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

- Wenjing Song
^{1, 2}Email author and - Wenjie Gao
^{1}

**2011**:569191

**DOI: **10.1155/2011/569191

© W. Song and W. Gao. 2011

**Received: **10 January 2011

**Accepted: **9 March 2011

**Published: **14 March 2011

## Abstract

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

with being a continuous function.

with , .

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 [1–14]. The one-sided Nagumo-type condition brings some difficulties in studying this kind of problem, as it can be seen in [15–18].

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.

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

(i) , ,

(ii) , ,

(iii) .

Remark 2.2.

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.

Lemma 2.4.

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

for and every , one has .

Proof.

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

for .

Proof.

Step 1.

for , for some independent of .

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

Step 2.

with independent of .

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

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 .

has at least one solution in .

Step 4.

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

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.

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

does not satisfy the two-sided Nagumo condition.

## Declarations

### Acknowledgments

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.

## Authors’ Affiliations

## References

- Gupta CP: Existence and uniqueness theorems for the bending of an elastic beam equation.
*Applicable Analysis*1988, 26(4):289-304. 10.1080/00036818808839715View ArticleMathSciNet - Gupta CP: Existence and uniqueness theorems for a fourth order boundary value problem of Sturm-Liouville type.
*Differential and Integral Equations*1991, 4(2):397-410.MathSciNet - Minhós F, Gyulov T, Santos AI: Existence and location result for a fourth order boundary value problem.
*Discrete and Continuous Dynamical Systems. Series A*2005, (supplement):662-671. - 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*2003, 285(1):174-190. 10.1016/S0022-247X(03)00388-3View ArticleMathSciNet - Cabada A, Pouso RL: Extremal solutions of strongly nonlinear discontinuous second-order equations with nonlinear functional boundary conditions.
*Nonlinear Analysis: Theory, Methods & Applications*2000, 42(8):1377-1396. 10.1016/S0362-546X(99)00158-3View ArticleMathSciNet - Cabada A, Pouso RL:Existence results for the problem
with nonlinear boundary conditions.
*Nonlinear Analysis: Theory, Methods & Applications*1999, 35(2):221-231. 10.1016/S0362-546X(98)00009-1View ArticleMathSciNet - de Coster C: La méthode des sur et sous solutions dans l'étude de problèmes aux limites. Départment de Mathématique, Faculté des Sciences, Université Catholique de Louvain, Février 1994
- de Coster C, Habets P: Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results. In
*Non-Linear Analysis and Boundary Value Problems for Ordinary Differential Equations (Udine), CISM Courses and Lectures*.*Volume 371*. Springer, Vienna, Austria; 1996:1-78.View Article - do Rosário Grossinho M, Minhós FM: Existence result for some third order separated boundary value problems.
*Nonlinear Analysis: Theory, Methods & Applications*2001, 47(4):2407-2418. 10.1016/S0362-546X(01)00364-9View ArticleMathSciNet - Grossinho MR, Minhós F: Upper and lower solutions for higher order boundary value problems.
*Nonlinear Studies*2005, 12(2):165-176.MathSciNet - Grossinho MR, Minhós F: Solvability of some higher order two-point boundary value problems.
*Proceedings of International Conference on Differential Equations and Their Applications (Equadiff 10), August 2001, Prague, Czechoslovak*183-189. CD-ROM papers - Kiguradze IT, Shekhter BL: Singular boundary value problems for second-order ordinary differential equations.
*Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh.*1987, 30: 105-201. English translation in*Journal of Soviet Mathematics*, vol. 43, no. 2, pp. 2340–2417, 1988MathSciNet - Minhós FM, Santos AI: Existence and non-existence results for two-point boundary value problems of higher order. In
*Proceedings of International Conference on Differential Equations (Equadiff 2003)*. World Sci. Publ., Hackensack, NJ, USA; 2005:249-251. - Nagumo M:Ueber die Differentialgleichung
.
*Proceedings of the Physico-Mathematical Society of Japan*19, 1937(3):861-866. - Cabada A: An overview of the lower and upper solutions method with nonlinear boundary value conditions.
*Boundary Value Problems*2011, 2011:-18. - Grossinho MR, Minhós FM, Santos AI: A third order boundary value problem with one-sided Nagumo condition.
*Nonlinear Analysis: Theory, Methods & Applications*2005, 63(5–7):247-256.View Article - 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*2005, 309(1):271-283. 10.1016/j.jmaa.2005.01.030View ArticleMathSciNet - Grossinho MR, Minhós FM, Santos AI: Solvability of some third-order boundary value problems with asymmetric unbounded nonlinearities.
*Nonlinear Analysis: Theory, Methods & Applications*2005, 62(7):1235-1250. 10.1016/j.na.2005.04.029View ArticleMathSciNet

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