An Explicit Criterion for the Existence of Positive Solutions of the Linear Delayed Equation 𝑥̇(𝑡)=−𝑐(𝑡)𝑥(𝑡−𝜏(𝑡))

Abstract

The paper investigates an equation with single delay 𝑥̇(𝑡)=−𝑐(𝑡)𝑥(𝑡−𝜏(𝑡)), where 𝜏∶[𝑡₀−𝑟,∞)→(0,𝑟], 𝑟>0, 𝑡₀∈ℝ, and 𝑐∶[𝑡₀−𝑟,∞)→(0,∞) are continuous functions, and the difference 𝑡−𝜏(𝑡) is an increasing function. Its purpose is to derive a new explicit integral criterion for the existence of a positive solution in terms of 𝑐 and 𝜏. An overview of known relevant criteria is provided, and relevant comparisons are also given.

1. Introduction

Let us consider an equation 𝑥̇(𝑡)=−𝑐(𝑡)𝑥(𝑡−𝜏(𝑡)),(1.1) where 𝜏∶[𝑡₀−𝑟,∞)→ℝ⁺ is a continuous function, 𝑡₀∈ℝ, 0<𝑟=const  ℝ⁺=(0,∞), 𝜏(𝑡)≤𝑟 if 𝑡∈[𝑡₀−𝑟,∞), and 𝑐∶[𝑡₀−𝑟,∞)→ℝ⁺ is a continuous function. The symbol “⋅” denotes the right-hand derivative.

A function 𝑥 is called a solution of (1.1) corresponding to an initial point 𝑡^∗∈[𝑡₀,∞) if 𝑥 is defined and is continuous on [𝑡^∗−𝑟,∞) and differentiable on [𝑡^∗,∞) and satisfies (1.1) for 𝑡≥𝑡^∗. We denote by 𝑥(𝑡^∗,𝜑)(𝑡) a solution of (1.1) corresponding to an initial point 𝑡^∗∈[𝑡₀,∞) generated by a continuous initial function 𝜑∶[𝑡^∗−𝑟,𝑡^∗]→ℝ. In the case of a linear equation (1.1), the solution 𝑥(𝑡^∗,𝜑)(𝑡) is unique on its maximal existence interval [𝑡^∗,∞) (see, e.g., [1]). If in the paper an initial point is not indicated, 𝑡₀ is assumed.

As is customary, a solution of (1.1) corresponding to an initial point 𝑡^∗∈[𝑡₀,∞) is called oscillatory if it has arbitrarily large zeros. Otherwise it is called nonoscillatory. A nonoscillatory solution 𝑥 of (1.1) corresponding to an initial point 𝑡^∗ is called positive (negative) if 𝑥(𝑡)>0 (𝑥(𝑡)<0) on [𝑡^∗−𝑟,∞). A nonoscillatory solution 𝑥 of (1.1) corresponding to an initial point 𝑡^∗ is called eventually positive (eventually negative) if there exists 𝑡^∗∗>𝑡^∗ such that 𝑥(𝑡)>0 (𝑥(𝑡)<0) on [𝑡^∗∗,∞). Instead of the terms “eventually positive” (“eventually negative”), the phrase “positive for 𝑡→∞” (“negative for 𝑡→∞”) is very often used.

In order to simplify the formulation of results in the paper, we do not mention all the technical conditions. Typically, we do not mention that inequalities are valid on an interval [𝑡₀,∞), tacitly assuming that 𝑡₀ is so large that the necessary assumptions hold on the interval in question, nor do we mention an initial point 𝑡^∗ when we characterize a property of a solution.

In the paper, we derive a new explicit criterion for the existence of a positive solution of the scalar differential equation with delay (1.1). Although, by its form, (1.1) is a simple equation, it plays an important role in the theory of differential equations with delay. One of the reasons is that, because of its simplicity, it is often used for testing new results and comparing them with previous ones.

In the literature there are numerous criteria for positivity even for equations more general than (1.1).

Below, we give an overview of previous results (we assume that above assumptions are true in spite of the fact that some of criteria are valid even under weaker assumptions).

In [2] the following criterion for the existence of a nonoscillatory solution is given.

Theorem 1.1. If limsup_𝑡→∞^𝑡_{𝑡−𝜏(𝑡)}𝑐(𝑠)d𝑠<1e,(1.2) then (1.1) has a nonoscillatory solution.

In addition to this, according to a result in [2], we have the following.

Theorem 1.2. If ^𝑡_{𝑡−𝜏(𝑡)}𝑐(𝑠)d𝑠≤1e,(1.3) for all sufficiently large 𝑡, then (1.1) has a nonoscillatory solution.

In [3], the authors consider (1.1) in the form𝑥̇(𝑡)=−1e+𝑎(𝑡)𝑥(𝑡−1),𝑡≥1,(1.4) and prove the following.

Theorem 1.3. Assume that 𝑎(𝑡)≤18e𝑡²,(1.5) for all sufficiently large 𝑡. Then (1.4) has a solution 𝑥(𝑡)≥√𝑡e^−𝑡.(1.6)

In [4], Theorem 1.3 is improved to the following theorem.

Theorem 1.4. Assume that 𝑎(𝑡)≤18e𝑡²1+1ln²𝑡,(1.7) for all sufficiently large 𝑡. Then (1.4) has a solution 𝑥(𝑡)≥√𝑡ln𝑡e^−𝑡.(1.8)

The authors then demonstrate that, in terms of the values of the coefficient of the equation itself, inequalities (1.5), (1.7) are sharp in a sense because the following result holds.

Theorem 1.5 (see [4]). Let 𝜏(𝑡)=𝜏=const in (1.1). Assume that liminf_𝑡→∞𝑐(𝑡)=1𝜏e,liminf_𝑡→∞𝑐(𝑡)−1𝜏e𝑡²=𝜏8e,liminf_𝑡→∞𝑐(𝑡)−1𝜏e𝑡²−𝜏8eln²𝑡=𝐶>𝜏8e.(1.9) Then all solutions of (1.1) oscillate.

Motivated by Theorems 1.3–1.5, the second author gave in [5] a generalization of these results. To formulate them we need to introduce the concept of an iterated logarithm.

Definition 1.6. Let one calls the expression ln_𝑘𝑡, 𝑘≥1, defined by the formula ln_𝑘𝑡=lnln⋯ln𝑡_{𝑘},𝑘≥1,(1.10) a 𝑘th iterated logarithm if 𝑡>exp_𝑘−21, where exp_𝑘𝑡≡(exp(exp(⋯exp𝑡)))_{𝑘},𝑘≥1,(1.11)exp₀𝑡≡𝑡, and exp₋₁𝑡≡0. Moreover, let us define ln₀𝑡≡𝑡. Instead of expressions ln₀𝑡, ln₁𝑡, we will write only 𝑡 and ln𝑡 in the sequel.

Theorem 1.7. Let 𝜏(𝑡)=𝜏=const in (1.1).(A)Let one assumes that 𝑐(𝑡)≤1e𝜏+𝜏8e𝑡²+𝜏8e(𝑡ln𝑡)²+𝜏8e𝑡ln𝑡ln₂𝑡²+⋯+𝜏8e𝑡ln𝑡ln₂𝑡⋯ln_𝑘𝑡²(1.12) for 𝑡→∞ and an integer 𝑘≥0. Then there is a positive solution 𝑥=𝑥(𝑡) of (1.1). Moreover, 𝑥(𝑡)<e^−𝑡/𝜏√𝑡ln𝑡ln₂𝑡⋯ln_𝑘𝑡(1.13) as 𝑡→∞.(B)Let one assumes that 𝑐(𝑡)≥1e𝜏+𝜏8e𝑡²+𝜏8e(𝑡ln𝑡)²+⋯+𝜏8e𝑡ln𝑡ln₂𝑡⋯ln_𝑘−1𝑡²+𝜃𝜏8e𝑡ln𝑡ln₂𝑡⋯ln_𝑘𝑡²(1.14) for 𝑡→∞, an integer 𝑘≥0, and a constant 𝜃>1. Then all solutions of (1.1) oscillate.

Recently, Theorem 1.7 has been generalized for variable delay in [6].

Theorem 1.8. (A) Let 0≤𝜏(𝑡)≤𝑟 for 𝑡→∞, and let condition (A) of Theorem 1.7 hold. Then (1.1) has a nonoscillatory solution.
(B) Let 𝜏(𝑡)≥𝑟>0 for 𝑡→∞, and let condition (B) of Theorem 1.7 hold. Then all solutions of (1.1) oscillate.

For further criteria on the existence of positive solutions, we refer, for example, to papers [6–14], books [15–18], and relevant references therein.

2. An Integral Positivity Criterion, Comparisons, and Open Problems

Theorems 1.3–1.8 in part 1 are formulated in terms of the values of the coefficients 𝑐 or 𝑎 itself whereas Theorems 1.1 and 1.2 are formulated in terms of the average of the coefficient 𝑐. In accordance with the opinion presented, for example, in [4], we observe that, sometimes, conditions of the first of the types mentioned yield stronger results. However, conditions of the second type are, in general, preferable.

The following criterion is of an integral type (i.e., of the second type) and uses the term “average” of coefficient 𝑐 of the form ^𝑡_{𝑡−𝜏(𝑡)}𝑐(𝑠)𝜔(𝑠)d𝑠(2.1) with an appropriate weight function 𝜔.

Theorem 2.1. Let one assumes that ^𝑡_{𝑡−𝜏(𝑡)}𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠≤1e−𝜏(𝑡)2e𝑡,(2.2) for 𝑡∈[𝑡₀,∞). Then there exists a positive solution 𝑥=𝑥(𝑡) of (1.1) on [𝑡₀,∞). Moreover, 𝑥(𝑡)<exp⎛⎜⎝−e^𝑡_𝑡0𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠⎞⎟⎠(2.3) for 𝑡∈[𝑡₀,∞).

The proof of this criterion, given in part 3, is not difficult and uses the retract technique. Now we consider a simple example of (1.1) to show that known criteria cannot be applied. For simplicity, a constant delay 𝜏(𝑡)=𝜏=const is assumed. The symbol 𝒪 (big “O”) below stands for the Landau order symbol.

Example 2.2. Let 𝑐(𝑡)=1𝜏e+12𝜏ee^𝜏/(2𝑡)sin2𝜋𝑡𝜏(2.4) be chosen in (1.1); that is, we consider an equation 𝑥̇(𝑡)=−1𝜏e+12𝜏ee^𝜏/(2𝑡)sin2𝜋𝑡𝜏𝑥(𝑡−𝜏).(2.5) Let us apply Theorem 2.1. Then the integral on the left-hand side of inequality (2.2) equals ^𝑡_𝑡−𝜏𝑐(𝑠)e^{−𝜏/(2𝑠)}d𝑠=^𝑡_𝑡−𝜏1𝜏e+12𝜏ee^𝜏/(2𝑠)sin2𝜋𝑠𝜏e^{−𝜏/(2𝑠)}d𝑠=𝐼₁(𝑡)+𝐼₂(𝑡),(2.6) where 𝐼₁(𝑡)=1𝜏e^𝑡_𝑡−𝜏e^{−𝜏/(2𝑠)}d𝑠,𝐼₂(𝑡)=12𝜏e^𝑡_𝑡−𝜏sin2𝜋𝑠𝜏d𝑠.(2.7) Obviously 𝐼₂(𝑡)=0. It remains to estimate 𝐼₁(𝑡) in order to show that inequality (2.2) holds. We develop an asymptotic decomposition for 𝐼₁(𝑡) for 𝑡→∞ with the necessary degree of accuracy. We obtain 𝐼₁(𝑡)=1𝜏e^𝑡_𝑡−𝜏e^{−𝜏/(2𝑠)}d𝑠=1𝜏e^𝑡_𝑡−𝜏1−𝜏2𝑠+𝜏²8𝑠²+𝒪1𝑠³d𝑠=1e−12eln11−(𝜏/𝑡)−𝜏8e1𝑡−1𝑡−𝜏+𝒪1𝑡³=1e−12eln1+1𝜏+1𝜏²+𝒪1𝑡³−𝜏8e𝑡1−11−(𝜏/𝑡)+𝒪1𝑡³=1e−12e𝜏𝑡+𝜏²2𝑡²+𝜏²8e𝑡²+𝒪1𝑡³=1e−𝜏2e𝑡−𝜏²8e𝑡²+𝒪1𝑡³.(2.8) This means that ^𝑡_𝑡−𝜏𝑐(𝑠)e^{−𝜏/(2𝑠)}d𝑠=1e−𝜏2e𝑡−𝜏²8e𝑡²+𝒪1𝑡³<1e−𝜏2e𝑡(2.9) and inequality (2.2) holds. By Theorem 2.1, (2.5) has a positive solution.

We use this example to compare Theorem 2.1 with known criteria mentioned in Section 1.

Comparison with Theorems 1.1 and 1.2
We show that the results mentioned cannot be applied to Example 2.2. For this, we will analyse the integral ^𝑡_𝑡−𝜏𝑐(𝑠)d𝑠=^𝑡_𝑡−𝜏1𝜏e+12𝜏ee^𝜏/(2𝑠)sin2𝜋𝑠𝜏d𝑠.(2.10) For an arbitrarily large integer 𝑘, we set 𝑡_𝑘=10𝑘𝜏. Then 𝑡_𝑘−𝜏=(10𝑘−1)𝜏 and ^𝑡𝑘_{𝑡𝑘−𝜏}𝑐(𝑠)d𝑠=^10𝑘𝜏_{(10𝑘−1)𝜏}1𝜏e+12𝜏ee^𝜏/(2𝑠)sin2𝜋𝑠𝜏d𝑠=𝐽₁(𝑘)+𝐽₂(𝑘),(2.11) where 𝐽₁(𝑘)=^10𝑘𝜏_{(10𝑘−1)𝜏}1𝜏ed𝑠=1e,𝐽₂(𝑘)=12𝜏e^10𝑘𝜏_{(10𝑘−1)𝜏}e^𝜏/(2𝑠)sin2𝜋𝑠𝜏d𝑠.(2.12) Then 𝐽₂(𝑘)=12𝜏e^{(10𝑘−(1/2))𝜏}_{(10𝑘−1)𝜏}e^𝜏/(2𝑠)sin2𝜋𝑠𝜏d𝑠+12𝜏e^10𝑘𝜏_{(10𝑘−(1/2))𝜏}e^𝜏/(2𝑠)sin2𝜋𝑠𝜏d𝑠>12𝜏ee^{1/(20𝑘−1)}^{(10𝑘−(1/2))𝜏}_{(10𝑘−1)𝜏}sin2𝜋𝑠𝜏d𝑠+12𝜏ee^{1/(20𝑘−1)}^10𝑘𝜏_{(10𝑘−(1/2))𝜏}sin2𝜋𝑠𝜏d𝑠=12𝜏ee^{1/(20𝑘−1)}^10𝑘𝜏_{(10𝑘−1)𝜏}sin2𝜋𝑠𝜏d𝑠=0.(2.13) We conclude that, for every 𝑘, the inequality ^𝑡𝑘_{𝑡𝑘−𝜏}𝑐(𝑠)d𝑠=𝐽₁(𝑘)+𝐽₂(𝑘)>1e(2.14) holds. Since 𝑘 can be sufficiently large with lim_𝑘→∞𝑡_𝑘=∞, inequalities (1.2) and (1.3) in Theorems 1.1 and 1.2 are not valid for all sufficiently large 𝑡 because at least the values 𝑡=𝑡_𝑘 must be excluded. We conclude that Theorems 1.1 and 1.2 are not applicable to (2.5).

Comparison with Theorems 1.3–1.8
Using Example 2.2, we show again that the results mentioned cannot be applied. We will demonstrate the nonapplicability of Theorem 1.7. The same arguments can be used for the remaining theorems.
For an arbitrarily large integer 𝑝, we set 𝑡_𝑝=(10𝑝−(3/4))𝜏. Then,𝑐𝑡_𝑝=1𝜏e+12𝜏ee^{𝜏/(2𝑡𝑝)}sin2𝜋𝑡_𝑝𝜏=1𝜏e+12𝜏ee^{1/(20𝑝−(3/4))}sin20𝑝−32𝜋=1𝜏e+12𝜏ee^{1/(20𝑝−(3/4))}.(2.15) Let 𝑡=𝑡_𝑝. Then inequality (1.14) becomes 𝑐𝑡_𝑝=1𝜏e+12𝜏ee^{1/(20𝑝−(3/4))}≤1e𝜏+𝜏8e𝑡²_𝑝+𝜏8e𝑡_𝑝ln𝑡_𝑝²+𝜏8e𝑡_𝑝ln𝑡_𝑝ln₂𝑡_𝑝²+⋯+𝜏8e𝑡_𝑝ln𝑡_𝑝ln₂𝑡_𝑝⋯ln_𝑘𝑡_𝑝².(2.16) As 𝑝→∞, we have 𝑡_𝑝→∞. Inequality (2.16) does not hold because, for 𝑝→∞, the left-hand side and the right-hand side tend to 3/(2e𝜏) and 1/(e𝜏), respectively, but the inequality 32e𝜏≤1e𝜏(2.17) does not hold.

The sharpness of Theorem 2.1 will be illustrated by the following example.

Example 2.3. Let 𝜏(𝑡)=𝜏=const and 𝑐(𝑡)=1𝜏e+𝜏𝜀8e𝑡²,(2.18) where a parameter 𝜀<1 is chosen in (1.1); that is, we consider an equation 𝑥̇(𝑡)=−1𝜏e+𝜏𝜀8e𝑡²𝑥(𝑡−𝜏).(2.19) Let us apply Theorem 2.1. Then the integral on the left-hand side of inequality (2.2) equals 𝐼(𝑡)=^𝑡_𝑡−𝜏𝑐(𝑠)e^{−𝜏/(2𝑠)}d𝑠=^𝑡_𝑡−𝜏1𝜏e+𝜏𝜀8e𝑡²e^{−𝜏/(2𝑠)}d𝑠.(2.20) We develop an asymptotic decomposition for 𝐼(𝑡) for 𝑡→∞ with the necessary degree of accuracy. Applying some similar asymptotic decompositions obtained by calculating the integral 𝐼₁(𝑡) in Example 2.2, we get 𝐼(𝑡)=^𝑡_𝑡−𝜏1𝜏e+𝜏𝜀8e𝑠²e^{−𝜏/(2𝑠)}d𝑠=^𝑡_𝑡−𝜏1𝜏e+𝜏𝜀8e𝑠²1−𝜏2𝑠+𝜏²8𝑠²+𝒪1𝑠³d𝑠=^𝑡_𝑡−𝜏1𝜏e+𝜏𝜀8e𝑠²1−𝜏2𝑠+𝜏²8𝑠²+𝒪1𝑠³d𝑠=^𝑡_𝑡−𝜏1𝜏e−12e𝑠+𝜏²(1+𝜀)8e𝑠²d𝑠=1e−12e𝜏𝑡+𝜏²2𝑡²+𝜏²(1+𝜀)8e𝑡²+𝒪1𝑡³=1e−𝜏2e𝑡+𝜏²(𝜀−1)8e𝑡²+𝒪1𝑡³.(2.21) This means that Theorem 2.1 will be applicable to (2.19) if inequality (2.2), that is, the inequality ^𝑡_𝑡−𝜏𝑐(𝑠)e^{−𝜏/(2𝑠)}d𝑠=1e−𝜏2e𝑡+𝜏²(𝜀−1)8e𝑡²+𝒪1𝑡³<1e−𝜏2e𝑡,(2.22) holds. This is true since 𝜀<1. We finish this example by concluding that, in the case of a constant delay, Theorem 2.1 gives a result not equivalent to that given by Theorems 1.3–1.8. Nevertheless, as can be seen from the form of the function 𝑐 in (2.18), which almost coincides with the first two terms of the auxiliary comparison function on the right-hand side of inequality (1.12) in Theorem 1.7, the result of Theorem 2.1 is not so far from a “sharp” criterion.

At the end of this part, we will formulate open problem not answered in this paper as a mathematical challenge for further research in this field.

Open Problem 1. Is it possible to improve the result of Theorem 2.1 in such a way that the new result completely covers the parts (A) of Theorems 1.7 and 1.8?

3. Proof of Theorem 2.1

Let 𝐶_𝑟([−𝑟,0],ℝ), be the Banach space of continuous functions from the interval [−𝑟,0] into ℝ equipped with the supremum norm. If 𝜎∈ℝ, 𝐴≥0, and 𝑥∶[𝜎−𝑟,𝜎+𝐴]→ℝ is a continuous function, then, for each 𝑡∈[𝜎,𝜎+𝐴], we define the function 𝑥_𝑡∈𝐶_𝑟 by 𝑥_𝑡(𝜃)∶=𝑥(𝑡+𝜃), 𝜃∈[−𝑟,0]. Let us consider a retarded functional differential equation 𝑥̇(𝑡)=𝑓𝑡,𝑥_𝑡,(3.1) where 𝑓∶Ω↦ℝ,Ω⊆ℝ×𝐶_𝑟 is a continuous quasi-bounded map which satisfies a local Lipschitz condition with respect to the second argument. We assume that the derivatives in the system (3.1) are at least right-sided.

In accordance with [1], a function 𝑥 is said to be a solution of (3.1) on [𝜎−𝑟,𝜎+𝐴) with 𝐴>0 if 𝑥∶[𝜎−𝑟,𝜎+𝐴)→ℝ is a continuous function, (𝑡,𝑥_𝑡)∈Ω for 𝑡∈[𝜎,𝜎+𝐴), and 𝑥 satisfies (3.1) on [𝜎,𝜎+𝐴). In view of the above conditions, each element (𝜎,𝜑)∈Ω determines a unique noncontinuable solution 𝑥(𝜎,𝜑) of system (3.1) through (𝜎,𝜑)∈Ω on its maximal existence interval. This solution depends continuously on the initial data [19].

For continuously differentiable functions 𝜌,𝛿∶[𝑡₀−𝑟,∞)→ℝ with 𝜌(𝑡)<𝛿(𝑡) for 𝑡∈[𝑡₀−𝑟,∞), we introduce the sets Ω∶=𝑡₀−𝑟,∞×𝐶_𝑟,𝜔∶=(𝑡,𝑥)∶𝑡≥𝑡₀−𝑟,𝜌(𝑡)<𝑥<𝛿(𝑡).(3.2)

In the sequel, we employ the following particular case of Theorem 1.1 from [10]. Its proof uses the topological (Ważewski’s) retract principle known in the theory of ordinary differential equations. The relevant references can be found, for example, in [10].

Theorem 3.1. Let for 𝑡≥𝑡₀, 𝜙∈𝐶_𝑟, and (𝑡+𝜃,𝜙(𝜃))∈𝜔, 𝜃∈[−𝑟,0), the inequalities 𝛿^(𝑡)<𝑓(𝑡,𝜙)𝑤ℎ𝑒𝑛𝜙(0)=𝛿(𝑡),(3.3)𝜌^(𝑡)>𝑓(𝑡,𝜙)𝑤ℎ𝑒𝑛𝜙(0)=𝜌(𝑡)(3.4) hold. Then there exists a solution of (3.1) on [𝑡₀−𝑟,∞) such that 𝜌(𝑡)<𝑥(𝑡)<𝛿(𝑡),𝑡∈𝑡₀−𝑟,∞.(3.5)

Proof of Theorem 2.1. In our case 𝑓(𝑡,𝑥_𝑡)∶=−𝑐(𝑡)𝑥(𝑡−𝜏(𝑡)) and 𝑓(𝑡,𝜙)∶=−𝑐(𝑡)𝜙(−𝜏(𝑡)). We will apply Theorem 3.1 with 𝑓(𝑡,𝜙)=−𝑐(𝑡)𝜙(−𝜏(𝑡)),𝜌(𝑡)=0,𝛿(𝑡)=exp⎛⎜⎝−e^𝑡_𝑡0𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠⎞⎟⎠,𝜔=⎧⎨⎩(𝑡,𝑥)∶𝑡≥𝑡₀−𝑟,0<𝑥<exp⎛⎜⎝−e^𝑡_𝑡0𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠⎞⎟⎠⎫⎬⎭.(3.6) First we verify inequality (3.3). We get 𝛿^(𝑡)−𝑓(𝑡,𝜙)=⎡⎢⎣exp⎛⎜⎝−e^𝑡_𝑡0𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠⎞⎟⎠⎤⎥⎦^+𝑐(𝑡)𝜙(−𝜏(𝑡))=−e⋅𝑐(𝑡)e^{−𝜏(𝑡)/(2𝑡)}exp⎛⎜⎝−e^𝑡_𝑡0𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠⎞⎟⎠+𝑐(𝑡)𝜙(−𝜏(𝑡))=(∗).(3.7) Since, in accordance with the assumptions of the theorem, (𝑡+𝜃,𝜙(𝜃))∈𝜔 if 𝜃∈[−𝑟,0), we can estimate the last term 𝜙(−𝜏(𝑡)): 𝜙(−𝜏(𝑡))<𝛿(𝑡−𝜏(𝑡))=exp⎛⎜⎝−e^{𝑡−𝜏(𝑡)}_𝑡0𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠⎞⎟⎠.(3.8) Continuing, we have (∗)<−e⋅𝑐(𝑡)e^{−𝜏(𝑡)/(2𝑡)}exp⎛⎜⎝−e^𝑡_𝑡0𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠⎞⎟⎠+𝑐(𝑡)exp⎛⎜⎝−e^{𝑡−𝜏(𝑡)}_𝑡0𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠⎞⎟⎠=𝑐(𝑡)exp⎛⎜⎝−e^{𝑡−𝜏(𝑡)}_𝑡0𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠⎞⎟⎠×−e⋅e^{−𝜏(𝑡)/(2𝑡)}exp−e^𝑡_{𝑡−𝜏(𝑡)}𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠+1=(∗∗).(3.9) Using inequality (2.2), we can obtain the estimate exp−e^𝑡_{𝑡−𝜏(𝑡)}𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠≥e⁻¹⋅e^{𝜏(𝑡)/(2𝑡)}.(3.10) Therefore, (∗∗)≤𝑐(𝑡)exp⎛⎜⎝−e^{𝑡−𝜏(𝑡)}_𝑡0𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠⎞⎟⎠−e⋅e^{−𝜏(𝑡)/(2𝑡)}⋅e⁻¹⋅e^{𝜏(𝑡)/(2𝑡)}+1=0,(3.11) and inequality (3.3) holds because 𝛿^(𝑡)−𝑓(𝑡,𝜙)<0. It is much easier is to verify inequality (3.4) since 𝜌^(𝑡)−𝑓(𝑡,𝜙)=0−(−𝑐(𝑡)𝜙(−𝜏(𝑡)))=𝑐(𝑡)𝜙(−𝜏(𝑡))>0.(3.12) All assumptions of Theorem 3.1 are fulfilled, and, therefore, by (3.5), there exists a solution 𝑥∶[𝑡₀−𝑟,∞)→ℝ^∗ such that 0<𝑥(𝑡)<exp⎛⎜⎝−e^𝑡_𝑡0𝑐(𝑠)e^{−𝜏(𝑠)/(2𝑠)}d𝑠⎞⎟⎠.(3.13)

Acknowledgments

This research was supported by Grant P201/10/1032 of the Czech Grant Agency (Prague) and by the Council of Czech Government MSM 00216 30529. This research was also supported by Grant P201/11/0768 of the Czech Grant Agency (Prague) and by Project FEKT-S-11-2(921).