Some Existence Results of Positive Solution to Second-Order Boundary Value Problems

and Applied Analysis 3 3.1. Case I: ξ, η ∈ (0, 1) u(0)=u(1)=u(0)=u(1)=0. In this case we define the cone P by P = {u ∈ C [0, 1] : u (t) ⩾ 0, u (t) is increasing on [0, 1] and u (t) ⩾ t ‖u‖ , t ∈ [0, 1]} . (6) Then P is a normal cone of E. Define the operatorT : P → E by


Introduction
Boundary value problems for ordinary differential equations play a very important role in both theory and applications.They are used to describe a large number of physical, biological, and chemical phenomena.In recent years many papers have been devoted to second-order two-point boundary value problem.For a small sample of such work, we refer the reader to the monographs of Agarwal [1], Agarwal et al. [2], and Guo and Lakshmikantham [3], the papers of Avery et al. [4] and Henderson and Thompson [5], and references therein along this line.In the literature, many attempts have been made by researchers to develop criteria which guarantee the existence and uniqueness of positive solutions to ordinary differential equations; this subject has attracted a lot of interests; see, for example, Cid et al. [6], Ehme [7], Ehme and Lanz [8], Ibrahim and Momani [9], Kong [10], Ma and An [11], Zhang and Liu [12], Zhang et al. [13], and Zhong and Zhang [14].
In a recent paper [15], by applying a fixed point theorem by Avery et al. [16], Sun studied the existence of monotone positive solutions to problem (1).In this paper we will prove some new existence results for problem (1) by using the new fixed point theorem of cone expansion and compression of functional type by Avery et al. [17].This paper is organized as follows.In Section 2 we present some notations, definitions, and lemmas.In Section 3 we establish some sufficient conditions which guarantee the existence of positive solutions to problem (1).In Section 4 we give four examples to illustrate the effectiveness and applications of the results presented in Section 3.

Preliminary Results
For the convenience of the reader, we present here the necessary definitions and background results.We also state the fixed point theorem of cone expansion and compression of functional type by Avery, Henderson, and O'Regan.Definition 1.Let  be a real Banach space.A nonempty closed convex set  ⊂  is called a  of  if it satisfies the following two conditions: (1)  ∈ ,  ⩾ 0, implies  ∈ ; (2)  ∈ , − ∈ , implies  = 0.
Every cone  ⊂  induces an ordering in  given by  ⩽ V if and only if V −  ∈ .Definition 2. Let  be a real Banach space.An operator  :  →  is said to be completely continuous if it is continuous and maps bounded sets into precompact sets.Definition 3. A map  is said to be a nonnegative continuous concave functional on a cone  of a real Banach space  if  :  → [0, +∞) is continuous and ( Similarly we said the map  is a nonnegative continuous convex functional on a cone  of a real Banach space  if  :  → [0, +∞) is continuous and We say that the map  is sublinear functional if All the concepts discussed above can be found in [3].To prove our results, we will need the following fixed point theorem, which is presented by Avery et al. [17].

Main Results
In this section, we will apply Theorem 4 to study the existence of positive and monotonic solution to problem (1).
Abstract and Applied Analysis It is clear that () ⩽ () for all  ∈ .

Case II: 𝜉,𝜂 ∈ (1,∞).
In this case we define the cone  by and Then  is a normal cone of .Define the operator  :  →  by (7).Then by Lemma 7 and Ascoli-Arzela Theorem we know that () ⊆  and  is a completely continuous operator.Let us define two continuous functionals  and  on the cone  by It is clear that () ⩽ () for all  ∈ .

Examples
At the end of the paper, we present some examples to illustrate the usefulness of our main results.