Stability of a Functional Differential System with a Finite Number of Delays

Abstract

The paper is devoted to the study of asymptotic properties of a real two-dimensional differential system with unbounded nonconstant delays. The sufficient conditions for the stability and asymptotic stability of solutions are given. Used methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of Lyapunov-Krasovskii functional. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one or more constant delays or one nonconstant delay were studied.

1. Introduction

There are a lot of papers dealing with the stability and asymptotic behaviour of 𝑛-dimensional real vector equations with delay. Among others we should mention the recent results [1–13]. Since the plane has special topological properties different from those of 𝑛-dimensional space, where 𝑛≥3 or 𝑛=1, it is interesting to study asymptotic behaviour of two-dimensional systems by using tools which are typical and effective for two-dimensional systems. The convenient tool is the combination of the method of complexification and the method of Lyapunov-Krasovskii functional. Using these techniques we obtain new and easy applicable results on stability, asymptotic stability, or boundedness of solutions of real two-dimensional differential system 𝑥^′(𝑡)(𝑡)𝑥(𝑡)+=𝖠_𝑚𝑘=1𝖡_𝑘(𝑡)𝑥𝜃_𝑘(𝑡)+ℎ𝑡,𝑥(𝑡),𝑥𝜃₁(𝑡),…,𝑥𝜃_𝑚(𝑡), (1) where 𝜃_𝑘(𝑡) are real functions, 𝖠(𝑡)=(𝑎_𝑖𝑗(𝑡)), 𝖡_𝑘(𝑡)=(𝑏_𝑖𝑗𝑘(𝑡))(𝑖,𝑗=1,2;𝑘=1,…,𝑚) are real square matrices, and ℎ(𝑡,𝑥,𝑦)=(ℎ₁(𝑡,𝑥,𝑦₁,…,𝑦_𝑚), ℎ₂(𝑡,𝑥,𝑦₁,…,𝑦_𝑚)) is a real vector function, 𝑥=(𝑥₁,𝑥₂), 𝑦_𝑘=(𝑦_1𝑘,𝑦_2𝑘). It is supposed that the functions 𝜃_𝑘, 𝑎_𝑖𝑗 are locally absolutely continuous on [𝑡₀,∞), 𝑏_𝑖𝑗𝑘 are locally Lebesgue integrable on [𝑡₀,∞), and the function ℎ satisfies Carathéodory conditions on [𝑡₀,∞)×ℝ^2(𝑚+1).

Delayed differential equations recently gain more importance in applications in science and real world. They can be found in applications in medicine (control of drug therapies and neurological, physiological, and epidemiological models), biology (predator-prey models and blowflies lifecycle), chemistry (chemical kinetics), physics (private communication and signal masking), and engineering (machining operation on a lathe). Equation (1) represents a generalization of many of these models. Particularly, (1) in this general form has an application in modeling of multiple regenerative effect in tool chatter. Obtained results on stability give the possibility to find the best spindle speeds and depth-of-cut for the machines for chatter-free high-productivity operation. For more details, see [14].

The main idea of the investigation, the combination of the method of complexification and the method of Lyapunov-Krasovskii functional, was introduced for ordinary differential equations in the paper by Ráb and Kalas [15]. The principle was transferred to differential equations with delay by Kalas and Baráková [16]. The results for several constant delays can be found in papers by Rebenda [17, 18]. Differential equations with one nonconstant delay are studied by Kalas [19] and Rebenda [20].

We extend such type of results to differential equations with a finite number of nonconstant delays. We introduce the transformation of the considered real system to one equation with complex-valued coefficients. We present sufficient conditions for stability and asymptotic stability of a solution and the conditions under which all solutions tend to zero. The applicability of the results is demonstrated with an example.

At the end of this introduction we append an overview of notations used in the paper and the transformation of the real system to one equation with complex-valued coefficients.

Consider the following: ℝ: set of all real numbers, ℝ₊: set of all positive real numbers,  ℝ⁰₊: set of all nonnegative real numbers,  ℝ₋: set of all negative real numbers,  ℝ⁰₋: set of all nonpositive real numbers,  ℂ: set of all complex numbers,  𝒞: class of all continuous functions [−𝑟,0]→ℂ,  𝐴𝐶_loc(𝐼,𝑀): class of all locally absolutely continuous functions 𝐼→𝑀,  𝐿_loc(𝐼,𝑀): class of all locally Lebesgue integrable functions 𝐼→𝑀,  𝐾(𝐼×Ω,𝑀): class of all functions 𝐼×Ω→𝑀 satisfying Carathéodory conditions on 𝐼×Ω,   Re𝑧: real part of 𝑧,  Im𝑧: imaginary part of 𝑧,  𝑧: complex conjugate of 𝑧.

Introducing complex variables 𝑧=𝑥₁+𝑖𝑥₂, 𝑤₁=𝑦₁₁+𝑖𝑦₁₂,…,𝑤_𝑚=𝑦_𝑚1+𝑖𝑦_𝑚2, we can rewrite system (1) into an equivalent equation with complex-valued coefficients: 𝑧^′(𝑡)=𝑎(𝑡)𝑧(𝑡)+𝑏(𝑡)𝑧(𝑡)+^𝑚_𝑘=1𝐴_𝑘(𝑡)𝑧𝜃_𝑘(𝑡)+𝐵_𝑘(𝑡)𝑧𝜃_𝑘(𝑡)+𝑔𝑡,𝑧(𝑡),𝑧𝜃₁(𝑡),…,𝑧𝜃_𝑚(𝑡), (2) where 𝜃_𝑘∈𝐴𝐶_loc(𝐽,ℝ) for 𝑘=1,…,𝑚, 𝐴_𝑘,𝐵_𝑘∈𝐿_loc(𝐽,ℂ), 𝑎,𝑏∈𝐴𝐶_loc(𝐽,ℂ), 𝑔∈𝐾(𝐽×ℂ^𝑚+1,ℂ), 𝐽=[𝑡₀,∞).

The relations between the functions are as follows: 𝑎(𝑡)=12𝑎₁₁(𝑡)+𝑎₂₂(𝑡)+𝑖2𝑎₂₁(𝑡)−𝑎₁₂(𝑡),𝑏(𝑡)=12𝑎₁₁(𝑡)−𝑎₂₂(𝑡)+𝑖2𝑎₂₁(𝑡)+𝑎₁₂(𝑡),𝐴_𝑘(𝑡)=12𝑏_11𝑘(𝑡)+𝑏_22𝑘(𝑡)+𝑖2𝑏_21𝑘(𝑡)−𝑏_12𝑘(𝑡),𝐵_𝑘(𝑡)=12𝑏_11𝑘(𝑡)−𝑏_22𝑘(𝑡)+𝑖2𝑏_21𝑘(𝑡)+𝑏_12𝑘(𝑡),𝑔𝑡,𝑧,𝑤₁,…,𝑤_𝑚=ℎ₁1𝑡,2(𝑧+𝑧),1(2𝑖𝑧−𝑧),12𝑤₁+𝑤₁1,…,𝑤2𝑖^𝑚−𝑤_𝑚+𝑖ℎ₂1𝑡,2(𝑧+𝑧),1(2𝑖𝑧−𝑧),12𝑤₁+𝑤₁,1𝑤2𝑖¹−𝑤₁1,…,𝑤2𝑖^𝑚−𝑤_𝑚. (3)

Conversely, putting 𝑎₁₁(𝑡)[𝑎(𝑡)(𝑡,𝑎=Re+𝑏)]₁₂(𝑡)[𝑏(𝑡)(𝑡,𝑎=Im−𝑎)]₂₁(𝑡)[𝑎(𝑡)(𝑡,𝑎=Im+𝑏)]₂₂(𝑡)[𝑎(𝑡)(𝑡,𝑏=Re−𝑏)]_11𝑘(𝑡)=Re𝐴_𝑘(𝑡)+𝐵_𝑘(𝑡,𝑏_12𝑘)(𝑡)=Im𝐵_𝑘(𝑡)−𝐴_𝑘(𝑡,𝑏_21𝑘)(𝑡)=Im𝐴_𝑘(𝑡)+𝐵_𝑘(𝑡,𝑏_22𝑘)(𝑡)=Re𝐴_𝑘(𝑡)−𝐵_𝑘(𝑡,ℎ₁)𝑡,𝑥,𝑦₁,…,𝑦_𝑚=Re𝑔𝑡,𝑥₁+𝑖𝑥₂,𝑦₁₁+𝑖𝑦₁₂,…,𝑦_𝑚1+𝑖𝑦_𝑚2,ℎ₂𝑡,𝑥,𝑦₁,…,𝑦_𝑚=Im𝑔𝑡,𝑥₁+𝑖𝑥₂,𝑦₁₁+𝑖𝑦₁₂,…,𝑦_𝑚1+𝑖𝑦_𝑚2, (4) equation (2) can be written in real form (1) as well.

2. Preliminaries

We consider (2) in the case when liminf_𝑡→∞(|Im𝑎(𝑡)|−|𝑏(𝑡)|)>0 (5) and study the behavior of solutions of (2) under this assumption. This situation corresponds to the case when the equilibrium 0 of the autonomous homogeneous system 𝑥^′=𝐀𝑥, (6) where 𝐀 is supposed to be regular constant matrix, is a centre or a focus.

This case is included in the case liminf_𝑡→∞(|𝑎(𝑡)|−|𝑏(𝑡)|)>0 considered in [21], but in this special case, we are able to derive more useful results as we will see later in an example. The idea is based on the well-known result that the condition |𝑎|>|𝑏| in an autonomous equation 𝑧^′=𝑎𝑧+𝑏𝑧 ensures that zero is a focus, a centre, or a node while under the condition |Im𝑎|>|𝑏| zero can be just a focus or a centre. Details are found in [15].

A simple example shows that, in some cases, the results of this paper can be applied more suitably than those given in [21].

Regarding (5) and since the delay functions 𝜃_𝑘 satisfy lim_𝑡→∞𝜃_𝑘(𝑡)=∞, there are numbers 𝑇₁≥𝑡₀, 𝑇≥𝑇₁, and 𝜇>0 such that |Im𝑎(𝑡)|>|𝑏(𝑡)|+𝜇for𝑡≥𝑇₁,𝑡≥𝜃_𝑘(𝑡)≥𝑇₁for𝑡≥𝑇(𝑘=1,…,𝑚). (7)

Denote 𝛾(𝑡)=Im𝑎(𝑡)+(Im𝑎(𝑡))²−|𝑏(𝑡)|²((𝑡,𝑐̃(𝑡)(𝑡).sgnIm𝑎))=−𝑖𝑏 (8)

Notice that, unlike the function 𝛾 introduced in [21], the previously defined function 𝛾 need not be positive.

Since |𝛾(𝑡)|>|Im𝑎(𝑡)| and |𝑐(̃𝑡)|=|𝑏(𝑡)|, the inequality 𝛾(𝑡)>|𝑐̃(𝑡)|+𝜇 (9) is valid for 𝑡≥𝑇₁. It can be easily verified that 𝛾,𝑐̃∈𝐴𝐶_loc([𝑇₁,∞),ℂ).

For the rest of this section, denote that 𝜗(𝑡)=Re𝛾(𝑡)𝛾^′(𝑡)−𝑐̃(𝑡)𝑐^′̃(𝑡)−𝛾(𝑡)𝑐^′̃(𝑡)−𝛾^′(𝑡)𝑐̃(𝑡)𝛾²(𝑡)−|𝑐̃(𝑡)|². (10)

The stability and asymptotic stability are studied under the following assumptions.(i) The numbers 𝑇₁≥𝑡₀, 𝑇≥𝑇₁, and 𝜇>0 are such that (7) holds. (ii) There exist functions 𝜘, 𝜅_𝑘, 𝜚̃∶[𝑇,∞)→ℝ such that 𝛾(𝑡)𝑔𝑡,𝑧,𝑤₁,…,𝑤_𝑚+𝑐̃(𝑡)𝑔𝑡,𝑧,𝑤₁,…,𝑤_𝑚≤𝜘(𝑡)𝛾(𝑡)𝑧+𝑐̃(𝑡)𝑧+^𝑚_𝑘=1𝜅_𝑘(𝑡)𝛾𝜃_𝑘(𝑡)𝑤_𝑘+𝑐̃𝜃_𝑘(𝑡)𝑤_𝑘+𝜚̃(𝑡), (11) for 𝑡≥𝑇, 𝑧, 𝑤_𝑘∈ℂ(𝑘=1,…,𝑚), where 𝜘, 𝜚̃∈𝐿_loc([𝑇,∞),ℝ). (iii)𝛽∈𝐴𝐶_loc([𝑇,∞),ℝ⁰₊) is a function satisfying 𝜃^′_𝑘(𝑡)𝛽(𝑡)≥𝜆_𝑘(𝑡)a.e.on[𝑇,∞), (12) where 𝜆_𝑘 is defined for 𝑡≥𝑇 by 𝜆_𝑘(𝑡)=𝜅_𝑘(𝑡)+𝐴_𝑘(𝑡)+𝐵_𝑘(𝑡×𝛾(𝑡+|𝑐̃(𝑡)))|𝛾𝜃_𝑘(𝑡)−𝑐̃𝜃_𝑘(𝑡). (13)(iv) There exists a function Λ∈𝐿_loc([𝑇,∞),ℝ) which satisfies the inequalities 𝛽^′(𝑡)≤Λ(𝑡)𝛽(𝑡), Θ(𝑡)≤Λ(𝑡) for almost all 𝑡∈[𝑇,∞), where the function Θ is defined by Θ(𝑡)=Re𝑎(𝑡)+𝜗(𝑡)+𝜘(𝑡)+𝑚𝛽(𝑡). (14)

If 𝐴_𝑘, 𝐵_𝑘, 𝜅_𝑘, 𝜃^′_𝑘 are locally absolutely continuous on [𝑇,∞) and 𝜆_𝑘(𝑡)≥0,𝜃^′_𝑘(𝑡)>0 on [𝑇,∞), the choice 𝛽(𝑡)=max_{𝑘=1,…,𝑚}[𝜆_𝑘(𝑡)(𝜃^′_𝑘(𝑡))⁻¹] is admissible in (iii).

From the assumption (i), it follows that 𝜗≤Re𝛾𝛾^′−𝑐𝑐̃^′̃+𝛾𝑐^′̃−𝛾^′𝑐̃𝛾²−|𝑐|̃²≤𝛾^′+𝑐^′̃𝛾+|𝑐|̃𝛾²−|𝑐|̃²=𝛾^′+𝑐^′̃𝛾−|𝑐|̃≤1𝜇𝛾^′+𝑐^′̃; (15) hence, the function 𝜗 is locally Lebesgue integrable on [𝑇,∞). Moreover, if 𝛽∈𝐴𝐶_loc([𝑇,∞),ℝ₊) and 𝜘∈𝐿_loc([𝑇,∞), then we can choose Λ(𝑡)Θ(𝑡),𝛽=max^′(𝑡)𝛽(𝑡), (16) in (iv).

Finally, if 𝜚̃(𝑡)≡0 in (ii), then (2) has the trivial solution 𝑧(𝑡)≡0. Notice that in this case the condition (ii) implies that the functions 𝜘(𝑡), 𝜅_𝑘(𝑡) are nonnegative on [𝑇,∞) for 𝑘=1,…,𝑚, and due to this, 𝜆_𝑘(𝑡)≥0 on [𝑇,∞). The case 𝜚̃(𝑡)<0 is omitted since it can be replaced by 𝜚̃(𝑡)≡0.

3. Main Results

The aim is to generalize the results for ordinary differential equations published in [15] as well as the results contained in [16] (one constant delay), [18] (a finite number of constant delays), and [20] (one nonconstant delay). In the proof of the crucial theorem, we use the following auxiliary result.

Lemma 1. Let 𝑎₁, 𝑎₂, 𝑏₁, 𝑏₂∈ℂ and |𝑎₂|>|𝑏₂|. Then 𝑎Re¹𝑧+𝑏₁𝑧𝑎₂𝑧+𝑏₂𝑧≤Re𝑎₁𝑎₂−𝑏₁𝑏₂+𝑎₁𝑏₂−𝑎₂𝑏₁𝑎₂²−𝑏₂², (17) for 𝑧∈ℂ, 𝑧≠0.

The proof of Lemma 1 can be found, for example, in [15, page 131] or [17, page 101].

Theorem 2. Let the conditions (i), (ii), (iii), and (iv) hold and 𝜚̃(𝑡)≡0. (a) If limsup_𝑡→∞^𝑡Λ(𝑠)𝑑𝑠<∞, (18) then the trivial solution of (2) is stable on [𝑇,∞). (b) If lim_𝑡→∞^𝑡Λ(𝑠)𝑑𝑠=−∞, (19) then the trivial solution of (2) is asymptotically stable on [𝑇,∞).

Proof. Choose arbitrary 𝑡₁≥𝑇. Let 𝑧(𝑡) be any solution of (2) satisfying the condition 𝑧(𝑡)=𝑧₀(𝑡) for 𝑡∈[𝑇₁,𝑡₁], where 𝑧₀(𝑡) is a continuous complex-valued initial function defined on 𝑡∈[𝑇₁,𝑡₁]. Consider the Lyapunov functional 𝑉(𝑡)=𝑈(𝑡)+𝛽(𝑡)^𝑚_𝑘=1^𝑡_{𝜃𝑘(𝑡)}𝑈(𝑠)𝑑𝑠, (20) where 𝑈(𝑡)=𝛾(𝑡)𝑧(𝑡)+𝑐̃(𝑡)𝑧(𝑡). (21)
To simplify the computations, denote that 𝑤_𝑘(𝑡)=𝑧(𝜃_𝑘(𝑡)) and write the functions of variable 𝑡 without brackets, for example, 𝑧 instead of 𝑧(𝑡).
From (20) we get 𝑉^′=𝑈^′+𝛽^′𝑚_𝑘=1^𝑡_{𝜃𝑘(𝑡)}𝑈(𝑠)𝑑𝑠+𝑚𝛽𝛾𝑧+𝑐𝑧̃−^𝑚_𝑘=1𝜃^′_𝑘𝛽𝛾𝜃_𝑘𝑤_𝑘+𝑐̃𝜃_𝑘𝑤_𝑘, (22) for almost all 𝑡≥𝑡₁ for which 𝑧(𝑡) is defined and 𝑈^′(𝑡) exists.
Denote that 𝒦={𝑡≥𝑡₁∶𝑧(𝑡)exists,𝑈(𝑡)≠0} and ℳ={𝑡≥𝑡₁∶𝑧(𝑡)exists,𝑈(𝑡)=0}. It is clear that the derivative 𝑈^′(𝑡) exists for almost all 𝑡∈𝒦; hence, we focus on the set ℳ.
In view of (9) we have 𝑧(𝑡)=0 for 𝑡∈ℳ. For almost all 𝑡∈ℳ, we compute 𝑈^′_±(𝑡)=lim_{𝜏→𝑡±}𝑈(𝜏)−𝑈(𝑡)𝜏−𝑡=lim_{𝜏→𝑡±}𝑈(𝜏)𝜏−𝑡=lim_{𝜏→𝑡±}𝛾(𝜏)[𝑧(𝜏)−𝑧(𝑡)]−𝑐̃(𝜏)[𝑧(𝜏)−𝑧(𝑡)]𝛾(𝑡)𝑧𝜏−𝑡=±^′(𝑡)+𝑐̃(𝑡)𝑧^′(𝑡𝛾(𝑡)𝑔)=±^∗(𝑡)+𝑐̃(𝑡)𝑔^∗(𝑡), (23) where 𝑔^∗(𝑡)=^𝑚_𝑘=1𝐴_𝑘(𝑡)𝑤_𝑘(𝑡)+𝐵_𝑘(𝑡)𝑤_𝑘(𝑡)+𝑔𝑡,0,𝑤₁(𝑡),…,𝑤_𝑚(𝑡). (24)
Hence, 𝑈 has one-sided derivatives a.e. in ℳ. According to [22, Chapter IX., Theorem (1.1)] or [23], the set of all 𝑡 such that 𝑈^′₊(𝑡)≠𝑈^′₋(𝑡) can be at most countable; thus, the derivative 𝑈^′ exists for almost all 𝑡∈ℳ, and for these 𝑡, 𝑈^′(𝑡)=0.
In particular, the derivative 𝑈^′ exists for almost all 𝑡≥𝑡₁ for which 𝑧(𝑡) is defined; thus, (22) holds for almost all 𝑡≥𝑡₁ for which 𝑧(𝑡) is defined.
Now return the attention to the set 𝒦. For almost all 𝑡∈𝒦, it holds that 𝑈𝑈^′=𝑈((𝛾𝑧+𝑐𝑧̃)(𝛾𝑧+𝑐𝑧̃))^′=Re[(𝛾𝑧+𝑐𝑧̃)(𝛾^′𝑧+𝛾𝑧^′+𝑐^′̃𝑧+𝑐𝑧̃^′)]. As 𝑧(𝑡) is a solution of (2), we have 𝑈𝑈^′=Re𝛾𝑧+̃×𝛾𝑐𝑧_′𝑧+𝑐^′̃𝑧+𝛾𝑎𝑧+𝑏𝑧+^𝑚_𝑘=1𝐴_𝑘𝑤_𝑘+𝐵_𝑘𝑤_𝑘+𝑐̃+𝑔𝑎𝑧+𝑏𝑧+^𝑚_𝑘=1𝐴_𝑘𝑤_𝑘+𝐵_𝑘𝑤_𝑘+𝑔=Re𝛾𝑧+̃×𝛾𝑐𝑧_′𝑧+𝑐^′̃𝑧+𝛾𝑎+𝑐𝑏̃𝑧+𝛾𝑏+𝑐𝑎̃𝑧+𝛾^𝑚_𝑘=1𝐴_𝑘𝑤_𝑘+𝐵_𝑘𝑤_𝑘+𝑐̃+𝑔_𝑚𝑘=1𝐴_𝑘𝑤_𝑘+𝐵_𝑘𝑤_𝑘+𝑔, (25) for almost all 𝑡∈𝒦.
Short computation gives (𝛾𝑎+𝑐𝑏̃)𝑐̃=(𝛾𝑏+𝑐𝑎̃)𝛾, and from this we get 𝑈𝑈^′≤Re𝛾𝑧+𝑐𝑧̃𝛾𝑎+𝑐𝑏̃𝑐̃𝑧+𝛾𝑧+Re𝛾𝑧+𝑐𝑧̃𝛾^𝑚_𝑘=1𝐴_𝑘𝑤_𝑘+𝐵_𝑘𝑤_𝑘+𝑐̃^𝑚_𝑘=1𝐴_𝑘𝑤_𝑘+𝐵_𝑘𝑤_𝑘+Re𝛾𝑧+𝑐𝑧̃𝛾𝑔+𝑐𝑔̃+Re𝛾𝑧+𝑐𝑧̃𝛾^′𝑧+𝑐^′̃𝑧≤𝑈²𝑐̃Re𝑎+𝛾𝑏+𝑈𝛾+|𝑐|̃^𝑚_𝑘=1𝐴_𝑘𝑤_𝑘+𝐵_𝑘𝑤_𝑘+𝑈𝛾𝑔+𝑐𝑔̃+𝑈²𝛾R𝑒^′𝑧+𝑐^′̃𝑧𝛾𝑧+𝑐𝑧̃, (26) for almost all 𝑡∈𝒦.
Applying Lemma 1 to the last term, we obtain 𝛾Re^′𝑧+𝑐^′̃𝑧𝛾𝑧+𝑐𝑧̃≤𝜗. (27)
Using this inequality together with (13), assumption (ii), and the relation Re(𝑎+(𝑐/̃𝛾)𝑏)=Re𝑎, we obtain 𝑈𝑈^′≤𝑈²Re𝑎+𝜗+𝜘+𝑈^𝑚_𝑘=1𝜅_𝑘𝛾𝜃_𝑘𝑤_𝑘+𝑐̃𝜃_𝑘𝑤_𝑘𝛾+|𝑐|̃×+𝑈_𝑚𝑘=1𝐴_𝑘𝑤_𝑘+𝐵_𝑘𝑤_𝑘𝛾𝜃_𝑘−𝑐̃𝜃_𝑘𝛾𝜃_𝑘−𝑐̃𝜃_𝑘≤𝑈²Re𝑎+𝜗+𝜘+𝑈^𝑚_𝑘=1𝜅_𝑘+𝐴_𝑘+𝐵_𝑘𝛾+|𝑐|̃𝛾𝜃_𝑘−𝑐̃𝜃_𝑘×𝛾𝜃_𝑘𝑤_𝑘+𝑐̃𝜃_𝑘𝑤_𝑘≤𝑈²Re𝑎+𝜗+𝜘+𝑈^𝑚_𝑘=1𝜆_𝑘𝛾𝜃_𝑘𝑤_𝑘+𝑐̃𝜃_𝑘𝑤_𝑘, (28) for almost all 𝑡∈𝒦.
Consequently, 𝑈^′≤𝑈Re𝑎+𝜗+𝜘+^𝑚_𝑘=1𝜆_𝑘𝛾𝜃_𝑘𝑤_𝑘+𝑐̃𝜃_𝑘𝑤_𝑘, (29) for almost all 𝑡∈𝒦.
Recalling that 𝑈^′(𝑡)=0 for almost all 𝑡∈ℳ, we can see that inequality (29) is valid for almost all 𝑡≥𝑡₁ for which 𝑧(𝑡) is defined.
From (29) we have 𝑉^′𝛽+≤𝑈Re𝑎+𝜗+𝜘+𝑚_𝑚𝑘=1𝜆_𝑘−𝜃^′_𝑘𝛽𝛾𝜃_𝑘𝑤_𝑘+𝑐̃𝜃_𝑘𝑤_𝑘+𝛽^′𝑚_𝑘=1^𝑡_{𝜃𝑘(𝑡)}𝛾(𝑠)𝑧(𝑠)+𝑐̃(𝑠)𝑧(𝑠)𝑑𝑠. (30)
As 𝛽(𝑡) fulfills condition (12), we obtain 𝑉^′(𝑡)(𝑡)Θ(𝑡)+𝛽≤𝑈_′(𝑡)^𝑚_𝑘=1^𝑡_{𝜃𝑘(𝑡)}𝛾(𝑠)𝑧(𝑠)+𝑐̃(𝑠)𝑧(𝑠)𝑑𝑠, (31) Hence, 𝑉^′(𝑡)−Λ(𝑡)𝑉(𝑡)≤0, (32) for almost all 𝑡≥𝑡₁ for which the solution 𝑧(𝑡) exists.
Notice that, with respect to (9), 𝑉(𝑡)≥𝛾(𝑡)−|𝑐̃(𝑡)||𝑧(𝑡)|≥𝜇|𝑧(𝑡)|, (33) for all 𝑡≥𝑡₁ for which 𝑧(𝑡) is defined.
Suppose that condition (18) holds, and choose arbitrary 𝜀>0. Put Δ=max_{𝑇𝑠∈1,𝑡1}𝛾(𝑠)+|𝑐̃(𝑠)|,𝐿=sup_{𝑇≤𝑡<∞}^𝑡_𝑇Λ(𝑠)𝑑𝑠,𝛿=𝜇𝜀Δ⁻¹1+𝑚𝛽𝑡₁𝑡₁−𝑇₁⁻¹exp^𝑡1_𝑇Λ(𝑠)𝑑𝑠−𝐿, (34) where 𝜇>0, 𝑇₁≥𝑡₀, and 𝑇≥𝑇₁ are the numbers from condition (i).
If the initial function 𝑧₀(𝑡) of the solution 𝑧(𝑡) satisfies max_{𝑠∈[𝑇1,𝑡1]}|𝑧₀(𝑠)|<𝛿, then the multiplication of (32) by exp{−∫^𝑡_𝑡1Λ(𝑠)𝑑𝑠} and the integration over [𝑡₁,𝑡] yield 𝑉(𝑡)exp−^𝑡_𝑡1Λ(𝑠)𝑑𝑠−𝑉𝑡₁≤0, (35) for all 𝑡≥𝑡₁ for which 𝑧(𝑡) is defined. From (33) and (35) we get 𝜇|𝑧(𝑡)|≤𝑉(𝑡)≤𝑉𝑡₁exp^𝑡_𝑡1Λ(𝑠)≤𝛾𝑡𝑑𝑠₁+𝑐̃𝑡₁𝑧𝑡₁+𝛽𝑡₁max_{𝑇𝑠∈1,𝑡1}|𝑧(𝑠×)|_{𝑚𝑘=1𝑡1𝜃𝑘𝑡1}𝛾(𝑠)+|𝑐̃(𝑠)|𝑑𝑠×exp^𝑡_𝑡1Λ(𝑠)≤𝑑𝑠Δmax_{𝑠∈[𝑇1,𝑡1]}𝑧₀(𝑠+𝛽𝑡₁)max_{𝑇𝑠∈1,𝑡1}𝑧₀(𝑠Δ×_𝑚𝑘=1)𝑡₁−𝜃_𝑘𝑡₁×exp^𝑡_𝑡1Λ(𝑠)𝑑𝑠; (36) that is, 𝜇|𝑧(𝑡)|≤Δmax_{𝑇𝑠∈1,𝑡1}𝑧₀(𝑠)1+𝑚𝛽𝑡₁𝑡₁−𝑇₁×exp𝐿−^𝑡1_𝑇Λ(𝑠)𝑑𝑠<𝜇𝜀. (37)
Thus, we have |𝑧(𝑡)|<𝜀 for all 𝑡≥𝑡₁, and we conclude that the trivial solution of (2) is stable.
Now suppose that condition (19) is valid. Then, in view of the first part of Theorem 2, for 𝐾>0, there is a 𝜌>0 such that max_{𝑠∈[𝑇1,𝑡1]}|𝑧₀(𝑠)|<𝜌 implies that the solution 𝑧(𝑡) of (2) exists for all 𝑡≥𝑡₁ and satisfies |𝑧(𝑡)|<𝐾, where 𝐾 is arbitrary real constant. Hence, from this and (33), we have |𝑧(𝑡)|≤𝜇⁻¹𝑉(𝑡)≤𝜇⁻¹𝑉𝑡₁exp^𝑡_𝑡1Λ(𝑠)𝑑𝑠, (38) for all 𝑡≥𝑡₁. This inequality, with condition (19), gives lim_𝑡→∞𝑧(𝑡)=0, (39) which completes the proof.

Remark 3. Theorem 2 represents a generalization of previous results.
If we take 𝐴₁(𝑡)=𝐴(𝑡), 𝐴_𝑘≡0, for 𝑘=2,…,𝑚, 𝐵₁(𝑡)=𝐵(𝑡), 𝐵_𝑘≡0, for 𝑘=2,…,𝑚, and 𝜃₁(𝑡)=𝑡−𝑟, where 𝑟>0, we get Theorem 4 from [16].
If we take 𝜃_𝑘(𝑡)=𝑡−𝑟_𝑘, where 𝑟_𝑘>0, 𝑘=1,…,𝑚, we obtain Theorem 1 from [18].
If we take 𝐴₁(𝑡)=𝐴(𝑡), 𝐴_𝑘≡0, for 𝑘=2,…,𝑚, 𝐵₁(𝑡)=𝐵(𝑡), 𝐵_𝑘≡0, for 𝑘=2,…,𝑚, and 𝜃₁(𝑡)=𝜃(𝑡), we get Theorem 2.2 from [20].

The next theorem involves the function 𝜚̃ in (ii); thus, it is more general than Theorem 2. A part of the proof of Theorem 2 is utilized in the proof of Theorem 4.

Theorem 4. Let the assumptions (i), (ii), (iii), and (iv) hold and 𝑉(𝑡)=𝛾(𝑡)𝑧(𝑡)+𝑐̃(𝑡)𝑧(𝑡+𝛽(𝑡)_{𝑚𝑘=1𝑡𝜃𝑘(𝑡)})𝛾(𝑠)𝑧(𝑠)+𝑐̃(𝑠)𝑧(𝑠)𝑑𝑠, (40) where 𝑧(𝑡) is any solution of (2) defined for 𝑡→∞. Then 𝜇|𝑧(𝑡)|≤𝑉(𝑠)exp^𝑡_𝑠Λ(𝜉)+𝑑𝜉_𝑡𝑠𝜚̃(𝜉)exp^𝑡_𝜉Λ(𝜎)𝑑𝜎𝑑𝜉, (41) for 𝑡≥𝑠≥𝑡₁, where 𝑡₁≥𝑇.

Proof. Following the proof of Theorem 2, we have 𝑉^′(𝑡)≤𝛾(𝑡)𝑧(𝑡)+𝑐̃(𝑡)𝑧(𝑡Θ(𝑡)+𝛽_′)(𝑡)^𝑚_𝑘=1^𝑡_{𝜃𝑘(𝑡)}𝛾(𝑠)𝑧(𝑠)+𝑐̃(𝑠)𝑧(𝑠+𝜚̃(𝑡)≤Λ(𝑡)𝑉(𝑡)+𝜚̃(𝑡),)𝑑𝑠 (42) a.e. on [𝑡₁,∞). Using this inequality, we get 𝑉^′(𝑡)−Λ(𝑡)𝑉(𝑡)≤𝜚̃(𝑡), (43) a.e. on [𝑡₁,∞). Multiplying (43) by exp(−∫^𝑡_𝑠Λ(𝜉)𝑑𝜉) gives 𝑉(𝑡)exp−^𝑡_𝑠Λ(𝜉)𝑑𝜉≤𝜚̃(𝑡)exp−^𝑡_𝑠Λ(𝜉)𝑑𝜉, (44) a.e. on [𝑡₁,∞). Integration over [𝑠,𝑡] yields 𝑉(𝑡)exp−^𝑡_𝑠Λ(𝜉)(𝑠)≤𝑑𝜉−𝑉_𝑡𝑠𝜚̃(𝜉)exp−^𝜉_𝑠Λ(𝜎)𝑑𝜎𝑑𝜉, (45) and multiplying (45) by exp(∫^𝑡_𝑠Λ(𝜉)𝑑𝜉), we obtain 𝑉(𝑡)≤𝑉(𝑠)exp^𝑡_𝑠Λ(𝜉)+𝑑𝜉_𝑡𝑠𝜚̃(𝜉)exp^𝑡_𝜉Λ(𝜎)𝑑𝜎𝑑𝜉. (46)
The statement now follows from (33).

Remark 5. Theorem 4 generalizes theorems contained in previous papers.
If we take 𝐴₁(𝑡)=𝐴(𝑡), 𝐴_𝑘≡0, for 𝑘=2,…,𝑚, 𝐵₁(𝑡)=𝐵(𝑡), 𝐵_𝑘≡0, for 𝑘=2,…,𝑚, and 𝜃₁(𝑡)=𝑡−𝑟, where 𝑟>0, we get Theorem 2 from [16].
If we take 𝜃_𝑘(𝑡)=𝑡−𝑟_𝑘, where 𝑟_𝑘>0, 𝑘=1,…,𝑚, we obtain Theorem 2 from [18].
If we take 𝐴₁(𝑡)=𝐴(𝑡), 𝐴_𝑘≡0, for 𝑘=2,…,𝑚, 𝐵₁(𝑡)=𝐵(𝑡), 𝐵_𝑘≡0, for 𝑘=2,…,𝑚, and 𝜃₁(𝑡)=𝜃(𝑡), we get Theorem 2.7 from [20].

The last of the main propositions gives the conditions under which all solutions of (2) tend to zero.

Theorem 6. Let the assumptions (i), (ii), (iii), and (iv) be satisfied. Let Λ(𝑡)≤0 a.e. on [𝑇^∗,∞), where 𝑇^∗∈[𝑇,∞). If lim_𝑡→∞^𝑡Λ(𝑠)𝑑𝑠=−∞,𝜚̃(𝑡)=𝑜Λ(𝑡), (47) then any solution 𝑧(𝑡) of (2) existing for 𝑡→∞ satisfies lim_𝑡→∞𝑧(𝑡)=0. (48)

Proof. Choose arbitrary 𝜀>0. According to (47), there is 𝑠≥𝑇^∗ such that 𝜚̃(𝑡)≤(1/2)𝜇𝜀|Λ(𝑡)| for 𝑡≥𝑠 and 𝜚̃^𝑡_𝑠(𝜏)exp^𝑡_𝜏Λ(𝜎)≤𝑑𝜎𝑑𝜏𝜇𝜀2^𝑡_𝑠−Λ(𝜏)exp^𝑡_𝜏Λ(𝜎)=𝑑𝜎𝑑𝜏𝜇𝜀2^𝑡_𝑠𝑑𝑑𝜏exp^𝑡_𝜏Λ(𝜎)=𝑑𝜎𝑑𝜏𝜇𝜀2exp^𝑡_𝜏Λ(𝜎)𝑑𝜎^𝑡_𝑠=𝜇𝜀21−exp^𝑡_𝑠Λ(𝜏)𝑑𝜏<𝜇𝜀2, (49) for 𝑡≥𝑠. From (47) we have exp(∫^𝑡_𝑠Λ(𝜏)𝑑𝜏)→0 as 𝑡→∞; hence, there is 𝑆≥𝑠 such that exp(∫^𝑡_𝑠Λ(𝜏)𝑑𝜏)<𝜇𝜀(2𝑉(𝑠))⁻¹ for 𝑡≥𝑆. Considering this fact and (41), we get 𝜇|𝑧(𝑡)|<𝑉(𝑠)𝜇𝜀(𝑠)+2𝑉𝜇𝜀2=𝜇𝜀, (50) for 𝑡≥𝑆. This completes the proof.

Remark 7. Theorem 6 is a generalization of results published in the papers [16, 18, 20].
If we take 𝐴₁(𝑡)=𝐴(𝑡), 𝐴_𝑘≡0, for 𝑘=2,…,𝑚, 𝐵₁(𝑡)=𝐵(𝑡), 𝐵_𝑘≡0, for 𝑘=2,…,𝑚, and 𝜃₁(𝑡)=𝑡−𝑟, where 𝑟>0, we get Theorem 3 from [16].
If we take 𝜃_𝑘(𝑡)=𝑡−𝑟_𝑘, where 𝑟_𝑘>0, 𝑘=1,…,𝑚, we obtain Theorem 3 from [18].
If we take 𝐴₁(𝑡)=𝐴(𝑡),𝐴_𝑘≡0, for 𝑘=2,…,𝑚, 𝐵₁(𝑡)=𝐵(𝑡),𝐵_𝑘≡0, for 𝑘=2,…,𝑚, and 𝜃₁(𝑡)=𝜃(𝑡), we get Theorem 2.14 from [20].

4. Corollaries and Examples

From Theorem 2 we easily obtain several corollaries. We give an example which shows that it is worth to consider the case (5).

Corollary 8. Let 𝑎(𝑡)≡𝑎∈ℂ, 𝑏(𝑡)≡𝑏∈ℂ, |Im𝑎|>|𝑏|. Suppose that lim_𝑡→∞𝜃_𝑘(𝑡)=∞, 𝜃_𝑘(𝑡)≤𝑡, for 𝑡≥𝑇₁, where 𝑇₁≥𝑡₀. Let 𝜌₀,𝜌₁,…,𝜌_𝑚∶[𝑇₁,∞)→ℝ be such that 𝑔𝑡,𝑧,𝑤₁,…,𝑤_𝑚≤𝜌₀(𝑡)|𝑧|+^𝑚_𝑘=1𝜌_𝑘(𝑡)𝑤_𝑘, (51) for 𝑡≥𝑇₁,|𝑧|<𝑅,|𝑤_𝑘|<𝑅,𝑘=1,…,𝑚,𝑅>0 and 𝜌₀∈𝐿_loc([𝑇₁,∞),ℝ).
Let 𝛽∈𝐴𝐶_loc([𝑇₁,∞),ℝ₊) satisfy 𝜃^′_𝑘(𝑡)𝛽(𝑡)≥|Im𝑎|+|𝑏||Im𝑎|−|𝑏|^1/2𝜌_𝑘(𝑡)+𝐴_𝑘(𝑡)+𝐵_𝑘(𝑡), (52) a.e. on [𝑇₁,∞) for 𝑘=1,…,𝑚. If limsup_𝑡→∞^𝑡||+|𝑏|maxRe𝑎+Im𝑎|Im𝑎|−|𝑏|^1/2𝜌₀(𝑠)𝛽(𝑠),𝛽+𝑚^′(𝑠)𝛽(𝑠)𝑑𝑠<∞, (53) then the trivial solution of (2) is stable. If lim_𝑡→∞^𝑡||+|𝑏|maxRe𝑎+Im𝑎|Im𝑎|−|𝑏|^1/2𝜌₀(𝑠)𝛽(𝑠),𝛽+𝑚^′(𝑠)𝛽(𝑠)𝑑𝑠=−∞, (54) then the trivial solution of (2) is asymptotically stable.

Proof. Choose 𝑇≥𝑇₁ such that 𝜃_𝑘(𝑡)≥𝑇₁ for 𝑡≥𝑇,𝑘=1,…,𝑚. Denote that 𝑧=𝑧(𝑡) and 𝑤_𝑘=𝑧(𝜃_𝑘(𝑡)) again. Since 𝑎,𝑏∈ℂ are constants, then also 𝛾 and 𝑐̃ are constants, and we have 𝜗(𝑡)≡0. Using condition (51) we get 𝛾𝑔𝑡,𝑧,𝑤₁,…,𝑤_𝑚+𝑐𝑔̃𝑡,𝑧,𝑤₁,…,𝑤_𝑚≤𝛾+|𝑐|̃𝜌₀(𝑡)|𝑧|+^𝑚_𝑘=1𝜌_𝑘(𝑡)𝑤_𝑘=𝛾+|𝑐|̃𝛾−|𝑐|̃𝛾−|𝑐|̃𝜌₀(𝑡)|𝑧|+^𝑚_𝑘=1𝜌_𝑘(𝑡)𝑤_𝑘≤𝛾+|𝑐|̃𝛾−|𝑐|̃𝜌₀(𝑡)𝛾𝑧+𝑐𝑧̃+^𝑚_𝑘=1𝜌_𝑘(𝑡)𝛾𝑤_𝑘+𝑐𝑤̃_𝑘, (55) and it follows that condition (ii) holds with 𝜘(𝑡)=𝛾+|𝑐|̃𝛾−|𝑐|̃𝜌₀(𝑡),𝜅_𝑘(𝑡)=𝛾+|𝑐|̃𝛾−|𝑐|̃𝜌_𝑘(𝑡), (56) and 𝜚̃(𝑡)≡0.
Condition (53) implies that Re𝑎≤0. Since 𝛾+|𝑐|̃𝛾−|𝑐|̃=|Im𝑎|+|Im𝑎|²−|𝑏|²+|𝑏||Im𝑎|+|Im𝑎|²−|𝑏|²−|𝑏|=|Im𝑎|+|𝑏||Im𝑎|−|𝑏|^1/2, (57) in view of (14) we obtain 𝜆_𝑘(𝑡)=|Im𝑎|+|𝑏||Im𝑎|−|𝑏|^1/2×𝜌_𝑘(𝑡)+𝐴_𝑘(𝑡)+𝐵_𝑘(𝑡,Θ(𝑡)𝛾+|𝑐|̃)=Re𝑎+𝛾−|𝑐|̃𝜌₀(𝑡)𝛽(𝑡)||+|𝑏|+𝑚=Re𝑎+Im𝑎|Im𝑎|−|𝑏|^1/2𝜌₀(𝑡)+𝑚𝛽(𝑡), (58) and the assertion follows from (16) and Theorem 2.

Now we show an example that, under certain circumstances, Corollary 8 is more useful than Corollary 1 from [21].

Example 9. Consider (2), where 𝑎(𝑡)≡−√5+2𝑖, 𝑏(𝑡)≡1, 𝐴_𝑘(𝑡)≡0, 𝐵_𝑘(𝑡)≡0, for 𝑘=1,…,𝑚, and 𝑔𝑡,𝑧,𝑤₁,…,𝑤_𝑚=2√3𝑒^𝑖𝑡𝑧+^𝑚_𝑘=1𝑘√2𝑚𝑡15−√14𝑒^−𝑡𝑤_𝑘. (59)

Assume that 𝑡₀=𝑚 and 𝑅=∞, 𝜃_𝑘(𝑡)=𝑘ln𝑡. Put 𝑇=𝑒^𝑡0=𝑒^𝑚. Then, 𝜌₀(𝑡)≡2/√3, 𝜌_𝑘(𝑡)=(𝑘/2𝑚𝑡)(√15−√14)𝑒^−𝑡. We have |𝑎|−|𝑏|max|𝑎||𝑎|+|𝑏|Re𝑎+|𝑎|−|𝑏|^1/2𝜌₀(𝑡)(𝑡),𝛽+𝑚𝛽^′(𝑡)𝛽(𝑡)−2=max3√5+√22√3(𝑡),𝛽+𝑚𝛽^′(𝑡)𝛽(𝑡)≥23√6−√5>0, (60) for 𝜃^′_𝑘(𝑡)𝛽(𝑡)=𝑘𝑡𝛽(𝑡)≥|𝑎|+|𝑏||𝑎|−|𝑏|^1/2×𝜌_𝑘(𝑡)+𝐴_𝑘(𝑡)+𝐵_𝑘(𝑡=𝑘)𝑚𝑡√2√15−√14𝑒^−𝑡, (61) where 𝑘∈{1,…,𝑚}; hence, we cannot apply Corollary 1 from [21].

On the other hand, if we use 𝜃^′_𝑘(𝑡)𝛽(𝑡)=𝑘𝑡𝛽(𝑡)=𝑘√3√2𝑚𝑡15−√14𝑒^−𝑡≥|Im𝑎|+|𝑏||Im𝑎|−|𝑏|^1/2𝜌_𝑘(𝑡)+𝐴_𝑘(𝑡)+𝐵_𝑘(𝑡), (62) where 𝑘∈{1,…,𝑚}, we have ||+|𝑏|maxRe𝑎+Im𝑎|Im𝑎|−|𝑏|^1/2𝜌₀(𝑡)𝛽(𝑡),𝛽+𝑚^′(𝑡)𝛽(𝑡)=max−√√5+2+𝑚3√2𝑚15−√14𝑒^−𝑡√,−1≤−√5+2+32√15−√14<−12100<0; (63) thus, Corollary 8 guarantees the stability and also asymptotic stability of the trivial solution of the considered equation.

The following corollary gives sufficient conditions for stability of the trivial solution of (2).

Corollary 10. Assume that the conditions (i), (ii), and (iii) are valid with 𝜚̃(𝑡)≡0. If 𝛽(𝑡) is monotone and bounded on [𝑇,∞) and if limsup_𝑡→∞^𝑡Θ(𝑠)₊𝑑𝑠<∞, (64) where [Θ(𝑡)]₊=max{Θ(𝑡),0}, then the trivial solution of (2) is stable.

Proof. Suppose firstly that 𝛽 is nonincreasing on [𝑇,∞). Then, 𝛽^′≤0 a.e. on [𝑇,∞).
If 𝛽(𝑇₂)=0 for some 𝑇₂≥𝑇, then 𝛽(𝑡)≡0 on [𝑇₂,∞). Consequently, Λ has to satisfy only the inequality Θ(𝑡)≤Λ(𝑡) a.e. on [𝑇₂,∞), so we may choose Λ(𝑡)=Θ(𝑡) on [𝑇₂,∞). It follows that Λ(𝑡)=Θ(𝑡)≤max{Θ(𝑡),0}=[Θ(𝑡)]₊.
On the other hand, if 𝛽(𝑡)>0 on [𝑇,∞), we may put Λ(𝑡)=max{Θ(𝑡),𝛽^′(𝑡)/𝛽(𝑡)}. Then, Λ(𝑡)Θ(𝑡),𝛽=max^′(𝑡)𝛽(𝑡)≤maxΘ(𝑡),0=Θ(𝑡)₊. (65)
In both cases, Λ satisfies condition (iv) and the inequality Λ(𝑡)≤[Θ(𝑡)]₊ on [𝑇₂,∞); hence, limsup_𝑡→∞^𝑡Λ(𝑠)𝑑𝑠≤limsup_𝑡→∞^𝑡Θ(𝑠)₊𝑑𝑠<∞. (66)
Now assume that 𝛽 is nondecreasing on [𝑇,∞). Then, 𝛽^′≥0 a.e. on [𝑇,∞).
If 𝛽(𝑡)≡0 on [𝑇,∞), we may treat it as previously mentioned.
Otherwise, there is some 𝑇₃≥𝑇 such that 𝛽(𝑡)>0 on [𝑇₃,∞), and we may choose Λ(𝑡)=max{Θ(𝑡),𝛽^′(𝑡)/𝛽(𝑡)} on [𝑇₃,∞). Clearly Λ satisfies condition (iv) on [𝑇₃,∞). Since 𝛽^′≥0 a.e. on [𝑇,∞), it follows that 𝛽^′/𝛽≥0 a.e. on [𝑇₃,∞). Hence, Λ(𝑡)Θ(𝑡),𝛽=max^′(𝑡)𝛽(𝑡)≤maxΘ(𝑡)₊,𝛽^′(𝑡)𝛽(𝑡)≤Θ(𝑡)₊+𝛽^′(𝑡)𝛽(𝑡), (67) and then limsup_𝑡→∞^𝑡Λ(𝑠)𝑑𝑠≤limsup_𝑡→∞^𝑡Θ(𝑠)₊𝑑𝑠+limsup_𝑡→∞^𝑡𝛽^′(𝑡)𝛽(𝑡)𝑑𝑠≤limsup_𝑡→∞^𝑡Θ(𝑠)₊𝑑𝑠+limsup_𝑡→∞𝛽(𝑡)𝛽𝑇ln−ln₃<∞, (68) since 𝛽 is bounded on [𝑇,∞).
The statement follows from Theorem 2.

We can derive several consequences from Theorem 4.

Corollary 11. Let the conditions (i), (ii), (iii), and (iv) be fulfilled and limsup_𝑡→∞^𝑡_𝑠𝜚̃(𝜉)exp−^𝜉_𝑠Λ(𝜎)𝑑𝜎𝑑𝜉<∞, (69) for some 𝑠≥𝑇.
If 𝑧(𝑡) is any solution of (2) existing for 𝑡→∞, then 𝑧(𝑡)=𝑂exp^𝑡_𝑠Λ(𝜉)𝑑𝜉. (70)

Proof. From the assumptions and (45) we can see that there are 𝐾>0 and 𝑆≥𝑠 such that for 𝑡≥𝑆 we have 𝑉(𝑡)exp−^𝑡_𝑠Λ(𝜉)(𝑠)≤𝑑𝜉−𝑉_𝑡𝑠𝜚̃(𝜉)exp−^𝜉_𝑠Λ(𝜎)𝑑𝜎𝑑𝜉≤𝐾<∞. (71)
Then, 𝜇|𝑧(𝑡)|≤𝑉(𝑡)≤(𝐾+𝑉(𝑠))exp^𝑡_𝑠Λ(𝜉)𝑑𝜉. (72)

Corollary 12. Let the assumptions (i), (ii), (iii), and (iv) hold, and let limsup_𝑡→∞Λ(𝑡)<∞,𝜚̃(𝑡)=𝑂𝑒^𝜂𝑡, (73) where 𝜂>limsup_𝑡→∞Λ(𝑡). If 𝑧(𝑡) is any solution of (2) existing for 𝑡→∞, then 𝑧(𝑡)=𝑂(𝑒^𝜂𝑡).

Proof. In view of (73), there are 𝐿>0, 𝜂^∗<𝜂, and 𝑠>𝑇 such that 𝜂^∗>Λ(𝑡) for 𝑡≥𝑠 and 𝜚̃(𝑡)𝑒^−𝜂𝑡<𝐿 for 𝑡≥𝑠. From (41) we get 𝜇|𝑧(𝑡)|≤𝑉(𝑠)𝑒^{𝜂∗(𝑡−𝑠)}+𝐿^𝑡_𝑠𝑒^𝜂𝜏𝑒^{𝜂∗(𝑡−𝜏)}(𝑠)𝑒𝑑𝜏≤𝑉^{𝜂∗(𝑡−𝑠)}+𝐿𝑒^𝜂∗𝑡𝑒^{(𝜂−𝜂∗)𝑡}−𝑒^{(𝜂−𝜂∗)𝑠}𝜂−𝜂^∗≤𝑉(𝑠)𝑒^{𝜂∗(𝑡−𝑠)}+𝐿𝜂−𝜂^∗𝑒^𝜂𝑡=𝑂𝑒^𝜂𝑡. (74)

Remark 13. If 𝜌(𝑡)≡0, we can take 𝐿=0 in the proof of Corollary 12, and taking inequalities (74) into account we obtain the following statement: there is an 𝜂^∗<𝜂₀<𝜂 such that 𝑧(𝑡)=𝑜(𝑒^𝜂0𝑡) holds for the solution 𝑧(𝑡) of (2).

5. Conclusion

We studied asymptotic behavior of real two-dimensional differential system with a finite number of nonconstant delays. We considered the case corresponding to the situation when the equilibrium point 0 of autonomous system (6) is a stable focus or a stable centre. We utilized the method of complexification and the method of Lyapunov-Krasovskii functional. Criteria for stability and asymptotic stability of the solutions as well as conditions ensuring that all solutions of (2) tend to zero are derived. At the end we supplied several corollaries and an example which shows that in some cases the criteria obtained in this paper are more applicable than the criteria presented in [21].

Acknowledgments

The first author was supported by the Project CZ.1.07/2.3.00/30.0039 of Brno University of Technology. The work of the second author was realised in CEITEC—Central European Institute of Technology—with research infrastructure supported by the Project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund and by the Project FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. This support is gratefully acknowledged.