Abstract
Keywords:
monotone positive solutions; multiplicity; higher order differential equations; nonlocal boundary conditions1 Introduction
In this paper, we are concerned with the existence of multiple monotone positive solutions for the higher order differential equation
subject to the following integral boundary conditions:
where , is continuous in which , A and B are right continuous on , left continuous at , and nondecreasing on , with ; and denote the RiemannStieltjes integrals of v with respect to A and B, respectively.
Boundary value problems (BVPs for short) for nonlinear differential equations arise in many areas of applied mathematics and physics. Many authors have discussed the existence of positive solutions for second order or higher order differential equations with boundary conditions defined at a finite number of points, for instance, [112]. In [2], Graef and Yang considered the following nthorder multipoint BVP:
where , is a parameter, g and f are continuous functions, , for and . The authors obtained the existence and nonexistence results of positive solutions by using Krasnosel’skii’s fixed point theorem in cones. In [5], we studied the following second order mpoint nonhomogeneous BVP:
where , (), , . The authors obtained the existence, nonexistence and multiplicity of positive solutions by using the Krasnosel’skiiGuo fixed point theorem, the upperlower solutions method and topological degree theory.
Boundary value problems with integral boundary conditions for ordinary differential equations represent a very interesting and important class of problems and arise in the study of various physical, biological and chemical processes [1315], such as heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics. They include two, three, multipoint and nonlocal BVPs as special cases. The existence and multiplicity of positive solutions for such problems have received a great deal of attention, see [1632] and the references therein. In [17], Feng, Ji and Ge considered the existence and multiplicity of positive solutions for a class of nonlinear boundary value problems of second order differential equations with integral boundary conditions in ordered Banach spaces
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 twopoint, threepoint, multipoint 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 [2] and some other known results to some degree. In this work we shall also utilize the following fixed point theorem in cones.
LetKbe a cone in a Banach spaceE. LetDbe an open bounded subset ofEwithand. Assume thatis a compact operator such thatfor. Then the following results hold.
(2) If there existssuch thatfor alland, then.
(3) LetUbe open inEsuch that. Ifand, thenAhas a fixed point in. The same result holds ifand.
2 Preliminary lemmas
Let , then E is a Banach space with the norm for each .
We make the following assumptions:
Lemma 2.1Assume that () holds. Then, for any, the BVP
has a unique solutionuthat can be expressed in the form
where
Proof Firstly, we prove that if u is a solution of BVP (2.1), then it will take the form of (2.2). Now, integrating differential equation (2.1) from 0 to t twice, we have
Substituting the boundary conditions of (2.1) and (2.5) into (2.4) yields
and, consequently,
Solving the above two equations for and , we have
and so
Hence, (2.2) follows from (2.6) and (2.7).
Next we prove that the u given by (2.2) satisfies the differential equation and boundary conditions of (2.1). From (2.2), we have
Direct differentiation of (2.8) gives . Also, from (2.2) we have
and, similarly,
Therefore, by solving the above two equations with the double integrals as unknowns, we have
and
Hence (2.6) follows from (2.2), (2.9) and (2.10), and thus , . This completes the proof. □
Defining
then is the Green function of BVP (1.1) and (1.2). Moreover, solving BVP (1.1) and (1.2) is equivalent to finding a solution of the following integral equation:
Remark 2.1 If () holds, then for any , it is easy to testify that
Lemma 2.2Let, then for any, , we have
Proof It is easy to show that , , . For , , we have
For any , we define . From Lemma 2.2, we know that
□
Lemma 2.3Assume that () holds. Ifsatisfies the boundary conditions (1.2) and
then
Thus
i.e.,
On the other hand,
and so
Now, , and is concave downward, so we have
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).
Let
Obviously, K is a cone in E. For any , let , and . Define an operator as follows:
Then u is a solution of BVP (1.1) and (1.2) if and only if u solves the operator equation .
Lemma 2.4Suppose that () and () hold, thenis completely continuous.
Proof For all , , by (), (2.11), (2.12) and (2.15), we have
and
Thus, further from the first inequality of (2.12), we have
Next by standard methods and the AscoliArzela theorem, one can prove that is completely continuous. So this is omitted. □
Let
Proceeding as for the proof of Lemma 2.5 in [33], we have the following.
Lemma 2.5has the following properties:
Now for convenience we introduce the following notations:
To prove our main results, we need the following lemmas.
Lemma 2.6Assume that (), () hold andfsatisfies
Proof For , we have and , , . Then by (2.16) we have, for ,
This implies that for . By the point (1) in Lemma 1.1, we have . □
Lemma 2.7Assume that (), () hold andfsatisfies
Proof Let , , then with . Next we prove that
In fact, if not, then there exist and such that . By (2.17) and the point (d) in Lemma 2.5, we have, for ,
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 . □
3 Main results
In the following, we shall give the main results on the existence of multiple positive solutions of BVP (1.1) and (1.2).
Theorem 3.1Suppose that () and () are satisfied. In addition, assume that one of the following conditions holds.
() There existwithandsuch that
Then BVP (1.1) and (1.2) has two nondecreasing positive solutions, inK. 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 3.1If there existssuch that one of the following conditions holds:
then BVP (1.1) and (1.2) has at least two nondecreasing positive solutions inK.
Proof We show that () implies (). It is easy to verify that implies that there exists such that . Let , by , there exists such that for , . Let
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 [2] to some degree.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
XH wrote the first manuscript and LL corrected and improved the final version. Both authors read and approved the final manuscript.
Acknowledgements
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).
References

Eloe, PW, Ahmad, B: Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions. Appl. Math. Lett.. 18, 521–527 (2005). Publisher Full Text

Graef, JR, Yang, B: Positive solutions to a multipoint higher order boundary value problem. J. Math. Anal. Appl.. 316, 409–421 (2006). Publisher Full Text

Graef, JR, Henderson, J, Wong, PJY, Yang, B: Three solutions of an nth order threepoint focal type boundary value problem. Nonlinear Anal.. 69, 3386–3404 (2008). Publisher Full Text

Hao, X, Liu, L, Wu, Y: Positive solutions for second order differential systems with nonlocal conditions. Fixed Point Theory. 13, 507–516 (2012)

Hao, X, Liu, L, Wu, Y: On positive solutions of mpoint nonhomogeneous singular boundary value problem. Nonlinear Anal.. 73, 2532–2540 (2010). Publisher Full Text

Henderson, J, Luca, R: On a system of secondorder multipoint boundary value problems. Appl. Math. Lett.. 25, 2089–2094 (2012). Publisher Full Text

Karaca, IY: Positive solutions of an nth order multipoint boundary value problem. J. Comput. Anal. Appl.. 14, 181–193 (2012)

Ma, R: Existence of positive solutions for superlinear semipositone mpoint boundaryvalue problems. Proc. Edinb. Math. Soc.. 46, 279–292 (2003). Publisher Full Text

Hao, X, Xu, N, Liu, L: Existence and uniqueness of positive solutions for fourthorder mpoint boundary value problems with two parameters. Rocky Mt. J. Math.. 43, 1161–1180 (2013). Publisher Full Text

Zhang, X: Eigenvalue of higherorder semipositone multipoint boundary value problems with derivatives. Appl. Math. Comput.. 201, 361–370 (2008). Publisher Full Text

Zhang, X, Liu, L: A necessary and sufficient condition of positive solutions for nonlinear singular differential systems with fourpoint boundary conditions. Appl. Math. Comput.. 215, 3501–3508 (2010). Publisher Full Text

Zhang, X, Liu, L: Positive solutions of fourorder multipoint boundary value problems with bending term. Appl. Math. Comput.. 194, 321–332 (2007). Publisher Full Text

Gallardo, JM: Second order differential operators with integral boundary conditions and generation of semigroups. Rocky Mt. J. Math.. 30, 1265–1292 (2000). Publisher Full Text

Karakostas, GL, Tsamatos, PC: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundaryvalue problems. Electron. J. Differ. Equ.. 2002, 1–17 (2002)

Lomtatidze, A, Malaguti, L: On an nonlocal boundaryvalue problems for second order nonlinear singular differential equations. Georgian Math. J.. 7, 133–154 (2000)

Boucherif, A: Secondorder boundary value problems with integral boundary conditions. Nonlinear Anal.. 70, 364–371 (2009). Publisher Full Text

Feng, M, Ji, D, Ge, W: Positive solutions for a class of boundaryvalue problem with integral boundary conditions in Banach spaces. J. Comput. Appl. Math.. 222, 351–363 (2008). Publisher Full Text

Hao, X, Liu, L, Wu, Y, Sun, Q: Positive solutions for nonlinear nthorder singular eigenvalue problem with nonlocal conditions. Nonlinear Anal.. 73, 1653–1662 (2010). Publisher Full Text

Hao, X, Liu, L, Wu, Y, Xu, N: Multiple positive solutions for singular nthorder nonlocal boundary value problem in Banach spaces. Comput. Math. Appl.. 61, 1880–1890 (2011). Publisher Full Text

Infante, G, Webb, JRL: Nonlinear nonlocal boundaryvalue problems and perturbed Hammerstein integral equations. Proc. Edinb. Math. Soc.. 49, 637–656 (2006). Publisher Full Text

Jiang, J, Liu, L, Wu, Y: Secondorder nonlinear singular SturmLiouville problems with integral boundary conditions. Appl. Math. Comput.. 215, 1573–1582 (2009). Publisher Full Text

Kang, P, Wei, Z: Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear secondorder ordinary differential equations. Nonlinear Anal.. 70, 444–451 (2008)

Kong, L: Second order singular boundary value problems with integral boundary conditions. Nonlinear Anal.. 72, 2628–2638 (2010). Publisher Full Text

Ma, H: Symmetric positive solutions for nonlocal boundary value problems of fourth order. Nonlinear Anal.. 68, 645–651 (2008). Publisher Full Text

Webb, JRL: Nonlocal conjugate type boundary value problems of higher order. Nonlinear Anal.. 71, 1933–1940 (2009). Publisher Full Text

Webb, JRL, Infante, G: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc.. 74, 673–693 (2006). Publisher Full Text

Webb, JRL, Infante, G: Positive solutions of nonlocal boundary value problems involving integral conditions. Nonlinear Differ. Equ. Appl.. 15, 45–67 (2008). Publisher Full Text

Liu, L, Hao, X, Wu, Y: Positive solutions for singular second order differential equations with integral boundary conditions. Math. Comput. Model.. 57, 836–847 (2013). Publisher Full Text

Yang, Z: Existence and nonexistence results for positive solutions of an integral boundary value problem. Nonlinear Anal.. 65, 1489–1511 (2006). Publisher Full Text

Yang, Z: Existence of nontrivial solutions for a nonlinear SturmLiouville problem with integral boundary conditions. Nonlinear Anal.. 68, 216–225 (2008). Publisher Full Text

Zhang, X, Feng, M, Ge, W: Symmetric positive solutions for pLaplacian fourthorder differential equations with integral boundary conditions. J. Comput. Appl. Math.. 222, 561–573 (2008). Publisher Full Text

Zhang, X, Han, Y: Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations. Appl. Math. Lett.. 25, 555–560 (2012). Publisher Full Text

Lan, KQ: Multiple positive solutions of semilinear differential equations with singularities. J. Lond. Math. Soc.. 63, 690–704 (2001). Publisher Full Text

Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Academic Press, New York (1988)