Existence and Nonexistence of Positive Solutions for a Higher-Order Three-Point Boundary Value Problem

Abstract

This paper is concerned with the existence and nonexistence of positive solutions for a nonlinear higher-order three-point boundary value problem. The existence results are obtained by applying a fixed point theorem of cone expansion and compression of functional type due to Avery, Henderson, and O’Regan.

1. Introduction

We consider the existence and nonexistence of positive solution to the nonlinear higher-order three-point boundary value problem (BVP for short): with , , are constants with . is continuous. Here, by a positive solution of BVP (1) we mean a function satisfying (1) and for . If , BVP (1) reduces to so-called boundary value problem which has been considered by many authors. For example, Agarwal et al. [1] considered the existence of positive solutions for the singular boundary value problem. Baxley and Houmand [2] considered the existence of multiple positive solutions for the boundary value problem. Yang [3] obtained some new upper estimates to positive solutions for the problem. Existence and nonexistence results for positive solutions of the problem were obtained by using the Krasnosel'skii fixed point theorem.

In recent years, the existence and multiplicity of positive solutions for nonlinear higher-order ordinary differential equations with three-point boundary conditions have been studied by several authors; we can refer to [4–19] and the references therein. For example, Eloe and Ahmad in [4] discussed the existence of positive solutions of a nonlinear th-order three-point boundary value problem where , . The existence of at least one positive solution if is either superlinear or sublinear was established by applying the fixed point theorem in cones due to Krasnosel'skii. Under conditions different from those imposed in [4], Graef and Moussaoui in [5] studied the existence of both sign changing solutions and positive solutions for BVP (2). Hao et al. [6] are devoted to the existence and multiplicity of positive solutions for BVP (2) under certain suitable weak conditions, where may be singular at and/or . The main tool used is also the Krasnosel'skii fixed point theorem. Graef et al. in [7, 8] considered the higher-order three-point boundary value problem where is an integer and is a constant. Sufficient conditions for the existence, nonexistence, and multiplicity of positive solutions of this problem are obtained by using Krasnosel'skii's fixed point theorem, Leggett-Williams' fixed point theorem, and the five-functional fixed point theorem.

Let be a real Banach space with norm and be a cone of . Recently, Zhang et al. [9] studied a higher order three-point boundary value problem where , , , and is the zero element of . By using the fixed-point principle in cone and the fixed-point index theory for strict-set-contraction operator, the authors obtained the existence, nonexistence, and multiplicity of positive solutions for the nonlinear three-point boundary value problems of th-order differential equations in ordered Banach spaces.

It is well known that fixed point theorems have been applied to various boundary value problems to show the existence and multiplicity of positive solutions. Fixed point theorems and their applications to nonlinear problems have a long history; the recent book by Agarwal et al. [20] contains an excellent summary of the current results and applications. Recently, Avery et al. [21] generalized the fixed point theorem of cone expansion and compression of norm type by replacing the norms with two functionals satisfying certain conditions to produce a fixed point theorem of cone expansion and compression of functional type, and then they applied the fixed point theorem to verify the existence of a positive solution to a second order conjugate boundary value problem.

Motivated greatly by the above-mentioned works, in this paper we will try using this new fixed point theorem to consider the existence of monotone positive solution to BVP (1). The methods used to prove the existence results are standard; however, their exposition in the framework of BVP (1) is new. This paper is organized as follows. In Section 2, we present some definitions and background results on cones and completely continuous operators. We also state the fixed point theorem of cone expansion and compression of functional type due to Avery, Henderson, and O'Regan. Expression and properties of Green's function will be given in Section 3. The main results will be given in Section 4.

2. Preliminaries

In this section, for the convenience of the reader, we present some definitions and background results on cones and completely continuous operators. We also state a fixed point theorem of cone expansion and compression of functional type due to Avery, Henderson, and O'Regan.

Definition 1. Let be a real Banach space. A nonempty closed convex set is called a cone 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 the map is sublinear functional if

All the concepts discussed above can be found in [22].

Property A1. Let be a cone in a real Banach space and be 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, if , and ;(b) is sublinear, if , and ;(c) is concave and unbounded.

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

The approach used in proving the existence results in this paper is the following fixed point theorem of cone expansion and compression of functional type due to Avery et al. [21].

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

3. Expression and Properties of Green's Function

In this section we present the expression and properties of Green's function associated with BVP (1). We will consider the Banach space equipped with norm . In arriving our result, we need the following two preliminary lemmas.

Lemma 5. Let , then boundary value problem has a unique solution where

Proof. One can reduce equation to an equivalent integral equation for some . By , it follows . Thus, Now we solve for by . It follows from that from which we get Therefore, substituting into (14), one has where is defined by (10). This completes the proof.

Lemma 6. Suppose . Then Green's function defined by (10) has the following properties: (a) is continuous on ;(b), for all ;(c), for all ;(d); (e)(f).

Proof. (a) This conclusion holds obviously, so we omit the proof here.
(b) First we prove In fact, If , obviously, . If , from (11) we know Now we prove In fact, if , obviously, (21) holds. If , one has which implies that (21) is also true. Therefore, By (10), (20), and (21), we find The proof of (b) is completed.
(c) From (b) we know that for any . Thus is increasing in , so On the other hand, if , then from (11), we have If , then from (11), we have Thus from (25) and (26), we obtain that From (10) and (27), one has The equations (d), (e), and (f) can be verified by direct calculations. Then the proof is completed.

Define the cone by Then is a normal cone of . Define the operator by Thus, to solve BVP (1), we only need to find a fixed point of the operator in .

Lemma 7. The operator defined in (30) is completely continuous and satisfies .

Proof. By Lemma 6(b) and (c), . is completely continuous by an application of Arzela-Ascoli theorem.

Finally, let us define two continuous functionals and on the cone by It is clear that for all .

4. Main Results

Consider firstly the existence of positive solution for BVP (1).

Theorem 8. Suppose that there exists positive numbers with such that the following conditions are satisfied: (A1), for all ,(A2), for all . Then BVP (1) admits a positive solution such that

Proof. Let we have that , , and being bounded open subsets of . Let , then we have which implies that ; that is, , thus .
Claim 1. If , then .
To see this let , then . Thus it follows from (A1), Lemma 6(f), and (30), one has
Claim 2. If , then .
To see this let , then . Thus condition (A2) and Lemma 6(d) yield
Clearly satisfies Property A1(c) and satisfies Property A2(a). Therefore the hypothesis (K1) of Theorem 4 is satisfied, and hence has at least one fixed point ; that is, BVP (1) has at least one positive solution such that This completes the proof.

Theorem 9. Suppose that there exist positive numbers with such that the following conditions are satisfied: (A3),  for ,(A4),  for . Then BVP (1) has at least one positive solution satisfying

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 (A3) and Lemma 6(d) yield
Claim 2. If , then .
To see this let , then . Thus it follows from (A4) and Lemma 6(f), one has
Clearly satisfies Property A1(c) and satisfies Property A2(a). Therefore the hypothesis (K2) of Theorem 4 is satisfied, and hence has at least one fixed point ; that is, BVP (1) has at least one positive solution such that This completes the proof.

Now we discuss nonexistence of positive solutions of BVP (1).

Theorem 10. Suppose that satisfies the condition Then BVP (1) does not admit positive solutions.

Proof. Assume to the contrary that is a positive solution of BVP (1), then from Lemma 6(d) and (43) we get This is a contradiction and completes the proof.

Theorem 11. Suppose that satisfies the condition then BVP (1) does not admit positive solutions.

Proof. Assume to the contrary that is a positive solution of BVP (1), then by the definition of the cone , we have for . Thus from Lemma 6(e) and (45) we get This is a contradiction and completes the proof.

Conflict of Interests

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

Acknowledgments

The authors gratefully acknowledge the referee for his/her valuable comments and suggestions. This work was supported financially by the Natural Science Foundation of Zhejiang Province of China (Y12A01012), the National Natural Science Foundation of China (11371221, 11071141), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.