Green's function related to a n order linear differential equation coupled to arbitrary linear non local boundary conditions

In this paper we obtain the explicit expression of the Green's function related to a general $n$ order differential equation coupled to non-local linear boundary conditions. In such boundary conditions, a $n$ dimensional parameter dependence is also assumed. Moreover, some comparison principles are obtained. The explicit expression depends on the value of the Green's function related to the two-point homogeneous problem, that is, we are assuming that when all the parameters involved on the boundary conditions take the value zero then the problem has a unique solution which is characterized by the corresponding Green's function $g$. The expression of the Green's function $G$ of the general problem is given as a function of $g$ and the real parameters considered at the boundary conditions. It is important to show that, in order to ensure the uniqueness of solutions of the linear considered problem, we must assume a non resonant additional condition on the considered problem, which depends on the non local conditions and the corresponding parameters. We point out that the assumption of the uniqueness of solutions of the two-point homogeneous problem is not a necessary condition to ensure the solution of the general case. Of course, in this situation the expression we are looking for must be obtained in a different manner. To show the applicability of the obtained results, a particular example is given.


Introduction
Most of the real phenomena that appear in fields as, among others, Physics, Engineering, Biology or Medicine, are modeled by Ordinary Differential Equations coupled to suitable boundary conditions located at some given set of the interval of definition. The majority of them take values at the extremes of the interval, they are known as two-point boundary value problems. There are a long tradition in studying these kind of problems and a lot of works in this direction have been developed to ensure the existence, uniqueness or multiplicity of solutions, coupled to their stability or instability (see, for instance, [1,6]).
To allow, on the boundary conditions, suitable dependence at some fixed points (or sets) of the interval, that are not the extreme ones, permits the study of a wider set of problems that model suitable real phenomena. Therefore, the so-called non-local conditions allow us to deal with more complicated problems that model more difficult real phenomena. In the non resonance case, such kind of problems can be studied as an equivalent integral equation of the type where r is a continuous function, B : C([a, b]) → R is a continuous linear functional, k is the Green's function related to the considered problem, and f (t, x) is the nonlinear part of the considered equation.
This kind of equations cover different non-local situations as, for instance, to model the steady-state of a heated bar of length b − a subject to a thermostat, where a controller in one end adds or removes heat accordingly to the temperature measured by a sensor at a point of the bar. This type of heat-flow problem has been studied in several works on the literature, see [8,10,11,13] and references therein.
An important part of the used methods to ensure the existence of solutions are mainly related to the theory of lower and upper solutions [6], degree theory [3,10,11] or monotone iterative techniques [2]. In all of these cases it is fundamental to ensure the constant sign (in the whole square of definition or in a suitable subset) of the Green's function related to the considered problem. In many situations, this study is not trivial and requires many tedious and complicate calculations. Such difficulty increases with the non-local operators on the boundary. One of the most common non-local boundary conditions are given as integral equations (some of them in the Stieltjes sense) and has been applied to different situations as fourth order beam equations [4], second order problems [9] or fractional equations [5,7].
To be concise, in this paper we will consider the following n-th order linear boundary value problem with parameter dependence: where Here σ and a k are continuous functions for all k = 0, . . . , n − 1, M ∈ R and δ i ∈ R for all i = 1, . . . , n. C i : C(I) → R is a linear continuous operator and B i covers the general two point linear boundary conditions, i.e.: being α i j , β i j real constants for all i = 1, . . . , n, j = 0, . . . , n − 1.
Remark 1.1. Examples of operator C i can be the integral operator or the multi-point operator We point out that Problem (1) covers any n-th order differential equation and that on the choice of δ i any of them could vanish, so it may be though as a perturbation of a two-point boundary value problem.
So, by considering the following homogeneous problem related to the general equation (1): we will obtain the explicit expression of the Green's function related to the non-local problem (1) under the assumption that the corresponding homogeneous Problem (2) has only the trivial solution. Moreover we will characterize the spectrum of Problem (1) as a function of the value of the non-local operators over functions related to the Green's function of Problem (1). We notice that the non-local linear operators depend only on the values of the function that we are looking for, but they could depend on any of its derivatives, and by using analogous reasoning, the result that we could obtain would be similar.
The paper is organized as follows. In next section we obtain the expression of the Green's function related to Problem (1) and characterize its spectrum. In Section 3 we present an example where the formula is used to obtain the corresponding expression and to describe the exact set of parameters for which its Green's function has constant sign on I × I.

Explicit expression of the solution of Problem (1)
This section is devoted to deduce the explicit expression of the solution of general Problem (1). To this end, we assume that the homogeneous Problem (2) has as unique solution the trivial one. In such a case, it is very well known that Problem (1), with δ i = 0, i = 0, . . . , n, has a unique solution for any σ ∈ C(I) given. Moreover, such solution is given by Here g M denotes the Green's function related to Problem (2), which exists and is unique (see, for details, [2,12]). Now, we enunciate the following particular case of the result proved in [2, page 35]: Theorem 2.1. The following boundary value problem has a unique solution for any σ ∈ C(I) and where u 1 , . . . , u n is any set of linearly independent solutions of T n [M ] u(t) = 0.
is a necessary (but not sufficient) condition to ensure the uniqueness of solution of Problem (4).
As a direct consequence of Theorem 2.1 we deduce the following result Lemma 2.5. There exists the unique Green's function related to Problem (2), g M , if and only if for any i ∈ {1, · · · , n}, the following problem has a unique solution, that we denote as ω i (t), t ∈ I.
In the following result, under suitable assumptions concerning the spectrum of the considered problem, we prove the existence and uniqueness of the solution of Problem (1). Moreover, the expression of its related Green's function is obtained.
Theorem 2.6. Assume that Problem (2) has u = 0 as its unique solution and let g M be its related Green's function. Let σ ∈ C (I), and δ i , i = 1, . . . , n, be such that with I n the identity matrix of order n and A = (a ij ) n×n ∈ M n×n given by Then Problem (1) has a unique solution u ∈ C n (I), given by the expression where with ω j defined on Lemma 2.5 and Proof. Since Problem (1) has a unique solution when δ i = 0 for all i = 1, . . . , n, from Lemma 2.5 we know that any solution of (1) satisfies the following expression with v given by (3). Applying linear continuous operators C j on both sides of (11) we infer that from which we deduce that Therefore, we arrive at the following systems of equations From previous equality we deduce that and substituting this expression in (11) we obtain that To calculate C j (v) we use the fact that C j is linear and continuous, so we get that Using the previous equality, we have that So, we have proved that under assumption (8) coupled to the uniqueness of solution of Problem (2), Problem (1) has at least one solution given by expression (9).
To conclude the proof, we must show the uniqueness of the solution. To this end, suppose that u and v are two different solutions of Problem (1). Then, As a consequence, we have that Applying operator C j in both sides again, we have that or, which is the same, Condition (8) implies that C i (u − v) = 0 for i = 1, . . . , n. Hence, form (13) we deduce that u − v is a solution of the homogeneous problem Since this problem has only the trivial solution, we deduce that u = v on I, and the proof is concluded.
Remark 2.7. We notice that in previous result, we assume that there is a unique Green's function related to Problem (2). Such condition does not depend on δ i or operators C i , i = 1, . . . , n. It is obvious that this condition is fundamental to construct function G on (10). However, such condition is not necessary in order to deduce the existence and uniqueness of solution of Problem (1). In practical situation our hypotheses ensure the existence of a unique solution of Problem (1) provided for any parameters (M, δ 1 , . . . , δ n ), such that M is not an eigenvalue of Problem (2). But, as we will see in next section, this condition is not necessary and Problem (2) could have a unique solution for some choice of (M, δ 1 , . . . , δ n ), with M an eigenvalue of Problem (2).
Moreover, we assume the non spectral condition (8), which is equivalent to assume that 1 is not an eigenvalue of matrix A. When such condition fails we have that Problem (1) has not a unique solution. So, this non spectral condition characterizes the uniqueness of solution of Problem (1) provided the existence of g M is assumed. In case of M being an eigenvalue of Problem (2), condition (8) has no sense because ω i do not exist.
We will show now the particular case of considering that all the functionals at boundary conditions are the same (that is, there is some linear continuous operator C such that C i = C for i = 1, . . . , n). In this case, since we have only C(u) as a unique unknown variable, system (12) reduces to the one dimensional equation and condition (8) reduces to In which case, it is obvious that As a direct consequence, we obtain the following result for this particular case.
Corollary 2.8. Assume that Problem (2) has u = 0 as its unique solution and let g M be its unique Green's function. Let σ ∈ C (I), and δ i , i = 1, . . . , n, be such that (15) holds. Then problem has a unique solution u ∈ C n (I), given by the expression Proof. It is enough to show that in this case expression (10) can be rewritten as (18). Indeed, since we have a unique functional C (and so, the sum in j reduces to a unique term), it is clear that we can argue as in the proof of Theorem 2.6, by denoting As a consequence, we deduce that expression (10) is rewritten in this case as (·, s)) .
Example 2.9. If operator C is given by As a direct consequence of expression (18), we deduce the following comparison result: Corollary 2.11. Assume that Problem (2) has u = 0 as its unique solution and let g M be its unique Green's function. Assume that condition (15) holds and let G be the Green's function related to Problem (17). Suppose that the following hypotheses are satisfied: Remark 2.12. As we will see in next section, the conditions of previous corollary are sufficient but not necessary to ensure the positiveness of the related Green's function. Now, given M ∈ R and δ j , j = k be fixed, by differentiating equality (18) with respect to δ k we deduce that (19) Thus, we can study the monotony of the Green's function related to Problem (17) with respect to any parameter δ k .

First order periodic problem
This section is devoted to show the applicability of the expression (18) obtained in previous section. Moreover, we show the validity of the assumptions of Theorem 2.6 and Corollary 2.11.
To be concise, we study the sign of the Green's function related to the following perturbed first order periodic problem.
It is immediate to verify that the spectrum of Problem (20) is given by In particular, when we consider the homogeneous periodic problem (δ = 0): we have that M = 0 is the unique eigenvalue of the considered problem. That is, there is a unique g M if and only if M = 0. Moreover, see [2], it is immediate to verify that the expression of the Green's function of Problem (21) is given by the following expression Using the notations on Lemma 2.5, it is not difficult to verify that, see [2], that As a consequence, in this case, condition (15) is written as δ = M , M = 0. Thus, formula (18) can be applied to this set of (M, δ). We point out that, in this case, it is valid for all the values (M, δ) that are not on the spectrum of (20) except the ones given by (0, δ), with δ = 0. The expression for this last situation must be studied separately.
First, we deduce the following symmetric property of the Green's function related to Problem (20). It is immediate to verify that v(t) := u(1−t) is the unique solution of the following problem: As a direct consequence, we deduce that On the other hand, we have Therefore, the equality (23) is fulfilled directly by identifying the two previous equalities.
Therefore it is enough to study the sign of the Green's function G (t, s, δ, M ) for M > 0 and δ = M (the case M = 0 and δ = 0 will be considered further).
In our case, the expression (18) is given by the following formula: Thus, we arrive at the following explicit expression of the Green's function G: Remark 3.2. We point out that, since for all s ∈ (0, 1) it is verified that lim t→s + g M (t, s) ≡ g M (s + , s) = 1 + lim t→s − g M (t, s) ≡ 1 + g M (s − , s), we can define g M (s, s) as g M (s + , s) or g M (s − , s) at our convenience. This is valid too for function G(t, s, δ, M ) and thus equality (23) in Lemma 3.1 must be interpreted in this sense.
As we will see in the sequel, this fact has no influence on the sign of the Green's function.