Existence Results for a Nonlinear Fourth Order Ordinary Di erential Equation with Four-Point Boundary Value Conditions

The aim of this paper is to study the more accurate existence results of positive solution for a nonlinear fourth order ordinary di erential equation (for short NLFOODE) using four-point boundary value conditions (for short BVCs). The upper and lower solution method and Schauder's xed point theorem have been applied to demonstrate the obtained existence results. First, the Green's function of the corresponding linear boundary value problem (for short BVP) has been constructed and then it is used to solve the considered BVP of this paper. An example has also been included at the end of this paper to support the analytic proof.


Introduction
It is well established that the xed point technique and upper and lower solution method are most important techniques for checking the existence of positive solutions of ordinary dierential equations using various BVCs. Boundary value problems (for short BVPs) for fourth order ordinary dierential equations (for short ODEs) are used to describe a huge number of physical, biological and chemical phenomena, see for instance [1,11,13,22,23,27] and references therein. In the last few decades, positive solution of two-point, three-point and four-point boundary value problems for second order, third order, fourth order as well as higher order has extensively been studied by using various techniques, see for instance [2,3,4,5,8,9,10,14,15,19,24,25,28] and references therein. Inspiring by the above-mentioned works, we have interested to check the existence of positive solutions of a four-point BVP for NLFOODE by applying upper and lower solution method [10] and Schauderâs xed point theorem [20]. For brevity, here we only described the most recent analogous literature about the existence of positive solutions of four-point BVP for NLFOODE. Although, literature may contain some more general results on BVPs for nonlinear fourth order dierential equation, for instance we may refer [6,7,16,17,18,26,29], but for the better correction of some previously developed results sometimes we have to reconsider some fewer general results. From this point of view, here we reconsider a fewer general result of Chen et al. [9].
In 2006, Chen et al. [9] checked the existence of positive solutions of following four-point BVP for NLFOODE by applying upper and lower solution method and Schauder's xed point theorem: where a, b, c, d are nonnegative constants satisfying, ad + bc + ac( Unfortunately, this lemma is incorrect. Now, we provide a counter example to demonstrate it. Counter example to the Lemma 1.1(Lemma 2.2 of Chen et al. [9]). Let u(t) = 1 6 t 4 − 1 3 t 3 + 15 196 t 2 + 7 64 , ξ 1 = 5 8 , ξ 2 = 8 15 and a, b, c, d are four positive constants such that a = b, and c = d. Then we have But, u 1 16 = 225 6272 > 0, which means that the Lemma 2.2 of Chen et al. [9] is not correct. Therefore, the results of Chen et al. [9] should be reconsidered. From this ground, here again we considered the fourth order four-point BVP given by (1) and establish the existence result of positive solutions of this BVP by applying upper and lower solution method [10] and Schauder's xed point theorem [20].
The considered BVP of this paper relates to the classical bending theory of exible elastic beams on a nonlinear basis. If we put f (t, u(t)) = p(t)g(u(t)) in the considered BVP given by (1), then it will refer as the beam equation and further physical interpretation of the beam equation can be found in the work of Zill and Cullen ( [27], pp. 237-243).
Rest of this paper has been arranged as follows: Section 2 is used to introduce some preliminaries facts. In Section 3, we state and prove our main result and verify it by a particular example. Finally, in Section 4 we conclude this paper.

Preliminaries Notes
In this section, we give some denitions, lemmas, and state Schauder's xed point theorem which are crucial to establish our main result. Denition 2.1. (See [10]) A function α(t) is said to be a lower solution of the BVP given by (1), if it belongs to C 4 [0, 1] and satises is said to be an upper solution of the BVP given by (1) is said to be a solution of the BVP given by (1), if it is both lower and upper solutions of that BVP.
has a unique solution is the Green's function of the linear BVP given by Proof. Here rst we solve the BVP (3) by using Green's function.
The general solution of (3) is Using the boundary conditions of (3), we obtain A = B = 0. Hence (4) yields only trivial solution u(t) = 0. Therefore, the unique Greenâs function exists for BVP (3) and is given by Now, by the properties of Greenâs function, we have Solving (6), (7), (8) and (9), we obtain Putting the values of a 1 , a 2 , b 1 , and b 2 in (5), we obtain the unique Green's function Therefore, the unique solution of BVP given by (3) is and this solution ensure that the BVP given by (2) has a unique solution and which is This completes the lemma.

Lemma 2.2. (See Lemma 2.3 of Chen et al. [9]). If
. We end this section by stating the Schauder's xed point theorem [20], which will be needed to establish our main result.
Proof. We will prove our theorem by three major steps.
In this step, we will prove that the functions α(t) = m 1 p(t) and β(t) = m 2 p(t) are lower and upper solutions of BVP given by (1) respectively, where By Lemma 2.1, we have Now, we noting that Combining the conclusion of Lemma 2.2 and (18), we get According to the assumption and from (19), we obtain and Thus, from assumption (A 2 ) and (19) to (22), we have and The inequalities (23) and (24), lead to Hence, α(t) = m 1 p(t) and β(t) = m 2 p(t) satises the BVP given by (1). Therefore, α(t) = m 1 p(t) and β(t) = m 2 p(t) are lower and upper solutions of BVP given by (1), respectively.
Now, we consider the operator T : (s − r)p(s, u(s))dsdr, where G 1 (t, r) is as in (11). It is clear that the operator T is continuous in C[0, 1]. Since, according to the assumption the function f (t, u(t)) is non-decreasing in u, and we know that, for any u(t) ∈ C[0, 1], Hence, there exists a positive constant M such that | p(t, u(t)) |≤ M for any u(t) ∈ C[0, 1], which implies that the operator T is uniformly bounded. Moreover, for all u(t) ∈ C[0, 1] and 0 ≤ t 1 < t 2 ≤ 1, we have which implies that the operator T is equicontinuous. So, by the well-known Arzela-Ascoli theorem [12,21], we can say that the operator T is compact. Consequently, by Theorem 2.3 (Schauder's xed point theorem [20]), the operator T must have a xed point and this ensure that the BVP given by (26) has a solution. This completes the step-2. Step-3.
In this step, we will prove that BVP given by (1) has a positive solution. Let u + (t) be a solution of the BVP given by (26). Since, according to assumption the function f (t, u(t)) is non-decreasing in u and we know that, Hence, if we apply Remark 2.1 and Remark 2.3 in the following BVP then we have, β(t) − u + (t) = v(t) ≥ 0, but v (t) 0, i.e., u + (t) ≤ β(t) for all t ∈ [0, 1]. Similarly, we can prove that α(t) ≤ u + (t) for t ∈ [0, 1]. Therefore u + (t) is a positive solution of the BVP given by (1). This completes the proof.
The Theorem 3.1 leads the following corollary: Corollary 3.1. If the assumption (A 2 ) of Theorem 3.1 is satised and we replace the BVCs of BVP given by (1) by the Lidstone BVCs then the BVP given by (1) has at least one positive solution. Now we give an example to justify our Theorem 3.1.

Conclusion
In this study, we established the existence result of positive solution for a NLFOODE with four-point BVCs by the help of Upper and Lower solution method and Schauder's xed point theorem (Theorem 2.3).
Here we improved the result of Chen et al. [9] by correcting their key lemma (Lemma 2.2 of Chen et al. [9]) and we made this correction by using Remark 2.1, Remark 2.2 and Remark 2.3. As the considered NLFOODE with four-point BVCs of this paper represents a beam equation, so we can conclude that the result of this paper will play a vital role to check the existence of positive solution of this type of beam equations. A justifying example also discussed here.

Competing Interests
The author declares that he has no any competing interests.