A Note on Existence Results for a Class of Three-Point Nonlinear BVPs

This article deals with a computational iterative technique for the following second order three point boundary value problem y(t) + f(t, y, y) = 0, 0 < t < 1, y(0) = 0, y(1) = δy(η), where f(I × R,R), I = [0, 1], 0 < η < 1, δ > 0. We consider simple iterative scheme and develop a monotone iterative technique. Some examples are constructed to show the accuracy of the present method. We show that our technique is quite powerful and some user friendly packages can be developed by using this technique to compute the solutions of the nonlinear three point BVPs whose close form solutions are not known.


Introduction
In several real life problems, e.g., [19,21], the boundary value conditions do not rely only at the end points, but also at the interior points of the interval. Such problems are known as multipoint boundary value problems. Several results are available in literature related to multipoint and nonlocal BVPs, e.g., [1,4,5,6,8,9,10,11,12,15,16,19,20,21]. S. Roman along with A.Štikonas [13,14,17,18] established results related to construction of Green's function for nonlocal and multipoint boundary value problems.
Li et al. [5,6] studied the existence and uniqueness of solutions of second order 3 point boundary value problems with upper and lower solutions in reverse order. They used monotone iterative method. In a recent work Bao et al. [1] and Singh et al. [15,16] prove some new results for similar 3 point boundary value problems. The functional version of these problems is considered by Kiguradze and Puza [3] and Lomtatidze and Vodstrch [7].
In this paper we present some new existence results for second order non linear 3 point boundary value problem with Dirichlet type boundary condition y (t) + f (t, y, y ) = 0, 0 < t < 1, (1.1) The result of this paper is an improvement over a recent result due to Bao et al. [1]. They assume two conditions f (t, 0, 0) = 0 and yf (t, y, y ) ≥ 0 for y ≥ 0. Consider −y = h(t) + y which is linear but f (t, 0, 0) = 0. So f (t, 0, 0) = 0 fails. Another simple example is −y = sin y. Here y sin y will change its sign for y ≥ 0, so the condition yf (t, y, y ) ≥ 0 for y ≥ 0 fails. But for both these problems the results of this paper are applicable.
Here we are looking for a simple monotone iterative scheme and propose the following

Preliminary
Here we consider the linear 3 point BVP. We prove maximum principle and also prove existence of some differential inequalities. Consider the corresponding nonhomogeneous linear 3 point BVP where h ∈ C(I), & b any constant.
Case I: λ > 0. Let us assume We can easily verify that there is a range of λ, which support (H 0 ) (see Figure 1).
The Green's function of the 3 point BVP Ly = 0, y(0) = 0, y(1) = δy(η) for λ > 0, is Proof. The Green's function for the 3 point BVP Ly = 0, y(0) = 0, y(1) = δy(η) for λ > 0, is defined as The unknown variables a 1 , a 2 , a 3 and a 4 are found with the help of the definition of Green's function, for any s ∈ [0, η], we have and thus Then by using the 3 point boundary value condition, we have The values of a 1 , a 2 , a 3 and a 4 are given by Similarly, for any s ∈ [η, 1], we have Consequently, we can get the Green's function G(t, s), and lemma is proved. We can easily prove that the constant sign of Green's function will be nonpositive when (H 0 ) holds.
We can easily verify that, there is a range of λ < 0, which support (H 0 ) (see Figure 6).

Lemma 3. The Green's function of the 3 point BVP Ly
is given by the following equation

Existence of some differential inequalities
In this section we prove existence of some differential inequalities which govern the range of λ and also they ensure that if these inequalities are true the solutions generated by iterative scheme are monotonic.
Proof. The function is non-decreasing for all t ∈ [0, 1] and satisfy the following inequality, By using the assumptions it is easy to verify (i).
Using the properties of sin, cos and assumptions, we can easily see that for all t ∈ [0, 1], Hence (ii) is verified.
The right hand side of the above inequality will be non-positive for all t ∈ [0, 1], if This complete the part (i) of the lemma.
Using the assumptions and the properties of sinh and cosh, we can easily see that for all t ∈ [0, 1] (M + λ) sinh |λ|t + N (t) |λ| cosh |λ|t is a non-increasing function, which proves part (ii).    Proof. By Lemma 6 and Lemma 7 we arrive at Lemma 8.

Maximum principle
We conclude the following two results.

Nonlinear Point BVP
In this section we consider the nonlinear 3 point BVP. We show that it is possible to find out a range of λ = 0 on λ axis so that the iterative scheme Our proof is based on uniform convergence of the sequences and for that we use Arzela-Ascoli theorem. To implement this we need equicontinuity and equiboundedness of {y n } and {y n }. Equicontinuity and equiboundedness of y n and y n can be proved by continuity of the Green's function and continuity of the solution on [0, 1] and continuity of the nonlinear term f (t, y, y ). Equiboundedness of {y n } is established by the following two lemmas.  Proof. The proof can be divided in two parts. Case (i). If solution is not monotone in [0, 1], then consider the interval (t 0 , t] ⊂ (0, 1) such that y (t 0 ) = 0 and y (t) > 0 for t > t 0 . Integrating (3.1) from t 0 to t we get

Priori bound
From (H P ) we can choose R > 0 such that which gives Now we consider the case in which y (t) < 0 for t < t 0 , y (t 0 ) = 0, and proceeding in the similar way we get − y (t) ≤ R, and the result follows.
By uniqueness of the limit and monotonicity of the sequences (α n ) n and (β n ) n , we have α n → v and β n → u.
We write the solution of iterative scheme (1.3)-(1.4) for both (α n ) and (β n ) by using Lemma 2, where h(t) is in terms of nonlinear term f . Now by using uniform convergence, one can easily conclude the existence of the solution of nonlinear 3 point BVP. This completes the proof.
Proof. Proof is same as Theorem 1.

Numerical Illustrations
To verify our results, we consider two examples for both λ > 0, λ < 0 and show that it is possible to compute a range of λ so that iterative scheme generates monotone sequences which converge to the solution of nonlinear problem.  Here f (t, y, y ) = (e t −1)

Conclusion
In this paper we have considered an iterative scheme which is simple enough for computational point of view. We did not consider λ as function of t. The method developed in this paper can be coded to generate a user friendly package which can be efficiently used to compute solutions of the nonlinear 3 point BVP whose close form solutions is not known.
We have constructed two examples one for each case λ > 0 and λ < 0 and show that derived sufficient conditions can generate solutions for a class of nonlinear 3 point BVPs. Mainly it is initial iterates (upper and lower solutions) choice of which matters and success of the method depends on them. If initial iterates are chosen properly then it is guaranteed that sequences will converge to the solutions of the nonlinear BVP. In Figure 11 we also observe that if λ does not belong to the range sequences are not monotone.
We also observe that Remark 5.4 of Cherpion et al. [2] seems to be true even in case of 3 point BVP with Dirichlet type boundary condition.     Figure 11. Non-monotonicity for λ = 2.