Positive solutions of three-point nonlinear second-order boundary value problem, Electron

In this paper we apply a cone theoretic xed point theorem and obtain conditions for the existence of positive solutions to the three-point nonlinear second order boundary value problem u 00 (t) + a (t)f(u(t)) = 0; t 2 (0; 1) u(0) = 0; u ( ) = u(1); where 0 < < 1 and 0 < < 1 : AMS Subject Classications: 34B20.


Introduction
In this paper, we are concerned with determining values for λ so that the three-point nonlinear second order boundary value problem u (t) + λ a(t)f (u(t)) = 0, t ∈ (0, 1) (1.1) where 0 < η < 1, (A1) the function f : [0, ∞) → [0, ∞) is continuous, (A2) a : [0, 1] → [0, ∞) is continuous and does not vanish identically on any subinterval, (L1) lim x = L with 0 < L < ∞ has positive solutions.In the case λ = 1, Ruyun Ma [11] showed the existence of positive solutions of (1.1)-(1.2) when f is superlinear (l = 0 and L = ∞), or f is sublinear (l = ∞ and L = 0).In this research it is not required that f be either sublinear or superlinear.As in [8] and [11], the arguments that we present here in obtaining the existence of a positive solution of (1.1)-(1.2),rely on the fact that solutions are concave downward.In arriving at our results, we make use of Krasnosel'skii fixed point theorem [10].The existence of positive periodic solutions of nonlinear functional differential equations have been studied extensively in recent years.For some appropriate references we refer the reader to [1], [2], [3], [4], [5], [6], [8], [9], [12], [13], [14], [15], [16] and the references therein.In section 2, we state some known results and Krasnosel'skii fixed point theorem [10].In section 3, we construct the cone of interest and present a lemma, four theorems and a corollary.In each of the theorems and the corollary, an open interval of eigenvalues is determined, which in return, imply the existence of a positive solution of (1.1)-(1.2) by appealing to Krasnosel'skii fixed point theorem.We say that u(t) is a solution of (1.1)
In arriving at our results, we need to state four preliminary Lemmas.Consider the boundary value problem u (t) + y(t) = 0, t ∈ (0, 1), (I) (2.1) The proof of (2.1) follows along the lines of the proof that is given in [7] in the case λ = 1, and hence we omit it.
The proofs of the next three lemmas can be found in [11].

Main Results
Assuming (A1) and (A2), it follows from Lemmas 2.3 and 2.4, that (1.1)-(1.2) has a non-negative solution if and only if α < 1 η .Therefore, throughout this paper we assume that α < Define a cone, P, by ).Note that, for 0 < α < 1/η, the first two terms on the right of (3.1) are less than or equal to zero.We seek a fixed point of T in the cone P.

Define an integral operator
For the sake of simplicity, we let and Lemma 3.1 Assume that (A1) and (A2) hold.If T is given by (3.1), then T : P → P and is completely continuous.
In view of A1, given an > 0 there exists a δ > 0 such that for ||φ − ψ|| < δ we have sup Using (3.1) we have for t ∈ (0, 1), Thus, T is continuous.Notice from Lemma 2.3 that, for u ∈ P, T u(t) ≥ 0 on [0, 1].Also, by Lemma 2.5, T P ⊂ P. Thus, we have shown that T : P → P. Next, we show that f maps bonded sets into bounded sets.Let D be a positive constant and define the set Since A1 holds, for any x, y ∈ K, there exists a δ > 0 such that if ||x − y|| < δ, implies We choose a positive integer N so that δ > D N .For As a consequence, we have Thus, f maps bounded sets into bounded sets.It follows from the above inequality and (3.1), that Next, for t ∈ (0, 1), we have Hence, Thus, the set {(T x) : x ∈ P, ||x|| ≤ D} is a family of uniformly bounded and equicontinuous functions on the set t ∈ [0, 1].By Ascoli-Arzela Theorem, the map T is completely continuous.This completes the proof.Proof: We construct the sets Ω 1 and Ω 2 in order to apply Theorem 2.1.Let λ be given as in (3.4), and choose > 0 such that .
Proof: We construct the sets Ω 1 and Ω 2 in order to apply Theorem 2.1.Let λ be given as in (3.7), and choose > 0 such that .
The construction of Ω 2 follows along the lines of the construction of Ω 2 in Theorem 3. Proof: Assume (L5) holds.Then, we may take the set Ω 1 to be the one obtained for Theorem 3.1.That is, Ω 1 = {y ∈ P : y < H 1 }.