- Research Article
- Open Access
© J. Zhengxian and H. Wenhua. 2009
- Received: 14 June 2009
- Accepted: 10 August 2009
- Published: 19 August 2009
- Hilbert Space
- Unique Solution
- Ordinary Differential Equation
- Functional Equation
- Uniqueness Theorem
Minimax theorems are one of powerful tools for investigation on the solution of differential equations and differential systems. The investigation on the solution of differential equations and differential systems with non- perturbation term using minimax theorems came into being in the paper of Stepan A.Tersian in 1986 . Tersian proved that the equation exists exactly one generalized solution under the operators related to the perturbation term being selfadjoint and commuting with the operator and some other conditions in . Huang Wenhua extended Tersian's theorems in  in 2005 and 2006, respectively, and studied the existence and uniqueness of solutions of some differential equations and differential systems with non- perturbation term [2–4], the conditions attached to the non- perturbation term are that the operator related to the term is self-adjoint and commutes with the operator (where is a selfadjoint operator in the equation ). Recently, by further research, we observe that the conditions imposed upon can be weakened, the self-adjointness of can be removed and is not necessarily commuting with the operator .
In this note, we consider a two-point boundary value problem of a class of Duffing-type systems with non- perturbation term and present a result as the operator related to the perturbation term is not necessarily a selfadjoint and commuting with the operator . We obtain several valuable results in the present paper under the weaker conditions than those in [2–4].
Let be a real Hilbert space with inner product and norm , respectively, let and be two orthogonal closed subspaces of such that . Let denote the projections from to and from to , respectively. The following theorem will be employed to prove our main theorem.
Theorem 2.1 ().
We also need the following lemma in the present work. To the best of our knowledge, the lemma seems to be new.
Let and be two diagonalization matrices, let and be the eigenvalues of and , respectively, where each eigenvalue is repeated according to its multiplicity. If commutes with , that is, , then is a diagonalization matrix and are the eigenvalues of .
The proof of Lemma 2.2 is fulfilled.
Now, we consider the boundary value problem
where , is a real constant diagonalization matrix with real eigenvalues (each eigenvalue is repeated according to its multiplicity), is a potential Carathéodory vector-valued function , is continuous, , , .
where , commutes with and is possessed of real eigenvalues . In the light of Lemma 2.2, is a diagonalization matrix with real eigenvalues (each eigenvalue is repeated according to its multiplicity). Assume that there exist positive integers such that for
and the norm induced by the inner product is
Now, we state and prove the following theorem concerning the solution of problem (2.11).
By (3.3), Clearly, and are nonincreasing. Now, all the conditions in the Theorem 2.1 are satisfied. By virtue of Theorem 2.1, there exists a unique such that and where is a functional defined implicity in (2.24) and . is just a unique solution of (2.12) and is exactly a unique solution of (2.11). The proof of Theorem 3.1 is completed.
Replace the condition (2.14) by (3.10) and replace (2.21), (2.22), (3.1), and (3.2) by (3.11), (3.12), (3.13), and (3.14), respectively. Using the similar proving techniques in the Theorem 3.1, we can prove the following theorem.
Assume that there exists a real diagonalization matrix with real eigenvalues satisfying (2.13) and (3.10) and commuting with . If the functions and defined in (3.13) and (3.14) satisfy (3.3), problem (2.11) has a unique solution , and satisfies and (3.4).
Using the similar techniques of the present paper, we can also investigate the two-point boundary value problem
The special case of and in problem (3.20) has been studied by Zhou Ting and Huang Wenhua . Zhou and Huang adopted the techniques different from this paper to achieve their research.
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