Homoclinic solutions for a kind of prescribed mean curvature Duffing-type equation
© Liang and Lu; licensee Springer. 2013
Received: 11 June 2013
Accepted: 27 August 2013
Published: 30 September 2013
In this paper, by using Mawhin’s continuation theorem and some analysis methods, the existence of a set with -periodic solutions for a kind of prescribed mean curvature Duffing-type equation is studied, and then a homoclinic solution is obtained as a limit of a certain subsequence of the above set.
where , , is a given constant.
As is well known, a solution of Eq. (1.1) is named homoclinic (to 0) if and as . In addition, if , then u is called a nontrivial homoclinic solution.
A prescribed mean curvature equation and its modified forms derived from differential geometry and physics have been widely researched in many papers. For example, combustible gas dynamics [1–3]. In recent years, many papers about periodic solutions for the prescribed mean curvature equation and its modified forms have appeared. For example, by using an approach based on the Leray-Schauder degree, Benevieri et al. in  studied the periodic solutions for nonlinear equations with mean curvature-like operators. And in  Benevieri et al. extended the results obtained in  to the N-dimensional case.
By using Mawhin’s continuation theorem in the coincidence degree theory, and given some sufficient conditions, the authors obtained that Eq. (1.2) has at least one periodic solution. From the first equation of (1.3), we can see that a T-periodic function must satisfy , hence the open and bounded set Ω of Mawhin’s continuation theorem must satisfy . But in , the authors obtained , there is no proof about . A similar problem also occurred in  and .
is a constant independent of k. The existence of -periodic solutions to Eq. (1.4) is obtained by using Mawhin’s continuation theorem . We obtain , where , by which we overcome the problem in [6–8]. The rest of this paper organized as follows. In Section 2, we provide some necessary background definitions and lemmas. In Section 3, we give the results that we have obtained.
In order to use Mawhin’s continuation theorem , we first recall it.
ImL is a closed subset of Y,
is a relative compact set of X,
is a bounded set of Y,
where we define , . Then we have the decompositions , . Let , be continuous linear projectors (meaning and ), and .
Lemma 2.1 
Let X and Y be two Banach spaces with norms , , respectively, and let Ω be an open and bounded set of X. Let be a Fredholm operator of index zero, and let be L-compact on . In addition, if the following conditions hold:
(H1) , ;
(H2) , ;
(H3) , where is just any homeomorphism, then Lv=Nv has at least one solution in .
This lemma is Corollary 2.1 in .
Lemma 2.3 
where , and are constants independent of . Then there exists a function such that for each interval , there is a subsequence of with uniformly on .
Let and , where the norm with and . It is obvious that and are Banach spaces.
then , , obviously , , thus , and it is also easy to prove that . So, L is a Fredholm operator of index zero.
For all Ω such that , we have is a relative compact set of , is a bounded set of , so the operator N is L-compact in .
3 Main results
For the sake of convenience, we list the following conditions.
(A1) There exist constants , such that and , .
(A2) is a bounded function with and , where .
Remark 3.1 From (1.5) we see that . So, if assumption (A2) holds, for each , and .
where . For each and all , let Σ represent the set of all the -periodic solutions to the above system.
Obviously, is a constant which is independent of k and λ. From (3.9), (3.12), (3.13) and (3.14), we know , , and are constants independent of k and λ. Hence the conclusion of Theorem 3.1 holds. □
where , , , are constants defined by Theorem 3.1.
Hence the conclusion of Theorem 3.2 holds. □
Theorem 3.3 Suppose that the conditions in Theorem 3.1 hold, then Eq. (1.1) has a nontrivial homoclinic solution.
Clearly, is a constant independent of . By using Lemma 2.3, we see that there is a function such that for each interval , there is a subsequence of with uniformly on . Below we show that is just a homoclinic solution to Eq. (1.1).
Considering that a, b are two arbitrary constants with , it is easy to see that , is a solution to system (1.1).
which contradicts (3.21), thus (3.23) holds. Clearly, , otherwise , which contradicts assumption (A2). Hence the conclusion of Theorem 3.3 holds. □
Research supported by the NNSF of China (No. 11271197) and the key NSF of Education Ministry of China (No. 207047).
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