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

Authors: J. Diblík; M. Kúdelčíková
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Solutions of the equationy˙(t)=  −f(t,yt) 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.


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## Body

## 1. Introduction

LetC([a,b],ℝn), where a,b∈ℝ, a<b, be the Banach space of the continuous mappings from the interval [a,b] into ℝn equipped with the supremum norm (1.1)‖ψ‖C=sup⁡θ∈[a,b]‖ψ(θ)‖,ψ∈C([a,b],Rn),
where ∥·∥ is the maximum norm in ℝn. In the case of a=-r<0 and b=0, we will denote this space as Crn, that is, (1.2)Crn:=C([-r,0],Rn).Ifσ∈ℝn, A≥0, and y∈C([σ-r,σ+A],ℝn), then, for each t∈[σ,σ+A], we define yt∈Crn by yt(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 equation(1.3)ẏ(t)=-f(t,yt),
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×Cr1 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:(1.4)CA(t)yt(-r)≤f(t,yt)≤CB(t)yt(-r),
where (t,yt)∈Ω, and CA,CB:[t0-r,∞)→ℝ+:=(0,∞), t0∈ℝ are continuous functions satisfying(1.5)0<CA(t)≤CB(t)≤1(re),t∈[t0-r,∞),(1.6)∫t0-r∞CB(t)dt<1.
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 equations(1.7)ẏ(t)=F(t,yt),
where F:Ω*↦ℝn 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×Crn. We recall that the functional F is quasibounded if F is bounded on every set of the form [t1,t2]×CrLn⊂Ω*, where t1<t2, CrLn:=C([-r,0],L) and L is a closed bounded subset of ℝn (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),ℝn), (t,yt)∈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 yσ(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 Iσ,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 if(1.8)yi(σ,φ)>0
on [σ-r,σ]∪Iσ,φ for each i=1,2,…,n. A nontrivial solution y(σ,φ) of the system (1.7) is said to be oscillatory on Iσ,φ (under condition Iσ,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 setω⊂ℝ×ℝn, then intω and ∂ω denote the interior and the boundary of ω, respectively.Definition 1.1 (see [16]).
Let the continuously differentiable functionsli(t,y),i=1,2,…,p and mj(t,y), j=1,2,…,q, p2+q2>0 be defined on some open set ω0⊂ℝ×ℝn. The set
(1.9)ω*={(t,y)∈ω0:li(t,y)<0,mj(t,y)<0,i=1,…,p,j=1,…,q}
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×Crn such that (t+θ,π(θ))∈ω* for θ∈[-r,0), we have (t,π)∈Ω*.(β)
For alli=1,2,…,p, all (t,y)∈∂ω* for which li(t,y)=0, and all π∈Crn for which π(0)=y and (t+θ,π(θ))∈ω*, θ∈[-r,0). It follows that Dli(t,y)>0, where
(1.10)Dli(t,y)≡∑k=1n∂li(t,y)∂ykfk(t,π)+∂li(t,y)∂t.(γ)
For allj=1,2,…,q, all (t,y)∈∂ω* for which mj(t,y)=0, and all π∈Crn for which π(0)=y and (t+θ,π(θ))∈ω*,θ∈[-r,0). It follows that Dmj(t,y)<0, where
(1.11)Dmj(t,y)≡∑k=1n∂mj(t,y)∂ykfk(t,π)+∂mj(t,y)∂t.
The elements(t,π)∈ℝ×Crn in the sequel are assumed to be such that (t,π)∈Ω*.In the following definition, a setω* is an arbitrary set without any connection with a regular polyfacial set ω* defined by (1.9) in Definition 1.1.Definition 1.2.
Asystem of initial functionspA,ω* with respect to the nonempty sets A and ω*, where A⊂ω*¯⊂ℝ×ℝn is defined as a continuous mapping p:A→Crn such that (α) and (β) below hold.  (α)
Ifz=(t,y)∈A∩intω*, then (t+θ,p(z)(θ))∈ω* for θ∈[-r,0].(β)
Ifz=(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]).
If𝒜⊂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.
Letω*⊂ω0 be a regular polyfacial set with respect to the system (1.7), and let W be defined as follows:
(1.12)W={(t,y)∈∂ω*:mj(t,y)<0,j=1,2,…,q}.
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 pZ,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
(1.13)(t,y(σ0,p(z0))(t))∈ω*
for each t∈Dσ0,p(z0).Remark 1.5.
When Lemma1.4 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(1.14)ρ*=(ρ1*,ρ2*,…,ρn*),δ*=(δ1*,δ2*,…,δn*):[t0-r,∞)⟶Rn,
with ρ*(t)≪δ*(t) for t∈[t0-r,∞) (the symbol ≪ here and below means that ρi*(t)<δi*(t) for all i=1,2,…,n), continuously differentiable on [t0,∞), we define the set (1.15)ω*:={(t,y):t∈[t0,∞),ρ*(t)≪y≪δ*(t)}.
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 ap∈{0,…,n} such that (i)
ift≥t0, ϕ∈Crn and (t+θ,ϕ(θ))∈ω* for any θ∈[-r,0), then
(1.16)(δ*i)′(t)<Fi(t,ϕ),whenϕi(0)=δ*i(t),(ρ*i)′(t)>Fi(t,ϕ),whenϕi(0)=ρ*i(t)
for any i=1,2,…,p, (If p=0, this condition is omitted.)(ii)
ift≥t0, ϕ∈Crn and (t+θ,ϕ(θ))∈ω* for any θ∈[-r,0) then
(1.17)(ρ*i)′(t)<Fi(t,ϕ),whenϕi(0)=ρ*i(t),(δ*i)′(t)>Fi(t,ϕ),whenϕi(0)=δ*i(t)
for any i=p+1,p+2,…,n. (If p=n, this condition is omitted.)
Then, there exists an uncountable set𝒴 of solutions of (1.7) on [t0-r,∞) such that each y∈𝒴 satisfies
(1.18)ρ*(t)≪y(t)≪δ*(t),t∈[t0-r,∞).


### 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 form(1.19)ẋ(t)=-c(t)x(t-τ(t))
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
(1.20)∫∞c(t)dt<∞.Remark 1.8.
Tracing the proof of Theorem1.7, we conclude that a positive solution x=x(t) of (1.19) with nonzero limit exists on [t0-r,∞) if
(1.21)∫t0-r∞c(t)dt<1.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 xd and xs of (1.19) on [t0-r,∞) satisfying the relation
(1.22)lim⁡t→∞xs(t)xd(t)=0
such that every solution x=x(t) of (1.19) on [t0-r,∞) can be represented by the formula
(1.23)x(t)=Kxd(t)+O(xs(t)),
where the constant K depends on x.The symbolO, 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 xd(t) and xs(t) in (1.23) by another pairs of solutions x̃d(t) and x̃s(t) if(1.24)lim⁡t→∞x̃s(t)x̃d(t)=0
as given in the following theorem.Theorem 1.10.
Letx̃d(t) and x̃s(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
(1.25)x(t)=K*x̃d(t)+O(x̃s(t)),
where the constant K* depends on x.The next definition is based on the properties of solutionsxd, x̃d, xs, and x̃s described in Theorems 1.9 and 1.10.Definition 1.11 (see [20, Definition 2]).
Suppose that the positive solutionsxd and xs of (1.19) on [t0-r,∞) satisfy the relation (1.22). Then, we call the solution xd a dominant solution and the solution xs 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 x̃d(t), x̃s(t) in Theorem 1.10.


## 1.1. Ważewski’s Principle

In this section we introduce Ważewski’s principle for a system of retarded functional differential equations(1.7)ẏ(t)=F(t,yt),
where F:Ω*↦ℝn 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×Crn. We recall that the functional F is quasibounded if F is bounded on every set of the form [t1,t2]×CrLn⊂Ω*, where t1<t2, CrLn:=C([-r,0],L) and L is a closed bounded subset of ℝn (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),ℝn), (t,yt)∈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 yσ(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 Iσ,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 if(1.8)yi(σ,φ)>0
on [σ-r,σ]∪Iσ,φ for each i=1,2,…,n. A nontrivial solution y(σ,φ) of the system (1.7) is said to be oscillatory on Iσ,φ (under condition Iσ,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 setω⊂ℝ×ℝn, then intω and ∂ω denote the interior and the boundary of ω, respectively.Definition 1.1 (see [16]).
Let the continuously differentiable functionsli(t,y),i=1,2,…,p and mj(t,y), j=1,2,…,q, p2+q2>0 be defined on some open set ω0⊂ℝ×ℝn. The set
(1.9)ω*={(t,y)∈ω0:li(t,y)<0,mj(t,y)<0,i=1,…,p,j=1,…,q}
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×Crn such that (t+θ,π(θ))∈ω* for θ∈[-r,0), we have (t,π)∈Ω*.(β)
For alli=1,2,…,p, all (t,y)∈∂ω* for which li(t,y)=0, and all π∈Crn for which π(0)=y and (t+θ,π(θ))∈ω*, θ∈[-r,0). It follows that Dli(t,y)>0, where
(1.10)Dli(t,y)≡∑k=1n∂li(t,y)∂ykfk(t,π)+∂li(t,y)∂t.(γ)
For allj=1,2,…,q, all (t,y)∈∂ω* for which mj(t,y)=0, and all π∈Crn for which π(0)=y and (t+θ,π(θ))∈ω*,θ∈[-r,0). It follows that Dmj(t,y)<0, where
(1.11)Dmj(t,y)≡∑k=1n∂mj(t,y)∂ykfk(t,π)+∂mj(t,y)∂t.
The elements(t,π)∈ℝ×Crn in the sequel are assumed to be such that (t,π)∈Ω*.In the following definition, a setω* is an arbitrary set without any connection with a regular polyfacial set ω* defined by (1.9) in Definition 1.1.Definition 1.2.
Asystem of initial functionspA,ω* with respect to the nonempty sets A and ω*, where A⊂ω*¯⊂ℝ×ℝn is defined as a continuous mapping p:A→Crn such that (α) and (β) below hold.  (α)
Ifz=(t,y)∈A∩intω*, then (t+θ,p(z)(θ))∈ω* for θ∈[-r,0].(β)
Ifz=(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]).
If𝒜⊂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.
Letω*⊂ω0 be a regular polyfacial set with respect to the system (1.7), and let W be defined as follows:
(1.12)W={(t,y)∈∂ω*:mj(t,y)<0,j=1,2,…,q}.
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 pZ,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
(1.13)(t,y(σ0,p(z0))(t))∈ω*
for each t∈Dσ0,p(z0).Remark 1.5.
When Lemma1.4 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(1.14)ρ*=(ρ1*,ρ2*,…,ρn*),δ*=(δ1*,δ2*,…,δn*):[t0-r,∞)⟶Rn,
with ρ*(t)≪δ*(t) for t∈[t0-r,∞) (the symbol ≪ here and below means that ρi*(t)<δi*(t) for all i=1,2,…,n), continuously differentiable on [t0,∞), we define the set (1.15)ω*:={(t,y):t∈[t0,∞),ρ*(t)≪y≪δ*(t)}.
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 ap∈{0,…,n} such that (i)
ift≥t0, ϕ∈Crn and (t+θ,ϕ(θ))∈ω* for any θ∈[-r,0), then
(1.16)(δ*i)′(t)<Fi(t,ϕ),whenϕi(0)=δ*i(t),(ρ*i)′(t)>Fi(t,ϕ),whenϕi(0)=ρ*i(t)
for any i=1,2,…,p, (If p=0, this condition is omitted.)(ii)
ift≥t0, ϕ∈Crn and (t+θ,ϕ(θ))∈ω* for any θ∈[-r,0) then
(1.17)(ρ*i)′(t)<Fi(t,ϕ),whenϕi(0)=ρ*i(t),(δ*i)′(t)>Fi(t,ϕ),whenϕi(0)=δ*i(t)
for any i=p+1,p+2,…,n. (If p=n, this condition is omitted.)
Then, there exists an uncountable set𝒴 of solutions of (1.7) on [t0-r,∞) such that each y∈𝒴 satisfies
(1.18)ρ*(t)≪y(t)≪δ*(t),t∈[t0-r,∞).


## 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 form(1.19)ẋ(t)=-c(t)x(t-τ(t))
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
(1.20)∫∞c(t)dt<∞.Remark 1.8.
Tracing the proof of Theorem1.7, we conclude that a positive solution x=x(t) of (1.19) with nonzero limit exists on [t0-r,∞) if
(1.21)∫t0-r∞c(t)dt<1.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 xd and xs of (1.19) on [t0-r,∞) satisfying the relation
(1.22)lim⁡t→∞xs(t)xd(t)=0
such that every solution x=x(t) of (1.19) on [t0-r,∞) can be represented by the formula
(1.23)x(t)=Kxd(t)+O(xs(t)),
where the constant K depends on x.The symbolO, 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 xd(t) and xs(t) in (1.23) by another pairs of solutions x̃d(t) and x̃s(t) if(1.24)lim⁡t→∞x̃s(t)x̃d(t)=0
as given in the following theorem.Theorem 1.10.
Letx̃d(t) and x̃s(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
(1.25)x(t)=K*x̃d(t)+O(x̃s(t)),
where the constant K* depends on x.The next definition is based on the properties of solutionsxd, x̃d, xs, and x̃s described in Theorems 1.9 and 1.10.Definition 1.11 (see [20, Definition 2]).
Suppose that the positive solutionsxd and xs of (1.19) on [t0-r,∞) satisfy the relation (1.22). Then, we call the solution xd a dominant solution and the solution xs 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 x̃d(t), x̃s(t) in Theorem 1.10.


## 2. Main Results

Let us consider two auxiliary linear equations:(2.1)ẋ(t)=-CA(t)x(t-r),(2.2)ż(t)=-CB(t)z(t-r),
where r∈ℝ+ and CA, CB 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 xd(t), xs(t) for (2.1) and zd(t), zs(t) for (2.2), respectively, such that(2.3)lim⁡t→∞xs(t)xd(t)=0,lim⁡t→∞zs(t)zd(t)=0.
Without loss of generality, we can suppose that xs(t)<xd(t) and zs(t)<zd(t) on [t0-r,∞).


### 2.1. Auxiliary Linear Result

The next lemma states that ifzd(t), zs(t) are dominant and subdominant solutions for (2.2), then there are dominant and subdominant solutions xd*(t), xs*(t) for (2.1) satisfying certain inequalities.Lemma 2.1.
Let (1.5) be valid. Let zd(t), zs(t) be dominant and subdominant solutions for (2.2). Then there are positive solutions xs*(t), xd*(t) of (2.1) on [t0-r,∞) such that: (a)
xs*(t)<zs(t), t∈[t0-r,∞),(b)
zd(t)<xd*(t),  t∈[t0-r,∞),(c)
xd*(t) and xs*(t) are dominant and subdominant solutions for (2.1).Proof.
(a) To prove the part (a), we employ Theorem1.6 with p=n=1; that is, we apply the case (i). Consider (2.1), set F(t,ϕ):=-CA(t)ϕ(-r), ρ*(t):=0, δ*(t):=zs(t), and assume (see the case (i)):
(2.4)0<ϕ(θ)<zs(t+θ),θ∈[-r,0),ϕ(0)=zs(t),t≥t0.
Now we have to verify the inequalities (1.16), that is, in our case:
(2.5)F(t,ϕ)-(δ*)′(t)=-CA(t)ϕ(-r)-(δ*)′(t)=-CA(t)ϕ(-r)-zs′(t)=-CA(t)ϕ(-r)+CB(t)zs(t-r)≥(weuse(1.5))≥-CB(t)ϕ(-r)+CB(t)zs(t-r)>CB(t)[zs(t-r)-zs(t-r)]=0
and F(t,ϕ)>(δ*)′(t) if t∈[t0,∞). Further, we have
(2.6)-F(t,ϕ)+(ρ*)′(t)=CA(t)ϕ(-r)+0=CA(t)ϕ(-r)>0
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 xs*(t) of (2.1) on [t0-r,∞) such that xs*(t)<zs(t) for t∈[t0-r,∞).
(b) To prove the part (b), we consider a solutionx=xd*(t) of the following initial problem:
(2.7)ẋ(t)=-CA(t)x(t-r),t∈[t0-r,∞),(2.8)x(t)=zd(t),t∈[t0-r,t0].
Now, let us define a function
(2.9)W(t,x)=zd(t)-x(t),t∈[t0-r,∞).
We find the sign of the full derivative of W along the trajectories of (2.7) if t∈[t0,t0+r]:
(2.10)dW(t,x)dt|t∈[t0,t0+r]=-CB(t)zd(t-r)+CA(t)x(t-r)=(dueto(2.8))=-CB(t)zd(t-r)+CA(t)zd(t-r)≤[CA(t)-CB(t)]zd(t-r)≤(dueto(1.5))≤0.
It means that function W is nonincreasing and it holds
(2.11)W(t0,x(t0))=zd(t0)-x(t0)=zd(t0)-zd(t0)=0≥W(t0+ɛ,x(t0+ɛ))=zd(t0+ɛ)-x(t0+ɛ),ɛ∈[0,r],
and hence zd(t0+ɛ)≤x(t0+ɛ). 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 pointt=t** such that zd(t**)>x(t**). Then there exists a point t*∈[t0,t**) such that zd(t*)<x(t*), otherwise zd(t)≡x(t) on [t0,t**]. Without loss of generality, we can suppose that x(t)≡zd(t) on [t0,t***] with a t***∈[t0,t*) and x(t)>zd(t) on (t***,t*). Then, there exists a point t⋄∈(t***,t*) such that x(t⋄)=Kzd(t⋄) for a constant K>1 and
(2.12)Kzd(t)>x(t),fort∈[t0,t⋄).
Hence, for a function W*(t,x) defined as W*(t,x):=Kzd(t)-x(t), t∈[t0,t⋄], we get
(2.13)dW*(t,x)dt|t=t⋄=K(-CB(t)zd(t-r))+CA(t)x(t-r)<(dueto(2.12))<K(-CB(t)zd(t-r))+CA(t)Kzd(t-r)=Kzd(t-r)[CA(t)-CB(t)]≤(by(1.5))≤0.
It means that Kzd(t)<x(t) on a right-hand neighborhood of t⋄. This is a contradiction with inequality
(2.14)zd(t)<Kzd(t)<x(t),
hence it is proved that the existence of a solution xd*(t) satisfies zd(t)<xd*(t) on [t0-r,∞).
(c) To prove the part (c), we considerlimt→∞xs*(t)/xd*(t). Due to (a) and (b), we get
(2.15)0≤lim⁡t→∞xs*(t)xd*(t)≤lim⁡t→∞zs(t)zd(t)=0,
and xd*(t) and xs*(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,∞)×Cr1. In the following, we will use our main supposition (1.4); that is, we assume that for (t,ϕ)∈Ω inequalities,(2.16)CA(t)ϕ(-r)≤f(t,ϕ)≤CB(t)ϕ(-r)
hold, where ϕ is supposed to be positive.Theorem 2.2.
Letf: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
(2.17)x(t)<y(t)<z(t)
for t∈[t0-r,∞).Proof.
To prove this theorem, we employ Theorem1.6 with p=n=1; that is, we apply the case (i). Set F(t,yt):=-f(t,yt), ρ*(t):=x(t), δ*(t):=z(t); hence, the set ω* will be defined as
(2.18)ω*:={(t,y):t∈[t0-r,∞),x(t)<y(t)<z(t)}.
Now, we have to verify the inequalities (1.16). In our case
(2.19)F(t,ϕ)-(δ*)′(t)=-f(t,ϕ)-(δ*)′(t)=-f(t,ϕ)-z′(t)=-f(t,ϕ)+CB(t)z(t-r)≥(weuse(2.16))≥-CB(t)ϕ(-r)+CB(t)z(t-r)>(weuse(2.18):ϕ(-r)<z(t-r))>CB(t)[z(t-r)-z(t-r)]=0,-F(t,ϕ)+(ρ*)′(t)=f(t,ϕ)+(ρ*)′(t)=f(t,ϕ)+x′(t)=f(t,ϕ)-CA(t)x(t-r)≥(weuse(2.16))≥CA(t)ϕ(-r)-CA(t)x(t-r)>(weuse(2.18):ϕ(-r)>x(t-r))>CA(t)[x(t-r)-x(t-r)]=0.
Therefore,
(2.20)F(t,ϕ)-(δ*)′(t)>0,-F(t,ϕ)+(ρ*)′(t)>0.
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 Lemma2.1, xs(t) be the subdominant solution of (2.1), and let zs(t) be the subdominant solution of (2.2), that is, xs(t)<zs(t) on [t0-r,∞). Then, there exists an uncountable set 𝒴s of positive solutions of (1.3) on [t0-r,∞) such that each solution ys∈𝒴s satisfies
(2.21)xs(t)<ys(t)<zs(t).If inequality (1.6) holds, then dominant solutions xd(t) of (2.1) and zd(t) of (2.2) have finite positive limits (2.22)Cx:=lim⁡t→∞xd(t),Cx>0,Cz:=lim⁡t→∞zd(t),Cz>0.
This is a simple consequence of positivity of solutions xd(t), zd(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 xd(t), zd(t) of both equations such that zd(t)<xd(t) on [t0-r,∞). In the following lemma, we without loss of generality suppose that xd(t) and zd(t) are such solutions and their initial functions are nonincreasing on initial interval [t0-r,t0]. We will need constants M and L satisfying (2.23)M>M*:=xd(t0-r)Cz,L>L*:=Mzd(t0-r)Cx.Lemma 2.4.
Letf: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 xd(t), t∈[t0-r,∞) be a dominant solution of (2.1), nonincreasing on [t0-r,t0], and let zd(t), t∈[t0-r,∞) be a dominant solution of (2.2), nonincreasing on [t0-r,t0], such that zd(t)<xd(t), t∈[t0-r,∞). Then there exists another dominant solution zd*(t) of (2.2) and a positive solution y=yd(t) of (1.3) on [t0-r,∞) such that it holds that
(2.24)xd(t)<yd(t)<zd*(t)
for t∈[t0-r,∞) and zd*(t)=Mzd(t).Proof.
Both dominant solutionsxd(t) and zd(t), of (2.1) and (2.2), respectively, have nonzero positive limits Cx and Cz. 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
(2.25)zd*(t0-r)=Mzd(t0-r)≥Mzd(t)=zd*(t)>MCz>xd(t0-r)≥xd(t),
where t∈[t0-r,∞).
Now, we define the setω* in the same way as (2.18) in the proof of Theorem 2.2, but with xd(t) instead of x(t) and with zd*(t) instead of z(t), that is,
(2.26)ω*:={(t,y):t∈[t0-r,∞),xd(t)<y(t)<zd*(t)}.
According to the Theorem 2.2 (with xd(t) instead of x(t) and with zd*(t) instead of z(t)), it is visible that there exists a positive solution y=yd(t) of (1.3) satisfying
(2.27)xd(t)<yd(t)<zd*(t),
where t∈[t0,∞); that is, inequalities (2.24) hold.Theorem 2.5.
Let all suppositions of Lemma2.4 be valid, and let yd(t) be a solution of (1.3) satisfying inequalities (2.24). Then, there exists a positive solution xd**(t) of (2.1) on [t0-r,∞) satisfying
(2.28)zd(t)<yd(t)<xd**(t),
where xd**(t)=Lxd(t) and t∈[t0-r,∞).Proof.
Multiplying solutionxd(t) by the constant L, we have
(2.29)Lxd(t)>LCx>Mzd(t0-r).
Using (2.29) and (2.24), we get
(2.30)xd**(t)=Lxd(t)>Mzd(t0-r)=zd*(t0-r)>zd*(t)>yd(t)>xd(t)>zd(t),
where t∈[t0-r,∞). Hence, there exists a solution yd(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 Theorem2.5, we can define a class of positive solutions 𝒴d of (1.3) such that every solution yd∈𝒴d is defined on [t0-r,∞) and satisfies(2.31)Czd(t)<yd(t)<Cxd**(t),
where t∈[t0-r,∞) for a positive constant C and, for every positive constant C, there exists a solution yd∈𝒴d satisfying (2.31) on [t0-r,∞).The following theorem states that positive solutionsys(t) and yd(t) of (1.3) have a different order of vanishing.Theorem 2.6.
Let all the assumptions of Corollary2.3 and Theorem 2.5 be met. Then there exist two classes 𝒴s and 𝒴d of positive solutions of (1.3) described by inequalities (2.21) and (2.31). Every two solutions ys, yd, such that ys∈𝒴s and yd∈𝒴d, have asymptotically different behavior, that is,
(2.32)lim⁡t→+∞ys(t)yd(t)=0.Proof.
Let the solutionys(t) be the one specified in Corollary 2.3 and the solution yd(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
(2.33)0≤lim⁡t→+∞ys(t)yd(t)≤lim⁡t→+∞zs(t)Czd(t)=0
in accordance with (1.22), since zs(t) and zd(t) are positive (subdominant and dominant) solutions of linear equation (2.2).Another final statement, being a consequence of Lemma2.1 and Theorems 2.2 and 2.5, is the following.Theorem 2.7.
Letf: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 solutionsxd(t), xs(t) of (2.1),(b)
dominant and subdominant solutionszd(t), zs(t) of (2.2),(c)
solutionsyd(t), ys(t) of (1.3)such that
(2.34)0<xs(t)<ys(t)<zs(t)<zd(t)<yd(t)<xd(t),(2.35)lim⁡t→∞xs(t)xd(t)=lim⁡t→∞zs(t)zd(t)=lim⁡t→∞ys(t)yd(t)=0.Example 2.8.
Let (1.3) be reduced to
(2.36)ẏ(t)=-f(t,yt):=-3texp⁡⁡(-3t+12cos⁡(ty(t-1)))⋅y(t-1),
and let auxiliary linear equations (2.1) and (2.2) be reduced to
(2.37)ẋ(t)=-4texp⁡⁡(2-4t)⋅x(t-1),(2.38)ż(t)=-2texp⁡⁡(1-2t)⋅z(t-1),
that is,
(2.39)CA(t):=4texp⁡(2-4t),CB(t):=2texp⁡(1-2t),r=1.
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 xd(t) of (2.37) and zd(t) of (2.38) such that
(2.40)lim⁡t→∞xd(t)=11,lim⁡t→∞zd(t)=2,zd(t)<xd(t),t∈[t0-1,∞).
Moreover, there exist subdominant solutions xs(t) of (2.37) and zs(t) of (2.38) such that xs(t)<zs(t), t∈[t0-1,∞) which are defined as
(2.41)xs(t):=exp⁡(-2t2),zs(t):=exp⁡(-t2).
By Theorem 2.7, we conclude that there exist solutions ys(t) and yd(t) of (2.36) satisfying inequalities (2.34), and (without loss of generality) inequalities
(2.42)0<xs(t)=exp⁡(-2t2)<ys(t)<zs(t)=exp⁡(-t2)<1≤zd(t)<yd(t)<10≤xd(t)
hold on [t0-1,∞).


## 2.1. Auxiliary Linear Result

The next lemma states that ifzd(t), zs(t) are dominant and subdominant solutions for (2.2), then there are dominant and subdominant solutions xd*(t), xs*(t) for (2.1) satisfying certain inequalities.Lemma 2.1.
Let (1.5) be valid. Let zd(t), zs(t) be dominant and subdominant solutions for (2.2). Then there are positive solutions xs*(t), xd*(t) of (2.1) on [t0-r,∞) such that: (a)
xs*(t)<zs(t), t∈[t0-r,∞),(b)
zd(t)<xd*(t),  t∈[t0-r,∞),(c)
xd*(t) and xs*(t) are dominant and subdominant solutions for (2.1).Proof.
(a) To prove the part (a), we employ Theorem1.6 with p=n=1; that is, we apply the case (i). Consider (2.1), set F(t,ϕ):=-CA(t)ϕ(-r), ρ*(t):=0, δ*(t):=zs(t), and assume (see the case (i)):
(2.4)0<ϕ(θ)<zs(t+θ),θ∈[-r,0),ϕ(0)=zs(t),t≥t0.
Now we have to verify the inequalities (1.16), that is, in our case:
(2.5)F(t,ϕ)-(δ*)′(t)=-CA(t)ϕ(-r)-(δ*)′(t)=-CA(t)ϕ(-r)-zs′(t)=-CA(t)ϕ(-r)+CB(t)zs(t-r)≥(weuse(1.5))≥-CB(t)ϕ(-r)+CB(t)zs(t-r)>CB(t)[zs(t-r)-zs(t-r)]=0
and F(t,ϕ)>(δ*)′(t) if t∈[t0,∞). Further, we have
(2.6)-F(t,ϕ)+(ρ*)′(t)=CA(t)ϕ(-r)+0=CA(t)ϕ(-r)>0
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 xs*(t) of (2.1) on [t0-r,∞) such that xs*(t)<zs(t) for t∈[t0-r,∞).
(b) To prove the part (b), we consider a solutionx=xd*(t) of the following initial problem:
(2.7)ẋ(t)=-CA(t)x(t-r),t∈[t0-r,∞),(2.8)x(t)=zd(t),t∈[t0-r,t0].
Now, let us define a function
(2.9)W(t,x)=zd(t)-x(t),t∈[t0-r,∞).
We find the sign of the full derivative of W along the trajectories of (2.7) if t∈[t0,t0+r]:
(2.10)dW(t,x)dt|t∈[t0,t0+r]=-CB(t)zd(t-r)+CA(t)x(t-r)=(dueto(2.8))=-CB(t)zd(t-r)+CA(t)zd(t-r)≤[CA(t)-CB(t)]zd(t-r)≤(dueto(1.5))≤0.
It means that function W is nonincreasing and it holds
(2.11)W(t0,x(t0))=zd(t0)-x(t0)=zd(t0)-zd(t0)=0≥W(t0+ɛ,x(t0+ɛ))=zd(t0+ɛ)-x(t0+ɛ),ɛ∈[0,r],
and hence zd(t0+ɛ)≤x(t0+ɛ). 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 pointt=t** such that zd(t**)>x(t**). Then there exists a point t*∈[t0,t**) such that zd(t*)<x(t*), otherwise zd(t)≡x(t) on [t0,t**]. Without loss of generality, we can suppose that x(t)≡zd(t) on [t0,t***] with a t***∈[t0,t*) and x(t)>zd(t) on (t***,t*). Then, there exists a point t⋄∈(t***,t*) such that x(t⋄)=Kzd(t⋄) for a constant K>1 and
(2.12)Kzd(t)>x(t),fort∈[t0,t⋄).
Hence, for a function W*(t,x) defined as W*(t,x):=Kzd(t)-x(t), t∈[t0,t⋄], we get
(2.13)dW*(t,x)dt|t=t⋄=K(-CB(t)zd(t-r))+CA(t)x(t-r)<(dueto(2.12))<K(-CB(t)zd(t-r))+CA(t)Kzd(t-r)=Kzd(t-r)[CA(t)-CB(t)]≤(by(1.5))≤0.
It means that Kzd(t)<x(t) on a right-hand neighborhood of t⋄. This is a contradiction with inequality
(2.14)zd(t)<Kzd(t)<x(t),
hence it is proved that the existence of a solution xd*(t) satisfies zd(t)<xd*(t) on [t0-r,∞).
(c) To prove the part (c), we considerlimt→∞xs*(t)/xd*(t). Due to (a) and (b), we get
(2.15)0≤lim⁡t→∞xs*(t)xd*(t)≤lim⁡t→∞zs(t)zd(t)=0,
and xd*(t) and xs*(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,∞)×Cr1. In the following, we will use our main supposition (1.4); that is, we assume that for (t,ϕ)∈Ω inequalities,(2.16)CA(t)ϕ(-r)≤f(t,ϕ)≤CB(t)ϕ(-r)
hold, where ϕ is supposed to be positive.Theorem 2.2.
Letf: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
(2.17)x(t)<y(t)<z(t)
for t∈[t0-r,∞).Proof.
To prove this theorem, we employ Theorem1.6 with p=n=1; that is, we apply the case (i). Set F(t,yt):=-f(t,yt), ρ*(t):=x(t), δ*(t):=z(t); hence, the set ω* will be defined as
(2.18)ω*:={(t,y):t∈[t0-r,∞),x(t)<y(t)<z(t)}.
Now, we have to verify the inequalities (1.16). In our case
(2.19)F(t,ϕ)-(δ*)′(t)=-f(t,ϕ)-(δ*)′(t)=-f(t,ϕ)-z′(t)=-f(t,ϕ)+CB(t)z(t-r)≥(weuse(2.16))≥-CB(t)ϕ(-r)+CB(t)z(t-r)>(weuse(2.18):ϕ(-r)<z(t-r))>CB(t)[z(t-r)-z(t-r)]=0,-F(t,ϕ)+(ρ*)′(t)=f(t,ϕ)+(ρ*)′(t)=f(t,ϕ)+x′(t)=f(t,ϕ)-CA(t)x(t-r)≥(weuse(2.16))≥CA(t)ϕ(-r)-CA(t)x(t-r)>(weuse(2.18):ϕ(-r)>x(t-r))>CA(t)[x(t-r)-x(t-r)]=0.
Therefore,
(2.20)F(t,ϕ)-(δ*)′(t)>0,-F(t,ϕ)+(ρ*)′(t)>0.
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 Lemma2.1, xs(t) be the subdominant solution of (2.1), and let zs(t) be the subdominant solution of (2.2), that is, xs(t)<zs(t) on [t0-r,∞). Then, there exists an uncountable set 𝒴s of positive solutions of (1.3) on [t0-r,∞) such that each solution ys∈𝒴s satisfies
(2.21)xs(t)<ys(t)<zs(t).If inequality (1.6) holds, then dominant solutions xd(t) of (2.1) and zd(t) of (2.2) have finite positive limits (2.22)Cx:=lim⁡t→∞xd(t),Cx>0,Cz:=lim⁡t→∞zd(t),Cz>0.
This is a simple consequence of positivity of solutions xd(t), zd(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 xd(t), zd(t) of both equations such that zd(t)<xd(t) on [t0-r,∞). In the following lemma, we without loss of generality suppose that xd(t) and zd(t) are such solutions and their initial functions are nonincreasing on initial interval [t0-r,t0]. We will need constants M and L satisfying (2.23)M>M*:=xd(t0-r)Cz,L>L*:=Mzd(t0-r)Cx.Lemma 2.4.
Letf: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 xd(t), t∈[t0-r,∞) be a dominant solution of (2.1), nonincreasing on [t0-r,t0], and let zd(t), t∈[t0-r,∞) be a dominant solution of (2.2), nonincreasing on [t0-r,t0], such that zd(t)<xd(t), t∈[t0-r,∞). Then there exists another dominant solution zd*(t) of (2.2) and a positive solution y=yd(t) of (1.3) on [t0-r,∞) such that it holds that
(2.24)xd(t)<yd(t)<zd*(t)
for t∈[t0-r,∞) and zd*(t)=Mzd(t).Proof.
Both dominant solutionsxd(t) and zd(t), of (2.1) and (2.2), respectively, have nonzero positive limits Cx and Cz. 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
(2.25)zd*(t0-r)=Mzd(t0-r)≥Mzd(t)=zd*(t)>MCz>xd(t0-r)≥xd(t),
where t∈[t0-r,∞).
Now, we define the setω* in the same way as (2.18) in the proof of Theorem 2.2, but with xd(t) instead of x(t) and with zd*(t) instead of z(t), that is,
(2.26)ω*:={(t,y):t∈[t0-r,∞),xd(t)<y(t)<zd*(t)}.
According to the Theorem 2.2 (with xd(t) instead of x(t) and with zd*(t) instead of z(t)), it is visible that there exists a positive solution y=yd(t) of (1.3) satisfying
(2.27)xd(t)<yd(t)<zd*(t),
where t∈[t0,∞); that is, inequalities (2.24) hold.Theorem 2.5.
Let all suppositions of Lemma2.4 be valid, and let yd(t) be a solution of (1.3) satisfying inequalities (2.24). Then, there exists a positive solution xd**(t) of (2.1) on [t0-r,∞) satisfying
(2.28)zd(t)<yd(t)<xd**(t),
where xd**(t)=Lxd(t) and t∈[t0-r,∞).Proof.
Multiplying solutionxd(t) by the constant L, we have
(2.29)Lxd(t)>LCx>Mzd(t0-r).
Using (2.29) and (2.24), we get
(2.30)xd**(t)=Lxd(t)>Mzd(t0-r)=zd*(t0-r)>zd*(t)>yd(t)>xd(t)>zd(t),
where t∈[t0-r,∞). Hence, there exists a solution yd(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 Theorem2.5, we can define a class of positive solutions 𝒴d of (1.3) such that every solution yd∈𝒴d is defined on [t0-r,∞) and satisfies(2.31)Czd(t)<yd(t)<Cxd**(t),
where t∈[t0-r,∞) for a positive constant C and, for every positive constant C, there exists a solution yd∈𝒴d satisfying (2.31) on [t0-r,∞).The following theorem states that positive solutionsys(t) and yd(t) of (1.3) have a different order of vanishing.Theorem 2.6.
Let all the assumptions of Corollary2.3 and Theorem 2.5 be met. Then there exist two classes 𝒴s and 𝒴d of positive solutions of (1.3) described by inequalities (2.21) and (2.31). Every two solutions ys, yd, such that ys∈𝒴s and yd∈𝒴d, have asymptotically different behavior, that is,
(2.32)lim⁡t→+∞ys(t)yd(t)=0.Proof.
Let the solutionys(t) be the one specified in Corollary 2.3 and the solution yd(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
(2.33)0≤lim⁡t→+∞ys(t)yd(t)≤lim⁡t→+∞zs(t)Czd(t)=0
in accordance with (1.22), since zs(t) and zd(t) are positive (subdominant and dominant) solutions of linear equation (2.2).Another final statement, being a consequence of Lemma2.1 and Theorems 2.2 and 2.5, is the following.Theorem 2.7.
Letf: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 solutionsxd(t), xs(t) of (2.1),(b)
dominant and subdominant solutionszd(t), zs(t) of (2.2),(c)
solutionsyd(t), ys(t) of (1.3)such that
(2.34)0<xs(t)<ys(t)<zs(t)<zd(t)<yd(t)<xd(t),(2.35)lim⁡t→∞xs(t)xd(t)=lim⁡t→∞zs(t)zd(t)=lim⁡t→∞ys(t)yd(t)=0.Example 2.8.
Let (1.3) be reduced to
(2.36)ẏ(t)=-f(t,yt):=-3texp⁡⁡(-3t+12cos⁡(ty(t-1)))⋅y(t-1),
and let auxiliary linear equations (2.1) and (2.2) be reduced to
(2.37)ẋ(t)=-4texp⁡⁡(2-4t)⋅x(t-1),(2.38)ż(t)=-2texp⁡⁡(1-2t)⋅z(t-1),
that is,
(2.39)CA(t):=4texp⁡(2-4t),CB(t):=2texp⁡(1-2t),r=1.
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 xd(t) of (2.37) and zd(t) of (2.38) such that
(2.40)lim⁡t→∞xd(t)=11,lim⁡t→∞zd(t)=2,zd(t)<xd(t),t∈[t0-1,∞).
Moreover, there exist subdominant solutions xs(t) of (2.37) and zs(t) of (2.38) such that xs(t)<zs(t), t∈[t0-1,∞) which are defined as
(2.41)xs(t):=exp⁡(-2t2),zs(t):=exp⁡(-t2).
By Theorem 2.7, we conclude that there exist solutions ys(t) and yd(t) of (2.36) satisfying inequalities (2.34), and (without loss of generality) inequalities
(2.42)0<xs(t)=exp⁡(-2t2)<ys(t)<zs(t)=exp⁡(-t2)<1≤zd(t)<yd(t)<10≤xd(t)
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 Lemma2.4 and Theorems 2.5–2.7 we used the convergence assumption (1.6) being, without loss of generality, equivalent to
(3.1)∫∞CB(t)dt<∞.
It is an open question whether similar results could be proved if the integral is divergent, that is, if
(3.2)∫∞CB(t)dt=∞.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 yd(t) and ys(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
(3.3)y(t)=Kyd(t)+O(ys(t)),
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]).



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