Periodic Solutions of Second-Order Differential Equations in Hilbert Spaces

Abstract

We prove the existence of periodic solutions of some infinite-dimensional systems by the use of the lower/upper solutions method. Both the well-ordered and non-well-ordered cases are treated, thus generalizing to systems some well-established results for scalar equations.

1 Introduction

The use of lower and upper solutions in boundary value problems dates back to the pioneering papers of Peano [20] in 1885 and Picard [21] in 1893. Later, Scorza-Dragoni [23] in 1931 and Nagumo [18] in 1937 were those who provided the main contributions toward a modern theory for scalar second-order ordinary differential equations with separated boundary conditions. The first results for the periodic problem were obtained by Knobloch [15] in 1963. There is nowadays a large literature on this subject, dealing with different types of boundary conditions for ordinary and partial differential equations of elliptic or parabolic type (see, e.g., [5, 7] and the references therein).

In this paper we consider the periodic problem

$$\begin{aligned} (P)\quad {\left\{ \begin{array}{ll} \ddot{x}=f(t,x),\\ x(0)=x(T),\quad \dot{x}(0)=\dot{x}(T). \end{array}\right. } \end{aligned}$$

In the scalar case when $f:[0,T]\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is continuous, the ${\mathcal {C}}^2$-functions $\alpha ,\beta :[0,T]\rightarrow {\mathbb {R}}$ are said to be lower/upper solutions of problem (P), respectively, if

$$\begin{aligned} \ddot{\alpha }(t)\ge f(t,\alpha (t)),\qquad \ddot{\beta }(t)\le f(t,\beta (t)). \end{aligned}$$

for every $t\in [0,T]$, and

$$\begin{aligned} \alpha (0)=\alpha (T),\quad \beta (0)=\beta (T),\quad \dot{\alpha }(0)\ge \dot{\alpha }(T),\quad \dot{\beta }(0)\le \dot{\beta }(T). \end{aligned}$$

We say that $(\alpha ,\beta )$ is a well-ordered pair of lower/upper solutions if $\alpha \le \beta$. It is well known that, when such a pair exists, problem (P) has a solution x such that $\alpha \le x\le \beta$.

When the inequality $\alpha \le \beta$ does not hold, we say that the lower and upper solutions are non-well-ordered. In this case, with the aim of obtaining existence results, some further conditions have to be added to avoid resonance with the positive eigenvalues of the differential operator $-\ddot{x}$ with T-periodic conditions (recall that 0 is an eigenvalue, and all the other eigenvalues are positive). Starting with the paper by Amann, Ambrosetti and Mancini [1] in 1978, there have been several improvements in the existence and localization of the solutions by Omari [19] in 1988, Gossez and Omari [13] in 1994, Habets and Omari [14] in 1996 and De Coster and Henrard [6] in 1998 (see also [12] for an abstract setting of the results).

The aim of this paper is to extend those classical existence results for scalar equations to systems, both in a finite-dimensional and in an infinite-dimensional setting.

Bebernes and Schmitt [3] generalized the scalar well-ordered case to a system of type (P), with $f:[0,T]\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$. Their result is reported in Sect. 2 below, in a slightly more general version. We are not aware of any results for systems in the non-well-ordered case, not even in the finite-dimensional case.

In Sect. 3, we provide an existence result for a system in ${\mathbb {R}}^N$ when the components of the lower/upper solutions can be both well-ordered and non-well-ordered. To avoid resonance with the higher part of the spectrum, for simplicity we ask the function f to be globally bounded in the non-well-ordered components, even if such an assumption could certainly be weakened (see the remarks in Sect. 5).

The case of a system in an infinite-dimensional Banach space E has been analyzed by Schmitt and Thompson [22] in 1975 for boundary value problems of Dirichlet type. However, when facing the periodic problem, they needed to assume E to be finite dimensional, concluding their paper by saying: “Whether the results of this section [...] remain true in case E is infinite dimensional is not known at this time”. We are not aware of any progress in this direction till now. In this paper we will try to give a partial answer to this question.

In Sect. 4, we extend our existing result of Sect. 3 to an infinite-dimensional separable Hilbert space. The lack of compactness is recovered by assuming the lower and upper solutions to take their values in a Hilbert cube. Moreover, we ask the function f to be globally bounded and completely continuous in the non-well-ordered components. These assumptions are reminiscent of an infinite-dimensional version of the Poincaré–Miranda Theorem as given in [16].

The study of periodic solutions for infinite-dimensional Hamiltonian systems has been already faced by several authors, see, e.g., [2, 4, 8, 9, 11]. Our approach does not need a Hamiltonian structure and could be applied also to systems with nonlinearity depending on the derivative of x, provided some Nagumo-type condition is assumed. Such kind of systems were studied, e.g., in [22]. In Sect. 5 we will discuss on these and other extensions and generalizations of our results, possibly also to partial differential equations of elliptic or parabolic type.

2 Well-Ordered Lower and Upper Solutions for Systems

In this section and the next one, we consider the problem

$$\begin{aligned} (P)\quad {\left\{ \begin{array}{ll} \ddot{x}=f(t,x),\\ x(0)=x(T),\quad \dot{x}(0)=\dot{x}(T), \end{array}\right. } \end{aligned}$$

where $f:[0,T]\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$ is a continuous function. We are thus in a finite-dimensional setting. Let us recall a standard procedure to reduce the search of solutions of (P) to a fixed point problem in Banach space. We define the set

$$\begin{aligned} {\mathcal {C}}^2_T=\{x\in {\mathcal {C}}^2([0,T],{\mathbb {R}}^N):x(0)=x(T),\,\dot{x}(0)=\dot{x}(T)\}, \end{aligned}$$

and the linear operator

$$\begin{aligned} \mathcal {L}:{\mathcal {C}}^2_T\rightarrow {\mathcal {C}}([0,T],{\mathbb {R}}^N),\quad \mathcal {L}x=-\ddot{x}+x, \end{aligned}$$

which is invertible and has a bounded inverse. We consider as well the Nemytskii operator

$$\begin{aligned} \mathcal {N}:{\mathcal {C}}([0,T],{\mathbb {R}}^N)\rightarrow {\mathcal {C}}([0,T],{\mathbb {R}}^N),\quad (\mathcal {N}x)(t)= x(t)-f(t,x(t)). \end{aligned}$$

Problem (P) is thus equivalent to the fixed point problem in ${\mathcal {C}}([0,T],{\mathbb {R}}^N)$

$$\begin{aligned} x=\mathcal {L}^{-1}\mathcal {N} x. \end{aligned}$$

Notice that $\mathcal {L}^{-1}\mathcal {N}:{\mathcal {C}}([0,T],{\mathbb {R}}^N)\rightarrow {\mathcal {C}}([0,T],{\mathbb {R}}^N)$ is completely continuous.

Here, we recall and slightly generalize [3, Theorem 4.1].

Definition 1

Given two ${\mathcal {C}}^2$-functions $\alpha ,\beta :[0,T]\rightarrow {\mathbb {R}}^N$, we say that $(\alpha ,\beta )$ is a well-ordered pair of lower/upper solutions of problem (P) if, for every $j\in \{1,\dots ,N\}$ and $t\in [0,T]$,

$$\begin{aligned} \alpha _j(t)\le & {} \beta _j(t),\\ \alpha _j(0)= & {} \alpha _j(T),\quad \beta _j(0)=\beta _j(T),\quad \dot{\alpha }_j(0)\ge \dot{\alpha }_j(T),\quad \dot{\beta }_j(0)\le \dot{\beta }_j(T), \end{aligned}$$

and, for every $x\in \prod _{m=1}^N[\alpha _m(t),\beta _m(t)]$,

$$\begin{aligned} \begin{aligned}&\ddot{\alpha }_j(t)\ge f_j(t,x_1,\dots ,x_{j-1},\alpha _j(t),x_{j+1},\dots ,x_N),\\&\ddot{\beta }_j(t)\le f_j(t,x_1,\dots ,x_{j-1},\beta _j(t),x_{j+1},\dots ,x_N). \end{aligned} \end{aligned}$$

Theorem 2

(Bebernes–Schmitt) If there exists a well-ordered pair of lower/upper solutions $(\alpha ,\beta )$, then problem (P) has a solution x(t) such that

$$\begin{aligned} \alpha _j(t)\le x_j(t)\le \beta _j(t),\quad \hbox { for every}\,\, j\in \{1,\dots ,N\}\; \hbox {and }\,t\in [0,T]. \end{aligned}$$

(1)

Proof

Step 1. Define the functions $\gamma _j:[0,T]\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ as

$$\begin{aligned} \gamma _j(t,s)=\left\{ \begin{array}{ll} \alpha _j(t) &{}\quad \hbox {if}\, s< \alpha _j(t),\\ s &{}\quad \hbox {if}\, \alpha _j(t)\le s\le \beta _j(t),\\ \beta _j(t) &{}\quad \hbox {if}\, s> \beta _j(t), \end{array}\right. \end{aligned}$$

and the functions $\Gamma ,\bar{f}:[0,T]\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$ as

$$\begin{aligned} \Gamma (t,x)=(\gamma _1(t,x_1),\dots ,\gamma _N(t,x_N)),\quad \bar{f}(t,x)=f(t,\Gamma (t,x)) . \end{aligned}$$

Consider the auxiliary problem

$$\begin{aligned} (P')\quad {\left\{ \begin{array}{ll} \ddot{x}=\bar{f}(t,x)+x-\Gamma (t,x),\\ x(0)=x(T),\quad \dot{x}(0)=\dot{x}(T), \end{array}\right. } \end{aligned}$$

and the corresponding Nemytskii operator

$$\begin{aligned} \widetilde{\mathcal {N}}:{\mathcal {C}}([0,T],{\mathbb {R}}^N)\rightarrow {\mathcal {C}}([0,T],{\mathbb {R}}^N),\quad ( \widetilde{\mathcal {N}}x)(t)=\Gamma (t,x(t))-\bar{f}(t,x(t)). \end{aligned}$$

Problem $(P')$ can then be equivalently written as a fixed point problem in ${\mathcal {C}}([0,T],{\mathbb {R}}^N)$, namely

$$\begin{aligned} x=\mathcal {L}^{-1}\widetilde{\mathcal {N}}x. \end{aligned}$$

By Schauder Theorem, since $\mathcal {L}^{-1}\widetilde{\mathcal {N}}:{\mathcal {C}}([0,T],{\mathbb {R}}^N)\rightarrow {\mathcal {C}}([0,T],{\mathbb {R}}^N)$ is completely continuous and has a bounded image, it has a fixed point, so that $(P')$ has a solution x(t).

Step 2. Let us show that (1) holds for every solution of ($P'$), thus proving the theorem. By contradiction, assume that there is a $j\in \{1,\dots ,N\}$ and a $t_j\in [0,T]$ for which $x_j(t_j)\notin [\alpha _j(t_j),\beta _j(t_j)]$. For instance, let $x_j(t_j)<\alpha _j(t_j)$ (the case $x_j(t_j)>\beta _j(t_j)$ being similar). Set $v_j(t)=\alpha _j(t)-x_j(t)$, and let $\hat{t}_j\in [0,T]$ be such that $v_j(\hat{t}_j)=\max \{v_j(t):t\in [0,T]\}$. We distinguish two cases.

Case 1 $\hat{t}_j\in \,]0,T[$. In this case, surely $\ddot{v}_j(\hat{t}_j)\le 0$. On the other hand,

$$\begin{aligned} \begin{aligned} \ddot{v}_j(\hat{t}_j)&=\ddot{\alpha }_j(\hat{t}_j)-\ddot{x}_j(\hat{t}_j)\\&= \ddot{\alpha }_j(\hat{t}_j)-\bar{f}_j(\hat{t}_j,x(\hat{t}_j))-x_j(\hat{t}_j)+\gamma _j(\hat{t}_j,x_j(\hat{t}_j))\\&>\ddot{\alpha }_j(\hat{t}_j)-f_j(\hat{t}_j,\gamma _1(\hat{t}_j,x_1(\hat{t}_j)),\dots , \alpha _j(\hat{t}_j),\dots ,\gamma _N(\hat{t}_j,x_N(\hat{t}_j)))\ge 0, \end{aligned} \end{aligned}$$

leading to a contradiction.

Case 2 $\hat{t}_j=0$ or $\hat{t}_j=T$. Assume for instance that $\hat{t}_j=0$ (the other situation being similar). Then,

$$\begin{aligned} 0\ge \dot{v}_j(0)=\dot{\alpha }_j(0)-\dot{x}_j(0)\ge \dot{\alpha }_j(T)-\dot{x}_j(T)=\dot{v}_j(T), \end{aligned}$$

so that, with $v_j(T)=v_j(0)$ being the maximum value of $v_j(t)$ over [0, T], it has to be that $\dot{v}_j(T)=0$, and hence also $\dot{v}_j(0)=0$. Now, since $v_j(0)>0$, there is a small $\delta >0$ such that $v_j(s)>0$, for every $s\in [0,\delta ]$. Then, if $t\in [0,\delta ]$, we have that $x_j(s)<\alpha _j(s)$, for every $s\in [0,t]$; hence,

$$\begin{aligned} \begin{aligned} \dot{v}_j(t)&=\dot{v}_j(0)+\int _0^t\ddot{v}_j(s)\,\mathrm {d}s\\&=\int _0^t\Big (\ddot{\alpha }_j(s)-\ddot{x}_j(s)\Big )\,\mathrm {d}s\\&=\int _0^t\Big (\ddot{\alpha }_j(s)-\bar{f}_j(s,x(s))-x_j(s)+\gamma _j(s,x_j(s))\Big )\,\mathrm {d}s\\&>\int _0^t\Big (\ddot{\alpha }_j(s)-f_j(s,\gamma _1(s,x_1(x)),\dots ,\alpha _j(s), \dots ,\gamma _N(s,x_N(x)))\Big )\,\mathrm {d}s\ge 0, \end{aligned} \end{aligned}$$

a contradiction, since 0 is a maximum point for $v_j(t)$. The proof is thus completed. $\square$

We now provide some illustrative examples.

Example 3

Let, for every $j\in \{1,\dots ,N\}$,

$$\begin{aligned} f_j(t,x)=a_jx_j^3+h_j(t,x), \end{aligned}$$

for some constants $a_j>0$, and assume that there is a $c>0$ such that

$$\begin{aligned} |h(t,x)|\le c,\quad \hbox { for every }(t,x)\in [0,T]\times {\mathbb {R}}^N. \end{aligned}$$

(2)

Then, taking the constant functions $\alpha _j=-\root 3 \of {c/a_j}$, $\beta _j=\root 3 \of {c/a_j}$, we see that Theorem 2 applies, and hence (P) has a solution.

Example 4

Let us consider, for every $j\in \{1,\dots ,N\}$,

$$\begin{aligned} f_j(t,x)= x_j^2 \sin x_j + h_j(t,x), \end{aligned}$$

and assume that there is a $c>0$ such that (2) holds. Then, for every $\ell \in \mathbb Z$ with $|\ell |$ sufficiently large, taking the constant functions $\alpha _j=-\pi /2+2\ell \pi$, $\beta _j=\pi /2+2\ell \pi$, we see that Theorem 2 applies, and we conclude that (P) admits an infinite number of solutions.

To work with Leray–Schauder degree, we need to introduce the notions of strict lower/upper solutions.

Definition 5

The well-ordered pair of lower/upper solutions $(\alpha ,\beta )$ of problem (P) is said to be strict if $\alpha _j(t) < \beta _j(t)$ for every $j\in \{1,\dots ,N\}$ and $t\in [0,T]$, and the following property holds: if x(t) is a solution of (P) satisfying (1), then

$$\begin{aligned} \alpha _j(t)< x_j(t)< \beta _j(t),\quad \hbox { for every}\; j\in \{1,\dots ,N\}\; \hbox {and }\;t\in [0,T]. \end{aligned}$$

When we have a well-ordered pair of strict lower/upper solutions, the previous theorem provides some additional information.

Theorem 6

If $(\alpha ,\beta )$ is a strict well-ordered pair of lower/upper solutions of problem (P), then

$$\begin{aligned} d(I-\mathcal {L}^{-1}{\mathcal {N}},\Omega )=1, \end{aligned}$$

where

$$\begin{aligned} \Omega :=\big \{x\in \mathcal {C}([0,T],{\mathbb {R}}^N) \,:\, \alpha _j(t)< x_j(t) < \beta _j(t),\, \text { for every } j\in \{1,\dots ,N\} \text { and } t\in [0,T]\big \}. \end{aligned}$$

Proof

Arguing as in Step 1 of the proof of Theorem 2, we can introduce the modified problem $(P')$ and we know, by Schauder Theorem, that

$$\begin{aligned} d(I-\mathcal {L}^{-1}\widetilde{\mathcal {N}}, B_R)=1, \end{aligned}$$

where $B_R$ is an open ball in $\mathcal {C}([0,T],{\mathbb {R}}^N)$ centered at the origin with a sufficiently large radius $R>0$. In particular, we may assume that $\Omega \subseteq B_R$. By the argument in Step 2 of the same proof and the fact that the pair of lower/upper solutions is strict, we have that all the solutions of $(P')$ belong to $\Omega$. In other words, there are no zeroes of $I-\mathcal {L}^{-1}\widetilde{\mathcal {N}}$ in the set $B_R \setminus \overline{\Omega }$. Then, by the excision property of the degree,

$$\begin{aligned} d(I-\mathcal {L}^{-1}\widetilde{\mathcal {N}}, \Omega )=1. \end{aligned}$$

Finally, since ${\mathcal {N}}$ and $\widetilde{\mathcal {N}}$ coincide on the set $\Omega$, the conclusion follows. $\square$

3 Non-well-Ordered Lower and Upper Solutions for Systems

In this section, we still consider problem (P) in the finite-dimensional space ${\mathbb {R}}^N$. We will treat the case in which we can find lower and upper solutions which are not well-ordered. To this aim, we need to distinguish the components which are well-ordered from the others.

We will say that the couple $(\mathcal {J},\mathcal {K})$ is a partition of the set of indices $\{1,\dots ,N\}$ if and only if $\mathcal {J} \cap \mathcal {K} = \varnothing$ and $\mathcal {J} \cup \mathcal {K} = \{1,\dots ,N\}$. Correspondingly, we can decompose a vector

$$\begin{aligned} x=(x_1,\dots ,x_N)=(x_n)_{n\in \{1,\dots ,N\}}\in {\mathbb {R}}^N \end{aligned}$$

as $x=(x_{\mathcal {J}}, x_{\mathcal {K}}),$ where $x_{\mathcal {J}}=(x_j)_{j\in \mathcal {J}}\in {\mathbb {R}}^{\#\mathcal {J}}$ and $x_{\mathcal {K}}=(x_k)_{k\in \mathcal {K}}\in {\mathbb {R}}^{\#\mathcal {K}}$. Here, $\#\mathcal {J}$ and $\#\mathcal {K}$ denote, respectively, the cardinality of the sets $\mathcal {J}$ and $\mathcal {K}$.

Similarly, every function $\mathcal {F}: \mathcal {A} \rightarrow {\mathbb {R}}^N$ can be written as $\mathcal {F}(x)=\big (\mathcal {F}_{\mathcal {J}}(x),\mathcal {F}_{\mathcal {K}}(x)\big ),$ where $\mathcal {F}_{\mathcal {J}}: \mathcal {A} \rightarrow {\mathbb {R}}^{\#\mathcal {J}}$ and $\mathcal {F}_{\mathcal {K}}: \mathcal {A} \rightarrow {\mathbb {R}}^{\#\mathcal {K}}$.

Definition 7

Given two $\mathcal {C}^2$-functions $\alpha ,\beta :[0,T]\rightarrow {\mathbb {R}}^N,$ we will say that $(\alpha ,\beta )$ is a pair of lower/upper solutions of (P) related to the partition $(\mathcal {J},\mathcal {K})$ of $\{1,\dots ,N\}$ if the following four conditions hold:

• 1. for any $j\in \mathcal {J}$, $\alpha _j(t)\le \beta _j(t)$ for every $t\in [0,T]$;

• 2. for any $k\in \mathcal {K}$, there exists $t^0_k\in [0,T]$ such that $\alpha _k(t^0_k)> \beta _k(t^0_k)$;

• 3. for any $n\in \{1,\dots ,N\}$ we have

  $$\begin{aligned}&\ddot{\alpha }_n(t) \ge f_n(t,x_1,\dots ,x_{n-1},\alpha _n(t),x_{n+1},\dots ,x_N), \end{aligned}$$

  (3)
  $$\begin{aligned}&\ddot{\beta }_n(t) \le f_n(t,x_1,\dots ,x_{n-1},\beta _n(t),x_{n+1},\dots ,x_N), \end{aligned}$$

  (4)

  for every $(t,x)\in \mathcal {E}$, where

  $$\begin{aligned} \mathcal {E} :=\left\{ (t,x)\in [0,T]\times {\mathbb {R}}^N \,:\, x=(x_{\mathcal {J}},x_{\mathcal {K}}), x_{\mathcal {J}}\in \prod _{j\in \mathcal {J}} [\alpha _j(t),\beta _j(t)] \right\} . \end{aligned}$$

• 4. for any $n\in \{1,\dots ,N\}$,

  $$\begin{aligned} \alpha _n(0)=\alpha _n(T), \qquad \beta _n(0)=\beta _n(T), \\ \dot{\alpha }_n(0)\ge \dot{\alpha }_n(T), \qquad \dot{\beta }_n(0)\le \dot{\beta }_n(T). \end{aligned}$$

Definition 8

The pair $(\alpha ,\beta )$ of lower/upper solutions of (P) is said to be strict with respect to the j -th component, with $j\in \mathcal {J}$, if $\alpha _j(t)<\beta _j(t)$ for every $t\in [0,T]$, and for every solution x of (P) we have

$$\begin{aligned}&\big (\forall t\in [0,T] ,\, \alpha _j(t)\le x_j(t)\le \beta _j(t) \big ) \Rightarrow \big (\forall t\in [0,T] ,\, \alpha _j(t)< x_j(t)<\beta _j(t)\big )\,;\nonumber \\ \end{aligned}$$

(5)

it is said to be strict with respect to the k -th component, with $k\in \mathcal {K}$, if for every solution x of (P) we have

$$\begin{aligned}&\big (\forall t\in [0,T] ,\, x_k(t)\ge \alpha _k(t) \big ) \Rightarrow \big (\forall t\in [0,T] ,\, x_k(t)>\alpha _k(t)\big ), \end{aligned}$$

(6)

$$\begin{aligned}&\big (\forall t\in [0,T] ,\, x_k(t)\le \beta _k(t)\big ) \Rightarrow \big (\forall t\in [0,T] ,\, x_k(t)<\beta _k(t)\big ). \end{aligned}$$

(7)

The following proposition provides a sufficient condition to guarantee the strictness property of a pair of lower/upper solutions of (P) with respect to a certain component.

Proposition 9

Given a pair $(\alpha ,\beta )$ of lower/upper solutions of (P),

• 1. if, for any $n\in \mathcal {J}$, both (3) and (4) hold with strict inequalities, then (5) holds for $n=j$;

• 2. if, for any $n\in \mathcal {K}$, (3) holds with strict inequality, then (6) holds for $n=k$;

• 3. if, for any $n\in \mathcal {K}$, (4) holds with strict inequality, then (7) holds for $n=k$.

The proof can be easily adapted from the corresponding scalar result in [5, Proposition III-1.1] and is omitted.

We are able to prove the existence of a solution of (P) in presence of a pair of lower/upper solutions $(\alpha ,\beta )$ provided that we ask the strictness property when the components $\alpha _k,\beta _k$ are non-well-ordered.

Theorem 10

Let $(\alpha ,\beta )$ be a pair of lower/upper solutions of (P) related to the partition $(\mathcal {J},\mathcal {K})$ of $\{1,\dots ,N\}$, and assume that it is strict with respect to the k-th component, for every $k\in \mathcal {K}$. Assume moreover the existence of a constant $C>0$ such that

$$\begin{aligned} |f_{\mathcal {K}}(t,x)| \le C ,\quad \hbox { for every }(t,x)\in \mathcal {E}. \end{aligned}$$

Then, (P) has a solution x with the following property: for any $(j,k)\in \mathcal {J}\times \mathcal {K}$,

${(W_j)}$:

$\alpha _j(t)\le x_j(t)\le \beta _j(t)$, for every $t \in [0,T]$ ;

${(NW_k)}$:

there exist $t_k^1,t_k^2\in [0,T]$ such that $x_k(t_k^1)<\alpha _k(t_k^1)$ and $x_k(t_k^2)>\beta _k(t_k^2)$.

In Sect. 3.2 we will provide a generalization of the above result, removing the strictness assumption on one of the components $\kappa \in \mathcal {K}$. Let us now present two illustrative examples.

Example 11

Assume $\mathcal {J}=\varnothing$ and let, for every $k\in \mathcal {K}$,

$$\begin{aligned} f_k(t,x)=-\frac{a_kx_k}{1+|x_k|}+h_k(t,x), \end{aligned}$$

for some $a_k>0$, with

$$\begin{aligned} \Vert h_k\Vert _\infty :=\sup \big \{|h_k(t,x)|:(t,x)\in [0,T]\times {\mathbb {R}}^N\big \}< a_k. \end{aligned}$$

(8)

Then, taking the constant functions

$$\begin{aligned} \alpha _k=\frac{\Vert h_k\Vert _\infty }{a_k-\Vert h_k\Vert _\infty }+1,\qquad \beta _k=-\frac{\Vert h_k\Vert _\infty }{a_k-\Vert h_k\Vert _\infty }-1, \end{aligned}$$

we see that Theorem 10 applies. The same would be true if $\mathcal {J}\ne \varnothing$, assuming for $j\in \mathcal {J}$, e.g., a situation like in Examples 3 and 4.

Example 12

Let

$$\begin{aligned} f_n(t,x)=-a_n\sin x_n+h_n(t,x), \end{aligned}$$

with $a_n>0$ and $h_n$ satisfying (8) with $k=n$. For every $n\in \{1,\dots ,N\}$ we have constant lower and upper solutions

$$\begin{aligned} \alpha _n\in \left\{ \frac{\pi }{2}+2m\pi :m\in \mathbb Z\right\} ,\qquad \beta _n\in \left\{ -\frac{\pi }{2}+2m\pi :m\in \mathbb Z\right\} . \end{aligned}$$

Then, for each equation we have both well-ordered and non-well-ordered pairs of lower/upper solutions. Let us fix, e.g.,

$$\begin{aligned} \alpha _n=\frac{\pi }{2},\qquad \beta ^\iota _n=\frac{\pi }{2}+\iota \pi ,\quad \hbox { with }\;\iota \in \{-1,1\}. \end{aligned}$$

Choosing $\mathbf {\iota }=(\iota _1,\dots ,\iota _N)\in \{-1,1\}^N$, and defining $(\alpha ,\beta )$ with $\beta _n=\beta ^{\iota _n}_n$, by Theorem 10 we get the existence of at least $2^N$ solutions $x^{\mathbf {\iota }}$ of problem (P), whose components are such that

$$\begin{aligned}&\iota _n=1\Rightarrow \forall t\in [0,T], \;x^{\mathbf {\iota }}_n(t)\in \left[ \frac{\pi }{2},\frac{3\pi }{2}\right] ,\\&\iota _n=-1\Rightarrow \exists \bar{t}_n\in [0,T], \;x^{\mathbf {\iota }}_n(\bar{t}_n)\in \left[ -\frac{\pi }{2},\frac{\pi }{2}\right] . \end{aligned}$$

We notice that, even if the function $h(t,x_1,\dots ,x_n)$ is $2\pi$-periodic in each variable $x_n$, the solutions we find are indeed geometrically distinct. We thus get a generalization of a result obtained for the scalar equation in [17].

3.1 Proof of Theorem 10

Notice that the case $\mathcal {K}=\varnothing$ reduces to Theorem 2. We thus assume $\mathcal {K}\ne \varnothing$ and, without loss of generality, we take either $\mathcal {J}=\varnothing$, or $\mathcal {J}=\{1,\dots ,M\}$ and $\mathcal {K}=\{M+1,\dots ,N\}$ for a certain $M\in \{1,\dots ,N\}$. Indeed, mixing the coordinates of $x=(x_1,\dots ,x_N)$, we can always reduce to such a situation. We continue the proof in the case $\mathcal {J}\ne \varnothing$. (The case $\mathcal {J}=\varnothing$ can be treated essentially in the same way.)

We need to suitably modify problem (P). For every $r>0$, we consider the problem

$$\begin{aligned} (P_r)\quad {\left\{ \begin{array}{ll} \ddot{x}=g_r(t,x),\\ x(0)=x(T),\quad \dot{x}(0)=\dot{x}(T), \end{array}\right. } \end{aligned}$$

where $g_r:[0,T]\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$, with

$$\begin{aligned} g_r(t,x)=\big (g_{r,1}(t,x), \dots ,g_{r,M}(t,x), g_{r,M+1}(t,x),\dots , g_{r,N}(t,x)\big ), \end{aligned}$$

is defined as follows.

We first introduce the functions $\bar{f}:[0,T]\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$ and $\Gamma :[0,T]\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$ as

$$\begin{aligned} \bar{f}(t,x)&= f(t,\Gamma (t,x)),\\ \Gamma (t,x)&= \big (\gamma _{1}(t,x_{1}),\dots ,\gamma _{M}(t,x_{M}),x_{M+1},\dots ,x_N \big ), \end{aligned}$$

where, for $j\in \mathcal {J}$,

$$\begin{aligned} \gamma _j(t,s)= {\left\{ \begin{array}{ll} \alpha _j(t), &{} \quad \text {if }\ s<\alpha _j(t),\\ s, &{}\quad \text {if }\ \alpha _j(t)\le s\le \beta _j(t),\\ \beta _j(t), &{}\quad \text {if }\ s>\beta _j(t). \end{array}\right. } \end{aligned}$$

Now we define, for every index $j\in \mathcal {J}$,

$$\begin{aligned} g_{r,j}(t,x)=\bar{f}_j(t,x)+x_j-\gamma _j(t,x_j), \end{aligned}$$

and for every index $k\in \mathcal {K}$,

$$\begin{aligned} g_{r,k}(t,x)= {\left\{ \begin{array}{ll} \bar{f}_k(t,x) &{}\quad \text {if }\ |x_k|\le r,\\ \displaystyle (|x_k|-r)\,C\frac{x_k}{|x_k|}+(1+r-|x_k|) \bar{f}_k(t,x) &{}\quad \text {if }\ r<|x_k|<r+1,\\ \displaystyle C\frac{x_k}{|x_k|} &{}\quad \text {if }\ |x_k|\ge r+1. \end{array}\right. } \end{aligned}$$

Notice that, for the indices $j\in \mathcal {J}$, the value $r>0$ does not affect the definition of the components $g_{r,j}$.

Proposition 13

If x is a solution of $(P_r)$, then $\alpha _j(t)\le x_j(t)\le \beta _j(t)$ for every $j\in \mathcal {J}$ and $t\in [0,T]$.

The proof follows from a classical reasoning and can be easily adapted from Step 2 of the proof of Theorem 2.

Proposition 14

There is a constant $K>0$ such that, if x is a solution of $(P_r)$, for any $r>0$, which satisfies ${(NW_k)}$ for a certain index $k\in \mathcal {K}$, then $\Vert x_k\Vert _{{\mathcal {C}}^2}\le K$.

Proof

Notice that

$$\begin{aligned} |g_{r,k}(t,x)|\le C,\quad \hbox { for every }(t,x)\in [0,T]\times {\mathbb {R}}^N, k\in \mathcal {K} \text { and } r>0. \end{aligned}$$

(9)

Fix any $k\in \mathcal {K}$. If x(t) is a solution of $(P_r)$, multiplying the k-th equation by $\tilde{x}_k$ and integrating, we have that

$$\begin{aligned} \Vert \tilde{x}_k\Vert _2^2 \le \left( \frac{T}{2\pi }\right) ^2 \Vert \dot{x}_k\Vert ^2_2\le \left( \frac{T}{2\pi }\right) ^2 C\sqrt{T}\Vert \tilde{x}_k\Vert _2. \end{aligned}$$

So, by a classical reasoning, there is a constant $C_1>0$ such that $\Vert \tilde{x}_k\Vert _{H^1}\le C_1$, and there is a constant $C_0>0$ such that $\Vert \tilde{x}_k\Vert _\infty \le C_0$, for every solution x of $(P_r)$. Define

$$\begin{aligned} u_k(t)=\min \{\alpha _k(t),\beta _k(t)\},\quad {\mathcal {U}}_k(t)=\max \{\alpha _k(t),\beta _k(t)\}. \end{aligned}$$

(10)

Since ${(NW_k)}$ holds, there is a $\tau _0\in [0,T]$ such that

$$\begin{aligned} u_k(\tau _0)\le x_k(\tau _0)\le {\mathcal {U}}_k(\tau _0). \end{aligned}$$

(11)

Then, if x is a solution of $(P_r)$,

$$\begin{aligned} |x_k(t)|= & {} \left| x_k(\tau _0)+\int _{\tau _0}^t\dot{x}_k(s)\,\mathrm {d}s\right| \le |x_k(\tau _0)|\\&+\int _0^T|\dot{x}_k(s)|\,\mathrm {d}s\le |x_k(\tau _0)|+\sqrt{T}\Vert \dot{x}_k\Vert _2\\\le & {} \max \{\Vert \alpha \Vert _\infty ,\,\Vert \beta \Vert _\infty \} + \sqrt{T} C_1 =: K_0, \end{aligned}$$

hence $\Vert x_k\Vert _\infty \le K_0$. Moreover, by periodicity, there is a $\tau _1\in [0,T]$ such that $\dot{x}_k(\tau _1)=0$, hence by (9)

$$\begin{aligned} |\dot{x}_k(t)|= & {} \left| \dot{x}_k(\tau _1)+\int _{\tau _1}^t\ddot{x}_k(s)\,\mathrm {d}s\right| = \left| \int _{\tau _1}^t g_{r,k}(s,x(s))\,\mathrm {d}s\right| \\\le & {} \int _{0}^T |g_{r,k}(s,x(s))|\,\mathrm {d}s \le CT, \end{aligned}$$

so that $\Vert \dot{x}_k\Vert _\infty \le CT$. Then,

$$\begin{aligned} \Vert x_k\Vert _{{\mathcal {C}}^2}=\Vert x_k\Vert _\infty +\Vert \dot{x}_k\Vert _\infty +\Vert \ddot{x}_k\Vert _\infty \le K_0+CT +C =: K, \end{aligned}$$

thus proving the proposition. $\square$

From now on, we fix $r>\max \{K, \Vert \alpha \Vert _\infty ,\,\Vert \beta \Vert _\infty \}$, where K is given by Lemma 14. Problem $(P_r)$ is equivalent to the fixed point problem

$$\begin{aligned} x=\mathcal {L}^{-1}\mathcal {N}_r x,\qquad x\in {\mathcal {C}}([0,T],{\mathbb {R}}^N), \end{aligned}$$

where we have introduced the Nemytskii operator

$$\begin{aligned} \mathcal {N}_r:{\mathcal {C}}([0,T],{\mathbb {R}}^N)\rightarrow {\mathcal {C}}([0,T],{\mathbb {R}}^N),\quad (\mathcal {N}_rx)(t)= x(t)-g_r(t,x(t)). \end{aligned}$$

Since we are looking for zeros of

$$\begin{aligned} {\mathcal {T}_r}x := (I-\mathcal {L}^{-1}\mathcal {N}_r)(x), \end{aligned}$$

we compute the Leray–Schauder degree on a family of open sets. Let us define the constant functions

$$\begin{aligned} \hat{\alpha } = -r-1, \qquad \hat{\beta } = r+1, \end{aligned}$$

as well as the functions

$$\begin{aligned} \check{\alpha }_j(t)=\alpha _j(t)-1, \quad \text {and} \quad \check{\beta }_j(t)=\beta _j(t) +1, \end{aligned}$$

for every $j\in \mathcal {J}$.

We define, for every multi-index $\mu =(\mu _{M+1},\dots ,\mu _N)\in \{1,2,3,4\}^{N-M}$, the open set

$$\begin{aligned} \Omega _\mu :=\{ x\in \mathcal {C}([0,T],{\mathbb {R}}^N) : ({\mathcal {O}}_j^0)\,\text { and}\,({\mathcal {O}}_k^{\mu _k}) \text { hold for every }\,j\in \mathcal {J}\,\text { and}\, k\in \mathcal {K} \big \}, \end{aligned}$$

(12)

where the conditions $({\mathcal {O}}_j^0)$ and $({\mathcal {O}}_k^{\mu _k})$ read as

$({\mathcal {O}}^0_j)$:

$\check{\alpha }_j(t)< x_j(t)<\check{\beta }_j(t)$, for every $t\in [0,T]$,

$({\mathcal {O}}^1_k)$:

$\hat{\alpha }< x_k(t) < \hat{\beta }$, for every $t\in [0,T]$,

$({\mathcal {O}}^2_k)$:

$\hat{\alpha }< x_k(t) < \beta _k(t)$, for every $t\in [0,T]$,

$({\mathcal {O}}^3_k)$:

$\alpha _k(t)< x_k(t) < \hat{\beta }$, for every $t\in [0,T]$,

$({\mathcal {O}}^4_k)$:

$\hat{\alpha }< x_k(t) < \hat{\beta }$, for every $t\in [0,T]$, and there are $t_k^1,t_k^2\in [0,T]$ such that $x(t_k^1)<\alpha _k(t_k^1)$ and $x(t_k^2)>\beta _k(t_k^2)$.

Proposition 15

The Leray–Schauder degree $d({\mathcal {T}_r},\Omega _\mu )$ is well defined for every $\mu \in \{1,2,3,4\}^{N-M}$.

Proof

Assume by contradiction that there is $x\in \partial \Omega _\mu$ such that ${\mathcal {T}_r}x=0$, i.e., x is a solution of $(P_r)$. All the several different situations which may arise lead back to the following four cases.

Case A For some index $j\in \mathcal {J}$, $\check{\alpha }_j(t)\le x_j(t)\le \check{\beta }_j(t)$, for every $t\in [0,T]$, and $\check{\alpha }_j(\tau )= x_j(\tau )$ for a certain $\tau \in [0,T]$ (the case when $x_j(\tau )=\check{\beta }_j(\tau )$ is similar). We can prove that

$$\begin{aligned} \ddot{\check{\alpha }}_j(t) > g_{r,j}(t,x_1(t),\dots ,x_{j-1}(t),\check{\alpha }_j(t),x_{j+1}(t),\dots ,x_N(t)), \quad \text {for every } t\in [0,T], \end{aligned}$$

so that arguing as in Step 2 of the proof of Theorem 2 we obtain a contradiction.

Case B For some index $k\in \mathcal {K}$, $\hat{\alpha } \le x_k(t) \le \hat{\beta }$, for every $t\in [0,T]$, and $\hat{\alpha } = x_k(\tau )$ for a certain $\tau \in [0,T]$ (the case when $x_k(\tau )=\hat{\beta }$ is similar). Since

$$\begin{aligned} g_{r,k}(t,x_1(t),\dots ,x_{k-1}(t),\hat{\alpha },x_{k+1}(t),\dots ,x_N(t))=-C<0, \quad \text {for every } t\in [0,T], \end{aligned}$$

we easily get a contradiction as before.

Case C For some index $k\in \mathcal {K}$, $\hat{\alpha } < x_k(t) \le \beta _k(t)$, for every $t\in [0,T]$, and $x_k(\tau )=\beta _k(\tau )$ for a certain $\tau \in [0,T]$. Such a situation cannot arise since (7) holds by assumption.

Case D For some index $k\in \mathcal {K}$, $\alpha _k(t) \le x_k(t) < \hat{\beta }$, for every $t\in [0,T]$, and $x_k(\tau )=\alpha _k(\tau )$ for a certain $\tau \in [0,T]$. Such a situation cannot arise since (6) holds by assumption. $\square$

Proposition 16

For every multi-index $\mu \in \{1,2,3\}^{N-M},$ we have $d({\mathcal {T}_r},\Omega _\mu )=1$.

Proof

In this case, it can be verified by the arguments of the previous proof that the definition of the set $\Omega _\mu$ provides us a well-ordered pair of strict lower/upper solutions of problem $(P_r)$. The conclusion is then an immediate consequence of Theorem 6. $\square$

For any multi-index $\hat{\mu }\in \{1,2,3\}^{N-M-1},$ we can consider, for every $\ell \in \{1,2,3,4\}$, the multi-index

$$\begin{aligned} (\ell ,\hat{\mu })=(\ell ,\mu _{M+2},\dots ,\mu _{N})\in \{1,2,3,4\}^{N-M}. \end{aligned}$$

We can verify that $\Omega _{(2,\hat{\mu })},\Omega _{(3,\hat{\mu })},\Omega _{(4,\hat{\mu })}$ are pairwise disjoint and all contained in $\Omega _{(1,\hat{\mu })}$ so that

$$\begin{aligned} \Omega _{(4,\hat{\mu })} = \Omega _{(1,\hat{\mu })}\setminus \overline{\Omega _{(2,\hat{\mu })}\cup \Omega _{(3,\hat{\mu })}}. \end{aligned}$$

(13)

Proposition 17

For every multi-index $\hat{\mu }\in \{1,2,3\}^{N-M-1},$ we have $d({\mathcal {T}_r},\Omega _{(4,\hat{\mu })})=-1.$

Proof

By Proposition 16 and (13),

$$\begin{aligned} 1&= d({\mathcal {T}_r},\Omega _{(1,\hat{\mu })})\\&= d({\mathcal {T}_r},\Omega _{(2,\hat{\mu })}) + d({\mathcal {T}_r},\Omega _{(3,\hat{\mu })}) + d({\mathcal {T}_r},\Omega _{(4,\hat{\mu })})\\&=2+d({\mathcal {T}_r},\Omega _{(4,\hat{\mu })}) \end{aligned}$$

and the conclusion follows. $\square$

Arguing similarly, we can prove by induction the following result.

Proposition 18

For every $K\in \{1,\dots ,N-M\}$ and every multi-index $\mu \in \{4\}^K\times \{1,2,3\}^{N-M-K}$, we have

$$\begin{aligned} d({\mathcal {T}_r},\Omega _{\mu })=(-1)^K. \end{aligned}$$

Proof

We proceed by induction. The validity of the statement for $K=1$ follows by Proposition 17. So, we fix $K\ge 2$ and assume that

$$\begin{aligned} d({\mathcal {T}_r}, \Omega _\mu )=(-1)^{K-1}, \;\text {for every } \mu \in \{4\}^{K-1}\times \{1,2,3\}^{N-M-K+1}. \end{aligned}$$

Consider the multi-index $\mu =(4,\dots ,4,\mu _{M+K},\mu _{M+K+1},\dots ,\mu _N)\in \{4\}^{K-1}\times \{1,2,3\}^{N-M-K+1}$ and define for every $\ell \in \{1,2,3,4\}$, the multi-index

$$\begin{aligned} \bar{\mu }^\ell =(4,\dots ,4,\ell ,\mu _{M+K+1},\dots ,\mu _N). \end{aligned}$$

We then see that

$$\begin{aligned} (-1)^{K-1}&= d({\mathcal {T}_r},\Omega _{\bar{\mu }^1})\\&= d({\mathcal {T}_r},\Omega _{\bar{\mu }^2}) + d({\mathcal {T}_r},\Omega _{\bar{\mu }^3}) +d({\mathcal {T}_r},\Omega _{\bar{\mu }^4}) \\&=2\cdot (-1)^{K-1}+d({\mathcal {T}_r},\Omega _{\bar{\mu }^4}) , \end{aligned}$$

yielding $d({\mathcal {T}_r},\Omega _{\bar{\mu }^4})=(-1)^K$. The proof is complete. $\square$

By the previous proposition, we conclude that

$$\begin{aligned} d({\mathcal {T}_r},\Omega _{(4,\dots ,4)})=(-1)^{N-M}. \end{aligned}$$

(14)

As a consequence, there is a solution x of problem $(P_r)$ in the set $\Omega _{(4,\dots ,4)}$. Recalling the a priori bounds in Propositions 13 and 14, we see that the solution x is indeed a solution of problem (P) and satisfies ${(W_j)}$ and ${(NW_k)}$, for every $j\in \mathcal {J}$ and $k\in \mathcal {K}$. The proof is thus completed.

3.2 An Extension of Theorem 10

The existence of a solution of (P) can be obtained also removing from the assumptions of Theorem 10 the strictness assumption on one of the components.

Theorem 19

Let $(\alpha ,\beta )$ be a pair of lower/upper solutions of (P) related to the partition $(\mathcal {J}, \mathcal {K})$ of $\{1,\dots ,N\}$. Fix $\kappa \in \mathcal {K}$ and assume that $(\alpha ,\beta )$ is strict with respect to the k-th component, for every $k\in \mathcal {K}\setminus \{\kappa \}$. Assume moreover the existence of a constant $C>0$ such that

$$\begin{aligned} |f_{\mathcal {K}}(t,x)| \le C ,\quad \hbox { for every }(t,x)\in \mathcal {E}. \end{aligned}$$

Then, (P) has a solution x such that ${(W_j)}$ and ${(NW_k)}$ hold for every $(j,k)\in \mathcal {J}\times (K\setminus \{\kappa \})$, and

${(\widetilde{NW}_\kappa )}$:

there exist $t_\kappa ^1,t_\kappa ^2\in [0,T]$ such that $x_\kappa (t_\kappa ^1)\le \alpha _\kappa (t_\kappa ^1)$ and $x_\kappa (t_\kappa ^2)\ge \beta _\kappa (t_\kappa ^2)$.

Proof

Without loss of generality, we can choose $\mathcal {J}=\{1,\dots ,M\}$, $\mathcal {K}=\{M+1,\dots ,N\}$ and $\kappa =N$. We can follow the proof of Theorem 10 step by step in the first part, noticing that Proposition 14 holds with the same constant when we assume $(\widetilde{NW}_N)$. Moreover, since we do not ask the strictness assumption with respect to the N-th component, when we introduce the sets $\Omega _\mu$ as in (12), we can consider only multi-indices with the last component frozen to 1, i.e., $\mu =(\mu _{M+1},\dots ,\mu _{N-1},1)\in \{1,2,3,4\}^{N-M-1}\times \{1\}$. Indeed, with this new choice of the multi-indices we can still guarantee that the Leray–Schauder degree is well defined.

Then, arguing as in Propositions 16, 17 and 18, we have

• $d({\mathcal {T}_r}, \Omega _\mu )=1$ for every $\mu \in \{1,2,3\}^{N-M-1}\times \{1\}$,

• $d({\mathcal {T}_r}, \Omega _\mu )=-1$ for every $\mu \in \{4\}\times \{1,2,3\}^{N-M-2}\times \{1\}$,

• for every $K\in \{1,\dots ,N-M-1\}$, $d({\mathcal {T}_r},\Omega _\mu )=(-1)^K$ for every multi-index $\mu \in \{4\}^K\times \{1,2,3\}^{N-M-K-1} \times \{1\}$.

However, we cannot conclude the proof saying that the Leray–Schauder degree is different from zero in $\Omega _{(4,\dots ,4)}$ as in (14), since we cannot ensure that it is well defined in the sets $\Omega _{(4,\dots ,4,\ell )}$ with $\ell =2,3,4$.

Anyhow, at this step of the proof, we can follow the classical reasoning adopted in the scalar case in presence of non-well-ordered lower/upper solutions, cf. [5, Theorem III-3.1]. If there exists $x\in \partial \Omega _{(4,\dots ,4,2)}$ such that ${\mathcal {T}_r}x=0$, then we can easily see that x must be a solution of $(P_r)$ such that $x_N(t)\le \beta _N(t)$ for every $t\in [0,T]$ and $x_N(\tau )=\beta _N(\tau )$ for a certain $\tau \in [0,T]$. Since the components $\alpha _N$, $\beta _N$ are non-well-ordered, we have $\alpha _N(t^0_N)>\beta _N(t^0_N)\ge x_N(t^0_N)$ for some $t_0^N\in [0,T]$. So $(\widetilde{NW}_N)$ holds, thus giving us that x is a solution of $(P_r)$ satisfying all the required assumptions.

We can argue similarly if there exists $x\in \partial \Omega _{(4,\dots ,4,3)}$ such that ${\mathcal {T}_r}x=0$.

If the previous situations do not occur, we can compute the degree both in $\Omega _{(4,\dots ,4,2)}$ and $\Omega _{(4,\dots ,4,3)}$. As in (13), we have

$$\begin{aligned} \Omega _{(4,\dots ,4,4)} = \Omega _{(4,\dots ,4,1)}\;{\setminus }\; \overline{\Omega _{(4,\dots ,4,2)}\cup \Omega _{(4,\dots ,4,3)}}, \end{aligned}$$

(15)

so that the degree is well defined also for $\Omega _{(4,\dots ,4,4)}$. Performing the same computation adopted in Propositions 17 and 18, we can conclude that $d({\mathcal {T}_r},\Omega _{(4,\dots ,4)})=(-1)^{N-M}$, thus finding also in this case a solution x with the desired properties. The proof is thus completed. $\square$

4 Lower and Upper Solutions for Infinite-Dimensional Systems

We now focus our attention on a system defined in a separable Hilbert space H with scalar product $\langle \cdot ,\cdot \rangle$ and corresponding norm $|\cdot |$. We study the problem

$$\begin{aligned} (P)\quad {\left\{ \begin{array}{ll} \ddot{x}=f(t,x),\\ x(0)=x(T),\quad \dot{x}(0)=\dot{x}(T), \end{array}\right. } \end{aligned}$$

where $f:[0,T]\times H\rightarrow H$ is a continuous function. In what follows, we extend the results of Sect. 3 to an infinite-dimensional setting, trying to maintain similar notations.

Let $\mathbb N_+=\{1,2,3,\dots \}$. Choosing a Hilbert basis $(e_n)_{n\in {\mathbb {N}}_+}$, every vector $x\in H$ can be written as $x=\sum _{n\in {\mathbb {N}}_+} x_ne_n$, or $x=(x_n)_{n\in {\mathbb {N}}_+}=(x_1,x_2,\dots )$. Similarly, for the function f, we will write

$$\begin{aligned} f(t,x)=(f_1(t,x),f_2(t,x),\dots ). \end{aligned}$$

We will sometimes identify H with $\ell ^2$.

As in the finite-dimensional case, we will say that the couple $(\mathcal {J}, \mathcal {K})$ is a partition of ${\mathbb {N}}_+$ if and only if $\mathcal {J}\cap \mathcal {K} = \varnothing$ and $\mathcal {J}\cup \mathcal {K} = {\mathbb {N}}_+$. Correspondingly, we can decompose the Hilbert space as $H=H_{\mathcal {J}}\times H_{\mathcal {K}}$, where every $x\in H$ can be written as $x=(x_{\mathcal {J}}, x_{\mathcal {K}})$ with $x_{\mathcal {J}}=(x_j)_{j\in \mathcal {J}}\in H_{\mathcal {J}}$ and $x_{\mathcal {K}}=(x_k)_{k\in \mathcal {K}}\in H_{\mathcal {K}}$.

Similarly, every function $\mathcal {F}: \mathcal {A} \rightarrow H$ can be written as $\mathcal {F}(x)=\big (\mathcal {F}_{\mathcal {J}}(x),\mathcal {F}_{\mathcal {K}}(x)\big ),$ where $\mathcal {F}_{\mathcal {J}}: \mathcal {A} \rightarrow H_{\mathcal {J}}$ and $\mathcal {F}_{\mathcal {K}}: \mathcal {A} \rightarrow H_{\mathcal {K}}$.

We rewrite Definition 7 in this context.

Definition 20

Given two $\mathcal {C}^2$-functions $\alpha ,\beta :[0,T]\rightarrow H,$ we say that $(\alpha ,\beta )$ is a pair of lower/upper solutions of (P) related to the partition $(\mathcal {J}, \mathcal {K})$ of ${\mathbb {N}}_+$ if the four conditions of Definition 7 hold replacing $\{1,\dots ,N\}$ by ${\mathbb {N}}_+$ and the inequalities (3), (4) by

$$\begin{aligned}&\ddot{\alpha }_n(t) \ge f_n(t,x_1,\dots ,x_{n-1},\alpha _n(t),x_{n+1},\dots ), \end{aligned}$$

(16)

$$\begin{aligned}&\ddot{\beta }_n(t) \le f_n(t,x_1,\dots ,x_{n-1},\beta _n(t),x_{n+1},\dots ). \end{aligned}$$

(17)

Moreover, it is said to be strict with respect to the n -th component, with $n\in {\mathbb {N}}_+$, if the conditions of Definition 8 hold.

We recall the definition of the set

$$\begin{aligned} \mathcal {E} :=\left\{ (t,x)\in [0,T]\times {\mathbb {R}}^N \,:\, x=(x_{\mathcal {J}},x_{\mathcal {K}}), x_{\mathcal {J}}\in \prod _{j\in \mathcal {J}} [\alpha _j(t),\beta _j(t)] \right\} . \end{aligned}$$

Here is our result in this infinite-dimensional setting.

Theorem 21

Let $(\alpha ,\beta )$ be a pair of lower/upper solutions of (P) related to the partition $(\mathcal {J}, \mathcal {K})$ of ${\mathbb {N}}_+$, and assume the following conditions:

• there exists a sequence $(d_n)_{n\in {\mathbb {N}}_+}\in \ell ^2$ such that

  $$\begin{aligned} -d_n \le \alpha _n(t) \le d_n \quad \text {and}\quad -d_n \le \beta _n(t) \le d_n , \quad \text {for every}\,\, n\in {\mathbb {N}}_+\,\text {and}\,\, t\in [0,T]; \end{aligned}$$

• $(\alpha ,\beta )$ is strict with respect to the k-th component, for every $k\in \mathcal {K}$;

• there exists a constant $C>0$ such that

  $$\begin{aligned} |f_{\mathcal {K}}(t,x)| \le C ,\quad \hbox { for every }(t,x)\in \mathcal {E}\,; \end{aligned}$$

• for every bounded set $\mathcal {B} \subset \mathcal {E}$, the set $f_{\mathcal {K}}(\mathcal {B})$ is precompact.

Then, (P) has a solution x with the following property: for any $(j,k)\in \mathcal {J}\times \mathcal {K}$,

${(W_j)}$:

$\alpha _j(t)\le x_j(t)\le \beta _j(t)$, for every $t \in [0,T]$ ;

${(\widetilde{NW}_k)}$:

there exist $t_k^1,t_k^2\in [0,T]$ such that $x_k(t_k^1)\le \alpha _k(t_k^1)$ and $x_k(t_k^2)\ge \beta _k(t_k^2)$.

The proof of the theorem is carried out in Sect. 4.2.

Remark 22

As in Theorem 19, we can drop the strictness assumption for a certain index $\kappa \in \mathcal {K}$.

As an immediate consequence of Theorem 21, taking $\alpha$ and $\beta$ constant functions, we have the following.

Corollary 23

Let there exist two sequences $(p_n)_{n\in {\mathbb {N}}_+}$ and $(q_n)_{n\in {\mathbb {N}}_+}$ in $\ell ^2$, with $p_n<q_n$ for every $n\in {\mathbb {N}}_+$, and a partition $(\mathcal {J}, \mathcal {K})$ of ${\mathbb {N}}_+$, such that, for every $(t,x)\in [0,T] \times \prod _{j\in \mathcal {J}}[p_j,q_j]\times H_{\mathcal {K}}$,

$$\begin{aligned} j\in \mathcal {J}\Rightarrow & {} f_j(t,x_1,\dots ,x_{j-1},p_j,x_{j+1},\dots ) \le 0\le f_j(t,x_1,\dots ,x_{j-1},q_j,x_{j+1},\dots )\,;\nonumber \\ \end{aligned}$$

(18)

$$\begin{aligned} k\in \mathcal {K}\Rightarrow & {} f_k(t,x_1,\dots ,x_{k-1},p_k,x_{k+1},\dots )>0> f_k(t,x_1,\dots ,x_{k-1},q_k,x_{k+1},\dots ).\nonumber \\ \end{aligned}$$

(19)

Furthermore, let there exist a sequence $(C_k)_{k\in \mathcal {K}}\in \ell ^2$ such that, for every $k\in \mathcal {K}$,

$$\begin{aligned} |f_k(t,x)| \le C_k,\quad \hbox { for every }(t,x)\in [0,T] \times \prod _{j\in \mathcal {J}}[p_j,q_j]\times H_{\mathcal {K}}. \end{aligned}$$

(20)

Then, (P) has a solution x(t) such that, for every $j\in \mathcal {J}$, $k\in \mathcal {K}$,

$$\begin{aligned}&\{x_j(t):t\in [0,T]\}\subseteq [p_j,q_j]\,; \end{aligned}$$

(21)

$$\begin{aligned}&\{x_k(t):t\in [0,T]\}\cap [p_k,q_k]\ne \varnothing . \end{aligned}$$

(22)

We now give some examples of applications, with $H=\ell ^2$, where we implicitly assume all functions to be continuous.

Example 24

Let, for every $j\in {\mathbb {N}}_+$,

$$\begin{aligned} f_j(t,x)=x_j^3+h_j(t,x), \end{aligned}$$

and assume that there is a $c>0$ such that

$$\begin{aligned} |h_j(t,x)|\le \frac{c}{j^3},\quad \hbox { for every }(t,x)\in [0,T]\times \ell ^2. \end{aligned}$$

(23)

Then, $f:[0,T]\times \ell ^2 \rightarrow \ell ^2$ is well defined and taking $q_j=-p_j=\root 3 \of {c}/j$, we see that both $(p_j)_j,(q_j)_j$ belong to $\ell ^2$, and (18) is satisfied, so that Corollary 23 applies with $\mathcal {K}=\varnothing$.

Example 25

Let us consider, for every $j\in {\mathbb {N}}_+$,

$$\begin{aligned} f_j(t,x)= x_j^2 \sin x_j + h_j(t,x), \end{aligned}$$

and assume that there is a $c>0$ such that (23) holds. Then, $f:[0,T]\times \ell ^2 \rightarrow \ell ^2$ is well defined. Since $x^2\sin x \ge \frac{1}{2} x^3$ in the interval $[0,\pi /2]$, taking $q_j=-p_j=\root 3 \of {2c}/j$, we see that both $(p_j)_j,(q_j)_j$ belong to $\ell ^2$, and (18) is satisfied, so that Corollary 23 applies with $\mathcal {K}=\varnothing$.

Furthermore, for every $\ell \in \mathbb Z$ with $|\ell |$ sufficiently large, we can see that the constants $p^\ell =-\pi /2+2\ell \pi$, $q^\ell =\pi /2+2\ell \pi$ satisfy (18), for every $j\in {\mathbb {N}}_+$. Thus, we can replace a finite number of couples $(p_j,q_j)$ with some couples $(p^\ell ,q^\ell )$. Such a replacement must be performed only for a finite number of indices $j\in {\mathbb {N}}_+,$ since we need to guarantee that the new sequences $(p_j)_{j}$ and $(q_j)_{j}$ remain in $\ell ^2$. Recalling that the so found solution of problem (P) must satisfy (22), then we conclude that (P) admits an infinite number of solutions.

Example 26

Let, for every $k\in {\mathbb {N}}_+$,

$$\begin{aligned} f_k(t,x)=-\frac{x_k}{1+k|x_k|}+h_k(t,x), \end{aligned}$$

and assume that there is a $c\in \,]0,1[$ such that

$$\begin{aligned} |h_k(t,x)|\le \frac{c}{k},\quad \hbox { for every }(t,x)\in [0,T]\times \ell ^2. \end{aligned}$$

Then, $f:[0,T]\times \ell ^2 \rightarrow \ell ^2$ is well defined and taking $q_k=-p_k=\frac{c}{(1-c)k}$, we see that both $(p_k)_k,(q_k)_k$ belong to $\ell ^2$, and (19) is verified, so that Corollary 23 applies with $\mathcal {J}=\varnothing$.

Example 27

Let $(a_n)_n$ and $(\sigma _n)_n$ be sequences of positive numbers in $\ell ^2$ and let, for every $n\in {\mathbb {N}}_+$,

$$\begin{aligned} f_n(t,x)=-a_n\sin \left( \frac{2\pi x_n}{\sigma _n}\right) +h_n(t,x). \end{aligned}$$

If $h_n$ satisfies

$$\begin{aligned} \sup \big \{|h_n(t,x)|:(t,x)\in [0,T]\times \ell ^2\big \}< a_n, \end{aligned}$$

(24)

we see that, for every $n\in \{1,\dots ,N\}$, it is possible to find pairs of constant lower and upper solutions

$$\begin{aligned} \alpha _n\in \left\{ \frac{\sigma _n}{4}+m\sigma _n:m\in \mathbb Z\right\} ,\qquad \beta _n\in \left\{ -\frac{\sigma _n}{4}+m\sigma _n:m\in \mathbb Z\right\} . \end{aligned}$$

Then, for each equation, we have both well-ordered and non-well-ordered pairs of lower/upper solutions. Applying Corollary 23 we thus get the existence of infinitely many solutions of problem (P). By the same argument in Example 12, we notice that, even if the function $h(t,x_1,x_2,\dots )$ is $\sigma _n$-periodic in each variable $x_n$, the solutions we find are indeed geometrically distinct.

Remark 28

This result should be compared with the ones in [4, 11], where one or two geometrically distinct solutions were found assuming a Hamiltonian structure of the problem, i.e.,

$$\begin{aligned} h_n(t,x)=\frac{\partial \mathcal {V}}{\partial x_n}(t,x), \end{aligned}$$

for some function $\mathcal {V}(t,x_1,x_2,\dots )$ which is $\sigma _n$-periodic in each variable $x_n$. It was said in the final section of [11] that it remained an open problem to know if the existence of more than two T-periodic solutions could be proved, and in [4] that “it would be natural to conjecture the existence of infinitely many T-periodic solutions”. It is interesting to notice that even in [4, 11], to recover some compactness, it was assumed that the sequence of the periods $(\sigma _n)_n$ belongs to $\ell ^2$.

Remark 29

For any choice of a partition $(\mathcal {J}, \mathcal {K})$ of ${\mathbb {N}}_+$, we can consider functions f satisfying the requirements of Examples 24, 25 or 27 for every $j\in \mathcal {J}$ and of Examples 26 or 27 for every $k\in \mathcal {K}$. Corollary 23 applies also in this case.

In the next section, we provide some preliminary lemmas, which will be used in order to prove Theorem 21.

4.1 Some Compactness Lemmas

For every sequence $\tau =(\tau _n)_{n\in {\mathbb {N}}_+}$ contained in [0, T] and every function $u\in {\mathcal {C}}([0,T],H)$, define the function $P_{\tau }u:[0,T]\rightarrow H$ as

$$\begin{aligned} (P_{\tau }u)_n(t)=\int _{\tau _n}^t u_n(s)\,\mathrm {d}s,\quad n\in {\mathbb {N}}_+. \end{aligned}$$

We will need the following extension of [11, Lemma 3.2].

Lemma 30

Let $E\subseteq {\mathcal {C}}([0,T],H)$ be such that the set

$$\begin{aligned} A=\left\{ u(t):u\in E,\,t\in [0,T] \right\} \end{aligned}$$

is precompact in H. Then the set

$$\begin{aligned} \Sigma =\left\{ P_{\tau } u : \tau \in [0,T]^{{\mathbb {N}}_+},u\in E\right\} \end{aligned}$$

is precompact in ${\mathcal {C}}([0,T],H)$. As a consequence, the set

$$\begin{aligned} \Xi =\left\{ P_{\tau } u(t) : \tau \in [0,T]^{{\mathbb {N}}_+},u\in E,\,t\in [0,T]\right\} \end{aligned}$$

is precompact in H.

Proof

Fix $\varepsilon >0$. Since A is precompact, there exist $v_1,\dots , v_m$ in H such that

$$\begin{aligned} A\subseteq \bigcup _{\iota =1}^m B(v_\iota ,\varepsilon ). \end{aligned}$$

(25)

Let $V=\mathrm{Span} (v_1,\dots ,v_m)$, and denote by $Q:H\rightarrow V$ the corresponding orthogonal projection. We first prove that the set

$$\begin{aligned} {\mathcal {R}}=\left\{ P_{\tau }(Qu):u\in E,\ \tau \in [0,T]^{{\mathbb {N}}_+}\right\} \end{aligned}$$

is precompact in ${\mathcal {C}}([0,T],V)$.

The set Q(A) is precompact in V and hence bounded; there exists a real constant D such that

$$\begin{aligned} |Qu(t)|<D, \quad \text {for all } u\in E \,\text { and }t \in [0,T]. \end{aligned}$$

(26)

Moreover, for every $u\in E$, $\tau \in [0,T]^{{\mathbb {N}}_+}$ and $t\in [0,T]$,

$$\begin{aligned} |(P_\tau (Q u))_n(t)|= \left|\int _{\tau _n}^t(Q u)_n(s)\,\mathrm {d}s \right|\le \left|\int _{\tau _n}^t|(Q u)_n(s)|\,\mathrm {d}s \right|,\quad n\in {\mathbb {N}}_+ \end{aligned}$$

and consequently

$$\begin{aligned} \begin{aligned} |P_{\tau }(Q u)(t)|^2&= \sum _{n=1}^{\infty }|(P_{\tau }(Q u))_n(t)|^2\le \sum _{n=1}^{\infty }\left|\int _{\tau _n}^t|(Q u)_n(s)|\,\mathrm {d}s \right|^2\\&\le \sum _{n=1}^{\infty }\left( \int _{0}^T|(Q u)_n(s)|\,\mathrm {d}s \right) ^2; \end{aligned} \end{aligned}$$

by the Hölder Inequality and the use of the Monotone Convergence Theorem, recalling (26),

$$\begin{aligned} \begin{aligned} \sum _{n=1}^{\infty }\left( \int _{0}^T|(Q u)_n(s)|\,\mathrm {d}s \right) ^2&\le T\sum _{n=1}^{\infty }\int _0^T|(Q u)_n(s)|^2\,\mathrm {d}s\\&=T\int _0^T\sum _{n=1}^{\infty }|(Q u)_n(s)|^2\,\mathrm {d}s\\&=T\int _0^T|Qu(s)|^2\,\mathrm {d}s<T^2D^2, \end{aligned} \end{aligned}$$

and then

$$\begin{aligned} |P_{\tau }(Q u)(t)|\le TD . \end{aligned}$$

Since V is finite dimensional, the set ${\mathcal {S}}=\left\{ w(t):w\in {\mathcal {R}}\right\} \subseteq V$ is precompact. On the other hand, for every $u\in E$, $\tau \in [0,T]^{{\mathbb {N}}_+}$ and every $t_1,t_2\in [0,T]$ with $t_1< t_2$, we have

$$\begin{aligned} |P_{\tau }(Qu)(t_1)-P_{\tau }(Qu)(t_2)|= \left|\int _{t_1}^{t_2}(Qu)(s)\,\mathrm {d}s\right|\le \int _{t_1}^{t_2}|(Qu)(s)|\,\mathrm {d}s\le D(t_1-t_2), \end{aligned}$$

so that ${\mathcal {R}}$ is equi-uniformly continuous as a subset of ${\mathcal {C}}([0,T],V)$. By the Ascoli–Arzelà Theorem, the set ${\mathcal {R}}$ is precompact in ${\mathcal {C}}([0,T],V)$.

Consequently, there exist $f_1,\dots ,f_\ell$ in ${\mathcal {C}}([0,T],V)$ such that

$$\begin{aligned} {\mathcal {R}}\subseteq \bigcup _{\iota =1}^\ell B(f_\iota ,\varepsilon ). \end{aligned}$$

(27)

Now, for every $u\in E$, $\tau \in [0,T]^{{\mathbb {N}}_+}$ and $t\in [0,T]$, by (25),

$$\begin{aligned} \begin{aligned} |P_{\tau }u(t)-P_{\tau }(Qu)(t)|^2&=\sum _{n=1}^\infty |(P_{\tau }u)_n(t)-(P_{\tau }(Qu))_n(t)|^2\\&\le \sum _{n=1}^\infty \left|\int _{\tau _n}^t|u_n(s)-(Q u)_n(s)|\,\mathrm {d}s \right|^2\\&\le \sum _{n=1}^\infty T\int _{0}^T|u_n(s)-(Q u)_n(s)|^2\,\mathrm {d}s\\&=T\int _{0}^T\sum _{n=1}^\infty |u_n(s)-(Q u)_n(s)|^2\,\mathrm {d}s\\&=T\int _0^T|u(s)-(Qu)(s)|^2\,\mathrm {d}s \le T^2\varepsilon ^2, \end{aligned} \end{aligned}$$

and so

$$\begin{aligned} |P_{\tau }u(t)-P_{\tau }(Qu)(t)|\le T\varepsilon . \end{aligned}$$

On the other hand, since $P_{\tau }(Qu)\in \mathcal {R}$, by (27) there exists $\bar{\iota }$ such that

$$\begin{aligned} \Vert P_{\tau }(Qu)-f_{\bar{\iota }}\Vert _\infty <\varepsilon ; \end{aligned}$$

hence

$$\begin{aligned} |P_{\tau }u(t)-f_{\bar{\iota }}(t)|\le & {} |P_{\tau }u(t)-P_{\tau }(Qu)(t)|+ |P_{\tau }(Qu)(t)-f_{\bar{\iota }}(t)|\\\le & {} \varepsilon T+\varepsilon =\varepsilon (T+1). \end{aligned}$$

We have thus shown that, given $\varepsilon >0$, there are $f_1,\dots ,f_\ell$ in ${\mathcal {C}}([0,T],H)$ such that

$$\begin{aligned} \Sigma \subseteq \bigcup _{\iota =1}^\ell B(f_\iota ,(T+1)\varepsilon ), \end{aligned}$$

hence proving that $\Sigma$ is precompact.

The fact that $\Xi$ is precompact in H now follows again from the Ascoli–Arzelà Theorem, recalling that this theorem gives a necessary and sufficient condition for precompactness. $\square$

Let us denote by $\Pi _N:H\rightarrow H$ the projection

$$\begin{aligned} \Pi _N(x)=(x_1,\dots , x_N,0,0,\dots ). \end{aligned}$$

(28)

Lemma 31

Let A be a compact subset of H. Then, for every $\varepsilon >0$, there is a $M\ge 1$ such that, for every $a=(a_n)_{n\in {\mathbb {N}}_+}$ in A,

$$\begin{aligned} \sum _{n=M}^\infty |a_n|^2\le \varepsilon ^2. \end{aligned}$$

In particular, $\lim _{N\rightarrow \infty }(\Pi _N-\mathrm{Id}) x=0$ uniformly for $x\in A$.

Proof

By contradiction, let there exist an $\varepsilon >0$ such that, for every $M\ge 1$, there is $a^M=(a^M_n)_{n\in {\mathbb {N}}_+}\in A$ such that $\sum _{n=M}^\infty |a_n^M|^2>\varepsilon ^2$. By compactness, the sequence $(a^M)_{M\in {\mathbb {N}}_+}$ has a subsequence, for which we keep the same notation, such that $a^M\rightarrow a^*$, for some $a^*\in A$. Let $M_*$ be any positive integer. Then, taking $M\ge M_*$ sufficiently large,

$$\begin{aligned} \begin{aligned} \left( \sum _{n=M_*}^\infty |a_n^*|^2\right) ^{1/2}&\ge \left( \sum _{n=M}^\infty |a_n^*|^2\right) ^{1/2}\\&\ge \left( \sum _{n=M}^\infty |a_n^M|^2\right) ^{1/2}- \left( \sum _{n=M}^\infty |a_n^M-a_n^*|^2\right) ^{1/2}\\&\ge \varepsilon -\Vert a^M-a^*\Vert _{\ell ^2}\ge \frac{\varepsilon }{2}. \end{aligned} \end{aligned}$$

We thus get a contradiction with the fact that $a^*\in H$. $\square$

As an immediate consequence, we find the following compactness property.

Lemma 32

Let A be a compact subset of H. Then, the set

$$\begin{aligned} A^{\mathcal {P}}:=\bigcup _{N\in {\mathbb {N}}_+} \Pi _N A \end{aligned}$$

is precompact in H.

Proof

Let us consider a sequence $(x_n)_{n\in {\mathbb {N}}_+}$ contained in $A^{\mathcal {P}}$.

If there exists $N_0\in {\mathbb {N}}_+$ and a subsequence $(x_{n_\ell })_\ell$ such that $x_{n_\ell }\in \Pi _{N_0} A$ for every $\ell$, then the conclusion is reached since $\Pi _{N_0} A$ is compact.

If the previous situation does not arise, then we can find a diverging sequence $(N_\ell )_\ell \subset {\mathbb {N}}_+$ and a subsequence $(x_{n_\ell })_\ell$ such that $x_{n_\ell }\in \Pi _{N_\ell } A$ for every $\ell$. So, there is a sequence $(y_{n_\ell })_\ell \subseteq A$ such that $x_{n_\ell }=\Pi _{N_\ell }y_{n_\ell }$. Since A is compact, then, up to a subsequence, we have $y_{n_\ell }\rightarrow \bar{y}\in A$. Hence,

$$\begin{aligned} |x_{n_\ell }-\bar{y}| \le |x_{n_\ell }-y_{n_\ell }|+|y_{n_\ell }-\bar{y}| \le |(\Pi _{N_\ell }-\mathrm{Id})y_{n_\ell }|+|y_{n_\ell }-\bar{y}| \rightarrow 0, \end{aligned}$$

where Lemma 31 has been applied. $\square$

Remark 33

The above statements have been formulated for a Hilbert space H. We will apply them also treating the previously introduced Hilbert spaces $H_{\mathcal {K}}$ and $H_{\mathcal {J}}$.

4.2 Proof of Theorem 21

We consider, for every $N\in {\mathbb {N}}_+$, the auxiliary system

$$\begin{aligned} \left\{ \begin{aligned}&\ddot{x}_1=f_1(t,x_1,\dots ,x_N,\alpha _{N+1}(t),\alpha _{N+2}(t), \dots )\\&\quad \vdots \\&\ddot{x}_N=f_N(t,x_1,\dots ,x_N,\alpha _{N+1}(t),\alpha _{N+2}(t), \dots )\\&\ddot{x}_{N+1}=0\\&\ddot{x}_{N+2}=0\\&\quad \vdots \end{aligned} \right. \end{aligned}$$

We recall the projections $\Pi _N$, introduced in (28), and define the function

$$\begin{aligned}&\widehat{\Pi }_N: \mathcal {C}([0,T],H) \rightarrow \mathcal {C}([0,T],H) \end{aligned}$$

(29)

$$\begin{aligned}&\widehat{\Pi }_N x(t)=(x_1(t),\dots ,x_N(t), \alpha _{N+1}(t),\alpha _{N+2}(t),\dots ). \end{aligned}$$

(30)

The auxiliary problem can then be written as

$$\begin{aligned} (\widehat{P}_N)\quad {\left\{ \begin{array}{ll} \ddot{x}=\Pi _N f(t,\widehat{\Pi }_Nx(t)),\\ x(0)=x(T),\quad \dot{x}(0)=\dot{x}(T). \end{array}\right. } \end{aligned}$$

Notice that

$$\begin{aligned} (t,\widehat{\Pi }_N x^N(t))\in \mathcal {E}, \qquad \text {for every } N\in {\mathbb {N}}_+ \text { and } t\in [0,T]. \end{aligned}$$

(31)

By Theorem 10, for every $N\in {\mathbb {N}}_+$, there is a solution $x^N(t)$ of $(\widehat{P}_N)$ such that ${(W_j)}$ and ${(NW_k)}$ hold for every $j\in \mathcal {J}\cap [1,N]$ and $k\in \mathcal {K}\cap [1,N]$. We impose

$$\begin{aligned} x^N_n(t)=0 ,\quad \hbox { for every}\, n>N\,\hbox { and }\,t\in [0,T]. \end{aligned}$$

(32)

Arguing as in the proof of Proposition 14, cf. (10) and (11), we conclude that $x^N$ satisfies

$$\begin{aligned}&\{x^N_j(t):t\in [0,T]\}\subseteq [-d_j,d_j],\\&\{x^N_k(t):t\in [0,T]\}\cap [-d_k,d_k]\ne \varnothing , \end{aligned}$$

for every $k\in \mathcal {K}$ and $j\in \mathcal {J}$. Concerning the indices $j\in \mathcal {J},$ we thus have

$$\begin{aligned} x^N_\mathcal {J}(t) \in \mathcal {D}_{\mathcal {J}}:= \prod _{j\in \mathcal {J}} [-d_j,d_j], \end{aligned}$$

(33)

for every $N\in {\mathbb {N}}_+$ and $t\in [0,T]$.

Now, we repeat the arguments of Proposition 14 with a slight modification. Given the solution $x^N$ of $(\widehat{P}_N)$, we can compute

$$\begin{aligned} \Vert \tilde{x}^N_{\mathcal {K}}\Vert _2^2 \le \left( \frac{T}{2\pi }\right) ^2 \Vert \dot{x}^N_{\mathcal {K}}\Vert ^2_2\le \left( \frac{T}{2\pi }\right) ^2 C\sqrt{T}\Vert \tilde{x}^N_{\mathcal {K}}\Vert _2, \end{aligned}$$

so that $\Vert \tilde{x}^N_{\mathcal {K}}\Vert _{H^1}\le C_1$ and $\Vert \tilde{x}^N_{\mathcal {K}}\Vert _\infty \le C_0$ for some constants $C_1$ and $C_0$.

Recalling the validity of (33), we can find a sequence $\tau ^N_{\mathcal {K}}=(\tau ^N_k)_{k\in \mathcal {K}}\subset [0,T]$ such that

$$\begin{aligned} |x^N_k(\tau ^N_k)| \le d_k ,\qquad \text { for every } k\in \mathcal {K}. \end{aligned}$$

(34)

Then, we can prove that the sequence $(x^N_{\mathcal {K}})_{N\in {\mathbb {N}}_+}$ is uniformly bounded. Indeed,

$$\begin{aligned} |x^N_{\mathcal {K}}(t)|^2&= \sum _{k\in \mathcal {K}} |x_k^N(t)|^2 = \sum _{k\in \mathcal {K}} \left| x^N_k(\tau ^N_k) + \int _{\tau ^N_k}^t \dot{x}_k^N(s)\, \mathrm{d}s \right| ^2\\&\le 2 \sum _{k\in \mathcal {K}} \left( |x^N_k(\tau ^N_k)|^2 + \left| \int _{\tau ^N_k}^t \dot{x}_k^N(s)\, \mathrm{d}s \right| ^2\right) \\&\le 2 \sum _{k\in \mathcal {K}} d_k^2 + 2T \Vert \dot{x}^N_{\mathcal {K}}\Vert _2^2 \le 2 \sum _{k\in \mathcal {K}} d_k^2 + 2TC_1^2 =:\varrho ^2 , \end{aligned}$$

Then, choosing $\mathcal {B}=\{(t,x)\in \mathcal {E} \,:\, |x_{\mathcal {K}}|\le \varrho \}$ and recalling (31) and that $f_{\mathcal {K}}$ is completely continuous in $\mathcal {E}$, we notice that the set $A=\{f_{\mathcal {K}}(t,\widehat{\Pi }_N x^N(t))\,:\, N\in {\mathbb {N}}_+, t\in [0,T]\}\subseteq f_{\mathcal {K}}(\mathcal {B})$ is precompact. Then, using Lemma 32, we deduce that the set $\{\ddot{x}_{\mathcal {K}}^N(t)\,:\, N\in {\mathbb {N}}_+, t\in [0,T]\}$ is precompact. By periodicity, there exists a sequence $t^N_{\mathcal {K}}=(t^N_k)_{k\in \mathcal {K}}$ such that $\dot{x}_k^N(t^N_k)=0$ for every $k\in \mathcal {K}$. Writing

$$\begin{aligned} \dot{x}^N_k(t)=\dot{x}^N_k(t^N_k)+\int _{t^N_k}^t\ddot{x}^N_k(s)\,\mathrm {d}s=\int _{t^N_k}^t\ddot{x}^N_k(s)\,\mathrm {d}s = \Big ( P_{t^N_{\mathcal {K}}} \ddot{x}^N_{\mathcal {K}} \Big )(t), \end{aligned}$$

we deduce from Lemma 30 that the set $\{\dot{x}_{\mathcal {K}}^N(t):N\in {\mathbb {N}}_+ , t\in [0,T]\}$ is precompact.

Finally, we prove that also the set $\{x_{\mathcal {K}}^N(t)\,:\, N\in {\mathbb {N}}_+, t\in [0,T]\}$ is precompact. Recalling the sequence $\tau ^N_{\mathcal {K}}=(\tau ^N_k)_{k\in \mathcal {K}}$ in (34), we can write using the notation of Sect. 4.1,

$$\begin{aligned} x_{\mathcal {K}}^N(t)=\xi _{\mathcal {K}}^N+\Big (P_{\tau _{\mathcal {K}}^N}\dot{x}_{\mathcal {K}}^N\Big )(t), \qquad \text {where } \xi _{\mathcal {K}}^N:=(x^N_k(\tau ^N_k))_{k\in \mathcal {K}}. \end{aligned}$$

By construction $\xi _{\mathcal {K}}^N\in \mathcal {D}_{\mathcal {K}}:= \prod _{k\in \mathcal {K}} [-d_k,d_k]$, so that, by Lemma 30, we conclude that both the addenda are in a compact set. Hence there is a compact set $\widehat{\mathcal {D}}_{\mathcal {K}}$ such that

$$\begin{aligned} x_{\mathcal {K}}^N(t)\in \widehat{\mathcal {D}}_{\mathcal {K}}, \qquad \text {for every}\, N\in {\mathbb {N}}_+\,\text { and }\,t\in [0,T]. \end{aligned}$$

(35)

We can now prove similar properties for the components of $x^N(t)$, and their derivatives, with indices $j\in \mathcal {J}$. At this step, the continuity of $f_{\mathcal {J}}$ is sufficient. Indeed, from (33) and (35), the compactness of $\{f_{\mathcal {J}}(t,\widehat{\Pi }_N x^N(t))\,:\, N\in {\mathbb {N}}_+, t\in [0,T]\}$ follows. Then, arguing as above, we can prove that both $\{\ddot{x}_{\mathcal {J}}^N(t)\,:\, N\in {\mathbb {N}}_+, t\in [0,T]\}$ and $\{\dot{x}_{\mathcal {J}}^N(t)\,:\, N\in {\mathbb {N}}_+, t\in [0,T]\}$ are precompact.

Consider now the sequence $(u^N)_{N\in {\mathbb {N}}_+}$ of functions $u^N:[0,T]\rightarrow H\times H$ defined by

$$\begin{aligned} u^N(t)=(x^N(t),\dot{x}^N(t)). \end{aligned}$$

By the above arguments, the sequence $(u^N)_{N\in {\mathbb {N}}_+}$ takes its values in a compact set, and it is equi-uniformly continuous. By the Ascoli–Arzelà Theorem there exists a subsequence, for which we keep the same notation, which uniformly converges to some $u^*: [0,T]\rightarrow H\times H$. Writing $u^*(t)=(x^*(t),y^*(t))$, we have that $(x^N,\dot{x}^N)$ uniformly converges to $(x^*,y^*)$. In particular $x^*(0)=x^*(T)$, $y^*(0)=y^*(T)$. Rewriting the differential equation in $(\widehat{P}_N)$ as a planar system, we have

$$\begin{aligned} (\widehat{Q}_N) \qquad {\left\{ \begin{array}{ll} \dot{x}=y,\\ \dot{y}=\Pi _N f(t,\widehat{\Pi }_N x(t)), \end{array}\right. } \end{aligned}$$

or equivalently

$$\begin{aligned} \dot{u}=F^N(t,u), \end{aligned}$$

where $F^N(t,x,y)=(y,\Pi _N f(t,\widehat{\Pi }_N x(t)))$. The corresponding integral formulation is then

$$\begin{aligned} u(t)=u(0)+\int _0^t F^N(s,u(s))\,\mathrm {d}s. \end{aligned}$$

(36)

System $(\widehat{Q}_N)$ has a solution $u^N=(x^N,\dot{x}^N)$ such that $u^N(0)=u^N(T)$ for every $N\in {\mathbb {N}}_+$. We want to show that

$$\begin{aligned} F^N(t,u^N(t))\rightarrow F(t,u^*(t)),\quad \hbox { uniformly in } t\in [0,T], \end{aligned}$$

(37)

where $F(t,x,y)=(y,f(t,x))$. Fix $\varepsilon >0$; for N sufficiently large, we have

$$\begin{aligned} \begin{aligned}&|F^N(t,u^N(t))-F(t,u^*(t))| \le |y^N(t)-y^*(t)|+|\Pi _N f(t,\widehat{\Pi }_N x^N(t))-f(t,x^*(t))|\\&\quad \le \varepsilon +|\Pi _N f(t,\widehat{\Pi }_N x^N(t))- f(t,\widehat{\Pi }_N x^N(t)) | +| f(t,\widehat{\Pi }_N x^N(t))-f(t,x^*(t))|. \end{aligned} \end{aligned}$$

Since $\{\widehat{\Pi }_N x^N(t)\,:\, N\in {\mathbb {N}}_+, t\in [0,T]\}$ is precompact, cf. (33) and (35), then by continuity $\{ f(t,\widehat{\Pi }_N x^N(t))\,:\, N\in {\mathbb {N}}_+, t\in [0,T]\}$ is precompact, too. So, by Lemma 31, for N sufficiently large,

$$\begin{aligned} |\Pi _N f(t,\widehat{\Pi }_N x^N(t))- f(t,\widehat{\Pi }_N x^N(t))|= |(\Pi _N-\mathrm{Id}) f(t, \widehat{\Pi }_N x^N(t))|\le \varepsilon . \end{aligned}$$

Moreover,

$$\begin{aligned} |\widehat{\Pi }_N x^N(t)-\Pi _N x^N(t)|&=|(0,\dots ,0,\alpha _{N+1}(t),\alpha _{N+2}(t),\dots )| \end{aligned}$$

(38)

$$\begin{aligned}&\le \sum _{n=N}^\infty d_n^2 \rightarrow 0, \qquad \text { as}\, N\rightarrow \infty . \end{aligned}$$

(39)

Then, applying Lemma 31,

$$\begin{aligned} |\widehat{\Pi }_N x^N(t)-x^*|\le & {} |\widehat{\Pi }_N x^N(t)-\Pi _N x^N(t)| + |\Pi _N x^N(t)- x^N(t)|\\&+|x^N(t)-x^*(t)| \rightarrow 0, \end{aligned}$$

as $N\rightarrow \infty$, so that by continuity, for N large enough,

$$\begin{aligned} |f(t,\widehat{\Pi }_N x^N(t))-f(t,x^*(t))|\le \varepsilon . \end{aligned}$$

Summing up, if N is large, then

$$\begin{aligned} |F^N(t,u^N(t))-F(t,u^*(t))|\le 3\varepsilon , \qquad \text { for every } t\in [0,T], \end{aligned}$$

thus proving (37). Passing to the limit in (36), we get

$$\begin{aligned} u^*(t)=u^*(0)+\int _0^t F(s,u^*(s))\,\mathrm {d}s, \end{aligned}$$

and so $x^*(t)$ is a solution of (P). The conditions ${(W_j)}$ and ${(\widetilde{NW}_k)}$ are easily seen to be preserved in the limit process. The proof is thus completed. $\square$

5 Final Remarks

In this final section, we briefly outline some possible extensions of the previous results.

• 1. The boundedness assumption on the function $f_{\mathcal {K}}(t,x)$ could be replaced by a nonresonance condition with respect to the higher part of the spectrum of the differential operator $-\ddot{x}$ with T-periodic conditions. For instance, denoting by $\lambda _2$ the first positive eigenvalue $(2\pi /T)^2$, one could assume that

  $$\begin{aligned} -f_{\mathcal {K}}(t,x)=\gamma _{\mathcal {K}}(t,x)x+r_{\mathcal {K}}(t,x), \end{aligned}$$

  where $\gamma _{\mathcal {K}}(t,x)\le c<\lambda _2$ and $r_{\mathcal {K}}(t,x)$ is bounded. Or, more generally, one could assume an asymmetric behaviour of the type

  $$\begin{aligned} -f_{\mathcal {K}}(t,x)=\mu _{\mathcal {K}}(t,x)x^+ -\nu _{\mathcal {K}}(t,x)x^-+r_{\mathcal {K}}(t,x), \end{aligned}$$

  where $(\mu _{\mathcal {K}}(t,x),\nu _{\mathcal {K}}(t,x))$ lie below the first curve of the Fučík spectrum (here, as usual, $x^+=\max \{x,0\}$ and $x^-=\max \{-x,0\}$).

• 2. One could deal with nonlinearities of the type $f(t,x,\dot{x})$, depending also on the derivative of x, assuming some type of Nagumo growth condition (see [5]). Such a situation has already been studied in the infinite-dimensional setting, e.g., in [22], in the well-ordered case.

• 3. In this paper we defined the lower and upper solutions as $C^2$-functions. However, this regularity could be weakened, and different definitions could be adopted. We do not enter into the details, for briefness, and we refer to the book [5] for further possible developments.

• 4. The results of this paper hold the same for the Neumann problem

  $$\begin{aligned} {\left\{ \begin{array}{ll} \ddot{x}=f(t,x),\\ \dot{x}(0)=0=\dot{x}(T), \end{array}\right. } \end{aligned}$$

  with almost identical proofs. Concerning the Dirichlet problem

  $$\begin{aligned} {\left\{ \begin{array}{ll} \ddot{x}=f(t,x),\\ x(0)=0=x(T), \end{array}\right. } \end{aligned}$$

  some modifications are needed in the non-well-ordered case. Both problems have their partial differential equations analogues. We will provide in [10] an extension of Theorem 10 in a finite-dimensional abstract setting including the case of elliptic and parabolic type systems with different types of boundary conditions, thus generalizing the results in [6, 7, 12]. However, an infinite-dimensional extension in the PDE case remains an open problem.