# Nonoscillation, maximum principles, and exponential stability of second order delay differential equations without damping term

- Alexander Domoshnitsky
^{1}Email author

**2014**:361

https://doi.org/10.1186/1029-242X-2014-361

© Domoshnitsky; licensee Springer. 2014

**Received: **23 May 2014

**Accepted: **9 September 2014

**Published: **24 September 2014

## Abstract

Delays, arising in nonoscillatory and stable ordinary differential equations, can induce oscillation and instability of their solutions. That is why the traditional direction in the study of nonoscillation and stability of delay equations is to establish a smallness of delay, allowing delay differential equations to preserve these convenient properties of ordinary differential equations with the same coefficients. In this paper, we find cases in which delays, arising in oscillatory and asymptotically unstable ordinary differential equations, induce nonoscillation and stability of delay equations. We demonstrate that, although the ordinary differential equation ${x}^{\u2033}(t)+c(t)x(t)=0$ can be oscillating and asymptotically unstable, the delay equation ${x}^{\u2033}(t)+a(t)x(t-h(t))-b(t)x(t-g(t))=0$, where $c(t)=a(t)-b(t)$, can be nonoscillating and exponentially stable. Results on nonoscillation and exponential stability of delay differential equations are obtained. On the basis of these results on nonoscillation and stability, the new possibilities of non-invasive (non-evasive) control, which allow us to stabilize a motion of single mass point, are proposed. Stabilization of this sort, according to common belief, requires a damping term in the second order differential equation. Results obtained in this paper refute this delusion.

**MSC:**34K20.

## Keywords

## 1 Introduction

*f*. We try to construct a control which makes this correction automatically. A standard approach is to construct a feedback control $u(t)$ which depends on the difference $X(t)-Y(t)$ or more exactly on the difference $X(t-\tau (t))-Y(t-\tau (t))$ since in real systems delay in receiving signal and in reaction on this signal arises. The control terms like

*i.e.*${Y}^{\u2033}(t)=f(t)$, and subtracting this equality from (1.3), we get the following system:

for the difference $Z(t)=X(t)-Y(t)$.

${p}_{i}$ and ${\tau}_{i}$ are measurable essentially bounded functions (below we write this as ${p}_{i}\in {L}_{\mathrm{\infty}}$) ($i=1,\dots ,m$).

We understand the solution of (1.7) in the traditional sense like in the book [1], *i.e.* as a function $x:[0,+\mathrm{\infty})\to (-\mathrm{\infty},+\mathrm{\infty})$ with absolutely continuous derivative ${x}^{\prime}$ on every finite interval such that ${x}^{\u2033}\in {L}_{\mathrm{\infty}}$, satisfying (1.7).

*p*and

*τ*(see [[4], Chapter III, Section 16, pp.105-106]). Note that the degree of instability of (1.6) (the maximum among positive real parts of the roots to (1.11)) tends to zero when $\tau \to 0$ and $\tau \to +\mathrm{\infty}$ [4]. The results on existence of unbounded solutions in the case of variable coefficients and delays were obtained on the basis of a growth of the Wronskians in [5] and developed in [6, 7]. One of the results can be formulated as follows [6]: if there exists a positive constant

*ε*such that $\tau (t)>\epsilon $ and $p(t)>\epsilon $, then there exist unbounded solutions to (1.7). In the paper [8] results on the instability of the system

where $\omega >0$ and $P(t)$ is an *ω*-periodic symmetric matrix function, were obtained.

*α*were obtained in the paper [9]. The asymptotic formulas of solutions to the second order equation (1.6) are presented in [[4], Chapter III, Section 16] and to (1.7) with a delay $\tau (t)$ summable on the semiaxis in [10, 11]. In [6] it was found that all solutions of (1.7) with positive nondecreasing and bounded coefficient $p(t)$ and nondecreasing $h(t)\equiv t-\tau (t)$ are bounded if and only if

where ${a}_{j}$, *q* and *λ* are constants, was proposed in [12] and developed in [13], where the equations with a combination of delayed and advanced arguments are considered. A study of advanced equations (see, for example, (1.7), where $\tau (t)\le 0$) can be found in the paper [14] in which results on boundedness, stability, and asymptotic representations of solutions are obtained.

with constant delay *τ* and coefficients ${p}_{1}$ and ${p}_{2}$ and without damping term were obtained in the paper [15]. These results were based on Pontryagin’s technique for analysis of the roots of quasi-polynomials [22] and could not be used in the case of (1.8) with variable coefficients and/or delays. Note also that the assumption as regards the absence of the delay in the first term does not allow us to use this result of [15] in the stabilization in the model of the motion described in (1.1) by the delay control (1.2). Let us try to imagine situations in which variable delays and coefficients arising in the delayed feedback control may be important: (1) the case of control for missiles, where the delay depends on their distance from the controller and is variable; (2) spending of fuel implies the change of the mass of the missiles, which leads to variable coefficients in the delay system. As a conclusion, it should be stressed that there are no results for the exponential stability of the second order equation (1.8) without damping term in the case of variable coefficients and/or delays.

It will be demonstrated that the control of the form (1.2), where $m\ge 2$, can stabilize system (1.1). We obtain results as regards the exponential stability of (1.8), which are based on the maximum principles for the second order delay differential equation (1.8). Denoting $(Mx)(t)={x}^{\u2033}(t)+{\sum}_{i=1}^{m}{p}_{i}(t)x(t-{\tau}_{i}(t))$, we can formulate the maximum inequalities principle [23] in the form: it follows from the inequalities $(My)(t)\ge (Mx)(t)$, $y(0)\ge x(0)$, ${y}^{\prime}(0)\ge {x}^{\prime}(0)$ that $y(t)\ge x(t)$ for all $t\in [0,\mathrm{\infty})$. Using the solutions’ representation (see (2.5) below), we can see that this principle is reduced to positivity of the Cauchy (fundamental) function of (1.8) and positivity of its nontrivial solution with nonnegative initial conditions. Nonoscillation of solutions in many cases leads to this property of the solution of second order equation (1.8). The positivity of the Cauchy function will open a way to the analysis of the asymptotic stability of nonlinear delay differential equations on the basis of the known schemes of upper and lower functions. The maximum boundedness principle [23] can be described in the form: there exists a constant Ω such that the solution of the initial value problem $(Mx)(t)=f(t)$, $x(0)=\alpha $, ${x}^{\prime}(0)=\beta $ satisfies the inequality $|x(t)|\le \mathrm{\Omega}(|f(t)|+|\alpha |+|\beta |)$ for $t\in [0,\mathrm{\infty})$. A generalization of the Perron theorem (see, for example, Theorem 3.5 in [1]) claims the exponential stability in this case.

The paper is organized as follows. After formulation of stabilization problem in Section 1, we discuss formulas of solution representations in Section 2. Simple results on stability of delay differential equations in the case $m=2$, which will be proven on the basis of general assertions obtained in Section 5, are formulated in Section 3. Auxiliary results used in the proof of our main assertions can be found in Section 4. Results on the positivity of the Cauchy function $C(t,s)$, nonoscillation and exponential stability are obtained in Sections 5 and 6. Discussion and open problems are presented in Section 7.

## 2 About representation of solutions for second order delay differential equations

*f*, ${p}_{i}$,

*φ*, ${\tau}_{i}$ ($i=0,1,\dots ,n-1$), and ${\tau}_{i}(t)\ge 0$ for $t\ge 0$. In the traditional approach (see, for example, [4, 24, 25]), the homogeneous equation is considered as the equation

*φ*, the space of its solutions is infinite-dimensional. In [26] a homogeneous equation was defined as (2.3) with the zero initial function

This definition of the homogeneous equation allows researchers to build an analog of the classical general theory of ordinary differential equations (ODEs) for functional differential equations using the key notions of the classical ODEs theory, and it is based on the fact that the space of the solutions of the second order equation (2.3), (2.4) becomes two-dimensional. Note that (2.3), (2.2) can be written as (2.1), (2.4), where $f(t)=-{\sum}_{i=1}^{m}{p}_{i}(t)\phi (t-{\tau}_{i}(t))\{1-\chi (t-{\tau}_{i}(t))\}$, $\chi (t)=1$ for $t\ge 0$, and $\chi (t)=0$ for $t<0$. Thus (2.1), (2.4) is a nonhomogeneous one according to the approach of the book [1].

*s*the function $C(t,s)$ as a function of the variable

*t*satisfies the equation [1, 26]

and the initial conditions $C(s,s)=0$, ${C}_{t}^{\prime}(s,s)=1$. Note that we will use this definition of the Cauchy function for the construction of $C(t,s)$ below. The behavior of the fundamental system ${x}_{1}$, ${x}_{2}$ of solutions of (2.3), (2.4) determines the existence and uniqueness of solutions of the boundary value problems for this equation. Positivity of the Cauchy function $C(t,s)$ of (2.1), (2.4) is a basis of various theorems about differential inequalities (under corresponding conditions, the solution of an inequality is greater than the solution of an equation). These theorems help essentially in the study of stability for delay differential equations and their natural generalization - functional differential equations (FDEs) [1].

Let us formulate several definitions concerning stability.

**Definition 2.1**Equation (2.3) is uniformly exponentially stable if there exist $N>0$ and $\alpha >0$, such that the solution of (2.3), (2.9), where

where *N* and *α* do not depend on ${t}_{0}$.

**Definition 2.2**We say that the Cauchy function $C(t,s)$ of (2.1) satisfies the exponential estimate if there exist positive

*N*and

*α*such that

It is well known that for (2.1) with bounded delays these two definitions are equivalent [1].

## 3 Tests of positivity of the Cauchy functions and stability

can be oscillating and asymptotically unstable, the delay equation (2.3), (2.4) under corresponding conditions on the coefficients ${p}_{i}(t)$ and delays ${\tau}_{i}(t)$ is nonoscillating and exponentially stable. The basic idea of our approach is to avoid the condition on the nonnegativity of the coefficients ${p}_{i}(t)$ for all $i=1,\dots ,m$, and to allow terms with positive and terms with negative coefficients ${p}_{i}(t)$ to compensate each other.

**Theorem 3.1**

*Assume that*$0\le \tau (t)\le \theta (t)$, $0\le b(t)\le a(t)$,

*Then*

- (1)
*the Cauchy function*$C(t,s)$*of*(3.2)*is nonnegative for*$0\le s<t<+\mathrm{\infty}$; - (2)
*if there exists such positive**ε**such that*$a(t)-b(t)\ge \epsilon ,$(3.7)

*then the Cauchy function*$C(t,s)$

*of*(3.2)

*satisfies the exponential estimate*(2.11)

*and the integral estimate*

- (3)
*if there exists*${lim}_{t\to \mathrm{\infty}}\{a(t)-b(t)\}=k$,*with*$k>0$,*then*$\underset{t\to \mathrm{\infty}}{lim}{\int}_{0}^{t}C(t,s)\phantom{\rule{0.2em}{0ex}}ds=\frac{1}{k}.$(3.9)

**Corollary 3.1**

*Assume that the delays*$\tau (t)\equiv \tau $, $\theta (t)\equiv \theta $

*are constants and*

*Then the assertions of Theorem * 3.1 *are true*.

*a*, and noted above in Section 1 sufficient conditions for existence of unbounded solutions from the paper [6]). Together with the fact of the oscillation [27] of all solutions this leads to chaos in behavior of solutions in the sense that small errors in a measurement of $x({t}_{0})$ and ${x}^{\prime}({t}_{0})$ can imply unpredictable changes in $x({t}_{0}+\omega )$ and ${x}^{\prime}({t}_{0}+\omega )$. Thus (3.12) is unstable. To stabilize its solution $x(t)$ to the given ‘trajectory’ $y(t)$ satisfying this equation, we choose the control in the form

for the difference $z(t)=x(t)-y(t)$. The stabilization has been reduced to the stability of this equation. A possible algorithm to construct this stabilizing control is clear now: first we choose the delay *θ* close to *τ* such that condition (3.11) is fulfilled, then we choose $b(t)$ close to $a(t)$ such that condition (3.10) is fulfilled.

**Example 3.1**Stabilizing (3.12), where $a(t)\equiv a$, let us choose the control in the form (3.13) with constant coefficient $b(t)\equiv b$. We come to the study of the exponential stability of the equation

**Example 3.2**The equation

In the following assertion we assume the smallness of the difference of the delays $\theta -\tau $ and do not assume the smallness of the delay *θ*.

with constant delays *τ* and *θ* and positive coefficients $a(t)$ and $b(t)$.

**Theorem 3.2**

*Assume that*$0<\tau <\theta $,

*there exists a positive*

*ε*

*such that*

*and*

*is fulfilled*.

*Then*

- (1)
*the Cauchy function*$C(t,s)$*of*(3.19)*is nonnegative for*$0\le s<t<+\mathrm{\infty}$; - (2)
*the solutions*${x}_{1}(t)$, ${x}_{2}(t)$*of*(3.19),*satisfying initial conditions*(2.6),*are nonegative for*$0<t<+\mathrm{\infty}$; - (3)
*the Cauchy function*$C(t,s)$*of*(3.19)*satisfies the exponential estimate*(2.11)*and the integral estimate*(3.8); - (4)
*if there exists*${lim}_{t\to \mathrm{\infty}}\{a(t)-b(t)\}=k$,*then equality*(3.9)*is fulfilled*.

**Example 3.3**Consider the equation

which Cauchy function is $C(t,s)=sin(t-s)$. Let us demonstrate that condition (3.6) is essential for positivity of the Cauchy function $C(t,s)$ in Theorem 3.1, assuming that the numbers *b* and *θ* are chosen such that all other conditions of Theorem 3.1 are fulfilled. If $\pi <\theta $, then in the triangle $0\le s\le t<\theta $, its Cauchy function $C(t,s)=sin(t-s)$ changes its sign. The same example demonstrates that condition (3.22) is essential in Theorem 3.2 for the positivity of $C(t,s)$. Note that in the case $\frac{\pi}{2}<\theta $, the solution ${x}_{1}(t)=cost$ satisfying conditions (2.6) changes its sign on the interval $[0,\theta ]$. It is clear that inequality (3.22) implies $\frac{\pi}{2}>\theta $.

**Remark 3.1**Note the closely related results for the first order delay differential equation

is nonoscillating and exponentially stable if $b>0$, $0<(a-b)\theta \le \frac{1}{e}$, $0<a(\theta -\tau )\le \frac{1}{e}$.

## 4 Auxiliary results

Let us formulate several results on systems of functional differential equations with Volterra operators. We mean the Volterra operators according to the following Tikhonov definition. Assume that $B:{C}_{[0,\mathrm{\infty})}\to {L}_{[0,\mathrm{\infty})}^{\mathrm{\infty}}$ is a linear bounded operator, where ${C}_{[0,\mathrm{\infty})}$ and ${L}_{[0,\mathrm{\infty})}^{\mathrm{\infty}}$ are the spaces of continuous and essentially bounded functions $x:[0,+\mathrm{\infty})\to (-\mathrm{\infty},+\mathrm{\infty})$ respectively.

**Definition 4.1** We say that an operator $B:{C}_{[0,\mathrm{\infty})}\to {L}_{[0,\mathrm{\infty})}^{\mathrm{\infty}}$ is a Volterra one if for every two functions ${y}_{1}$ and ${y}_{2}$ and every *ω*, the equality ${y}_{1}(t)={y}_{2}(t)$ for $t\in [0,\omega ]$ implies the equality $(B{y}_{1})(t)=(B{y}_{2})(t)$ for $t\in [0,\omega ]$.

where $K(t,s)={\{{K}_{ij}(t,s)\}}_{i,j=1,2}$ is called the Cauchy matrix of system (4.1), $x=col\{{x}_{1},{x}_{2}\}$, $f=col\{{f}_{1},{f}_{2}\}$.

**Lemma 4.1**

*Let the Cauchy function*$k(t,s)$

*of the scalar equation*

*be positive in*$0\le s\le t<+\mathrm{\infty}$, ${B}_{12}$

*and*$(-{B}_{21})$

*be positive operators*.

*Then the following assertions are equivalent*:

- (1)
${K}_{21}(t,s)\ge 0$

*and*${K}_{22}(t,s)>0$*for*$0\le s\le t<+\mathrm{\infty}$; - (2)
*there exists a vector function*$v=col\{{v}_{1},{v}_{2}\}$*with absolutely continuous components and essentially bounded derivatives*${v}^{\prime}=col\{{v}_{1}^{\prime},{v}_{2}^{\prime}\}$*such that*${v}_{2}(t)>0,\phantom{\rule{2em}{0ex}}({M}_{i}v)(t)\le 0,\phantom{\rule{1em}{0ex}}i=1,2,t\in [0,+\mathrm{\infty}),\phantom{\rule{2em}{0ex}}{v}_{1}(0)\le 0.$(4.4)

This lemma follows from Theorem 16.6 for the case $n=2$ (see [[28], p.409]).

In order to formulate the conditions of positivity of the Cauchy function $k(t,s)$ of the scalar first order equation (4.3), let us introduce the function $H:[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ which describes the ‘size’ of the memory of the operator ${B}_{11}$.

**Definition 4.2** Let us define the function $H(t):[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ as the maximal possible value on $[0,t]$ for which the equality ${y}_{1}(s)={y}_{2}(s)$ for $s\in [H(t),+\mathrm{\infty})$ for every two continuous functions ${y}_{1}$ and ${y}_{2}:[0,+\mathrm{\infty})\to (-\mathrm{\infty},+\mathrm{\infty})$ implies the equality $({B}_{11}{y}_{1})(s)=({B}_{11}{y}_{2})(s)$ for almost every $s\in [t,+\mathrm{\infty})$.

**Lemma 4.2**

*Let*${B}_{11}:{C}_{[0,\mathrm{\infty})}\to {L}_{[0,\mathrm{\infty})}^{\mathrm{\infty}}$

*be a positive linear Volterra operator and*

*then nontrivial solutions of the homogeneous equation*

*have no zeros for* $t\in [0,+\mathrm{\infty})$ *and its Cauchy function* $C(t,s)>0$ *for* $0\le s\le t<+\mathrm{\infty}$.

This lemma follows from Theorem 15.7 (see [[28], p.358]).

Denote ${H}^{\ast}={esssup}_{t\ge 0}\{t-H(t)\}$.

Denoting $\parallel {B}_{11}\parallel $ the norm of the operator ${B}_{11}:{C}_{[0,\mathrm{\infty})}\to {L}_{[0,\mathrm{\infty})}^{\mathrm{\infty}}$, as a corollary of this assertion we get the following.

**Lemma 4.3** *If* $\parallel {B}_{11}\parallel {H}^{\ast}\le \frac{1}{e}$, *then the Cauchy function* $C(t,s)$ *of* (4.3) *is positive for* $0\le s\le t<+\mathrm{\infty}$.

**Remark 4.1**Let us consider the second order scalar functional differential equation

then ${K}_{21}(t,0)=z(t)$, ${K}_{11}(t,0)={z}^{\prime}(t)$.

## 5 Main results

**Theorem 5.1**

*Let*${(-1)}^{i+1}{p}_{i}(t)\u2a7e0$

*for*$i=1,\dots ,2m$, ${p}_{2i-1}(t)+{p}_{2i}(t)\ge 0$

*for*$i=1,\dots ,m$, $t\in [0,+\mathrm{\infty})$,

*and the Cauchy function of the first order equation*

*be positive for*$0\le s\le t<+\mathrm{\infty}$.

*Then the following assertions are equivalent*:

- (1)
*there exists a function**v**with absolutely continuous and essentially bounded derivative*${v}^{\prime}$*and essentially bounded derivative*${v}^{\u2033}$*such that*$v(t)>0,\phantom{\rule{2em}{0ex}}{v}^{\prime}(t)\le 0,\phantom{\rule{2em}{0ex}}(Mv)(t)\le 0,\phantom{\rule{1em}{0ex}}t\in [0,+\mathrm{\infty});$(5.4) - (2)
*there exists a bounded absolutely continuous function**u**with essentially bounded derivative*${u}^{\prime}$*such that*$\begin{array}{r}u(t)\ge 0,\\ {u}^{2}(t)-{u}^{\prime}(t)+\sum _{i=1}^{2m}{p}_{i}(t)\chi (t-{\tau}_{i}(t),0)exp\{{\int}_{t-{\tau}_{i}(t)}^{t}u(s)\phantom{\rule{0.2em}{0ex}}ds\}\le 0,\phantom{\rule{1em}{0ex}}t\in [0,+\mathrm{\infty});\end{array}$(5.5) - (3)
*the Cauchy function*$C(t,s)$*of*(5.1)*is nonnegative for*$0\le s<t<+\mathrm{\infty}$,*and solutions*${x}_{1}(t)$, ${x}_{2}(t)$*of*(5.2),*satisfying initial conditions*(2.6),*are such that*${x}_{1}(t)>0$, ${x}_{2}(t)\ge 0$*for*$0<t<+\mathrm{\infty}$.

The proof of Theorem 5.1 is based on several auxiliary results.

**Lemma 5.1** *Assertions* (1) *and* (2) *of Theorem * 5.1 *are equivalent*.

*Proof*(2) ⇒ (1). In order to prove this implication, let us choose the function $v(t)$ in the condition (1) in the form

It is clear now that inequalities (5.5) imply that $(Mv)(t)\le 0$. The implication (2) ⇒ (1) has been proven.

(1) ⇒ (2). Denote $u(t)=-\frac{{v}^{\prime}(t)}{v(t)}$, where the function $v(t)$ from the assertion (1), and demonstrate that this function satisfies the assertion (2). Obviously the function $u(t)$ is nonnegative. The function $v(t)$ satisfies the equation ${v}^{\prime}(t)+u(t)v(t)=0$ and consequently it can be represented in the form (5.6). The derivatives of the function $v(t)$ are defined by (5.7) and (5.8). From the inequality $(Mv)(t)\le 0$ we get inequality (5.9) and consequently condition (5.5) is fulfilled.

The implication (1) ⇒ (2) has been proven. □

*Proof of Theorem 5.1*(1) ⇒ (3). Let us rewrite (5.1) in the form

Under the conditions ${(-1)}^{i+1}{p}_{i}(t)\u2a7e0$, ${p}_{2i-1}(t)+{p}_{2i}(t)\ge 0$, ${\tau}_{2i-1}(t)\le {\tau}_{2i}(t)$ of Theorem 5.1, the operators ${B}_{11}$, ${B}_{12}$, and $(-{B}_{21})$ are positive. Note that, according to the condition of Theorem 5.1, the Cauchy function of the first order equation (5.3) is positive for $0\le s\le t\le +\mathrm{\infty}$. The vector function $col\{{v}_{1}(t),{v}_{2}(t)\}$, where ${v}_{1}(t)={v}^{\prime}(t)$, ${v}_{2}(t)=v(t)$, satisfies all conditions of assertion (2) of Lemma 4.1. According to Lemma 4.1, we get the positivity of the element ${K}_{22}(t,s)$ and the nonnegativity of the element ${K}_{21}(t,s)$ of the Cauchy matrix of system (5.11) for $0\le s\le t\le +\mathrm{\infty}$. Remark 4.1 completes now the proof of the implication (1) ⇒ (3).

Noting that the equivalence (1) ⇔ (2) was obtained in Lemma 5.1, we can conclude that Theorem 5.1 has been proven. □

Denote ${H}^{\ast}={esssup}_{t\ge 0}{max}_{1\le i\le 2m}{\tau}_{i}(t)$. For bounded ${\tau}_{i}(t)$ for $i=1,\dots ,m$, we have ${H}^{\ast}<+\mathrm{\infty}$.

**Remark 5.1**The inequality

where we set ${p}_{2i}(s)=0$ for $s<0$, implies the positivity of the Cauchy function of the first order equation (5.3) (see Theorem 15.7, p.358 in [28]).

**Remark 5.2** There is a corresponding inconvenience in choosing the functions $v(t)$ because of the equality $v(t-{\tau}_{i}(t))=0$ for $t-{\tau}_{i}(t)<0$. Trying to avoid this inconvenience, we can make the following trick.

where the parameter *β* will be defined below in the formulation of Theorem 5.2. Define the norm of this operator as $\parallel B\parallel =max\{\beta ,{esssup}_{t\ge 0}{\sum}_{i=1}^{m}|{p}_{2i}(t)|[{\tau}_{2i}(t)-{\tau}_{2i-1}(t)]\}$.

with the same *β* as in (5.18), which will be defined below.

It is clear from the definition of the Cauchy function by (2.7), (2.8) that the Cauchy functions of (5.1) and (5.19) coincide in $0\le s\le t<+\mathrm{\infty}$.

**Theorem 5.2**

*Assume that*${(-1)}^{i+1}{p}_{i}(t)\u2a7e0$

*for*$i=1,\dots ,2m$, ${p}_{2i-1}(t)+{p}_{2i}(t)\ge 0$, ${\tau}_{2i-1}(t)\le {\tau}_{2i}(t)$

*for*$i=1,\dots ,m$

*and*$t\in [0,+\mathrm{\infty})$

*and there exists a real positive number*

*α*

*such that*

- (a)
$\parallel B\parallel {H}^{\ast}\le \frac{1}{e}$,

*where*$\beta =\alpha $*in*(5.18)*defining the operator**B*; - (b)
*the inequality*${\alpha}^{2}+\sum _{i=1}^{2m}{p}_{i}(t)exp\{\alpha {\tau}_{i}(t)\}\le 0,\phantom{\rule{1em}{0ex}}t\in [0,+\mathrm{\infty})$(5.20)

*is fulfilled*.

*Then*

- (1)
*the Cauchy function*$C(t,s)$*of*(5.19)*is nonnegative for*$0\le s<t<+\mathrm{\infty}$; - (2)
*if in addition there exists a positive**ε**such that*$\sum _{i=1}^{2m}{p}_{i}(t)\ge \epsilon ,$(5.21)

*then the Cauchy function*$C(t,s)$

*of*(5.1),

*its derivative*${C}_{t}^{\prime}(t,s)={K}_{11}(t,s)$

*and solutions of the homogeneous equation*$Mx=0$

*satisfy the exponential estimate and the estimate*

*is true*.

*Proof of Theorem 5.2* Let us start with the proof of the assertion (1). We set $v(t)=exp\{-\alpha (t-{H}^{\ast})\}$ for $t\in [-{H}^{\ast},+\mathrm{\infty})$ in the assertion (1) of the following reformulation of the assertion (1) ⇒ (3) of Theorem 5.1 for (5.19).

**Lemma 5.2**

*Let*${(-1)}^{i+1}{p}_{i}(t)\u2a7e0$

*for*$1=1,\dots ,2m$, ${p}_{2i-1}(t)+{p}_{2i}(t)\ge 0$, ${\tau}_{2i-1}(t)\le {\tau}_{2i}(t)$

*for*$i=1,\dots ,m$, $t\in [0,+\mathrm{\infty})$

*and the Cauchy function of the first order equation*

*where*$\beta =\alpha $,

*be positive for*$-{H}^{\ast}\le s\le t<+\mathrm{\infty}$.

*Then the assertion*(2)

*follows from the assertion*(1),

*where*

- (1)
*there exists a function**v**with absolutely continuous and essentially bounded derivative*${v}^{\prime}$*and essentially bounded derivative*${v}^{\u2033}$*such that*$v(t)>0,\phantom{\rule{2em}{0ex}}{v}^{\prime}(t)\le 0,\phantom{\rule{2em}{0ex}}(Mv)(t)\le 0,\phantom{\rule{1em}{0ex}}t\in [-{H}^{\ast},+\mathrm{\infty});$ - (2)
*the Cauchy function*$C(t,s)$*of*(5.19)*is nonnegative for*$0\le s<t<+\mathrm{\infty}$,*and the solutions*${x}_{1}(t)$, ${x}_{2}(t)$*of the homogeneous equation*$(Mx)(t)\equiv {x}^{\u2033}(t)+q(t){x}^{\prime}(t)+\sum _{i=1}^{2m}{p}_{i}(t)x(t-{\tau}_{i}(t))=0,\phantom{\rule{1em}{0ex}}t\in [-{H}^{\ast},+\mathrm{\infty}),$

*satisfying the initial conditions* (2.6), *are such that* ${x}_{1}(t)>0$, ${x}_{2}(t)\ge 0$ *for* $-{H}^{\ast}<t<+\mathrm{\infty}$.

*M*defined by (5.19), we obtain the following two inequalities on $[-{H}^{\ast},0]$ and $[0,+\mathrm{\infty})$. The first one is

*β*in the definition of the operator

*B*by (5.18) and the coefficient $q(t)$ in (5.19). The second inequality is

and, after carrying the exponent out of the brackets, we get inequality (5.20). Thus inequalities (5.24) and (5.25) are results of a choice of the corresponding function $v(t)$ in assertion (1) of Lemma 5.2.

where the operator *B* is defined by (5.18). The condition (a) implies, according to Lemma 4.3, the positivity of the Cauchy function of the first order equation (5.23).

Now we see that all conditions of Lemma 5.2 and assertion (1) are fulfilled. According to Lemma 5.2, we obtain assertion (1) of Theorem 5.2.

*f*such that $|f(t)|\le 1$ for $t\in [0,+\mathrm{\infty})$. It means that the function

*f*measurable and bounded on the semiaxis $t\in [0,+\mathrm{\infty})$. Each of the solutions of (5.19) can be presented on $[0,+\mathrm{\infty})$ as a solution of the equation

*x*of (5.36) is bounded and consequently the solution of (5.19) is bounded for every measurable bounded function $f(t)$, $t\in [-{H}^{\ast},+\mathrm{\infty})$ and

is bounded for every measurable bounded function *f*.

*M*defined by (5.19) and $f(t)=(My)(t)$. It is clear that this function

*f*is bounded on the semiaxis $t\in [-{H}^{\ast},+\mathrm{\infty})$. The solution

*y*of (5.38), (5.39) can be written in the form

It follows now from the boundedness of the integral (5.37) for every bounded *f* that ${x}_{2}(t)$ is bounded for $t\in [-{H}^{\ast},+\mathrm{\infty})$.

*α*satisfies inequality (5.20), the functions ${v}_{1}(t)=-\alpha exp\{-\alpha (t+{H}^{\ast})\}$, ${v}_{2}(t)=exp\{-\alpha (t+{H}^{\ast})\}$ satisfy the assertion (2) of Lemma 4.1. The condition (a) implies that all other conditions of Lemma 4.1 are also fulfilled. Now, according to Lemma 4.1, the entries in the second row of the Cauchy matrix $K(t,s)={\{{K}_{ij}(t,s)\}}_{i,j=1}^{2}$ of system (5.41) satisfy the inequalities ${K}_{21}(t,s)\ge 0$, ${K}_{22}(t,s)>0$ for $0\le s\le t<+\mathrm{\infty}$. Let us substitute the constant vector $z=col\{{z}_{1},{z}_{2}\}$, where ${z}_{1}=-1$ and ${z}_{2}=\frac{1+\parallel B\parallel}{\epsilon}$ instead of ${y}_{1}(t)$ and ${y}_{2}(t)$, respectively, into system (5.41). Condition (5.21) implies that $({M}_{1}y)(t)\ge 1$, $({M}_{2}y)(t)=1$. Consider now system (5.41), where ${f}_{1}(t)\equiv ({M}_{1}z)(t)$, ${f}_{2}(t)\equiv 1$ for $t\in [-{H}^{\ast},+\mathrm{\infty})$. From representation (4.2) of the solution we get for the second component of the solution vector

satisfying conditions (5.27). Now it follows from the boundedness of the integral ${\int}_{-{H}^{\ast}}^{t}{K}_{21}(t,s)\phantom{\rule{0.2em}{0ex}}ds$ and solutions ${x}_{1}$, ${x}_{2}$ that the integral ${\int}_{-{H}^{\ast}}^{t}{K}_{22}(t,s)\phantom{\rule{0.2em}{0ex}}ds$ is bounded, and, using the positivity of ${K}_{22}(t,s)$, we can claim that the integral ${\int}_{-{H}^{\ast}}^{t}{K}_{22}(t,s)f(s)\phantom{\rule{0.2em}{0ex}}ds$ is bounded for every measurable bounded $f(t)$, $t\in [-{H}^{\ast},+\mathrm{\infty})$. We see that the component ${y}_{2}$ of the solution vector to system (5.41) is bounded for every measurable bounded ${f}_{1}$ and ${f}_{2}$. The positivity of the Cauchy function of the first order equation (5.25) together with condition (5.21) imply the boundedness of the component ${y}_{1}$ of solution vector to system (5.41). Now a generalization of the Perron theorem (see, for example, Theorem 3.5 in [1]) claims that all entries of the Cauchy matrix $K(t,s)={\{{K}_{ij}(t,s)\}}_{i,j=1}^{2}$ satisfy the exponential estimate (2.11). Each of the solutions of (5.1) coincides with a corresponding solution of (5.19) with the measurable essentially bounded right-hand side $\psi (t)$ such that $\psi (t)=f(t)$ for $t\ge {H}^{\ast}$. Thus for every bounded right-hand side, the solution of (5.1) is bounded, and the exponential estimate (2.11) of the Cauchy function of (5.1), its derivative ${C}_{t}^{\prime}(t,s)={K}_{11}(t,s)$, and the solutions of the homogeneous equation $Mx=0$ satisfy the exponential estimate (2.11). □

**Corollary 5.1**

*If under the conditions of Theorem*5.2

*there exists a positive limit*

*then*

The proof of Corollary 5.1 can be obtained from equality (5.31) and the fact of the exponential estimate of all elements of the Cauchy matrix of system (5.41).

**Remark 5.3**The positivity of

*ε*in condition (5.21) is essential, as the following example demonstrates. Consider the equation

Here $\epsilon =0$ and one of solutions is a constant and does not tend to zero when $t\to +\mathrm{\infty}$.

${q}_{j}$, ${\theta}_{j}$, ${p}_{i}$ and ${\tau}_{i}$ are measurable essentially bounded functions for $j=1,\dots ,n$, $i=1,\dots ,m$.

**Theorem 5.3** *Let all assumptions of Theorem * 5.2 *be fulfilled*.

*Then*

- (1)
*if*${q}_{j}(t)\le 0$*for*$t\in [0,+\mathrm{\infty})$,*then the Cauchy function*$C(t,s)$*of*(5.47)*is positive for*$0\le s<t<+\mathrm{\infty}$; - (2)
*if there exist positive*${\epsilon}_{0}$*and**ε**such that*$\sum _{i=1}^{2m}{p}_{i}(t)\ge \epsilon ,\phantom{\rule{2em}{0ex}}\epsilon -{\epsilon}_{0}\ge \sum _{j=1}^{n}|{q}_{j}(t)|,\phantom{\rule{1em}{0ex}}t\in [0,+\mathrm{\infty}),$(5.50)

*then the Cauchy function*$C(t,s)$

*of*(5.47)

*satisfies the exponential estimate*(2.11)

*and the integral estimate*

*Proof*Using the formula of the solutions’ representation (2.5), we can conclude that the solution of the initial problem (5.47), (5.49), (5.52), where

*K*in the case of nonpositive ${q}_{j}(t)$ is positive. If we consider this operator

*K*on every finite interval $[0,\omega ]$ its spectral radius $\rho (K)$ is zero, and one can write

If $f\ge 0$, then $\psi \ge 0$ and consequently $x\ge 0$. Reference to (2.5) completes the proof of the assertion (1).

In order to prove the assertion (2), we have to demonstrate that the solution *x* is bounded for every bounded right-hand side *f* and then the extension of the Bohl-Perron implies the exponential estimate (2.11) of the Cauchy matrix (see Theorem 3.5 in the book [1]). According to Theorem 5.2, the Cauchy function $W(t,s)$ of (5.1) satisfies the exponential estimate (2.11). This implies the boundedness of *ψ* for every bounded *f*. If the norm of the operator $K:{C}_{[0,\mathrm{\infty})}\to {C}_{[0,\mathrm{\infty})}$ is less than one, then the bounded operator ${(I-K)}^{-1}$ exists. We have only to prove that $\parallel K\parallel <1$.

This, according to our observations above, completes the proof of exponential estimate (2.11) of the Cauchy function of (5.47).

*x*of the initial problem (5.47), (5.49), (5.52), we have two representations

This completes the proof of the assertion (2) of Theorem 5.3. □

## 6 Proofs of Theorems 3.1 and 3.2

*Proof of Theorem 3.1*For (3.2), inequality (5.20) is of the following form:

The inequality in the condition (a) of Theorem 5.2 is fulfilled if inequality (3.6) is satisfied.

Now Theorem 3.1 follows from Theorem 5.2 and Corollary 5.1. □

**Remark 6.1** We have not yet used the assertion (2) of Theorem 5.1 to obtain sufficient conditions for the positivity of the Cauchy function $C(t,s)$ of (3.2). Let us build the function $u(t)$, proving the following assertion.

*Proof of Theorem 3.2*To construct the function $u(t)$ on the interval $[0,\tau ]$ we solve the equation ${u}^{\prime}(t)={u}^{2}(t)$. Its solution is

For positivity of the function $u(t)$ for $t\in [\tau ,\theta ]$, we have to assume that ${c}_{2}>-\tau $, which follows from inequality (3.22).

where $u(\tau )$ is defined by (6.10), (6.12).

we have built the positive absolutely continuous function $u(t)$ satisfying the condition (2) of Theorem 5.1. Now reference to Theorem 5.1 completes the proof of the assertions (1) and (2) of Theorem 3.2. To prove assertions (3) and (4) we actually repeat the proof of the assertion (2) of Theorem 5.2 (in the case ${H}^{\ast}=0$).

The proof of Theorem 3.2 have been completed. □

## 7 Open problems

- 1.To obtain results on the exponential stability of the equation${x}^{\u2033}(t)+\sum _{j=1}^{n}{q}_{j}(t){x}^{\prime}(t-{\theta}_{j}(t))+\sum _{i=1}^{m}{p}_{i}(t)x(t-{\tau}_{i}(t))=f(t),\phantom{\rule{1em}{0ex}}t\in [0,+\mathrm{\infty}),$(7.1)

- 2.To obtain results as regards stabilization of the equation ${x}^{(n)}(t)=f(t)$, where $n>2$, to the trajectory $y(t)$ by the control of the form$u(t)=-\sum _{i=1}^{m}{p}_{i}(t)\{x(t-{\tau}_{i}(t))-y(t-{\tau}_{i}(t))\},\phantom{\rule{1em}{0ex}}t\in [0,+\mathrm{\infty}),$(7.2)

*i.e.*to obtain results about the exponential stability of the equation

- 3.
To obtain results on oscillation/nonoscillation, existence of solutions tending to zero or tending to infinity for the second order equation (2.3) without the assumption of the nonnegativity of the coefficients.

- 4.To obtain results on stabilization of system (1.1) to the trajectory $Y(t)$ by the control of the form$u(t)=-\sum _{i=1}^{m}{P}_{i}(t)\{X(t-{\tau}_{i}(t))-Y(t-{\tau}_{i}(t))\},\phantom{\rule{1em}{0ex}}t\in [0,+\mathrm{\infty}),$(7.4)

*i.e.*to obtain results on the exponential stability of the system

- 5.
To obtain results as regards the exponential stability of systems of functional differential equations of different orders.

## Declarations

## Authors’ Affiliations

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