SOLVABILITY OF HIGHER-ORDER BVPS IN THE HALF-LINE WITH UNBOUNDED NONLINEARITIES*

. This work presents suﬃcient conditions for the existence of unbounded solutions of a Sturm-Liouville type boundary value problem on the half-line. One-sided Nagumo condition plays a special role because it allows an asymmetric unbounded behavior on the nonlinearity. The arguments are based on ﬁxed point theory and lower and upper solutions method. An example is given to show the applicability of our results.

Higher order boundary value problems on infinite intervals appear in several real phenomena such as the gas pressure in a semi-infinite porous medium, or theoretical results as, for example, the study of radially symmetric solutions of nonlinear elliptic equations. For these and other applications see, for example, [1]. Third-order differential equations, in general, arise in many areas, such as the deflection of a curved beam having a constant or a varying cross-section, three layer beam, electromagnetic waves or gravity-driven flows, ([9]). In infinite intervals, third order boundary value problems can describe the evolution of physical phenomena, for example some draining or coating fluid-flow problems, (see [4,15,16]).
By the non-compactness of the interval, the discussion about sufficient conditions for the solvability of boundary value problems on the half-line is more delicate. In the existent literature the main methods to obtain existence results are the extension of continuous solutions on the corresponding finite intervals under a diagonalization process and fixed point theorems in special Banach spaces (see [2,3,11,17] and the references therein.) Lower and upper solutions method is an useful technique to deal with boundary value problems as it provides not only the existence of solution but also its localization and, from 842 FELIZ MINHÓS AND HUGO CARRASCO that, it can be obtained some qualitative data about solutions variation and behavior (see [5,8,12,13,14]). An important tool of this technique is the Nagumo condition, useful to obtain a priori estimates on some derivatives of the solution. As it can be seen in the references above, the usual growth condition of the Nagumo type is a bilateral one. However the same estimations hold with a similar one-sided assumption, allowing that the boundary value problems can include unbounded nonlinearities. In this way it generalizes the two-sided condition, as it is proved in [7,10].
The paper is organized as it follows: In Section 2 is defined the space, the weighted norms and the unilateral Nagumo conditions to be used. As far as we know, it is the first time where such conditions are assumed for boundary value problems defined on the half-line.
Section 3 contains the main result: an existence and localization theorem, where it is proved the existence of a solution, and some bounds on the first and second derivatives as well.
Finally, an example shows the applicability of the theorem and, moreover, it is emphasized that the nonlinearity considered verifies the one-sided Nagumo condition but not the bilateral one.
2. Definitions and auxiliary results. In this section it is proved an a priori estimate on the derivative u under an unilateral Nagumo-type condition. With that aim, let us introduce the following definitions: Let w i (t) = 1 + t 2−i , i = 0, 1, 2 and define the space , for i = 0, 1, 2.
Therefore (X, . ) is a Banach space.
In the first case, by (3), we can take R > r such that If condition (4) holds, then by (7) there are t * , t + ∈ [0, +∞) such that t * < t + , u (t * ) = r and u (t) > r, ∀t ∈ (t * , t + ]. Therefore So u (t + ) < R and as t * and t + are arbitrary in (0, +∞), we have u (t) < R, ∀t ∈ [0, +∞). Similarly, it can be proved the case where there are t − , t * ∈ [0, +∞) such that t − < t * and . Now consider that f verifies (5). By (7), consider that there are t − , t * ∈ [0, +∞) such that t − < t * and u (t * ) = r, u (t) > r, ∀t ∈ [t − , t * ). Therefore, following similar steps as before So u (t − ) < R and by the arbitrariness of t − and t * in [0, +∞), we have The exact solution for the associated linear problem can be obtained by simple calculations: has a unique solution in X. Moreover, this solution can be expressed as The lack of compactness is overcome by the following lemma which gives a general criterium for relative compactness, (see [1]) : (iii) all functions from M are equiconvergent at infinity, that is, for any given > 0, there exists a t > 0 such that for all t > t , u ∈ M and i = 0, 1, 2.

Main Result.
In this section we prove the existence of at least one solution for the problem (1),(2) applying lower and upper solutions method and, moreover, some data on its behavior and variation are given.

STEP 2:
If u is a solution of the modified problem (13),(2) then there exists R > 0, not depending on u, such that u 2 < R. By the previous step, all solutions of equation (13) are solutions of (1), and as f verifies the one-sided Nagumo condition (4), or (5), this claim is a direct application of Lemma 2.3. STEP 3: Problem (13),(2) has at least one solution.
By Lemma 2.4, the fixed points of T are solutions of problem (13), (2). So it is enough to prove that T has a fixed point.

HIGHER-ORDER BVPS DEFINED ON UNBOUNDED DOMAINS 847
Therefore T u ∈ X.
Claim 2. T is continuous. Consider a convergent sequence u n → u in X. Then there exists r 1 > 0 such that u n < r 1 and

Claim 3. T is compact.
Let B ⊂ X be any bounded subset. Therefore there is R > 0 such that u < R, ∀u ∈ B.
Claim 3.1. T B is uniformly bounded.
As T is completely continuous then by Schauder Fixed Point Theorem, T has at least one fixed point u ∈ X.
Moreover, from the localization part of the theorem, we can precise some qualitative properties of this solution: it is nonpositive, nonincreasing and, as C = 0, this solution is unbounded.