New Fixed Point Approach For a Fully Nonlinear Fourth Order Boundary Value Problem

abstract: In this paper we propose a method for investigating the solvability and iterative solution of a nonlinear fully fourth order boundary value problem. Namely, by the reduction of the problem to an operator equation for the righthand side function we establish the existence and uniqueness of a solution and the convergence of an iterative process. Our method completely differs from the methods of other authors and does not require the condition of boundedness or linear growth of the right-hand side function on infinity. Many examples, where exact solutions of the problems are known or not, demonstrate the effectiveness of the obtained theoretical results.

Dang Quang A and Ngo Thi Kim Quy The special case of equation (1.1), where f does not contain derivative terms u ′ and u ′′′ , i.e.
In 1997 Ma et al. [9] and in 2004 Bai et al. [4] by the monotone method in the presence of lower and upper solutions constructed two monotone sequences of functions converging to the extremal solutions of the problem under some monotone condition of f .The idea of Bai et al. is used in a recent work of Li [7].Except for the existence Li successfully investigated the uniqueness of the problem.It should be emphasised that in the monotone method the assumption of the presence of lower and upper solutions is always needed and the finding of them is not easy.
Differently from the approaches to the problem (1.3)-(1.2) of the authors mentioned above and the approaches to other nonlinear fourth order differential equations with various boundary conditions including nonlocal equations and nonlocal boundary conditions, e.g., in [2,3,10,11], where the problem is led to integral operators for the unknown function u(x), in [6] we reduce the original problem to an operator equation for the right-hand side function.This idea was used by ourselves first in a previous paper [5] when studying the Neumann problem for a biharmonic type equation.
For the fully fourth order nonlinear boundary value problem (1.1)-(1.2), in 2013, Li and Liang [8] established the existence of solution for the problem under the restriction of the linear growth of the function f (x, u, y, v, z) in each variable on the infinity.In the present paper by a completely different approach, namely by the approach of [6] we free this restriction.Due to the reduction of the problem to an operator equation for the right hand side function, which will be proved to be contractive, we establish the existence and uniqueness of a solution and the convergence of an iterative method for finding the solution.Some examples demonstrate the applicability of our approach and the efficiency of the proposed iterative method.

The existence and uniqueness of a solution
For investigating the problem (1.1)-(1.2) we set Then the problem is reduced to the two second order problems Fixed Point Approach For a Fully Nonlinear Fourth Order Problem 211 Clearly, the solutions v and u of the above problems depend on ϕ, that is, v = v ϕ (x), u = u ϕ (x).Therefore, for ϕ we have the equation where A is a nonlinear operator defined by with We shall prove that under some conditions A is contractive operator.For each number M > 0 denote  (2.7) for any (x, u, y, v, z), (x, Then, the operator A defined by (2.5), where v ϕ , u ϕ are the solutions of the problems (2.2), (2.3) where G(x, t) is the Green function for the differential operator −u ′′ with homogeneous Dirichlet boundary condition (2.12) Therefore, for the solution of the problem (2.3) we have the estimate ( (2.16) Using the representation of the solutions v i and u i via the Green function and the estimates (2.10) and (2.11) we obtain (2.17) Now from (2.5) and (2.8) it follows Using the estimate (2.17) we obtain and satisfies the Lipschitz condition (2.8).Then, the operator A defined by (2.5), where v ϕ , u ϕ are the solutions of the problems (2.2),(2.3)maps the strip S M into itself.Moreover, if (2.9) is satisfied then A is contractive operator in S M .Therefore, the problem (1.1)-(1.2) has a unique nonnegative solution.
Proof: The proof of the theorem is similar to that of Lemma 2.1 and Theorem 2.1, where instead of the ball we consider the strip S M .✷

Iterative method
Consider the following iterative process: We obtain the following result.
Theorem 3.1.Under the assumptions of Lemma 2.1 the above iterative method converges with the rate of geometric progression and there hold the estimates where u is the exact solution of the problem (1.1)-(1.2).
Proof: Notice that the above iterative method is the successive iteration method for finding the fixed point of the operator A with the initial approximation (3.1) belonging to B[O, M ].Therefore, it converges with the rate of geometric progression and there is the estimate Combining this estimate with those of the type (2.17) we obtain (3.5), and the theorem is proved.✷ For numerical realization of the iterative method we use the difference schemes of fourth order accuracy for the Dirichlet problems (3.2), (3.3) on uniform grids ω h = {x i = ih, i = 0, 1, ..., N ; h = 1/N }.Namely, for the typical second order problem we use the Numerov difference scheme For grid functions on ω h we use the norm follows for brevity we omit the subscript ω h .The iterations are performed until e k = u k −u k−1 ≤ 10 −16 .In the tables of results of computation n is the number of grid points, where u d is the exact solution.

Examples
In this section we consider some examples for demonstrating the applicability of the obtained theoretical results.
First, we consider an example for the case of known exact solution.
The exact solution of the problem is It is easy to see that the function f (x, u, y, v, z) does not satisfy the conditions of [8, Theorem 1], so this theorem cannot guarantee the existence of a solution.Below, using the obtained theoretical results in Section 2 we show that the problem has a unique solution and the iterative method is very efficient for finding the solution.
First, choose M such that |f (x, u, y, v, z)| ≤ M .This number M may be defined from the inequality Clearly, M = 5 is a suitable choice.Then in the domain D 5 , since All the conditions of Theorem 2.1 are satisfied.Hence, the problem has a unique solution, and the iterative method converges.
The convergence of the iterative method for Example 4.1 is given in Table 1 and Figure 1.  1 we observe that the convergence of the iterative method does not depend on the grid size.
As in the previous example, obviously, that the function f (x, u, y, v, z) does not satisfy the conditions of [8, Theorem 1], so this theorem cannot guarantee the existence of a solution.Analogously as in Example 4.1 we can choose M = 4 and therefore, it is easy to verify that in the trip S 4 all the conditions of Theorem 2.2 are satisfied with 0 ≤ f ≤ 4 and c 0 = 0.03, c 1 = 0.5, c 2 = 0.75, c 3 = 0.083, q ≈ 0.167 < 1.Hence, the problem has a unique nonnegative solution, and the iterative method converges.
The numerical experiment for n = 100 shows that with the above stopping criterion after k = 8 iterations the iterative process stops and e 8 = 6.5919e − 17.
The convergence of the iterative method for Example 4.2 is given in Figure 2 and the graph of the approximate solution is depicted in Figure 3.The numerical experiment for n = 100 shows that with the above stopping criterion after k = 13 iterations the iterative process stops and e 13 = 6.9389e − 18.
The convergence of the iterative method for Example 4.3 is given in Figure 4 and the graph of the approximate solution is depicted in Figure 5.Moreover, below we show theoretically that this solution is nonnegative.Indeed, consider the domain and the strip Fixed Point Approach For a Fully Nonlinear Fourth Order Problem 219 Therefore, it is easy to see that in D + 4 we have and all the conditions of Theorem 2.2 are satisfied.Hence, the problem has a unique nonnegative solution.
Example 4.4.Consider the boundary value problem where Firstly, we prove that the f (x, u, y, v, z) satisfies the Lipschitz condition with respect to u, y, v, z.For this reason we need the following claim.Proof: Since the role of x, y is the same it sufficies to consider the following cases of location of x, y.
Case 2. If −A ≤ y ≤ x ≤ 0 then from Case 1 we have Thus, in all cases we can choose L = αA α−1 , and Claim 1 is proved.✷ Now, return to the function f given by (4.1).Using Claim 1 we have Thus, the function f satisfies the Lipschitz condition (2.8) and the quantity q given by (2.9) has the value Then, it is not difficult to choose M > 0, so that q < 1.Moreover, this M is such that |f (x, u, y, v, z)| ≤ M .Indeed, set K = min{k i , i = 0, 1, 2, 3} ≥ 1.Then for any (x, u, y, v, z) ∈ D M we have So, since q < 1, K ≥ 1 we have |f (x, u, y, v, z)| ≤ M. Thus, all the conditions of Theorem 2.1 are satisfied.Hence, the problem has a unique solution.
Remark 4.1.In all Examples 4.1-4.4 the right-hand side functions do not satisfy the condition of linear growth at infinity, therefore, [8, Theorem 1] cannot ensure the existence of a solution of the problems.But as seen above using the theory in Section 2 and 3 we have established the existence and uniqueness of a solution and the convergence of the iterative method.This convergence is also confirmed by numerical experiments.
We can verify that the function u(x) = sin(πx) is an exact solution of the problem.It is interesting that for this example the starting approximation is ϕ 0 (x) = π 4 sin πx, and solving the problems at the 0−iteration we obtain u 0 (x) = sin πx, which coincides with the exact solution.2 and the graph of the approximate solution is depicted in Figure 6.

Conclusion
In this paper we have established the existence and uniqueness of a solution of a fully fourth order nonlinear boundary value problem.Differently from the approaches of the other authors, we have reduced the problem to an operator equation for the right-hand side function.The investigation of the resulting operator equation does not require any condition for the right-hand side function with respect to all variables on infinity.The convergence of an iterative method has proved.Many examples have confirmed the validity of the obtained theoretical results and the wide applicability of the iterative method.
The proposed method can be used for some other nonlinear boundary value problems for ordinary and partial differential equations.This is the direction of our research in the future.
) and by B[O, M ] we denote the closed ball centered at O with the radius M in the space of continuous functions C[0, 1] with the norm ϕ = max 0≤x≤1 |ϕ(x)|.

Figure 1 :
Figure 1: The graph of e k in Example 4.1 for n = 100

Example 4 . 2 .
Consider the boundary value problem

Figure 2 :
Figure 2: The graph of e k in Example 4.2

Figure 4 :
Figure 4: The graph of e k in Example 4.3 .

Figure 5 :
Figure 5: The graph of the approximate solution in Example 4.3

Figure 6 :
Figure 6: The graph of the approximate solution in Example 4.6 for n = 200 , maps the closed ball B[O, M ] into itself.Moreover, if .18) Proof: It is easy to see that the solution of the problem (1.1)-(1.2) is the function u(x) obtained from the problems (2.2),(2.3),where ϕ is the unique fixed point of A. The estimates (2.18) indeed are the estimates (2.12)-(2.14).
Now consider a particular case of Theorem 2.1.Let us denote Fixed Point Approach For a Fully Nonlinear Fourth Order Problem 215 Example 4.1.Consider the boundary value problem

Table 1 :
Fixed Point Approach For a Fully Nonlinear Fourth Order Problem 221 Remark 4.2.Theorem 2.1 gives only sufficient conditions for the problem (1.1), (1.2) to have a unique solution and the iterative method (3.1)-(3.4) to be convergent.When these conditions are not met, in some cases the problem may have a solution and the iterative method may be convergent, too.Below, we show two examples, for one of them an exact solution is known, and for the other, Theorem 2.1 gives no information of its solution, but in the both examples the iterative method converges.

Table 2 :
The convergence in Example 4.6