## Complex Boundary Value Problems of Nonlinear Differential Equations 2014

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

**Academic Editor:**Xinan Hao

#### Abstract

We study the existence of positive and monotone solution to the boundary value problem , , , , where . The main tool is the fixed point theorem of cone expansion and compression of functional type by Avery, Henderson, and O’Regan. Finally, four examples are provided to demonstrate the availability of our main results.

#### 1. 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 this paper, we study the existence of positive and monotone solution for the second-order two-point boundary value problem where is continuous function and are two constants. The boundary conditions in problem (1) are closely related to some other boundary conditions. If , the boundary conditions in problem (1) reduce to periodic boundary conditions. If , the boundary conditions in problem (1) reduce to antiperiodic boundary conditions. If , problem (1) reduces to second-order right focal boundary value problem. 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.

#### 2. 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), , implies ;(2), , implies .

Every cone induces an ordering in given by if and only if .

*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 is continuous and
Similarly we said the map is a nonnegative continuous convex functional on a cone of a real Banach space if is continuous and

We say that the map is sublinear functional if All the concepts discussed above can be found in [3].

*Property A1*. Let be a cone in a real Banach space and a bounded open subset of with . Then a continuous functional is said to satisfy Property A1 if one of the following conditions holds:(a) is convex, , and if and ,(b) is sublinear, , and if and ,(c) is concave and unbounded.

*Property A2*. Let be a cone in a real Banach space and a bounded open subset of with . Then a continuous functional is said to satisfy Property A2 if one of the following conditions hold:(a) is convex, , and if ,(b) is sublinear, , and if ,(c) for all , , and if .

To prove our results, we will need the following fixed point theorem, which is presented by Avery et al. [17].

Theorem 4. *Let and be two bounded open sets in a Banach space such that and and is a cone in . Suppose that is a completely continuous operator, and are nonnegative continuous functional on , and one of the two conditions* *(K1) satisfies Property A1 with , for all , and satisfies Property A2 with , for all ; or* *(K2) satisfies Property A2 with , for all , and satisfies Property A1 with , for all **is satisfied. Then has at least one fixed point in .*

To study problem (1), we need the following lemmas (see [15]).

Lemma 5. *Green’s function for the BVP is given by
*

Lemma 6. *Suppose that . Then Green’s function defined by (5) has the following properties:*(a)*, , ;*(b)*, ;*(c)*;
*(d)* .*

Lemma 7. *Suppose that , . Then Green’s function defined by (5) has the following properties: *(a)*, , ;*(b)*, ;*(c)*;
*(d)* .*

#### 3. Main Results

In this section, we will apply Theorem 4 to study the existence of positive and monotonic solution to problem (1).

##### 3.1. Case I:

In this case we define the cone by Then is a normal cone of . Define the operator by Then by Lemma 6 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 .

Theorem 8. *Suppose that and there exist , with such that the following conditions are satisfied: * *(A1) , for all ;* *(A2) , for all .**Then problem (1) admits a positive and increasing solution such that
*

*Proof. *Let
it is easy to see that , and and are bounded open subsets of . Let ; then we have
Thus ; that is, , so . *Claim **1* (if , then ). To see this let ; then , . It follows from and Lemmas 6 and (7) that
*Claim **2* (if , then ). To see this let ; then , . Thus condition and Lemma 6(c) yield that
Clearly satisfies Property A1(c) and satisfies Property A2(a). Therefore hypothesis of Theorem 4 is satisfied and hence has at least one fixed point ; that is, problem (1) has at least one positive solution such that
This completes the proof.

Theorem 9. *Suppose that and there exist with such that the following conditions are satisfied: * *(A3) , for ;* *(A4) , for .**Then problem (1) admits a positive and increasing solution such that
*

*Proof. *For all we have . Thus if we let
we have that and , with and being bounded open subsets of .*Claim **1* (if , then ). To see this let ; then , . Thus condition and Lemma 6 yield that
*Claim **2* (if , then ). To see this let ; then , . Thus it follows from and Lemma 6 that
Clearly satisfies Property A1(c) and satisfies Property A2(a). Therefore hypothesis of Theorem 4 is satisfied and hence has at least one fixed point ; that is, problem (1) has at least one positive solution such that
This completes the proof.

##### 3.2. Case II:

In this case we define the cone by 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 .

Theorem 10. *Suppose that and there exist , with such that the following conditions are satisfied: **(B1) , for all ;**(B2) , for all .**Then problem (1) admits a positive and decreasing solution such that
*

*Proof. *Letting
then , , and are bounded open subsets of . Let ; then we have
Thus ; that is, , so .*Claim **1* (if , then ). To see this let ; then , . Thus it follows from and Lemmas 7 and (7) that
*Claim **2* (if , then ). To see this let ; then , . Thus condition and Lemma 7 yield
Clearly satisfies Property A1(c) and satisfies Property A2(a). Therefore hypothesis of Theorem 4 is satisfied and hence has at least one fixed point ; that is, problem (1) has at least one positive solution such that
This completes the proof.

Theorem 11. *Suppose that , and there exist with such that the following conditions are satisfied: * *(B3) , for ;* *(B4) , for .**Then problem (1) admits a positive and decreasing solution such that
*

*Proof. *For all we have . Thus if we let
we have that and , with and being bounded open subsets of .*Claim **1* (if , then ). To see this let ; then , . Thus condition and Lemma 7 yield
*Claim **2* (if , then ). To see this let ; then , . Thus it follows from and Lemma 7 that
Clearly satisfies Property A1(c) and satisfies Property A2(a). Therefore hypothesis of Theorem 4 is satisfied and hence has at least one fixed point ; that is, problem (1) has at least one positive solution such that
This completes the proof.

#### 4. Examples

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

*Example 1. *Consider the second-order boundary value problem
In this case, . Let , , ; observe that . Setting
then , and for , we have
For , we have
Clearly, all the assumptions of Theorem 8 hold and consequently problem (33) has at least one positive and increasing solution such that

*Example 2. *Consider the second-order boundary value problem
In this case, and . Let , , , and
Then , and for ,
For ,
Hence, by Theorem 9, problem (38) has at least one positive and increasing solution such that

*Example 3. *Consider the second-order boundary value problem
In this case, . Let , , ; it is evident that . Set
Then , and for , we have
For , we have
So all conditions of Theorem 10 are satisfied and consequently problem (43) has at least one positive and decreasing solution such that

*Example 4. *Consider the second-order boundary value problem
In this problem, , and . Let , , , and
Then , and for ,
For ,
Hence, by Theorem 11, problem (48) has at least one positive and decreasing solution such that

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors thank the referee for her/his careful reading of the paper and useful suggestions. This work was supported by Zhejiang Provincial Natural Science Foundation of China (no. Y12A01012).

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#### Copyright

Copyright © 2014 Shuhong Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.