Periodic Solutions for Second-Order Ordinary Differential Equations with Linear Nonlinearity

By using minimax methods in critical point theory, we obtain the existence of periodic solutions for second-order ordinary differential equations with linear nonlinearity.


Theorem 1. Suppose that (A) and (8) hold and
(i) where  is a parameter and satisfies  > 1/2; Then problem (1) has at least one solution.
Theorem 2. Suppose that (A), ( 8) and (i) hold and Then problem (1) has at least one solution.
Remark 3. (i) It is worth noting that, in the case of  = 0, one solution was obtained by Tang [9] and Han [15] under the sublinear nonlinearity condition.
(ii) It is also worth noting that the sublinear nonlinearity condition in [15,18] is different from that of [9].

Proof of Main Results
Let for any  ∈  1  .It follows from assumption () that the functional  on  1  is continuously differentiable; moreover we obtain (∇ (,  ()) , V ())  (18) for any , V ∈  1  .It is well known that the solutions of problem (1) correspond to the critical points of  (see [6]).