Monotone iterative technique for ( k , n − k ) conjugate boundary value problems

In this paper, a comparison result for (k, n − k) conjugate boundary value problems is established. By using the monotone iterative technique and the method of upper and lower solutions, we investigate the existence of extremal solutions for a nonlinear differential equation with (k, n− k) conjugate boundary value problems. As an application, an example is presented to illustrate the main results.


Introduction
We consider the existence of solution of the following (k, n − k) conjugate boundary value problems for nonlinear ordinary differential equations, using the method of upper and lower solutions and its associated monotone iterative technique where n ≥ 2 and k ≥ 1 are fixed integers.
The method of upper and lower solutions coupled with the monotone iterative technique plays a very important role in investigating the existence of solutions to ordinary differential equation problems, for example [3,8,11].However, as far as we know, there are no papers dealing with the existence of solutions for (k, n − k) conjugate boundary value problems, by means of the lower and upper solutions method.
The aims of this paper are to establish comparison result for (k, n − k) conjugate boundary value problems and to investigate the existence of extremal solutions of problem (1.1).
The rest of this paper is organized as follows: in Section 2, we present some useful preliminaries and lemmas.The main results are given in Section 3. In Section 4, examples are presented to illustrate the main results.
Throughout this paper, we shall use the following notation: It is well known from the papers [10,17] that G(t, s) is the Green's function of the following homogeneous boundary value problem: Lemma 2.1 ([14, 19]).The function G(t, s) defined as above has the following properties: In the rest of this paper, we also make the following assumptions: , in which we may take I i (t) = J j (t) ≡ 0 for i ∈ I + ∪ I − and j ∈ J + ∪ J − .Moreover, if can be explicitly given by where Example 2.5 ([15]).When n = 4, k = 2, the unique solution of = d can be explicitly given by where Example 2.6.When n = 5, k = 3, the unique solution of can be explicitly given by where is called an upper solutions of (k, n − k) conjugate boundary value problem if the above inequalities are reversed.
For example, u is a lower solution of (3, 2) conjugate boundary value problem if where M is a nonnegative constant and where α is given in Lemma 2.1 and B(t, s) denotes the Beta function, then (2.1) has a unique solution x, which can be expressed by where ψ(t) is given in Remark 2.2, ) All functions G n (t, s), H(t, s), Q(t, s) are continuous on [0, 1] × [0, 1] and the series on the right-hand side of (2.4) Proof.It follows from the paper [10]

1) if and only if
x ∈ C[0, 1] is a solution of the following operator equation We shall prove r(T) < 1, where r(T) denotes the spectral radius of operator T. Actually, for x ∈ C[0, 1], by Lemma 2.1, we have Hence, we have By the induction method, we have This yields that the unique solution of operator equation (2.5) is given by Substituting (2.6) into the above equality, we get (2.3) and the proof is complete. ) ) and Then σ(t) ≥ 0 and By Lemma 2.9, (2.3) holds in which ψ(t) ≥ 0 for t ∈ [0, 1].It follows from the expression of G m (t, s) that G m (t, s) ≤ 0 when m is odd and G m (t, s) ≥ 0 when m is even.Thus, we obtain for m = 3, 5, . .., by using Lemma 2.1, Consequently, we have Thus, by (2.8), we have that x(t) ≥ 0 for t ∈ [0, 1], and the lemma is proved.

Main results
In this section, we prove the existence of extremal solutions of differential equation (1.1).