1 Introduction

We study the following second-order non-linear coupled system,

$$\begin{aligned} \left\{ \begin{array}{c} z^{\prime \prime }(t)=f\left( t, z(t), w(t), z'(t), w'(t)\right) \\ w^{\prime \prime }(t)=g\left( t, z(t), w(t), z'(t), w'(t)\right) \end{array} \right. , \end{aligned}$$
(1)

for \(t\in \left[ 0,T\right] \), \(T>0\), with continuous functions \(f,g:\left[ 0,T\right] \times \mathbb {R}^{4}\rightarrow \mathbb {R}\), and the periodic boundary conditions

$$\begin{aligned} \begin{array}{ccc} &{}z(0) = z(T),\qquad &{}z'(0)=z'(T),\\ &{}w(0) = w(T),\qquad &{}w'(0)=w'(T). \end{array} \end{aligned}$$
(2)

Coupled ordinary differential systems arise in many domains of life, namely Chemistry [1], Biology [2], Physiology [3], Mechanics [4], Economy [5], Electronics [6], among others.

Finding periodic solutions in second-order differential equations or systems can be delicate, specially when the non-linearities are non-periodic.

The solvability of second-order complete systems has been studied by several authors with different type of boundary conditions. As an example, we mention [7,8,9] for non-linear boundary conditions, and [10,11,12] for the periodic case.

The most common argument for obtaining periodic solutions is based on the assumptions of periodicity in the non-linearities. As an example, we refer to the work [12], where the following second-order differential equation is studied,

$$\begin{aligned} \chi ''(t) + \Psi (\chi (t))\,\chi '(t) + \phi (t) \chi ^m(t) - \frac{a(t)}{\chi ^{\mu }(t)} + \frac{b(t)}{\chi ^{y}(t)}=0, \end{aligned}$$
(3)

where \(\Psi \in C((0,+\infty ), \mathbb {R})\), \(\phi \), a and b are T-periodic, and in \(L([0,T], \mathbb {R})\), m, \(\mu \) and y are constants with \(m \ge 0\) and \(\mu \ge y > 0\). Here, asymptotic arguments are a key argument to obtain periodic solutions.

The above problem, (3), requires periodicity for the non-linearities. In [11], the authors study the second-order differential problem,

$$\begin{aligned} \left\{ \begin{aligned}&\ddot{x}(t) = \Theta (t, x),\\&x(0) = x(T), \\&\dot{x}(0) = \dot{x}(T), \end{aligned} \right. \end{aligned}$$
(4)

without requiring periodicity for the non-linearities. However, the function \(\Theta \) does not depend on the first derivative, thus making problem (4) a particular case of problem (1). Moreover, the obtained results require an order relation (for the well and inverse order cases) between lower and upper solutions.

Other attempts to find periodic solutions in ordinary differential problems may not present periodicity requirements but, typically, only approximate solutions are found, see [13,14,15].

To overcome the above constraints, in this work we present an original methodology for finding periodic solutions in generalized second-order coupled systems.

Motivated by previous works existent in the literature, (see, for instance, [16,17,18,19]), it is the first time, to the best of our knowledge, where sufficient conditions are given to prove the existence of periodic solutions for coupled systems, without assuming periodicity in the non-linearities nor any order requirement between lower and upper solutions. Therefore, the set of admissible functions for possible lower and upper solutions has a wider range.

Within this line of research, the present work is somehow a continuation of [20, 21], which aim to provide a complete study on the existence and localization of periodic solutions in first-order generalized coupled systems, with and without impulses, using this technique of orderless lower and upper solutions, where the lack of order is overcome with adequate translations.

The strategy employed consists of proving the existence of at least a periodic solution (z(t), w(t)) applying the Topological Degree Theory and to localize the existing solution in a strip bounded by lower and upper functions.

This paper is organized as follows. In Sect. 2 all the required definitions and lemmas are presented. In Sect. 3 the existence and localization result is formulated as the main theorem, together with the respective proof in four claims. A numerical example shows the applicability of the main theorem. In Sect. 4 we apply our methodology to the problem of two coupled Van der Pol oscillators with forcing terms. As all the requisites of the main theorem in Sect. 3 are verified, we suggest possible localizing functions for the existing periodic solutions of this problem. We draw the conclusions of our study in Sect. 5.

2 Definitions

Consider the Banach space \(X:=C^1[0,T]\), equipped with the norm

$$\begin{aligned} \Vert u \Vert _X:= \max \{ \Vert u \Vert , \Vert u' \Vert \}, \qquad \Vert u \Vert := \max _{t \in [0,T]} \{ \vert u(t) \vert \}, \end{aligned}$$

and the vectorial space \(X^2:=(C^1[0,T])^2\), with the norm

$$\begin{aligned} \Vert (u,v) \Vert _{X^2} = \max \{ \Vert u \Vert _X, \Vert v \Vert _X \}. \end{aligned}$$

In second-order differential equations it is important to have a control on the first derivatives, which is achieved using a Nagumo-type condition:

Definition 1

Consider \(C^1\) continuous functions \(\gamma _i, \Gamma _i: [0,T]\rightarrow \mathbb {R}, i=1,2\), and the set

$$\begin{aligned} S = \{ (t, z_0, w_0, z_1, w_1) \in [0,T]\times \mathbb {R}^4: \gamma _1(t)\le z_0 \le \Gamma _1(t), \gamma _2(t)\le w_0 \le \Gamma _2(t) \}. \end{aligned}$$

The continuous functions \(f,g: [0,T]\times \mathbb {R}^4\rightarrow \mathbb {R}\) satisfy a Nagumo-type condition relative to the intervals \([\gamma _1(t), \Gamma _1(t)]\) and \([\gamma _2(t), \Gamma _2(t)]\), for all \(t \in [0,T]\), if there exist continuous functions \(\varphi , \psi : [0, +\infty [ \rightarrow ]0, +\infty [\) verifying

$$\begin{aligned} \int _{0}^{+\infty } \frac{\textrm{d}s}{\varphi (\vert s \vert )} = + \infty , \qquad \int _{0}^{+\infty } \frac{\textrm{d}s}{\psi (\vert s \vert )} = + \infty , \end{aligned}$$
(5)

such that

$$\begin{aligned} \begin{aligned} \vert f(t, z_0, w_0, z_1, w_1) \vert \le \varphi (\vert z_1 \vert ), \quad \forall (t, z_0, w_0, z_1, w_1) \in S, \\ \vert g(t, z_0, w_0, z_1, w_1) \vert \le \psi (\vert w_1 \vert ), \quad \forall (t, z_0, w_0, z_1, w_1) \in S. \end{aligned} \end{aligned}$$
(6)

The a priori estimates for the first derivatives are given by the following lemma.

Lemma 1

Suppose that the continuous functions \(f,g:[0,T]\times \mathbb {R}^4 \rightarrow \mathbb {R}\) satisfy a Nagumo-type condition relative to the intervals \([\gamma _1(t), \Gamma _1(t)]\) and \([\gamma _2(t), \Gamma _2(t)]\), for all \(t \in [0,T]\).

Then, for every solution \((z(t),w(t)) \in X^2\) of (1), (2) verifying

$$\begin{aligned} \gamma _1(t) \le z(t) \le \Gamma _1(t), \quad \text {and} \quad \gamma _2(t) \le w(t) \le \Gamma _2(t), \quad \forall t \in [0,T], \end{aligned}$$
(7)

there are \(N_1, N_2 > 0 \) such that

$$\begin{aligned} \Vert z' \Vert \le N_1, \qquad \Vert w' \Vert \le N_2. \end{aligned}$$
(8)

Proof

Let (z(t), w(t)) be a solution of (1), (2) verifying (7).

By (2) and by Rolle’s Theorem, there exists \(t_0\in [0,T]\) such that \(z'(t_0)=0\).

Consider \(N_i>0, i=1,2\) such that

$$\begin{aligned} \int _0^{N_1} \frac{\textrm{d}s}{\varphi (\vert s \vert )}> T, \qquad \int _0^{N_2} \frac{\textrm{d}s}{\psi (\vert s \vert )} > T, \end{aligned}$$
(9)

and assume, without loss of generality, there exist \(t_1, t_2 \in [0,T]\) such that \(z'(t_1) \le 0\) and \(z'(t_2)>0\).

By continuity of \(z'(t)\), there exists \(t_3 \in [t_1, t_2]\) such that \(z'(t_3)=0\). Using a convenient change of variables, and by (1) and (6),

$$\begin{aligned} \begin{aligned} \int ^{z'(t_2)}_{z'(t_3)} \frac{\textrm{d}s}{\varphi (\vert s \vert )}&= \int ^{t_2}_{t_3} \frac{z''(t)}{\varphi (\vert z'(t) \vert )} \textrm{d}t \le \int _0^T \frac{\vert z''(t) \vert }{\varphi (\vert z'(t) \vert )} \textrm{d}t \\&= \int _0^T \frac{\vert f(t,z(t),w(t),z'(t),w'(t)) \vert }{\varphi (\vert z'(t) \vert )} \textrm{d}t \le \int _0^T \frac{\varphi (\vert z'(t) \vert )}{\varphi (\vert z'(t) \vert )} \textrm{d}t = T. \end{aligned} \end{aligned}$$

By (9),

$$\begin{aligned} \int ^{z'(t_2)}_{z'(t_3)} \frac{\textrm{d}s}{\varphi (\vert s \vert )} = \int ^{N_1}_{0} \frac{\textrm{d}s}{\varphi (\vert s \vert )} \le T < \int ^{N_1}_{0} \frac{\textrm{d}s}{\varphi (\vert s \vert )}, \end{aligned}$$
(10)

and, therefore, \(z'(t_2)<N_1\). As \(t_2\) is chosen arbitrarily, then \(z'(t)<N_1\), for all \(t \in [0,T]\).

The case where \(z'(t_1)>0\) and \(z'(t_2)\le 0\) follows similar arguments. Therefore, \(\Vert z' \Vert \le N_1\), for all \(t \in [0,T]\). Likewise, \(\Vert w'\Vert \le N_2\), for all \(t \in [0,T]\).

Remark that the constant \(N_1\) depends only on \(\gamma _i\), \(\Gamma _i\), \(\varphi \) and T. In the same way, \(N_2\) depends only on \(\gamma _i\), \(\Gamma _i\), \(\psi \) and T. \(\square \)

We present below the definition of upper and lower solutions.

Definition 2

The pair of real functions \(\left( \alpha _{1},\alpha _{2}\right) \in (C^1[0,T])^2\) is a lower solution of the periodic problem (1), (2) if

$$\begin{aligned} \begin{aligned}&\alpha _1''(t) \ge f(t, \alpha _1^0(t), \alpha _2^0(t), \alpha _1'(t), w_1), \forall w_1 \in \mathbb {R},\\&\alpha _2''(t) \ge g(t, \alpha _1^0(t), \alpha _2^0(t), z_1, \alpha _2'(t)), \forall z_1 \in \mathbb {R}, \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \alpha _i^0(t):= \alpha _i(t)-\Vert \alpha _i \Vert , \end{aligned}$$

and

$$\begin{aligned} \alpha _i(0) = \alpha _i(T), \qquad \alpha '_i(0) \ge \alpha '_i(T). \end{aligned}$$

The pair of real functions \(\left( \beta _{1},\beta _{2}\right) \in (C^1[0,T])^2\) is an upper solution of the periodic problem (1), (2) if

$$\begin{aligned} \begin{aligned}&\beta _1''(t) \le f(t, \beta _1^0(t), \beta _2^0(t), \beta _1'(t), w_1), \forall w_1 \in \mathbb {R},\\&\beta _2''(t) \le g(t, \beta _1^0(t), \beta _2^0(t), z_1, \beta _2'(t)), \forall z_1 \in \mathbb {R}, \end{aligned} \end{aligned}$$
(11)

with

$$\begin{aligned} \beta _i^0(t):= \beta _i(t)+\Vert \beta _i \Vert , \end{aligned}$$
(12)

and

$$\begin{aligned} \beta _i(0) = \beta _i(T), \qquad \beta _i'(0) \le \beta _i'(T). \end{aligned}$$
(13)

3 Main result

The following theorem is an existence and localization result using orderless lower and upper solutions.

Theorem 3

Let \(f,g:[0,T] \times \mathbb {R}^4 \rightarrow \mathbb {R}\) be continuous functions. Suppose \((\alpha _1,\alpha _2)\), \((\beta _1,\beta _2)\) are lower and upper solutions of problem (1), (2), as in Definition 2.

Suppose f, g satisfy a Nagumo-type condition, according to Definition 1 relative to the intervals \([\alpha _1^0(t), \beta _1^0(t)]\) and \([\alpha _2^0(t), \beta _2^0(t)]\), for all \(t \in [0,T]\), with

$$\begin{aligned}{} & {} f(t, z_0,w_0,z_1,w_1) \text { non-increasing in }w_0, \text { for } t \in [0,T], z_0 \in \mathbb {R}\text { fixed}, \end{aligned}$$
(14)
$$\begin{aligned}{} & {} \min \left\{ \min _{t\in [0,T]} \alpha _1'(t), \min _{t\in [0,T]} \beta _1'(t) \right\} \le z_1 \le \max \left\{ \max _{t\in [0,T]} \alpha _1'(t), \min _{t\in [0,T]} \beta _1'(t) \right\} , \nonumber \\ \end{aligned}$$
(15)

and

$$\begin{aligned}{} & {} g(t, z_0,w_0,z_1,w_1) \text { non-increasing in }z_0, \text { for } t \in [0,T], w_0 \in \mathbb {R}\text { fixed},\\{} & {} \min \left\{ \min _{t\in [0,T]} \alpha _2'(t), \min _{t\in [0,T]} \beta _2'(t) \right\} \le w_1 \le \max \left\{ \max _{t\in [0,T]} \alpha _2'(t), \min _{t\in [0,T]} \beta _2'(t) \right\} . \end{aligned}$$

Then, there exists at least one pair \((z(t), w(t)) \in (C^2[0,T])^2\), solution of problem (1), (2) and, moreover,

$$\begin{aligned} \alpha _1^0(t) \le z(t) \le \beta _1^0(t), \qquad \alpha _1^0(t) \le w(t) \le \beta _1^0(t), \qquad \forall t \in [0,T]. \end{aligned}$$

Proof

Define the truncated functions \(\delta _i:[0,T]\times \mathbb {R}\rightarrow \mathbb {R}\),

$$\begin{aligned} \begin{array}{ccc} &{}\delta _1(t,z)= \left\{ \begin{aligned} &{}\beta _1^0(t), \quad z>\beta _1^0(t) \\ &{}z, \quad \alpha _1^0(t)\le z \le \beta _1^0(t) \\ &{}\alpha _1^0(t), \quad z<\alpha _1^0(t) \\ \end{aligned} \right. , \quad &{}\delta _2(t,w)= \left\{ \begin{aligned} &{}\beta _2^0(t), \quad w>\beta _2^0(t) \\ &{}w, \quad \alpha _2^0(t)\le w \le \beta _2^0(t) \\ &{}\alpha _2^0(t), \quad w<\alpha _2^0(t) \\ \end{aligned} \right. . \end{array} \nonumber \\ \end{aligned}$$
(16)

For \(\lambda , \mu \in [0,1]\), consider the truncated, perturbed and homotopic auxiliary problem

$$\begin{aligned} \left\{ \begin{aligned}&z''(t)-z(t)= \lambda \, f(t, \delta _1(t,z(t)), \delta _2(t,w(t)), z'(t), w'(t))-\lambda \, \delta _1(t, z(t)) \\&w''(t)-w(t)= \mu \, g(t, \delta _1(t,z(t)), \delta _2(t,w(t)), z'(t), w'(t))-\mu \, \delta _2(t, w(t)) \end{aligned} \right. , \nonumber \\ \end{aligned}$$
(17)

for all \(t \in [0,T]\), with the boundary conditions (2).

Consider \(r_i>0, i=1,2\), such that, for all \(\lambda , \mu \in [0,1]\), every \(t \in [0,T]\), and every \(N_1^*,N_2^* >0\) given by (8),

$$\begin{aligned} -r_i< \alpha _i^0(t)\le \beta _i^0(t)<r_i, \end{aligned}$$
(18)

with

$$\begin{aligned} \begin{aligned}&\lambda \, f(t, \beta _1^0(t), \beta _2^0(t), 0, w'(t))- \beta _1^0(t)+r_1>0, \quad \text { for } \Vert w' \Vert \le N_2^*,\\&\mu \, g(t, \beta _1^0(t), \beta _2^0(t), z'(t), 0)- \beta _2^0(t)+r_2 >0, \quad \text { for } \Vert z' \Vert \le N_1^*,\\&\lambda \, f(t, \alpha _1^0(t), \alpha _2^0(t), 0, w'(t))- \alpha _1^0(t)-r_1<0,\quad \text { for } \Vert w' \Vert \le N_2^*,\\&\mu \, g(t, \alpha _1^0(t), \alpha _2^0(t), z'(t), 0)- \alpha _2^0(t)-r_2 <0, \quad \text { for } \Vert z' \Vert \le N_1^*. \end{aligned} \end{aligned}$$
(19)

Claim 1

Every solution of (17), (2) verifies \(\vert z(t) \vert < r_1\) and \(\vert w(t) \vert < r_2\), for all \(t \in [0,T]\), independently of \(\lambda , \mu \in [0,1]\).

Assume, by contradiction, that there exist \(\lambda \in [0,1]\), a pair (z(t), w(t)), solution of problem (17), (2), and \(t \in [0,T]\) such that \(\vert z(t) \vert \ge r_1\). If \(z(t) \ge r_1\), define

$$\begin{aligned} \max _{t\in [0,T]} z(t):=z(t_0). \end{aligned}$$
(20)

If \(t\in ]0,T[\) and \(\lambda \in ]0,1]\), then \(z'(t)=0\) and \(z''(t)\le 0\). By (18), (14) and (19), the following contradiction holds:

$$\begin{aligned} \begin{aligned} 0\ge z''(t_0)&= \lambda \, f(t_0, \delta _1(t_0,z(t_0)), \delta _2(t_0,w(t_0)), z'(t_0), w'(t_0))-\lambda \, \delta _1(t_0, z(t_0)) + z(t_0)\\&= \lambda \, f(t_0, \beta _1^0(t_0), \delta _2(t_0,w(t_0)), 0, w'(t_0))-\lambda \, \beta _1^0(t_0) + z(t_0)\\&\ge \lambda \, f(t_0, \beta _1^0(t_0), \delta _2(t_0,w(t_0)), 0, w'(t_0))-\beta _1^0(t_0) + z(t_0)\\&\ge \lambda \, f(t_0, \beta _1^0(t_0), \delta _2(t_0,w(t_0)), 0, w'(t_0))-\beta _1^0(t_0) + r_1 >0 \end{aligned} \end{aligned}$$

If \(t_0=0\) or \(t_0=T\), then, by (2),

$$\begin{aligned} 0 \ge z'(0)=z'(T) \ge 0. \end{aligned}$$

So, \(z'(0)=z'(T)=0\), \(z''(0)\le 0\) and \(z''(T)\le 0\). Therefore, the arguments follow the previous case.

If \(\lambda =0\), we obtain the contradiction

$$\begin{aligned} 0 \ge z''(t_0)=z(t_0)\ge r_1>0. \end{aligned}$$

Then, \(z(t)<r_1, \forall t \in [0,T]\), regardless of \(\lambda \).

The same arguments can be made to prove that \(z(t)>-r_1\). Therefore, \(\vert z(t) \vert < r_1\), for \(t \in [0,T]\), independently of \(\lambda \).

Likewise, \(\vert w(t) \vert < r_2\), for \(t \in [0,T]\), independently of \(\mu \).

Claim 2

For every solution (z(t), w(t)) of (17), (2), there are \(N_1^*, N_2^*>0\) such that \(\vert z'(t) \vert < N_1^*\) and \(\vert w'(t) \vert < N_2^*\), \(\forall t \in [0,T]\), independently of \(\lambda , \mu \in [0,1]\).

Define the continuous functions

$$\begin{aligned} \begin{aligned}&F_{\lambda }(t,z_0,w_0,z_1,w_1)):= \lambda \, f(t,\delta _1(t,z_0),\delta _2(t,w_0),z_1,w_1) - \lambda \, \delta _1(t,z_0) + z_0,\\&G_{\mu }(t,z_0,w_0,z_1,w_1):= \mu \, f(t,\delta _1(t,z_0),\delta _2(t,w_0),z_1,w_1) - \mu \, \delta _2(t,w_0) + w_0, \end{aligned} \end{aligned}$$

and, as \(f,g:[0,T]\times \mathbb {R}^4\rightarrow \mathbb {R}\) satisfy a Nagumo-type condition relative to the intervals \([\alpha _1^0(t),\beta _1^0(t)]\) and \([\alpha _2^0(t),\beta _2^0(t)]\), then,

$$\begin{aligned} \begin{aligned}&\vert F_{\lambda }(t,z(t),w(t),z'(t),w'(t)) \vert \\&\qquad \le \vert f(t,\delta _1(t,z(t)),\delta _2(t,w(t)),z'(t),w'(t)) \vert +\vert \delta _1(t,z(t)) \vert + \vert z(t) \vert \\&\qquad< \varphi (\vert z' \vert ) + 2r_1,\\&\vert G_{\mu }(t,z(t),w(t),z'(t),w'(t)) \vert \\&\qquad \le \vert f(t,\delta _1(t,z(t)),\delta _2(t,w(t)),z'(t),w'(t)) \vert +\vert \delta _2(t,w(t)) \vert + \vert w(t) \vert \\&\qquad < \psi (\vert w' \vert ) + 2r_2. \end{aligned} \end{aligned}$$

Therefore, for continuous positive functions \(\varphi ^*,\psi ^*:[0,+\infty [\rightarrow ]0,+\infty [\) given by

$$\begin{aligned} \varphi ^*(\vert z' \vert ):=\varphi (\vert z' \vert ) + 2r_1, \qquad \psi ^*(\vert w' \vert ):=\psi (\vert w' \vert ) + 2r_2, \end{aligned}$$

\(F_{\lambda }\) and \(G_{\mu }\) satisfy a Nagumo-type condition in

$$\begin{aligned} E=\{ (t,z,w,z',w') \in [0,T]\times \mathbb {R}^4:\vert z \vert<r_1, \vert w \vert <r_2 \}, \end{aligned}$$

as, by (5), we have

$$\begin{aligned}{} & {} \int _0^{+\infty }\frac{\textrm{d}s}{\varphi ^*(\vert z' \vert )} = \int _0^{+\infty }\frac{\textrm{d}s}{\varphi (\vert z' \vert )+2r_1}=+\infty , \nonumber \\{} & {} \int _0^{+\infty }\frac{\textrm{d}s}{\psi ^*(\vert w' \vert )} = \int _0^{+\infty }\frac{\textrm{d}s}{\psi (\vert w' \vert )+2r_2}=+\infty . \end{aligned}$$
(21)

Therefore, by Lemma 1, there are \(N_1^*,N_2^*>0\) such that \(\vert z'(t) \vert < N_1^*\) and \(\vert w'(t) \vert < N_2^*\), \(\forall t \in [0,T]\), independently of \(\lambda , \mu \in [0,1]\).

Claim 3

Problem (17), (2), for \(\lambda =\mu =1\), has at least one solution (z(t), w(t)).

Define the operators

$$\begin{aligned} \mathcal {L}: (C^2[0,T])^2 \subset (C^1[0,T])^2 \rightarrow (C[0,T])^2 \times \mathbb {R}^4, \end{aligned}$$

given by

$$\begin{aligned} \mathcal {L} (z, w):= (z''(t)-z(t), w''(t)-w(t), z(0), w(0), z'(0), w'(0)), \end{aligned}$$

and

$$\begin{aligned} \mathcal {N}_{(\lambda , \mu )}: (C^1[0,T])^2 \rightarrow (C[0,T])^2 \times \mathbb {R}^4, \end{aligned}$$

given by

$$\begin{aligned} \mathcal {N}_{(\lambda ,\mu )} (z, w):= (Z_{\lambda }(t), W_{\mu }(t), z(T), w(T), z'(T), w'(T)), \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&Z_{\lambda }(t):= \lambda \, f(t,\delta _1(t,z(t)),\delta _2(t,w(t)),z'(t),w'(t)) - \lambda \, \delta _1(t,z(t)),\\&W_{\mu }(t):= \mu \, f(t,\delta _1(t,z(t)),\delta _2(t,w(t)),z'(t),w'(t)) - \mu \, \delta _2(t,w(t)). \end{aligned} \end{aligned}$$

We define the continuous operator

$$\begin{aligned} \mathcal {T}: (C^1[0,T])^2 \rightarrow (C^1[0,T])^2, \end{aligned}$$

given by

$$\begin{aligned} \mathcal {T}_{(\lambda ,\mu )} (z, w):= \mathcal {L}^{-1}\mathcal {N}_{(\lambda , \mu )}(z, w). \end{aligned}$$

For \(M:=\max \{ r_1, r_2, N_1^*, N_2^* \}\), consider the set

$$\begin{aligned} \Omega = \{ (z,w) \in (C^1([0,T]))^2: \Vert (z,w) \Vert _{X^2} < M \}. \end{aligned}$$
(22)

By Claims 1 and 2, for all \(\lambda , \mu \in [0,1]\), the degree \(d(I-\mathcal {T}_{(0,0)}, \Omega , 0)\) is well defined and, by homotopy invariance,

$$\begin{aligned} d(I-\mathcal {T}_{(0,0)}, \Omega , 0) = d(I-\mathcal {T}_{(1,1)}, \Omega , 0). \end{aligned}$$
(23)

As the equation \((z,w)=\mathcal {T}_{(0,0)}(z,w)\), that is, the homogeneous system

$$\begin{aligned} \left\{ \begin{aligned}&z''(t)-z(t)=0\\&w''(t)-w(t)=0, \end{aligned} \right. \end{aligned}$$

with the periodic conditions (2), admits only the null solution, then \(\mathcal {L}^{-1}\) is well-defined and, by degree theory,

$$\begin{aligned} d(I-\mathcal {T}_{(0,0)}, \Omega , 0)=\pm 1. \end{aligned}$$

By (23), \(d(I-\mathcal {T}_{(1,1)}, \Omega , 0)=\pm 1\). Therefore, the equation \((z,w)=\mathcal {T}_{(1,1)}(z,w)\) has at least one solution, that is, the auxiliary problem with \(\lambda =\mu =1\),

$$\begin{aligned} \left\{ \begin{aligned}&z''(t)-z(t)=f(t,\delta _1(t,z(t)),\delta _2(t,w(t)),z'(t),w'(t)) - \delta _1(t,z(t))\\&w''(t)-w(t)=g(t,\delta _1(t,z(t)),\delta _2(t,w(t)),z'(t),w'(t)) - \delta _2(t,w(t)), \end{aligned} \right. \end{aligned}$$

together with the boundary conditions (2), has at least one solution \((z_*(t), w_*(t))\).

Claim 4

The pair \((z_*(t),w_*(t)) \in X^2\), solution of the auxiliary problem (17), (2), for \(\lambda =\mu =1\), is also a solution of the original problem (1), (2).

This Claim is proven if the solution \((z_*(t),w_*(t))\) satisfies

$$\begin{aligned} \alpha _1^0(t) \le z_*(t) \le \beta _1^0(t), \quad \alpha _2^0(t) \le w_*(t) \le \beta _2^0(t), \quad \forall t \in [0,T]. \end{aligned}$$
(24)

Suppose, by contradiction, that there is \(t\in [0,T]\) such that \(z_*(t)>\beta _1^0(t)\), and define

$$\begin{aligned} \max _{t\in [0,T]} \{ z_*(t)-\beta _1^0(t) \}:=z_*(t_0)-\beta _1^0(t_0)>0. \end{aligned}$$
(25)

If \(t_0 \in ]0,T[\), then \((z_*-\beta _1^0)'(t_0)=0\) and \((z_*-\beta _1^0)''(t_0)\le 0\). Then, by (17), (25), (14) and (11), the following contradiction holds,

$$\begin{aligned} 0\ge & {} z''_*(t_0) - (\beta _1^0)''(t_0) \nonumber \\= & {} f(t_0, \delta _1(t_0, z_*(t_0)), \delta _2(t_0, w(t_0)), z'_*(t_0), w'(t_0)) - \delta _1(t_0, z_*(t_0)) + z_*(t_0) - \beta _1''(t_0) \nonumber \\= & {} f(t_0, \beta _1^0(t_0), \delta _2(t_0, w(t_0)), \beta _1'(t_0), w'(t_0)) - \beta _1^0(t_0) + z_*(t_0) - \beta _1''(t_0) \nonumber \\> & {} f(t_0, \beta _1^0(t_0), \delta _2(t_0, w(t_0)), \beta _1'(t_0), w'(t_0)) -\beta _1''(t_0) \nonumber \\\ge & {} f(t_0, \beta _1^0(t_0), \beta _2^0(t_0), \beta _1'(t_0), w'(t_0)) -\beta _1''(t_0) \ge 0. \end{aligned}$$
(26)

If \(t_0=0\) or \(t_0=T\), then \((z_*-\beta _1^0)'(0)\le 0\) and \((z_*-\beta _1^0)'(T)\ge 0\). By (12), (2) and (13),

$$\begin{aligned} \begin{aligned} 0&\ge (z_*-\beta _1^0)'(0) = (z_*-\beta _1)'(0) = z_*'(T)-\beta _1'(0) \\&\ge z_*'(T)-\beta _1'(T) = (z_*-\beta _1^0)'(T) \ge 0, \end{aligned} \end{aligned}$$

so,

$$\begin{aligned} (z_*-\beta _1^0)'(0) = (z_*-\beta _1^0)'(T) = 0, \end{aligned}$$
(27)

and \((z_*-\beta _1^0)''(t_0) \le 0\). Therefore, we can apply the previous arguments to obtain a similar contradiction, and so, \(z_*(t) \le \beta _1^0(t)\).

Similar arguments can be used to prove the other inequalities in (24).

\(\square \)

Example 1

Consider the following system, for \(t\in [0,1],\)

$$\begin{aligned} \left\{ \begin{aligned}&z''(t) = 2 z^3(t) - w(t) + 3 z'(t) - \frac{2}{1+(w'(t))^2} -10 t,\\&w''(t) = - z(t) + 10 w^3(t) - e^{-(z'(t))^2} - 3 w'(t) - 12 t, \end{aligned} \right. \end{aligned}$$
(28)

together with the periodic boundary conditions (2).

The functions \(\alpha _{i},\beta _{i}:\left[ 0,1\right] \rightarrow \mathbb {R},\) \(i=1,2,\) given by

$$\begin{aligned} \begin{array}{lll} &{}\alpha _{1}(t) = 1+2t^2-2t^3, \qquad &{}\beta _{1}(t) = 6/5 - 2t^2 + 2t^3, \\ &{}\alpha _{2}(t) = 2t - 2t^2, \qquad &{}\beta _{2}(t) = 1 -3t + 3t^2, \end{array} \end{aligned}$$
Fig. 1
figure 1

Orderless \(\alpha _i, \beta _i\) functions, with \(i=1,2\)

Full size image

are, respectively, lower and upper solutions of problem (28), (2), according to Definition 2, with

$$\begin{aligned} \begin{array}{lll} &{}\alpha _{1}^0(t) = -8/27 + 2t^2 - 2t^3, &{}\qquad \beta _{1}^0(t) = 12/5 - 2t^2 + 2t^3, \\ &{}\alpha _{2}^0(t) = -1/2 +2t - 2t^2, &{}\qquad \beta _{2}^0(t) = 2 -3t + 3t^2. \end{array} \end{aligned}$$

Remark that the lower and upper functions, shown in Fig. 1, are not ordered, as it is usual in the literature.

The above problem is a particular case of (1), (2), with \(T=1\), and where the non-linearities, according to Definition 1, satisfy a Nagumo-type condition in the set

$$\begin{aligned} \widetilde{S} = \left\{ \begin{aligned}&(t, z_0, w_0, z_1, w_1) \in [0,1]\times \mathbb {R}^4:\\&\qquad \qquad -8/27 + 2t^2 - 2t^3 \le z_0 \le 12/5 - 2t^2 + 2t^3, \\&\qquad \qquad -1/2 +2t - 2t^2 \le w_0 \le 2 -3t + 3t^2 \end{aligned} \right\} , \end{aligned}$$

such that

$$\begin{aligned} \left| f(t,z,w,z',w') \right|\le & {} 2 \left| z\right| ^3 + w + 3 \left| z'\right| + \left| \frac{2}{1+(w')^2}\right| + 10\left| t \right| \\\le & {} 2 \times \frac{12}{5} + 2 + 3 \left| z'\right| + 2 + 10 \\= & {} \frac{94}{5} + 3\left| z'\right| := \widetilde{\varphi }(\left| z' \right| ),\\ \left| g(t,z,w,z',w') \right|\le & {} \left| z \right| + 10 \left| w \right| ^3 + \left| e^ {(z')^2}\right| + 3\left| w' \right| + 12\left| t \right| \\\le & {} \frac{12}{5} + 10 \times 2^3 + 1 + 3\left| w' \right| + 12\\= & {} \frac{477}{5} + 3\left| w' \right| := \widetilde{\psi }(\left| w' \right| ). \end{aligned}$$

It is clear that functions \(\widetilde{\varphi }\) and \(\widetilde{\psi }\) satisfy (5).

As the assumptions of Theorem 3 are verified, then the system (28), (2) has, at least, a solution \((z^*(t),w^*(t))\in (C^2[0,1])^{2}\) such that

$$\begin{aligned} -8/27 + 2t^2 - 2t^3\le & {} z^*(t)\le 12/5 - 2t^2 + 2t^3, \nonumber \\ -1/2 +2t - 2t^2\le & {} w^*(t) \le 2 -3t + 3t^2,\quad \forall t\in [0,1], \end{aligned}$$
(29)

as shown in Fig. 2.

Fig. 2
figure 2

Shifted functions, \(\alpha _i^0, \beta _i^0, i=1,2\), localizing the solution pair \((z^*(t), w^*(t))\)

Full size image

4 Coupled forced Van der Pol oscillators

The equation for the damped harmonic motion is

$$\begin{aligned} x''(t) + \mu x'(t) + x(t) = 0. \end{aligned}$$

Balthazar Van der Pol (1889–1959) modified the damped harmonic oscillator by considering a negative quadratic term for the friction term to obtain self-sustained oscillations. This modification resulted in the Van der Pol oscillator [22, 23], which can be represented by the following equation,

$$\begin{aligned} x''(t) - \epsilon (1 - x^2(t)) x'(t) + x(t) = 0, \end{aligned}$$

where x(t) is the time-dependent variable and \(-\epsilon (1 - x^2(t))\) is the non-linear damping term, with \(\epsilon > 0\).

A variant of this problem can be thought of, by coupling two Van der Pol oscillators. We consider the following system, for \(t\in [0,T]\),

$$\begin{aligned} \left\{ \begin{aligned}&z''(t) = z'(t)(A_1 - B_1 z^2(t)) - C_1 z(t) + D_1 \tanh (E_1 z(t) - F_1 w(t)) + G_1 \cos (t)\\&w''(t) = w'(t)(A_2 - B_2 w^2(t)) - C_2 w(t) + D_2 \arctan (E_2 w(t) - F_2 z(t)) + G_2 \cos (t) \end{aligned} \right. \nonumber \\ \end{aligned}$$
(30)

with \(A_i, B_i, C_i, D_i, F_i > 0\) and \(E_i, G_i \in \mathbb {R}\), \(i=1,2\), together with the periodic boundary conditions (2).

The terms \(D_1 \tanh (E_1 z(t) - F_1 w(t))\) and \(D_2 \arctan (E_2 w(t) - F_2 z(t))\) correspond, respectively, to each coupling and \(G_i \cos (t)\) is a time-dependent periodic forcing. We chose \(T=1\) and the parameter set

$$\begin{aligned} \begin{array}{ccccccc} A_1 = 1 &{}\quad B_1 = 0.5 &{}\quad C_1 = 1 &{}\quad D_1 = 6 &{}\quad E_1 = 3 &{}\quad F_1 = 2 &{}\quad G_1 = 1,\\ A_2 = 1 &{}\quad B_2 = 0.5 &{}\quad C_2 = 1 &{}\quad D_2 = 8 &{}\quad E_2 = 2 &{}\quad F_2 = 1 &{}\quad G_2 = -1. \end{array} \nonumber \\ \end{aligned}$$
(31)

The functions \(\alpha _{i},\beta _{i}:\left[ 0,1\right] \rightarrow \mathbb {R},\) \(i=1,2,\) given by

$$\begin{aligned} \begin{array}{lll} &{}\alpha _{1}(t) = -1 + t - t^2, &{}\qquad \beta _{1}(t) = 1 - t^2/2 + t^3/2, \\ &{}\alpha _{2}(t) = -3/4 + t - t^2, &{}\qquad \beta _{2}(t) = 1 -t^2 + t^3, \end{array} \end{aligned}$$

are, respectively, lower and upper solutions of problem (30), (2), with the numerical values (31) according to Definition 2, with

$$\begin{aligned} \begin{array}{lll} &{}\alpha _{1}^0(t) = -5/4 + t - t^2, &{}\qquad \beta _{1}^0(t) = 2 -t^2/2 + t^3/2, \\ &{}\alpha _{2}^0(t) = -1 + t - t^2, &{}\qquad \beta _{2}^0(t) = 2 -t^2 + t^3. \end{array} \end{aligned}$$

The functions f and g verify the monotonicity requirements and the Nagumo-type condition of Definition 1 in the set

$$\begin{aligned} S^* =\left\{ \begin{aligned}&(t, z_0, w_0, z_1, w_1) \in [0,1]\times \mathbb {R}^4:\\&\qquad \qquad -5/4 + t - t^2 \le z_0 \le 2 -t^2/2 + t^3/2,\\&\qquad \qquad -1 + t - t^2 \le w_0 \le 2 -t^2 + t^3 \end{aligned} \right\} , \end{aligned}$$

with

$$\begin{aligned} f(t, z_0, w_0, z_1, w_1)\le & {} |z_1| \, (1+ 0.5 |z_0|^2) + |z_0| + 6 |\tanh (3z_0 - 2w_0)| + |\cos (t)|\\\le & {} 3|z_1| + 9:= \varphi ^*(|z_1|),\\ g(t, z_0, w_0, z_1, w_1)\le & {} |w_1| \, (1+ 0.5 |w_0|^2) + |w_0| + 8 |\arctan (2w_0 - z_0)| + |\cos (t)|\\\le & {} 3|w_1| + 3 + 4\pi := \psi ^*(|w_1|). \end{aligned}$$
Fig. 3
figure 3

Shifted functions, \(\alpha _i^0, \beta _i^0, i=1,2\), localizing the solution pair \((z^*(t), w^*(t))\)

Full size image

As all the assumptions of Theorem 3 are verified, then there is at least a periodic solution for the system of two coupled forced Van der Pol oscillators (30) with the periodic boundary conditions (2), and the parameter set of values (31). Moreover, remark that, from (30) and (31), this solution is a non-trivial one.

Figure 3 shows the shifted functions that localize existing periodic solutions in the system (30), (2), with parameter values (31).

5 Conclusions

When reviewing the literature, one realizes that there are several attempts to find periodic solutions in second-order differential problems. However, either periodicity requirements are made [10, 12], or approximated solutions are presented [13,14,15], or the problem does not present complete non-linearities [11].

In this work we present a general second-order differential coupled system, with dependencies on both variables, z, w, their derivatives, \(z',w'\), and on time, t. We provide sufficient conditions to prove the existence of periodic solutions for problem (1), (2) without assuming any type of periodicity in the non-linearities.

To localize an existing periodic solution, no order requirement is made between lower and upper solutions. Therefore, we increase the range of possibilities when considering well-ordered and not well-ordered localizing functions \(\alpha _i, \beta _i\), see Example 1.

Our methodology is successfully applied to a system of coupled Van der Pol oscillators with forcing terms. Under the conditions of the main result, Theorem 3, we prove the existence of a periodic solution for problem (1), (2), and localize it within a strip bounded by shifted lower and upper solutions \(\alpha _i^0, \beta _i^0\), whose requirements follow Definition 2.

When applying Theorem 3 to a specific second-order system, one must verify that the non-linearities satisfy the Nagumo-type condition of Definition 1, as well as the required monotonicities. Note that Theorem 3 does not guarantee the existence of a non-trivial solution. That guarantee arises with the nature of the system in study, i.e., whether if it allows for constant solutions or not.

As we deal with a generalized system, our study has applicability in several real-case scenarios in Nature, and it can be an important tool for other mathematical problems.