In this section, we firstly consider the stability of equation (1) with the delay function *τ*(*u*(*t*)) which has an upper bound for all *t* ∈ ℝ^{+}, i.e., there exist at least one *M*_{1} ∈ ℝ such that 0 < *τ*(*u*(*t*)) < *M*_{1} for all *t* ∈ ℝ^{+}.

In this case, the value of delay of equation (1) varies in interval (0, *M*_{1}) while *t* is varying. The independent parameter *h* of the characteristic equation (2) takes values in the interval (0, *M*_{1}). As a part of the D-partition method, we have

$${C}_{*}:{A}_{0}+{A}_{1}=0\text{\hspace{0.17em}}\text{for\hspace{0.17em}\hspace{0.17em}}\lambda \text{=0}$$(10)

this straight line is a line forming the boundary of the D-partition and is denoted by *C*_{*}. Substituting λ = *iω* and equating to zero the real and imaginary parts in characteristic equation (2), we find the following equations

$${A}_{0}+{A}_{1}\mathrm{cos}(\omega h)=0$$(11)

$$\omega +{A}_{1}\mathrm{sin}(\omega h)=0.$$(12)

Solving the above equations for *A*_{0} and *A*_{1}, the following parametric curve equations are obtained

$${A}_{0}(\omega ,h)=-\frac{\omega \mathrm{cos}(\omega h)}{\mathrm{sin}(\omega h)}$$(13)

$${A}_{1}(\omega ,h)=\frac{\omega}{\mathrm{sin}(\omega h)}.$$(14)

Since *A*_{0}(*ω*, h/ and *A*_{1}(*ω*, *h*) are even with respect to *ω*, it is sufficient to take *ω* ∈ (0, ∞). Equations (13)-(14) define a family of curves since h is not a constant. Holding *h* fixed, these define *A*_{0}(*ω*, *h*) and *A*_{1}(*ω*, *h*) as function of *ω*, providing a parametric representation of a curve. Different values of *h* give different curves in the family. Since equations (13)-(14) have singularity for *ωh* = *kπ*, we introduce intervals ${J}_{k}=(\frac{k\pi}{h},\frac{(k+1)\pi}{h})$ and denote by *C*_{k}(*h*) the curve in the parameter space (*A*_{0}, *A*_{1}) for *ω* ∈ *J*_{k}.

*C*_{0}(*h*) contains the limit point for *ω* → 0

$$(\underset{\omega \to 0}{\mathrm{lim}}{A}_{0}(\omega ,h),\underset{\omega \to 0}{\mathrm{lim}}{A}_{1}(\omega ,h))=(-\frac{1}{h},\frac{1}{h}).$$(14)

In addition, the following limits can be obtained for *k* ∈ ℕ – {0}

$$\begin{array}{l}\underset{\omega \to (\frac{(2k-1)\pi}{h})-}{\mathrm{lim}}{A}_{0}(\omega ,h)=\underset{\omega \to (\frac{(2k-1)\pi}{h})-}{\mathrm{lim}}{A}_{1}(\omega ,h)=\underset{\omega \to (\frac{(2k\pi )}{h})-}{\mathrm{lim}}{A}_{0}(\omega ,h)=\underset{\omega \to (\frac{(2k\pi )}{h})+}{\mathrm{lim}}{A}_{1}(\omega ,h)=+\infty \\ \underset{\omega \to (\frac{(2k-1)\pi}{h})+}{\mathrm{lim}}{A}_{0}(\omega ,h)=\underset{\omega \to (\frac{(2k-1)\pi}{h})+}{\mathrm{lim}}{A}_{1}(\omega ,h)=\underset{\omega \to (\frac{(2k\pi )}{h})+}{\mathrm{lim}}{A}_{0}(\omega ,h)=\underset{\omega \to (\frac{(2k\pi )}{h})+}{\mathrm{lim}}{A}_{1}(\omega ,h)=-\infty \end{array}$$

*The curves C*_{0}(*h*) *intersect C*_{*} *exactly once at* $(-\frac{1}{h},\frac{1}{h})$ *for each positive number h. Moreover, C*_{k}(*h*) *do not intersect C*_{*} *for k* ∈ ℕ – {0}:

*Proof*. Intersection of *C*_{0}(*h*) and *C*_{*} is obvious from (15). For the second part of Lemma 3.1, suppose that if *C*_{k}(*h*) and *C*_{*} has intersection points there exist *ω* ∈ *J*_{k} for equations (13)-(14) which satisfies equation (10). By using equations (13)-(14) in equation (10) we have

$$\frac{\omega \mathrm{cos}(\omega h)}{\mathrm{sin}(\omega h)}=\frac{\omega}{\mathrm{sin}(\omega h)}.$$

There is no solution *ω* ∈ *J*_{k} for *k* ∈ ℕ – {0} which is a contradiction. □

*The curves C*_{k}(*h*_{0}) *do not intersect each other for h*_{0} ∈ ℝ^{+} .

*Proof*. Suppose that there exist an intersection point. It means that, there exist *ω*_{1} ≠ *ω*_{2} ∈ ℝ^{+} such that *A*_{0}(*ω*_{1}, *h*_{0}) = *A*_{0}(*ω*_{2}, *h*_{0}) and *A*_{1}(*ω*_{1}, *h*_{0}) = *A*_{1}(*ω*_{2}, *h*_{0}). These equalities imply that

$$\frac{{\omega}_{1}}{\mathrm{sin}({\omega}_{1}{h}_{0})}=\frac{{\omega}_{2}}{\mathrm{sin}({\omega}_{2}{h}_{0})}\frac{{\omega}_{1}\mathrm{cos}({\omega}_{1}{h}_{0})}{\mathrm{sin}({\omega}_{1}{h}_{0})}=\frac{{\omega}_{2}\mathrm{cos}({\omega}_{2}{h}_{0})}{\mathrm{sin}({\omega}_{2}{h}_{0})}$$(16)

from equation (13) and (14). For *n* ∈ ℕ, *ω*_{1}*h*_{0} ≠ *ω*_{2}*h*_{0} + 2*nπ* is obtained from the left equality in (16) because of *ω*_{1} ≠ *ω*_{2}: In addition, left and right equalities in (16) lead to cos(*ω*_{1}*h*_{0}) = cos(*ω*_{2}*h*_{0}) which is a contradiction. □

*The curve C*_{k}(*h*_{0}) *intersects the line A*_{0} = 0 *exactly once for h*_{0} ∈ ℝ^{+}*. Moreover, the intersection point* (0, *P*_{k}) *satisfies the following inequalities*

$$\begin{array}{l}{P}_{k}<{P}_{k+2}\text{\hspace{0.17em}}for\text{\hspace{0.17em}}=2n,\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\in \mathbb{N}\hfill \\ {P}_{k+2}<{P}_{k}\text{\hspace{0.17em}}for\text{\hspace{0.17em}}k=2n+1,\text{\hspace{0.17em}}n\in \mathbb{N}.\hfill \end{array}$$

*Proof*. When *ω* ∈ *J*_{k}, the equation *A*_{0}(*ω*, *h*_{0}) = 0 implies $\omega =\frac{\pi +2k\pi}{2{h}_{0}}.$Hence,

$${P}_{k}=\{\begin{array}{l}\frac{\pi +2k\pi}{2{h}_{0}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for\hspace{0.17em}\hspace{0.17em}}k=2n,\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\in \mathbb{N}\hfill \\ -\frac{\pi +2k\pi}{2{h}_{0}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for\hspace{0.17em}\hspace{0.17em}}k=2n+1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\in \mathbb{N}.\hfill \end{array}$$

is obtained by substituting $\omega =\frac{\pi +2k\pi}{2{h}_{0}}.$ This completes the proof. □

*The solution of equation*

$${u}^{\prime}(t)=-{A}_{0}u(t)-{A}_{1}u(t-h),\text{\hspace{0.17em}}for\text{\hspace{0.17em}}h\in {\mathbb{R}}^{+}$$(17)

*is asymptotically stable, i.e., all the roots of equation*

$$\lambda +{A}_{0}+{A}_{1}{e}^{-\lambda h}=0$$(18)

*have negative real parts, if and only if*

(a) $-\frac{1}{h}<{A}_{0}$

(b) $-{A}_{0}<{A}_{1}<\frac{\omega}{\mathrm{sin}(\omega h)}$ *where ω is the root of* ${A}_{0}=-\frac{\omega \mathrm{cos}(\omega h)}{\mathrm{sin}(\omega h)}$ *such that ωh* ∈ (0, *π*)

*Proof*. When *A*_{0} > 0 and *A*_{1} = 0, the solution of equation is clearly asymptotically stable. The stability region which includes half line *A*_{0} > 0 and *A*_{1} = 0, lies above *C*_{*} and below *C*_{0}(*h*) because of Lemmaa 3.1, 3.2 and 3.3. The conditions (a)–(b) are algebraic representation of this region in parameter space (*A*_{0}, *A*_{1}).

To find the number of roots with positive real parts in each region in the parameter space determined by the D-curves, we use the following the ideas from [38]. Writing *λ* = *μ* + *iω* with *μ*, ω ∈ ℝ in characteristic equation g(*λ*, *A*_{0}, *A*_{1}), we find two real equations

$${G}_{1}(\mu ,\omega ,{A}_{0},{A}_{1}):=\mathrm{Re}(g(\mu ,\omega ,{A}_{0},{A}_{1}))=0$$(19)

$${G}_{2}(\mu ,\omega ,{A}_{0},{A}_{1}):=\mathrm{Im}(g(\mu ,\omega ,{A}_{0},{A}_{1}))=0$$(20)

for the real and imaginary parts of *λ*. Direction of movement of an element is determined by the following proposition, using Jacobian matrix *J* defined by

$$J={\left[\begin{array}{ll}\frac{\partial {G}_{1}}{\partial {A}_{0}}\hfill & \frac{\partial {G}_{1}}{\partial {A}_{1}}\hfill \\ \frac{\partial {G}_{2}}{\partial {A}_{0}}\hfill & \frac{\partial {G}_{2}}{\partial {A}_{1}}\hfill \end{array}\right]}_{\mu =0}$$

*The pure imaginary roots enter the right half-plane for parameters sets in the* (*A*_{0}, *A*_{1}) *parameter region to the left of the D-curves, when we follow this curve in the direction of increasing ω, whenever* det(*J*) < 0 *and to the right when* det(*J*) > 0 *[38]*.

Since the determinant of Jacobian matrix of equation (18) satisfies the following inequalities

$$\begin{array}{l}\mathrm{det}(J)<0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\omega \in {J}_{2k}\hfill \\ \mathrm{det}(J)>0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\omega \in {J}_{2k+1}\hfill \end{array},$$

the pure imaginary roots move into the right half-plane when moving away in the parameter space to the left of *C*_{2k}(*h*) and to the right of *C*_{2k+1}(*h*), with "left" and "right" as determined w.r.t. a counter clock-wise tracking of *C*_{2k}(*h*) and a clock-wise tracking of *C*_{2k+1}(*h*) respectively.

In Fig. 1 these results are illustrated for *h* = 1. The curves (13) and (14) and the straight line (10) form the D-partition are shown and the number of roots in the right half plane is indicated for each region.

Fig. 1 The member of the D-curves family *C*_{k}(*h*) in the parameter space (*A*_{0}, *A*_{1}) for *h* = 1. The arrows along the curves refer to the direction of increasing *ω*. The numbers s in the different regions bordered by the curves indicate the number of roots in the right half plane.

Until Theorem 3.4, parameter is taken as a real constant. Now we determine how stability region varies when h is varying.

*The members of family of curves C*_{k}(*h*) *do not intersect each other for k* ∈ ℕ*. Proof*. Suppose that there exists an intersection point for *h*_{1} < *h*_{2}. It means that, there exist *ω*_{1}*h*_{1}, *ω*_{2}*h*_{2} ∈ (*kπ*, (*k* + 1)*π*) such that

$$\frac{{\omega}_{1}\mathrm{cos}({\omega}_{1}{h}_{1})}{\mathrm{sin}({\omega}_{1}{h}_{1})}=\frac{{\omega}_{2}\mathrm{cos}({\omega}_{2}{h}_{2})}{\mathrm{sin}({\omega}_{2}{h}_{2})}$$(21)

$$\frac{{\omega}_{1}}{\mathrm{sin}({\omega}_{1}{h}_{1})}=\frac{{\omega}_{2}}{\mathrm{sin}({\omega}_{2}{h}_{2})}.$$(22)

Cos(*ω*_{1}*h*_{1}) = cos(*ω*_{2}*h*_{2}) is obtained by using (21) and (22) which implies that *ω*_{1}*h*_{1} = *ω*_{2}*h*_{2} because of *ω*_{1}*h*_{1}, *ω*_{2}*h*_{2} ∈ (*kπ*, (*k* + 1)*π*). Therefore we have *ω*_{1} ≠ *ω*_{2} from the assumption *h*_{1} < *h*_{2} which contradicts (22).

*If h*_{1} < *h*_{2}*, C*_{0}(*h*_{2}) *lies below C*_{0}(*h*_{1}) *in parameter space* (*A*_{0}, *A*_{1}).

*Proof*. Taking the derivative of (13) and (14) with respect to *h*, we obtain

$$\begin{array}{l}\frac{\partial {A}_{0}}{\partial h}=\frac{{\omega}^{2}}{{\mathrm{sin}}^{2}(\omega h)}\hfill \\ \frac{\partial {A}_{1}}{\partial h}=\frac{-{\omega}^{2}\mathrm{cos}(\omega h)}{{\mathrm{sin}}^{2}(\omega h)}.\hfill \end{array}$$

It implies that, *A*_{0}(*ω*, *h*) is a monotone increasing function and *A*_{1}(*ω*, *h*) is a monotone decreasing function for $\omega h\in (0,\frac{\pi}{2})$. For each *ωh* value, functions *A*_{0}(*ω*, *h*) and *A*_{1}(*ω*, *h*) represent coordinates of points on *C*_{0}(*h*). Therefore, if *h*_{1} < *h*_{2}, the point (*A*_{0}(*ω*, *h*_{2}), *A*_{1}(*ω*, *h*_{2})) lies below the point (*A*_{0}(*ω*, *h*_{1}), *A*_{1}(*ω*, *h*_{1})) for *ωh*_{1}, $\omega {h}_{2}\in (0,\frac{\pi}{2})$. Suppose that a part of *C*_{0}(*h*_{2}) lies above *C*_{0}(*h*_{1}) for *ωh*_{1}, $\omega {h}_{2}\in [\frac{\pi}{2},\pi )$, then there exists at least one intersection point of *C*_{0}(*h*_{2}) and *C*_{0}(*h*_{1}), which contradicts Lemma 3.6. □

The curves *C*_{0}(*h*) are shown for *h* = 0:25; 0:75, 1, 1:3 in Fig. 2.

Fig. 2 The members of the D-curves family *C*_{0}(*h*) in the parameter space (*A*_{0}, *A*_{1}) for *h* = 0:25, 0:5, 0:75, 1 and 1:3

*Let’s define the set S*_{h} as follows

$${S}_{h}=\{({A}_{0},{A}_{1})|{A}_{0},{A}_{1}\in \mathbb{R}\text{\hspace{0.17em}}and\text{\hspace{0.17em}\hspace{0.17em}}satisfy\text{\hspace{0.17em}\hspace{0.17em}}the\text{\hspace{0.17em}\hspace{0.17em}}conditions\text{\hspace{0.17em}}(a)\text{\hspace{0.17em}}and\text{\hspace{0.17em}}(b)\text{\hspace{0.17em}}for\text{\hspace{0.17em}}h\in {\mathbb{R}}^{+}.\}$$

*If h*_{1} < *h*_{2} *then S*_{h}_{1} ⊂ *S*_{h}_{2}.

*Proof*. It is clear from Lemma 3.7

*The solution of equation* (1) *with delay term τ*(*u*(*t*)) > 0 *which satisfies the condition* 0 < *τ*(*u*(*t*)) < *M*_{1} *for all t* ∈ ℝ^{+} *is asymptotically stable if and only if the following conditions are satisfied*:

(ã) $-\frac{1}{{M}_{1}}<{A}_{0}$

$(\tilde{b})-{A}_{0}<{A}_{1}<\frac{\omega}{\mathrm{sin}(\omega {M}_{1})}$ *where ω is the root of* ${A}_{0}=-\frac{\omega \mathrm{cos}(\omega {M}_{1})}{\mathrm{sin}(\omega {M}_{1})}$ *such that ωM*_{1} ∈ (0, *π*)

*Proof*. It is obvious form Theorem 2.1 and Proposition 3.8 that if the conditions. (ã) and $(\tilde{b})$ are satisfied, all roots of characteristic equation of equation (1) have negative real parts. □

*A delay value of DDE is called critical delay if DDE has pure imaginary or zero eigenvalues at this delay value*.

Critical delays of an equation are the values at which the qualitative behavior of the system changes. Between any two successive critical values, the behavior of the solution does not change.

Now, by using transformation (5) we rewrite the critical delay values of equation (17) in terms of parameter.

*λ* = *iω is a root of equation* (18) *for some h if and only if λ* = *iω is also a root of*

$$T{\lambda}^{2}+(1+{A}_{0}T-{A}_{1}T)\lambda +{A}_{0}+{A}_{1}=0$$(23)

*for some* T ≥ 0

*Proof*. Let *λ* = *iω* be a root of equation (18). By using transformation (5) in equation (18), we obtain

$$i\omega +{A}_{0}+{A}_{1}\frac{1-i\omega T}{1+i\omega T}=0.$$

Multiplying this equation by 1 + *iωT* and arranging properly, we get

$$T{(i\omega )}^{2}+(1+{A}_{0}T-{A}_{1}T)i\omega +{A}_{0}+{A}_{1}=0$$

which implies that *λ* = *iω* is a root of equation (23) for $h=\frac{2}{\omega}(\mathrm{arctan}(\omega T)+p\pi )$. Moreover, the singular cases of Rekasius transform (5) are satisfied for equation (16) and equation (21).

As a result, $T=\frac{1}{{A}_{1}-{A}_{0}},\omega =\pm \sqrt{{A}_{1}^{2}-{A}_{0}^{2}}$ and critical delays

$${h}_{p}=\frac{2}{\sqrt{{A}_{1}^{2}-{A}_{0}^{2}}}(\mathrm{arctan}(\frac{\sqrt{{A}_{1}^{2}-{A}_{0}^{2}}}{{A}_{1}-{A}_{0}}+p\pi )),p\in \text{\hspace{0.17em}\hspace{0.17em}}\mathbb{Z}$$

are obtained under the condition |*A*_{0}| < *A*_{1}. Let *h*_{n} denote least *h*_{p} value which is greater than 0. Therefore, the solution of equation (17) is stable for *h* ∈ (0, *h*_{n}) when 0 < *A*_{0} < *A*_{1}. Hence we can state the following result about stability of the solution for the equation (1).

*The solution of the equation* (1) *with delay τ*(*u*(*t*)) > 0 *such that* 0 < *τ*(*u*(*t*)) < *M*_{1} *for all t* ∈ ℝ^{+} *is asymptotically stable under the conditions |A*_{0}| < *A*_{1} *and M*_{1} < *h*_{n}.

Now, we give a stability criterion which is independent from delay for equation (1) by using D-partition method.

*If* |*A*_{1}| < *A*_{0}*, then the solution of equation (1) is asymptotically stable*.

*Proof*. It is obvious from (11) that, |*A*_{1}(*ω*)| ≥ |*A*_{0}(*ω*)| for all *ω* ∈ *J*_{k}. Therefore, all of the D-curves is in this region, i.e., there is no D-curve in the region described by |*A*_{1}| < |*A*_{0}|. Moreover, the half line *A*_{0} > 0 and *A*_{1} = 0 on which the solution equation (1) is asymptotically stable, is in the region described by |*A*_{1}| < *A*_{0}.

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