Abstract
We prove some multiplicity results for Neumann-type boundary value problems associated with a Hamiltonian system. Such a system can be seen as the weak coupling of two systems, the first of which has some periodicity properties in the Hamiltonian function, the second one presenting the existence of a well-ordered pair of lower/upper solutions.
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1 Introduction
In the recent paper [12], the first author jointly with R. Ortega have obtained a multiplicity result for a two-point boundary value problem associated with a Hamiltonian system in \({\mathbb {R}}^{2N}\). For simplicity in the exposition, let us recall their result for a planar system
with Neumann-type boundary conditions
Theorem 1
(Fonda–Ortega [12]) Let \(H:[a,b]\times {\mathbb {R}}^{2}\rightarrow {\mathbb {R}}\) be a continuous function with continuous partial derivatives with respect to x and y. Assume moreover that H is \(\tau \)-periodic with respect to x, for some \(\tau >0\), and that all solutions of (1) starting with \(y(a)=0\) are defined on [a, b]. Then, problem (1)-(2) has at least two geometrically distinct solutions.
Let us take a moment to explain what we mean by geometrically distinct solutions. In view of the \(\tau \)-periodicity of H in the x-variable, we can define an equivalence relation in \(C^1([a,b])\times C^1([a,b])\) as follows:
We say that two solutions (x, y) and \(({{\hat{x}}},{{\hat{y}}})\) of (1) are geometrically distinct if they do not belong to the same equivalence class.
There are some similarities between Theorem 1 and the Poincaré–Birkhoff Theorem for the periodic problem associated with the Hamiltonian system (1). However, it is well known that the periodic problem requires an additional assumption, the so-called twist condition. Different versions of it have been proposed (see e.g. [16] for a deeper insight on this topic). Conversely, in the case of the Neumann-type boundary conditions (2), it has been shown in [12] that multiplicity results can be achieved without assuming any twist condition.
Although the Neumann problem for scalar second order equations has been widely studied, there are only few papers in the literature proving multiplicity results for Neumann-type problems associated with systems of ordinary differential equations. See, e.g., [2, 21].
We now consider a four-dimensional system of the form
with Neumann-type boundary conditions
Here \(H:[a,b]\times {\mathbb {R}}^{2}\rightarrow {\mathbb {R}}\) and \(P:[a,b]\times {\mathbb {R}}^4\rightarrow {\mathbb {R}}\) are continuous functions, with continuous partial derivatives with respect to the variables x, y, u, v ; the functions \(f:[a,b]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) and \(g:[a,b]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) are continuous, and \(\varepsilon \) is a small real parameter.
When \(\varepsilon =0\), problem (3)–(4) decouples into two planar problems, the first one being (1)–(2), the second one being
with the boundary conditions
Concerning problem (5)–(6), we will assume the existence of a pair of strict well-ordered lower/upper solutions, the definition of which will be recalled in Sect. 2.
Besides the regularity hypotheses stated above, here is the list of our assumptions.
- (A1):
-
The function \(H=H(t,x,y)\) is \(\tau \)-periodic in the variable x, for some \(\tau >0\) .
- (A2):
-
All solutions (x, y) of (1) starting with \(y(a)=0\) are defined on [a, b] .
- (A3):
-
The function \(P=P(t,x,y,u,v)\) is \(\tau \)-periodic in the variable x .
- (A4):
-
The function \(P=P(t,x,y,u,v)\) has a bounded gradient with respect to \(z=(x,y,u,v)\), i.e., there exists \(C>0\) such that
$$\begin{aligned} |\nabla _{z} P(t,z)|\le C\quad \hbox {for every }\,(t,z)\in [a,b]\times {\mathbb {R}}^4. \end{aligned}$$ - (A5):
-
There exist a strict lower solution \(\alpha \) and a strict upper solution \(\beta \) for problem (5)-(6) such that \(\alpha \le \beta \) .
- (A6):
-
The function f has continuous partial derivative with respect to the variable v and there exists \(\lambda >0\) such that
$$\begin{aligned} \partial _{v}f(t,\,v)\ge \lambda ,\hbox { for every }(t,v)\in [a,b]\times {\mathbb {R}}. \end{aligned}$$ - (A7):
-
\(\partial _{v}P\) is independent of x and y, and locally Lipschitz continuous in v.
Let us state our main result.
Theorem 2
Let assumptions \((A1)-(A7)\) hold true. Then, there exists \(\overline{\varepsilon }>0\) such that, when \(|\varepsilon |\le \overline{\varepsilon }\), problem (3)–(4) has at least two solutions (x, y, u, v) with \(\alpha \le u\le \beta \).
The proof is provided in Sect. 3. It relies on a Theorem by Szulkin [23], which can be seen as an infinite-dimensional extension of the classical Lusternik–Schnirelmann theory on the multiplicity of critical points.
Theorem 2 is the counterpart of the main result in [15] for the periodic problem associated with system (3). In that paper, a further twist condition was needed in order to apply an extension of the Poincaré–Birkhoff Theorem due to the first author and P. Gidoni [7]. Surprisingly enough, in Theorem 2, as for Theorem 1, no twist condition is needed.
In Sects. 4 and 5 we will extend Theorem 2 to higher dimensions. An analogue of system (3) will be considered in \({\mathbb {R}}^{2M}\times {\mathbb {R}}^{2L}\), assuming periodicity in the variables \(x_1,\dots ,x_M\). We will obtain the existence of at least \(M+1\) solutions to the related Neumann-type boundary value problem assuming either the existence of a pair of well-ordered vector valued lower/upper solutions (in Sect. 4), or a Hartman-type condition (in Sect. 5).
2 Lower and upper solutions
Let us recall the definitions of lower and upper solutions for planar systems (cf. [9, 13, 14]).
Definition 3
A \(C^{1}\)-function \(\alpha :[a,b]\rightarrow {\mathbb {R}}\) is a lower solution for problem (5)–(6) if there exists a \(C^{1}\)-function \(v_{\alpha }:[a,b]\rightarrow {\mathbb {R}}\) such that, for every \(t\in [a,b]\),
and
The lower solution \(\alpha \) is strict if the strict inequality holds in (8), for every \(t\in [a,b]\).
Definition 4
A \(C^{1}\)-function \(\beta :[a,b]\rightarrow {\mathbb {R}}\) is an upper solution for problem (5)–(6) if there exists a \(C^{1}\)-function \(v_{\beta }:[a,b]\rightarrow {\mathbb {R}}\) such that, for every \(t\in [a,b]\),
and
The upper solution \(\beta \) is strict if the strict inequality holds in (10), for every \(t\in [a,b]\).
We will consider the case when the pair of lower/upper solutions is well-ordered, i.e., such that \(\alpha \le \beta \). For an intuitive meaning of the previous definitions, see Fig. 1.
A visual illustration of hypotheses (7) and (9) from a dynamical point of view. Horizontal arrows represent the relative velocity \(u'\) of solutions of system (5) compared with \(\alpha '\) and \(\beta '\). The trajectories enter or exit the grey regions through the vertical lines depending on the direction of the arrows. Notice that dashed lines may move in time
Let us recall the following consequence of [13, Corollary 10].
Theorem 5
Let assumption (A6) hold true. If \(\alpha ,\beta \) is a well-ordered pair of lower/upper solutions for problem (5)–(6), then there exists a solution (u, v) of (5)–(6) such that \(\alpha \le u\le \beta \).
Remark 6
As an immediate consequence of (7) and (9) we have, respectively,
for every \(t\in [a,b]\). In view of Assumption (A6), these identities uniquely determine the functions \(v_\alpha \) and \(v_\beta \).
Remark 7
In the case when \(f(t,v)=v\), problem (5)–(6) is equivalent to the Neumann problem
In such a case a lower solution \(\alpha \) will satisfy the usual conditions
while for an upper solution \(\beta \) we will have
Indeed, in this case it is sufficient to choose \(v_\alpha =\alpha '\) and \(v_\beta =\beta '\). These are the classical definitions of lower/upper solutions dating back to the pioneering works [19, 20, 22] (for a historical account, see [3]).
3 The proof of Theorem 2
In this section we will prove Theorem 2. Precisely, in Sect. 3.1 we modify the problem and provide some useful lemmas. Then, in Sect. 3.2, we define the function spaces where the variational problem will be settled. In Sects. 3.3 and 3.4 we introduce the functional whose critical points correspond to the solutions of the modified problem, the existence of which will follow from the application of a theorem by Szulkin. Finally, we will show that such solutions are indeed solutions of the original problem.
Without loss of generality, from now on, we will assume \([a,b]=[0,\pi ]\). Moreover, it is not restrictive to look for \(\overline{\varepsilon }\) in \(\,]0,1]\).
3.1 Some preliminaries
In this section we provide some preliminary tools which will be useful for proving the main result. First of all we remark that, since the inequalities in (8) and (10) are assumed to be strict, by continuity there exists a \({{\bar{\delta }}}>0\) such that, if \(0<\delta \le {{\bar{\delta }}}\), then \(\alpha (t)+\delta \) and \(\beta (t)-\delta \) are still a well-ordered pair of lower/upper solutions for problem (5)–(6), with the same associated functions \(v_\alpha \) and \(v_\beta \). In what follows we replace \(\alpha (t)\) with \(\alpha (t)+\delta \) and \(\beta (t)\) with \(\beta (t)-\delta \).
Before stating the next lemma, we recall that, by assumption (A7), the function \(\partial _{v}P\) does not depend on x and y.
Lemma 8
For every \(\varepsilon \in {\mathbb {R}}\) there exist some \(C^1\)-functions \(\alpha _\varepsilon \) and \(\beta _\varepsilon \) such that
- (i):
-
\(f(t,v_{\alpha }(t))+\varepsilon \partial _{v}P(t,\alpha _\varepsilon (t),v_\alpha (t))=\alpha _{\varepsilon }'(t)\) ,
- (ii):
-
\(f(t,v_{\beta }(t))+\varepsilon \partial _{v}P(t,\beta _\varepsilon (t),v_\beta (t))=\beta _{\varepsilon }^{'}(t)\) ,
- (iii):
-
\(|\alpha _{\varepsilon }(t)-\alpha (t)|<\varepsilon C\pi \), and \(|\beta _{\varepsilon }(t)-\beta (t)|<\varepsilon C\pi \) ,
for every \(t\in [0,\,\pi ]\), where the constant C is defined in assumption (A4).
Proof
Let \(\Gamma :[0,\pi ]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) be the continuous function defined by
and let \(w_{\varepsilon }:[0,\pi ]\rightarrow {\mathbb {R}}\) be a solution of the Cauchy problem
Define
Then, recalling (11), we get
so that (i) is proved. Notice that
with the constant C provided by (A4). Hence, \(|w_{\varepsilon }(t)|\le \varepsilon C\pi \), for every \(t\in [0,\pi ]\), thus proving the first part of (iii). An analogous argument applies for proving the existence of the function \(\beta _\varepsilon \) satisfying (ii) and its property in (iii).
\(\square \)
Remark 9
Lemma 8 above is the analogue of [15, Lemma 3.1]. We observe however that in [15] a different approach was chosen, i.e., the functions \(\alpha \) and \(\beta \) were kept the same for every \(\varepsilon \), while \(v_\alpha \) and \(v_\beta \) varied. Our approach here permits to avoid some regularity assumptions needed in [15].
Here after, we are going to modify system (3). Set
and choose
We define \({\tilde{f}}:[0,\,\pi ]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) by
where
and \({\tilde{g}}_{\varepsilon }:[0,\,\pi ]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) by
where
In the above definition, \(\alpha _\varepsilon \) and \(\beta _\varepsilon \) are the functions introduced in Lemma 8.
Concerning the function H, by assumption (A2) there exists a constant \(D>0\) such that every solution (x, y) of the system (1), starting with \(y(0)=0\), satisfies
(see e.g. [5]). Let \(\zeta :{\mathbb {R}}\rightarrow {\mathbb {R}}\) be a \(C^{\infty }\)-function such that
Then consider the function \({\widetilde{H}}:[0,\pi ]\times {\mathbb {R}}^{2}\rightarrow {\mathbb {R}}\) defined as
so that the partial derivatives of \({\widetilde{H}}(t,x,y)\) with respect to x and y are bounded.
We can now introduce the modified system
which can also be written as
where
We will prove that, for \(|\varepsilon |\) small enough, the modified problem (17)–(4) has at least two geometrically distinct solutions. These solutions will indeed be the solutions of the original problem (3)–(4) we are looking for. In order to show this, we first need to prove some preliminary lemmas.
Lemma 10
There exists \(\overline{\varepsilon }>0\) such that, if (x, y, u, v) is a solution of problem (17)–(4), with \(|\varepsilon |\le \overline{\varepsilon }\), then \(|y(t)|\le D+1\), for every \(t\in [0,\pi ]\).
Proof
Assume, by contradiction, that, for all \(n\ge 1\), there exists a solution \((x_n,y_n,u_n,v_n)\) of problem (17)–(4), with \(\varepsilon =1/n\), satisfying \(\Vert y_n\Vert _\infty > D+1\).
By the periodicity of \({{\widetilde{H}}}\) and P in the variable x we can assume without loss of generality that \(x_n(0)\in [0,\tau ]\) for every n. Moreover, since, \({{\widetilde{H}}}\) and P have bounded gradients, the sequences \((x_n)_n\) and \((y_n)_n\) are uniformly bounded, together with their derivatives, hence, by the Ascoli–Arzelà Theorem, they uniformly converge, up to a subsequence, to some functions \(x_0\) and \(y_0\), respectively. We can write
and
The functions in the integrals are bounded, hence by the dominated convergence theorem we can take the limits and obtain
Therefore, \((x_0,y_0)\) is a solution of system (1), with H replaced by \({{\widetilde{H}}}\), starting with \(y_0(0)=0\), satisfying \(\Vert y_0\Vert _\infty \ge D+1\). Such a solution will solve the original system (1) on some maximal interval \([0,\omega ]\subseteq [0,\pi ]\). By the estimate in (15), it has to be \(\omega =\pi \) and \(\Vert y_0\Vert _\infty \le D\), a contradiction. \(\square \)
In what follows we will always assume \(0< \overline{\varepsilon }\le \frac{1}{C\pi }\), where the constant C is the one introduced in assumption (A4).
Lemma 11
Reducing if necessary the constant \(\overline{\varepsilon }\), if \(|\varepsilon |\le \overline{\varepsilon }\), then,
for every \((t,x,y,v)\in [0,\pi ]\times {\mathbb {R}}^{3}\).
Proof
We start by proving the second inequality in (20). Fix \(t\in [0,\pi ]\), and assume \(v>v_{\alpha }(t)\). We need to consider two cases: the first when \(v_{\alpha }(t)<v<d\), and the second when \(v\ge d\).
Case 1 : \(v_{\alpha }(t)<v<d\). Using in the order (14), Lemma 8(i), assumptions (A6) and (A4), we get
If \(v-v_{\alpha }(t)>\frac{2}{\lambda }\ge \frac{2C|\varepsilon |}{\lambda }\), then
Conversely, if \(0<v-v_{\alpha }(t)\le \frac{2}{\lambda }\) , then by assumption (A7) we get
where \({\widetilde{C}}\) is the Lipschitz constant such that
for every \((t,u)\in [0,\pi ]\times [A-1,B+1]\). Choosing \(|\varepsilon |<\lambda /{\widetilde{C}}\), we get
Case 2 : \(v\ge d\). Similarly as above, if \(|\varepsilon |\) is small enough, we get
completing the proof of the second inequality in (20).
The proofs of the first inequality in (20) and of the two inequalities in (21) are carried out similarly.
We now establish the first inequality in (22), a similar argument holding for the proof of the latter. Recalling (8) and the fact that \(\alpha \) is a strict lower solution, let \(\overline{e}>0\) be such that
Reducing \(\overline{\varepsilon }\) if necessary, recalling (iii) in Lemma 8, by the continuity of g we have
when \(|\varepsilon |\le \overline{\varepsilon }\). Combining (23) and (24), we obtain
for all \(t\in [0,\pi ]\). So, for \(|\varepsilon |\) sufficiently small,
thus completing the proof. \(\square \)
Let us fix now \(\varepsilon \) satisfying \(|\varepsilon |\le \overline{\varepsilon }\). We define some open sets in the space \([0,\,\pi ]\times {\mathbb {R}}^2\) and study some invariance properties of them with respect to the solutions of system (17). We set (see Fig. 2)
The following three lemmas are consequences of Lemma 11. We avoid giving the detailed proofs as they essentially follow the arguments in [13]; see also [8, 9, 15].
Lemma 12
Let (x, y, u, v) be a solution of (17) defined at a point \(t_0\in [0,\pi ]\). We have:
- (i):
-
if \((t_0,u(t_0),v(t_0))\in A_{NE}\,\), then \((t,u(t),v(t))\in A_{NE}\) for all \(t>t_0;\)
- (ii):
-
if \((t_0,u(t_0),v(t_0))\in A_{SE}\,\), then \((t,u(t),v(t))\in A_{SE}\) for all \(t<t_0;\)
- (iii):
-
if \((t_0,u(t_0),v(t_0))\in A_{SW\,}\), then \((t,u(t),v(t))\in A_{SW}\) for all \(t>t_0;\)
- (iv):
-
if \((t_0,u(t_0),v(t_0))\in A_{NW}\,\), then \((t,u(t),v(t))\in A_{NW}\) for all \(t<t_0\) .
Lemma 13
Let (x, y, u, v) be a solution of (17) defined at a point \(t_0\in [0,\pi ]\). We have:
- (i):
-
if \(u(t_0)>\beta _\varepsilon (t_0)\) and \(v(t_0)=v_\beta (t_0)\), then \(v'(t_0)>v_\beta '(t_0)\,;\)
- (ii):
-
if \(u(t_0)<\alpha _\varepsilon (t_0)\) and \(v(t_0)=v_\alpha (t_0)\), then \(v'(t_0)<v_\alpha '(t_0)\) .
Lemma 14
Let (x, y, u, v) be a solution of problem (17)-(4). Then, \(\alpha _\varepsilon (t) \le u(t)\le \beta _\varepsilon (t)\), for every \(t\in [0,\pi ]\).
By Lemma 10, the definition of M in (12), and the choice \(\overline{\varepsilon }\le \frac{1}{C\pi }\), we can finally state the following a priori bound (Fig. 2).
Proposition 15
Let (x, y, u, v) be a solution of problem (17)–(4). Then,
for every \(t\in [0,\pi ]\). As a consequence, (x, y, u, v) is also a solution of problem (3)–(4).
Recalling now the preliminary remark at the beginning of this subsection, going back to the original lower and upper solutions \(\alpha \) and \(\beta \), by (iii) in Lemma 8 we can conclude that, if \(\varepsilon C\pi \le \delta \), then \(\alpha \le u\le \beta \).
3.2 The function spaces
In this section we provide the functional spaces needed in our variational setting. We refer to [12] for a detailed exposition.
For any \(\mu \in \,]0,1[\) , we define \(X_{\mu }\) as the set of those real valued functions \({\tilde{x}}\in L^{2}(0,\pi )\) such that
where \(({\tilde{x}}_{m})_{m\ge 1}\) is a sequence in \({\mathbb {R}}\) satisfying
The space \(X_{\mu }\) is endowed with the inner product and the norm
We denote by \({{\widetilde{C}}}^1\left( [0,\pi ]\right) \) the set of \(C^1\)-functions having zero mean in \([0,\pi ]\).
Proposition 16
The space \(X_\mu \) is continuously embedded in \(L^2(0,\pi )\) and is made of functions with zero mean on \([0,\pi ]\). The set \({{\widetilde{C}}}^1([0,\pi ])\) is a dense subset of \(X_\mu \).
For any \(\nu \in \,]0,1[\,\), we define \(Y_{\nu }\) as the set of those real valued functions \(y\in L^{2}(0,\pi )\) such that
where \((y_{m})_{m\ge 1}\) is a sequence in \({\mathbb {R}}\) satisfying
The space \(Y_{\nu }\) is endowed with the inner product and the norm
We denote by \(C^1_0([0,\pi ])\) the set of \(C^1\)-functions y satisfying \(y(0)=0=y(\pi )\).
Proposition 17
The space \(Y_\nu \) is continuously embedded in \(L^2(0,\pi )\) and if \(\nu >\tfrac{1}{2}\) it is continuously embedded in \(C([0,\pi ])\). The set \(C^1_0([0,\pi ])\) is a dense subset of \(Y_\nu \).
We will look for solutions of problem (17)–(4) by decomposing them as
where
We choose two positive numbers \(\mu<\frac{1}{2}<\nu \) such that \(\mu +\nu =1\), and consider the space \(E=X_\mu \times Y_\nu \times ({\mathbb {R}}\times X_\mu )\times Y_\nu \). It is a separable Hilbert space endowed with the scalar product
and the corresponding norm
Recalling that the function \({{\widetilde{K}}}_\varepsilon \) in (19) is \(\tau \)-periodic in x, we can assume \(\overline{x}\in S^1={\mathbb {R}}/(\tau {\mathbb {Z}})\) and look for critical points
of a suitable functional \(\varphi : E \times S^1\rightarrow {\mathbb {R}}\).
Let us briefly describe the rest of the proof of Theorem 2, to be carried out in the next sections. In Sect. 3.4 we will introduce a bounded selfadjoint invertible operator \(L\in {{{\mathcal {L}}}}(E)\) so to define the functional
where \(\psi :E\times S^{1}\rightarrow {\mathbb {R}}\) is given by
In Sect. 3.3 we will prove that \(d\psi (E\times S^{1})\) is relatively compact. Then, in Sect. 3.4 we will verify that the critical points of \(\varphi \) are indeed solutions of problem (17)–(4). The existence of such critical points will be provided by the application of the following theorem, which is a particular case of [23, Theorem 3.8].
Theorem 18
(Szulkin) If \(\varphi : E\times S^{1}\rightarrow {\mathbb {R}}\) is as in (25), where \(d\psi (E\times S^{1})\) is relatively compact and \(L:E\rightarrow E\) is a bounded selfadjoint invertible operator, then there exist at least two critical points of \(\varphi \).
Finally, in view of Proposition 15, we will conclude that such solutions also solve problem (3)–(4), thus completing the proof of Theorem 2.
3.3 The functional \(\psi \)
With the aim of applying Szulkin’s Theorem, in this section we prove that the functional \(\psi \) defined in (26) is continuously differentiable, with Fréchet differential \(d\psi \), and the image \(d\psi (E\times S^1)\) is relatively compact in the dual space \({\mathcal {L}}(E\times S^1,{\mathbb {R}})\). The proof essentially follows the arguments of [12, Section 2.2]. For sake of simplicity, in this section we replace \(S^1\) with the linear space \({\mathbb {R}}\). The function \(\psi \) is defined in the same way.
Proposition 19
The functional \(\psi : E\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is continuously differentiable.
Proof
Fix any point \((z_0,\overline{x}_0)=\big (({{\tilde{x}}}_0,y_0,\overline{u}_0,{{\tilde{u}}}_0,v_0),\overline{x}_0)\big ) \in E\times {\mathbb {R}}\). For every \((z,\overline{x})=\big (({{\tilde{x}}},y,\overline{u},{{\tilde{u}}},v),\overline{x}\big )\in E\times {\mathbb {R}}\) we compute the directional derivative
(In the above computations, the dominated convergence theorem has been used, since all the quantities inside the integrals are uniformly bounded.) We verify now that the Gâteaux differential \(d_G\psi : E\times {\mathbb {R}}\rightarrow {\mathcal {L}}(E\times {\mathbb {R}},{\mathbb {R}})\) is continuous at \((z_0,\overline{x}_0)\). The function \({\mathcal {T}}: E\times {\mathbb {R}}\rightarrow [L^2(0,\pi )]^4\), defined by
is continuous, as the spaces \(X_\mu \) and \(Y_\nu \) are continuously embedded into \(L^2(0,\pi )\). The Nemytskii operator \({\mathcal {N}}: [L^2(0,\pi )]^4 \rightarrow [L^2(0,\pi )]^4\), defined by
is continuous, since all the functions involved are continuous and bounded. Finally, the linear map \(\Phi :[L^2(0,\pi )]^4 \rightarrow {\mathcal {L}}(E\times {\mathbb {R}},{\mathbb {R}})\), defined by
is bounded, hence continuous. As \(d_G\psi =\Phi \circ {\mathcal {N}}\circ {\mathcal {T}}\), we conclude that \(d_G\psi \) is continuous and \(\psi \) is Fréchet differentiable, and \(d\psi =d_G\psi \). \(\square \)
We verify now that the set \(d\psi (E\times {\mathbb {R}})\) is relatively compact in \( {\mathcal {L}}(E\times {\mathbb {R}},{\mathbb {R}})\). We need to recall the Hausdorff–Young-type inequality proved in [12, Proposition 2.2].
Proposition 20
Assume that \(1<p\le 2 \le q\) verify \((1/p)+(1/q)=1\). Let \({{\widetilde{\Phi }}}\in L^p(0,\pi )\) be such that \(\int _0^\pi {{\widetilde{\Phi }}}(t)\,dt=0\), with
Then,
For all \(m\ge 1\), we set
and consider the orthonormal basis \({\mathcal {B}}\) in \(E\times {\mathbb {R}}\) defined by
We need the following result.
Proposition 21
For all \(\epsilon >0\) there exists \(m_0\ge 1\) such that, for all \((z_0,\overline{x}_0)\in E\times {\mathbb {R}}\), we have
Proof
Let R be a constant satisfying, for every \((t,\sigma )\in [0,\pi ]\times {\mathbb R}\),
Fix \((z_0,\overline{x}_0)\in E\times {\mathbb {R}}\) and expand the function
in a Fourier series as \(\Phi (t)\sim \Phi _0+\sum _{k=1}^\infty \Phi _k \cos (kt).\) We have
Pick \(\rho >2\) such that \(2\mu \rho >1\) and set
Let \(\rho '\) be the conjugate exponent of \(\rho \), satisfying \(1/\rho +1/\rho '=1\). By the Hölder inequality, we have
In the following computation we apply the Hausdorff–Young inequality of Proposition 20 to \({{\widetilde{\Phi }}}(t)=\Phi (t)-\Phi _0\). Moreover, we observe that \(|\Phi (t)-\Phi _0|\le 2R\), where R is the constant defined in (27). We get
Since \(\displaystyle \lim _{{m_0}\rightarrow \infty } S_{m_0}=0\) we conclude that there exists \({m_0}\) such that
for all \((z_0,\overline{x}_0)\in E\times {\mathbb {R}}\). Similar computations allow to take \({m_0}\) such that
hence the claim is proved. \(\square \)
Proposition 22
The image \(d\psi (E\times {\mathbb {R}})\) is relatively compact in \( {\mathcal {L}}(E\times {\mathbb {R}},{\mathbb {R}})\).
Proof
To verify that \(d\psi (E\times {\mathbb {R}})\) is bounded in \( {\mathcal {L}}(E\times {\mathbb {R}},{\mathbb {R}})\), take any \((z,\overline{x})=({{\tilde{x}}}, y,\overline{u}, {{\tilde{u}}}, v, \overline{x})\in E\times {\mathbb {R}}\) with unitary norm and compute
We note that both
and
are uniformly bounded. Moreover, by Proposition 21, the series in the right-hand side of (28) is also uniformly bounded, hence we conclude that \(d\psi (E\times {\mathbb {R}})\) is bounded.
To verify compactness, pick any sequence \(\big (d\psi ( z_0^n,\overline{x}_0^n)\big )_n\) in \(d\psi (E\times {\mathbb {R}})\), where \(z_0^n = ({{\tilde{x}}}_0^n,y_0^n,\overline{u}_0^n,{{\tilde{u}}}_0^n,v_0^n)\). Since \(d\psi (E\times {\mathbb {R}})\) is bounded, we may assume that the sequence weakly converges to some \(h\in {\mathcal {L}}(E\times {\mathbb {R}},{\mathbb {R}})\). We aim to prove that the sequence strongly converges to h.
We set \(h_{[1],m}= h(e_{[1],m})\), \(h_{[2],m}= h(e_{[2],m})\), \(h_{[3]}= h(e_{[3]})\), \(h_{[4],m}= h(e_{[4],m})\), \(h_{[5],m}= h(e_{[5],m})\) and \(h_{[6]}= h(e_{[6]})\), so that,
Fix \(\epsilon >0\). By Proposition 21, there is \(m_0>1\), such that, for all \((z_0,\overline{x}_0)\in E\times {\mathbb {R}}\), we have
By (29), we may further assume that
From the weak convergence of the sequence \(\big (d\psi ( z_0^n,\overline{x}_0^n)\big )_n\), we can take \(n_0\in {\mathbb {N}}\) large enough such that, for all \(n\ge n_0\), for all \(m\ge 1\), with \(m\le m_0-1\), we have
and furthermore
Then we compute, using (30), (31), (32), and (33),
We just proved that the sequence \(\big (d\psi ( z_0^n,\overline{x}_0^n)\big )_n\) strongly converges to h. This shows that the set \(d\psi (E\times {\mathbb {R}})\) is relatively compact in \({\mathcal {L}}(E\times {\mathbb {R}},{\mathbb {R}})\). \(\square \)
3.4 The operator L
In this section we are going to introduce the operator L which is in force in the application of Theorem 18.
Let us first introduce a continuous symmetric bilinear form \({\mathcal {B}}: D \times D \rightarrow {\mathbb {R}}\), where
Given \(z=({{\tilde{x}}}, y, \overline{u}, {{\tilde{u}}}, v)\) and \(Z=({{\widetilde{X}}}, Y, \overline{U}, {{\widetilde{U}}}, V)\) in D we define
which is equivalent to
recalling the boundary conditions \(y(0)=0=y(\pi )\), \(v(0)=0=v(\pi )\), \(Y(0)=0=Y(\pi )\), and \(V(0)=0=V(\pi )\).
Let us verify that the form \({\mathcal {B}}\) is continuous in \(D\times D\) with the topology induced by the topology of \(E\times E\). We can write
and compute
Similar inequalities hold for the other terms in the definition of \({\mathcal {B}}\). For example we compute, writing
Therefore we have
Since D is a dense subspace of E (see Propositions 16 and 17 ) we can extend \({\mathcal {B}}\) to a continuous bilinear symmetric form \({\mathcal {B}}: E \times E \rightarrow {\mathbb {R}}\).
Now, we can introduce the bounded selfadjoint operator \(L: E\rightarrow E\) generated by \({\mathcal {B}}\): we define L such that, for every \(z,Z\in E\),
Lemma 23
The operator L is invertible with continuous inverse.
Proof
At first notice that we can decompose \({\mathcal {B}}\) as follows
where
and
Consequently we will have
where
and
Arguing as in [12, Proposition 2.14] we can prove that
Now, we are going to prove that there are two constants \(c_1,c_2>0\) such that
for every \((\overline{u}, {{\tilde{u}}}, v)\in {{\mathbb {R}}\times X_\mu \times Y_\nu }\). To this aim, let \((\overline{p},{{\tilde{p}}},q)\in {\mathbb {R}}\times X_\mu \times Y_\nu \) be such that
for every \((\overline{U}, {{\widetilde{U}}}, V)\in {\mathbb {R}}\times X_\mu \times Y_\nu \). Setting
and choosing in (36) at first \(V=0\), and next \(\overline{U} + {{\widetilde{U}}}=0\) we get the identities
In particular
so that, using the Young inequality,
Hence, from the first identity in (37) we get
so that we can choose \(c_2=\pi \) in (35).
We now provide the value \(c_1\). At first notice that (38) in the case \(m=1\) reads as
For \(m\ge 2\), since \(( m^{-2\mu } - m^{-2\nu })\le m^{-2\mu }\le \left( \tfrac{1}{2}\right) ^{2\mu }\), from (38) we get, using again the Young inequality,
Finally we get, from the first estimate in (37) and (39),
providing the constant \(c_1= \frac{\pi }{2} \sqrt{1- (\tfrac{1}{2})^{2\mu }}\). Hence, (35) holds.
Summing up, from (34) and (35), since \(c_1\le \frac{\pi }{2} \le c_2\), we deduce that
in particular L is continuous and \(\ker L=\{0\}\). A classical reasoning (cf. [12, Proposition 2.14]) shows that the image of L is closed and, since L is selfadjoint, we conclude that it is bijective and admits a continuous inverse \(L^{-1}:E\rightarrow E\).
\(\square \)
Proposition 24
If \(\big ( ({{\tilde{x}}}_0\,,y_0 \,, \overline{u}_0\,, {{\tilde{u}}}_0, v_0)\,, \overline{x}_0\big )\) is a critical point of \(\varphi \), then
Proof
Let \(\big ( z_0\,, \overline{x}_0\big )= \big ( ({{\tilde{x}}}_0\,,y_0 \,, \overline{u}_0\,, {{\tilde{u}}}_0, v_0)\,, \overline{x}_0\big )\in E\times {\mathbb {R}}\) be a critical point of \(\varphi \). Then, for any \(\big ( z\,, \overline{x} \big )=\big ( ({{\tilde{x}}}\,,y \,, \overline{u}\,, {{\tilde{u}}}, v)\,, \overline{x}\big )\in E\times {\mathbb {R}}\), we have
Let us consider \(u\in C^1([0,\pi ])\) and write \(u={{\tilde{u}}} + \overline{u}\), with \(\overline{u} = \frac{1}{\pi } \int _0^\pi u(t)\, dt\). Choosing \(\big ( z\,, \overline{x} \big )=\big ( (0\,,0 \,, \overline{u}\,,{{\tilde{u}}}, 0)\,, 0\big )\) in (40), we obtain
that is, as \(u'={{\tilde{u}}}'\),
Therefore, in the sense of distributions, we have
which is the fourth equation in (18). In particular, \(v_0\in W^{1,2}(0,\pi )\) and therefore it is continuous. With a similar reasoning, choosing, respectively, \(\big ( z\,, \overline{x} \big )=\big ( (0\,,y \,,0\,, 0\,, 0)\,, 0\big )\), \(\big ( z\,, \overline{x} \big )=\big ( ({{\tilde{x}}}\,,0 \,, 0\,, 0 \,, 0)\,, 0\big )\) and \(\big ( z, \overline{x} \big )=\big ( (0,0 , 0, 0,v), 0\big )\) in formula (40), we see that the functions \(\overline{x}_0 +{{\tilde{x}}}_0\), \(y_0\) and \(\overline{u}_0 + {{\tilde{u}}}_0\) are continuous and, in the sense of distributions, satisfy the other three equations in (18). From the equations in (18), we also deduce that \((\overline{x}_0 + {{\tilde{x}}}_0\,,y_0 \,, \overline{u}_0+ {{\tilde{u}}}_0, v_0)\in C^1([0,\pi ])^4\), so that the equations are satisfied in the classical sense. Therefore \((\overline{x}_0 + {{\tilde{x}}}_0\,,y_0 \,, \overline{u}_0+ {{\tilde{u}}}_0, v_0)\) is a solution of problem (17). Since \(y_0, v_0\in Y_\nu \), the boundary conditions (4) are also satisfied and the conclusion follows. \(\square \)
4 The higher dimensional case
For \(z=(x,y,u,v)\in {\mathbb {R}}^{N}\), we write
We now consider the higher dimensional system
with Neumann-type boundary conditions
Here \(H:[a,b]\times {\mathbb {R}}^{2M}\rightarrow {\mathbb {R}}\), \(P:[a,b]\times {\mathbb {R}}^{2M+2L}\rightarrow {\mathbb {R}}\) and \(f_j:[a,b]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) are continuous functions, with continuous partial derivatives with respect to the variables x, y, u, v, for every \(j=1,\dots ,L\) ; the functions \(g_j:[a,b]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) are continuous, and \(\varepsilon \) is a small real parameter.
We recall the definition of lower and upper solution for the system
with Neumann-type boundary conditions
Definition 25
A \(C^{1}\)-function \(\alpha :[a,b]\rightarrow {\mathbb {R}}^L\) is a lower solution for problem (43)-(44) if there exists a \(C^{1}\)-function \(v_{\alpha }:[a,b]\rightarrow {\mathbb {R}}^L\) such that, for every \(t\in [a,b]\) and \(j=1, \dots , L\),
and
The lower solution is strict if the strict inequalities in (45) hold, for every \(t\in [a,b]\) and \(j=1, \dots , L\).
Definition 26
A \(C^1\)-function \(\beta :[a,b] \rightarrow {\mathbb {R}}^{L}\) is an upper solution for problem (43)–(44) if there exists a \(C^1\)-function \(v_{\beta }:[a,b]\rightarrow {\mathbb {R}}^{L}\) such that, for every \(t\in [a,b]\) and \(j=1, \dots , L\),
and
The upper solution is strict if the strict inequalities in (46) hold, for every \(t\in [a,b]\) and \(j=1, \dots , L\).
In the sequel, inequalities of n-tuples will be meant component-wise. Here is the list of our assumptions.
- \((A1')\):
-
The function \(H=H(t,x,y)\) is \(\tau _j\)-periodic in the variable \(x_j\), for some \(\tau _j>0\), for every \(j=1,\dots ,M\) .
- \((A2')\):
-
All solutions (x, y) of system
$$\begin{aligned} x'= \nabla _{y} H(t,x,y), \qquad y'=- \nabla _{x} H(t,x,y) \end{aligned}$$starting with \(y(a)=0\) are defined on [a, b] .
- \((A3')\):
-
The function \(P=P(t,x,y,u,v)\) is \(\tau _j\)-periodic in the variable \(x_j\), for every \(j=1,\dots ,M\).
- \((A4')\):
-
The function \(P=P(t,x,y,u,v)\) has a bounded gradient with respect to \(z=(x,y,u,v)\).
- \((A5')\):
-
There exist a strict lower solution \(\alpha \) and a strict upper solution \(\beta \) for problem (43)–(44) such that \(\alpha \le \beta \) .
- \((A6')\):
-
There exists \(\lambda >0\) such that \(\partial _{s}f_j(t,\,s)\ge \lambda \), for every \((t,s)\in [a,b]\times {\mathbb {R}}\) and \(j=1,\dots ,L\) .
- \((A7')\):
-
For every \(j=1,\dots ,L\), the partial derivative \(\partial _{v_j} P\) depends only on t, u and \(v_j\) and is locally Lipschitz continuous with respect to \(v_j\).
Let us state our main theorem.
Theorem 27
Let assumptions \((A1')\)–\((A7')\) hold true. Then, there exists \(\overline{\varepsilon }>0\) such that, when \(|\varepsilon |\le \overline{\varepsilon }\), problem (41)–(42) has at least \(M+1\) solutions (x, y, u, v) with \(\alpha \le u\le \beta \).
Proof
Arguing as in Lemma 8, for every \(\varepsilon \in {\mathbb {R}}\) and \(j=1,\dots ,L\) there exist some \(C^1\)-functions \(\alpha _{\varepsilon ,j}:[a,b]\rightarrow {\mathbb {R}}\) and \(\beta _{\varepsilon ,j}:[a,b]\rightarrow {\mathbb {R}}\) such that
for every \(t\in [a,\,b]\), where \(\Vert \nabla _z P\Vert _\infty \le C\), from assumption \((A4')\).
Proceeding with the same strategy as in Sect. 3.1, we get the following modified system associated with system (41),
In system (47),
-
for every \(j=1,\dots ,L\), \({\tilde{f}}_{j}:[a,b]\times {\mathbb {R}}^{L}\rightarrow {\mathbb {R}}\) is defined by
$$\begin{aligned} {\tilde{f}}_{j}(t,v_{j})={\left\{ \begin{array}{ll} f_{j}(t,-d_{j})+v_{j}+d_{j},&{}\quad \hbox {if}\,\,v_{j}\le -d_{j},\\ f_{j}(t,v_{j}),&{}\quad \hbox {if}\,\,|v_{j}|\le d_{j},\\ f_{j}(t,d_{j})+v_{j}-d_{j},&{}\quad \hbox {if}\,\,v_{j}\ge d_{j}, \end{array}\right. } \end{aligned}$$where \(d=(d_{1},\dots ,d_{j})\) is defined similarly as in (13).
-
for every \(j=1,\dots ,L\), \({\tilde{g}}_{\varepsilon ,j}:[a,b]\times {\mathbb {R}}^{L}\rightarrow {\mathbb {R}}\) is defined by
$$\begin{aligned} {\tilde{g}}_{\varepsilon ,j}(t,u_{j})={\left\{ \begin{array}{ll} g_{j}(t,\alpha _{\varepsilon ,j}(t))-\alpha _{\varepsilon ,j}(t)+u_{j},&{}\quad \hbox {if}\,\,u_{j}\le \alpha _{\varepsilon ,j}(t),\\ g_{j}(t,u_{j}),&{}\quad \hbox {if}\,\,\alpha _{\varepsilon ,j}(t)\le u_{j}\le \beta _{\varepsilon ,j}(t),\\ g_{j}(t,\beta _{\varepsilon ,j}(t))-\beta _{\varepsilon ,j}(t)+u_{j},&{}\quad \hbox {if}\,\,u_{j}\ge \beta _{\varepsilon ,j}(t). \end{array}\right. } \end{aligned}$$ -
\({\widetilde{H}}:[a,b]\times {\mathbb {R}}^{2M}\rightarrow {\mathbb {R}}\) is defined by
$$\begin{aligned} {\widetilde{H}}(t,x,y)=\zeta (|y|)H(t,x,y), \end{aligned}$$where \(\zeta \) is given in (16).
System (47) can also be written as
where
Arguing as in Lemma 11, we can verify that \(\alpha _\varepsilon \) and \(\beta _\varepsilon \) are indeed lower and upper solutions for the modified problem.
We will consider functions x and y belonging to the spaces
Propositions 16 and 17 which are taken from [12] hold here also.
The existence of \(M+1\) solutions of problem (41) with Neumann boundary conditions (42) will be given through the application of the following theorem.
Theorem 28
(Szulkin) If \(\varphi : E\times {\mathbb {T}}^{M}\rightarrow {\mathbb {R}}\) is as in (25), where \(d\psi (E\times {\mathbb {T}}^{M})\) is relatively compact and \(L:E\rightarrow E\) is a bounded selfadjoint invertible operator, then there exist at least \(M+1\) critical points of \(\varphi \).
In the above theorem, \({\mathbb {T}}^{M}\) denotes the torus
We will apply it with L defined with the same strategy adopted in Sect. 3.4, the functionals \(\varphi \) and \(\psi \) defined by (25) and (26), respectively. All the hypotheses of Szulkin’s theorem are verified, providing the existence of \(M+1\) solutions of the modified problem.
The lemmas stated in Sect. 3.1 are also true for the higher dimensional situation. Specifically, Proposition 15 in the higher dimensional setting assures that all the \(M+1\) distinct solutions of (47)–(42) are also solutions of problem (41)–(42). This completes the proof. \(\square \)
5 A further result in higher dimension
Finally we want to deal with a system of a different type, i.e.,
with Neumann-type boundary conditions
Here \(H:[a,b]\times {\mathbb {R}}^{2M}\rightarrow {\mathbb {R}}\), \(P:[a,b]\times {\mathbb {R}}^{2M+L}\rightarrow {\mathbb {R}}\) and \(G:[a,b]\times {\mathbb {R}}^L\rightarrow {\mathbb {R}}\) are continuous functions, with continuous partial derivatives with respect to the variables x, y, u. Here is our result.
Theorem 29
Let assumptions \((A1')\)–\((A4')\) hold true. Moreover, let \(R>0\) be such that
Then, there exists \(\overline{\varepsilon }>0\) such that, when \(|\varepsilon |\le \overline{\varepsilon }\), problem (49)–(50) has at least \(M+1\) solutions (x, y, u, v) with \(|u(t)|\le R\), for every \(t\in [a,b]\).
Proof
We modify the function H exactly as above. Moreover, we also modify G as follows. From the Hartman’s condition (51) and the continuity of the inner product, there exists \({{\bar{e}}}>0\) and \(\lambda >0\) such that
Without loss of generality we can assume that
Define the function \({\widetilde{G}}:[a,b]\times {\mathbb {R}}^{L}\rightarrow {\mathbb {R}}\) by
where \(\eta :{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a \(C^{\infty }\)-function such that
and
We consider the modified system
We are in force to apply Szulkin’s Theorem 28, which provides us at least \(M+1\) solutions for problem (56)–(50).
We now need to show that the solutions of problem (56)–(50) satisfy
so that they are also solutions of problem (49)–(50). In order to show this, we argue by contradiction. Suppose there is \(t_0\in [a,b]\) such that
Let \(f(t)=|u(t)|^{2}\); we have \(f'(t)=2\langle u(t),u'(t)\rangle \), and
Assume first that \(t_0\in \,]a,b[\) . Then, since f(t) has a maximum point at \(t=t_0\), we have
On the other hand, if \(t_0=a\), then necessarily \(f'(a)=2\langle u(a),v(a)\rangle =0\), hence also in this case it has to be \(f''(a)\le 0\). The same if \(t_0=b\); it will be \(f'(b)=2\langle u(b),v(b)\rangle =0\), hence \(f''(b)\le 0\). We thus conclude that (58) holds in any case of \(t_0\in [a,b]\).
Let us now analyze two distinct cases.
The first case is when \(|u(t_0)|\ge R+{{\bar{e}}}\). From (57) and (54), if we apply the Cauchy–Schwarz inequality and the fact that \(|\nabla _{u}P(t,x(t),y(t),u(t))|\le C\), we get
when \(|\varepsilon |\) is small, a contradiction.
The other case is when \(R<|u(t_0)|<R+{{\bar{e}}}\). Then, we compute
From (53) and (55), we have that \(E_{1}\ge 0\). Again from (55), it follows that \(E_{3}\le 0\). From (52) and \(|u(t_0)|>R\), we have
Finally,
Combining all the above facts, for \(|\varepsilon |\) sufficiently small we get \(f''(t_0)>0\), a contradiction.
This completes the proof. \(\square \)
Remark 30
The assumption (51) was introduced by Hartman [17] for the periodic problem (see also [4]). Notice that, when \(L=1\), it is equivalent to asking that the constant functions \(\alpha =-R\) and \(\beta =R\) are a strict lower solution and a strict upper solution, respectively.
6 Examples and final remarks
As an illustrative example of application of Theorem 2, we consider the problem
where the functions \(h,g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) are continuous and \(P:[0,\pi ]\times {\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is continuous and has bounded continuous partial derivatives \(\partial _x P(t,x,u)\) and \(\partial _u P(t,x,u)\). The functions h and P are \(2\pi \)-periodic in x, with \(\int _0^{2\pi } h(s)\, ds =0\). Concerning the function g we assume the existence of some constants \(\alpha < \beta \) such that \(g(\alpha )<0<g(\beta )\).
A typical example for the function h in the first equation in (59) might be \(h(x)=-\sin x\), in which case we have a perturbed pendulum equation. Another choice would be the saw-tooth function \(h(x)=\arcsin (\sin x)\). Concerning the second equation, possible examples for the function g are
For the higher dimensional cases, similar examples can be constructed.
Let us now mention some possible further developments and open problems.
-
1.
In assumption (A5) we require that the lower and upper solutions are strict. It would be interesting to obtain the same multiplicity result without such a strictness assumption.
-
2.
The possibility of considering systems like (3) or (41) without a small parameter \(\varepsilon \) has been analyzed in [18] in presence of constant lower and upper solutions. The case of non-constant lower and upper solutions remains open.
-
3.
We wonder whether assumptions (A6) and (A7), and their corresponding higher dimensional versions, could be weakened.
-
4.
We have treated here only the case when the lower and upper solutions are well-ordered. It would be interesting to know if the results may be extended to the non-well-ordered case.
-
5.
In this paper we dealt with \(C^1\)-smooth lower and upper solutions. Following the ideas developed in [9], one might consider weaker regularity assumptions.
-
6.
In view of the results in [11], concerning the radial solutions for an elliptic problem with Neumann boundary conditions, one could try to deal with a coupled system, where the fourth equation in (3) is replaced by
$$\begin{aligned} (t^{n-1} v)'=t^{n-1}\big [g(t,u) - \varepsilon \partial _u P(t,x,y,u,v) \big ], \qquad t\in [0,R]. \end{aligned}$$ -
7.
It would be interesting to extend the results of this paper to an infinite-dimensional setting. A version of the Poincaré–Birkhoff theorem for the periodic case in this setting has been proposed in [1]. On the other hand, the lower and upper solutions technique for second order differential equations has been extended to infinite dimensional systems in [10], also in the non-well-ordered case. We expect that the approaches in [1, 10] could be combined in order to obtain similar existence results for coupled systems as those considered in this paper.
References
Boscaggin, A., Fonda, A., Garrione, M.: An infinite-dimensional version of the Poincaré–Birkhoff theorem on the Hilbert cube. Ann. Sc. Norm. Super. Pisa Cl. Sci. 20, 751–770 (2020)
Castro, A.: Periodic solutions of the forced pendulum equation. In: Ahmad, S., Keener, M., Lazer, A.C. (eds.) Differential Equations, pp. 149–160. Academic Press, New York (1980)
De Coster, C., Habets, P.: Two-Point Boundary Value Problems, Lower and Upper Solutions. Elsevier, Amsterdam (2006)
Feltrin, G., Zanolin, F.: Bound sets for a class of \(\phi \)-Laplacian operators. J. Differ. Equ. 297, 508–535 (2021)
Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer, Dordrecht (1988)
Fonda, A., Garzón, M., Sfecci, A.: An extension of the Poincaré–Birkhoff Theorem coupling twist with lower and upper solutions. J. Math. Anal. Appl. 528, 127599 (2023)
Fonda, A., Gidoni, P.: Coupling linearity and twist: an extension of the Poincaré–Birkhoff Theorem for Hamiltonian systems. NoDEA - Nonlinear Differ. Equ. Appl. 27, 1–26 (2020)
Fonda, A., Klun, G., Obersnel, F., Sfecci, A.: On the Dirichlet problem associated with bounded perturbations of positively-\((p, q)\)-homogeneous Hamiltonian systems. J. Fixed Point Theory Appl. 24, 66 (2022)
Fonda, A., Klun, G., Sfecci, A.: Well-ordered and non-well-ordered lower and upper solutions for periodic planar systems. Adv. Nonlinear Stud. 21, 397–419 (2021)
Fonda, A., Klun, G., Sfecci, A.: Periodic solutions of second order differential equations in Hilbert spaces. Mediterr. J. Math. 18, 223 (2021)
Fonda, A., Mamo, N.G., Obersnel, F., Sfecci, A.: A lower/upper solutions result for generalised radial \(p\)-Laplacian boundary value problems. Mediterr. J. Math. 20, 152 (2023)
Fonda, A., Ortega, R.: A two-point boundary value problem associated with Hamiltonian systems on a cylinder. Rend. Circ. Mat. Palermo 72, 3931–3947 (2023)
Fonda, A., Sfecci, A., Toader, R.: Two-point boundary value problems for planar systems: a lower and upper solutions approach. J. Differ. Equ. 308, 507–544 (2022)
Fonda, A., Toader, R.: A dynamical approach to lower and upper solutions for planar systems. Discret. Contin. Dynam. Syst. 41, 3683–3708 (2021)
Fonda, A., Ullah, W.: Periodic solutions of Hamiltonian systems coupling twist with generalized lower/upper solutions. J. Differ. Equ. 379, 148–174 (2024)
Fonda, A., Ureña, A.J.: A higher dimensional Poincaré–Birkhoff theorem for Hamiltonian flows. Ann. Inst. H. Poincaré Anal. Non Linéaire 34, 679–698 (2017)
Hartman, P.: On boundary value problems for systems of ordinary, nonlinear, second order differential equations. Trans. Am. Math. Soc. 96, 493–509 (1960)
Mamo, N.G.: Neumann-type boundary value problem associated with Hamiltonian systems. Topol. Methods Nonlinear Anal., to appear
Nagumo, M.: Über die Differentialgleichung \(y^{\prime \prime }=f(t, y, y^{\prime })\). Proc. Phys-Math. Soc. Jpn. 19, 861–866 (1937)
Picard, E.: Sur l’application des méthodes d’approximations successives à l’étude de certaines équations différentielles ordinaires. J. Math. Pures Appl. 9, 217–271 (1893)
Rabinowitz, P.H.: On a class of functionals invariant under a \({\mathbb{Z}}^n\) action. Trans. Am. Math. Soc. 310, 303–311 (1988)
Scorza Dragoni, G.: Il problema dei valori ai limiti studiato in grande per le equazioni differenziali del secondo ordine. Math. Ann. 105, 133–143 (1931)
Szulkin, A.: A relative category and applications to critical point theory for strongly indefinite functionals. Nonlinear Anal. 15, 725–739 (1990)
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Fonda, A., Mamo, N.G., Obersnel, F. et al. Multiplicity results for Hamiltonian systems with Neumann-type boundary conditions. Nonlinear Differ. Equ. Appl. 31, 31 (2024). https://doi.org/10.1007/s00030-023-00913-4
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DOI: https://doi.org/10.1007/s00030-023-00913-4