- Research Article
- Open Access
Periodic Problem with a Potential Landesman Lazer Condition
© Petr Tomiczek. 2010
- Received: 6 January 2010
- Accepted: 22 September 2010
- Published: 26 September 2010
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.
- Nonlinear Problem
- Strong Convergence
- Type Condition
- Variational Approach
- Previous Equality
where , the nonlinearity is a Caratheodory function and .
To state an existence result to (1.1) Amster  assumes that is a nondecreasing function (see also ). 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 .
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.
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.
Let the condition (1.2) be satisfied. Then any critical point of the functional is a solution to (1.1).
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.
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 .
Under the assumptions (1.2), (3.2), (3.3), problem (1.1) has at least one solution.
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.
This work was supported by Research Plan MSM 4977751301.
- 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.Google Scholar
- Amster P, Mariani MC: A second order ODE with a nonlinear final condition. Electronic Journal of Differential Equations 2001,2001(75):1.MathSciNetGoogle Scholar
- 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 ArticleGoogle Scholar
- 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 ArticleGoogle Scholar
- 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 ArticleGoogle Scholar
- 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 ArticleGoogle Scholar
- 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 ArticleGoogle Scholar
- Tomiczek P: Potential Landesman-Lazer type conditions and the Fučík spectrum. Electronic Journal of Differential Equations 2005,2005(94):1-12.MathSciNetGoogle Scholar
- 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.Google Scholar
- Rabinowitz PH: Some minimax theorems and applications to nonlinear partial differential equations. In Nonlinear Analysis. Academic Press, New York, NY, USA; 1978:161-177.Google Scholar
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