Existence and Asymptotic Behavior of Positive Solutions of Functional Differential Equations of Delayed Type

Abstract

Solutions of the equation 𝑦(̇𝑡)=  −𝑓(𝑡,𝑦_𝑡) are considered for 𝑡→∞. The existence of two classes of positive solutions which are asymptotically different is proved using the retract method combined with Razumikhin's technique. With the aid of two auxiliary linear equations, which are constructed using upper and lower linear functional estimates of the right-hand side of the equation considered, inequalities for both types of positive solutions are given as well.

1. Introduction

Let 𝐶([𝑎,𝑏],ℝ^𝑛), where 𝑎,𝑏∈ℝ, 𝑎<𝑏, be the Banach space of the continuous mappings from the interval [𝑎,𝑏] into ℝ^𝑛 equipped with the supremum norm ‖𝜓‖_𝐶=sup_{𝜃∈[𝑎,𝑏]}‖𝜓(𝜃)‖,𝜓∈𝐶([𝑎,𝑏],ℝ^𝑛),(1.1) where ‖⋅‖ is the maximum norm in ℝ^𝑛. In the case of 𝑎=−𝑟<0 and 𝑏=0, we will denote this space as C^𝑛𝑟, that is, C^𝑛𝑟∶=𝐶([−𝑟,0],ℝ^𝑛).(1.2)

If 𝜎∈ℝ^𝑛, 𝐴≥0, and 𝑦∈𝐶([𝜎−𝑟,𝜎+𝐴],ℝ^𝑛), then, for each 𝑡∈[𝜎,𝜎+𝐴], we define 𝑦_𝑡∈C^𝑛𝑟 by 𝑦_𝑡(𝜃)=𝑦(𝑡+𝜃), 𝜃∈[−𝑟,0].

The present article is devoted to the problem of the existence of two classes of asymptotically different positive solutions of the delayed equation𝑦̇(𝑡)=−𝑓𝑡,𝑦_𝑡,(1.3) for 𝑡→+∞, where 𝑓∶Ω→ℝ is a continuous quasibounded functional that satisfies a local Lipschitz condition with respect to the second argument and Ω is an open subset in ℝ×C^1𝑟 such that conditions which use 𝑓 are well defined.

The main supposition of our investigation is that the right-hand side of (1.3) can be estimated as follows:𝐶_𝐴(𝑡)𝑦_𝑡(−𝑟)≤𝑓𝑡,𝑦_𝑡≤𝐶_𝐵(𝑡)𝑦_𝑡(−𝑟),(1.4) where (𝑡,𝑦_𝑡)∈Ω, and 𝐶_𝐴,𝐶_𝐵∶[𝑡₀−𝑟,∞)→ℝ⁺∶=(0,∞), 𝑡₀∈ℝ are continuous functions satisfying0<𝐶_𝐴(𝑡)≤𝐶_𝐵(𝑡)≤1(𝑟e),𝑡∈𝑡⁰−𝑟,∞,(1.5)_{∞𝑡0−𝑟}𝐶_𝐵(𝑡)d𝑡<1.(1.6) Quite lots of investigations are devoted to existence of positive solutions of different classes of equations (we mention at least monographs [1–6] and papers [7–12]). The investigation of two classes of asymptotically different solutions of (1.3) has been started in the paper [13] using a monotone iterative technique and a retract principle. Assumptions of results obtained are too cumbersome and are applied to narrow classes of equations. In the presented paper we derive more general statements under weaker conditions. This progress is related to more general inequalities (1.4) for the right-hand side of (1.3) which permit to omit utilization of properties of solutions of transcendental equations used in [13].

1.1. Ważewski’s Principle

In this section we introduce Ważewski’s principle for a system of retarded functional differential equations𝑦̇(𝑡)=𝐹𝑡,𝑦_𝑡,(1.7) where 𝐹∶Ω^∗↦ℝ^𝑛 is a continuous quasibounded map which satisfies a local Lipschitz condition with respect to the second argument and Ω^∗ is an open subset in ℝ×C^𝑛𝑟. We recall that the functional 𝐹 is quasibounded if 𝐹 is bounded on every set of the form [𝑡₁,𝑡₂]×C^𝑛𝑟𝐿⊂Ω^∗, where 𝑡₁<𝑡₂, C^𝑛𝑟𝐿∶=𝐶([−𝑟,0],𝐿) and 𝐿 is a closed bounded subset of ℝ^𝑛 (compare [2, page 305]).

In accordance with [14], a function 𝑦(𝑡) is said to be a solution of system (1.7) on [𝜎−𝑟,𝜎+𝐴) if there are 𝜎∈ℝ and 𝐴>0 such that 𝑦∈𝐶([𝜎−𝑟,𝜎+𝐴),ℝ^𝑛), (𝑡,𝑦_𝑡)∈Ω^∗, and 𝑦(𝑡) satisfies the system (1.7) for 𝑡∈[𝜎,𝜎+𝐴). For a given 𝜎∈ℝ, 𝜑∈𝐶, we say 𝑦(𝜎,𝜑) is a solution of the system (1.7) through (𝜎,𝜑)∈Ω^∗ if there is an 𝐴>0 such that 𝑦(𝜎,𝜑) is a solution of the system (1.7) on [𝜎−𝑟,𝜎+𝐴) and 𝑦_𝜎(𝜎,𝜑)=𝜑. In view of the above conditions, each element (𝜎,𝜑)∈Ω^∗ determines a unique solution 𝑦(𝜎,𝜑) of the system (1.7) through (𝜎,𝜑)∈Ω^∗ on its maximal interval of existence 𝐼_𝜎,𝜑=[𝜎,𝑎), 𝜎<𝑎≤∞ which depends continuously on initial data [14]. A solution 𝑦(𝜎,𝜑) of the system (1.7) is said to be positive if𝑦_𝑖(𝜎,𝜑)>0(1.8) on [𝜎−𝑟,𝜎]∪𝐼_𝜎,𝜑 for each 𝑖=1,2,…,𝑛. A nontrivial solution 𝑦(𝜎,𝜑) of the system (1.7) is said to be oscillatory on 𝐼_𝜎,𝜑 (under condition 𝐼_𝜎,𝜑=[𝜎,∞)) if (1.8) does not hold on any subinterval [𝜎₁,∞)⊂[𝜎,∞), 𝜎₁≥𝜎.

As a method of proving the existence of positive solutions of (1.3), we use Ważewski's retract principle which was first introduced by Ważewski [15] for ordinary differential equations and later extended to retarded functional differential equations by Rybakowski [16] and which is widely applicable to concrete examples. A summary of this principle is given below.

As usual, if a set 𝜔⊂ℝ×ℝ^𝑛, then int𝜔 and 𝜕𝜔 denote the interior and the boundary of 𝜔, respectively.

Definition 1.1 (see [16]). Let the continuously differentiable functions 𝑙_𝑖(𝑡,𝑦),𝑖=1,2,…,𝑝 and 𝑚_𝑗(𝑡,𝑦), 𝑗=1,2,…,𝑞, 𝑝²+𝑞²>0 be defined on some open set 𝜔₀⊂ℝ×ℝ^𝑛. The set 𝜔^∗=(𝑡,𝑦)∈𝜔₀∶𝑙_𝑖(𝑡,𝑦)<0,𝑚_𝑗(𝑡,𝑦)<0,𝑖=1,…,𝑝,𝑗=1,…,𝑞(1.9) is called a regular polyfacial set with respect to the system (1.7), provided that it is nonempty, if (𝛼) to (𝛾) below hold. (𝛼) For (𝑡,𝜋)∈ℝ×C^𝑛𝑟 such that (𝑡+𝜃,𝜋(𝜃))∈𝜔^∗ for 𝜃∈[−𝑟,0), we have (𝑡,𝜋)∈Ω^∗. (𝛽) For all 𝑖=1,2,…,𝑝, all (𝑡,𝑦)∈𝜕𝜔^∗ for which 𝑙_𝑖(𝑡,𝑦)=0, and all 𝜋∈C^𝑛𝑟 for which 𝜋(0)=𝑦 and (𝑡+𝜃,𝜋(𝜃))∈𝜔^∗, 𝜃∈[−𝑟,0). It follows that 𝐷𝑙_𝑖(𝑡,𝑦)>0, where 𝐷𝑙_𝑖(𝑡,𝑦)≡^𝑛_𝑘=1𝜕𝑙_𝑖(𝑡,𝑦)𝜕𝑦_𝑘𝑓_𝑘(𝑡,𝜋)+𝜕𝑙^𝑖(𝑡,𝑦)𝜕𝑡.(1.10)(𝛾) For all 𝑗=1,2,…,𝑞, all (𝑡,𝑦)∈𝜕𝜔^∗ for which 𝑚_𝑗(𝑡,𝑦)=0, and all 𝜋∈C^𝑛𝑟 for which 𝜋(0)=𝑦 and (𝑡+𝜃,𝜋(𝜃))∈𝜔^∗,𝜃∈[−𝑟,0). It follows that 𝐷𝑚_𝑗(𝑡,𝑦)<0, where 𝐷𝑚_𝑗(𝑡,𝑦)≡^𝑛_𝑘=1𝜕𝑚_𝑗(𝑡,𝑦)𝜕𝑦_𝑘𝑓_𝑘(𝑡,𝜋)+𝜕𝑚^𝑗(𝑡,𝑦)𝜕𝑡.(1.11)
The elements (𝑡,𝜋)∈ℝ×C^𝑛𝑟 in the sequel are assumed to be such that (𝑡,𝜋)∈Ω^∗.

In the following definition, a set 𝜔^∗ is an arbitrary set without any connection with a regular polyfacial set 𝜔^∗ defined by (1.9) in Definition 1.1.

Definition 1.2. A system of initial functions 𝑝_𝐴,𝜔∗ with respect to the nonempty sets 𝐴 and 𝜔^∗, where 𝐴⊂𝜔^∗⊂ℝ×ℝ^𝑛 is defined as a continuous mapping 𝑝∶𝐴→C^𝑛𝑟 such that (𝛼) and (𝛽) below hold. (𝛼) If 𝑧=(𝑡,𝑦)∈𝐴∩int𝜔^∗, then (𝑡+𝜃,𝑝(𝑧)(𝜃))∈𝜔^∗ for 𝜃∈[−𝑟,0].(𝛽) If 𝑧=(𝑡,𝑦)∈𝐴∩𝜕𝜔^∗, then (𝑡+𝜃,𝑝(𝑧)(𝜃))∈𝜔^∗ for 𝜃∈[−𝑟,0) and (𝑡,𝑝(𝑧)(0))=𝑧.

Definition 1.3 (see [17]). If 𝒜⊂ℬ are subsets of a topological space and 𝜋∶ℬ→𝒜 is a continuous mapping from ℬ onto 𝒜 such that 𝜋(𝑝)=𝑝 for every 𝑝∈𝒜, then 𝜋 is said to be a retraction of ℬ onto 𝒜. When a retraction of ℬ onto 𝒜 exists, 𝒜 is called a retract of ℬ.

The following lemma describes the main result of the paper [16].

Lemma 1.4. Let 𝜔^∗⊂𝜔₀ be a regular polyfacial set with respect to the system (1.7), and let W be defined as follows: 𝑊=(𝑡,𝑦)∈𝜕𝜔^∗∶𝑚_𝑗(𝑡,𝑦)<0,𝑗=1,2,…,𝑞.(1.12) Let 𝑍⊂𝑊∪𝜔^∗ be a given set such that 𝑍∩𝑊 is a retract of 𝑊 but not a retract of 𝑍. Then for each fixed system of initial functions 𝑝_𝑍,𝜔∗, there is a point 𝑧₀=(𝜎₀,𝑦₀)∈𝑍∩𝜔^∗ such that for the corresponding solution 𝑦(𝜎₀,𝑝(𝑧₀))(𝑡) of (1.7), one has 𝑡,𝑦𝜎₀,𝑝𝑧₀(𝑡)∈𝜔^∗(1.13) for each 𝑡∈𝐷_{𝜎0,𝑝(𝑧0)}.

Remark 1.5. When Lemma 1.4 is applied, a lot of technical details should be fulfilled. In order to simplify necessary verifications, it is useful, without loss of generality, to vary the first coordinate 𝑡 in definition of the set 𝜔^∗ in (1.9) within a half-open interval open at the right. Then the set 𝜔^∗ is not open, but tracing the proof of Lemma 1.4, it is easy to see that for such sets it remains valid. Such possibility is used below. We will apply similar remark and explanation to sets of the type Ω, Ω^∗ which serve as domains of definitions of functionals on the right-hand sides of equations considered.

For continuous vector functions 𝜌^∗=𝜌^∗1,𝜌^∗2,…,𝜌^∗𝑛,𝛿^∗=𝛿^∗1,𝛿^∗2,…,𝛿^∗𝑛∶𝑡₀−𝑟,∞⟶ℝ^𝑛,(1.14) with 𝜌^∗(𝑡)≪𝛿^∗(𝑡) for 𝑡∈[𝑡₀−𝑟,∞) (the symbol ≪ here and below means that 𝜌^∗𝑖(𝑡)<𝛿^∗𝑖(𝑡) for all 𝑖=1,2,…,𝑛), continuously differentiable on [𝑡₀,∞), we define the set 𝜔^∗∶=(𝑡,𝑦)∶𝑡∈𝑡₀,∞,𝜌^∗(𝑡)≪𝑦≪𝛿^∗(𝑡).(1.15) In the sequel, we employ the following result from [18, Theorem 1], which is proved with the aid of the retract technique combined with Razumikhin's approach.

Theorem 1.6. Let there be a 𝑝∈{0,…,𝑛} such that (i)if 𝑡≥𝑡₀, 𝜙∈C^𝑛𝑟 and (𝑡+𝜃,𝜙(𝜃))∈𝜔^∗ for any 𝜃∈[−𝑟,0), then 𝛿𝜌^∗𝑖^(𝑡)<𝐹^𝑖(𝑡,𝜙),when𝜙^𝑖(0)=𝛿^∗𝑖(𝑡),_∗𝑖^(𝑡)>𝐹^𝑖(𝑡,𝜙),when𝜙^𝑖(0)=𝜌^∗𝑖(𝑡)(1.16) for any 𝑖=1,2,…,𝑝, (If 𝑝=0, this condition is omitted.) (ii)if 𝑡≥𝑡₀, 𝜙∈C^𝑛𝑟 and (𝑡+𝜃,𝜙(𝜃))∈𝜔^∗ for any 𝜃∈[−𝑟,0) then 𝜌𝛿^∗𝑖^(𝑡)<𝐹^𝑖(𝑡,𝜙),when𝜙^𝑖(0)=𝜌^∗𝑖(𝑡),_∗𝑖^(𝑡)>𝐹^𝑖(𝑡,𝜙),when𝜙^𝑖(0)=𝛿^∗𝑖(𝑡)(1.17) for any 𝑖=𝑝+1,𝑝+2,…,𝑛. (If 𝑝=𝑛, this condition is omitted.)
Then, there exists an uncountable set 𝒴 of solutions of (1.7) on [𝑡₀−𝑟,∞) such that each 𝑦∈𝒴 satisfies 𝜌^∗(𝑡)≪𝑦(𝑡)≪𝛿^∗(𝑡),𝑡∈𝑡₀−𝑟,∞.(1.18)

1.2. Structure of Solutions of a Linear Equation

In this section we focus our attention to structure of solutions of scalar linear differential equation of the type (1.3) with variable bounded delay of the form𝑥̇(𝑡)=−𝑐(𝑡)𝑥(𝑡−𝜏(𝑡))(1.19) with continuous functions 𝑐∶[𝑡₀−𝑟,∞)→ℝ⁺ and 𝜏∶[𝑡₀,∞)→(0,𝑟].

In accordance with above definitions of positive or oscillatory solutions, we call a solution of (1.19) oscillatory if it has arbitrary large zeros, otherwise it is called nonoscillatory (positive or negative).

Let us mention properties of (1.19) which will be used later. Theorem 13 from [19] describes sufficient conditions for existence of positive solutions of (1.19) with nonzero limit.

Theorem 1.7 (see [19, Theorem 13]). Linear equation (1.19) has a positive solution with nonzero limit if and only if ^∞𝑐(𝑡)d𝑡<∞.(1.20)

Remark 1.8. Tracing the proof of Theorem 1.7, we conclude that a positive solution 𝑥=𝑥(𝑡) of (1.19) with nonzero limit exists on [𝑡₀−𝑟,∞) if ^∞𝑡_0−𝑟𝑐(𝑡)d𝑡<1.(1.21)

The following theorem is a union of parts of results from [20] related to the structure formulas for solutions of (1.19).

Theorem 1.9. Suppose the existence of a positive solution of (1.19) on [𝑡₀−𝑟,∞). Then there exist two positive solutions 𝑥_𝑑 and 𝑥_𝑠 of (1.19) on [𝑡₀−𝑟,∞) satisfying the relation lim_𝑡→∞𝑥_𝑠(𝑡)𝑥_𝑑(𝑡)=0(1.22) such that every solution 𝑥=𝑥(𝑡) of (1.19) on [𝑡₀−𝑟,∞) can be represented by the formula 𝑥(𝑡)=𝐾𝑥_𝑑(𝑡)+𝑂𝑥_𝑠(𝑡),(1.23) where the constant 𝐾 depends on 𝑥.

The symbol 𝑂, applied in (1.23) and below, is the Landau order symbol frequently used in asymptotic analysis.

Moreover, Theorem 9 in [20] gives a possibility to replace the pair of solutions 𝑥_𝑑(𝑡) and 𝑥_𝑠(𝑡) in (1.23) by another pairs of solutions 𝑥̃_𝑑(𝑡) and 𝑥̃_𝑠(𝑡) iflim_𝑡→∞𝑥̃_𝑠(𝑡)𝑥̃_𝑑(𝑡)=0(1.24) as given in the following theorem.

Theorem 1.10. Let 𝑥̃_𝑑(𝑡) and 𝑥̃_𝑠(𝑡) be positive solutions of (1.19) on [𝑡₀−𝑟,∞) such that (1.24) holds. Then every solution 𝑥=𝑥(𝑡) of (1.19) on [𝑡₀−𝑟,∞) can be represented by the formula 𝑥(𝑡)=𝐾^∗𝑥̃_𝑑(𝑡)+𝑂𝑥̃_𝑠(𝑡),(1.25) where the constant 𝐾^∗ depends on 𝑥.

The next definition is based on the properties of solutions 𝑥_𝑑, 𝑥̃_𝑑, 𝑥_𝑠, and 𝑥̃_𝑠 described in Theorems 1.9 and 1.10.

Definition 1.11 (see [20, Definition 2]). Suppose that the positive solutions 𝑥_𝑑 and 𝑥_𝑠 of (1.19) on [𝑡₀−𝑟,∞) satisfy the relation (1.22). Then, we call the solution 𝑥_𝑑 a dominant solution and the solution 𝑥_𝑠 a subdominant solution.

Due to linearity of (1.19), there are infinitely many dominant and subdominant solutions. Obviously, another pair of a dominant and a subdominant solutions is the pair 𝑥̃_𝑑(𝑡), 𝑥̃_𝑠(𝑡) in Theorem 1.10.

2. Main Results

Let us consider two auxiliary linear equations:𝑥𝑧̇̇((𝑡𝑡))==−−𝐶𝐶_𝐴𝐵(𝑡)𝑥(𝑡−𝑟),(2.1)(𝑡)𝑧(𝑡−𝑟),(2.2) where 𝑟∈ℝ⁺ and 𝐶_𝐴, 𝐶_𝐵 are positive continuous functions on [𝑡₀−𝑟,∞), 𝑡₀∈ℝ. According to the Theorems 1.7 and 1.9, both (2.1) and (2.2) have two types of positive solutions (subdominant and dominant). Let us denote them 𝑥_𝑑(𝑡), 𝑥_𝑠(𝑡) for (2.1) and 𝑧_𝑑(𝑡), 𝑧_𝑠(𝑡) for (2.2), respectively, such thatlim_𝑡→∞𝑥_𝑠(𝑡)𝑥_𝑑(𝑡)=0,lim^𝑡→∞𝑧_𝑠(𝑡)𝑧_𝑑(𝑡)=0.(2.3) Without loss of generality, we can suppose that 𝑥_𝑠(𝑡)<𝑥_𝑑(𝑡) and 𝑧_𝑠(𝑡)<𝑧_𝑑(𝑡) on [𝑡₀−𝑟,∞).

2.1. Auxiliary Linear Result

The next lemma states that if 𝑧_𝑑(𝑡), 𝑧_𝑠(𝑡) are dominant and subdominant solutions for (2.2), then there are dominant and subdominant solutions 𝑥^∗𝑑(𝑡), 𝑥^∗𝑠(𝑡) for (2.1) satisfying certain inequalities.

Lemma 2.1. Let (1.5) be valid. Let 𝑧_𝑑(𝑡), 𝑧_𝑠(𝑡) be dominant and subdominant solutions for (2.2). Then there are positive solutions 𝑥^∗𝑠(𝑡), 𝑥^∗𝑑(𝑡) of (2.1) on [𝑡₀−𝑟,∞) such that: (a)𝑥^∗𝑠(𝑡)<𝑧_𝑠(𝑡), 𝑡∈[𝑡₀−𝑟,∞), (b)𝑧_𝑑(𝑡)<𝑥^∗𝑑(𝑡), 𝑡∈[𝑡₀−𝑟,∞), (c)𝑥^∗𝑑(𝑡) and 𝑥^∗𝑠(𝑡) are dominant and subdominant solutions for (2.1).

Proof. (a) To prove the part (a), we employ Theorem 1.6 with 𝑝=𝑛=1; that is, we apply the case (i). Consider (2.1), set 𝐹(𝑡,𝜙)∶=−𝐶_𝐴(𝑡)𝜙(−𝑟), 𝜌^∗(𝑡)∶=0, 𝛿^∗(𝑡)∶=𝑧_𝑠(𝑡), and assume (see the case (i)): 0<𝜙(𝜃)<𝑧_𝑠(𝑡+𝜃),𝜃∈[−𝑟,0),𝜙(0)=𝑧_𝑠(𝑡),𝑡≥𝑡₀.(2.4) Now we have to verify the inequalities (1.16), that is, in our case: 𝐹(𝑡,𝜙)−𝛿^∗^(𝑡)==−−𝐶𝐶_𝐴𝐴(𝑡)𝜙(−𝑟)−𝛿^∗^(𝑡)=−𝐶_𝐴(𝑡)𝜙(−𝑟)−𝑧^𝑠(𝑡)≥≥(−w𝐶e_𝐵u(𝑡s)e𝜙((1−.5𝑟)))+𝐶_𝐵(𝑡)𝑧_𝑠(𝑡−𝑟)>𝐶_𝐵(𝑡)𝜙(−𝑟)+𝐶_𝐵(𝑡)𝑧_𝑠(𝑡−𝑟)(𝑡)𝑧_𝑠(𝑡−𝑟)−𝑧_𝑠(𝑡−𝑟)=0(2.5) and 𝐹(𝑡,𝜙)>(𝛿^∗)^(𝑡) if 𝑡∈[𝑡₀,∞). Further, we have −𝐹(𝑡,𝜙)+𝜌^∗′(𝑡)=𝐶_𝐴(𝑡)𝜙(−𝑟)+0=𝐶_𝐴(𝑡)𝜙(−𝑟)>0(2.6) and 𝐹(𝑡,𝜙)<(𝜌^∗)^(𝑡) if 𝑡∈[𝑡₀,∞). Since both inequalities are fulfilled and all assumptions of Theorem 1.6 are satisfied for the case in question, there exists a solution 𝑥^∗𝑠(𝑡) of (2.1) on [𝑡₀−𝑟,∞) such that 𝑥^∗𝑠(𝑡)<𝑧_𝑠(𝑡) for 𝑡∈[𝑡₀−𝑟,∞).
(b) To prove the part (b), we consider a solution 𝑥=𝑥^∗𝑑(𝑡) of the following initial problem: 𝑥̇(𝑡)=𝑥−(𝑡𝐶)_𝐴=(𝑧𝑡)_𝑑𝑥(𝑡−𝑟),𝑡∈𝑡₀−𝑟,∞,(2.7)(𝑡),𝑡∈𝑡₀−𝑟,𝑡₀.(2.8) Now, let us define a function 𝑊(𝑡,𝑥)=𝑧_𝑑(𝑡)−𝑥(𝑡),𝑡∈𝑡₀−𝑟,∞.(2.9) We find the sign of the full derivative of 𝑊 along the trajectories of (2.7) if 𝑡∈[𝑡₀,𝑡₀+𝑟]: d𝑊(𝑡,𝑥)d𝑡|||_{𝑡∈[𝑡0,𝑡0+𝑟]}===(−−d𝐶𝐶ue_𝐵𝐵(to𝑡)𝑧(2_𝑑.(8𝑡)−)𝑟)+𝐶_𝐴(𝑡)𝑥(𝑡−𝑟)≤𝐶(𝑡)𝑧_𝑑(𝑡−𝑟)+𝐶_𝐴(𝑡)𝑧_𝑑(𝑡−𝑟)_𝐴(𝑡)−𝐶_𝐵(𝑡)𝑧_𝑑(𝑡−𝑟)≤(dueto(1.5))≤0.(2.10) It means that function 𝑊 is nonincreasing and it holds 𝑊𝑡₀,𝑥𝑡₀==0𝑧_𝑑≥𝑡𝑊₀−𝑡₀𝑥𝑡₀=𝑧_𝑑𝑡₀−𝑧_𝑑𝑡₀+𝜀,𝑥𝑡₀+𝜀=𝑧_𝑑𝑡₀+𝜀−𝑥𝑡₀+𝜀,𝜀∈[0,𝑟],(2.11) and hence 𝑧_𝑑(𝑡₀+𝜀)≤𝑥(𝑡₀+𝜀). It will be showed that this inequality holds also for every 𝑡>𝑡₀+𝑟.
On the contrary, let us suppose that the inequality is not true, that is, there exists a point 𝑡=𝑡^∗∗ such that 𝑧_𝑑(𝑡^∗∗)>𝑥(𝑡^∗∗). Then there exists a point 𝑡^∗∈[𝑡₀,𝑡^∗∗) such that 𝑧_𝑑(𝑡^∗)<𝑥(𝑡^∗), otherwise 𝑧_𝑑(𝑡)≡𝑥(𝑡) on [𝑡₀,𝑡^∗∗]. Without loss of generality, we can suppose that 𝑥(𝑡)≡𝑧_𝑑(𝑡) on [𝑡₀,𝑡^∗∗∗] with a 𝑡^∗∗∗∈[𝑡₀,𝑡^∗) and 𝑥(𝑡)>𝑧_𝑑(𝑡) on (𝑡^∗∗∗,𝑡^∗). Then, there exists a point 𝑡^⋄∈(𝑡^∗∗∗,𝑡^∗) such that 𝑥(𝑡^⋄)=𝐾𝑧_𝑑(𝑡^⋄) for a constant 𝐾>1 and 𝐾𝑧_𝑑(𝑡)>𝑥(𝑡),for𝑡∈𝑡₀,𝑡^⋄.(2.12) Hence, for a function 𝑊^∗(𝑡,𝑥) defined as 𝑊^∗(𝑡,𝑥)∶=𝐾𝑧_𝑑(𝑡)−𝑥(𝑡), 𝑡∈[𝑡₀,𝑡^⋄], we get d𝑊^∗(𝑡,𝑥)d𝑡||||_𝑡=𝑡⋄<<=(𝐾𝐾due−−t𝐶𝐶o_𝐵𝐵((2𝑡).1𝑧2_𝑑))(𝑡−𝑟)+𝐶_𝐴(𝑡)𝑥(𝑡−𝑟)=𝐾𝑧_𝑑(𝑡)𝑧_𝑑(𝑡−𝑟)+𝐶_𝐴(𝑡)𝐾𝑧_𝑑(𝑡−𝑟)(𝑡−𝑟)𝐶_𝐴(𝑡)−𝐶_𝐵(𝑡)≤(by(1.5))≤0.(2.13) It means that 𝐾𝑧_𝑑(𝑡)<𝑥(𝑡) on a right-hand neighborhood of 𝑡^⋄. This is a contradiction with inequality 𝑧_𝑑(𝑡)<𝐾𝑧_𝑑(𝑡)<𝑥(𝑡),(2.14) hence it is proved that the existence of a solution 𝑥^∗𝑑(𝑡) satisfies 𝑧_𝑑(𝑡)<𝑥^∗𝑑(𝑡) on [𝑡₀−𝑟,∞).
(c) To prove the part (c), we consider lim_𝑡→∞𝑥^∗𝑠(𝑡)/𝑥^∗𝑑(𝑡). Due to (a) and (b), we get 0≤lim_𝑡→∞𝑥^∗𝑠(𝑡)𝑥^∗𝑑(𝑡)≤lim^𝑡→∞𝑧_𝑠(𝑡)𝑧_𝑑(𝑡)=0,(2.15) and 𝑥^∗𝑑(𝑡) and 𝑥^∗𝑠(𝑡) are (by Definition 1.11) dominant and subdominant solutions for (2.1).

2.2. Existence of Positive Solutions of (1.3)

The next theorems state that there exist two classes of positive solutions of (1.3) such that graphs of each solution of the first class are between graphs of dominant solutions of (2.1) and (2.2), and graphs of each solution of the second class are between graphs of subdominant solutions of (2.1) and (2.2), respectively. It means that we prove there are two classes of asymptotically different positive solutions of (1.3). Without loss of generality (see Remark 1.5), we put Ω∶=[𝑡₀,∞)×C^1𝑟. In the following, we will use our main supposition (1.4); that is, we assume that for (𝑡,𝜙)∈Ω inequalities,𝐶_𝐴(𝑡)𝜙(−𝑟)≤𝑓(𝑡,𝜙)≤𝐶_𝐵(𝑡)𝜙(−𝑟)(2.16) hold, where 𝜙 is supposed to be positive.

Theorem 2.2. Let 𝑓∶Ω→ℝ be a continuous quasibounded functional. Let inequality (1.5) be valid, and (2.16) holds for any (𝑡,𝜙)∈Ω with 𝜙(𝜃)>0, 𝜃∈[−𝑟,0]. Let 𝑥(𝑡) be a positive solution of (2.1) on [𝑡₀−𝑟,∞), and let 𝑧(𝑡) be a positive solution of (2.2) on [𝑡₀−𝑟,∞) such that 𝑥(𝑡)<𝑧(𝑡) on [𝑡₀−𝑟,∞). Then there exists an uncountable set 𝒴 of positive solutions of (1.3) on [𝑡₀−𝑟,∞) such that each solution 𝑦∈𝒴 satisfies 𝑥(𝑡)<𝑦(𝑡)<𝑧(𝑡)(2.17) for 𝑡∈[𝑡₀−𝑟,∞).

Proof. To prove this theorem, we employ Theorem 1.6 with 𝑝=𝑛=1; that is, we apply the case (i). Set 𝐹(𝑡,𝑦_𝑡)∶=−𝑓(𝑡,𝑦_𝑡), 𝜌^∗(𝑡)∶=𝑥(𝑡), 𝛿^∗(𝑡)∶=𝑧(𝑡); hence, the set 𝜔^∗ will be defined as 𝜔^∗∶=(𝑡,𝑦)∶𝑡∈𝑡₀−𝑟,∞,𝑥(𝑡)<𝑦(𝑡)<𝑧(𝑡).(2.18) Now, we have to verify the inequalities (1.16). In our case 𝐹(𝑡,𝜙)−𝛿^∗^(𝑡)==−−𝑓𝑓((𝑡𝑡,,𝜙𝜙))−−𝑧𝛿^∗_^(𝑡)=−𝑓(𝑡,𝜙)+𝐶_𝐵(𝑡)≥≥(−w𝐶e_𝐵use(2.16)()𝑡)𝑧(𝑡−𝑟)>>(𝐶w_𝐵eus(e𝑡)(𝜙2(.1−8𝑟))∶+𝜙𝐶(_𝐵−(𝑟𝑡))𝑧<(𝑡𝑧−(𝑡𝑟−)𝑟))−𝐹(𝑡,𝜙)+𝜌_∗(𝑡)[𝑧(𝑡−𝑟)−𝑧(𝑡−𝑟)]=0,^(𝑡)==𝑓𝑓((𝑡𝑡,,𝜙𝜙))++𝑥𝜌^∗_^(𝑡)=𝑓(𝑡,𝜙)−𝐶_𝐴(𝑡)≥≥𝐶(w_𝐴euse(2.1(6𝑡)))𝑥(𝑡−𝑟)>>(𝐶w_𝐴e(𝑡u)s𝜙e((−2𝑟.1)8−)𝐶∶_𝐴𝜙(𝑡()−𝑥𝑟)(𝑡>−𝑥𝑟)(𝑡−𝑟))(𝑡)[𝑥(𝑡−𝑟)−𝑥(𝑡−𝑟)]=0.(2.19) Therefore, −𝐹𝐹(𝑡(,𝑡,𝜙𝜙))−+𝛿𝜌^∗_∗^(𝑡)>0,^(𝑡)>0.(2.20) Both inequalities (1.16) are fulfilled, and all assumptions of Theorem 1.6 are satisfied for the case in question. There exists class of positive solutions 𝒴 of (1.3) on [𝑡₀−𝑟,∞) that for each solution 𝑦∈𝒴 from this class it is satisfied that 𝑥(𝑡)<𝑦(𝑡)<𝑧(𝑡) for 𝑡∈[𝑡₀−𝑟,∞).

Corollary 2.3. Let, in accordance with Lemma 2.1, 𝑥_𝑠(𝑡) be the subdominant solution of (2.1), and let 𝑧_𝑠(𝑡) be the subdominant solution of (2.2), that is, 𝑥_𝑠(𝑡)<𝑧_𝑠(𝑡) on [𝑡₀−𝑟,∞). Then, there exists an uncountable set 𝒴_𝑠 of positive solutions of (1.3) on [𝑡₀−𝑟,∞) such that each solution 𝑦_𝑠∈𝒴_𝑠 satisfies 𝑥_𝑠(𝑡)<𝑦_𝑠(𝑡)<𝑧_𝑠(𝑡).(2.21)

If inequality (1.6) holds, then dominant solutions 𝑥_𝑑(𝑡) of (2.1) and 𝑧_𝑑(𝑡) of (2.2) have finite positive limits 𝐶𝐶_𝑥∶=lim_𝑡→∞𝑥_𝑑(𝑡),𝐶_𝑥>0,_𝑧∶=lim_𝑡→∞𝑧_𝑑(𝑡),𝐶_𝑧>0.(2.22) This is a simple consequence of positivity of solutions 𝑥_𝑑(𝑡), 𝑧_𝑑(𝑡) and properties of dominant and subdominant solutions (see Theorem 1.7, Remark 1.8, Theorem 1.9, formulas (1.22)–(1.25) and (2.3)). Then, due to linearity of (2.1) and (2.2), it is clear that there are dominant solutions 𝑥_𝑑(𝑡), 𝑧_𝑑(𝑡) of both equations such that 𝑧_𝑑(𝑡)<𝑥_𝑑(𝑡) on [𝑡₀−𝑟,∞). In the following lemma, we without loss of generality suppose that 𝑥_𝑑(𝑡) and 𝑧_𝑑(𝑡) are such solutions and their initial functions are nonincreasing on initial interval [𝑡₀−𝑟,𝑡₀]. We will need constants 𝑀 and 𝐿 satisfying 𝑀>𝑀^∗∶=𝑥^𝑑𝑡₀−𝑟𝐶_𝑧𝐿>𝐿_∗,∶=𝑀𝑧^𝑑𝑡₀−𝑟𝐶_𝑥.(2.23)

Lemma 2.4. Let 𝑓∶Ω→ℝ be a continuous quasibounded functional. Let inequalities (1.5) and (1.6) be valid, and (2.16) holds for any (𝑡,𝜙)∈Ω with 𝜙(𝜃)>0, 𝜃∈[−𝑟,0]. Let 𝑥_𝑑(𝑡), 𝑡∈[𝑡₀−𝑟,∞) be a dominant solution of (2.1), nonincreasing on [𝑡₀−𝑟,𝑡₀], and let 𝑧_𝑑(𝑡), 𝑡∈[𝑡₀−𝑟,∞) be a dominant solution of (2.2), nonincreasing on [𝑡₀−𝑟,𝑡₀], such that 𝑧_𝑑(𝑡)<𝑥_𝑑(𝑡), 𝑡∈[𝑡₀−𝑟,∞). Then there exists another dominant solution 𝑧^∗𝑑(𝑡) of (2.2) and a positive solution 𝑦=𝑦_𝑑(𝑡) of (1.3) on [𝑡₀−𝑟,∞) such that it holds that 𝑥_𝑑(𝑡)<𝑦_𝑑(𝑡)<𝑧^∗𝑑(𝑡)(2.24) for 𝑡∈[𝑡₀−𝑟,∞) and 𝑧^∗𝑑(𝑡)=𝑀𝑧_𝑑(𝑡).

Proof. Both dominant solutions 𝑥_𝑑(𝑡) and 𝑧_𝑑(𝑡), of (2.1) and (2.2), respectively, have nonzero positive limits 𝐶_𝑥 and 𝐶_𝑧. From linearity of (2.1) and (2.2), it follows that solutions multiplied by an arbitrary constant are also solutions of (2.1) and (2.2), respectively. It holds that 𝑧^∗𝑑𝑡₀−𝑟=𝑀𝑧_𝑑𝑡₀−𝑟≥𝑀𝑧_𝑑(𝑡)=𝑧^∗𝑑(𝑡)>𝑀𝐶_𝑧>𝑥_𝑑𝑡₀−𝑟≥𝑥_𝑑(𝑡),(2.25) where 𝑡∈[𝑡₀−𝑟,∞).
Now, we define the set 𝜔^∗ in the same way as (2.18) in the proof of Theorem 2.2, but with 𝑥_𝑑(𝑡) instead of 𝑥(𝑡) and with 𝑧^∗𝑑(𝑡) instead of 𝑧(𝑡), that is, 𝜔^∗∶=(𝑡,𝑦)∶𝑡∈𝑡₀−𝑟,∞,𝑥_𝑑(𝑡)<𝑦(𝑡)<𝑧^∗𝑑(𝑡).(2.26) According to the Theorem 2.2 (with 𝑥_𝑑(𝑡) instead of 𝑥(𝑡) and with 𝑧^∗𝑑(𝑡) instead of 𝑧(𝑡)), it is visible that there exists a positive solution 𝑦=𝑦_𝑑(𝑡) of (1.3) satisfying 𝑥_𝑑(𝑡)<𝑦_𝑑(𝑡)<𝑧^∗𝑑(𝑡),(2.27) where 𝑡∈[𝑡₀,∞); that is, inequalities (2.24) hold.

Theorem 2.5. Let all suppositions of Lemma 2.4 be valid, and let 𝑦_𝑑(𝑡) be a solution of (1.3) satisfying inequalities (2.24). Then, there exists a positive solution 𝑥^𝑑∗∗(𝑡) of (2.1) on [𝑡₀−𝑟,∞) satisfying 𝑧_𝑑(𝑡)<𝑦_𝑑(𝑡)<𝑥^𝑑∗∗(𝑡),(2.28) where 𝑥^𝑑∗∗(𝑡)=𝐿𝑥_𝑑(𝑡) and 𝑡∈[𝑡₀−𝑟,∞).

Proof. Multiplying solution 𝑥_𝑑(𝑡) by the constant 𝐿, we have 𝐿𝑥_𝑑(𝑡)>𝐿𝐶_𝑥>𝑀𝑧_𝑑𝑡₀−𝑟.(2.29) Using (2.29) and (2.24), we get 𝑥^𝑑∗∗(𝑡)=𝐿𝑥_𝑑(𝑡)>𝑀𝑧_𝑑𝑡₀−𝑟=𝑧^∗𝑑𝑡₀−𝑟>𝑧^∗𝑑(𝑡)>𝑦_𝑑(𝑡)>𝑥_𝑑(𝑡)>𝑧_𝑑(𝑡),(2.30) where 𝑡∈[𝑡₀−𝑟,∞). Hence, there exists a solution 𝑦_𝑑(𝑡) of (1.3) such that inequalities (2.28) hold.

2.3. Asymptotically Different Behavior of Positive Solutions of (1.3)

Somewhat reformulating the statement of Theorem 2.5, we can define a class of positive solutions 𝒴_𝑑 of (1.3) such that every solution 𝑦_𝑑∈𝒴_𝑑 is defined on [𝑡₀−𝑟,∞) and satisfies𝐶𝑧_𝑑(𝑡)<𝑦_𝑑(𝑡)<𝐶𝑥^𝑑∗∗(𝑡),(2.31) where 𝑡∈[𝑡₀−𝑟,∞) for a positive constant 𝐶 and, for every positive constant 𝐶, there exists a solution 𝑦_𝑑∈𝒴_𝑑 satisfying (2.31) on [𝑡₀−𝑟,∞).

The following theorem states that positive solutions 𝑦_𝑠(𝑡) and 𝑦_𝑑(𝑡) of (1.3) have a different order of vanishing.

Theorem 2.6. Let all the assumptions of Corollary 2.3 and Theorem 2.5 be met. Then there exist two classes 𝒴_𝑠 and 𝒴_𝑑 of positive solutions of (1.3) described by inequalities (2.21) and (2.31). Every two solutions 𝑦_𝑠, 𝑦_𝑑, such that 𝑦_𝑠∈𝒴_𝑠 and 𝑦_𝑑∈𝒴_𝑑, have asymptotically different behavior, that is, lim_𝑡→+∞𝑦_𝑠(𝑡)𝑦_𝑑(𝑡)=0.(2.32)

Proof. Let the solution 𝑦_𝑠(𝑡) be the one specified in Corollary 2.3 and the solution 𝑦_𝑑(𝑡) specified by (2.31) with a positive constant 𝐶. Now let us verify that (2.32) holds. With the aid of inequalities (2.21) and (2.31), we get 0≤lim_𝑡→+∞𝑦_𝑠(𝑡)𝑦_𝑑(𝑡)≤lim^𝑡→+∞𝑧_𝑠(𝑡)𝐶𝑧_𝑑(𝑡)=0(2.33) in accordance with (1.22), since 𝑧_𝑠(𝑡) and 𝑧_𝑑(𝑡) are positive (subdominant and dominant) solutions of linear equation (2.2).

Another final statement, being a consequence of Lemma 2.1 and Theorems 2.2 and 2.5, is the following.

Theorem 2.7. Let 𝑓∶Ω→ℝ be a continuous quasibounded functional. Let inequalities (1.5) and (1.6) be valid, and (2.16) holds for any (𝑡,𝜙)∈Ω with 𝜙(𝜃)>0, 𝜃∈[−𝑟,0]. Then on [𝑡₀−𝑟,∞) there exist (a)dominant and subdominant solutions 𝑥_𝑑(𝑡), 𝑥_𝑠(𝑡) of (2.1), (b)dominant and subdominant solutions 𝑧_𝑑(𝑡), 𝑧_𝑠(𝑡) of (2.2), (c)solutions 𝑦_𝑑(𝑡), 𝑦_𝑠(𝑡) of (1.3) such that 0<𝑥li_𝑠m(_𝑡→∞𝑡)<𝑦_𝑠(𝑡)<𝑧_𝑠(𝑡)<𝑧_𝑑(𝑡)<𝑦_𝑑(𝑡)<𝑥_𝑑(𝑡),(2.34)𝑥_𝑠(𝑡)𝑥_𝑑(𝑡)=lim^𝑡→∞𝑧_𝑠(𝑡)𝑧_𝑑(𝑡)=lim^𝑡→∞𝑦_𝑠(𝑡)𝑦_𝑑(𝑡)=0.(2.35)

Example 2.8. Let (1.3) be reduced to 𝑦̇(𝑡)=−𝑓𝑡,𝑦_𝑡∶=−3𝑡exp−3𝑡+12cos(𝑡𝑦(𝑡−1))⋅𝑦(𝑡−1),(2.36) and let auxiliary linear equations (2.1) and (2.2) be reduced to 𝑥𝑧̇̇((𝑡𝑡))==−−24𝑡𝑡eexxpp((21−−42𝑡𝑡))⋅⋅𝑧𝑥((𝑡𝑡−−11)),,((22..3378)) that is, 𝐶_𝐴(𝑡)∶=4𝑡exp(2−4𝑡),𝐶_𝐵(𝑡)∶=2𝑡exp(1−2𝑡),𝑟=1.(2.39) Let 𝑡₀ be sufficiently large. Inequalities (1.5), (1.6), and (2.16) hold. In view of linearity and by Remark 1.8, we conclude that there exist dominant solutions 𝑥_𝑑(𝑡) of (2.37) and 𝑧_𝑑(𝑡) of (2.38) such that lim_𝑡→∞𝑥_𝑑(𝑡)=11,lim_𝑡→∞𝑧_𝑑(𝑡)=2,𝑧_𝑑(𝑡)<𝑥_𝑑(𝑡),𝑡∈𝑡₀−1,∞.(2.40) Moreover, there exist subdominant solutions 𝑥_𝑠(𝑡) of (2.37) and 𝑧_𝑠(𝑡) of (2.38) such that 𝑥_𝑠(𝑡)<𝑧_𝑠(𝑡), 𝑡∈[𝑡₀−1,∞) which are defined as 𝑥_𝑠(𝑡)∶=exp−2𝑡²,𝑧_𝑠(𝑡)∶=exp−𝑡².(2.41) By Theorem 2.7, we conclude that there exist solutions 𝑦_𝑠(𝑡) and 𝑦_𝑑(𝑡) of (2.36) satisfying inequalities (2.34), and (without loss of generality) inequalities 0<𝑥_𝑠(𝑡)=exp−2𝑡²<𝑦_𝑠(𝑡)<𝑧_𝑠(𝑡)=exp−𝑡²<1≤𝑧_𝑑(𝑡)<𝑦_𝑑(𝑡)<10≤𝑥_𝑑(𝑡)(2.42) hold on [𝑡₀−1,∞).

3. Conclusions and Open Problems

The following problems were not answered in the paper and present interesting topics for investigation.

Open Problem 3.1. In Lemma 2.4 and Theorems 2.5–2.7 we used the convergence assumption (1.6) being, without loss of generality, equivalent to ^∞𝐶_𝐵(𝑡)d𝑡<∞.(3.1) It is an open question whether similar results could be proved if the integral is divergent, that is, if ^∞𝐶_𝐵(𝑡)d𝑡=∞.(3.2)

Open Problem 3.2. Dominant and subdominant solutions are used for representation of family of all solutions of scalar linear differential delayed equation, for example, by formula (1.25). Investigation in this line of the role of solutions 𝑦_𝑑(𝑡) and 𝑦_𝑠(𝑡) of (1.3) (see Theorems 2.6 and 2.7) is an important question. Namely, it seems to be an interesting question to establish sufficient conditions for the right-hand side of (1.3) such that its every solution 𝑦=𝑦(𝑡) can be represented on [𝑡₀−𝑟,∞) by the formula 𝑦(𝑡)=𝐾𝑦_𝑑(𝑡)+𝑂𝑦_𝑠(𝑡),(3.3) where the constant 𝐾 depends only on 𝑦(𝑡).

Open Problem 3.3. The notions dominant and subdominant solutions are in the cited papers defined for scalar differential delayed equations only. It is a rather interesting question if the results presented can be enlarged to systems of differential delayed equations.

Remark 3.4. Except for papers and books mentioned in this paper we refer, for example, to sources [21–23], treating related problems as well. Note that the topic is connected with similar questions for discrete equations (e.g., [24–27]).

Acknowledgments

This research was supported by the Grant no. 1/0090/09 of the Grant Agency of Slovak Republic (VEGA) and by the Project APVV-0700-07 of Slovak Research and Development Agency.