Research Article | Open Access

# On Six Solutions for *m*-Point Differential Equations System with Two Coupled Parallel Sub-Super Solutions

**Academic Editor:**Rudong Chen

#### Abstract

Under the assumption of two coupled parallel subsuper solutions, the existence of at least six solutions for a kind of second-order *m*-point differential equations system is obtained using the fixed point index theory. As an application, an example to demonstrate our result is given.

#### 1. Introduction

In this paper, we consider the following second-order -point boundary value problems of nonlinear equations system where are continuous and satisfying with for and .

Multipoint boundary value problems arise in many applied sciences for example, the vibrations of a guy wire composed of parts with a uniform cross-section throughout, but different densities in different parts can be set up as a multipoint boundary value problems (see [1]). Many problems in the theory of elastic stability can be modelled by multipoint boundary value problems (see [2]). The study of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev [3]. Subsequently, Gupta [4] studied certain three-point boundary value problems for nonlinear second-order ordinary differential equations. Since then, the solvability of more general nonlinear multipoint boundary value problems has been discussed by several authors using various methods. We refer the readers to [5â€“12] and the references therein.

In the recent years, many authors have studied existence and multiplicity results for solutions of multipoint boundary value problems via the well-ordered upper and lower solutions method, see [8, 13, 14] and the references therein. However, only in very recent years, some authors considered the multiplicity of solutions under conditions of non-well-ordered upper and lower solutions. For some abstract results concerning conditions of non-well-ordered upper and lower solutions, the readers are referred to recent works [15â€“18].

In [19], Xu et al. considered the following second-order three-point boundary value problem where . He obtained the following result. First, let us give the following condition to be used later.

There exists such that Let the function be for .

Theorem 1.1. *Suppose that holds, and are two strict lower solutions of (1.2), and are two strict upper solutions of (1.2), and . Moreover, assume
**
for some . Then, the three-point boundary value problem (1.2) has at least six solutions.*

We would also like to mention the result of Yang [20], in [20]. Yang studied the following integral boundary value problem where and , and denote the Riemann-Stieltjes integrals of with respect to and , respectively. Some sufficient conditions for the existence of either none, or one, or more positive solutions of the problem (1.5) were established. The main tool used in the proofs of existence results is a fixed point theorem in a cone, due to Krasnoselskii and Zabreiko.

At the same time, we note that Webb and Lan [21] have considered the first eigenvalue of the following linear problem they also investigated the existence and multiplicity of positive solutions of several related nonlinear multipoint boundary value problems. Furthermore, Ma and O'Regan [22] studied the spectrum structure of the problem (1.6), and the authors obtained the concrete computational method and the corresponding properties of real eigenvalue of (1.6) by constructing an auxiliary function. Their work is very fundamental to further study for multipoint boundary value problems. By extending and improving the work in [22], Rynne [23] showed that the associated Sturm-Liouville problem consisting of (1.6) has a strictly increasing sequence of simple eigenvalues with eigenfunctions .

Very recently, Kong et al. [24] were concerned with the general boundary value problem with a variable By relating (1.7) to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, the existence and nonexistence of nodal solutions of (1.7) were obtained. We also point out that Webb [25] made the excellent remark on some existence results of symmetric positive solutions obtained in some recent papers and the author also corrected the values of the principle eigenvalue previously given in some examples.

In this paper, by means of two coupled parallel subsuper solutions, we obtain some sufficient conditions for the existence of six solutions for (1.1) and our main tool is based on the fixed point index theory. At the end of this paper, we will give an example which illustrates that our work is true. Our method stems from the paper [18].

#### 2. Preliminaries and a Lemma

In the section, we shall give some preliminaries and a lemma which are fundamental to prove our main result.

Let be an ordered Banach space in which the partial ordering â‰¤ is induced by a cone . A cone is said to be normal if there exists a constant , such that implies , the smallest is called the normal constant of . is called solid, if int, that is, has nonempty interior. Every cone in defines a partial ordering in given by if and only if . If and , we write ; if cone is solid and int, we write . is called total if . Let be a bounded linear operator. is said to be positive if . An operator is strongly increasing, that is, implies . If is a linear operator, is strongly increasing implying is strongly positive.

Let be an ordered Banach space, a total cone in , the partial ordering â‰¤ induced by . is a positive completely continuous linear operator. Let the spectral radius of , the conjugated operator of , and the conjugated cone of . Since is a total cone (i.e., ), according to the famous Krein-Rutman theorem (see [26]), we infer that if , then there exist and , such that Fixed such that (2.1) holds. For , let then is also a cone in . One can refer [26â€“28] for definition and properties about the cones.

*Definition 2.1 (see [29]). *Let be a positive linear operator. The operator is said to satisfy condition , if there exist , and such that (2.1) holds, and maps into .

Lemma 2.2 (see [10]). *Suppose that . Then, the BVP
**
has Green's function
**
where
*

For convenience, we list the following hypotheses which will be used in our main result.â€‰â€‰ * are strictly increasing*;* there exist constants ** and ** such that **,*, *, and* (iv) *, where *; there exist constants , such that, for all , we have , ,, and;
â€‰
uniformly for , * uniformly for **, where ** is the first eigenvalue of the following boundary value problem:*

It is well known that , where linear operator is defined as .

#### 3. Main Results

Theorem 3.1. *Assume , hold, then BVP (1.1) has at least six distinct continuous solutions.*

*Proof. *It is easy to check that BVP (1.1) is equivalent to the following integral equation systems:
where is defined as in Lemma 2.2. By , we know that .

Let , define the norm in as . Then, is a Banach space with this norm. Let ,â€‰â€‰. Then, is a normal and solid cone. Set , such that
it is clear that the solutions of (1.1) are equivalent to the fixed points of .

Set , let
where , then . It is easy to see that is a strongly positive completely continuous operator, and it follows from and the continuity of that is a strongly positive completely continuous operator. Since are strictly increasing continuous functions, we know that is a strictly increasing continuous bounded operator. By , we can prove that is completely continuous. We infer from the increasing properties of and that is increasing.

Let , then satisfy
By (iii) and (i), we have
It follows from (ii),(iv), and the increasing property of that
Equations (3.2), (3.5), and (3.6) imply that
Similarly, by , we obtain

By [20, Lemma 3], we get that satisfies condition **H**. Therefore, there exist , such that
By the definition of spectral radius of completely continuous operator, we have , and combining (3.9), we infer that
Let , then . According to the proof in [18], we can get that satisfies condition .

By condition , we obtain that there exists , such that
Equations (3.12)â€“(3.15) imply
Since are continuous in , so they are bounded, then there exists such that
By virtue of (3.18) and the increasing properties of and , one shows

In addition, if satisfy , then it follows from (3.12), (3.18), and the increasing property of that
where
Similarly, if satisfy , then combining the increasing property of with (3.14) and (3.18), we know that
where . Let . By (3.21) and (3.22), we get that if or , it is obvious that
It follows from (3.16), (3.19), and (3.23) that
In a similar way, from (3.12) and (3.14), we can show that there exists such that
Let . It follows from (3.24) and (3.25) that if or , then
In a similar way, from (3.13) and (3.15), we can prove that there exists constant such that if or , then
Let , , , , , then or , ; therefore, in virtue of expression of , and (3.26), we have
where . Since , one can show

This implies that there exist and such that
Similarly, we get by (3.27) that there exist and such that
We get by (3.7) that
Let . Since is normal, then is bounded (see [28]). Choose such that
Let , then and is a bounded open set. By the proof of Theorem 2.1 in [18], we can show that
where satisfies .

Equation (3.34) implies that has no fixed point on . It is easy to prove that is a retract of , which together with (3.32) implies that the fixed point index over with respect to is well defined, and a standard proof yields
Set , then for any , we have . It follows from (3.34) that , and by (3.35) and the homotopy invariance of the fixed point index, we get
Let . By means of usual method (see [30]), we get that
It is evident that has no fixed point on , by (3.36), (3.37), and the additivity of the fixed point index, we have
Set , and choose such that
Let
Similarly to the proof of (3.37) and (3.38), we get that
Choose such that
Set
By virtue of (3.31) and the same method as that for (3.34), we have
By (3.44), similarly to the proof of (3.38), we can prove that
Equations (3.37)â€“(3.41), (3.45) imply that has at least six distinct fixed points, that is, the system of differential equations (1.1) has at least six solution in .

#### 4. An Example

In this section, we present a simple example to explain our results.

Consider the following second-order three-point BVP for nonlinear equations system: where , , , , , are strictly increasing continuous functions, and condition is satisfied. Choose . Some direct calculations show Therefore, condition is satisfied.

Choosing , it is easy to check that Therefore, condition is satisfied. At last, we will check condition , by the method of [9, 31], and we consider the linear eigenvalue problem Let . By the paper [31], we know that the sequence of positive eigenvalue of (4.4) is exactly given by , where is the sequence of positive solutions of . In [31], Han obtained that , moreover . It is easy to know that uniformly for , uniformly for . Therefore, condition is also satisfied. Consequently, all conditions of Theorem 3.1 are satisfied, and we get the system of differential equations (4.1) has at least six solutions in .

#### Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. This project is supported by the National Natural Science Foundation of China (10971046), the University Science and Technology Foundation of Shandong Provincial Education Department (J10LA62), the Natural Science Foundation of Shandong Province (ZR2009AM004, ZR2010AL014), and the Doctor of Scientific Startup Foundation for Shandong University of Finance (08BSJJ32).

#### References

- M. Moshinsky, â€śSobre los problems de condiciones a la frontiera en una dimension de caracteristicas discontinuas,â€ť
*BoletĂn de la Sociedad MatemĂˇtica Mexicana*, vol. 7, pp. 1â€“25, 1950. View at: Google Scholar - S. P. Timoshenko,
*Theory of Elastic Stability*, McGraw-Hill, New York, NY, USA, 1961. - V. A. Ilâ€™in and E. I. Moiseev, â€śonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects,â€ť
*Journal of Difference Equations*, vol. 23, pp. 803â€“810, 1987. View at: Google Scholar - C. P. Gupta, â€śSolvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation,â€ť
*Journal of Mathematical Analysis and Applications*, vol. 168, no. 2, pp. 540â€“551, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH - C. P. Gupta, â€śA generalized multi-point boundary value problem for second order ordinary differential equations,â€ť
*Applied Mathematics and Computation*, vol. 89, no. 1â€“3, pp. 133â€“146, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH - N. Aykut Hamal and F. Yoruk, â€śPositive solutions of nonlinear
*m*-point boundary value problems on time scales,â€ť*Journal of Computational and Applied Mathematics*, vol. 231, no. 1, pp. 92â€“105, 2009. View at: Publisher Site | Google Scholar - G. Infante and P. Pietramala, â€śExistence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations,â€ť
*Nonlinear Analysis*, vol. 71, no. 3-4, pp. 1301â€“1310, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. A. Khan and J. R. L. Webb, â€śExistence of at least three solutions of a second-order three-point boundary value problem,â€ť
*Nonlinear Analysis*, vol. 64, no. 6, pp. 1356â€“1366, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Ma, â€śExistence of solutions of nonlinear
*m*-point boundary-value problems,â€ť*Journal of Mathematical Analysis and Applications*, vol. 256, no. 2, pp. 556â€“567, 2001. View at: Publisher Site | Google Scholar - R. Ma and D. O'Regan, â€śSolvability of singular second order
*m*-point boundary value problems,â€ť*Journal of Mathematical Analysis and Applications*, vol. 301, no. 1, pp. 124â€“134, 2005. View at: Publisher Site | Google Scholar - F. Xu, Z. Chen, and F. Xu, â€śMultiple positive solutions for nonlinear second-order
*m*-point boundary-value problems with sign changing nonlinearities,â€ť*Electronic Journal of Differential Equations*, pp. 1â€“12, 2008. View at: Google Scholar - Z. Yang, â€śPositive solutions to a system of second-order nonlocal boundary value problems,â€ť
*Nonlinear Analysis*, vol. 62, no. 7, pp. 1251â€“1265, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. Xian, â€śThree solutions for three-point boundary value problems,â€ť
*Nonlinear Analysis*, vol. 62, no. 6, pp. 1053â€“1066, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z. Zhang and J. Wang, â€śThe upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems,â€ť
*Journal of Computational and Applied Mathematics*, vol. 147, no. 1, pp. 41â€“52, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Sun and K. Zhang, â€śOn the number of fixed points of nonlinear operators and applications,â€ť
*Journal of Systems Science and Complexity*, vol. 16, no. 2, pp. 229â€“235, 2003. View at: Google Scholar | Zentralblatt MATH - J. Sun and Y. Cui, â€śMultiple solutions for nonlinear operators and applications,â€ť
*Nonlinear Analysis*, vol. 66, no. 9, pp. 1999â€“2015, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. Xian, â€śSolutions for an operator equation under the conditions of pairs of paralleled lower and upper solutions,â€ť
*Nonlinear Analysis*, vol. 69, no. 7, pp. 2251â€“2266, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - K. M. Zhang and J. X. Sun, â€śMultiple solutions for superlinear operator equations in Banach spaces and applications,â€ť
*Acta Mathematica Sinica*, vol. 48, no. 1, pp. 99â€“108, 2005 (Chinese). View at: Google Scholar | Zentralblatt MATH - X. Xu, D. O'Regan, and J. Sun, â€śMultiplicity results for three-point boundary value problems with a non-well-ordered upper and lower solution condition,â€ť
*Mathematical and Computer Modelling*, vol. 45, no. 1-2, pp. 189â€“200, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z. Yang, â€śExistence and nonexistence results for positive solutions of an integral boundary value problem,â€ť
*Nonlinear Analysis*, vol. 65, no. 8, pp. 1489â€“1511, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. R. L. Webb and K. Q. Lan, â€śEigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type,â€ť
*Topological Methods in Nonlinear Analysis*, vol. 27, no. 1, pp. 91â€“115, 2006. View at: Google Scholar | Zentralblatt MATH - R. Ma and D. O'Regan, â€śNodal solutions for second-order
*m*-point boundary value problems with nonlinearities across several eigenvalues,â€ť*Nonlinear Analysis*, vol. 64, no. 7, pp. 1562â€“1577, 2006. View at: Publisher Site | Google Scholar - B. P. Rynne, â€śSpectral properties and nodal solutions for second-order,
*m*-point, boundary value problems,â€ť*Nonlinear Analysis*, vol. 67, no. 12, pp. 3318â€“3327, 2007. View at: Publisher Site | Google Scholar - L. Kong, Q. Kong, and J. S. W. Wong, â€śNodal solutions of multi-point boundary value problems,â€ť
*Nonlinear Analysis*, vol. 72, no. 1, pp. 382â€“389, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. R. L. Webb, â€śRemarks on a non-local boundary value problem,â€ť
*Nonlinear Analysis*, vol. 72, no. 2, pp. 1075â€“1077, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - K. Deimling,
*Nonlinear Functional Analysis*, Springer, Berlin, Germany, 1985. - D. J. Guo and V. Lakshmikantham,
*Nonlinear Problems in Abstract Cones*, Academic Press, Boston, Mass, USA, 1988. - D. Guo,
*Nonlinear Functional Analysis*, Shandong Science and Technology Press, Jinan, China, 2nd edition, 2001. - J. X. Sun, â€śNontrivial solutions of superlinear Hammerstein integral equations and their applications,â€ť
*Chinese Annals of Mathematics. Series A*, vol. 7, no. 5, pp. 528â€“535, 1986 (Chinese). View at: Google Scholar - H. Amann, â€śFixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,â€ť
*SIAM Review*, vol. 18, no. 4, pp. 620â€“709, 1976. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X.-l. Han, â€śPositive solutions of a three-point boundary value problem,â€ť
*Journal of Mathematical Research and Exposition*, vol. 27, no. 3, pp. 497â€“504, 2007. View at: Google Scholar | Zentralblatt MATH

#### Copyright

Copyright Â© 2012 Jian Liu 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.