Solutions for perturbed fractional Hamiltonian systems without coercive conditions
 Xionghua Wu^{1} and
 Ziheng Zhang^{1}Email author
Received: 26 June 2015
Accepted: 31 July 2015
Published: 28 August 2015
Abstract
In this paper we are concerned with the existence of solutions for the following perturbed fractional Hamiltonian systems: \({}_{t}D^{\alpha}_{\infty}({}_{\infty}D^{\alpha}_{t}u(t)) L(t)u(t)+\nabla W(t,u(t))=f(t)\), \(u\in H^{\alpha}(\mathbb{R},\mathbb{R}^{n})\) (PFHS), where \(\alpha\in(1/2,1)\), \(t\in\mathbb{R}\), \(u\in\mathbb{R}^{n}\), \(L\in C(\mathbb{R},\mathbb{R}^{n^{2}})\) is a symmetric and positive definite matrix for all \(t\in\mathbb{R}\), \(W\in C^{1}(\mathbb{R}\times \mathbb{R}^{n},\mathbb{R})\), and \(\nabla W(t,u)\) is the gradient of \(W(t,u)\) at u, \(f\in C(\mathbb{R},\mathbb{R}^{n})\) and belongs to \(L^{2}(\mathbb{R},\mathbb{R}^{n})\). The novelty of this paper is that, assuming \(L(t)\) is bounded in the sense that there are constants \(0<\tau_{1}<\tau_{2}<\infty\) such that \(\tau_{1} u^{2}\leq(L(t)u,u)\leq\tau_{2} u^{2}\) for all \((t,u)\in\mathbb{R}\times\mathbb{R}^{n}\) and \(W(t,u)\) satisfies the AmbrosettiRabinowitz condition and some other reasonable hypotheses, \(f(t)\) is sufficiently small in \(L^{2}(\mathbb{R},\mathbb{R}^{n})\), we show that (PFHS) possesses at least two nontrivial solutions. Recent results are generalized and significantly improved.
Keywords
MSC
1 Introduction
Fractional differential equations both ordinary and partial ones are applied in mathematical modeling of process in physics, mechanics, control theory, biochemistry, bioengineering and economics. Therefore, the theory of fractional differential equations is an area intensively developed during the last decades [1, 2]. The monographs [3–5] enclose a review of methods of solving fractional differential equations, which are an extension of procedures from differential equations theory.
Recently, also equations including both left and right fractional derivatives are discussed. Apart from their possible applications, equations with left and right derivatives is an interesting and new field in fractional differential equations theory. In this topic, many results are obtained dealing with the existence and multiplicity of solutions of nonlinear fractional differential equations by using techniques of nonlinear analysis, such as fixed point theory (including the LeraySchauder nonlinear alternative) [6], topological degree theory (including coincidence degree theory) [7] and comparison method (including upper and lower solutions and monotone iterative method) [8] and so on.
It should be noted that critical point theory and variational methods have also turned out to be very effective tools in determining the existence of solutions for integer order differential equations. The idea behind them is to try to find solutions of a given boundary value problem by looking for critical points of a suitable energy functional defined on an appropriate function space. In the last 30 years, critical point theory has become a wonderful tool in studying the existence of solutions to differential equations with variational structures; we refer the reader to Mawhin and Willem [9], Rabinowitz [10], Schechter [11], and the references listed therein.
Assuming that \(L(t)\) and \(W(t,u)\) are independent of t or periodic in t, many authors have studied the existence of homoclinic solutions for (HS); see for instance [15–17] and the references therein and some more general Hamiltonian systems are considered in [18, 19]. In this case, the existence of homoclinic solutions can be obtained by going to the limit of periodic solutions of approximating problems. If \(L(t)\) and \(W(t,u)\) are neither autonomous nor periodic in t, the existence of homoclinic solutions of (HS) is quite different from the periodic systems, because of the lack of compactness of the Sobolev embedding, such as in [14, 17, 20] and the references mentioned there.
 (L):

\(L(t)\) is a positive definite symmetric matrix for all \(t\in \mathbb {R}\) and there exists an \(l\in C(\mathbb {R},(0,\infty))\) such that \(l(t)\rightarrow\infty\) as \(t\rightarrow\infty\) and$$ \bigl(L(t)u,u\bigr)\geq l(t)u^{2} \quad\mbox{for all } t\in \mathbb {R}\mbox{ and } u\in \mathbb {R}^{n}. $$(1.1)
 (W_{1}):

\(W\in C^{1}(\mathbb {R}\times \mathbb {R}^{n},\mathbb {R})\) and there is a constant \(\theta>2\) such that$$0< \theta W(t,u)\leq\bigl(\nabla W(t,u),u\bigr) \quad\mbox{for all } t\in \mathbb {R}\mbox{ and } u\in \mathbb {R}^{n}\backslash\{0\}. $$
 (W_{2}):

\(\nabla W(t,u)=o(u)\) as \(u\rightarrow0\) uniformly with respect to \(t\in \mathbb {R}\).
 (W_{3}):

There exists \(\overline{W}\in C(\mathbb {R}^{n},\mathbb {R})\) such that$$\biglW(t,u)\bigr+\bigl\nabla W(t,u)\bigr\leq\bigl\overline{W}(u)\bigr \quad\mbox{for every } t\in \mathbb {R}\mbox{ and } u\in \mathbb {R}^{n}. $$
 (L)′:

\(L\in C(\mathbb {R},\mathbb {R}^{n^{2}})\) is a symmetric and positive definite matrix for all \(t\in \mathbb {R}\) and there are constants \(0<\tau _{1}<\tau_{2}<\infty\) such that$$\tau_{1}u^{2}\leq\bigl(L(t)u,u\bigr)\leq \tau_{2}u^{2} \quad\mbox{for all } (t,u)\in \mathbb {R}\times \mathbb {R}^{n}. $$
 (W_{2})′:

there exists some positive continuous function \(a:\mathbb {R}\rightarrow \mathbb {R}\) withsuch that$$ \lim_{t\rightarrow\infty}a(t)=0 $$(1.2)$$\bigl\nabla W(t,u)\bigr\leq a(t)u^{\theta1} \quad\mbox{for all } (t,u)\in \mathbb {R}\times \mathbb {R}^{n}. $$
 (W_{ f }):

\(\varrho<\frac{1}{2C_{2}}\) and \(f:\mathbb {R}\rightarrow \mathbb {R}^{n}\) is a continuous square integrable function such thatwhere \(C_{2}\) and \(C_{\infty}\) are defined in Section 2.$$ \f\_{L^{2}}< \frac{1}{C_{\infty}} \biggl(\frac{1}{2C_{2}} \varrho C_{2} \biggr), $$(1.3)
Theorem 1.1
Suppose that (L)′, (W_{1}), (W_{2})′ and (W_{ f }) are satisfied, then (PFHS) possesses at least two nontrivial solutions.
Remark 1.2
Note that in (L)′, we assume that \(L(t)\) is bounded. Therefore, the coercive condition (L) is not satisfied. Thus the results in [12] are generalized and improved significantly.
Moreover, as mentioned above, the coercive condition (L) is used to establish some compact embedding theorems to guarantee that the (PS) condition (or the other weak compactness conditions) holds, which is the essential step to obtain the existence of homoclinic solutions of (PFHS) via critical point theory and variational methods. In the present paper, we assume that \(L(t)\) satisfies (L)′ and could not obtain some compact embedding theorem. Therefore, the main difficulty is to adapt some new technique to overcome this difficulty and test that the (PS) condition is verified; see Lemmas 3.1 and 3.2 below.
The remaining part of this paper is organized as follows. Some preliminary results are presented in Section 2. Section 3 is devoted to accomplishing the proof of Theorem 1.1.
2 Preliminary results
Let \(C(\mathbb {R},\mathbb {R}^{n})\) denote the space of continuous functions from \(\mathbb {R}\) into \(\mathbb {R}^{n}\). Then we obtain the following lemma.
Lemma 2.1
([21], Theorem 2.1)
Remark 2.2
Similar to Lemma 2.1 in [21], we have the following conclusion. Its proof is just the repetition of Lemma 2.1 of [21], so we omit the details.
Lemma 2.3
Suppose \(L(t)\) satisfies (L)′, then \(X^{\alpha}\) is continuously embedded in \(H^{\alpha}\).
Remark 2.4
Proposition 2.5
([18], Fact 2.1)
 (i)
\(W(t,u)\leq W(t,\frac{u}{u})u^{\theta}\) for \(t\in \mathbb {R}\) and \(0<u\leq1\);
 (ii)
\(W(t,u)\geq W(t,\frac{u}{u})u^{\theta}\) for \(t\in \mathbb {R}\) and \(u\geq1\).
Now we introduce some more notations and necessary definitions. Let \(\mathcal{B}\) be a real Banach space, \(I\in C^{1}(\mathcal{B},\mathbb {R})\) means that I is a continuously Fréchetdifferentiable functional defined on \(\mathcal{B}\). Recall that \(I\in C^{1}(\mathcal{B},\mathbb {R})\) is said to satisfy the (PS) condition if any sequence \(\{u_{n} \}_{n\in \mathbb {N}}\subset \mathcal{B}\), for which \(\{I(u_{n}) \}_{n\in \mathbb {N}}\) is bounded and \(I'(u_{n})\rightarrow0\) as \(n\rightarrow\infty\), possesses a convergent subsequence in \(\mathcal{B}\).
Moreover, let \(B_{r}\) be the open ball in \(\mathcal{B}\) with the radius r and centered at 0 and \(\partial B_{r}\) denotes its boundary. Under the conditions of Theorem 1.1, we obtain the existence of the first solution of (PFHS) by using the following wellknown mountain pass theorem; see [10].
Lemma 2.6
([10], Theorem 2.2)
 (A1)
there are constants ρ, \(\alpha>0\) such that \(I_{\partial B_{\rho}}\geq\alpha\), and
 (A2)
there is an \(e\in\mathcal{B}\setminus\overline {B}_{\rho}\) such that \(I(e)\leq0\).
As far as the second one is concerned, we obtain it by a minimizing method, which is concerned with a small ball centered at 0; see Step 4 in proof of Theorem 1.1.
3 Proof of Theorem 1.1
Lemma 3.1
Proof
Lemma 3.2
Under the conditions of Theorem 1.1, I satisfies the (PS) condition.
Proof
Now we are in the position to give the proof of Theorem 1.1. We divide its proof into four steps.
Proof
Step 1. It is clear that \(I(0)=0\) and \(I\in C^{1}(X^{\alpha}, \mathbb {R})\) satisfies the (PS) condition by Lemma 3.2.
Step 2. We now show that there exist constants \(\rho>0\) and \(\alpha>0\) such that I satisfies condition (A1) of Lemma 2.6. Let \(\rho=\frac{1}{C_{\infty}}\), where \(C_{\infty}\) is defined in (2.5). Assume that \(u\in E\) with \(\u\_{X^{\alpha} }\leq\rho\), then \(\u\_{\infty}\leq1\).
Declarations
Acknowledgements
Project supported by the National Natural Science Foundation of China (Grant No. 11101304).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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