HIGHER ORDER TWO-POINT BOUNDARY VALUE PROBLEMS WITH ASYMMETRIC GROWTH

. In this work it is studied the higher order nonlinear equation with n ∈ N such that n ≥ 2 , f : [ a,b ] × R n → R a continuous function, and the two-point boundary conditions From one-sided Nagumo-type condition, allowing that f can be unbounded, it is obtained an existence and location result, that is, besides the existence, given by Leray-Schauder topological degree, some bounds on the solution and its derivatives till order ( n − 2) are given by well ordered lower and upper solutions.Anapplication to a continuous model of human-spine, via beam theory, will be presented.

As it can be seen in [5], for some second order two-point boundary value problems this type of restriction is a necessary condition to obtain a solution. In this work the bound restriction defined by the bilateral condition (4) is replaced by a weaker one-sided Nagumo condition, f (x, y 0 , ..., y n−1 ) ≤ ϕ E (|y n−1 |), or f (x, y 0 , ..., y n−1 ) ≥ − ϕ E (|y n−1 |), (7) for every (x, y 0 , ..., y n−1 ) ∈ E, allowing an unbounded behaviour on f as it is suggested, for third order problems, in [3,4]. As an example, it is mentioned that function f : [0, 1] × R n → R, n ≥ 2, given by with ϕ E * (x n−1 ) := k > 0, but condition (4) does not hold in E * . In fact, if there was a function ϕ E * ∈ C R + 0 , R + verifying (4) and (5) then, in particular, −f (x, y 0 , . . . , y n−1 ) ≤ ϕ E * (|y n−1 |), ∀(x, y 0 , . . . , y n−1 ) ∈ E * and, for x ∈ [0, 1], y 0 = 0, y n−2 = 0 and y n−1 ∈ R, − f (x, 0, y 1 , . . . , y n−3 , 0, y n−1 ) = [(n − 2)!] 2 + y 2p+4 n−1 ≤ ϕ E * (|y n−1 |) and the following contradiction is obtained The arguments make use of lower and upper solutions technique and topological degree theory to obtain existence and location results, meaning that it is proved not only the existence of solutions, but some information about its localization and of some derivatives is also attained. The existence of positive (negative) solutions is a particular case of the above results because it is enough to consider a nonnegative lower solution or a non positive upper one.
Last section provides an application of these kind of problems, for n = 4, to continuous models of the human-spine to estimate, by beam theory, its lateral displacement under some loading forces. It is pointed out that, in these models, lower and upper solutions method is highly recommended due to the location part. In fact, it is estimated not only the total displacement of the column but also its torsion and curvature.
2. Definitions and a priori bound. This section will provide some definitions to be used forward and an a priori estimation for the derivative u (n−1) , which is one of the key points of the Nagumo-type condition. In fact, the growth restriction on the nonlinear part is given by the one sided Nagumo-type assumption, in spite of the usual bilateral one, as it is precised in the following definition: (6) and (5).
The a priori bound on the (n − 1) th derivative is obtained in the following way: and define r ≥ 0 such that Then, there exists R > 0 such that every u(x) solution of (1) verifying and satisfies u (n−1) ∞ < R. Remark 1. Observe that: (6) is replaced by (7) the a priori estimation given by Lemma 1 still holds if it is assumed, instead (9), Proof. Let u be a solution of the differential equation (1) satisfying (9) and (10).
Suppose that u (n−1) (x) > r, for every x ∈ ]a, b[. In the case u (n−1) (x) > r, for all x ∈ ]a, b[, the following contradiction is obtained If In the first case consider, by (9), Then, with a convenient change of variable, by (6) and (11), it is obtained, for arbitrary Therefore, u (n−1) (x 2 ) < R and, as x 1 is arbitrary as long as u (n−1) (x) > r, then By a similar way, it can be proved that The method of lower and upper solutions is an important tool to obtain existence and location results in next sections. The functions used in this technique are defined as follows: for x ∈ [a, b], and A function β ∈ C n ([a, b])is said to be an upper solution of problem (1)-(2), if it verifies the reversed inequalities.
3. Existence and location result. The main result is an existence and location theorem, as is usual in lower and upper solutions technique. In this case some data on the location of the derivatives until (n − 2) th -order are also given.
) are lower and upper solutions of problem (1)- (2), respectively, such that and define the set Let f : [a, b]×R n → R be a continuous function verifying the one-sided Nagumo-type condition (6) and for fixed x, y n−2 , y n−1 and Then problem (1)-(2) has at least one solution u ∈ C n ([a, b]) such that for every x ∈ [a, b] and i = 0, . . . , n − 2.
In the case x 0 = b the arguments are analogous. Thus, u (n−2) (x) < r * , for every x ∈ [a, b]. In a similar way it can be proved that u (n−2) (x) > −r * , for every x ∈ [a, b].
As r * and ϕ do not depend on λ, the estimate u (n−1) (x) < R, is independent of λ.
Step 4: The function u 1 (x) is also a solution of the initial problem.

4.
Applications. The case n = 4 will be studied in this section. A "natural" application to a fourth order differential equation is in beam theory, where twopoint boundary conditions mean the different types of support at the endpoints. For example, the problem u(a) = u ′ (a) = u ′′′ (a) = u ′′′ (b) = 0 models the bending of an elastic beam, cantilevered at the left endpoint and with null shear force (vertical) at both endpoints. A not so common application is the study of some mechanical properties of the human spine by a continuous beam model, illustrated in Fig 1, as it is done in [10,11] and the references therein. In short, the total lateral displacement, y(x), of the beam-column is the sum of the initial displacement, y 0 (x), (known) and the lateral displacement due to the axial and transverse loads, y 1 (x), i.e., y(x) = y 0 (x) + y 1 (x). This unknown displacement y 1 (x) is modelled ( see [11]) by the problem composed by the differential equation where EI is the flexural rigidity of the beam-column, P the axial load and Q(x) the transverse load (Figure 1), and the boundary conditions f (x, z 1 , z 2 , z 3 , z 4 ) = P EI z 3 + P y ′′ 0 (x) + Q(x) EI , assuming that the initial lateral displacement y 0 ∈ C 2 (J) and the function Q ∈ C(J) verify, for x ∈ J, b E I − P x 2 2 − P L 2 ≤ P y ′′ 0 (x) + Q(x) ≤ a E I − P for every x ∈ J.