Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations

We study a three-point nonlinear boundary value problem with higher-order p-Laplacian. We show that there exist countable many positive solutions by using the fixed point index theorem for operators in a cone.

In recent years, because of the wide mathematical and physical backgrounds [7,8], the existence of positive solutions for nonlinear boundary value problems with p-Laplacian received wide attention.Especially, when p = 2, the existence of positive solutions for nonlinear singular boundary value problems has been obtained (see [5,6,10]); when p = 2 and the nonlinearities are continuous, many results of the existence of positive solutions 2 Positive solutions of a three-point BVP have been obtained [1][2][3][4]9] by using comparison results, topological degree theorem, respectively.Recently, on the existence of positive solutions of multipoint boundary value problems for second-order ordinary differential equation, some authors have obtained the existence results (see [5][6][7][8]10]).However, all of the above-mentioned references dealt with the case of the nonlinearity without singularities.For the singular case of multipoint boundary value problems, to our acknowledge, no one has studied the existence of positive solutions in this case.
Very recently, Kaufmann and Kosmatov [3] established the result of countably many positive solutions for the two-point boundary value problems with infinitely many singularities of the following form: where a ∈ L p [0,1], p ≥ 1, and a(t) can have countably singularities on [0,1/2).Lian and Ge in [4] investigated the following boundary value problem: where φ p (s) = |s| p−2 s, p > 1, α,β,γ,δ ≥ 0, αγ + αδ + γβ > 0 and obtained that the problem has at least one positive solution by using the fixed point theorem of the compression and expansion of norm in the cone.Motivated by the results mentioned above, in this paper, we extend the results obtained in [4] to the more general three-point boundary value problems (1.1)-(1.2) which are generalization of problems (1.4).We would stress that the results presented in this paper complement and improve those obtained in [3,4], since we allow nonlinearity to have infinitely many singularities and the boundary value conditions are more general.We will show that the problems (1.1)-(1.2) have infinitely many solutions if g and f satisfy some suitable conditions.

Preliminaries and lemmas
We denote and the norm Our main tool of this paper is the following fixed point theorem of cone expansion and compression of norm type.
Lemma 2.1 [1].Suppose E is a banach space, → K be completely continuous.Suppose that one of the following two conditions holds:

Now we define a mapping
where w(t) is given by 4 Positive solutions of a three-point BVP where δ is a solution of the equation y 0 (x) = y 1 (x), here (2.4) Obviously, y 0 (x) is a nondecreasing continuous function defined on [0,1] with y 0 (0) = 0 and y 1 (x) is a nonincreasing continuous function defined on [0,1] with y 1 (1 are solutions of the equation y 0 (x) = y 1 (x), then we have (2.6) Obviously, we can obtain the following results: where θ ∈ (0,1/2) is a given constant.We can easily get the following lemmas.
6 Positive solutions of a three-point BVP

The main result
In this section, we present our main results, and also provide an example of family of functions a(t) that satifies condition (H 3 ).For convenience, we set and for each natural number k, assume that f satisfy

.13)
For each natural number k, assume that f satisfies

.18)
For each natural number k, assume that f satisfies Then, the boundary value problem (1.1), (1.2) has infinitely many solutions {u k } ∞ k=1 such that