Initial functions defining dominant positive solutions of a linear differential equation with delay
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Linear differential equation
where is a positive continuous function and delay r is a positive constant, is considered for . It is proved that, under certain assumptions on the function and delay r, a class of positive linear initial functions defines dominant positive solutions with positive limit for .
Keywordslinear differential equation with delay initial function retract method dominant solution asymptotic behavior of solution
with a positive continuous function on the set , , in the non-oscillatory case. The following results on the asymptotic behavior of solutions, needed in the following analysis, are taken from  (see  as well).
Theorem 1 (Theorem 18 in )
where and a coefficient depends on y.
In  it is shown that in representation (3) an arbitrary couple and of two positive solutions of (1) satisfying (2) can be used, i.e., the following theorem holds.
Theorem 2 Assume that and are two positive solutions of (1) on satisfying (2). Then every solution y of (1) on is represented by formula (3), where and a coefficient depends on y.
This is the reason for introducing the following definition.
Definition 1 
Let and be fixed positive solutions of (1) on with property (2). Then is called a pair of dominant and subdominant solutions on .
We note that in the literature one can find numerous criteria of positivity of solutions not only to (1), but more complicated, as well as lots of properties of such solutions and explanation of their importance (see, e.g., books [3, 4, 5, 6, 7, 8, 9], papers [1, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], and the references therein). They are formulated as implicit criteria (simultaneously both sufficient and necessary) or as explicit sufficient criteria. In the paper we employ the following explicit criterion (assumptions are slightly modified to restrict the criterion to the considered case).
Theorem 3 
for , then (1) has a non-oscillatory solution on .
In this paper we prove that every positive linear initial function given on the initial interval and satisfying certain restrictions, defines a positive solution of (1) on . Moreover, we show that this positive solution is a dominant solution and its limit is positive.
The paper is organized as follows. The main result (Theorem 5 below) in Section 3 is proved by the sensitive and flexible retract method. It is shortly described in Section 2. Its applicability is performed via Theorem 4, where an important role is played by a system of initial functions (see Definition 3). Proper choice of such a system of initial functions together with the application of Theorem 4 form the mainstay of the proof of Theorem 5.
2 Preliminaries - Ważewski’s retract principle
If , , and , then, for each , we define by , .
We recall that the functional F is quasi-bounded if F is bounded on every set of the form , where , and L is a closed bounded subset of (see [, p.305]).
In accordance with , a function is said to be a solution of the system (4) on if there are Open image in new window and such that Open image in new window , and satisfies the system (4) for . For given Open image in new window , , we say is a solution of the system (4) through if there is an such that is a solution of the system (4) on and . In view of the above conditions, each element determines a unique solution of the system (4) through on its maximal interval of existence , which depends continuously on initial data . A solution of the system (4) is said to be positive if on for each .
As usual, if a set Open image in new window , then intω and ∂ω denote the interior and the boundary of ω, respectively.
Definition 2 
is called a regular polyfacial set with respect to the system (4), provided it is nonempty and the conditions (α) to (γ) below hold:
(α) For Open image in new window such that for , we have .
The elements Open image in new window in the sequel are assumed to be such that .
If , then for .
If , then for and .
Definition 4 
If are subsets of a topological space and is a continuous mapping from ℬ onto such that for every , then π is said to be a retraction of ℬ onto . When a retraction of ℬ onto exists, is called a retract of ℬ.
The following lemma describes the main result of the paper .
for each .
Remark 1 When Lemma 1 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 the definition of the set in (5) within a half-open interval open at the right. Then the set is not open, but tracing the proof of Lemma 1, it is easy to see that for such sets it remains valid. Such possibility is used below. Similar remark and explanation can be applied to sets of the type Ω, which serve as domains of definitions of functionals on the right-hand sides of equations considered.
In the sequel we employ the result from [, Theorem 1].
- (i)If , and for any , then(6)
- (ii)If , and for any , then
for any . (If , this condition is omitted.)
The original Theorem 1 is in  proved using the retract technique combined with Razumikhin-type ideas known in the theory of stability of retarded functional differential equations.
3 Main result
The first condition (8), in accordance with Theorem 3, guarantees the existence of a positive solution on the interval . Then Theorems 1 and 2 are valid and equation (1) has two different positive solutions (dominant and subdominant) and on the interval . Condition (9), as will be seen from the explanation below, implies that the dominant solution has a positive limit for .
Due to positivity of on , we have .
where is a positive constant depending on the choice of the initial linear function .
if , . Consequently, (i) in Definition 3 holds.
If , then inequalities (12) hold if and (ii) is also valid.
due to positivity of .
Since the statement of the theorem holds for initial functions with , we can also conclude that due to linearity of equation (1), every constant positive initial function defines a positive solution.
We finish the proof with the conclusion that the existence of positive limit is proved. □
Theorem 6 Let all assumptions of Theorem 5 be valid. Then the solution of equation (1) is a positive dominant solution.
which is obviously false. The possibility (15) remains. Then, by Definition 1, a solution of equation (1) is a dominant solution on . □
Remark 2 It is well known [, Theorem 3.3.1] that every continuous initial function φ, defined on the interval , such that , , , defines a positive solution on if the assumptions of Theorem 5 hold. But it is not known if such a solution is dominant or subdominant or if its limit for is positive or equals zero. The statements of Theorems 5 and 6 give new results in this direction since, for a class of linear initial positive functions (not fully covered by known results), positivity of generated solutions (including positivity of their limits) is established together with dominant character of their asymptotical behavior. It is a problem for future investigation to find values of positive limits of solutions considered in the paper (e.g., by methods used in [25, 26, 27]) or to enlarge the presented method to more general classes of equations and initial functions.
This research was supported by the Grant No 1/0090/09 of the Grant Agency of Slovak Republic (VEGA).
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.