On the Dirichlet problem associated with bounded perturbations of positively-(p, q)- homogeneous Hamiltonian systems

Abstract

The existence of solutions for the Dirichlet problem associated with bounded perturbations of positively-(p, q)-homogeneous Hamiltonian systems is considered both in nonresonant and resonant situations. To deal with the resonant case, the existence of a couple of lower and upper solutions is assumed. Both the well-ordered and the non-well-ordered cases are analysed. The proof is based on phase-plane analysis and topological degree theory.

1 Introduction

Let $H:{\mathbb {R}}^2\rightarrow {\mathbb {R}}$ be a continuously differentiable positively-(p, q)-homogeneous positive-definite function. By this we mean that, for some $p>1$ and $q>1$ with $(1/p)+(1/q)=1$,

$$\begin{aligned} H(\lambda ^q x, \lambda ^p y)= \lambda ^{p+q} H(x,y)>0\,, \end{aligned}$$

(1)

for every $\lambda >0$ and $(x,y)\ne (0,0)$. We are interested in Dirichlet problems of the type

$$\begin{aligned} {\left\{ \begin{array}{ll} x'=\displaystyle \frac{\partial H}{\partial y}(x,y) + \phi (t,x,y)\,,\quad y'=\displaystyle -\frac{\partial H}{\partial x}(x,y) + \psi (t,x,y)\,,\\ x(a)=0=x(b)\,, \end{array}\right. } \end{aligned}$$

(2)

where the functions $\phi ,\psi :[a,b]\times {\mathbb {R}}^2\rightarrow {\mathbb {R}}$ are continuous and bounded.

The autonomous system

$$\begin{aligned} x'=\displaystyle \frac{\partial H}{\partial y}(x,y)\,,\quad y'=\displaystyle -\frac{\partial H}{\partial x}(x,y) \end{aligned}$$

(3)

is isochronous (see [25]). More precisely, (0, 0) is a global center, and all solutions are periodic of the same period, which we denote by $\tau$. Moreover, if $y_0>0$, for all solutions (x, y) starting with $(x(0),y(0))=(0,y_0)$ there is a first time $\tau _+>0$ for which $x(\tau _+)=0$, while $x(t)>0$ for every $t\in \,]0,\tau _+[$ , and this time $\tau _+$ is independent of $y_0>0$. Symmetrically, if $y_0<0$, there is a first time $\tau _->0$ for which $x(\tau _-)=0$, while $x(t)<0$ for every $t\in \,]0,\tau _-[$ . Clearly enough, $\tau _++\tau _-=\tau$.

First of all, let us state the following nonresonance result.

Theorem 1

Assume that there is an integer $n\ge 0$ such that one of the following alternatives hold

$$\begin{aligned} n\tau&< b-a<n\tau +\min \{\tau _-,\tau _+\}\,, \end{aligned}$$

(4)

$$\begin{aligned} n\tau +\max \{\tau _-,\tau _+\}&<b-a<(n+1)\tau \,. \end{aligned}$$

(5)

Then, problem (2) has a solution.

Its proof relies on the so called shooting method, following the ideas presented in [10] (see also [3, 23]), and will be given in Section 3.

As a possible example we may consider the function

$$\begin{aligned} H(x,y)=\frac{1}{q}\Big (\delta (y^+)^q+\gamma (y^-)^q\Big )+\frac{1}{p}\Big (\mu (x^+)^p+\nu (x^-)^p\Big )\,, \end{aligned}$$

for some positive constants $\delta ,\gamma ,\mu ,\nu$ (we use the standard notation where $f^+=\max \{f,0\}$, $f^-=\max \{-f,0\}$). If we choose $\delta =\gamma =1$ and $\phi \equiv 0$, our problem is then equivalent to

$$\begin{aligned} {\left\{ \begin{array}{ll} (|x'|^{p-2}x')'+\mu |x^+|^{p-2}x^+-\nu |x^-|^{p-2}x^-=h(t,x,x') \,,\\ x(a)=0=x(b)\,, \end{array}\right. } \end{aligned}$$

(6)

with $h(t,x,v)=-\psi (t,x,|v|^{p-2}v)$. In this case, $\tau _+=\pi _p\mu ^{-1/p}$ and $\tau _-=\pi _p\nu ^{-1/p}$, cf. [18], where

$$\begin{aligned} \pi _p=\frac{2(p-1)^{1/p}}{p\sin (\pi /p)}\,\pi \,. \end{aligned}$$

If $p=2$, the differential equation in (6) models an asymmetric oscillator, with $\tau _+=\pi /\sqrt{\mu }$, $\tau _-=\pi /\sqrt{\nu }$, and it is well known that some care on the choice of $\mu ,\nu$ must be taken to avoid resonance phenomena; when h is bounded, the existence of a solution depends on the position of $(\mu ,\nu )$ with respect to the Fučík spectrum, cf. [10]. In Fig. 1, the assumption of Theorem 1 can be visualized, when $b-a=\pi _p$, taking the values $(\mu ^{-1/p},\nu ^{-1/p})$ in the white regions.

Fig. 1

The Fučík spectrum

A different way of avoiding resonance would be the assumption of the existence of a well-ordered couple of lower/upper solutions $\alpha \le \beta$, together with some Nagumo-type conditions. This method goes back to the pioneering papers [20,21,22]. Some more care is needed in the non-well-ordered case $\alpha \not \le \beta$, see [1, 8, 9, 16, 17]. We refer to the book [7] for an extensive exposition on the theory of lower and upper solutions for scalar second order differential equations.

The concept of lower and upper solution has been recently extended to planar systems in [15] for the periodic problem (see also [12]), and in [14] for Sturm–Liouville type problems, including the Dirichlet problem. We recall in Section 4 the main definitions in this case.

Here is our result for problem (2), in the well-ordered case.

Theorem 2

Assume H to be a continuously differentiable positively-(p, q)-homogeneous positive-definite function, for some $p>1$ and $q>1$ with $(1/p)+(1/q)=1$. Let $\phi ,\psi$ be uniformly bounded continuous functions, and let $(\alpha ,\beta )$ be a well-ordered pair of lower/upper solutions of problem (2). Then, there exists a solution (x, y) of (2) such that $\alpha \le x\le \beta$.

The proof will be given in Section 5. We will first need some properties of positively-(p, q)-homogeneous Hamiltonian systems, which we provide in Section 2. Then, the main issue will be the construction of some guiding curves in the phase plane so to enter the framework of [14, Theorem 11].

In the sequel, we denote by $C^{j,\ell }_{loc}$ the space of $C^j$-smooth real functions whose jth derivative is locally $\ell$-Hölder continuous, with $\ell >0$. Moreover, defining the function

$$\begin{aligned} \varphi _1(t)=\sin \Big (\frac{t-a}{b-a}\,\pi \Big )\,, \end{aligned}$$

we introduce the following order relation: for any continuous function $x:[a,b]\rightarrow {\mathbb {R}}$, we write

$$\begin{aligned} x\gg 0 \end{aligned}$$

(7)

if and only if

$$\begin{aligned} \hbox { there exists }\epsilon >0\hbox { such that } x(t)\ge \epsilon \, \varphi _1(t)\,, \hbox { for every }t\in [a,b]\,. \end{aligned}$$

We will write either $u\gg v$ or $v\ll u$ when $u-v \gg 0$.

Concerning the non-well-ordered case, we recall that the existence of a pair of lower/upper solutions does not guarantee the existence of a solution, since resonance phenomena can occur with respect to the higher part of the spectrum. However, at least for scalar second-order equations, it is well known that resonance can be handled with respect to the first eigenvalue, cf. [7]. This observation leads us to assume, in the non-well-ordered case, that

$$\begin{aligned} b-a \le \min \{\tau _+,\tau _-\}\,. \end{aligned}$$

(8)

Here is our result, in the non-well-ordered case.

Theorem 3

Let $\ell >0$ and assume H to be a $C^{1,\ell }_{loc}$-smooth positively-(p, q)-homogeneous positive-definite function, for some $p>1$ and $q>1$ with $(1/p)+(1/q)=1$. Let $\phi ,\psi$ be uniformly bounded $C^{0,\ell }_{loc}$-smooth functions, and let $(\alpha ,\beta )$ be a non-well-ordered pair of lower/upper solutions of (2). Assume moreover that

$$\begin{aligned} \frac{\partial H}{\partial y}(x_0,\cdot ) \text{ is } \text{ a } \text{ strictly } \text{ increasing } \text{ function, } \text{ for } \text{ every } \ x_0\in {\mathbb {R}}. \end{aligned}$$

(9)

Then, if (8) holds, there exists a solution (x, y) of (2) such that $\alpha \not \ll x$ and $x\not \ll \beta$.

The above theorem generalizes [14, Theorem 19]. Its proof is provided in Section 6 by the use of topological degree techniques, which require the above regularity assumptions. It would be interesting to know whether the result still holds when the functions $\phi ,\psi$ are only assumed to be continuous. Notice that assumption (9) is surely verified for problems like (6), where $\frac{\partial H}{\partial y}(x,y)= |y|^{q-2}y$.

In the final section of the paper, we extend the previous results to systems in ${\mathbb {R}}^{2N}$ which can be considered as weakly coupled planar systems of the above type. The different planar systems involved could have either well-ordered or non-well-ordered lower and upper solutions. We are able to deal with this mixed type of situations, still obtaining an existence result, thus carrying out the investigation opened in [14]. However, for the non-well-ordered case, we need to ask the lower/upper solutions to be strict, a concept we will introduce in Section 7.

2 Elementary properties of positively-(p, q)-homogeneous Hamiltonian systems

For the autonomous system (3) the origin (0, 0) is an isochronous center, all solutions having minimal period $\tau >0$. We denote by $S(t)=(S_1(t),S_2(t))$ the periodic solution such that $S_1(0)=0$ and $S_2(0)>0$, with $H(S_1(t),S_2(t))=1$ for every t. Then, the periodic solutions of system (3) having energy $H(x,y)=E$ can be written as

$$\begin{aligned} \big (x(t),y(t)\big )=\big (E^{\frac{1}{p}} S_1(t+\sigma ), E^{\frac{1}{q}} S_2(t+\sigma ) \big )\,, \end{aligned}$$

for some $\sigma \in {\mathbb {R}}$. We will use the notations

$$\begin{aligned} \overline{S}&= \max \{ |S_1(t)|+|S_2(t)|:t\in {\mathbb {R}}\}\,,\nonumber \\ {\overline{S'}}&= \max \{|S'_1(t)|+|S'_2(t)|:t\in {\mathbb {R}}\}\,. \end{aligned}$$

(10)

For any function $u=(x,y):[a,b]\rightarrow {\mathbb {R}}^2\setminus \{(0,0)\}$, we can introduce the generalized polar coordinates

$$\begin{aligned} {\left\{ \begin{array}{ll} x(t) = r(t)^{\frac{1}{p}} S_1(\theta (t))\,,\\ y(t) = r(t)^{\frac{1}{q}} S_2(\theta (t))\,, \end{array}\right. } \end{aligned}$$

(11)

where $r(t)\ge 0$. If $u(t)=(0,0)$, we set $r(t)=0$, while $\theta (t)$ is not defined. Let

$$\begin{aligned} {\mathcal {N}}_p(u)= \Vert r\Vert _\infty ^{1/p}=\sup \{r(t)^{\frac{1}{p}}:t\in [a,b]\}\,. \end{aligned}$$

For every $D>0$, one has that

$$\begin{aligned} {\mathcal {N}}_p(u)\le D \quad \Rightarrow \quad \Vert x\Vert _\infty \le D\, \overline{S}\quad \hbox {and}\quad \Vert y\Vert _\infty \le D^{\frac{p}{q}}\, \overline{S}\,. \end{aligned}$$

Condition (1) can be rewritten as

$$\begin{aligned} H(\lambda ^{\frac{1}{p}} x, \lambda ^{\frac{1}{q}} y)= \lambda H(x,y)>0\,, \end{aligned}$$

for every $\lambda >0$ and $(x,y)\ne (0,0)$. Then,

$$\begin{aligned} \frac{\partial H}{\partial x}(\lambda ^q x,\lambda ^p y)&=\lambda ^{q(p-1)} \frac{\partial H}{\partial x}(x,y) =\lambda ^{p} \frac{\partial H}{\partial x}(x,y)\,, \end{aligned}$$

(12)

$$\begin{aligned} \frac{\partial H}{\partial y}(\lambda ^q x, \lambda ^p y)&=\lambda ^{p(q-1)} \frac{\partial H}{\partial y}(x,y) =\lambda ^{q} \frac{\partial H}{\partial y}(x,y)\,. \end{aligned}$$

(13)

We can write the generalized Euler formula

$$\begin{aligned} \left( \frac{x}{p}\, , \frac{y}{q} \right) \cdot \nabla H(x,y)= H(x,y)\,. \end{aligned}$$

(14)

We can rewrite (12) and (13) as

$$\begin{aligned} \frac{\partial H}{\partial x}(x, \mu y)&=\mu \frac{\partial H}{\partial x}(\mu ^{-\frac{q}{p}} x,y)\,, \end{aligned}$$

(15)

$$\begin{aligned} \frac{\partial H}{\partial y}( x, \mu y)&=\mu ^{\frac{q}{p}} \frac{\partial H}{\partial y}(\mu ^{-\frac{q}{p}} x,y)\,, \end{aligned}$$

(16)

for every $\mu >0$. We will need the following property.

Lemma 4

For every $L>0$, we have that

$$\begin{aligned} \lim _{y\rightarrow \pm \infty } \frac{\partial H}{\partial y}(x,y)=\pm \infty \text{ uniformly } \text{ with } \text{ respect } \text{ to } \ x\in [-L, L]. \end{aligned}$$

Proof

We first prove that there are $c_0>0$ and $y_0\ge 1$ such that

$$\begin{aligned} |x|\le L \quad \Rightarrow \quad \frac{\partial H}{\partial y}(x,y_0)>c_0\quad \text{ and } \quad \frac{\partial H}{\partial y}(x,-y_0)<-c_0\,. \end{aligned}$$

From (14), we have both

$$\begin{aligned} \frac{\partial H}{\partial y}(0,1)>0\quad \text{ and } \quad \frac{\partial H}{\partial y}(0,-1)<0\,. \end{aligned}$$

Hence, we can find $\delta _0>0$ and $c_0>0$ such that

$$\begin{aligned} |u|\le \delta _0 \quad \Rightarrow \quad \frac{\partial H}{\partial y}(u,1)>c_0\quad \text{ and } \quad \frac{\partial H}{\partial y}(u,-1)<-c_0 \,. \end{aligned}$$

(17)

If we set

$$\begin{aligned} y_0=\max \left\{ \left( {L}/{\delta _0}\right) ^{\frac{p}{q}},\,1 \right\} , \end{aligned}$$

then, for every $x\in [-L,L]$ we get, using (16) and (17),

$$\begin{aligned} \frac{\partial H}{\partial y}(x,y_0)&= y_0^{\frac{q}{p}}\frac{\partial H}{\partial y}\left( y_0^{-\frac{q}{p}} x,1\right) > y_0^{\frac{q}{p}} c_0 \ge c_0\,,\\ \frac{\partial H}{\partial y}(x,-y_0)&= y_0^{\frac{q}{p}} \frac{\partial H}{\partial y}\left( y_0^{-\frac{q}{p}} x,-1\right) < - y_0^{\frac{q}{p}} c_0 \le -c_0\,. \end{aligned}$$

So, for any $x\in [-L, L]$ and $\mu >1$, from (16) we obtain

$$\begin{aligned} \frac{\partial H}{\partial y}(x,\mu y_0)&= \mu ^{\frac{q}{p}} \frac{\partial H}{\partial y} \left( \mu ^{-\frac{q}{p}} x,y_0\right) \ge \mu ^{\frac{q}{p}} c_0\,,\\ \frac{\partial H}{\partial y}(x,-\mu y_0)&= \mu ^{\frac{q}{p}} \frac{\partial H}{\partial y}\left( \mu ^{-\frac{q}{p}} x,-y_0\right) \le - \mu ^{\frac{q}{p}} c_0\,. \end{aligned}$$

The conclusion follows. $\square$

3 Proof of Theorem 1

To prove Theorem 1, let us first assume the validity of the alternative (4).

Introducing generalized polar coordinates (11), we can compute (cf. [25, Section 2])

$$\begin{aligned} \theta '(t)&= 1 + \frac{1}{r(t)}\left[ \frac{1}{q} y(t) \phi (t,x(t),y(t)) - \frac{1}{p} x(t) \psi (t,x(t),y(t))\right] ,\\ r'(t)&= \psi (t,x(t),y(t)) r^{\frac{1}{p}}(t) S_1'(\theta (t)) - \phi (t,x(t),y(t)) r^{\frac{1}{q}}(t) S_2'(\theta (t))\,. \end{aligned}$$

Since the functions $\phi ,\psi :[a,b]\times {\mathbb {R}}^2\rightarrow {\mathbb {R}}$ are bounded, there is a constant K for which

$$\begin{aligned} |\phi (t,x,y)|\le K\,, \quad |\psi (t,x,y)|\le K\,, \quad \text{ for } \text{ every } (t,x,y)\in [a,b]\times {\mathbb {R}}^2. \end{aligned}$$

(18)

Setting $\omega =\min \{\frac{1}{p},\frac{1}{q}\}$ and using (10), we have

$$\begin{aligned} |\theta '(t) -1|&\le K\overline{S} \, r(t)^{-\omega }\,, \end{aligned}$$

(19)

$$\begin{aligned} |r'(t)|&\le 2K {\overline{S'}}\, r(t) \,, \end{aligned}$$

(20)

when $r(t) > 1$. As a consequence of (20), all the solutions of

$$\begin{aligned} x'=\displaystyle \frac{\partial H}{\partial y}(x,y) + \phi (t,x,y)\,,\quad y'=\displaystyle -\frac{\partial H}{\partial x}(x,y) + \psi (t,x,y) \end{aligned}$$

(21)

are globally defined. Since we are assuming (4), choosing $\delta >0$ sufficiently small, we have

$$\begin{aligned}&n\tau< (b-a)(1-\delta )< (b-a)(1+\delta )<n\tau +\min \{\tau _-,\tau _+\}\,. \end{aligned}$$

(22)

Correspondingly, we can fix $\overline{R}>1$ such that

$$\begin{aligned} K\overline{S} \, {\overline{R}}^{-\omega } < \delta \,. \end{aligned}$$

From (19), a solution of (21) satisfying $r(t)\ge \overline{R}$ for every $t\in [a,b]$ is such that

$$\begin{aligned} 1-\delta<\theta '(t)<1+\delta \,, \end{aligned}$$

for every $t\in [a,b]$. Hence, recalling assumption (4), from (22) we get

$$\begin{aligned} n\tau< \theta (b)-\theta (a) <n\tau +\min \{\tau _-,\tau _+\}\,. \end{aligned}$$

(23)

Finally, from (20) and Gronwall lemma, we can find $R_1>R_0>\overline{R}$ with the following property: if a solution of (21) satisfies $r(t_0)=R_0$ for some $t_0\in [a,b]$, then

$$\begin{aligned} \overline{R}< r(t) < R_1 \text{ for } \text{ every } t\in [a,b]\,. \end{aligned}$$

(24)

Let us set

$$\begin{aligned} y_0^+= R_0^{\frac{1}{q}} S_2(0)>0 \quad \text{ and }\quad y_0^-= R_0^{\frac{1}{q}} S_2(\tau _+)<0\,. \end{aligned}$$

For any $\sigma \in J:=[y_0^-,y_0^+]$, we consider the Cauchy problem

$$\begin{aligned} {\left\{ \begin{array}{ll} x'=\displaystyle \frac{\partial H}{\partial y}(x,y) + \phi (t,x,y)\,, \quad y'=\displaystyle -\frac{\partial H}{\partial x}(x,y) + \psi (t,x,y)\,,\\ x(a)=0\,, \qquad y(a)=\sigma \,. \end{array}\right. } \end{aligned}$$

(25)

We recall that all the solutions of (25) are defined in the interval [a, b]. Then, by [6, Corollary 2.3], there exists a closed connected set

$$\begin{aligned} {\mathcal {K}} \subseteq \{(\sigma ,z)\in J \times C([a,b],{\mathbb {R}}^2) \,:\, z=(x,y) \text{ is } \text{ a } \text{ solution } \text{ of }~(25) \} \end{aligned}$$

whose projection on J coincides with J.

So, we can find some $(y_0^-,(x^-,y^-))\in {\mathcal {K}}$ and $(y_0^+,(x^+,y^+))\in {\mathcal {K}}$. The solutions $(x^\pm ,y^\pm )$ satisfy (23) and

$$\begin{aligned} \overline{R}< r(t) < R_1 \text{ for } \text{ every } t\in [a,b]\,. \end{aligned}$$

Indeed, denoting by $(r^\pm ,\theta ^\pm )$ the modified polar coordinates of $(x^\pm ,y^\pm )$, we have $r^\pm (a)=R_0$ and so (24) holds. Then, if we consider the initial data $y^+(a)=y_0^+$, [resp. $y^-(a)=y_0^-$], we can choose $\theta ^+(a)=0$ [resp. $\theta ^-(a)=\tau _+$].

From (23), we get

$$\begin{aligned} n\tau< \theta ^+(b)<n\tau +\tau _+\,, \qquad n\tau +\tau _+< \theta ^-(b) <(n+1)\tau \,, \end{aligned}$$

so that

$$\begin{aligned} x^-(b) \cdot x^+(b) <0\,. \end{aligned}$$

(26)

Hence, defining the continuous function ${\mathcal {Z}}: {\mathcal {K}} \rightarrow {\mathbb {R}}$ as ${\mathcal {Z}} (\sigma , (x,y))=x(b)$, from (26) we easily deduce that

$$\begin{aligned} {\mathcal {Z}} (y_0^-,(x^-,y^-)) \cdot {\mathcal {Z}} (y_0^+,(x^+,y^+)) <0\,. \end{aligned}$$

By continuity, ${\mathcal {Z}}({\mathcal {K}})$ is an interval and we find the existence of $(\overline{\sigma },(\overline{x},\overline{y}))\in {\mathcal {K}}$ satisfying ${\mathcal {Z}}(\overline{\sigma },(\overline{x},\overline{y}))=0$. Hence, $(\overline{x},\overline{y})$ is the solution we were looking for.

The proof is thus completed if (4) holds. If (5) holds, the proof is similar: we need to replace (22) by

$$\begin{aligned}&n\tau +\max \{\tau _-,\tau _+\}< (b-a)(1-\delta )< (b-a)(1+\delta )< (n+1)\tau \,, \end{aligned}$$

and (23) by

$$\begin{aligned} n\tau +\max \{\tau _-,\tau _+\}< \theta (b)-\theta (a) < (n+1)\tau \,. \end{aligned}$$

Finally, we get

$$\begin{aligned} n\tau +\tau _+< \theta _n^+(b)< (n+1)\tau \,, \quad (n+1)\tau< \theta _n^-(b) < (n+1)\tau +\tau _+\,, \end{aligned}$$

which gives (26), permitting us to conclude as above.

The proof of Theorem 1 is now completed. $\square$

Remarks. Other types of boundary conditions can be considered, leading to similar results. The case of a Neumann-type problem, with boundary conditions $y(a)=0=y(b)$, is nothing but the previous Dirichlet-type problem by a simple switch in the variables . The mixed problem

$$\begin{aligned} {\left\{ \begin{array}{ll} x'=\displaystyle \frac{\partial H}{\partial y}(x,y) + \phi (t,x,y)\,,\quad y'=\displaystyle -\frac{\partial H}{\partial x}(x,y) + \psi (t,x,y)\,,\\ x(a)=0=y(b) \end{array}\right. } \end{aligned}$$

(27)

can be considered, as well. Going back to the solution $S=(S_1,S_2)$ of the unperturbed system (3), we can find the positive values $\tau _1,\tau _2,\tau _3,\tau _4$ such that

$$\begin{aligned}&S_1(0)=0\,,&S_2(0)>0\\&S_1(\tau _1)>0\,,&S_2(\tau _1)=0\,,\\&S_1(\tau _1+\tau _2)=0\,,&S_2(\tau _1+\tau _2)<0\,,\\&S_1(\tau _1+\tau _2+\tau _3)<0\,,&S_2(\tau _1+\tau _2+\tau _3)=0\,,\\&\tau _1+\tau _2+\tau _3+\tau _4=\tau \,, \end{aligned}$$

leading to the following.

Theorem 5

Assume that there is an integer $n\ge 0$ such that one of the following alternatives hold

$$\begin{aligned} n\tau&< b-a<n\tau +\min \{\tau _1,\tau _3\}\,,\\ n\tau +\tau _1+\tau _3+ \max \{\tau _2,\tau _4\}&<b-a<(n+1)\tau \,. \end{aligned}$$

Then, problem (27) has a solution.

Similarly, we can consider the boundary conditions $y(a)=0=x(b)$, as well, thus obtaining the corresponding existence result. Finally, Carathéodory conditions on $\phi$ and $\psi$ could be assumed. We avoid entering into details, for briefness.

4 The definition of lower and upper solutions

Let us consider the boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} x'=f(t,x,y)\,, \quad y'=g(t,x,y)\,,\\ x(a)=0=x(b)\,, \end{array}\right. } \end{aligned}$$

(28)

where $f,g:[a,b]\times {\mathbb {R}}^2\rightarrow {\mathbb {R}}$ are continuous functions, and recall the definitions introduced in [14].

Definition 6

A continuously differentiable function $\alpha :[a,b]\rightarrow {\mathbb {R}}$ is said to be a lower solution for problem (28) if there exists a continuously differentiable function $y_\alpha :[a,b]\rightarrow {\mathbb {R}}$ such that, for every $t\in [a,b]$,

$$\begin{aligned} {\left\{ \begin{array}{ll} y<y_\alpha (t)\quad \Rightarrow \quad f(t,\alpha (t),y)<\alpha '(t)\,,\\ y>y_\alpha (t)\quad \Rightarrow \quad f(t,\alpha (t),y)>\alpha '(t)\,, \end{array}\right. } y_\alpha '(t)\ge g(t,\alpha (t),y_\alpha (t))\,, \end{aligned}$$

and

$$\begin{aligned} \alpha (a)\le 0\,, \quad \alpha (b) \le 0\,. \end{aligned}$$

Definition 7

A continuously differentiable function $\beta :[a,b]\rightarrow {\mathbb {R}}$ is said to be an upper solution for problem (28) if there exists a continuously differentiable function $y_\beta :[a,b]\rightarrow {\mathbb {R}}$ such that, for every $t\in [a,b]$,

$$\begin{aligned} {\left\{ \begin{array}{ll} y<y_\beta (t)\quad \Rightarrow \quad f(t,\beta (t),y)<\beta '(t)\,,\\ y>y_\beta (t)\quad \Rightarrow \quad f(t,\beta (t),y)>\beta '(t)\,, \end{array}\right. } y_\beta '(t)\le g(t,\beta (t),y_\beta (t))\,, \end{aligned}$$

and

$$\begin{aligned} \beta (a)\ge 0\,, \quad \beta (b) \ge 0\,. \end{aligned}$$

We say that $(\alpha ,\beta )$ is a well-ordered pair of lower/upper solutions of problem (28), if $\alpha$ and $\beta$ are respectively a lower and an upper solution for problem (28) and they satisfy

$$\begin{aligned} \alpha (t)\le \beta (t) \,, \quad \text{ for } \text{ every } t\in [a,b]\,. \end{aligned}$$

On the other hand, if the above inequality does not hold, we say that the pair $(\alpha ,\beta )$ is non-well-ordered.

5 Proof of Theorem 2

Define

$$\begin{aligned} f(t,x,y)=\displaystyle \frac{\partial H}{\partial y}(x,y) + \phi (t,x,y)\,,\quad g(t,x,y)=-\frac{\partial H}{\partial x}(x,y) + \psi (t,x,y)\,, \end{aligned}$$

and set $m_\alpha =\min \alpha$ and $M_\beta =\max \beta$. To apply [14, Theorem 11], we need to construct some guiding curves $\gamma ^\pm _{1,2}:[m_\alpha ,M_\beta ]\rightarrow {\mathbb {R}}$ such that, for every $t\in [a,b]$ and $x\in [\alpha (t),\beta (t)]$,

$$\begin{aligned} \gamma ^-_i(x)< \min \{ y_\alpha (t),y_\beta (t)\}&\le \max \{ y_\alpha (t),y_\beta (t)\}< \gamma ^+_i(x)\,, \quad i=1,2\,,\ \nonumber \\ g\big (t,x,\gamma ^+_1(x)\big )&> f\big (t,x,\gamma ^+_1(x)\big ) (\gamma ^+_1)'(x)\,, \end{aligned}$$

(29)

$$\begin{aligned} g\big (t,x,\gamma ^+_2(x)\big )&< f\big (t,x,\gamma ^+_2(x)\big ) (\gamma ^+_2)'(x)\,, \end{aligned}$$

(30)

$$\begin{aligned} g\big (t,x,\gamma ^-_1(x)\big )&< f\big (t,x,\gamma ^-_1(x)\big ) (\gamma ^-_1)'(x)\,, \end{aligned}$$

(31)

$$\begin{aligned} g\big (t,x,\gamma ^-_2(x)\big )&> f\big (t,x,\gamma ^-_2(x)\big ) (\gamma ^-_2)'(x)\,. \end{aligned}$$

(32)

This can be obtained as an immediate consequence of the next two lemmas.

Lemma 8

For any $L>0$ and $y_0>0$ we can define two continuously differentiable functions $\gamma ^+_1,\gamma ^+_2:[-{L}, {L}]\rightarrow {\mathbb {R}}$ satisfying

$$\begin{aligned} \gamma ^+_i(x)\ge y_0 \,, \quad i=1,2\,, \quad \text{ for } \text{ all } \ x\in [-{L}, {L}]\,, \end{aligned}$$

and such that (29) and (30) hold for every $t\in [a,b]$ and $x\in [-L,L]$.

Proof

Since both $\phi$ and $\psi$ are bounded, let us consider $K>0$ as in (18). From Lemma 4 we can find $y_1\ge y_0$ such that

$$\begin{aligned} \frac{\partial H}{\partial y}(x,y)>K+1, \hbox { for all } (x,y)\in [-L,L]\times [y_1,+\infty [\,. \end{aligned}$$

(33)

Since H is $C^1$, we can find a positive constant $c_1$ such that

$$\begin{aligned} \left| \frac{\partial H}{\partial x}(x,y)\right| \le c_1\,, \quad \text{ for } \text{ every } (x,y)\in [-L,L]\times [y_1, y_1+2L]\,. \end{aligned}$$

(34)

Now, since $2-q<1$, we can find $M>1$ sufficiently large so to have

$$\begin{aligned} \big (c_1 M^{2-q} +1\big )(K+1)<M\,. \end{aligned}$$

(35)

We define $\gamma ^+_{1,2}:[-{L}, {L}]\rightarrow {\mathbb {R}}$ as

$$\begin{aligned} \gamma ^+_1(x)=M (y_1 + {L}-x)\,, \quad \gamma ^+_2(x)=M (y_1 + {L}+x)\,. \end{aligned}$$

(36)

Notice that $\gamma ^+_{1,2}(x)\ge y_1\ge y_0$, for every $x\in [-L,L]$. If we show that, for $i=1,2$,

$$\begin{aligned} \frac{\displaystyle \left| \frac{\partial H}{\partial x}(x,\gamma ^+_i(x)) \right| + K}{ \displaystyle \frac{\partial H}{\partial y}(x,\gamma ^+_i(x))-K}<M\,, \quad \text{ for } \text{ every } x\in [-L,L]\,, \end{aligned}$$

(37)

then, since from (36) and (33) the denominator in (37) is positive, we get both

$$\begin{aligned}&M \left( \frac{\partial H}{\partial y}(x,\gamma ^+_1(x)) + \phi (t,x,\gamma ^+_1(x)) \right) \ge M \left( \frac{\partial H}{\partial y}(x,\gamma ^+_1(x)) -K \right) \\&\quad >\left| \frac{\partial H}{\partial x}(x,\gamma ^+_1(x))\right| + K \ge \frac{\partial H}{\partial x}(x,\gamma ^+_1(x)) - \psi (t,x,\gamma ^+_1(x))\,, \end{aligned}$$

and

$$\begin{aligned}&M \left( \frac{\partial H}{\partial y}(x,\gamma ^+_2(x)) + \phi (t,x,\gamma ^+_2(x)) \right) \ge M \left( \frac{\partial H}{\partial y}(x,\gamma ^+_2(x)) -K \right) \\&\quad >\left| \frac{\partial H}{\partial x}(x,\gamma ^+_2(x))\right| + K \ge -\frac{\partial H}{\partial x}(x,\gamma ^+_2(x)) + \psi (t,x,\gamma ^+_2(x))\,, \end{aligned}$$

for every $t\in [a,b]$ and $x\in [-L,L]$ and the lemma will be proved. Hence, we need to prove the validity of (37).

Using equality (15) and the estimate (34), we have that

$$\begin{aligned}&\left| \frac{\partial H}{\partial x}(x,M v)\right| =M \left| \frac{\partial H}{\partial x}\left( M^{-\frac{q}{p}}x,v\right) \right| \le M c_1\,,\nonumber \\&\qquad \text{ for } \text{ every } \ x\in [-L,L] \ \mathrm{and} \ v\in [y_1,y_1+2L]\,. \end{aligned}$$

(38)

Moreover, using equality (16) and recalling (33), we have that

$$\begin{aligned}&M^{-\frac{q}{p}} \frac{\partial H}{\partial y}(x,Mv) = \frac{\partial H}{\partial y}\left( M^{-\frac{q}{p}}x,v\right) >1\,,\nonumber \\&\qquad \text{ for } \text{ every } \ x\in [-L,L] \ \mathrm{and} \ v\in [y_1,y_1+2L]\,. \end{aligned}$$

(39)

From (38) and (39),

$$\begin{aligned}&\left| \frac{\partial H}{\partial x}(x,M v)\right| \le M c_1 <c_1 M^{1-\frac{q}{p}} \frac{\partial H}{\partial y}(x,Mv) =c_1 M^{2-q} \frac{\partial H}{\partial y}(x,Mv)\,,\\&\qquad \text{ for } \text{ every } \ x\in [-L,L] \ \mathrm{and} \ v\in [y_1,y_1+2L]\,. \end{aligned}$$

Since, using (36), for all $x\in [-L,L]$ we have $\frac{1}{M} \gamma ^+_i(x)\in [y_1,y_1+2L]$, $i=1,2$, then we get

$$\begin{aligned} \left| \frac{\partial H}{\partial x}(x,\gamma ^+_i(x))\right| < c_1 M^{2-q} \frac{\partial H}{\partial y}(x,\gamma ^+_i(x))\,. \end{aligned}$$

Hence, recalling (33) and (36), we have

$$\begin{aligned} \frac{ \left| \frac{\partial H}{\partial x}(x,\gamma ^+_i(x))\right| + K}{\frac{\partial H}{\partial y}(x,\gamma ^+_i(x))-K}&< \frac{ c_1 M^{2-q} \frac{\partial H}{\partial y}(x,\gamma ^+_i(x)) +K}{ \frac{\partial H}{\partial y}(x,\gamma ^+_i(x))-K}\\&= \frac{ c_1 M^{2-q} +\frac{K}{\frac{\partial H}{\partial y}(x,\gamma ^+_i(x))} }{1- \frac{K}{\frac{\partial H}{\partial y}(x,\gamma ^+_i(x))}}\\&\le \big (c_1 M^{2-q} +1\big )(K+1) <M\,, \end{aligned}$$

where the last estimate is given by (35). We have thus proved (37), and so the proof of the lemma is completed. $\square$

Lemma 9

For any $L>0$ and $y_0>0$ we can define two continuously differentiable functions $\gamma ^-_1,\gamma ^-_2:[-{L}, {L}]\rightarrow {\mathbb {R}}$ satisfying

$$\begin{aligned} \gamma ^-_i(x)\le -y_0 \,, \quad i=1,2\,, \quad \text{ for } \text{ all } \ x\in [-{L}, {L}]\,, \end{aligned}$$

and such that (31) and (32) hold for every $t\in [a,b]$ and $x\in [-L,L]$.

The proof of this lemma is analogous to the previous one, so we omit it, for briefness.

Let us now conclude the proof of Theorem 2. Choosing

$$\begin{aligned} L=\max \{\Vert \alpha \Vert _\infty \,,\, \Vert \beta \Vert _\infty \}\,,\qquad y_0>\max \{\Vert y_\alpha \Vert _\infty \,,\, \Vert y_\beta \Vert _\infty \}\,, \end{aligned}$$

we can apply Lemmas 8 and 9 to get the existence of the needed curves $\gamma _i^\pm$. Moreover, for every $t\in [a,b]$,

$$\begin{aligned}&y\ge \min \{ \gamma ^+_1(0),\gamma ^+_2(0)\}\quad \Rightarrow \quad f(t,0,y)>0\,,\\&y\le \max \{ \gamma ^-_1(0),\gamma ^-_2(0)\}\quad \Rightarrow \quad f(t,0,y)<0\,. \end{aligned}$$

This is an immediate consequence of Lemma 4, choosing a larger value of $y_0$, if necessary. Then, [14, Theorem 11] applies, thus completing the proof of Theorem 2.

6 Proof of Theorem 3

As a first step, we need the following two lemmas, where the ordering relation defined in Introduction is used.

Lemma 10

([14, Lemma 17]). Given a continuous function $\varphi :[a,b]\rightarrow {\mathbb {R}}$, the sets

$$\begin{aligned} \{x\in C^1_0([a,b]) \,:\, \varphi \ll x\} \,, \quad \{x\in C^1_0([a,b]) \,:\, x\ll \varphi \} \end{aligned}$$

are open in $C^1_0([a,b])= \{x\in C^{1}([a,b]) \,:\, x(a)=0=x(b)\}$.

Lemma 11

([14, Lemma 18]). Given a continuously differentiable function $\varphi :[a,b]\rightarrow {\mathbb {R}}$, we have that

$$\begin{aligned} \max \{\varphi (a),\varphi (b)\}\le 0&\quad \Rightarrow \quad \exists {\hat{c}}>0 \, : \, \varphi \ll {\hat{c}} \varphi _1\,,\\ \min \{\varphi (a),\varphi (b)\}\ge 0&\quad \Rightarrow \quad \exists {\hat{c}}>0 \, : \, \varphi \gg -{\hat{c}} \varphi _1\,. \end{aligned}$$

We now use the notation introduced in Section 2, and set ${\widetilde{S}}_1(t)=S_1(t+\tau _+)$.

Lemma 12

Given a continuously differentiable function $\varphi :[0,\tau ^+]\rightarrow {\mathbb {R}}$, we have that

$$\begin{aligned} \max \{\varphi (0),\varphi (\tau ^+)\}\le 0&\quad \Rightarrow \quad \exists c>0 \, : \, \varphi \ll c S_1\,. \end{aligned}$$

Similarly, given a continuously differentiable function $\varphi :[0,\tau ^-]\rightarrow {\mathbb {R}}$, we have that

$$\begin{aligned} \min \{\varphi (0),\varphi (\tau ^-)\}\ge 0&\quad \Rightarrow \quad \exists c>0 \, : \, \varphi \gg c {\widetilde{S}}_1\,. \end{aligned}$$

Proof

To prove the former statement, we apply Lemma 11 with $[a,b]=[0,\tau ^+]$, recalling that $S_1\gg 0$. To prove the latter statement, we apply the same lemma with $[a,b]=[0,\tau ^-]$, recalling that ${\widetilde{S}}_1 \ll 0$. $\square$

Let us introduce a $C^\infty$-smooth cut-off function $\chi :{\mathbb {R}}\rightarrow [0,1]$ such that

$$\begin{aligned} \chi (s)= {\left\{ \begin{array}{ll} 1 &{} \text{ if } |s| \le 1\,,\\ 0 &{} \text{ if } |s| \ge 2\,, \end{array}\right. } \end{aligned}$$

(40)

and, for every $d\ge 1$, consider the modified problem

$$\begin{aligned} {\left\{ \begin{array}{ll} x'={\widehat{f}}_d(t,x,y)\,, \quad y'={\widehat{g}}_d(t,x,y)\,, \\ x(a)=0=x(b)\,, \end{array}\right. } \end{aligned}$$

(41)

where

$$\begin{aligned} {\widehat{f}}_d(t,x,y)&=\frac{d-1}{d} \,\frac{\partial H}{\partial y}(x,y) + {\widetilde{\phi }}_d(t,x,y)\,,\\ {\widehat{g}}_d(t,x,y)&=- \frac{d-1}{d} \,\frac{\partial H}{\partial x}(x,y) + {\widetilde{\psi }}_d(t,x,y)\,, \end{aligned}$$

with

$$\begin{aligned} {\widetilde{\phi }}_d(t,x,y)&= \chi \Big (\frac{x}{d\overline{S}}\Big )\chi \Big (\frac{y}{d^{p/q}{\overline{S}}}\Big )\left( \frac{1}{d} \frac{\partial H}{\partial y}(x,y)+ \phi (t,x,y) \right) ,\\ {\widetilde{\psi }}_d(t,x,y)&= \chi \Big (\frac{x}{d\overline{S}}\Big )\chi \Big (\frac{y}{d^{p/q}{\overline{S}}}\Big )\left( - \frac{1}{d} \frac{\partial H}{\partial x}(x,y) + \psi (t,x,y) \right) . \end{aligned}$$

Notice that (2) and (41) coincide in the set $[-d\overline{S},d\overline{S}]\times [-d^{\frac{p}{q}}\overline{S},d^{\frac{p}{q}} \overline{S}]$. In particular, if a solution $u=(x,y)$ of (41) is such that ${\mathcal {N}}_p(u)<d$, then u is necessarily a solution of (2).

Lemma 13

There exists $D>1$ with the following property: if $u=(x,y)$ is any solution of (41), with $d\in [D,+\infty ]$, satisfying and , then ${\mathcal {N}}_p(u) < D$.

Proof

Assume by contradiction that there exist a diverging sequence $(d_n)_n$ and some solutions $u_n=(x_n,y_n)$ of (41), with $d=d_n$, such that , and ${\mathcal {N}}_p(u_n) >n$. We introduce the functions

$$\begin{aligned} v_n = \frac{x_n}{{\mathcal {N}}_p(u_n)} \quad \text{ and }\quad w_n = \frac{y_n}{{\mathcal {N}}_p(u_n)^{p-1}}\,. \end{aligned}$$

In particular, ${\mathcal {N}}_p(v_n,w_n)=1$. Notice that, from (11) and (10), we have

$$\begin{aligned} \Vert v_n\Vert _\infty \le \overline{S} \quad \text{ and }\quad \Vert w_n\Vert _\infty \le \overline{S}\,. \end{aligned}$$

(42)

By (12) and (13), we see that $(v_n,w_n)$ solves

$$\begin{aligned} {\left\{ \begin{array}{ll} v_n'=\displaystyle \frac{d_n-1}{d_n} \,\frac{\partial H}{\partial y}(v_n,w_n) + \Phi _n(t,v_n,w_n)\,, \\ w_n'=\displaystyle - \frac{d_n-1}{d_n} \,\frac{\partial H}{\partial x}(v_n,w_n) +\Psi _n(t,v_n,w_n)\,,\\ v_n(a)=0=v_n(b)\,, \end{array}\right. } \end{aligned}$$

(43)

where

$$\begin{aligned} \Phi _n(t,v_n,w_n)&=\frac{1}{{\mathcal {N}}_p(u_n)} {\widetilde{\phi }}_{d_n}(t,{\mathcal {N}}_p(u_n)v_n,{\mathcal {N}}_p(u_n)^{p-1}w_n)\,,\\ \Psi _n(t,v_n,w_n)&=\frac{1}{{\mathcal {N}}_p(u_n)^{p-1}} {\widetilde{\psi }}_{d_n}(t,{\mathcal {N}}_p(u_n)v_n,{\mathcal {N}}_p(u_n)^{p-1}w_n)\,. \end{aligned}$$

Then, by (42), since $(v_n,w_n)$ solves (43), it is bounded in $C^1\times C^1$. By a standard compactness argument, a subsequence converges to some $(\overline{v},\overline{w})$ in $C^1\times C^1$, and

$$\begin{aligned} {\left\{ \begin{array}{ll} \overline{v}'=\displaystyle \frac{\partial H}{\partial y}(\overline{v},\overline{w})\,,\quad \overline{w}'=\displaystyle - \frac{\partial H}{\partial x}(\overline{v},\overline{w})\\ \overline{v}(a)=0=\overline{v}(b)\,. \end{array}\right. } \end{aligned}$$

Hence, we have either $(\overline{v}(t),\overline{w}(t))=S(t)$, or $(\overline{v}(t),\overline{w}(t))=S(t+\tau _+)$. More precisely, using (8), the first case is possible if and only if $b-a=\tau _+$, the second one if and only if $b-a=\tau _-$. In the first case, since $v_n$ $C^1$-converges to $S_1\gg 0$, we have that $v_n \gg \frac{1}{2} S_1$ when n is sufficiently large, so that, recalling Lemma 12, we get the contradiction

$$\begin{aligned} x_n = {\mathcal {N}}_p(u_n) v_n\gg \frac{1}{2} {\mathcal {N}}_p(u_n) S_1 \gg c S_1 \gg \alpha \,. \end{aligned}$$

In the second case, since $v_n$ $C^1$-converges to ${\widetilde{S}}_1= S_1(\,\cdot \,+\tau _+)\ll 0$, we have that $v_n \ll \frac{1}{2} {\widetilde{S}}_1$ when n is sufficiently large, thus providing the contradiction

$$\begin{aligned} x_n = {\mathcal {N}}_p(u_n) v_n\ll \frac{1}{2} {\mathcal {N}}_p(u_n) {\widetilde{S}}_1 \ll c {\widetilde{S}}_1 \ll \beta \,. \end{aligned}$$

The proof is thus completed. $\square$

We now fix $D>1$ as in Lemma 13, assuming also

$$\begin{aligned}&\max \{ \Vert \alpha \Vert _\infty \,,\, \Vert \beta \Vert _\infty \}<D \overline{S}\,, \nonumber \\&\max \{\Vert y_\alpha \Vert _\infty \,,\, \Vert y_\beta \Vert _\infty \}<D \overline{S}^{\,p/q}\,. \end{aligned}$$

(44)

From Lemma 13, if $u=(x,y)$ is any solution of (41) with $d\ge D$, satisfying both and , then u is a solution of (2), too.

Lemma 14

The functions $\alpha$ and $\beta$ are a lower and an upper solution of (41), respectively, provided that d is chosen large enough.

Proof

From Lemma 4 and the boundedness of $\phi$, we deduce the existence of a constant ${\widetilde{K}}>0$ such that, for every $d\ge 1$, we have both ${\widetilde{\phi }}_d(t,\alpha (t),y)\ge -{\widetilde{K}}$ when $y>0$, and ${\widetilde{\phi }}_d(t,\alpha (t),y)\le {\widetilde{K}}$ when $y<0$, for every $t\in [a,b]$. Moreover, using again Lemma 4, we can take d large enough so that, if $y>d^{p/q}\overline{S}$ then

$$\begin{aligned} \frac{d-1}{d} \frac{\partial H}{\partial y}(\alpha (t),y)+{\widetilde{\phi }}_d(t,\alpha (t),y)\ge \frac{d-1}{d} \frac{\partial H}{\partial y}(\alpha (t),y)-\widetilde{K}>\alpha '(t)\,, \end{aligned}$$

and if $y<-d^{p/q}\overline{S}$ then

$$\begin{aligned} \frac{d-1}{d} \frac{\partial H}{\partial y}(\alpha (t),y)+{\widetilde{\phi }}_d(t,\alpha (t),y)\le \frac{d-1}{d} \frac{\partial H}{\partial y}(\alpha (t),y)+\widetilde{K}<\alpha '(t)\,. \end{aligned}$$

On the other hand, if $y\in [-d^{p/q}\overline{S},d^{p/q}\overline{S}]$, then $(\alpha (t),y)$ belongs to the region where the problem has not been modified and the desired estimates hold.

The inequality for $y_\alpha '(t)$ still holds since, for every $t\in [a,b]$, also the point $(\alpha (t),y_\alpha (t))$ belongs to the region where the problem has not been modified. $\square$

Our aim now is to construct a lower solution ${\widehat{\alpha }}$ and an upper solution ${\widehat{\beta }}$ such that ${\widehat{\alpha }}\ll \beta$ and $\alpha \ll {\widehat{\beta }}$. To do so, let us consider, for every $d>1$, the slowed autonomous system

$$\begin{aligned} x'=\displaystyle \frac{d-1}{d} \, \frac{\partial H}{\partial y}(x,y)\,,\quad y'=\displaystyle -\frac{d-1}{d} \, \frac{\partial H}{\partial x}(x,y)\,. \end{aligned}$$

(45)

It is isochronous, all solutions being periodic with minimal period $\tau \frac{d}{d-1}>\tau$.

Lemma 15

For every $\xi >0$, the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} x'=\displaystyle \frac{d-1}{d} \, \frac{\partial H}{\partial y}(x,y)\,,\quad y'=\displaystyle -\frac{d-1}{d} \, \frac{\partial H}{\partial x}(x,y)\,,\\ x(a)=\xi =x(b)\,, \end{array}\right. } \end{aligned}$$

has a unique solution $(x_\xi ,y_\xi )$. Moreover, $x_\xi (t)>\xi$ for every $t\in \,]a,b[$. Similarly, the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} x'=\displaystyle \frac{d-1}{d} \, \frac{\partial H}{\partial y}(x,y)\,,\quad y'=\displaystyle -\frac{d-1}{d} \, \frac{\partial H}{\partial x}(x,y)\,,\\ x(a)=-\xi =x(b)\,, \end{array}\right. } \end{aligned}$$

has a unique solution $(x_{-\xi },y_{-\xi })$, satisfying $x_{-\xi }(t)<-\xi$ for every $t\in \,]a,b[$.

Proof

If we parametrize the solutions (x, y) with the energy $E=\frac{d-1}{d}H(x,y)$, we see that they cross the line ${\mathcal {L}}=\{(x,y)\in {\mathbb {R}}^2:x=\xi \}$ if and only if E is greater than a well determined value ${\overline{E}}_\xi >0$, and assumption (9) ensures that there are exactly two crossing points $(\xi , {\bar{y}}_\xi ^+)$ and $(\xi , {\bar{y}}_\xi ^-)$, with ${\bar{y}}_\xi ^+>{\bar{y}}_\xi ^-$. Since $b-a\le \tau _-<\tau _-d/(d-1)$, the solution (x, y) we are looking for cannot follow the path to the left of ${\mathcal {L}}$, hence, if it exists, it has to be $x_\xi (t)>\xi$ for every $t\in \,]a,b[\,$.

Let us denote by $T_\xi (E)$ the time needed to go from $(\xi , y_\xi ^+)$ to $(\xi , y_\xi ^-)$. We have thus defined a function $T_\xi :\,]{\overline{E}}_\xi ,+\infty [\,\rightarrow {\mathbb {R}}$ which is continuous, positive and strictly increasing. Surely $T_\xi (E)<b-a$ if E is in a small right neighbourhood of ${\overline{E}}_\xi$. Hence, since $\lim _{E\rightarrow +\infty }T_\xi (E)=\tau _+d/(d-1)>b-a$, there is a unique value $E_\xi >{\overline{E}}_\xi$ for which $T_\xi (E_\xi )=b-a$, and this value of the energy determines the solution we are looking for.

We have thus proved the first part of the lemma; the proof of the second part is similar.$\square$

We are now ready to define the lower and upper solutions ${\widehat{\alpha }}$ and ${\widehat{\beta }}$.

Lemma 16

Taking $d>D$ and $\xi >2d\overline{S}$, the functions ${\widehat{\alpha }}=x_{-\xi }$ and ${\widehat{\beta }}=x_\xi$ are a lower and an upper solution of (41), respectively. Moreover,

$$\begin{aligned} {\widehat{\alpha }}\ll \beta \quad \hbox { and }\quad \alpha \ll {\widehat{\beta }}\,. \end{aligned}$$

Proof

Let us show that ${\widehat{\beta }}=x_\xi$ is an upper solution, with associated function $y_{{\widehat{\beta }}}=y_\xi$. Recalling that ${\widetilde{\phi }}_d(t,x,y)$ and ${\widetilde{\psi }}_d(t,x,y)$ vanish when $x\notin [-2d{\overline{S}},2d{\overline{S}}]$, since $\xi >2d\overline{S}$, using assumption (9), we have

$$\begin{aligned}&y<y_\xi (t)\quad \Rightarrow \quad \frac{d-1}{d}\frac{\partial H}{\partial y}(x_\xi (t),y)<\frac{d-1}{d}\frac{\partial H}{\partial y}(x_\xi (t),y_\xi (t))=x_\xi '(t)\,,\\&y>y_\xi (t)\quad \Rightarrow \quad \frac{d-1}{d}\frac{\partial H}{\partial y}(x_\xi (t),y)>\frac{d-1}{d}\frac{\partial H}{\partial y}(x_\xi (t),y_\xi (t))=x_\xi '(t)\,. \end{aligned}$$

Moreover,

$$\begin{aligned} y_\xi '(t)\le -\frac{d-1}{d}\frac{\partial H}{\partial x}(x_\xi (t),y_\xi (t))\,, \end{aligned}$$

since equality holds. Finally, $x_\xi (a)\ge 0$ and $x_\xi (b)\ge 0$, thus proving that $x_\xi$ is an upper solution. Analogously one proves that $x_{-\xi }$ is a lower solution.

Now, since from (44) we have $\xi>2d\overline{S} > \max \{\Vert \alpha \Vert _\infty \,,\Vert \beta \Vert _\infty \}$, one has

$$\begin{aligned} x_{-\xi }(t)\le -\xi <\beta (t)\,,\quad x_\xi (t)\ge \xi >\alpha (t)\,,\quad \hbox { for every }t\in [a,b]\,, \end{aligned}$$

thus ending the proof of the lemma. $\square$

Let us now fix $d>D$ sufficiently large to ensure the validity of Lemma 14. We thus have three well-ordered pairs of lower/upper solutions of (41):

$$\begin{aligned} ({\widehat{\alpha }},{\widehat{\beta }})\,,\quad (\alpha ,{\widehat{\beta }})\,,\quad ({\widehat{\alpha }},\beta )\,. \end{aligned}$$

Let ${\bar{a}}=\min {\widehat{\alpha }}$ and ${\bar{b}}=\max {\widehat{\beta }}$. By Lemma 4, we can find $\overline{M}>0$ such that, if $t\in [a,b]$ and $x\in [{\bar{a}},{\bar{b}}]$, then

$$\begin{aligned} y\ge \overline{M}&\quad \Rightarrow \quad {\widehat{f}}_d(t,x,y)>0\,, \nonumber \\ y\le -\overline{M}&\quad \Rightarrow \quad {\widehat{f}}_d(t,x,y)<0\,. \end{aligned}$$

(46)

We now need to introduce the guiding curves.

Lemma 17

There are four continuously differentiable functions $\gamma _i^\pm :[{\bar{a}},{\bar{b}}]\rightarrow {\mathbb {R}}$, with $i=1,2$, satisfying

$$\begin{aligned}&{\widehat{g}}_d\big (t,x,\gamma ^+_1(x)\big )> \widehat{f}_d\big (t,x,\gamma ^+_1(x)\big ) (\gamma ^+_1)'(x)\,, \nonumber \\&{\widehat{g}}_d\big (t,x,\gamma ^+_2(x)\big )< \widehat{f}_d\big (t,x,\gamma ^+_2(x)\big ) (\gamma ^+_2)'(x)\,, \nonumber \\&{\widehat{g}}_d\big (t,x,\gamma ^-_1(x)\big ) < {\widehat{f}}_d\big (t,x,\gamma ^-_1(x)\big ) (\gamma ^-_1)'(x)\,, \nonumber \\&{\widehat{g}}_d\big (t,x,\gamma ^-_2(x)\big ) > \widehat{f}_d\big (t,x,\gamma ^-_2(x)\big ) (\gamma ^-_2)'(x)\,, \end{aligned}$$

(47)

and such that

$$\begin{aligned} \gamma _i^-(x)&<\min \{-\overline{M},y_\alpha (t),y_{{\widehat{\alpha }}}(t),y_\beta (t),y_{{\widehat{\beta }}}(t) \} \nonumber \\&\le \max \{\overline{M},y_\alpha (t),y_{,{\widehat{\alpha }}}(t),y_\beta (t),y_{{\widehat{\beta }}}(t) \}<\gamma _i^+(x)\,, \end{aligned}$$

(48)

for every $t\in [a,b]$ and $x\in [{\bar{a}},{\bar{b}}]$.

Proof

It follows the lines of the proofs of Lemmas 8 and 9. $\square$

Let us now introduce our functional setting for the problem (41).

Set ${\mathcal {I}}=[a,b]$, denote by $C^{0,\ell }({\mathcal {I}})$ the space of $\ell$-Hölder continuous functions and by $C^{1,\ell }({\mathcal {I}})$ the space of functions having derivative belonging to $C^{0,\ell }({\mathcal {I}})$. Moreover, define

$$\begin{aligned} C^{1,\ell }_0({\mathcal {I}}) = \{x\in C^{1,\ell }({\mathcal {I}}) \,:\, x(a)=0=x(b)\}\,. \end{aligned}$$

We consider the linear operator

$$\begin{aligned}&L: C_0^{1,\ell }({\mathcal {I}}) \times C^{1,\ell }({\mathcal {I}}) \rightarrow C^{0,\ell }({\mathcal {I}}) \times C^{0,\ell }({\mathcal {I}})\\&L(x,y)=(x'-y\,,\,y')\,, \end{aligned}$$

and the Nemitskii operator

$$\begin{aligned}&N_d: C^{1}_0({\mathcal {I}}) \times C^{1}({\mathcal {I}}) \rightarrow C^{0,\ell }({\mathcal {I}}) \times C^{0,\ell }({\mathcal {I}})\\&N_d(x,y)(t)= \left( {\widehat{f}}_d(t,x(t),y(t))-y(t) \,,\, \widehat{g}_d(t,x(t),y(t)) \right) . \end{aligned}$$

Problem (41) is then the same as

$$\begin{aligned} Lu=N_du\,, \end{aligned}$$

with

$$\begin{aligned} u\in X:= C_0^{1}({\mathcal {I}}) \times C^{1}({\mathcal {I}})\,. \end{aligned}$$

Lemma 18

The operator L is invertible with continuous inverse, and problem (41) is equivalent to $u=L^{-1}N_d u$. Moreover, the operator

$$\begin{aligned} L^{-1}N_d: X \rightarrow X \end{aligned}$$

is completely continuous.

Proof

It is rather standard, using the fact that $C^{1,\ell }({\mathcal {I}})$ is compactly imbedded in $C^1({\mathcal {I}})$. $\square$

Let us define the sets

$$\begin{aligned} {\mathcal {V}}_1={\mathcal {V}}({\widehat{\alpha }},{\widehat{\beta }})\,,\quad {\mathcal {V}}_2={\mathcal {V}}({\widehat{\alpha }},\beta )\,,\quad {\mathcal {V}}_3={\mathcal {V}}(\alpha ,{\widehat{\beta }})\,, \end{aligned}$$

with the notation

$$\begin{aligned} {\mathcal {V}}(\varphi ,\eta )=&\{(x,y)\in X \,:\, \varphi \ll x\ll \eta \hbox { and }\\&\qquad \gamma ^-(x(t))<y(t)<\gamma ^+(x(t))\,,\hbox { for every }t\in {\mathcal {I}}\}\,, \end{aligned}$$

where

$$\begin{aligned} \gamma ^-(x)=\min \{\gamma _i^-(x)\,,\, i=1,2\} \,,\qquad \gamma ^+(x)=\max \{\gamma _i^+(x)\,,\, i=1,2\}\,. \end{aligned}$$

(49)

These sets are open in $X=C_0^{1}({\mathcal {I}}) \times C^{1}({\mathcal {I}})$, by Lemma 10.

We say that an open set $\Omega \subset X$ is admissible if $L^{-1}N_d$ has no fixed points on $\partial \Omega$ and the set of fixed points of $L^{-1}N_d$ in $\Omega$ is bounded, i.e., it is contained in some open ball $B_\rho \,$. In this case, we can define

$$\begin{aligned} \deg (I-L^{-1}N_d,\Omega )=d_{LS}(I-L^{-1}N_d,\Omega \cap B_\rho )\,, \end{aligned}$$

where $d_{LS}$ denotes the Leray–Schauder degree. By excision, this definition does not depend on the choice of $\rho$.

Our aim is to prove that, if there are no solutions of (41) on $\partial {\mathcal {V}}_j$, then ${\mathcal {V}}_j$ is admissible and

$$\begin{aligned} \deg (I-L^{-1}N_d,{\mathcal {V}}_j)=1\,,\quad j=1,2,3. \end{aligned}$$

This fact will be proved in Lemma 22.

In the following we denote by $(\varphi ,\eta )$ any of the three pairs

$$\begin{aligned} ({\widehat{\alpha }},{\widehat{\beta }})\,, \quad (\alpha ,{\widehat{\beta }})\,,\quad ({\widehat{\alpha }},\beta )\,. \end{aligned}$$

We need to truncate the functions ${\widehat{f}}_d$ and ${\widehat{g}}_d$ so to modify system (41). Define, for any $\mu \le \nu$,

$$\begin{aligned} \zeta (s,\mu ,\nu )={\left\{ \begin{array}{ll} \mu \,, &{}\hbox { if }s<\mu \,,\\ s\,, &{}\hbox { if } \mu \le x\le \nu \,,\\ \nu \,, &{}\hbox { if } s>\nu \,. \end{array}\right. } \end{aligned}$$

(50)

Fix $\overline{Y}>0$ such that

$$\begin{aligned} -\overline{Y}\le \gamma ^-(x)\le \gamma ^+(x)\le \overline{Y}\,, \end{aligned}$$

for every $x\in [{\bar{a}},{\bar{b}}]$. Let

$$\begin{aligned} f_{\varphi ,\eta }(t,x,y)&={\widehat{f}}_d\big ( t,\zeta (x,\varphi (t),\eta (t)),\zeta (y,-\overline{Y},\overline{Y})\big )-\zeta (y,-\overline{Y},\overline{Y})\,,\\ g_{\varphi ,\eta }(t,x,y)&=\widehat{g}_d\big (t,\zeta (x,\varphi (t),\eta (t)),\zeta (y,-\overline{Y},\overline{Y})\big )-\zeta (x,\varphi (t),\eta (t))\,, \end{aligned}$$

and consider the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} x'=y+f_{\varphi ,\eta }(t,x,y)\,,\quad y'=x+g_{\varphi ,\eta }(t,x,y)\\ x(a)=0=x(b)\,, \end{array}\right. } \end{aligned}$$

(51)

which is equivalent to $Lu=N_{\varphi ,\eta }u$, with the appropriate Nemitskii operator. Notice that the functions $f_{\varphi ,\eta }$ and $g_{\varphi ,\eta }$ are bounded. Moreover, if (x, y) is a solution of (51) satisfying $\varphi \le x \le \eta$ and $-\overline{Y}< y < \overline{Y}$, then it is a solution of (41), too.

Lemma 19

Each $(\varphi ,\eta )$ is a well-ordered pair of lower/upper solutions for (51).

Proof

Assume for instance that $(\varphi ,\eta )=(\alpha ,{\widehat{\beta }})$. Then, if $y\in [-\overline{Y},\overline{Y}]$,

$$\begin{aligned}&y>y_\alpha (t) \Rightarrow y+f_{\alpha ,\eta }(t,\alpha (t),y)=\widehat{f}_d(t,\alpha (t),y)>\alpha '(t)\,,\\&y<y_\alpha (t) \Rightarrow y+f_{\alpha ,\eta }(t,\alpha (t),y)=\widehat{f}_d(t,\alpha (t),y)<\alpha '(t)\,, \end{aligned}$$

while

$$\begin{aligned} y>\overline{Y}&\Rightarrow y+f_{\alpha ,\eta }(t,\alpha (t),y)>\overline{Y}+f_{\alpha ,\eta }(t,\alpha (t),\overline{Y})>\alpha '(t)\,,\\ y<-\overline{Y}&\Rightarrow y+f_{\alpha ,\eta }(t,\alpha (t),y)<-\overline{Y}+f_{\alpha ,\eta }(t,\alpha (t),-\overline{Y})<\alpha '(t)\,. \end{aligned}$$

Since $-\overline{Y}<y_\alpha (t)<\overline{Y}$ for every $t\in [a,b]$ from the choice (48), the inequality for $y_\alpha '(t)$ still holds, since $(\alpha (t),y_\alpha (t))$ belongs to the region where the problem has not been modified. All the other cases can be treated similarly. $\square$

Lemma 20

Every solution $u=(x,y)$ of (51) satisfies $\varphi \le x\le \eta$.

Proof

Let us define the following regions

$$\begin{aligned} A_{NE}&= \{(t,x,y)\,:\, t\in [a,b]\,,\, x>\eta (t)\,,\, y>y_\eta (t) \}\,,\\ A_{SE}&= \{(t,x,y)\,:\, t\in [a,b]\,,\, x>\eta (t)\,,\, y<y_\eta (t) \}\,,\\ A_{SW}&= \{(t,x,y)\,:\, t\in [a,b]\,,\, x<\varphi (t)\,,\, y<y_\varphi (t) \}\,,\\ A_{NW}&= \{(t,x,y)\,:\, t\in [a,b]\,,\, x<\varphi (t)\,,\, y>y_\varphi (t) \}\,. \end{aligned}$$

As in [12] and [14], one can verify that, if $u=(x,y)$ is a solution of

$$\begin{aligned} x'=y+f_{\varphi ,\eta }(t,x,y)\,,\quad y'=x+g_{\varphi ,\eta }(t,x,y)\,, \end{aligned}$$

then, for any $t_0\in [a,b]$,

$$\begin{aligned} \begin{aligned}&(t_0,u(t_0))\in A_{SE} \Rightarrow (t,u(t)) \in A_{SE} \hbox { for every } t\in [a,t_0]\,,\\&(t_0,u(t_0))\in A_{NW} \Rightarrow (t,u(t)) \in A_{NW} \hbox { for every } t\in [a,t_0]\,,\\&(t_0,u(t_0))\in A_{NE} \Rightarrow (t,u(t)) \in A_{NE} \hbox { for every } t\in [t_0,b]\,,\\&(t_0,u(t_0))\in A_{SW} \Rightarrow (t,u(t)) \in A_{SW} \hbox { for every } t\in [t_0,b]\,. \end{aligned} \end{aligned}$$

Moreover, for any $t_0\in [a,b]$, if

$$\begin{aligned} x(t_0) < \varphi (t_0)\quad \hbox { and } \quad y(t_0)=y_\varphi (t_0)\,, \end{aligned}$$

then there exists $\delta >0$ such that

$$\begin{aligned} t_0\ne a\,,\, t\in \,]t_0-\delta ,t_0[\,&\Rightarrow (t,u(t))\in A_{NW}\,,\\ t_0\ne b\,,\, t\in \,]t_0,t_0+\delta [\,&\Rightarrow (t,u(t))\in A_{SW}\,. \end{aligned}$$

Similarly, if, for any $t_0\in [a,b]$,

$$\begin{aligned} x(t_0) > \eta (t_0)\quad \hbox { and } \quad y(t_0)=y_\eta (t_0)\,, \end{aligned}$$

then there exists $\delta >0$ such that

$$\begin{aligned} t_0\ne a\,,\, t\in \,]t_0-\delta ,t_0[\,&\Rightarrow (t,u(t))\in A_{SE}\,,\\ t_0\ne b\,,\, t\in \,]t_0,t_0+\delta [\,&\Rightarrow (t,u(t))\in A_{NE}\,. \end{aligned}$$

Indeed, by contradiction, let $u=(x,y)$ be a solution of (51) such that $x(t_0)<\varphi (t_0)$, for some $t_0\in [a,b]$. Since $x(a)=0\ge \varphi (a)$ and $x(b)=0\le \varphi (b)$, then $t_0\in \,]a,b[$ and by the above considerations it cannot be that $(t_0,u(t_0))\in A_{NW}\cup A_{SW}$. Hence, $y(t_0)=y_\varphi (t_0)$, and there exists $\delta >0$ such that $(t,u(t))\in A_{NW}$ for $t\in \,]t_0-\delta , t_0[$ , which leads to a contradiction. The case $x(t_0)>\eta (t_0)$ leads to a similar contradiction, as well. $\square$

Fig. 2

A section at a fixed time of the regions introduced in Lemmas 20 and 21, describing the dynamics of (51)

Lemma 21

Every solution $u=(x,y)$ of (51) satisfies

$$\begin{aligned} \gamma ^-(x(t))< y(t) < \gamma ^+(x(t))\,, \quad \hbox {for every } t\in [a,b]\,. \end{aligned}$$

Proof

We introduce the functions $G^\pm _i(t)=y(t)-\gamma ^\pm _i(x(t))$, with $i=1,2$. We first show that $y(a)<\gamma ^+(x(a))$ and $y(b)<\gamma ^+(x(b))$.

Let us prove that we cannot have $y(a)\ge \gamma ^+_1(x(a))$. At first, assume that $y(t)\ge \gamma ^+_1(x(t))$ for every $t\in [a,b]$. Then, from (48) we get $y(t)\ge \overline{M}$ for every $t\in [a,b]$, so that (46) gives $x(b)>0$, a contradiction. So, there exists $t_0\in [a,b[\,$ such that $G^+_1(t_0)=0$ and $G^+_1(t)<0$ in a right neighborhood of $t_0$. We can compute, recalling (47),

$$\begin{aligned}&(G^+_1)'(t_0)= y'(t_0)- (\gamma ^+_1)'(x(t_0))x'(t_0)\\&\quad = {\widehat{g}}_d \big (t_0,x(t_0),\gamma ^+_1(x(t_0))\big ) - (\gamma ^+_1)'(x(t_0)) \widehat{f}_d\big (t_0,x(t_0),\gamma ^+_1(x(t_0))\big )\\&\quad = {\widehat{g}}_d\big (t_0,x(t_0),\gamma ^+_1(x(t_0))\big ) - (\gamma ^+_1)'(x(t_0)) \widehat{f}_d\big (t_0,x(t_0),\gamma ^+_1(x(t_0))\big )>0\,, \end{aligned}$$

getting again a contradiction. Hence, $y(a)<\gamma ^+_1(x(a))\le \gamma ^+(x(a))$, recalling (49).

Similarly one shows that $y(b)<\gamma ^+_2(x(b))\le \gamma ^+(x(b))$, going backwards in time.

We now prove that $y(t)<\gamma ^+(x(t))$ for every $t\in [a,b]$. Assume by contradiction that there is $t_0\in \,]a,b[$ such that $y(t_0)\ge \gamma ^+(x(t_0))$. We distinguish two possibilities. First, if $x(t_0)\ge 0$, then the solution remains above $\gamma _1^+$ for all $t\in \,]t_0,b]$; hence, $y(t)>{\overline{M}}$ and $x'(t)>0$ for all $t\in \,]t_0,b]$, leading to $x(b)>0$, which is impossible. Second, if $x(t_0)<0$, then there must exist a $t_1\in [a,t_0[$ such that $y(t_1)\le {\overline{M}}$. But then the solution remains below $\gamma _2^+$ for all $t\in [t_1,b]$, in contradiction with the assumption.

Similarly, one proves that $y(t)>\gamma ^-(x(t))$ for every $t\in [a,b]$. $\square$

Lemma 22

If there are no solutions of (41) on $\partial {\mathcal {V}}_j$, then ${\mathcal {V}}_j$ is admissible and

$$\begin{aligned} \deg (I-L^{-1}N_d,{\mathcal {V}}_j)=1\,,\quad j=1,2,3. \end{aligned}$$

Proof

For any sufficiently large $\rho >0$, denoting by $B_\rho$ the open ball in X with radius $\rho$, centered at the origin, we claim that

$$\begin{aligned} \deg (I-L^{-1} N_{\varphi ,\eta },B_\rho )=1\,. \end{aligned}$$

Indeed, let us show that there is a $\rho >0$ such that, for every $\lambda \in [0,1]$, every solution of $Lu=\lambda N_{\varphi ,\eta }u$ satisfies $\Vert u\Vert _{C^1}<\rho$. By contradiction, if this is not true, there exist a sequence $(\lambda _n)_n$ in [0, 1] and some solutions $u_n=(x_n,y_n)$ of $Lu_n=\lambda _n N_{\varphi ,\eta }u_n$ such that $\Vert u_n\Vert _{C^1}\rightarrow \infty$. Let $v_n=x_n/\Vert u_n\Vert _{C^1}$ and $w_n=y_n/\Vert u_n\Vert _{C^1}$. By a standard argument it can be seen that, for a subsequence, $\lambda _n\rightarrow {\bar{\lambda }}\in [0,1]$, while $(v_n,w_n)\rightarrow ({\bar{v}},{\bar{w}})$ in X. Moreover, ${\bar{v}}'={\bar{w}}$, ${\bar{w}}'={\bar{\lambda }}{\bar{v}}$, so that ${\bar{v}}''={\bar{\lambda }}{\bar{v}}$, and since ${\bar{v}}(a)=0={\bar{v}}(b)$, this implies ${\bar{v}}\equiv 0$, hence also ${\bar{w}}\equiv 0$, a contradiction. By homotopy invariance, the degree is then equal to 1.

Fix $\rho >0$ as above. Since there are no solutions of (41) on $\partial {\mathcal {V}}(\varphi ,\eta )$, we also have that there are no solutions of (51) on $\partial {\mathcal {V}}(\varphi ,\eta )$. Hence, ${\mathcal {V}}(\varphi ,\eta )$ is admissible and, by excision,

$$\begin{aligned} \deg (I-L^{-1} N_{\varphi ,\eta },{\mathcal {V}}(\varphi ,\eta ))=\deg (I-L^{-1} N_{\varphi ,\eta },B_\rho )=1\,. \end{aligned}$$

Now, since $N_{\varphi ,\eta }=N_d$ on ${\mathcal {V}}(\varphi ,\eta )$, the result follows. $\square$

Lemma 23

There are no solutions of (41) on $\partial {\mathcal {V}}_1$.

Proof

We recall that in Lemma 16 we provided the definition of the functions ${\widehat{\alpha }}=x_{-\xi }$ and ${\widehat{\beta }}=x_\xi$ with the choice $\xi > 2d\overline{S}$. Let $u=(x,y)$ be a solution belonging to the closure of ${\mathcal {V}}_1$. Then ${\widehat{\alpha }}(t)\le x(t)\le {\widehat{\beta }}(t)$, for every $t\in {\mathcal {I}}$. Assume by contradiction that $x\not \ll {\widehat{\beta }}$. Since $x(a)=0<\xi ={\widehat{\beta }}(a)$ and $x(b)=0<\xi ={\widehat{\beta }}(b)$, there must be a $t_0\in \,]a,b[$ such that

$$\begin{aligned} x(t_0)={\widehat{\beta }}(t_0)>\xi \,,\quad x'(t_0)={\widehat{\beta }}'(t_0)\,. \end{aligned}$$

Hence, as $\xi > 2d\overline{S}$, both (x(t), y(t)) and $(x_\xi (t),y_\xi (t))$ solve the autonomous system (45) in a neighborhood of $t_0$. Since

$$\begin{aligned} x'(t_0)=\frac{d-1}{d}\frac{\partial H}{\partial y}(x(t_0),y(t_0))\,, \end{aligned}$$

and

$$\begin{aligned} x_\xi '(t_0)=\frac{d-1}{d}\frac{\partial H}{\partial y}(x_\xi (t_0),y_\xi (t_0))\,, \end{aligned}$$

being $x'(t_0)=x_\xi '(t_0)$ and $x(t_0)=x_\xi (t_0)$, by assumption (9) it has to be that $y(t_0)=y_\xi (t_0)$. Since autonomous planar Hamiltonian systems have the uniqueness property for Cauchy problems when the initial value is not an equilibrium (cf. [19, Theorem 1]), then the two solutions (x(t), y(t)) and $(x_\xi (t),y_\xi (t))$ coincide, as long as they remain in $[\xi ,+\infty [\,\times {\mathbb {R}}$, leading to a contradiction. Hence, $x\ll {\widehat{\beta }}$. Similarly, one proves that ${\widehat{\alpha }}\ll x$. So, there are no solutions of (41) on $\partial {\mathcal {V}}_1$. $\square$

Now, if there is a solution $u=(x,y)$ of (41) on $\partial {\mathcal {V}}_2$, then $x\le \beta$ and $x\not \ll \beta$. Since $\alpha \not \le \beta$, there is a $t_0$ such that $x(t_0)\le \beta (t_0)<\alpha (t_0)$, implying that $\alpha \not \ll x$. So, u is the solution of (28) we are looking for.

A similar argument shows that if $u=(x,y)$ is a solution of (41) on $\partial {\mathcal {V}}_3$, then u is the solution of (28) we are looking for.

Finally, if there are no solutions of (41) on $\partial {\mathcal {V}}_2\cup \partial {\mathcal {V}}_3$, then

$$\begin{aligned}&\deg (I-L^{-1}N_d,{\mathcal {V}}_1\setminus \overline{{\mathcal {V}}_2\cup {\mathcal {V}}_3})=\\&=\deg (I\!-\!L^{-1}N_d,{\mathcal {V}}_1)\!-\!\Big (\deg (I\!-\!L^{-1}N_d,{\mathcal {V}}_2)+\deg (I\!-\!L^{-1}N_d,{\mathcal {V}}_3)\Big )=-1\,. \end{aligned}$$

Then, there is a solution of (41) in ${\mathcal {V}}_1\setminus \overline{{\mathcal {V}}_2\cup {\mathcal {V}}_3}$, and this is the solution of (28) we are looking for.

The proof of Theorem 3 is thus completed.

7 Higher dimensional systems

Let us now introduce a higher dimensional version of problem (2). We will write $x=(x_1,\dots ,x_N)\in {\mathbb {R}}^N$, $y=(y_1,\dots ,y_N)\in {\mathbb {R}}^N$, and assume that the continuously differentiable function $H:{\mathbb {R}}^{2N}\rightarrow {\mathbb {R}}$ is of the type

$$\begin{aligned} H(x,y)=\sum _{n=1}^N H_n(x_n,y_n)\,. \end{aligned}$$

(52)

Moreover, for every $n\in \{1,\dots ,N\}$, we assume that there exist $p_n>1$ and $q_n>1$, with $(1/p_n)+(1/q_n)=1$, such that

$$\begin{aligned} H_n(\lambda ^{q_n} u, \lambda ^{p_n} v)= \lambda ^{p_n+q_n} H_n(u,v)>0\,, \end{aligned}$$

for every $\lambda >0$ and $(u,v)\ne (0,0)$. We consider the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} x'=\displaystyle \frac{\partial H}{\partial y}(x,y) + \phi (t,x,y)\,,\quad y'=\displaystyle -\frac{\partial H}{\partial x}(x,y) + \psi (t,x,y)\,,\\ x(a)=0=x(b)\,, \end{array}\right. } \end{aligned}$$

(53)

where the functions $\phi ,\psi :[a,b]\times {\mathbb {R}}^{2N}\rightarrow {\mathbb {R}}^N$ are continuous and bounded. Equivalently, writing $\phi =(\phi _1,\dots , \phi _N)$ and $\psi =(\psi _1,\dots , \psi _N)$,

$$\begin{aligned} {\left\{ \begin{array}{ll} x_n'=\displaystyle \frac{\partial H_n}{\partial y_n}(x_n,y_n) + \phi _n(t,x_1,\dots ,x_N,y_1,\dots ,y_N)\,,\\ y_n'=\displaystyle -\frac{\partial H_n}{\partial x_n}(x_n,y_n) + \psi _n(t,x_1,\dots ,x_N,y_1,\dots ,y_N)\,,\\ x_n(a)=0=x_n(b)\,,\quad n=1,\dots ,N\,. \end{array}\right. } \end{aligned}$$

For $\xi ,\upsilon \in {\mathbb {R}}^N$ we write $\xi \preceq \upsilon$ (or $\upsilon \succeq \xi$) if

$$\begin{aligned} \xi _n\le \upsilon _n \text{ for } \text{ every } n\in \{1,\dots ,N\}\,, \end{aligned}$$

and in this case we define

$$\begin{aligned} {\langle \!\langle }\xi ,\upsilon {\rangle \!\rangle }= \{u\in {\mathbb {R}}^N\,:\, \xi \preceq u \preceq \upsilon \}\,. \end{aligned}$$

We now adapt the definition of lower/upper solutions given in [14, Definition 31] to the higher dimensional problem

$$\begin{aligned} {\left\{ \begin{array}{ll} x'=f(t,x,y)\,,\qquad y'=g(t,x,y)\,,\\ x(a)=0=x(b)\,, \end{array}\right. } \end{aligned}$$

(54)

where $f,g:[a,b]\times {\mathbb {R}}^{2N}\rightarrow {\mathbb {R}}^N$ are continuous functions. As usual, we write $f=(f_1,\dots ,f_N)$ and $g=(g_1,\dots , g_N)$. Similarly for the vector-valued functions considered below.

Definition 24

Given two $C^1$-functions $\alpha ,\beta :[a,b]\rightarrow {\mathbb {R}}^N$, we say that $(\alpha ,\beta )$ is a well-ordered pair of lower/upper solutions of problem (54) if

$$\begin{aligned}&\alpha (t) \preceq \beta (t)\,,\quad \hbox { for every }t\in [a,b]\,,\\&\alpha (a)\preceq 0 \preceq \beta (a)\,,\qquad \alpha (b)\preceq 0 \preceq \beta (b)\,, \end{aligned}$$

and there exist two $C^1$-functions $y^\alpha ,y^\beta :[a,b]\rightarrow {\mathbb {R}}^N$ such that, for every $t\in [a,b]$, $x,y\in {\mathbb {R}}^N$ with $x\in {\langle \!\langle }\alpha (t) \,,\, \beta (t) {\rangle \!\rangle }$, and $n\in \{1,\dots ,N\}$, one has

$$\begin{aligned} {\left\{ \begin{array}{ll} f_n(t,x,y)<\alpha _n'(t) \quad \text{ when } x_n= \alpha _n(t) \text{ and } y_n<y_n^\alpha (t)\,, \\ f_n(t,x,y)>\alpha _n'(t) \quad \text{ when } x_n= \alpha _n(t) \text{ and } y_n>y_n^\alpha (t)\,, \end{array}\right. } \end{aligned}$$

(55)

$$\begin{aligned} {\left\{ \begin{array}{ll} f_n(t,x,y)<\beta _n'(t) \quad \text{ when } x_n= \beta _n(t) \text{ and } y_n<y_n^\beta (t)\,, \\ f_n(t,x,y)>\beta _n'(t) \quad \text{ when } x_n= \beta _n(t) \text{ and } y_n>y_n^\beta (t)\,, \end{array}\right. } \end{aligned}$$

(56)

and

$$\begin{aligned} (y^\alpha _n)'(t)\ge g_n(t,x,y) \quad \text{ when } x_n=\alpha _n(t) \text{ and } y_n= y_n^\alpha (t)\,, \end{aligned}$$

(57)

$$\begin{aligned} (y^\beta _n)'(t)\le g_n(t,x,y) \quad \text{ when } x_n=\beta _n(t) \text{ and } y_n= y_n^\beta (t)\,. \end{aligned}$$

(58)

Here is our result in the well-ordered case.

Theorem 25

Assume H to be as in (52), with components $H_n$ being positively-$(p_n,q_n)$-homogeneous positive-definite continuously differentiable functions, for some $p_n>1$ and $q_n>1$ with $(1/p_n)+(1/q_n)=1$. Let $\phi ,\psi$ be uniformly bounded continuous functions, and let $(\alpha ,\beta )$ be a well-ordered pair of lower/upper solutions of problem (53). Then, there exists a solution (x, y) of (53) such that $\alpha (t)\preceq x(t) \preceq \beta (t)$, for every $t\in [a,b]$.

Proof

One proceeds like in the proof of [14, Theorem 32]. The main difference here is that the functions $\phi$ and $\psi$ depend on all variables x and y. However, setting

$$\begin{aligned}&m_\alpha = \min \{\alpha _n(t):t\in [a,b],\,n=1,\dots ,N\}\,,\\&M_\beta = \max \{\beta _n(t):t\in [a,b],\,n=1,\dots ,N\}\,, \end{aligned}$$

the fact that $\phi$ and $\psi$ are bounded permits to recover, for every $n=1,\dots ,N$, the required guiding curves $\gamma _{1,n}^\pm ,\gamma _{2,n}^\pm :[m_\alpha ,M_\beta ]\rightarrow {\mathbb {R}}$, with $i=1,2$, and the result follows. $\square$

For the non-well-ordered case, we need to introduce the notion of strict lower and upper solutions. To this aim we will follow the ideas developed in [11], and distinguish the components which are well ordered from the others.

The couple $({\mathcal {J}},{\mathcal {K}})$ is a partition of $\{1,\dots ,N\}$ if and only if ${\mathcal {J}} \cap {\mathcal {K}} = \varnothing$ and ${\mathcal {J}} \cup {\mathcal {K}} = \{1,\dots ,N\}$. We denote by $\#{\mathcal {J}}$ and $\#{\mathcal {K}}$ the cardinality of the sets ${\mathcal {J}}$ and ${\mathcal {K}}$. For a given partition $(\mathcal J,{\mathcal {K}})$, a vector

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

can be decomposed as $x=(x_{{\mathcal {J}}}, x_{{\mathcal {K}}})$ where $x_{{\mathcal {J}}}=(x_j)_{j\in {\mathcal {J}}}\in {\mathbb {R}}^{\#{\mathcal {J}}}$ and $x_{{\mathcal {K}}}=(x_k)_{k\in {\mathcal {K}}}\in {\mathbb {R}}^{\#{\mathcal {K}}}$. Similarly, we can decompose every function ${\mathcal {F}}: {\mathcal {D}} \rightarrow {\mathbb {R}}^N$ as ${\mathcal {F}}(x)=\big ({\mathcal {F}}_{{\mathcal {J}}}(x),\mathcal F_{{\mathcal {K}}}(x)\big )$ with ${\mathcal {F}}_{{\mathcal {J}}}: {\mathcal {D}} \rightarrow {\mathbb {R}}^{\#{\mathcal {J}}}$ and ${\mathcal {F}}_{{\mathcal {K}}}: {\mathcal {D}} \rightarrow {\mathbb {R}}^{\#{\mathcal {K}}}$, for any domain ${\mathcal {D}}$. Moreover, for $\xi ,\upsilon \in {\mathbb {R}}^N$ we write

$$\begin{aligned} {\langle \!\langle }\xi , \upsilon {\rangle \!\rangle }_{{\mathcal {J}}} = \{ u=(u_{\mathcal J},u_{{\mathcal {K}}})\in {\mathbb {R}}^N \,:\, \xi _{{\mathcal {J}}} \preceq u_{\mathcal J} \preceq \upsilon _{{\mathcal {J}}} \}\,. \end{aligned}$$

Definition 26

Let $\alpha ,\beta :[a,b]\rightarrow {\mathbb {R}}^N$ be two $C^1$-functions. We will say that $(\alpha ,\beta )$ is a pair of lower/upper solutions of (54) related to the partition $(\mathcal J,{\mathcal {K}})$ of $\{1,\dots ,N\}$ if the following conditions hold:

• 1. $\alpha _j \le \beta _j$, for any $j\in {\mathcal {J}}\,;$

• 2. $\alpha _k \not \le \beta _k$, for any $k\in {\mathcal {K}}\,;$

• 3. $\alpha (a)\preceq 0 \preceq \beta (a)$ and $\alpha (b)\preceq 0 \preceq \beta (b)\,;$

• 4. there are two $C^1$-functions $y^\alpha ,y^\beta :[a,b]\rightarrow {\mathbb {R}}^N$ such that (55), (56), (57), and (58) hold for every $t\in [a,b]$, $x,y\in {\mathbb {R}}^N$ with $x\in {\langle \!\langle }\alpha (t) \,,\, \beta (t) {\rangle \!\rangle }_{{\mathcal {J}}}$, and $n\in \{1,\dots ,N\}.$

In the following definition, we will use the relation $\gg$ introduced in (7).

Definition 27

Let $(\alpha ,\beta )$ be a pair of lower/upper solutions of (54) related to the partition $({\mathcal {J}}, {\mathcal {K}})$. We will say that this pair of lower/upper solutions is strict with respect to the jth component, with $j\in {\mathcal {J}}$, if $\alpha _j \ll \beta _j$ and, for every solution (x, y) of (54),

$$\begin{aligned}&\alpha _j \le x_j \le \beta _j \quad \Rightarrow \quad \alpha _j \ll x_j \ll \beta _j\,. \end{aligned}$$

We will say that this pair of lower/upper solutions is strict with respect to the kth component, with $k\in {\mathcal {K}}$ if, for every solution (x, y) of (54),

$$\begin{aligned} \alpha _k \le x_k \;\Rightarrow \; \alpha _k \ll x_k\,, \quad \hbox {and }\quad x_k \le \beta _k \;\Rightarrow \; x_k \ll \beta _k \,. \end{aligned}$$

To prove the existence of a solution of (53), once a pair of lower/upper solutions $(\alpha ,\beta )$ is given, we need to ask the strictness property with respect to the non-well-ordered components $\alpha _k,\beta _k$. Moreover, we will need to ask more regularity on the functions $\phi$ and $\psi$: we will ask them to be locally $\ell$-Hölder continuous for a certain $\ell >0$. Here is our result.

Theorem 28

Let $\ell >0$ and assume $H:{\mathbb {R}}^{2N}\rightarrow {\mathbb {R}}$ to be as in (52), with components $H_n$ being $C^{1,\ell }_{loc}$-smooth positively-$(p_n,q_n)$-homogeneous positive-definite functions, for some $p_n>1$ and $q_n>1$ with $(1/p_n)+(1/q_n)=1$. Let $(\alpha ,\beta )$ be a pair of lower/upper solutions of (53) related to the partition $(\mathcal J,{\mathcal {K}})$ of $\{1,\dots ,N\}$ which is strict with respect to the kth component, for every $k\in {\mathcal {K}}$. Moreover, assume that for every $k\in {\mathcal {K}}$,

$$\begin{aligned} \frac{\partial H_k}{\partial y}(x_0,\cdot ) \text{ is } \text{ a } \text{ strictly } \text{ increasing } \text{ function, } \text{ for } \text{ every } \ x_0\in {\mathbb {R}}. \end{aligned}$$

(59)

Let $\phi ,\psi$ be uniformly bounded $C^{0,\ell }_{loc}$-smooth functions. If

$$\begin{aligned} b-a \le \min \{ \tau ^+_k, \tau ^-_k \, :\, k\in {\mathcal {K}}\}\,, \end{aligned}$$

then (53) has a solution (x, y) with the following properties:

(J):

for any $j\in {\mathcal {J}}$, $\alpha _j(t)\le x_j(t)\le \beta _j(t)$, for every $t \in [a,b]\,;$

(K):

for any $k\in {\mathcal {K}}$, there exist $t_k^1,t_k^2\in \,]a,b[\,$ such that $x_k(t_k^1)<\alpha _k(t_k^1)$ and $x_k(t_k^2)>\beta _k(t_k^2)$.

Proof

The case ${\mathcal {K}}=\varnothing$ has been already treated in Theorem 25. To simplify the exposition, we assume both ${\mathcal {J}}\ne \varnothing$ and $\mathcal K\ne \varnothing$, and that ${\mathcal {J}} = \{1,\dots ,M\}$ and ${\mathcal {K}}=\{M+1,\dots N\}$ for a certain $M\in \{1,\dots ,N-1\}$. The proof can be easily adapted to the case $\mathcal J=\varnothing$.

As in Section 2, for every $n\in \{1,\dots ,N\}$, let $S_n(t)=(S_{1,n}(t),S_{2,n}(t))$ be the periodic solution of the planar autonomous system

$$\begin{aligned} u'=\displaystyle \frac{\partial H_n}{\partial v}(u,v)\,,\quad v'=\displaystyle -\frac{\partial H_n}{\partial u}(u,v)\,, \end{aligned}$$

such that $S_{1,n}(0)=0$ and $S_{2,n}(0)>0$, with $H_n(S_{1,n}(t),S_{2,n}(t))=1$ for every t. We set

$$\begin{aligned} \overline{S}_n = \max \{ |S_{1,n}(t)|+|S_{2,n}(t)|:t\in {\mathbb {R}}\}\,, \end{aligned}$$

and

$$\begin{aligned} \overline{S} = \max \big \{\overline{S}_n:n\in \{1\dots ,N\}\big \}\,. \end{aligned}$$

(60)

For each $n\in \{1,\dots ,N\}$, we will mainly follow the procedure developed in Section 5 if $n\in {\mathcal {J}}$, and the one in Section 6 if $n\in {\mathcal {K}}$, so that the couple of variables $(x_n,y_n)$ will overshadow the remaining ones, which will essentially act as parameters. We first need to modify problem (53) both in the k-variables, following the lines of the proof of Theorem 3, and in the j-variables, to apply a topological degree argument, as in the proof of [11, Theorem 10].

Let us rewrite (53) as

$$\begin{aligned} {\left\{ \begin{array}{ll} x_j'=f_j(t,x,y)\,, \qquad y_j'=g_j(t,x,y)\,, \qquad j\in \{1,\dots ,M\}\,,\\ x_k'=f_k(t,x,y)\,, \qquad y_k'=g_k(t,x,y)\,, \qquad k\in \{M+1,\dots ,N\}\,,\\ x(a)=0=x(b)\,. \end{array}\right. } \end{aligned}$$

We introduce the functions

$$\begin{aligned} \zeta _j(t,x_j)= \zeta (x_j,\alpha _j(t),\beta _j(t))\,, \end{aligned}$$

where $\zeta$ was defined in (50), and

$$\begin{aligned} \Gamma (t,x)=\big ( \zeta _1(t,x_1)\,,\dots \,, \zeta _M(t,x_M)\,,x_{M+1}\,,\dots \,, x_N\big )\,. \end{aligned}$$

Then we set

$$\begin{aligned} {\widehat{f}}_j(t,x,y)= f_j(t,\Gamma (t,x),y)\,,\qquad \widehat{g}_j(t,x,y)= g_j(t,\Gamma (t,x),y)+ x_j - \zeta _j(t,x_j)\,. \end{aligned}$$

Concerning the non-well-ordered components, we need to consider a positive parameter d, which will be fixed later, following the construction of problem (41) in Section 6. We set

$$\begin{aligned} {\widehat{f}}_{k,d}(t,x,y)&=\frac{d-1}{d} \,\frac{\partial H_k}{\partial y_k}(x_k,y_k) + {\widetilde{\phi }}_{k,d}(t,x,y)\,,\\ {\widehat{g}}_{k,d}(t,x,y)&=- \frac{d-1}{d} \,\frac{\partial H_k}{\partial x_k}(x_k,y_k) + {\widetilde{\psi }}_{k,d}(t,x,y)\,, \end{aligned}$$

with

$$\begin{aligned} {\widetilde{\phi }}_{k,d}(t,x,y)&= \chi \Big (\frac{x_k}{d\overline{S}}\Big )\chi \Big (\frac{y_k}{d^{p_k/q_k}{\overline{S}}}\Big )\left( \frac{1}{d} \frac{\partial H_k}{\partial y_k}(x_k,y_k)+ \phi _k(t,\Gamma (t,x),y) \right) ,\\ {\widetilde{\psi }}_{k,d}(t,x,y)&= \chi \Big (\frac{x_k}{d\overline{S}}\Big )\chi \Big (\frac{y_k}{d^{p_k/q_k}{\overline{S}}}\Big )\left( - \frac{1}{d} \frac{\partial H_k}{\partial x_k}(x_k,y_k) + \psi _k(t,\Gamma (t,x),y) \right) , \end{aligned}$$

where $\overline{S}$ is defined in (60) and the cut-off function $\chi$ in (40). We thus are led to the modified problem

$$\begin{aligned} {\left\{ \begin{array}{ll} x_j'={\widehat{f}}_j(t,x,y)\,, \quad y_j'={\widehat{g}}_j(t,x,y)\,, \quad j\in \{1,\dots ,M\}\,,\\ x_k'={\widehat{f}}_{k,d}(t,x,y)\,, \quad y_k'={\widehat{g}}_{k,d}(t,x,y)\,, \quad k\in \{M+1,\dots ,N\}\,,\\ x(a)=0=x(b)\,. \end{array}\right. } \end{aligned}$$

(61)

We will now provide, working separately on every component, an indexed family of well-ordered pairs of lower/upper solutions of the modified problem (61), which will be strict in every component.

Let us first operate on a component $j\in \{1,\dots ,M\}$. We can argue as in Lemmas 8 and 9 so to find some guiding curves $\gamma ^\pm _{1,j}$ and $\gamma ^\pm _{2,j}$. To be more precise, for an illustrative purpose, the curve $\gamma ^+_{1,j}$ will satisfy the analogue of (29), i.e.,

$$\begin{aligned} {\widehat{g}}_j(t,x,y) > {\widehat{f}}_j(t,x,y) (\gamma ^+_{1,j})'(x_j)\,, \end{aligned}$$

for every $t\in [a,b]$, every $x\in {\mathbb {R}}^n$ with $\alpha _j(t)\le x_j \le \beta _j(t)$ and every $y\in {\mathbb {R}}^N$ with $y_j= \gamma ^+_{1,j}(x_j)$.

Then, we can follow the reasoning in [14, Theorem 11, Claim 3] and prove that all the solutions (x, y) of (61) satisfy

$$\begin{aligned} \alpha _j(t) \le x_j(t) \le \beta _j(t)\,, \end{aligned}$$

(62)

and

$$\begin{aligned} \min \{\gamma ^-_{1,j}(x(t)) \,,\, \gamma ^-_{2,j}(x(t)) \}< y_j(t) < \max \{\gamma ^+_{1,j}(x(t)) \,,\, \gamma ^+_{2,j}(x(t)) \}\,, \end{aligned}$$

(63)

for every $t\in [a,b]$. We then define the functions

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

and conclude that

(S1):

for every $j\in {\mathcal {J}}$, the functions $\check{\alpha }_j$ and $\check{\beta }_j$ satisfy

$$\begin{aligned} \check{\alpha }_j\ll \check{\beta }_j\,, \quad \check{\alpha }_j(a)<0< \check{\beta }_j(a)\,, \quad \check{\alpha }_j(b)<0< \check{\beta }_j(b)\,. \end{aligned}$$

Moreover, the conditions (55), (56), (57) and (58) hold with $n=j$, replacing $\alpha _j,\beta _j,f_j,g_j$ by $\check{\alpha }_j,\check{\beta }_j,\widehat{f}_j,{\widehat{g}}_j\,$, setting $y^{\check{\alpha }}_j=y^\alpha _j$ and $y^{\check{\beta }}_j=y^\beta _j$. Finally, every solution (x, y) of (61) satisfies $\check{\alpha }_j\ll x_j\ll \check{\beta }_j$ and (63).

Let us now focus our attention on a component $k\in {\mathcal {K}}$. Arguing as in Lemma 13 we can prove that

(S2):

there exists $D>1$ with the following property: if $u=(x,y)$ is a solution of (61), with $d\ge D$, such that and , then ${\mathcal {N}}_{p_k}(u_k) < D$.

We can surely take the same constant D for every $k\in {\mathcal {K}}$. Now, as in Lemma 14, enlarging D if necessary we can prove that

(S3):

for every $k\in {\mathcal {K}}$ and $d\ge D$, the functions $\alpha _k$ and $\beta _k$ still satisfy conditions (55), (56), (57) and (58), replacing $f_k,g_k$ by $\widehat{f}_{k,d},{\widehat{g}}_{k,d}$, respectively.

Then, Lemma 16 suggests us how to continue the proof once we have fixed $d>D$ sufficiently large:

(S4):

for every $k\in {\mathcal {K}}$, there are two couples of functions $({\widehat{\alpha }}_k, y^{{\widehat{\alpha }}}_k)$ and $({\widehat{\beta }}_k, y^{{\widehat{\beta }}}_k)$ such that $\widehat{\alpha }_k\ll {\widehat{\beta }}_k$, ${\widehat{\alpha }}_k \ll \beta _k$, and $\alpha _k \ll {\widehat{\beta }}_k$ satisfying the conditions (55), (56), (57) and (58), replacing in all formulas $\alpha _k$, $y^\alpha _k$, $\beta _k$, $y^\beta _k$, $f_k$, $g_k$ by ${\widehat{\alpha }}_k$, $y^{{\widehat{\alpha }}}_k$, ${\widehat{\beta }}_k$, $y^{{\widehat{\beta }}}_k$, ${\widehat{f}}_{k,d}$, ${\widehat{g}}_{k,d}\,$, respectively.

Now a further step is needed. We are going to prove that

(S5):

for every $k\in {\mathcal {K}}$, if (x, y) is a solution of (61) such that ${\widehat{\alpha }}_k\le x_k$, then ${\widehat{\alpha }}_k\ll x_k$. Analogously, if (x, y) is a solution of (61) such that $x_k\le {\widehat{\beta }}_k$, then $x_k\ll {\widehat{\beta }}_k$.

We prove the second assertion, the proof of the first one being similar. Following the proof of Lemma 23, assume by contradiction that there is a solution (x, y) of (61) such that $x_k\le {\widehat{\beta }}_k$ but . Since $x_k(a)=0<{\widehat{\beta }}_k(a)$ and $x_k(b)=0<{\widehat{\beta }}_k(b)$, there exists $t_0\in \,]a,b[\,$ such that $x_k(t_0)={\widehat{\beta }}_k(t_0)$ and $x_k'(t_0)={\widehat{\beta }}_k'(t_0)$. Using assumption (59), since

$$\begin{aligned} \frac{d-1}{d} \,\frac{\partial H_k}{\partial y_k}(x_k(t_0),y^{{\widehat{\beta }}}_k(t_0)) ={\widehat{\beta }}_k'(t_0)= x_k'(t_0)= \frac{d-1}{d} \,\frac{\partial H_k}{\partial y_k}(x_k(t_0),y_k(t_0))\,, \end{aligned}$$

we get $y^{{\widehat{\beta }}}_k(t_0)=y_k(t_0)$. Recalling that ${\widetilde{\phi }}_{k,d}=0$ and ${\widetilde{\psi }}_{k,d}=0$ in a neighborhood of $\{({\widehat{\beta }}_k, y^{{\widehat{\beta }}}_k)(t)\,:\, t\in [a,b]\}$, arguing as in the proof of Lemma 23, we conclude that $({\widehat{\beta }}_k, y^{{\widehat{\beta }}}_k)$ and $(x_k,y_k)$ coincide on [a, b], leading to a contradiction with the fact that $x_k(a)=x_k(b)=0$. Hence, (S5) is proved.

For every multi-index $\mu =(\mu _{M+1},\dots ,\mu _N)\in \{1,2,3\}^{N-M}$, we define the couple of functions $(\varphi ^\mu ,\eta ^\mu )$ by components: for every $j\in {\mathcal {J}}$, we set

$$\begin{aligned} (\varphi _j^\mu ,\eta _j^\mu )=(\check{\alpha }_j,\check{\beta }_j)\,, \end{aligned}$$

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

$$\begin{aligned} \text {if } \mu _k=1 \text { we set }&(\varphi _k^\mu ,\eta _k^\mu )= ({\widehat{\alpha }}_k \,, {\widehat{\beta }}_k)\,,\\ \text {if } \mu _k=2 \text { we set }&(\varphi _k^\mu ,\eta _k^\mu )= ({\widehat{\alpha }}_k \,, \beta _k)\,,\\ \text {if } \mu _k=3 \text { we set }&(\varphi _k^\mu ,\eta _k^\mu )= ( \alpha _k \,, {\widehat{\beta }}_k)\,. \end{aligned}$$

From (S1), (S3), (S4) and (S5), we can verify that, for every $\mu \in \{1,2,3\}^{N-M}$, the couple $(\varphi ^\mu ,\eta ^\mu )$ is a well-ordered pair of lower/upper solutions of problem (61) which is strict with respect to all its components. Let

$$\begin{aligned} \Xi =\left\{ (\varphi ^\mu ,\eta ^\mu ) \,:\, \mu \in \{1,2,3\}^{N-M} \right\} \,. \end{aligned}$$

As in Lemmas 8, 9 and 17, we can construct some guiding curves $\gamma ^\pm _{1,n}$, $\gamma ^\pm _{2,n}$, for every $n\in \{1,\dots ,N\}$. Then, for every couple $(\varphi ^\mu ,\eta ^\mu )\in \Xi$, we can modify system (61), only in the components $k\in \mathcal K$, exactly as we did to define problem (51), and obtain the new problem

$$\begin{aligned} {\left\{ \begin{array}{ll} x_j'={\widehat{f}}_j(t,x,y)\,, \quad y_j'={\widehat{g}}_j(t,x,y)\,, \quad j\in {\mathcal {J}}\,,\\ x_k'= y_k + (f_{k})_{\varphi ^\mu ,\eta ^\mu }(t,x,y)\,, \quad y_k'=x_k + (g_{k})_{\varphi ^\mu ,\eta ^\mu }(t,x,y)\,, \quad k\in {\mathcal {K}}\,,\\ x(a)=0=x(b)\,. \end{array}\right. } \end{aligned}$$

(64)

With the same procedure we can show that every couple $(\varphi ^\mu ,\eta ^\mu )$ is a well-ordered pair of lower/upper solutions for the new problem, too. Moreover, as in Lemmas 20 and 21, we can show that each solution of (64) satisfies, for every $t\in [a,b]$,

$$\begin{aligned} \varphi ^\mu (t) \preceq x(t) \preceq \eta ^\mu (t)\,, \end{aligned}$$

and, for every $n\in \{1,\dots ,N\}$,

$$\begin{aligned} \min \{\gamma ^-_{1,n}(x(t)) \,,\, \gamma ^-_{2,n}(x(t)) \}< y_n(t) < \max \{\gamma ^+_{1,n}(x(t)) \,,\, \gamma ^+_{2,n}(x(t)) \}\,. \end{aligned}$$

We need now to introduce the functional setting. We denote by ${\widehat{f}}_d$ and ${\widehat{g}}_d$ the functions

$$\begin{aligned} {\widehat{f}}_d = ( {\widehat{f}}_1, \dots , {\widehat{f}}_M, \widehat{f}_{M+1,d}, \dots , {\widehat{f}}_{N,d} )\,, \quad {\widehat{g}}_d = ( {\widehat{g}}_1, \dots , {\widehat{g}}_M, {\widehat{g}}_{M+1,d}, \dots , {\widehat{g}}_{N,d} )\,, \end{aligned}$$

which describe system (61). Recalling the notations introduced in Section 6, we consider the linear operator

$$\begin{aligned}&L: [C_0^{1,\ell }({\mathcal {I}})]^N \times [C^{1,\ell }({\mathcal {I}})]^{N} \rightarrow [C^{0,\ell }({\mathcal {I}})]^N \times [C^{0,\ell }({\mathcal {I}})]^{N}\\&L(x,y)=(x'-y\,,\, y')\,, \end{aligned}$$

and the Nemitskii operator

$$\begin{aligned}&N_d:[C^1_0({\mathcal {I}})]^{N}\times [C^1({\mathcal {I}})]^{N} \rightarrow [C^{0,\ell }({\mathcal {I}})]^N \times [C^{0,\ell }({\mathcal {I}})]^{N}\\&N_d(x,y)(t)= \left( {\widehat{f}} (t,x(t),y(t))-y(t) \,,\, {\widehat{g}} (t,x(t),y(t)) \right) . \end{aligned}$$

Then, the analogue of Lemma 18 holds and $u=(x,y)$ is a solution of (61) if and only if it solves

$$\begin{aligned} Lu=N_du\,, \end{aligned}$$

with

$$\begin{aligned} u\in X:=[C^1_0({\mathcal {I}})]^{N}\times [C^1({\mathcal {I}})]^{N}. \end{aligned}$$

For every $(\varphi ^\mu ,\eta ^\mu )\in \Xi$, we introduce the set

$$\begin{aligned}&{\mathcal {V}}^\mu =\big \{(x,y)\in X \,: \varphi ^\mu _n\ll x_n\ll \eta ^\mu _n\, \hbox { and }\\&\qquad \gamma ^-_n(x_n(t))<y_n(t)<\gamma ^+_n(x_n(t))\,,\hbox { for every }t\in [a,b]\,, n\in \{1,\dots ,N\}\big \}\,, \end{aligned}$$

where

$$\begin{aligned} \gamma ^-_n(x)=\min \{\gamma _{i,n}^-(x)\,,\, i=1,2\} \,,\qquad \gamma ^+_n(x)=\max \{\gamma _{i,n}^+(x)\,,\, i=1,2\}\,. \end{aligned}$$

Notice that ${\mathcal {V}}^\mu$ is open in X.

Fix any $(\varphi ^\mu ,\eta ^\mu )\in \Xi$. From (S1) and (S5), we see that there are no solutions of (61) on $\partial {\mathcal {V}}^\mu$. Then, arguing as in the proof of Lemma 22, the set ${\mathcal {V}}^\mu$ is admissible and

$$\begin{aligned} \deg \big (I-L^{-1}N_d,{\mathcal {V}}^\mu \big )=1\,. \end{aligned}$$

Then, we can follow the procedure in [11, Section 3.1] and define, for every multi-index $\mu =(\mu _{M+1},\dots ,\mu _N)\in \{1,2,3,4\}^{N-M}$, the open set

$$\begin{aligned}&\Omega ^\mu :=\big \{(x,y)\in X :\, ({\mathcal {A}}_j^0), ({\mathcal {A}}_k^{\mu _k}), ({\mathcal {A}}_j^\gamma ) \ \text {and} \ ({\mathcal {A}}_k^\gamma ) \\&\text { hold for every} \ j\in {\mathcal {J}} \ \text {and} \ k\in {\mathcal {K}} \big \}\,, \end{aligned}$$

where the conditions $({\mathcal {A}}_j^0)$, $({\mathcal {A}}_k^{\mu _k})$, and $({\mathcal {A}}_n^\gamma )$ read as

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

$\check{\alpha }_j \ll x_j \ll \check{\beta }_j\,;$

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

${\widehat{\alpha }}_k \ll x_k \ll {\widehat{\beta }}_k\,;$

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

${\widehat{\alpha }}_k \ll x_k\ll \beta _k\,;$

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

$\alpha _k \ll x_k \ll {\widehat{\beta }}_k\,;$

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

${\widehat{\alpha }}_k \ll x_k \ll {\widehat{\beta }}_k$, and there are $t_k^1,t_k^2\in \,]a,b[$ such that $x_k(t_k^1)<\alpha _k(t_k^1)$ and $x_k(t_k^2)>\beta _k(t_k^2)\,;$

$({\mathcal {A}}^\gamma _n)$:

$\gamma ^-_n(x_n(t))<y_n(t)<\gamma ^+_n(x_n(t))$, for every $t\in [a,b]$ .

Notice that

$$\begin{aligned} \Omega ^\mu = {\mathcal {V}}^\mu \,, \quad \text {for every } \mu \in \{1,2,3\}^{N-M}\,. \end{aligned}$$

Moreover, for any $\mu \in \{1,2,3,4\}^{N-M}$ and $\kappa \in \{M+1,\dots ,N\}$, using the notation

$$\begin{aligned} \Omega ^\mu _{\kappa ,i}=\Omega ^{(\mu _{M+1},\dots ,\mu _{\kappa -1},i,\mu _{\kappa +1},\dots ,\mu _N)},\quad i=1,2,3,4\,, \end{aligned}$$

the arguments in [14, Proposition 23] show that

$$\begin{aligned} \overline{ \Omega ^\mu _{\kappa ,4}} \cap \Omega ^\mu _{\kappa ,1} = [\Omega ^\mu _{\kappa ,1}\setminus \Omega ^\mu _{\kappa ,2}] \cap [\Omega ^\mu _{\kappa ,1}\setminus \Omega ^\mu _{\kappa ,3} ]\,, \end{aligned}$$

giving easily, passing to the complementary, $\Omega ^\mu _{\kappa ,1}\setminus \overline{ \Omega ^\mu _{\kappa ,4}} = \Omega ^\mu _{\kappa ,2} \cup \Omega ^\mu _{\kappa ,3}$ and consequently

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

Arguing as in [11, Propositions 15–18], we can prove that $\deg \big (I-L^{-1}N_d,\Omega ^\mu \big )$ is well defined for every $\mu \in \{1,2,3,4\}^{N-M}$, and it is equal to $(-1)^m$, where m is the number of times the number 4 appears in the multi-index $\mu$. In particular, we have that

$$\begin{aligned} \deg \big (I-L^{-1}N_d,\Omega ^{(4,4,\dots ,4,4)} \big ) = (-1)^{N-M} \ne 0\,. \end{aligned}$$

So, there exists a solution (x, y) of (61) belonging to $\Omega ^{(4,4,\dots ,4,4)}$. This solution satisfies $({\mathcal {A}}_k^4)$, for every $k\in {\mathcal {K}}$, hence and , for every $k\in {\mathcal {K}}$. Then, from (S2) we conclude that ${\mathcal {N}}_{p_k}(u_k)<D$, so that (x, y) is indeed a solution of the original problem (53), since the differential equation defining the two problems coincide in the set $\{u\in {\mathbb {R}}^{2N} \,:\, {\mathcal {N}}_{p_k}(u_k)<D\}$. Moreover, from (62), we have $\alpha _{{\mathcal {J}}}(t) \preceq x_{\mathcal J}(t) \preceq \beta _{{\mathcal {J}}}(t)$ for every $t\in [a,b]$. The proof is thus completed. $\square$

As an example of application, consider the Dirichlet problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (|x_n'|^{p-2}x_n')'+ \mu _n|x_n^+|^{p-2}x_n^+-\nu _n |x_n^-|^{p-2}x_n^-=h_n(t,x_1,\dots ,x_N,x_1',\dots ,x_N')\\ x_n(0)=0=x_n(\pi _p)\,,\qquad n=1,\dots ,N\,, \end{array}\right. } \nonumber \\ \end{aligned}$$

(65)

where $p>1$, $\mu _n>0$ and $\nu _n>0$. In this case, if $\alpha :[0,\pi _p]\rightarrow {\mathbb {R}}^N$ is a lower solution, then taking $y^\alpha _n=|\alpha _n'|^{p-2}\alpha _n'$, for every $n=1,\dots ,N$, conditions (55) are always satisfied, while (57) reads as

$$\begin{aligned}&(|\alpha _n'|^{p-2}\alpha _n')'(t)+|\alpha _n^+(t)|^{p-2}\alpha _n^+(t)-\nu _n |\alpha _n^-(t)|^{p-2}\alpha _n^-(t)\ge \nonumber \\&\ge h_n(t,x_1,\dots ,\alpha _n(t),\dots ,x_N,y_1,\dots ,|\alpha _n'(t)|^{p-2}\alpha _n'(t),\dots ,y_N)\,, \end{aligned}$$

(66)

for every $x\in {\mathbb {R}}^N$ and $y\in {\mathbb {R}}^N$. Similarly for an upper solution.

As an immediate consequence of Theorem 28, we have the following.

Corollary 29

Let $\ell >0$ and $h_n$ be a uniformly bounded $C^{0,\ell }$-smooth function, for every n. Assume that $(\alpha ,\beta )$ is a pair of lower/upper solutions of (65) related to the partition $({\mathcal {J}},{\mathcal {K}})$ of $\{1,\dots ,N\}$, which is strict with respect to the kth component, for every $k\in {\mathcal {K}}$. If $\max \{\mu _k\,, \nu _k \,:\, k\in {\mathcal {K}}\} \le 1$, then problem (65) has a solution x with the following properties:

(J):

for any $j\in {\mathcal {J}}$, $\alpha _j(t)\le x_j(t)\le \beta _j(t)$, for every $t \in [0,\pi _p]\,;$

(K):

for any $k\in {\mathcal {K}}$, there exist $t_k^1,t_k^2\in \,]0,\pi _p[\,$ such that $x_k(t_k^1)<\alpha _k(t_k^1)$ and $x_k(t_k^2)>\beta _k(t_k^2)$.

The above result should be compared with [5, Theorem 2.4] (see also [2] and the references therein) where some monotonicity assumptions on the nonlinearities were assumed.

Notice that, since we are dealing with second order differential equations, the strictness property asked in the statement can be verified by a standard argument, e.g., simply verifying that a strict inequality holds in (66).

As an illustrative example, we suggest the system

$$\begin{aligned} {\left\{ \begin{array}{ll} (|x_n'|^{p-2}x_n')'+ \mu _n|x_n^+|^{p-2}x_n^+-\nu _n |x_n^-|^{p-2}x_n^-\\ \qquad =\dfrac{s_n x_n}{1+|x_n|}+ {\widetilde{h}}_n(t,x_1,\dots ,x_N,x_1',\dots ,x_N')\,,\\ x_n(0)=0=x_n(\pi _p)\,,\qquad n=1,\dots ,N\,, \end{array}\right. } \end{aligned}$$

where $s_n\in \{-1,+1\}$ and $\Vert {\widetilde{h}}_n\Vert _\infty <1$ for every n. Moreover, for those components having $s_n=-1$, we ask that $\mu _n \le 1$ and $\nu _n\le 1$.

In this example, the pair of lower/upper solutions can be defined by components as follows. The set ${\mathcal {J}}$ is made of those n for which $s_n=+1$, while the set ${\mathcal {K}}$ consists of those n with $s_n=-1$. If $n\in {\mathcal {J}}$, we simply need to choose some sufficiently large constants $\beta _n=-\alpha _n>0$. If $n\in {\mathcal {K}}$, we can argue as in [14, Proposition 26], where the case $p=2$ is treated, to find $\alpha _n$ and $\beta _n$ such that $\alpha _n \not \le \beta _n$.

Remark 30

A generalization of Theorem 28 can be obtained removing the strictness assumption on one of the components $\kappa \in {\mathcal {K}}$. Indeed, the above proof can be easily adapted following the ideas in the proof of [11, Theorem 19]. In such a situation, the conclusion (K) of the statement must be slightly changed concerning the estimates on the $\kappa$th component: the solution will be such that