PERIODIC SOLUTIONS FOR SOME FULLY NONLINEAR FOURTH ORDER DIFFERENTIAL EQUATIONS

In this paper we present sufficient conditions for the existence of solutions to the periodic fourth order boundary value problem u(x) = f(x, u(x), u′(x), u′′(x), u′′′(x)) u(a) = u(b), i = 0, 1, 2, 3, for x ∈ [a, b], and f : [a, b] × R4 → R a continuous function. To the best of our knowledge it is the first time where this type of general nonlinearities is considered in fourth order equations with periodic boundary conditions. The difficulties in the odd derivatives are overcome due to the following arguments: the control on the third derivative is done by a Nagumo-type condition and the bounds on the first derivative are obtained by lower and upper solutions, not necessarily ordered. By this technique, not only it is proved the existence of a periodic solution, but also, some qualitative properties of the solution can be obtained.


Introduction.
In this paper we get sufficient conditions for the existence of solutions of the periodic fourth order boundary value problem composed by the fully equation u (4) (x) = f (x, u(x), u ′ (x), u ′′ (x), u ′′′ (x)) for x ∈ [a, b], and f : [a, b] × R 4 → R a continuous function, and the boundary conditions Higher order periodic boundary value problems have been studied by several authors in last decades, using different types of arguments and techniques.However, as far as we know, these methods were not able to deal with fully nonlinear differential equations, in particular with both odd derivatives.As examples, see [7,8,16] for variational methods, [1,2,3,5,14,15,17,18] for second and higher order equations, and [12] for a linear n th order periodic problem.
The above difficulties are overcome applying lower and upper solutions technique and topological degree, like it is suggested in [4,6,11].In short, the method is based in three key points:

FELIZ MINH ÓS
• A Nagumo-type condition ( [13]) assumed on the nonlinearity, which is useful to obtain an a priori estimation for the third derivative and to define an open and bounded set where the topological degree is well defined.• A new kind of definition of lower and upper solutions, required to deal with the absence of a definite order for the first derivatives.In fact lower and upper solutions, not necessarily ordered, are associated by translations with some ordered functions, used to define the sets where the solutions and their first derivatives are contained.• An adequate auxiliary and perturbed problem, where the truncations and the homotopy are extended to some boundary conditions of mixed type, allowing an invertible linear operator and the evaluation of the Leray-Schauder degree.Remark that, by [9], for periodic second order problems, the existence of lower and upper solutions is not sufficient to obtain a solution in the absence of Nagumo condition.At the moment this is an open issue for higher order boundary value problems.
This method is particularly well adapted to boundary value problems (see [10]) because it provides not only the existence of a periodic solution, without monotone assumptions on the nonlinearity, but also, some qualitative properties of this solution and its derivatives.
The last section contains an example where the location part is used to prove that the solution is nontrivial and to obtain some of its values.

Definitions and a priori bounds.
In this section it is precise how to control the odd derivatives.For the third derivative it is obtained an a priori estimate via a Nagumo-type growth condition.

Definition 2.1. A continuous function
with γ i (x) and Γ i (x) continuous functions such that, if there exists a real continuous function with Lemma 2.2.([11], Lemma 2) Suppose that f satisfies Nagumo-type condition as in Definition 2.1.Then, there exists R > 0 (depending on γ 2 , Γ 2 and h E ) such that every solution u (x) of (1) with Non ordered upper and lower solutions will be useful to define such set E and the strips where solutions and the derivatives will be located.
In the periodic case, lower and upper solutions and their first derivatives are translate to some auxiliary functions, which are well ordered: ) is an upper solution of problem ( 1)-( 2) if: 3. Existence of periodic solutions.Next theorem provides an existence and location result for problem (1)-( 2) in presence of non ordered lower and upper solutions.
Remark 2. The above remark shows that the order between lower and upper solutions and their first derivatives is not important to guarantee the solvability of the periodic problem (1)-( 2).
Remark 3. The proof that the solution is nontrivial can be obtained by the location part of the theorem.For example, if the lower solution is chosen such that α ′′ (x) > 0, or the upper solution such that β ′′ (x) < 0, for some x ∈ [a, b], then the periodic solution of ( 1)-( 2) is nontrivial.
The last section contains an example where this issue is applied.

Define the operators
) .
As L has a compact inverse then it can be considered the completely continuous operator T λ : By Steps 1 and 2, for every u solution of ( 13)-( 14), u / ∈ ∂Ω and so the degree d(I − T λ , Ω, 0) is well defined for every λ ∈ [0, 1] .By the invariance under homotopy has only the null solution then, by degree theory, So, the equation T 1 (x) = x has at least a solution, that is the corresponding problem has at least a solution u 1 (x) in Ω.
By Theorem 3.1 there is a solution u(x) of problem ( 24)-( 2), such that The proof that this solution is a nontrivial periodic solution can be done applying the location part of Theorem 3.1 in two ways: The only constant solution allowed by the equation (24) would be 3 √ 129, but this value is not in the set [α 0 , β 0 ] := {y(x) : α 0 (x) ≤ y(x) ≤ β 0 (x), x ∈ [0, 1]}.
Another possible argument is to remark that the solution u(x) can not be a constant because, by (25), ) .