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.
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 .
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 .
Theorem 2.2 (Saddle Point Theorem).
3. Main Result
Under the assumptions (1.2), (3.2), (3.3), problem (1.1) has at least one solution.
This work was supported by Research Plan MSM 4977751301.
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