Positive Solutions for a Multi-point Eigenvalue Problem Involving the One Dimensional

A multi-point boundary value problem involving the one dimensional p-Laplacian and depending on a parameter is studied in this paper and existence of positive solutions is established by means of a fixed point theorem for operators defined on Banach spaces with cones.

The study of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev [1,2].Since then there has been much current attention focused on the study of nonlinear multipoint boundary value problems, see [3,4,5,7].The methods include the Leray-Schauder continuation theorem, nonlinear alternatives of Leray-Schauder, coincidence degree theory, and fixed point theorem in cones.For example, In [6], R. Ma and N. Castaneda studied the following BVP        x (t) + a(t)f (x(t)) = 0, 0 ≤ t ≤ 1, The authors in [7] considered the multi-point BVP for one dimensional p-Laplacian Using a fixed point theorem in a cone, we provided sufficient conditions for the existence of multiple positive solutions to the above BVP.
In paper [8], we investigated the following more general multi-point BVPs The main tool is a fixed point theorem due to Avery and Peterson [9], we provided sufficient conditions for the existence of multiple positive solutions.
In view of the common concern about multi-point boundary value problems as exhibited in [6,7,8] and their references, it is of interests to continue the investigation and study the problem Motivated by the works of [7] and [8], the aim of this paper is to show the existence of positive Our main results will depend on the following Guo-Krasnoselskii fixed-point theorem .
Theorem A [10][11].Let E be a Banach space and let K ⊂ E be a cone in are open subsets of E with 0 ∈ Ω 1 , Ω 1 ⊂ Ω 2 , and let be a completely continuous operator such that either Then T has a fixed point in K ∩ (Ω 2 \ Ω 1 ).
In section 3, we shall present some sufficient conditions with λ belonging to an open interval to ensure the existence of positive solutions to problems (1.1) and (1.2).To the author's knowledge, no one has studied the existence of positive solutions for problems (1.1) and (1.2) using the Guo-Krasnoselskii fixed-point theorem.

The preliminary lemmas
Let has a unique solution where A x , B x satisfy Proof.Define Then, then we have The zero point theorem guarantees that there exists an x 0 ∈ [x, x] such that H(x 0 ) = 0.
If there exist two constants Obviously, there exists a unique x = 0 satisfying H(x) = 0. So, where As H(x 1 ) = 0 and x 1 = 0, so there must exist Thus, we get H(x) is strictly increasing on (0, +∞).Therefore, H(x) = 0 has a unique solution on (0, +∞).Combining case 1, case 2, we obtain , then there exists a unique real number A x that satisfies (2.4).Furthermore, A x is contained in the interval −λk Proof The equation is equivalent to By Lemma 2.2, we can easily obtain So the conclusion is obvious.
Proof: According to Lemmas 2.1 and 2.3, we first have If t ∈ (0, 1), we have Furthermore, it is easy to see that u (t 2 ) ≤ u (t 1 ) for any t 1 , t 2 ∈ [0, 1] with t 1 ≤ t 2 .Hence u (t) is a decreasing function on [0,1].This means that the graph of u (t) is concave down on (0,1).For and, by means of the boundary condition u(1) This completes the proof.
EJQTDE, 2009 No. 40, p. 7 Now we define where γ is defined in Lemma 2.5.For any λ > 0, define operator By Lemmas 2.1 and 2.3, we know T λ x is well defined.Furthermore, we have the following result.

Main results
We now give our results on the existence of positive solutions of BVP(1.1) and (1.2). where 2) has at least one positive solution.
(ii).Next, since min f ∞ > 0, there exists an H 2 > 0 such that for x > H 2 , Take EJQTDE, 2009 No. 40, p. 9 Therefore, Therefore, by the first part of Theorem A, T λ has a fixed point x * ∈ K ∩ (Ω 2 \ Ω 1 ) such that It is easily checked that x * (t) is a positive solution of problems (1.1) and (1.2).
The proof is complete.
Case 1. Suppose that f is bounded, i.e., there exists N > 0 such that f (t, x) ≤ φ p (N ) for t ∈ [0, 1] and 0 ≤ x < ∞.Define and Therefore, by the second part of Theorem A, T λ has a fixed point x * ∈ K ∩ (Ω 2 \ Ω 1 ) such that