α*) = 8 and *S*(*β*) = 12.

As for the organization of the paper, Section 2 will provide the main results of the paper. Concretely, in the Section 2.1 we will tackle the general case of admissible boundary conditions, in the Section 2.2 we will prove some additional results associated to *p*-alternate boundary conditions and in the Section 2.3 we will cover the strongly admissible boundary conditions. Finally in Section 3 we will elaborate some conclusions.
