The aim of this paper is to show certain properties of the Green's functions related to the Hill's equation coupled with different two point boundary value conditions. We will obtain the expression of the Green's function of Neumann, Dirichlet, Mixed and anti-periodic problems as a combination of the Green's function related to periodic ones.As a consequence we will prove suitable results in spectral theory and deduce some comparison results for the solutions of the Hill's equation with different boundary value conditions.

In this work we will consider integral equations defined on the whole real line and look for solutions which satisfy some certain kind of asymptotic behavior. To do that, we will define a suitable Banach space which, to the best of our knowledge, has never been used before. In order to obtain fixed points of the integral operator, we will consider the fixed point index theory and apply it to this new Banach space.Partially supported by Xunta de Galicia (Spain), project EM2014/032 and AIE Spain and FEDER, grants MTM2013-43014-P, MTM2016-75140-P.Supported by FPU scholarship, Ministerio de Educación, Cultura y Deporte, Spain.1 consequence of the lack of compactness of the operator. In all of the cited references the authors solve this problem by means of the following relatively compactness criterion (see [1,13]) which involves some stability condition at ±∞: 1 ([13, Theorem 1]). Let E be a Banach space and ( , E) the space of all bounded continuous functions x : → E. For a set D ⊂ ( , E) to be relatively compact, it is necessary and sufficient that:for any t ∈ ; 2. for each a > 0, the family D a := {x| [−a,a] , x ∈ D} is equicontinuous; 3. D is stable at ±∞, that is, for any ǫ > 0, there exists T > 0 and δ > 0 such that if x(T ) − y(T ) ≤ δ, then x(t) − y(t) ≤ ǫ for t ≥ T and if x(−T ) − y(−T ) ≤ δ, then x(t) − y(t) ≤ ǫ for t ≤ −T , where x and y are arbitrary functions in D.

In this paper, we obtain the explicit expression of the Green’s function related to a general n-th order differential equation coupled to non-local linear boundary conditions. In such boundary conditions, an 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 note that, in order to ensure the uniqueness of the solution of the considered linear 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 the solution of the two-point homogeneous problem is not a necessary condition to ensure the existence of 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.

In this paper, we prove the existence of solutions of nonlinear boundary value problems of arbitrary even order using the lower and upper solutions method. In particular, we point out the fact that the existence of a pair of lower and upper solutions of a considered problem could imply the existence of solution of another one with different boundary conditions. We consider Neumann, Dirichlet, mixed and periodic boundary conditions.

In this paper we will show several properties of the Green's functions related to various boundary value problems of arbitrary even order. In particular, we will write the expression of the Green's functions related to the general differential operator of order 2n coupled to Neumann, Dirichlet and mixed boundary conditions, as a linear combination of the Green's functions corresponding to periodic conditions on a different interval. This will allow us to ensure the constant sign of various Green's functions and to deduce spectral results.