# Periodic Problem with a Potential Landesman Lazer Condition

- Petr Tomiczek
^{1}Email author

**2010**:586971

**DOI: **10.1155/2010/586971

© Petr Tomiczek. 2010

**Received: **6 January 2010

**Accepted: **22 September 2010

**Published: **26 September 2010

## Abstract

We prove the existence of a solution to the periodic nonlinear second-order ordinary differential equation with damping , , . We suppose that , the nonlinearity satisfies the potential Landesman Lazer condition and prove that a critical point of a corresponding energy functional is a solution to this problem.

## 1. Introduction

where , the nonlinearity is a Caratheodory function and .

To state an existence result to (1.1) Amster [1] assumes that is a nondecreasing function (see also [2]). He supposes that the nonlinearity satisfies the growth condition , for , , where is the first eigenvalue of the problem , and there exist such that An interval is centered in with the radius where , and is a solution to the problem with .

In [3, 4] authors studied (1.1) with a constant friction term and results with repulsive singularities were obtained in [5, 6].

where , , and

To obtain our result we use variational approach even if the linearization of the periodic problem (1.1) is a non-self-adjoint operator.

## 2. Preliminaries

Notation 2.

We will use the classical space of functions whose th derivative is continuous and the space of measurable real-valued functions whose th power of the absolute value is Lebesgue integrable. We denote the Sobolev space of absolutely continuous functions such that and with the norm . By a solution to (1.1) we mean a function such that is absolutely continuous, satisfies the boundary conditions and (1.1) is satisfied a.e. in .

for all .

Now we prove that any critical point of the functional is a solution to (1.1) mentioned above.

Lemma 2.1.

Let the condition (1.2) be satisfied. Then any critical point of the functional is a solution to (1.1).

Proof.

Thus is a solution to (1.1).

We say that
satisfies the *Palais-Smale condition* (PS) if every sequence
for which
is bounded in
and
(as
possesses a convergent subsequence.

To prove the existence of a critical point of the functional we use the Saddle Point Theorem which is proved in Rabinowitz [9] (see also [10]).

Theorem 2.2 (Saddle Point Theorem).

Let , and . Let be a functional such that and

(a) there exists a bounded neighborhood of in and a constant such that ,

(b) there is a constant such that ,

(c) satisfies the Palais-Smale condition (PS).

Then, the functional has a critical point in .

## 3. Main Result

Theorem 3.1.

Under the assumptions (1.2), (3.2), (3.3), problem (1.1) has at least one solution.

Proof.

We verify that the functional satisfies assumptions of the Saddle Point Theorem 2.2 on , then has a critical point and due to Lemma 2.1 is the solution to (1.1).

It is easy to see that . Let then and .

a contradiction to (3.2). We proceed for the case Then assumption (a) of Theorem 2.2 is verified.

Since the function is strictly positive equality (3.10) implies that the functional is bounded from below.

Using (3.4), (3.10) we see that there exists a bounded neighborhood of in , a constant such that , and there is a constant such that .

Hence we obtain the strong convergence in . This shows that satisfies the Palais-Smale condition and the proof of Theorem 3.1 is complete.

## Declarations

### Acknowledgment

This work was supported by Research Plan MSM 4977751301.

## Authors’ Affiliations

## References

- Amster P:
**Nonlinearities in a second order ODE.**In*Proceedings of the USA-Chile Workshop on Nonlinear Analysis, 2001, San Marcos, Tex, USA, Electron. J. Differ. Equ. Conf.*.*Volume 6*. Southwest Texas State Univ.; 13-21. - Amster P, Mariani MC:
**A second order ODE with a nonlinear final condition.***Electronic Journal of Differential Equations*2001,**2001**(75):1.MathSciNet - Chen H, Li Y:
**Rate of decay of stable periodic solutions of Duffing equations.***Journal of Differential Equations*2007,**236**(2):493-503. 10.1016/j.jde.2007.01.023MATHMathSciNetView Article - Mawhin J, Ward JR:
**Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Liénard and Duffing equations.***The Rocky Mountain Journal of Mathematics*1982,**12**(4):643-654. 10.1216/RMJ-1982-12-4-643MATHMathSciNetView Article - Torres PJ:
**Bounded solutions in singular equations of repulsive type.***Nonlinear Analysis: Theory, Methods & Applications*1998,**32**(1):117-125. 10.1016/S0362-546X(97)00456-2MATHMathSciNetView Article - Li X, Zhang Z:
**Periodic solutions for damped differential equations with a weak repulsive singularity.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(6):2395-2399. 10.1016/j.na.2008.03.023MATHMathSciNetView Article - Lazer AC:
**On Schauder's fixed point theorem and forced second-order nonlinear oscillations.***Journal of Mathematical Analysis and Applications*1968,**21:**421-425. 10.1016/0022-247X(68)90225-4MATHMathSciNetView Article - Tomiczek P:
**Potential Landesman-Lazer type conditions and the Fučík spectrum.***Electronic Journal of Differential Equations*2005,**2005**(94):1-12.MathSciNet - Rabinowitz P:
*Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, no. 65*. American Mathematical Society, Providence, RI, USA; 1986. - Rabinowitz PH:
**Some minimax theorems and applications to nonlinear partial differential equations.**In*Nonlinear Analysis*. Academic Press, New York, NY, USA; 1978:161-177.

## Copyright

This article is published under license to BioMed Central Ltd. 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.