ftp ejde.math.txstate.edu (login: ftp) POSITIVE SOLUTIONS OF A NONLINEAR HIGHER ORDER BOUNDARY-VALUE PROBLEM

The authors consider the higher order boundary-value problem u (n) (t) = q(t)f(u(t)), 0 t 1, where n 4 is an integer, and p 2 (1/2,1) is a constant. Sucient conditions for the existence and nonexistence of positive solutions of this problem are obtained. The main results are illustrated with an example.

The importance of boundary-value problems in a wide variety of applications in the physical, biological and engineering sciences is now well documented in the literature, and in the last ten years this has become an extremely active area of research.The monographs of Agarwal [1] and Agarwal, O'Regan, and Wong [3] contain excellent surveys of known results.Recent contributions to the study of multipoint boundary-value problems can be found in the papers of Agarwal and Kiguradze [2], Anderson and Davis [4], Cao and Ma [5], Graef, Henderson and Yang [6], Graef, Qian, and Yang [7,8], Graef and Yang [9,10], Hu and Wang [12], Infante [13], Infante and Webb [14], Kong and Kong [15], Ma [17,18,19], Maroun [20], Raffoul [21], Wang [22], Webb [23,24], and Zhou and Xu [25].The three-point boundary conditions considered here, namely, conditions (1.2) above, have been used by many authors in the study of existence of positive solutions of second order problems.Here, we use these conditions but for problems involving higher order (n ≥ 4) differential equations. Let For n ≥ 4, we define Then, for n ≥ 4, G n (t, s) is the Green's function for the equation subject to the boundary conditions (1.2).Moreover, solving the problem (1.1)-(1.2) is equivalent to finding a solution to the integral equation
Throughout this paper, we let To prove our results, we will use the following fixed point theorem known as the Guo-Krasnosel'skii fixed point theorem [11,16].
Theorem 1.1.Let X be a Banach space over the reals, and let P ⊂ X be a cone in X. Assume that Ω 1 and Ω 2 are bounded open subsets of X with 0 ∈ Ω 1 ⊂ Ω 1 ⊂ Ω 2 , and let L : P ∩ ( Ω 2 − Ω 1 ) → P be a completely continuous operator such that, either one of the following two conditions hold.
The next section contains some preliminary lemmas; our main results appear in Sections 3 and 4.

Preliminary Lemmas
The following lemmas will be used in the proofs of our main results.
The next two lemmas give estimates on the growth of u(t).
2) and (2.1), then Proof.If we define (2.10) To prove the lemma, it suffices to show that h(t) ≥ 0 for 0 ≤ t ≤ 1.It is easy to see from (2.9) that By the Mean Value Theorem, in view of the fact that h(0) = h(1) = 0, there exists r 1 ∈ (0, 1) such that h (r 1 ) = 0.Because h (0) = h (r 1 ) = 0, there exists r 2 ∈ (0, r 1 ) such that h (r 2 ) = 0.If we continue this procedure, then we can find a sequence of numbers We can also see from (2.9) that Therefore, we have This means that h (n−4) (t) is nonincreasing.Since h (n−4) (r n−4 ) = 0, we have If we continue this procedure, we finally obtain (2.11) Combining (2.11) with the fact that h(0) = h(1) = 0 yields which completes the proof of the lemma.
The next theorem is a direct consequence of Lemmas 2.1, 2.2, and 2.3.
Note that Theorem 2.4 provides both an upper and a lower estimate to each positive solution to the problem (1.1)-(1.2).

Existence of Positive Solutions
We begin by introducing some notation.Define and let Obviously X is a Banach space and P is a positive cone of X. Define the operator T : P → X by By a standard argument we can show that T : P → X is a completely continuous operator.It is obvious that if u ∈ P , then u(1) = u .We see from Theorem 2.4 that if u(t) is a nonnegative solution to the problem (1.1)-(1.2),then u ∈ P .In a similar fashion to the proof of Theorem 2.4, we can show that T (P ) ⊂ P .To find a positive solution to the problem (1.1)-(1.2),we only need to find a fixed point u of T such that u ∈ P and u(1) = u > 0.
We now give our first existence result.
2) has at least one positive solution.
Proof.Choose ε > 0 such that (F 0 + ε)B ≤ 1.There exists H 1 > 0 such that For each u ∈ P with u = H 1 , we have Next we construct Ω 2 .Since 1 < Af ∞ , we can choose c ∈ (0, 1/4) and δ > 0 such that There exists Since the condition (K1) of Theorem 1.1 is satisfied, there exists a fixed point of T in P , and this completes the proof of the theorem.
2) has at least one positive solution.

Nonexistence Results and Example
In this section, we establish some nonexistence results for the positive solutions of the problem (1.1)-(1.2).Theorem 4.1.Suppose that (H1) and (H2) hold.If Bf (x) < x for all x > 0, then problem (1.1)-(1.2) has no positive solutions.
Proof.Assume to the contrary that u(t) is a positive solution of the problem (1.1)-(1.2).Then u ∈ P , u(t) > 0 for 0 < t ≤ 1, and which is a contradiction.
Similarly, we have the following result.The proof of Theorem 4.2 is quite similar to that of Theorem 4.1 and is therefore omitted.
In [6], the present authors considered this same boundary-value problem and obtained sufficient conditions for the existence of at least one positive solution and sufficient conditions for there to be no positive solutions.The approach used in [6] was an adaptation of the technique used in [10].The following example not only illustrates the main results in this paper but in fact shows that the results here are better than those obtained in [6].

Example 4 . 3 .
Consider the boundary-value problem u