A note on iterative solutions for a nonlinear fourth order ode ∗

which models bending equilibrium of elastic beams on nonlinear supports. Following Ginsberg [7] or Grossinho and Tersian [8], u represents the configuration of an elastic beam of length L, subject to a force f exerted by the foundation. Both ends are attached to fixed torsional springs represented by the functions g and h. Our objective is to show the existence of iterative solutions under local conditions on the functions f, g, h. Some numerical simulations are also presented. We refer the reader to [2,3,4,5,8,9] for other related works.


Introduction
In this work we are concerned with the boundary value problem u (iv) (t) = f (t, u, u ′ ), 0 < t < L (1) u ′′ (0) = g(u ′ (0)), u ′′ (L) = h(u ′ (L)), which models bending equilibrium of elastic beams on nonlinear supports.Following Ginsberg [7] or Grossinho and Tersian [8], u represents the configuration of an elastic beam of length L, subject to a force f exerted by the foundation.Both ends are attached to fixed torsional springs represented by the functions g and h.
Our objective is to show the existence of iterative solutions under local conditions on the functions f, g, h.Some numerical simulations are also presented.We refer the reader to [2,3,4,5,8,9] for other related works.

Iterative Solutions
Our existence result is the following.Theorem 2.1 Suppose that f, g, h are continuous functions and there exist constants A, B, C > 0 such that and Then if problem (1)-( 3) has at least a solution.
Theorem 2.2 Suppose the assumptions of Theorem 2.1 hold.Suppose further that there exist constants λ f , λ g , λ h > 0 such that and problem (1)-( 3) has an iterative solution u with u ′ ∞ ≤ R.
The proofs rely on fixed point theorems.We begin by rewriting problem (1)-(3) into a second order system.If v = u ′′ then we have The Green's function associated to the second order problem (12) is precisely and gives Analogously, from (13) we have Then, combining the above identities we get We can see that u is a solution of ( 1)-( 3) if and only if it is a solution of ( 14).Next we apply fixed point arguments to solve (14).In view of (2) we apply fixed point theorems on the Banach space Because u(0) = u(L) = 0, we see that and, in particular, the usual norm which will be adopted here.Then we note that Proof of Theorem 2.1 Let us define the operator T : E → E with (T u)(x) equal to the right hand side of (14).Then fixed points of T are solutions of problem (1)-(3).Next we show that T maps the closed ball B[0, R] of E into itself.Indeed, noting that we have from that for u ∈ B[0, R] and using ( 4)-( 7), Therefore with respect to the norm (15), To conclude the proof we note that T is completely continuous on B[0, R] (by Arzela-Ascoli theorem) and therefore it has a fixed point by the Schauder's fixed point theorem (e.g.[1]).
Proof of Theorem 2.2 Let u, v ∈ B[0, R].Then as before, but using ( 8)-( 10), From (11) we see that T is a contraction on B[0, R] and then it has a fixed point from the Banach's fixed point theorem (e.g.[1]).

Numerical Simulations
From Theorem 2.2 we obtain the iterative formulae u k+1 = T u k , were which converges to a solution of (1)-(3) for any initial approximation u 0 ∈ B[0, R].
We show two numerical simulations to illustrate the use of ( 16).In both examples, L = 1, u 0 = 0 and mesh size is 0.1.The integrals are approximated by trapezoidal method.
Example 1 First example we take After 10 iterations we get maximum error Other values are shown in the Table 1.
After 4 iterations we get maximum error