**2. Asymptotics of Solutions in the Case** *b >* **0 and** *d >* **0**

Firstly, we consider asymptotics of the solution to Equation (2) with initial conditions from *S*+. We solve our equation using the method of steps.

On the first step (on the segment *t* ∈ [0, *T*]), the function *u*(*t* − *T*) is greater than or equal to *pR*, which is why on this segment, Equation (2) has the form

$$
\dot{u} + \mathfrak{u} = \lambda d.\tag{4}
$$

Hence, on this time segment, the solution to Equation (2) has the form

$$
\mu(t) = p\_R e^{-t} + \lambda d(1 - e^{-t}).\tag{5}
$$

Because *d* > 0 and *λ* is sufficiently large, we obtain *u*(*t*) > *pR* on *t* ∈ [0, *T*]. Therefore, Equation (2) has the form of (4) on the next step *t* ∈ [*T*, 2*T*] and so on (Equation (2) has the form of (4); until then *u*(*t*) < *pR*). However, at *λ* 1, the condition *u*(*t*) = *pRe*−*<sup>t</sup>* + *<sup>λ</sup>d*(<sup>1</sup> − *<sup>e</sup>*−*<sup>t</sup>* ) < *pR* is not true for all *t* ≥ 0, so Equation (2) has the form of (4) for all *t* ≥ 0, and the solution has the form of (5) for all *t* ≥ 0 (see Figure 1).

**Figure 1.** Solution to Equation (2) with initial conditions from *S*+ in the case *b* > 0 and *d* > 0. Values of parameters: *<sup>λ</sup>* <sup>=</sup> 104, *<sup>T</sup>* <sup>=</sup> 5, *pL* <sup>=</sup> <sup>−</sup>1, *pR* <sup>=</sup> 2, *<sup>b</sup>* <sup>=</sup> 2, *<sup>d</sup>* <sup>=</sup> 3.

Secondly, we take initial conditions from *S*<sup>−</sup> and construct asymptotics for these initial conditions.

Then, on the first step (on the segment *t* ∈ [0, *T*]), the function *u*(*t* − *T*) is less than or equal to *pL*, which is why on this segment, Equation (2) has the form

$$
\ddot{\mu} + \mu = \lambda b.\tag{6}
$$

It follows from (6) that the solution has the form

$$u(t) = p\_L e^{-t} + \lambda b(1 - e^{-t}).\tag{7}$$

Therefore,

$$
\mu(T) = p\_L e^{-T} + \lambda b (1 - e^{-T}).\tag{8}
$$

**Lemma 1.** *The leading part of the asymptotics of the solution to Equation* (2) *on the segment t* ∈ [*T*, 2*T*] *coincides with the leading part of the asymptotics of the solution to the Cauchy problem* (4) *and* (8)*. The solution to Equation* (2) *in this interval has the form*

$$u(t) = \lambda b(1 - e^{-T})e^{-(t-T)} + \lambda d(1 - e^{-(t-T)}) + o(\lambda). \tag{9}$$

**Proof.** On the segment *t* ∈ [0, *T*], the solution to Equation (2) has the form of (7). This expression is an increasing function of *t* because *λb* > 0 and *pL* < 0. Therefore, (7) is greater than *pL* for all *t* ∈ [0, *T*]. It is easy to see that expression (7) is less than *pR* for all *t* ∈ [0, *δ*), where

$$\delta = \ln\left(1 + \frac{p\_R - p\_L}{\lambda b - p\_R}\right),\tag{10}$$

and is greater than *pR* for all *t* ∈ (*δ*, *T*]. Note that *δ* is asymptotically small by *λ* at *λ* → +∞ (it has order *O*(*λ*−1)).

It follows from the estimation of the expression (7) that on the segment *t* ∈ [*T*, *T* + *δ*], Equation (2) has the form

$$
\dot{u} + u = \lambda f(u(t - T)),
\tag{11}
$$

and on the interval *t* ∈ (*T* + *δ*, 2*T*], it has the form of (4).

At the segment *t* ∈ [*T*, *T* + *δ*], the exact solution to Equation (2) (which is Equation (11) in this interval) has the form

$$u(t) = (p\_L e^{-T} + \lambda b(1 - e^{-T}))e^{-(t - T)} + \lambda \int\_{T}^{t} e^{s - t} f(u(s - T)) ds. \tag{12}$$

Function *f* is bounded; therefore there exists a constant *M* such that | *f*(*u*(*s* − *T*))| < *M* for all *s* ∈ [*T*, *T* + *δ*]. Thus,

$$\left| \lambda \int\_{T}^{t} e^{s-t} f(u(s-T)) ds \right| \leq \lambda \int\_{T}^{t} |e^{s-t} f(u(s-T))| ds \leq \lambda \int\_{T}^{t} M ds \leq \lambda \int\_{T}^{T+\delta} M ds = \lambda \delta M \leq M\_{1}. \tag{13}$$

where *M*<sup>1</sup> is some constant. The last inequality is true because *δ* has order *O*(*λ*−1) at *λ* → +∞.

Note that on the interval *t* ∈ [*T*, *T* + *δ*]

$$0 \le \lambda d(1 - e^{-(t-T)}) \le \lambda d(1 - e^{-(T+\delta-T)}) \le M\_{2,\epsilon} \tag{14}$$

(where *<sup>M</sup>*<sup>2</sup> is some constant), *<sup>δ</sup>* has order *<sup>O</sup>*(*λ*−1) at *<sup>λ</sup>* → +∞. It follows from inequalities (13) and (14) that on the segment *t* ∈ [*T*, *T* + *δ*], the leading terms of asymptotics at *λ* → +∞ of expressions (12) and (9) coincide.

On the segment *t* ∈ [*T* + *δ*, 2*T*], the exact solution to Equation (2) (which is Equation (4) in this interval) has the form

$$u(t) = \lambda b(1 - e^{-T})e^{-(t-T)} + \lambda d(1 - e^{-(t-T)}e^{\delta}) + \\\lambda e^{T+\delta},$$

$$(p\_L e^{-(T+\delta)} + \lambda \int\_T^{T+\delta} e^{s - (T+\delta)} f(u(s-T)) ds)e^{-(t-(T+\delta))}.\tag{15}$$

Since *<sup>δ</sup>* = *<sup>O</sup>*(*λ*−1) at *<sup>λ</sup>* → +∞, then on the segment *<sup>t</sup>* ∈ [*<sup>T</sup>* + *<sup>δ</sup>*, 2*T*], the leading terms of asymptotics at *λ* → +∞ of expressions (15) and (9) coincide. Thus, the solution to Equation (2) has the form of (9) on the whole segment *t* ∈ [*T*, 2*T*].

The exact solution to the Cauchy problem (4), (8) has the form

$$u(t) = \lambda b(1 - e^{-T})e^{-(t-T)} + \lambda d(1 - e^{-(t-T)}) + p\_L e^{-t}.\tag{16}$$

It is easy to see that the leading terms of asymptotics at *λ* → +∞ of expressions (16) and (9) coincide on the whole segment *t* ∈ [*T*, 2*T*].

Thus, on this segment, the leading part of asymptotics of solution to Equation (2) coincides with the leading part of asymptotics of the solution to the Cauchy problem (4) and (8).

Expression (9) is greater than *pR* for all *t* ∈ [*T*, +∞). Therefore, Equation (2) has the form of (4) for all *t* ≥ *T* + *δ*, and the solution of Equation (2) has the form of (9) for all *t* ≥ *T* (see Figure 2).

**Figure 2.** Typical graphs of solutions to Equation (2) with initial conditions from *S*<sup>−</sup> in the case *b* > 0 and *<sup>d</sup>* <sup>&</sup>gt; 0 if (**a**) *<sup>d</sup>* <sup>&</sup>gt; *<sup>b</sup>* <sup>&</sup>gt; 0 and (**b**) *<sup>b</sup>* <sup>&</sup>gt; *<sup>d</sup>* <sup>&</sup>gt; 0. Values of parameters: *<sup>λ</sup>* <sup>=</sup> 104, *<sup>T</sup>* <sup>=</sup> 5, *pL* <sup>=</sup> <sup>−</sup>1, *pR* <sup>=</sup> 2, *b* = 2, (**a**) *d* = 3, and (**b**) *d* = 1.

Therefore, in the case *b* > 0 and *d* > 0, all solutions with initial conditions from sets *S*<sup>+</sup> and *S*<sup>−</sup> tend to the constant *λd* at *t* → +∞.
