Research Article | Open Access
Multiple Solutions for Generalized Asymptotical Linear Hamiltonian Systems Satisfying Bolza Boundary Conditions
This paper is devoted to multiple solutions of generalized asymptotical linear Hamiltonian systems satisfying Bolza boundary conditions. We classify the linear Hamiltonian systems by the index theory and obtain the existence and multiplicity of solutions for the Hamiltonian systems, based on an application of the classical symmetric mountain pass lemma.
This paper is concerned with a classical problem on the existence of solutions for Hamiltonian systems when a more general form of twist condition between the origin and infinity holds for the Hamiltonian function. More precisely consider the system where , and and satisfies the following:), as uniformly in ,(), as uniformly in , for , . Here and below, we use to denote the first derivative of with respect to .
A quantitative way to measure the twisting is given by the Maslov-type index. As in [1, 2], an index for the second-order and first-order linear Hamiltonian systems was defined and developed in  for the study of linear operator equation with infinite Morse index. In [4, 5], by Conley and Zehnder and Long, an index theory for symplectic path was defined. We refer to two excellent books [6, 7] for systematical treatment.
In , Dong discussed the classification of the linear Hamiltonian systems with Bolza boundary conditions as follows: where and , for any . That is, for any , he associated it with a pair of numbers . This pair of integers is called index and nullity of , respectively. And he defined the nullity as the dimension of the solution space of (1). Let be the integer such that . So that for all integer defined as in . In order to process the definition of , he defined . In particular, when , . We will introduce the definitions and properties of in detail in Section 2. After the discussion of the index theory, we will prove our main result in Section 3.
Throughout this paper, denotes the usual norm in . For any , , we write if is positively semidefinite, and we write if is positive definite. For any , , we write if for a.e. , and we write if and on a subset of with a nonzero measure.
Remark 1. Let , , , with , . Assume that , , and , . Then, we have , , and , for ; (1) has at least pairs of solutions.
We make use of the critical point theory [8, 9] to prove Theorem 8. The novelty of our result is that it suffices to assume that is convex. The twisting between origin and infinity is reflected in () and (). Thus, our results complement with Theorem 3.8 in  and Theorem 1.1 in . For other results, we refer to [11–15].
This paper is organized as follows. In Section 2, we introduce some preliminaries including index theory, and establish the -index theory which is needed in the proofs. In Section 3, we present the proofs of the results.
2. Index Theory for Linear Hamiltonian Systems Satisfying Bolza Boundary Value Conditions
First, we recall some definitions and propositions in . We consider the following system: Let is continuous on , satisfies (4), and with the norm , , , where , and let for any . Then, and are self-adjoint operators, and is bounded.
Definition 2 (see [2, Definitions 2.1, 2.3, 2.4, and 2.7]). For any , one defines that(1), (2) for ,(3) for satisfies .
Using spectral theory, a Morse-type index was established in . More precisely, for any , let with . Then, , is invertible, and the inverse is self-adjoint and compact. He put a quadratic form: where . Then, defines a Hilbert space structure on . is a self-adjoint and compact operator under this interior product. By the spectral theory, there is a basis , of , and a sequence in such that For any as , (5) can be rewritten as follows: Define that , , and are -orthogonal, and . Since as , and are two finite-dimensional subspaces.
Definition 3 (see [2, Definition 2.9]). For any , with , one defines that
One calls and -nullity and -index of , respectively.
Proposition 4 (see [2, Propositions 2.10, 2.13]). The index, relative Morse index, and -index have the following properties.(1) For any , , , with , one has (2) There exists , such that, for any , one gets In particular, if , one obtains for .(3) is a constant for satisfying , that is, for any other with . For any , one has for any with .
Remark 5. From this proposition, there is , such that and .
Lemma 6 (cf. Chang [9, Theorem ]). Assume that satisfies the (PS) condition, , , and (1) there are an m-dimensional subspace and a constant such that (2) there is a j-dimensional subspace such that where is a subset of such that . Then, has at least pairs of critical points if .
Lemma 7 (see [8, Theorem ]). Suppose that satisfies (C) condition, , even, and there exist two closed subspaces of , with , and two constants such that(a) for all ,(b) for all .
Then, if , possesses at least distinct pairs of critical points whose corresponding critical values belong to [, ].
3. Proof of Main Results
We state the main result in this paper. We further make the following assumptions. () There exists , such that with , . () There exists , such that with , . (), . () is convex.
Our main result reads as follows.
Theorem 8. Assume that (), (), (), (), (), () are satisfied, then (1) has at least pairs of solutions.
Theorem 9. If , then (1) has at least pairs of solutions.
If , let with , . Recalling that , one denotes . Then, one obtains Thus, and (by the Fenchel conjugate formula; see [6, Proposition II]). Hence, if and only if . Consider the functional defined by
In order to prove Theorem 9, we need the following lemmas.
Lemma 10. satisfies the (PS) condition.
Proof. Assume that is a sequence in such that is bounded and in . It suffices to prove that has convergent sequence. By (18), we have
for all . Hence, we get
Let , , and . If , then . Joining () with (17) and (20), we have
Using the preceding notations, we have
So is bounded in . We assume that in , and hence . From (22), there exists , such that
And we have . Taking the limit as in (22), we obtain . Let . We get
By assumption (), we have and , which is impossible. Since , is bounded. From (17) and (20), is bounded. Assume that in ; then . Let ; then . Fenchel conjugate formula gives in (by [6, Preposition II, Theorem 4]).
Proof. From () and (), we have that, for any , there exists such that
Let . By (16) and (), is strictly concave. Then, as . So, achieves its maximum at a unique point. Because of , we get that
And we can easily get . Hence, we have when . Otherwise, there exists , , , s.t. . But , . Hence, is bounded, and there exists , such that , , . Thus, . This contradicts the uniqueness, which yields there exists such that, for any , we have . So, for any , we have By (18), that is,
Let . Then, . For any , we have . By Proposition 4, for being small enough, .
Lemma 12. The assumption (2) of Lemma 6 is also satisfied.
Proof. From (), let . Then, for any , there exists a constant such that
Combining () and (31), there exists a constant such that
for all . By the definition of , we have
So, we infer
Let . We have and . So, we have
By Proposition 4, we have , and we can also let be small enough such that . This completes the proof.
Theorem 13. If , then (2) has at least pairs of solutions.
The author would like to thank the reviewers for their valuable comments which improved the paper. The paper is partially supported by the National Natural Science Foundation of China (10871095) and the Foundation for Innovative Program of Jiangsu Province (CXZZ12_0377, CXZZ12_0386).
- Y. Dong, “Index theory, nontrivial solutions, and asymptotically linear second-order Hamiltonian systems,” Journal of Differential Equations, vol. 214, no. 2, pp. 233–255, 2005.
- Y. Dong, “Maslov type index theory for linear Hamiltonian systems with Bolza boundary value conditions and multiple solutions for nonlinear Hamiltonian systems,” Pacific Journal of Mathematics, vol. 221, no. 2, pp. 253–280, 2005.
- Y. Dong, “Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations,” Calculus of Variations and Partial Differential Equations, vol. 38, no. 1-2, pp. 75–109, 2010.
- C. Conley and E. Zehnder, “Morse-type index theory for flows and periodic solutions for Hamiltonian equations,” Communications on Pure and Applied Mathematics, vol. 37, no. 2, pp. 207–253, 1984.
- Y. M. Long, “Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems,” Science in China Series A, vol. 33, no. 12, pp. 1409–1419, 1990.
- I. Ekeland, Convexity Methods in Hamiltonian Mechanics, vol. 19, Springer, Berlin, Germany, 1990.
- J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
- P. Bartolo, V. Benci, and D. Fortunato, “Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 7, no. 9, pp. 981–1012, 1983.
- K. C. Chang, Critical Point Theory and Its Application, Shanghai Kexue Jishu Chubanshe, Shanghai, China, 1986.
- Y. Shan, “Multiple solutions of generalized asymptotical linear Hamiltonian systems satisfying Sturm-Liouville boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 14, pp. 4809–4819, 2011.
- V. Benci, “On critical point theory for indefinite functionals in the presence of symmetries,” Transactions of the American Mathematical Society, vol. 274, no. 2, pp. 533–572, 1982.
- F. Dalbono and C. Rebelo, “Multiplicity of solutions of Dirichlet problems associated with second-order equations in ,” Proceedings of the Edinburgh Mathematical Society. Series II, vol. 52, no. 3, pp. 569–581, 2009.
- F. Dalbono, “Branches of index-preserving solutions to systems of second order ODEs,” Nonlinear Differential Equations and Applications, vol. 16, no. 5, pp. 569–595, 2009.
- G. H. Fei, “Relative Morse index and its application to Hamiltonian systems in the presence of symmetries,” Journal of Differential Equations, vol. 122, no. 2, pp. 302–315, 1995.
- D. Zhang, “Relative Morse index and multiple brake orbits of asymptotically linear Hamiltonian systems in the presence of symmetries,” Journal of Differential Equations, vol. 245, no. 4, pp. 925–938, 2008.
Copyright © 2013 Yuan Shan and Baoqing Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.