**1. Introduction**

Differential equations with delay

$$
\dot{\mathfrak{x}} = G(\mathfrak{x}, \mathfrak{x}(t - \tau)),
\tag{1}
$$

where *x* is from R*n*, *G* is some continuous function, and *τ* > 0 is a delay time, arise as mathematical models in different areas of science (see [1,2] and references therein). Many studies are devoted to the construction of solutions or the analysis of the stability of solutions to differential equations with delay [3–11].

Consider differential equation with delay

$$
\dot{\mu} + \nu u = \lambda F(\mu(t - T)).\tag{2}
$$

Here, *u* is a scalar real function, and parameters *ν* and *λ* and delay time *T* are positive. This equation plays an important role in mathematical modelling and is of great interest for fundamental research.

This equation simulates a process of production and destruction where the single state variable *u* decays with a rate *ν* proportional to *u* at the present and is produced with a rate dependent on the value of *u* some time in the past. Such processes arise in many biological applications, for example, in normal and pathological behaviour of control systems in the physiology of blood cell production and respiration and periodic or irregular activity in neural networks (see Table 1 in [12], paper [13] and references therein).

Equation (2) with compactly supported nonlinearity simulates an oscillator with nonlinear delayed feedback with an RC low-pass filter of the first order [14,15]. Additionally, Equation (2) with another nonlinear functions *F* occurs in laser optics [1,16] and in mathematical ecology [2,17].

There are many studies on the dynamics of this equation: its dynamics were studied in the case of piecewise constant [13], monotone [18,19], or compactly supported nonlinearity [20] in the case of positive and negative feedback [21]. Asymptotics of solutions [22]

**Citation:** Kashchenko, A. Asymptotics of Solutions to a Differential Equation with Delay and Nonlinearity Having Simple Behaviour at Infinity. *Mathematics* **2022**, *10*, 3360. https://doi.org/ 10.3390/math10183360

Academic Editor: Andrei D. Polyanin

Received: 29 July 2022 Accepted: 13 September 2022 Published: 16 September 2022

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**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

and the existence of periodic solutions [23] were studied in the case of a singularly perturbed equation:

$$
\varepsilon \dot{\iota} + \iota = F(\iota(t - T)), \quad (0 < \varepsilon \ll 1).
$$

In [24], the authors determined how dynamics of this differential equation when *ε* is small related with dynamics of this equation in the case of *ε* = 0. In [25,26], the authors proposed methods to reconstruct Equation (2) from time series.

For systems of two [27], three [28] and *N* > 3 [29]-coupled oscillators (2) with compactly supported nonlinearity *F* and *λ* 1, asymptotics of relaxation solutions were constructed.

Using simple renormalizations, we can obtain that the coefficient *ν* in (2) is equal to one. Therefore, without limiting generality, below, we consider case *ν* = 1.

In the present work, we analytically study behaviour at *t* → +∞ of solutions to Equation (2) with initial conditions from a wide subset of the phase space *C*[−*T*, 0] under conditions

*λ* 1

and

$$F(\mathbf{x}) = \begin{cases} b\_{\prime} & \mathbf{x} \le p\_{L\prime} \\ f(\mathbf{x})\_{\prime} & p\_{L} < \mathbf{x} < p\_{R\prime} \\ d\_{\prime} & \mathbf{x} \ge p\_{R\prime} \end{cases} \tag{3}$$

where *pL* < 0 < *pR*.

We assume that nonlinear function *f*(*x*) is bounded and piecewise-smooth. We consider positive, negative, and zero values of parameters *b* and *d* (but we assume that at least one of the parameters *b* or *d* is nonzero, because if *b* = *d* = 0, then *F* is a compactly supported function, and this case has been studied in [20]).

This type of function, *F*(*u*), is a generalization of two important applications [12,15] regarding types of nonlinearity: compactly supported and piecewise constant nonlinearities. The class of nonlinearity *F*(*u*) is broad because constants *pL* < 0, *pR* > 0, *b*, and *d* are arbitrary, and conditions on function *f* are quite general. Therefore, this type of nonlinearity may occur in many applied problems, and the results obtained in this paper can be directly applied to study dynamics of the mathematical models with certain nonlinear functions *F* (if function *F* satisfies conditions (3)).

We analytically draw a conclusion about qualitative and quantitative properties of solutions to Equation (2) with arbitrary function *F* (satisfying conditions (3)) with initial conditions from a wide subset of the phase space and give numerical illustrations of the obtained results. It is important to mention that it is impossible to obtain such a result using only numerical methods because it is impossible to iterate through all functions *F* from the considered class and through all the considered initial conditions.

The method of investigation in this paper is the following.

1. We select two sets of initial conditions: *S*<sup>−</sup> and *S*+. The set *S*<sup>−</sup> consists of continuous functions *u*(*s*), (*s* ∈ [−*T*, 0]), such that *u*(*s*) ≤ *pL* on *s* ∈ [−*T*, 0), and *u*(0) = *pL*. The set *S*<sup>+</sup> consists of continuous functions *u*(*s*), (*s* ∈ [−*T*, 0]), such that *u*(*s*) ≥ *pR* on *s* ∈ [−*T*, 0), and *u*(0) = *pR*.

2. We take initial conditions from sets *S*<sup>−</sup> and *S*<sup>+</sup> and construct asymptotics at *λ* → +∞ of all solutions to Equation (2) using the method of steps [30].

3. By the asymptotics of solutions, we draw conclusions about the behaviour of solutions at *t* → +∞.

In this paper we conclude that two types of behaviour at *t* → +∞ of solutions to Equation (2) with initial conditions from the set *S*<sup>+</sup> or *S*<sup>−</sup> are possible: (1) the solution tends to a constant at *t* → +∞, or (2) after the pre-period, the solution becomes a cycle.

The idea of the proof that after the pre-period, the solution becomes a cycle is the following: 1. it follows from the form of sets *S*<sup>−</sup> and *S*<sup>+</sup> and properties of function *F*(*u*), that on the first step (*t* ∈ [0, *T*]) all solutions from the set *S*<sup>−</sup> (*S*+) coincide with each other. Thus, all solutions with initial conditions from *S*<sup>−</sup> (*S*+) coincide with each other for all *t* ≥ 0; 2. if we take initial conditions from one of these sets (*S*<sup>−</sup> or *S*+) and if there exists

a time moment *t*<sup>∗</sup> such that *u*(*t*<sup>∗</sup> + *s*) (where *s* ∈ [−*T*, 0]) belongs to the chosen set, then there exists a periodic solution to Equation (2).

4. We generalize obtained results to the wide sets of initial conditions *u*(*s*) ≥ *pR* (or *u*(*s*) ≤ *pL*) at *s* ∈ [−*T*, 0].

The paper has the following structure: in Sections 2–5, we construct asymptotics of solutions to Equation (2) considering cases of different signs of *b* and *d* under condition *bd* = 0; in Sections 6 and 7 we consider cases *b* = 0 and *d* = 0; and in Sections 8 and 9, we consider cases *b* = 0 and *d* = 0. In Section 10, we generalize results of Sections 2–9 to wide sets of initial conditions *u*(*s*) ≥ *pR* (or *u*(*s*) ≤ *pL*) at *s* ∈ [−*T*, 0].
