On Coupled Systems of Lidstone-Type Boundary Value Problems

. This research concerns the existence and location of solutions for coupled system of diﬀerential equations with Lidstone-type boundary conditions. Methodology used utilizes three fundamental aspects: upper and lower solutions method, degree theory and nonlinearities with monotone conditions. In the last section an application to a coupled system composed by two fourth order equations, which models the bending of coupled suspension bridges or simply supported coupled beams, is presented. using the upper and lower solutions method. One

Lidstone-type boundary value problems have applications in real phenomena such as the study of bending of simply-supported beams or suspended bridges (see [13,14]). In [8], de Sousa and Minhós used Lidstone boundary conditions in a coupled system composed by two and fourth order differential equations, to model the bending of the main beam in suspension bridges. Likewise, Li and Gao [15], discuss models of a static bending elastic beam whose two ends are simply supported, given by Lidstone-type boundary conditions.
Over the course of several years, one can find several different approaches and techniques on problems or the family of problems with boundary value problems as Lidstone. For example, in [1], Agarwal and Wong deal with the existence of a positive solution of the complementary Lidstone boundary value problem (−1) m γ (2m+1) (t) =λF (t, γ(t), γ (t)) t ∈ (0, 1), where m ≥ 1, λ > 0, and F is continuous at least in the interior of the domain of interest; in [11] the authors apply a different methodology to a similar Lidstone problem, analyzing the existence of solution via bifurcation techniques; in [3] it is studied the existence, multiplicity and nonexistence results for nontrivial solutions to a nonlinear discrete fourth-order Lidstone boundary value problem; in [6] Cid et al. consider Lidstone boundary value problem, applying the monotone iterative technique with fixed point theorems of cone expansion or compression type; in [21] the authors deal with the existence and uniqueness of solution for a class of elliptic Lidstone boundary value problems; on [2,20], the authors study, respectively, Lidstone polynomials and boundary value problems and boundary layer phenomenon,...among others.
More recently on [5], Cabada and Somoza extensively study a family of problems involving Lidstone boundary conditions type. The existence of solution is proved through the usage of lower and upper solutions.
In addition, Lidstone boundary value problems can also be found on [18], where Minhós et al. prove an existence and location result for the fourth order fully nonlinear equation with the Lidstone boundary conditions where f : [0, 1] × R 4 → R is a continuous function satisfying a Nagumo type condition.
Consequently, Fialho and Minhós in [9], study the nonlinear fully equation The authors ellaborate on how different definitions of lower and upper solutions can generalize existence and location results for boundary value problems with Lidstone boundary data. An approach with Lidstone boundary conditions using system of differential equations can be found in [23]. The authors considered the existence of positive solutions for fourth-order nonlinear singular semipositone system The existence results were obtained by approximating the fourth-order system to a secondorder singular one and using a fixed point index theorem on cones.
Motivated by the works mentioned above, this paper concerns the study of the fourth order coupled system with Lidstone boundary conditions where f, h : [0, 1] × R 6 → R are continuous functions and A i , B i ∈ R, for i = 1, 2, 3.
The method applies lower and upper solutions and degree theory. The autors would like to point out, that to the best of their knowledge, it is the first time where fourth order coupled systems of differential equations with dependence of the first and second derivatives is considered subject to Lidstone boundary conditions (1.3).
This paper is outlined as follows. In Section 2, the key definitions and considerations are stated. The main theorem is presented in Section 3. Finally, in Section 4, an application to a system of coupled suspension bridges is proposed, as an example of (1.2)-(1.3), based on [16].

Preliminaries
For system (1.2)-(1.3) we consider coupled lower and upper solutions given by next definition: then from Definition 1, (iii), and from Definition 1, (ii), Below, a proposition of the degree theory that will be used to guarantee the existence of a solution for problem (1.2)-(1.3): [7,10]) Let X be a Banach space, Ω ⊂ X be a bounded open set such that 0 ∈ Ω and T : Ω → X be completely continuous linear operator such that I − T is an homeomorphism. Then |d(I − T, Ω)| = 1.

Main result
This section is dedicated to the main theorem which is an existence and location of solution for problem (1.2)-(1.3).
and moreover, if f and h satisfy Then there exists at least one nontrivial solution of (1.
As F, H have a compact inverses we can define the completely continuous operator where r i and r   Suppose by contradiction, that there is t ∈ [0, 1] such that α 1 (t) > u (t) and consider In fact t 0 = 0, as, by (1.3) and Definition 1 (ii), Analogously t 0 = 1.

Bending of crossed suspension bridges
In this section a coupled system composed by two suspension crossed bridges is considered. To comply with the conceptual meaning of the model, the authors will adopt x as independent variable for this section, instead of t, as previously, as in these models, as x represents displacement. Such suspension crossed bridges can be approached via a coupled system of two fourth order differential equations, following the same principle as the original model suggested in [12], assuming the adapted form , for x ∈ (0, L 1 ), where for i = 1, 2, H i , p i , are non negative forces, w i = w i (x) are the vertical displacement of the beams, y i = y i (x) are the position of the cable at rest, E i and I i are the elastic modulus of the material and the moment of inertia of the cross section respectively, combined they are the flexural rigidity related with the material used on each bridge. H i are the horizontal tensions in each cable, subject to some loads p i = p i (x) and L i are the lengths of the respective beams.
In addition the nonlinearities, (4.3) satisfy all the monotone assumptions, (3.1), therefore, by Theorem 1, the system (4.2) has at least one non trivial solution (w 1 , w 2 ) ∈ C 4 ([0, 1]) 2 , that is, The authors would like to highlight that this solution is in fact a non-trivial solution, as the null function does not satisfy (4.2). Moreover, we can estimate graphically the bending of both suspension bridges, as it is illustrated in the Figure 2.

Conclusions
In this paper the authors study and present an existence and location result for the problem (1.2)-(1.3) using the upper and lower solutions method. One  of the key features of the method used lies on the extra-information that can be obtained on the solution. In fact, in addition to the existence of solution, the upper and lower solution provide details on the solution's location and also on it's derivatives. This type of information can, therefore provide some qualitative insight on the solution.
To highlight the level of applicability of the results shown, an application to crossed suspended bridges is presented, following some adjustments on the models suggested in [12].
Our method can be adapted to fourth order coupled systems with fully differential equations, applying the Nagumo condition to control the third derivatives, following the method suggested in [19].
To generalize into a fully higher order Lidstone problem it remains to find how to define lower and upper solutions to have some relation (well order or reverse order) on the odd derivatives of lower and upper solutions.