1 Introduction and Main Results

In this paper we prove the existence of periodic solutions for the second order Hamiltonian systems

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d}{dt}\left( \nabla \Phi (\dot{q}(t))\right) +V_{q}(t,q(t))=\lambda W_{q}(t,q(t)),\,t\in [0,T],\\ q(0)-q(T)=\dot{q}(0)-\dot{q}(T)=0, \end{array}\right. } \end{aligned}$$
(1)

where \(V:\mathbb {R}\times \mathbb {R}^{n}\rightarrow \mathbb {R}\) and \(W:\mathbb {R}\times \mathbb {R}^{n}\rightarrow \mathbb {R}\) are \(C^{1}\)-smooth, T-periodic with respect to \(t\in \mathbb {R}\), \(n\ge 1\), \(T>0\), \(\lambda \) is a real small parameter and \(\Phi :\mathbb {R}^{n}\rightarrow [0,\infty )\) is a G-function in the sense of Trudinger, i.e. \(\Phi (0)=0\), \(\Phi \) is \(C^{1}\)-smooth, coercive, convex and symmetric, and \(\nabla \Phi \in C^{1}\left( \mathbb {R}^{n}\setminus \{0\},\mathbb {R}^{n}\right) \). Here and subsequently \(V_{q}:\mathbb {R}\times \mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) and \(W_{q}:\mathbb {R}\times \mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) denote the gradient maps of V and W, respectively, with respect to \(q\in \mathbb {R}^{n}\). From now on \((\cdot ,\cdot ):\mathbb {R}^{n}\times \mathbb {R}^{n}\rightarrow \mathbb {R}\) stands for the standard inner product in \(\mathbb {R}^{n}\) and \(|\cdot |:\mathbb {R}^{n}\rightarrow [0,\infty )\) is the Euclidean norm. We assume the conditions below:

(a):

there exists a constant \(\alpha >0\) such that

$$\begin{aligned} V(t,q)+\alpha |q|^{2}\le V(t,0) \end{aligned}$$

for all \(t\in [0,T]\) and \(q\in \mathbb {R}^{n}\);

\((\Delta _{2})\):

there is a constant \(L>0\) such that

$$\begin{aligned} \Phi (2q)\le L\Phi (q) \end{aligned}$$

for each \(q\in \mathbb {R}^{n}\);

\((\nabla _{2})\):

there exists a constant \(l>0\) such that

$$\begin{aligned} \Phi (lq)\ge 2\,l\Phi (q) \end{aligned}$$

for each \(q\in \mathbb {R}^{n}\).

Our assumptions imply that the action functional corresponding to the system (1) with \(\lambda =0\) satisfies the Palais–Smale condition (Lemma 2.1 in Sect. 2). Let us also remark that \(q\equiv 0\) is a solution of (1) for \(\lambda =0\). Our aim is to prove the existence of periodic solutions of (1) for \(|\lambda |\) small enough without any extra conditions on W.

Let us consider the Orlicz space

$$\begin{aligned} L^{\Phi }(0,T;\mathbb {R}^{n})=\left\{ q:\mathbb {R}\rightarrow \mathbb {R}^{n}:q \,\text {is { T}-periodic, measurable,} \int _{0}^{T}\Phi (q(t))dt<\infty \right\} \end{aligned}$$

with the Luxemburg norm

$$\begin{aligned} \Vert q\Vert _{\Phi }=\inf \left\{ v>0:\int _{0}^{T}\Phi \left( \frac{q(t)}{v}\right) dt\le 1\right\} . \end{aligned}$$

It is well-known that \(L^{\Phi }(0,T;\mathbb {R}^{n})\) is a Banach space (cf. [11]). As \(\Phi \) is \(\Delta _{2}\)-regular and \(\nabla _{2}\)-regular, \(L^{\Phi }(0,T;\mathbb {R}^{n})\) is separable and reflexive (cf. [1]). Moreover, it is not difficult to show that

$$\begin{aligned} \Vert q\Vert _{\Phi }\le 1+\int _{0}^{T}\Phi (q(t))dt,\,\,\, q\in L^{\Phi }(0,T;\mathbb {R}^{n}). \end{aligned}$$
(2)

Proposition 1.1

(cf. [3], Lem. 3.16) Let \(q_{k}\) be a sequence in \(L^{\Phi }(0,T;\mathbb {R}^{n})\) and \(q\in L^{\Phi }(0,T;\mathbb {R}^{n})\). If \(q_{k}\rightarrow q\) almost everywhere in (0, T) and \(\int _{0}^{T}\Phi (q_{k}(t))dt\rightarrow \int _{0}^{T}\Phi (q(t))dt\) then \(q_{k}\rightarrow q\) in \(L^{\Phi }(0,T;\mathbb {R}^{n})\).

The mixed Orlicz–Sobolev space \(W_{T}^{1,\Phi }\) is the space of functions \(q\in L^{2}(0,T;\mathbb {R}^{n})\) having a weak derivative \(\dot{q}\in L^{\Phi }(0,T;\mathbb {R}^{n})\). Let us recall that, if \(q\in W_{T}^{1,\Phi }\),

$$\begin{aligned} q(t)=\int _{0}^{t}\dot{q}(s)ds+c \end{aligned}$$

and \(q(0)=q(T)\). The norm over \(W_{T}^{1,\Phi }\) is defined by

$$\begin{aligned} \Vert q\Vert ^{2}=\Vert q\Vert _{2}^{2}+\Vert \dot{q}\Vert _{\Phi }^{2}, \end{aligned}$$

where

$$\begin{aligned} \Vert q\Vert _{2}=\left( \int _{0}^{T}|q(t)|^{2}dt\right) ^\frac{1}{2}. \end{aligned}$$

It is easy to verify that \(W_{T}^{1,\Phi }\) is a reflexive Banach space.

Proposition 1.2

(cf. [8], Prop. 2.1) There exists a positive constant \(C_{\Phi }\) such that for \(q\in W_{T}^{1,\Phi }\),

$$\begin{aligned} \Vert q\Vert _{\infty }\le C_{\Phi }\Vert q\Vert , \end{aligned}$$
(3)

where \(\Vert q\Vert _{\infty }=\max _{t\in [0,T]}|q(t)|\).

By Proposition 2.3 of [8], the imbedding of \(W_{T}^{1,\Phi }\) in \(C(0,T;\mathbb {R}^{n})\), with its natural norm \(\Vert \cdot \Vert _{\infty }\), is compact. We are now ready to state the announced result.

Theorem 1.3

Let V(t, q) and W(t, q) be \(C^{1}\)-smooth on \(\mathbb {R}\times \mathbb {R}^{n}\), T-periodic in t, and \(\Phi (q)\) be a G-function. Under the assumptions (a), \((\Delta _{2})\), \((\nabla _{2})\), the following assertions hold.

  1. (i)

    There is a positive number \(\lambda _{0}\) such that the system (1) has a solution \(q_{\lambda }\) when \(|\lambda |\le \lambda _{0}\).

  2. (ii)

    For any sequence \(\lambda _{j}\) converging to zero, along a subsequence \(q_{\lambda _{j}}\) converges to zero in \(W_{T}^{1,\Phi }\).

Let us emphasize that we mean by solution of (1) an absolutely continuous function in \(L^{2}(0,T;\mathbb {R}^{n})\) that satisfies (1) weakly. If we require that \(\Phi \) is not only convex but stricly convex, then \(q_{\lambda }\) has a classical first derivative. There are many important examples of \(\Phi \) satisfying our assumptions. If we set \(\Phi (q)=\frac{1}{2}|q|^{2}\), \(q\in \mathbb {R}^{n}\), we obtain the classical second order Hamiltonian systems. Applications of fundamental techniques of critical point theory to the existence of periodic solutions of second order Hamiltonian systems were presented e.g. in [9]. If we set \(\Phi (q)=\frac{1}{p}|q|^{p}\), \(q\in \mathbb {R}^{n}\), \(1<p<\infty \), we get the one-dimensional p-Laplacian. Nonlinear perturbations of this operator have been studied recently e.g. in [2, 5, 6]. Variational systems involving p-Laplacian occur naturally in a variety of settings in physics and engineering [2]. Moreover, let us remind an anisotropic example \(\Phi (q)=\sum _{i=1}^{n}a_{i}|q_{i}|^{p_{i}}\), \(1<p_{i}<\infty \), \(a_{i}>0\), \(q=(q_{1},q_{2},\ldots ,q_{n})\), which has been investigated e.g. in [4, 10].

2 Proof of Theorem 1.3

We shall prove Theorem 1.3. Our approach is based on Ekeland’s variational principle. For (1) with \(\lambda =0\), we define the Lagrangian functional by

$$\begin{aligned} I_{0}(q)=\int _{0}^{T}\left( \Phi (\dot{q}(t))-V(t,q(t))\right) dt, \end{aligned}$$
(4)

where \(\Phi \) and V satisfy our assumptions. Then \(I_{0}\) is well-defined in \(W_{T}^{1,\Phi }\) and becomes a \(C^{1}\)-functional (cf. [8], Prop. 2.10). Moreover, \(I_{0}\) is bounded from below. Using (a), we get

$$\begin{aligned} I_{0}(q)\ge \int _{0}^{T}-V(t,q(t))dt\ge \int _{0}^{T}-V(t,0)dt =: V_{0}. \end{aligned}$$
(5)

From an easy calculation, we also see that

$$\begin{aligned} I^{\prime }_{0}(q)v=\int _{0}^{T}\left( (\nabla \Phi (\dot{q}(t)),\dot{v}(t))-(V_{q}(t,q(t)),v(t))\right) dt, \end{aligned}$$
(6)

where \(q,v\in W_{T}^{1,\Phi }\).

Lemma 2.1

\(I_{0}\) satisfies the Palais–Smale condition.

Proof

Let \(q_{k}\) be any sequence in \(W_{T}^{1,\Phi }\) such that \(I_{0}(q_{k})\) is bounded and \(I_{0}^{\prime }(q_{k})\) converges to zero in \(\left( W_{T}^{1,\Phi }\right) ^{*}\). By (a) and (2), we obtain

$$\begin{aligned} I_{0}(q)\ge & {} \Vert \dot{q}\Vert _{\Phi }-1+\alpha \int _{0}^{T}|q(t)|^{2}dt+\int _{0}^{T}-V(t,0)dt \nonumber \\= & {} \Vert \dot{q}\Vert _{\Phi }-1+\alpha \Vert q\Vert _{2}^{2}+V_{0}. \end{aligned}$$
(7)

As \(I_{0}(q_{k})\) is bounded, there is \(C>0\) such that \(|I_{0}(q_{k})|\le C\) for each \(k\in \mathbb {N}\). We thus get

$$\begin{aligned} \Vert \dot{q}_{k}\Vert _{\Phi }-1+\alpha \Vert q_{k}\Vert _{2}^{2}+V_{0}\le C \end{aligned}$$
(8)

for each \(k\in \mathbb {N}\). Hence \(q_{k}\) is bounded in \(W_{T}^{1,\Phi }\). Since \(W_{T}^{1,\Phi }\) is reflexive, there is a subsequence of \(q_{k}\) that converges weakly to some \(q\in W_{T}^{1,\Phi }\). We keep denoting this subsequence by \(q_{k}\). By the compact imbedding, \(q_{k}\) converges to q in \(C(0,T;\mathbb {R}^{n})\) and, in consequence, \(q_{k}\) converges to q in \(L^{2}(0,T;\mathbb {R}^{n})\). Moreover, since the modulus function increases essentially more slowly than \(\Phi \) near infinity \(\dot{q}_{k}\) goes to \(\dot{q}\) in \(L^{1}(0,T;\mathbb {R})\), and hence, along a subsequence \(\dot{q}_{k}\) goes to \(\dot{q}\) almost everywhere in (0, T). Without loss of generality we denote this subsequence by \(q_{k}\). According to the above remarks, we have

$$\begin{aligned}{} & {} |I_{0}^{\prime }(q_{k})(q_{k}-q)|\le \Vert I_{0}^{\prime }(q_{k})\Vert _{\left( W_{T}^{1,\Phi }\right) ^{*}}\Vert q_{k}-q\Vert \rightarrow 0,\\{} & {} \int _{0}^{T}\left( V_{q}(t,q_{k}(t)),q_{k}(t)-q(t)\right) dt\rightarrow 0, \end{aligned}$$

and consequently,

$$\begin{aligned}&\int _{0}^{T}\left( \nabla \Phi (\dot{q}_{k}(t)),\dot{q}_{k}(t)-\dot{q}(t)\right) dt=I_{0}^{\prime }(q_{k})(q_{k}-q)\, \nonumber \\&\quad +\int _{0}^{T}\left( V_{q}(t,q_{k}(t)),q_{k}(t)-q(t)\right) dt\rightarrow 0 \end{aligned}$$
(9)

as \(k\rightarrow \infty \). As \(\Phi \) is convex,

$$\begin{aligned} \Phi (x)-\Phi (x-y)\le (\nabla \Phi (x),y) \end{aligned}$$

for each \(x,y\in \mathbb {R}^{n}\). From this it follows that

$$\begin{aligned}{} & {} \int _{0}^{T}\Phi (\dot{q}_{k}(t))dt-\int _{0}^{T}\Phi (\dot{q}(t))dt\le \int _{0}^{T}(\nabla \Phi (\dot{q}_{k}(t)),\dot{q}_{k}(t)-\dot{q}(t))dt,\\{} & {} \int _{0}^{T}\Phi (\dot{q}_{k}(t))dt\le \int _{0}^{T}\Phi (\dot{q}(t))dt+\int _{0}^{T}(\nabla \Phi (\dot{q}_{k}(t)),\dot{q}_{k}(t)-\dot{q}(t))dt. \end{aligned}$$

Letting \(k\rightarrow \infty \) we obtain

$$\begin{aligned} \limsup _{k\rightarrow \infty }\int _{0}^{T}\Phi (\dot{q}_{k}(t))dt\le \int _{0}^{T}\Phi (\dot{q}(t))dt. \end{aligned}$$

On the other hand, by Fatou’s lemma

$$\begin{aligned} \liminf _{k\rightarrow \infty }\int _{0}^{T}\Phi (\dot{q}_{k}(t))dt\ge \int _{0}^{T}\Phi (\dot{q}(t))dt. \end{aligned}$$

Therefore

$$\begin{aligned} \lim _{k\rightarrow \infty }\int _{0}^{T}\Phi (\dot{q}_{k}(t))dt=\int _{0}^{T}\Phi (\dot{q}(t))dt, \end{aligned}$$

and finally, by Proposition 1.1, \(\dot{q}_{k}\rightarrow \dot{q}\) in \(L^{\Phi }(0,T;\mathbb {R}^{n})\). Since \(q_{k}\rightarrow q\) in \(L^{2}(0,T;\mathbb {R}^{n})\) and \(\dot{q}_{k}\rightarrow \dot{q}\) in \(L^{\Phi }(0,T;\mathbb {R}^{n})\), we have \(q_{k}\rightarrow q\) in \(W_{T}^{1,\Phi }\), which completes the proof. \(\square \)

We now choose a function such that \(0\le h(x)\le 1\) in \(\mathbb {R}^{n}\), \(h(x)=1\) for \(|x|\le C_{\Phi }\) and \(h(x)=0\) for \(|x|\ge 2C_{\Phi }\), where \(C_{\Phi }\) is given by (3). We define

$$\begin{aligned} I_{\lambda }(q)=\int _{0}^{T}\left( \Phi (\dot{q}(t))-V(t,q(t))+\lambda h(q(t))W(t,q(t))\right) dt, \end{aligned}$$
(10)

where \(q\in W_{T}^{1,\Phi }\). Then a critical point of \(I_{\lambda }\) is a solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d}{dt}\left( \nabla \Phi (\dot{q}(t))\right) +V_{q}(t,q(t))=\lambda h(q(t))W_{q}(t,q(t))+\lambda \nabla h(q(t))W(t,q(t))\\ q(0)-q(T)=\dot{q}(0)-\dot{q}(T)=0. \end{array}\right. } \end{aligned}$$
(11)

Our plan to prove Theorem 1.3 is as follows. First, we find a critical point \(q_{\lambda }\) of \(I_{\lambda }\). Next, we show that \(\Vert q_{\lambda }\Vert _{\infty }\le C_{\Phi }\) for \(|\lambda |\) small enough. Then \(h(q_{\lambda })=1\), \(\nabla h(q_{\lambda })=0\) and therefore \(q_{\lambda }\) becomes a solution of (1). Set

$$\begin{aligned} C_{0}=\max \{W(t,q):t\in [0,T]\,\wedge \,|q|\le 2C_{\Phi }\}. \end{aligned}$$

We have

$$\begin{aligned} I_{\lambda }(q)=I_{0}(q)+\lambda \int _{0}^{T}h(q(t))W(t,q(t))dt\ge V_{0}-|\lambda |TC_{0}, \end{aligned}$$

and so \(I_{\lambda }\) is bounded from below. Using the same arguments as in Lemma 2.1 with the fact that h(q)W(t, q) and its gradient with respect to q are bounded, we get the next lemma.

Lemma 2.2

For each \(\lambda \in \mathbb {R}\), \(I_{\lambda }\) satisfies the Palais–Smale condition.

Applying Ekeland’s variational principle we conclude that \(I_{\lambda }\) has a minimum on \(W_{T}^{1,\Phi }\). It follows that there is \(q_{\lambda }\in W_{T}^{1,\Phi }\) such that

$$\begin{aligned} I_{\lambda }(q_{\lambda })=\inf _{q\in W_{T}^{1,\Phi }}I_{\lambda }(q)\,\,\wedge \,\,I_{\lambda }^{\prime }(q_{\lambda })=0. \end{aligned}$$

Since

$$\begin{aligned} I_{0}(q)-|\lambda |TC_{0}\le I_{\lambda }(q)\le I_{0}(q)+|\lambda |TC_{0} \end{aligned}$$

for each \(q\in W_{T}^{1,\Phi }\), we obtain \(I_{\lambda }(q_{\lambda })\rightarrow V_{0}\) as \(\lambda \rightarrow 0\).

Lemma 2.3

Let \(\lambda _{m}\) be a sequence converging to zero and let the functional \(I_{\lambda _{m}}\) reach a minimum at the point \(q_{\lambda _{m}}\). Then a subsequence of \(q_{\lambda _{m}}\) converges to zero in \(W_{T}^{1,\Phi }\).

Proof

By definition,

$$\begin{aligned} I_{\lambda _{m}}(q_{\lambda _{m}})=\inf _{q\in W_{T}^{1,\Phi }}I_{\lambda _{m}}(q)\,\,\wedge \,\,I_{\lambda _{m}}^{\prime }(q_{\lambda _{m}})=0, \end{aligned}$$

and hence \(q_{\lambda _{m}}\) is a solution of (11) with \(\lambda \) replaced by \(\lambda _{m}\). Using the same argument as in the proof of Lemma 2.1, by the boundedness of \(I_{\lambda _{m}}(q_{\lambda _{m}})\), we can conclude that \(q_{\lambda _{m}}\) is bounded in \(W_{T}^{1,\Phi }\) and a subsequence of \(q_{\lambda _{m}}\) converges to a limit \(q_{0}\) in \(W_{T}^{1,\Phi }\). Then \(q_{0}\) satisfies that \(I_{0}(q_{0})=V_{0}\) and \(I_{0}^{\prime }(q_{0})=0\), i.e. \(q_{0}\equiv 0\). \(\square \)

Lemma 2.4

There is \(\lambda _{0}>0\) such that for \(|\lambda |\le \lambda _{0}\) we have \(\Vert q_{\lambda }\Vert _{\infty }\le C_{\Phi }\).

Proof

Suppose on the contrary to our claim that there is a sequence \(\lambda _{m}\) converging to zero such that \(\Vert q_{\lambda _{m}}\Vert _{\infty }> C_{\Phi }\). By Lemma 2.3 it follows that there is a subsequence of \(q_{\lambda _{m}}\) going to zero in \(W_{T}^{1,\Phi }\). Without loss of generality we will denote this subsequence by \(q_{\lambda _{m}}\). Thus for m large enough, \(\Vert q_{\lambda _{m}}\Vert \le 1\), and consequently \(\Vert q_{\lambda _{m}}\Vert _{\infty }\le C_{\Phi }\), by (3). A contradiction occurs. \(\square \)

The lemma above will be used to find a solution of (1). We are now in a position to prove Theorem 1.3.

Proof (Proof of Theorem 1.3)

[Proof of Theorem 1.3] Choose \(\lambda _{0}>0\) that satisfies Lemma 2.4. Let \(I_{\lambda }\) reach a minimum at \(q_{\lambda }\) with \(|\lambda |\le \lambda _{0}\). Then \(\Vert q_{\lambda }\Vert _{\infty }\le C_{\Phi }\). For this reason \(h(q_{\lambda })=1\), \(\nabla h(q_{\lambda })=0\), and consequently \(q_{\lambda }\) becomes a solution of (1). Let \(\lambda _{j}\) be a sequence converging to zero. From Lemma 2.3 it follows that a subsequence of \(q_{\lambda _{j}}\) converges to zero in \(W_{T}^{1,\Phi }\), which completes the proof. \(\square \)

We conclude our work by explaining the regularity of solutions of (1) in case that \(\Phi \) is strictly convex. We set for \(|\lambda |\le \lambda _{0}\) and \(t\in [0,T]\),

$$\begin{aligned} x_{\lambda }(t)=\nabla \Phi \left( \dot{q}_{\lambda }(t)\right) . \end{aligned}$$

Let us note that

$$\begin{aligned} \dot{x}_{\lambda }(t)=\frac{d}{dt}\left( \nabla \Phi \left( \dot{q}_{\lambda }(t)\right) \right) =-V_{q}(t,q_{\lambda }(t))+\lambda W_{q}(t,q_{\lambda }(t)), \end{aligned}$$

and so it is continuously differentiable. It is known that if \(\Phi \) is strictly convex then \(\nabla \Phi :\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) is invertible and its inverse map \((\nabla \Phi )^{-1}=\nabla \Phi ^{*}\) is continuous (Corollary 4.1.3 in [7]), where \(\Phi ^{*}\) denotes the Fenchel transform of \(\Phi \) defined by

$$\begin{aligned} \Phi ^{*}(y)=\sup _{x\in \mathbb {R}^{n}}\left( (x,y)-\Phi (x)\right) . \end{aligned}$$

Hence \(\dot{q}_{\lambda }(t)=(\nabla \Phi )^{-1}(x_{\lambda }(t))\) is continuously differentiable too. Finally, if \(\nabla \Phi ^{*}\) is \(C^{1}\) then \(q_{\lambda }\) is \(C^{2}\), i.e. a classical solution. These additional assumptions are satisfied for \(\Phi (x)=\frac{1}{p}|x|^{p}\), \(1<p\le 2\).