- Open Access
Multiple monotone positive solutions for higher order differential equations with integral boundary conditions
© Hao and Liu; licensee Springer. 2014
- Received: 17 January 2014
- Accepted: 19 March 2014
- Published: 28 March 2014
This paper investigates the higher order differential equations with nonlocal boundary conditions
The existence results of multiple monotone positive solutions are obtained by means of fixed point index theory for operators in a cone.
- monotone positive solutions
- higher order differential equations
- nonlocal boundary conditions
where , is continuous in which , A and B are right continuous on , left continuous at , and nondecreasing on , with ; and denote the Riemann-Stieltjes integrals of v with respect to A and B, respectively.
where , (), , . The authors obtained the existence, nonexistence and multiplicity of positive solutions by using the Krasnosel’skii-Guo fixed point theorem, the upper-lower solutions method and topological degree theory.
The arguments are based upon a specially constructed cone and fixed point theory in a cone for strict set contraction operators.
Motivated by the works mentioned above, in this paper, we consider the existence of multiple monotone positive solutions for BVP (1.1) and (1.2). In comparison with previous works, our paper has several new features. Firstly, we consider higher order boundary value problems, and we allow the nonlinearity f to contain derivatives of the unknown function up to order. Secondly, we discuss the boundary value problem with integral boundary conditions, i.e., BVP (1.1) and (1.2), which includes two-point, three-point, multi-point and nonlocal boundary value problems as special cases. Thirdly, we consider the existence of multiple monotone positive solutions. To our knowledge, few papers have considered the monotone positive solutions for a higher order differential equation with integral boundary conditions. We shall emphasize here that with these new features our work improves and generalizes the results of  and some other known results to some degree. In this work we shall also utilize the following fixed point theorem in cones.
If , , then .
If there exists such that for all and , then .
Let U be open in E such that . If and , then A has a fixed point in . The same result holds if and .
Let , then E is a Banach space with the norm for each .
We make the following assumptions:
() is continuous.
Hence, (2.2) follows from (2.6) and (2.7).
Hence (2.6) follows from (2.2), (2.9) and (2.10), and thus , . This completes the proof. □
i.e., , therefore , and so .
From (2.14) and , we obtain (2.13). This completes the proof of Lemma 2.3. □
Remark 2.2 From Lemma 2.3, if u is a positive solution of BVP (1.1) and (1.2), then u is nondecreasing on , i.e., u is a monotone positive solution of BVP (1.1) and (1.2).
Then u is a solution of BVP (1.1) and (1.2) if and only if u solves the operator equation .
Lemma 2.4 Suppose that () and () hold, then is completely continuous.
Hence, and .
Next by standard methods and the Ascoli-Arzela theorem, one can prove that is completely continuous. So this is omitted. □
Proceeding as for the proof of Lemma 2.5 in , we have the following.
is open relative to K;
if and only if ;
if , then for .
To prove our main results, we need the following lemmas.
This implies that for . By the point (1) in Lemma 1.1, we have . □
This implies that , and so by the point (c) in Lemma 2.5, this is a contradiction. It follows from the point (2) of Lemma 1.1 that . □
In the following, we shall give the main results on the existence of multiple positive solutions of BVP (1.1) and (1.2).
Theorem 3.1 Suppose that () and () are satisfied. In addition, assume that one of the following conditions holds.
Then BVP (1.1) and (1.2) has two nondecreasing positive solutions , in K. Moreover, if in () is replaced by , then BVP (1.1) and (1.2) has a third nondecreasing positive solution .
Proof Assume that () holds. We show that either T has a fixed point or in . If for , by Lemmas 2.6 and 2.7, we have , , and . By Lemma 2.5(b), we have since . It follows from Lemma 1.1(3) that T has a fixed point . Similarly, T has a fixed point . The proof is similar when () holds. □
Corollary 3.1 If there exists such that one of the following conditions holds:
() , , for , ,
() , , for , ,
then BVP (1.1) and (1.2) has at least two nondecreasing positive solutions in K.
This implies that and () holds. Similarly, () implies (). This completes the proof. □
Remark 3.1 We establish the multiplicity of monotone positive solutions for a higher order differential equation with integral boundary conditions, and we allow the nonlinearity f to contain derivatives of the unknown function up to order, so our work improves and generalizes the results of  to some degree.
Research supported by the National Natural Science Foundation of China (11371221, 11201260), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20123705120004, 20123705110001), a Project of Shandong Province Higher Educational Science and Technology Program (J11LA06) and Foundation of Qufu Normal University (BSQD20100103).
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