The Existence of Periodic Solutions for Non-Autonomous Differential Delay Equations via Minimax Methods
© Rong Cheng. 2009
Received: 9 April 2009
Accepted: 19 October 2009
Published: 26 October 2009
By using variational methods directly, we establish the existence of periodic solutions for a class of nonautonomous differential delay equations which are superlinear both at zero and at infinity.
1. Introduction and Main Result
where is odd and is parameter. Since Jone's work in , there has been a great deal of research on problems of existence, multiplicity, stability, bifurcation, uniqueness, density of periodic solutions to (1.1) by applying various approaches. See [2, 4–23]. But most of those results concern scalar equations (1.1) and generally slowly oscillating periodic solutions. A periodic solution of (1.1) is called a "slowly oscillating periodic solution" if there exist numbers and such that for , for and for all .
where and are symmetric constant matrices. Before Guo and Yu's work, many authors generally first use the reduction technique introduced by Kaplan and Yorke in  to reduce the search for periodic solutions of (1.2) with and its similar ones to the problem of finding periodic solutions for a related system of ordinary differential equations. Then variational method was applied to study the related systems and the existence of periodic solutions of the equations is obtained.
When (1.2) satisfies (1.3), we can apply the twist condition between the zero and at infinity for to establish the existence of periodic solutions of (1.2). Under the superlinear conditions (1.5), there is no twist condition for , which brings difficulty to the study of the existence of periodic solutions of (1.4). But we can use minimax methods to consider the problem without twist condition for .
Throughout this paper, we assume that the following conditions hold.
Then our main result can be read as follows.
2. Proof of the Main Result
Then [25, Theorem ] can be stated as follows.
Theorem 2 A.
Case (ii). Let
From (1.5) and L'Hospital rules, we have (2.3) by a direct computation.
By reducing method, we have
Below we will construct a variational functional of (1.4) defined on a suitable Hilbert space such that finding -periodic solutions of (1.4) is equivalent to seeking critical points of the functional.
The proof is complete.
Let and denote the positive definite and negative definite subspace of in , respectively. Then . Letting , , we see that of Theorem A holds. Since is compact, of Theorem A holds. Now we establish of Theorem A by the following three lemmas.
By the above inequality, one has
Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1.
It is obviously that Theorem 1.1 holds from Lemmas 2.3, 2.4, 2.6, 2.7, 2.8, and 2.9 and Theorem A.
This work is supported by the Specialized Research Fund for the Doctoral Program of Higher Education for New Teachers and the Science Research Foundation of Nanjing University of Information Science and Technology (20070049).
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