SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Open Badges Research Article

A Fourth-Order Boundary Value Problem with One-Sided Nagumo Condition

Wenjing Song12* and Wenjie Gao1

Author Affiliations

1 Institute of Mathematics, Jilin University, Changchun 130012, China

2 Institute of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130017, China

For all author emails, please log on.

Boundary Value Problems 2011, 2011:569191  doi:10.1155/2011/569191

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/569191

Received:10 January 2011
Accepted:9 March 2011
Published:14 March 2011

© 2011 Wenjing Song and Wenjie Gao.

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.

The aim of this paper is to study a fourth-order separated boundary value problem with the right-hand side function satisfying one-sided Nagumo-type condition. By making a series of a priori estimates and applying lower and upper functions techniques and Leray-Schauder degree theory, the authors obtain the existence and location result of solutions to the problem.

1. Introduction

In this paper we apply the lower and upper functions method to study the fourth-order nonlinear equation


with being a continuous function.

This equation can be used to model the deformations of an elastic beam, and the type of boundary conditions considered depends on how the beam is supported at the two endpoints [1, 2]. We consider the separated boundary conditions


with , .

For the fourth-order differential equation


the authors in [3] obtained the existence of solutions with the assumption that satisfies the two-sided Nagumo-type conditions. For more related works, interested readers may refer to [114]. The one-sided Nagumo-type condition brings some difficulties in studying this kind of problem, as it can be seen in [1518].

Motivated by the above works, we consider the existence of solutions when satisfies one-sided Nagumo-type conditions. This is a generalization of the above cases. We apply lower and upper functions technique and topological degree method to prove the existence of solutions by making a priori estimates for the third derivative of all solutions of problems (1.1) and (1.2). The estimates are essential for proving the existence of solutions.

The outline of this paper is as follows. In Section 2, we give the definition of lower and upper functions to problems (1.1) and (1.2) and obtain some a priori estimates. Section 3 will be devoted to the study of the existence of solutions. In Section 4, we give an example to illustrate the conclusions.

2. Definitions and A Priori Estimates

Upper and lower functions will be an important tool to obtain a priori bounds on , , and . For this problem we define them as follows.

Definition 2.1.

The functions verifying


define a pair of lower and upper functions of problems (1.1) and (1.2) if the following conditions are satisfied:

(i), ,

(ii), ,


Remark 2.2.

By integration, from (iii) and (2.1), we obtain


that is, lower and upper functions, and their first derivatives are also well ordered.

To have an a priori estimate on , we need a one-sided Nagumo-type growth condition, which is defined as follows.

Definition 2.3.

Given a set , a continuous is said to satisfy the one-sided Nagumo-type condition in if there exists a real continuous function , for some , such that




Lemma 2.4.

Let satisfy


and consider the set


Let be a continuous function satisfying one-sided Nagumo-type condition in .

Then, for every , there exists an such that for every solution of problems (1.1) and (1.2) with



for and every , one has .


Let be a solution of problems (1.1) and (1.2) such that (2.7) and (2.8) hold. Define


Assume that , and suppose, for contradiction, that for every . If for every , then we obtain the following contradiction:


If for every , a similar contradiction can be derived. So there is a such that . By (2.4) we can take such that


If for every , then we have trivially . If not, then we can take such that or such that . Suppose that the first case holds. By (2.7) we can consider such that


Applying a convenient change of variable, we have, by (2.3) and (2.11),


Hence, . Since can be taken arbitrarily as long as , we conclude that for every provided that .

In a similar way, it can be proved that , for every if . Therefore,


Consider now the case , and take such that


In a similar way, we may show that


Taking , we have .

Remark 2.5.

Observe that the estimation depends only on the functions , , , and and it does not depend on the boundary conditions.

3. Existence and Location Result

In the presence of an ordered pair of lower and upper functions, the existence and location results for problems (1.1) and (1.2) can be obtained.

Theorem 3.1.

Suppose that there exist lower and upper functions and of problems (1.1) and (1.2), respectively. Let be a continuous function satisfying the one-sided Nagumo-type conditions (2.3) and (2.4) in


If verifies


for and


where means and , then problems (1.1) and (1.2) has at least one solution satisfying


for .


Define the auxiliary functions


For , consider the homotopic equation


with the boundary conditions


Take large enough such that, for every ,





Step 1.

Every solution of problems (3.6) and (3.7) satisfies


for , for some independent of .

Assume, for contradiction, that the above estimate does not hold for . So there exist , , and a solution of (3.6) and (3.7) such that . In the case define


If , then and . Then, by (3.2) and (3.10), for , the following contradiction is obtained:


For ,


If , then


and . If , then and so . Therefore, the above computations with replaced by 0 yield a contradiction. For , by (3.11), we get the following contradiction:


The case is analogous. Thus, for every . In a similar way, we may prove that for every .

By the boundary condition (3.7) there exists a , such that . Then by integration we obtain


Step 2.

There is an such that for every solution of problems (3.6) and (3.7)


with independent of .

Consider the set


and for the function given by


In the following we will prove that the function satisfies the one-sided Nagumo-type conditions (2.3) and (2.4) in independently of . Indeed, as verifies (2.3) in , then


So, defining in , we see that verifies (2.3) with and replaced by and , respectively. The condition (2.4) is also verified since


Therefore, satisfies the one-sided Nagumo-type condition in with replaced by , with independent of .

Moreover, for


every solution of (3.6) and (3.7) satisfies




The hypotheses of Lemma 2.4 are satisfied with replaced by . So there exists an , depending on and , such that for every . As and do not depend on , we see that is maybe independent of .

Step 3.

For , the problems (3.6) and (3.7) has at least one solution .

Define the operators




and for , by




Observe that has a compact inverse. Therefore, we can consider the completely continuous operator


given by


For given by Step 2, take the set


By Steps 1 and 2, degree is well defined for every and by the invariance with respect to a homotopy


The equation is equivalent to the problem


and has only the trivial solution. Then, by the degree theory,


So the equation has at least one solution, and therefore the equivalent problem


has at least one solution in .

Step 4.

The function is a solution of the problems (1.1) and (1.2).

The proof will be finished if the above function satisfies the inequalities


Assume, for contradiction, that there is a such that , and define


If , then and . Therefore, by (3.2) and Definition 2.1, we obtain the contradiction


If , then we have


By Definition 2.1 this yields a contradiction


Then and, by similar arguments, we prove that . Thus,


Using an analogous technique, it can be deduced that for every . So we have


On the other hand, by (1.2),


that is,


Applying the same technique, we have


and then by Definition 2.1 (iii), (3.44) and (3.46), we obtain


that is,


Since, by (3.44), is nondecreasing, we have by (3.49)


and, therefore, for every . By the monotonicity of ,


and so for every .

The inequalities and for every can be proved in the same way. Then is a solution of problems (1.1) and (1.2).

4. An Example

The following example shows the applicability of Theorem 3.1 when satisfies only the one-sided Nagumo-type condition.

Example 4.1.

Consider now the problem



with . The nonlinear function


is continuous in . If , then the functions defined by


are, respectively, lower and upper functions of (4.1) and (4.2). Moreover, define


Then satisfies condition (3.2) and the one-sided Nagumo-type condition with , in .

Therefore, by Theorem 3.1, there is at least one solution of Problem (4.1) and (4.2) such that, for every ,


Notice that the function


does not satisfy the two-sided Nagumo condition.


The authors would like to thank the referees for their valuable comments on and suggestions regarding the original manuscript. This work was supported by NSFC (10771085), by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education, and by the 985 Program of Jilin University.


  1. Gupta, CP: Existence and uniqueness theorems for the bending of an elastic beam equation. Applicable Analysis. 26(4), 289–304 (1988). Publisher Full Text OpenURL

  2. Gupta, CP: Existence and uniqueness theorems for a fourth order boundary value problem of Sturm-Liouville type. Differential and Integral Equations. 4(2), 397–410 (1991)

  3. Minhós, F, Gyulov, T, Santos, AI: Existence and location result for a fourth order boundary value problem. Discrete and Continuous Dynamical Systems. Series A.(supplement), 662–671 (2005)

  4. Cabada, A, Grossinho, MDR, Minhós, F: On the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions. Journal of Mathematical Analysis and Applications. 285(1), 174–190 (2003). Publisher Full Text OpenURL

  5. Cabada, A, Pouso, RL: Extremal solutions of strongly nonlinear discontinuous second-order equations with nonlinear functional boundary conditions. Nonlinear Analysis: Theory, Methods & Applications. 42(8), 1377–1396 (2000). PubMed Abstract | Publisher Full Text OpenURL

  6. Cabada, A, Pouso, RL: Existence results for the problem with nonlinear boundary conditions. Nonlinear Analysis: Theory, Methods & Applications. 35(2), 221–231 (1999). PubMed Abstract | Publisher Full Text OpenURL

  7. de Coster, C: La méthode des sur et sous solutions dans l'étude de problèmes aux limites.

  8. de Coster, C, Habets, P: Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results. Non-Linear Analysis and Boundary Value Problems for Ordinary Differential Equations (Udine), CISM Courses and Lectures, pp. 1–78. Springer, Vienna, Austria (1996)

  9. do Rosário Grossinho, M, Minhós, FM: Existence result for some third order separated boundary value problems. Nonlinear Analysis: Theory, Methods & Applications. 47(4), 2407–2418 (2001). PubMed Abstract | Publisher Full Text OpenURL

  10. Grossinho, MR, Minhós, F: Upper and lower solutions for higher order boundary value problems. Nonlinear Studies. 12(2), 165–176 (2005)

  11. Grossinho, MR, Minhós, F: Solvability of some higher order two-point boundary value problems. In: Proceedings of International Conference on Differential Equations and Their Applications (Equadiff 10), August 2001, Prague, Czechoslovak. 183–189

  12. Kiguradze, IT, Shekhter, BL: Singular boundary value problems for second-order ordinary differential equations. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh.. 30, 105–201 (1987) English translation in Journal of Soviet Mathematics, vol. 43, no. 2, pp. 2340–2417, 1988

  13. Minhós, FM, Santos, AI: Existence and non-existence results for two-point boundary value problems of higher order. Proceedings of International Conference on Differential Equations (Equadiff 2003), pp. 249–251. World Sci. Publ., Hackensack, NJ, USA (2005)

  14. Nagumo, M: Ueber die Differentialgleichung . Proceedings of the Physico-Mathematical Society of Japan. 1937(3), 861–866 (19)

  15. Cabada, A: An overview of the lower and upper solutions method with nonlinear boundary value conditions. Boundary Value Problems. 2011, (2011)

  16. Grossinho, MR, Minhós, FM, Santos, AI: A third order boundary value problem with one-sided Nagumo condition. Nonlinear Analysis: Theory, Methods & Applications. 63(5–7), 247–256 (2005). PubMed Abstract | Publisher Full Text OpenURL

  17. Grossinho, MR, Minhós, FM, Santos, AI: Existence result for a third-order ODE with nonlinear boundary conditions in presence of a sign-type Nagumo control. Journal of Mathematical Analysis and Applications. 309(1), 271–283 (2005). Publisher Full Text OpenURL

  18. Grossinho, MR, Minhós, FM, Santos, AI: Solvability of some third-order boundary value problems with asymmetric unbounded nonlinearities. Nonlinear Analysis: Theory, Methods & Applications. 62(7), 1235–1250 (2005). PubMed Abstract | Publisher Full Text OpenURL