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Advances in Difference Equations

, 2012:213 | Cite as

Initial functions defining dominant positive solutions of a linear differential equation with delay

  • Josef DiblíkEmail author
  • Mária Kúdelčíková
Open Access
Research Article
Part of the following topical collections:
  1. Progress in Functional Differential and Difference Equations

Abstract

Linear differential equation

y ˙ ( t ) = c ( t ) y ( t r ) , Open image in new window

where c ( t ) Open image in new window is a positive continuous function and delay r is a positive constant, is considered for t Open image in new window. It is proved that, under certain assumptions on the function c ( t ) Open image in new window and delay r, a class of positive linear initial functions defines dominant positive solutions with positive limit for t Open image in new window.

MSC:34K15, 34K25.

Keywords

linear differential equation with delay initial function retract method dominant solution asymptotic behavior of solution 

1 Introduction

This article is devoted to the problem of the asymptotic behavior of solutions of delayed equations of the type
y ˙ ( t ) = c ( t ) y ( t r ) Open image in new window
(1)

with a positive continuous function c ( t ) Open image in new window on the set [ t 0 r , ) Open image in new window, t 0 R Open image in new window, 0 < r = const Open image in new window in the non-oscillatory case. The following results on the asymptotic behavior of solutions, needed in the following analysis, are taken from [1] (see [2] as well).

Theorem 1 (Theorem 18 in [1])

Let there exist a positive solution y ˜ Open image in new window of (1) on [ t 0 r , ) Open image in new window. Then there are two positive solutions y 1 Open image in new window and y 2 Open image in new window of (1) on [ t 0 r , ) Open image in new window satisfying
lim t y 2 ( t ) y 1 ( t ) = 0 . Open image in new window
(2)
Moreover, every solution y of (1) on [ t 0 r , ) Open image in new window is represented by the formula
y ( t ) = K y 1 ( t ) + O ( y 2 ( t ) ) , Open image in new window
(3)

where t [ t 0 r , ) Open image in new window and a coefficient K R Open image in new window depends on y.

In [2] it is shown that in representation (3) an arbitrary couple y 1 Open image in new window and y 2 Open image in new window of two positive solutions of (1) satisfying (2) can be used, i.e., the following theorem holds.

Theorem 2 Assume that y 1 Open image in new window and y 2 Open image in new window are two positive solutions of (1) on [ t 0 r , ) Open image in new window satisfying (2). Then every solution y of (1) on [ t 0 r , ) Open image in new window is represented by formula (3), where t [ t 0 r , ) Open image in new window and a coefficient K R Open image in new window depends on y.

This is the reason for introducing the following definition.

Definition 1 [2]

Let y 1 Open image in new window and y 2 Open image in new window be fixed positive solutions of (1) on [ t 0 r , ) Open image in new window with property (2). Then ( y 1 , y 2 ) Open image in new window is called a pair of dominant and subdominant solutions on [ t 0 r , ) Open image in new window.

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 [17]

If
t r t c ( s ) d s 1 e Open image in new window

for t [ t 0 , ) Open image in new window, then (1) has a non-oscillatory solution on [ t 0 r , ) Open image in new window.

In this paper we prove that every positive linear initial function given on the initial interval [ t 0 r , t 0 ] Open image in new window and satisfying certain restrictions, defines a positive solution y = y ( t ) Open image in new window of (1) on [ t 0 r , ) Open image in new window. Moreover, we show that this positive solution is a dominant solution and its limit y ( ) Open image in new window 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

Let C ( [ a , b ] , R n ) Open image in new window, where a , b R Open image in new window, a < b Open image in new window, be the Banach space of the continuous mappings from the interval [ a , b ] Open image in new window into R n Open image in new window equipped with the supremum norm
ψ C = sup θ [ a , b ] ψ ( θ ) , ψ C ( [ a , b ] , R n ) , Open image in new window
where Open image in new window is the maximum norm in R n Open image in new window. In the case a = r < 0 Open image in new window and b = 0 Open image in new window, we shall denote this space as C r n Open image in new window, that is,
C r n : = C ( [ r , 0 ] , R n ) . Open image in new window

If σ R Open image in new window, A 0 Open image in new window, and y C ( [ σ r , σ + A ] , R n ) Open image in new window, then, for each t [ σ , σ + A ] Open image in new window, we define y t C r n Open image in new window by y t ( θ ) = y ( t + θ ) Open image in new window, θ [ r , 0 ] Open image in new window.

In this section we present Ważewski’s principle for a system of retarded functional differential equations
y ˙ ( t ) = F ( t , y t ) , Open image in new window
(4)

where Open image in new window is a continuous quasi-bounded map which satisfies a local Lipschitz condition with respect to the second argument and Ω Open image in new window is an open subset in Open image in new window .

The principle below was for the first time introduced by Ważewski [21] for ordinary differential equations and later extended to retarded functional differential equations by Rybakowski [22].

We recall that the functional F is quasi-bounded if F is bounded on every set of the form [ t 1 , t 2 ] × C r , L n Ω Open image in new window, where t 1 < t 2 Open image in new window, C r , L n : = C ( [ r , 0 ] , L ) Open image in new window and L is a closed bounded subset of R n Open image in new window (see [[5], p.305]).

In accordance with [23], a function y ( t ) Open image in new window is said to be a solution of the system (4) on [ σ r , σ + A ) Open image in new window if there are Open image in new window and A > 0 Open image in new window such that Open image in new window , ( t , y t ) Ω Open image in new window and y ( t ) Open image in new window satisfies the system (4) for t [ σ , σ + A ) Open image in new window. For given Open image in new window , φ C r n Open image in new window, we say y ( σ , φ ) Open image in new window is a solution of the system (4) through ( σ , φ ) Ω Open image in new window if there is an A > 0 Open image in new window such that y ( σ , φ ) Open image in new window is a solution of the system (4) on [ σ r , σ + A ) Open image in new window and y σ ( σ , φ ) = φ Open image in new window. In view of the above conditions, each element ( σ , φ ) Ω Open image in new window determines a unique solution y ( σ , φ ) Open image in new window of the system (4) through ( σ , φ ) Ω Open image in new window on its maximal interval of existence I σ , φ = [ σ , a ) Open image in new window, σ < a Open image in new window which depends continuously on initial data [23]. A solution y ( σ , φ ) Open image in new window of the system (4) is said to be positive if y i ( σ , φ ) > 0 Open image in new window on [ σ r , σ ] I σ , φ Open image in new window for each i = 1 , 2 , , n Open image in new window.

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 [22]

Let the continuously differentiable functions l i ( t , y ) Open image in new window, i = 1 , 2 , , p Open image in new window and m j ( t , y ) Open image in new window, j = 1 , 2 , , q Open image in new window, p 2 + q 2 > 0 Open image in new window be defined on some open set Open image in new window . The set
ω = { ( t , y ) ω 0 : l i ( t , y ) < 0 , m j ( t , y ) < 0 , i = 1 , , p , j = 1 , , q } Open image in new window
(5)

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 ( t + θ , π ( θ ) ) ω Open image in new window for θ [ r , 0 ) Open image in new window, we have ( t , π ) Ω Open image in new window.

(β) For all i = 1 , 2 , , p Open image in new window, all ( t , y ) ω Open image in new window for which l i ( t , y ) = 0 Open image in new window, and all π C r n Open image in new window for which π ( 0 ) = y Open image in new window and ( t + θ , π ( θ ) ) ω Open image in new window, θ [ r , 0 ) Open image in new window, it follows that D l i ( t , y ) > 0 Open image in new window, where
D l i ( t , y ) k = 1 n l i ( t , y ) y k f k ( t , π ) + l i ( t , y ) t . Open image in new window
(γ) For all j = 1 , 2 , , q Open image in new window, all ( t , y ) ω Open image in new window for which m j ( t , y ) = 0 Open image in new window, and all π C r n Open image in new window for which π ( 0 ) = y Open image in new window and ( t + θ , π ( θ ) ) ω Open image in new window, θ [ r , 0 ) Open image in new window, it follows that D m j ( t , y ) < 0 Open image in new window, where
D m j ( t , y ) k = 1 n m j ( t , y ) y k f k ( t , π ) + m j ( t , y ) t . Open image in new window

The elements Open image in new window in the sequel are assumed to be such that ( t , π ) Ω Open image in new window.

Definition 3 A system of initial functions p A , ω Open image in new window with respect to the nonempty sets A and ω Open image in new window, where Open image in new window , is defined as a continuous mapping p : A C r n Open image in new window such that (i) and (ii) below hold:
  1. (i)

    If z = ( t , y ) A int ω Open image in new window, then ( t + θ , p ( z ) ( θ ) ) ω Open image in new window for θ [ r , 0 ] Open image in new window.

     
  2. (ii)

    If z = ( t , y ) A ω Open image in new window, then ( t + θ , p ( z ) ( θ ) ) ω Open image in new window for θ [ r , 0 ) Open image in new window and ( t , p ( z ) ( 0 ) ) = z Open image in new window.

     

Definition 4 [24]

If A B Open image in new window are subsets of a topological space and π : B A Open image in new window is a continuous mapping from ℬ onto A Open image in new window such that π ( p ) = p Open image in new window for every p A Open image in new window, then π is said to be a retraction of ℬ onto A Open image in new window. When a retraction of ℬ onto A Open image in new window exists, A Open image in new window is called a retract of ℬ.

The following lemma describes the main result of the paper [22].

Lemma 1 Let ω ω 0 Open image in new window be a regular polyfacial set with respect to the system (4), and let W be defined as follows:
W = { ( t , y ) ω : m j ( t , y ) < 0 , j = 1 , 2 , , q } . Open image in new window
Let Z W ω Open image in new window be a given set such that Z W Open image in new window is a retract of W but not a retract of Z. Then, for each fixed system of initial functions p Z , ω Open image in new window, there is a point z 0 = ( σ 0 , y 0 ) Z ω Open image in new window such that for the corresponding solution y ( σ 0 , p ( z 0 ) ) ( t ) Open image in new window of (4) we have
( t , y ( σ 0 , p ( z 0 ) ) ( t ) ) ω Open image in new window

for each t D σ 0 , p ( z 0 ) Open image in new window.

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 ω Open image in new window in (5) within a half-open interval open at the right. Then the set ω Open image in new window 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, Ω Open image in new window which serve as domains of definitions of functionals on the right-hand sides of equations considered.

Continuously differentiable functions l i ( t , y ) Open image in new window, i = 1 , 2 , , p Open image in new window and m j ( t , y ) Open image in new window, j = 1 , 2 , , q Open image in new window, p 2 + q 2 > 0 Open image in new window mentioned in Definition 2 are often used in the form:
l i ( t , y ) = ( y i ρ i ( t ) ) ( y i δ i ( t ) ) , i = 1 , 2 , , p , m j ( t , y ) = ( y j ρ j ( t ) ) ( y j δ j ( t ) ) , j = p + 1 , p + 2 , , n , m n + 1 ( t , y ) = t + t 0 r , Open image in new window
where ρ, δ are continuous vector functions
ρ = ( ρ 1 , ρ 2 , , ρ n ) , δ = ( δ 1 , δ 2 , , δ n ) : [ t 0 r , ) R n , Open image in new window
with ρ ( t ) δ ( t ) Open image in new window for t [ t 0 r , ) Open image in new window (the symbol ≪ here and below means ρ i ( t ) < δ i ( t ) Open image in new window for all i = 1 , 2 , , n Open image in new window), continuously differentiable on [ t 0 , ) Open image in new window. Hence, the shape of the regular polyfacial set ω Open image in new window from Definition 2 can be simplified to
ω : = { ( t , y ) : t [ t 0 r , ) , ρ ( t ) y δ ( t ) } . Open image in new window

In the sequel we employ the result from [[11], Theorem 1].

Theorem 4 Let there be a p { 0 , , n } Open image in new window such that:
  1. (i)
    If t t 0 Open image in new window, ϕ C r n Open image in new window and ( t + θ , ϕ ( θ ) ) ω Open image in new window for any θ [ r , 0 ) Open image in new window, then
    ( δ i ) ( t ) < F i ( t , ϕ ) when  ϕ i ( 0 ) = δ i ( t ) , Open image in new window
    (6)
     
( ρ i ) ( t ) > F i ( t , ϕ ) when  ϕ i ( 0 ) = ρ i ( t ) Open image in new window
(7)
for any i = 1 , 2 , , p Open image in new window. (If p = 0 Open image in new window, this condition is omitted.)
  1. (ii)
    If t t 0 Open image in new window, ϕ C r n Open image in new window and ( t + θ , ϕ ( θ ) ) ω Open image in new window for any θ [ r , 0 ) Open image in new window, then
    ( ρ i ) ( t ) < F i ( t , ϕ ) when  ϕ i ( 0 ) = ρ i ( t ) , ( δ i ) ( t ) > F i ( t , ϕ ) when  ϕ i ( 0 ) = δ i ( t ) Open image in new window
     

for any i = p + 1 , p + 2 , , n Open image in new window. (If p = n Open image in new window, this condition is omitted.)

Then, for each fixed system of initial functions p Z , ω Open image in new window, where the set Z is defined as
Z = { ( t 0 , y ) , y [ ρ ( t 0 ) , δ ( t 0 ) ] } , Open image in new window
there is a point z 0 = ( σ 0 , y 0 ) Z ω Open image in new window such that for the corresponding solution y ( σ 0 , p ( z 0 ) ) ( t ) Open image in new window of (4) we have
( t , y ( σ 0 , p ( z 0 ) ) ( t ) ) ω Open image in new window
for each t D σ 0 , p ( z 0 ) Open image in new window, i.e., then there exists an uncountable set Y Open image in new window of solutions of (4) on [ t 0 r , ) Open image in new window such that each y Y Open image in new window satisfies
ρ ( t ) y ( t ) δ ( t ) , t [ t 0 r , ) . Open image in new window

The original Theorem 1 is in [11] proved using the retract technique combined with Razumikhin-type ideas known in the theory of stability of retarded functional differential equations.

3 Main result

In this section we consider scalar differential equation (1), where r > 0 Open image in new window and c : [ t 0 r , ) R + = ( 0 , ) Open image in new window is a continuous function satisfying
t r t c ( s ) d s 1 e Open image in new window
(8)
for t [ t 0 r , ) Open image in new window and
c ( s ) d s < . Open image in new window
(9)

The first condition (8), in accordance with Theorem 3, guarantees the existence of a positive solution y = y ( t ) Open image in new window on the interval [ t 0 r , ) Open image in new window. Then Theorems 1 and 2 are valid and equation (1) has two different positive solutions (dominant and subdominant) y = y 1 ( t ) Open image in new window and y = y 2 ( t ) Open image in new window on the interval [ t 0 r , ) Open image in new window. Condition (9), as will be seen from the explanation below, implies that the dominant solution has a positive limit for t Open image in new window.

We set
C : = exp ( e t 0 r c ( t ) d t ) > 0 , Open image in new window
where the constant C is well defined due to (9), and
φ ( t ) : = exp ( e t 0 r t c ( s ) d s ) C > 0 , t [ t 0 r , ) . Open image in new window
Obviously, φ ( ) = 0 Open image in new window. Denote
m = min [ t 0 r , t 0 ] { | φ ( t ) | } = min [ t 0 r , t 0 ] { e c ( t ) exp ( e t 0 r t c ( s ) d s ) } . Open image in new window

Due to positivity of c ( t ) Open image in new window on [ t 0 r , t 0 ] Open image in new window, we have m > 0 Open image in new window.

Let φ K , μ C r 1 Open image in new window be a linear initial function defined on the interval [ t 0 r , t 0 ] Open image in new window as
φ K , μ ( t 0 + θ ) : = K + μ θ , θ [ r , 0 ] , Open image in new window
(10)
where K , μ R Open image in new window and | μ | m Open image in new window. The following theorem gives sufficient conditions for the property
y ( t 0 , φ K , μ ) ( t ) > 0 , t [ t 0 r , ) Open image in new window
together with
lim t + y ( t 0 , φ K , μ ) ( t ) = K ( φ K , μ ) , Open image in new window

where K ( φ K , μ ) Open image in new window is a positive constant depending on the choice of the initial linear function φ K , μ Open image in new window.

Theorem 5 Let inequalities (8), (9) be valid, a constant C 3 / 4 Open image in new window and φ K , μ C r 1 Open image in new window be defined by (10). Then the solution y ( t 0 , φ K , μ ) ( t ) Open image in new window, where φ K , μ Open image in new window is defined by (10), K , μ R Open image in new window and | μ | m Open image in new window, is positive including the value y ( t 0 , φ K , μ ) ( ) Open image in new window, i.e.
y ( t 0 , φ K , μ ) ( t ) > 0 , t [ t 0 r , ) Open image in new window
and
lim t + y ( t 0 , φ K , μ ) ( t ) = K ( φ K , μ ) > 0 . Open image in new window
Proof We will employ Theorem 4 with p = n = 1 Open image in new window, i.e., the case (i) only. Set
F ( t , ϕ ) : = c ( t ) ϕ ( r ) , ρ ( t ) : = φ 1 ( t ) , δ ( t ) : = φ 2 ( t ) , Open image in new window
where functions φ j : [ t 0 r , ) R Open image in new window are defined as
φ 2 ( t ) : = C + φ ( t ) , φ 1 ( t ) : = C φ ( t ) . Open image in new window
We have
lim t φ i ( t ) = C , i = 1 , 2 Open image in new window
(11)
and since C 3 / 4 > 1 / 2 Open image in new window (i.e., C > φ ( t 0 r ) Open image in new window): φ 2 ( t ) > φ 1 ( t ) > 0 Open image in new window on [ t 0 r , ) Open image in new window. Now, we define
ω : = { ( t , y ) : t [ t 0 r , ) , φ 1 ( t ) < y < φ 2 ( t ) } Open image in new window
and
Z : = { ( t 0 , y ) , y [ φ 1 ( t 0 ) , φ 2 ( t 0 ) ] } . Open image in new window
We verify inequality (6). For t t 0 Open image in new window, ϕ C r 1 Open image in new window, and ( t + θ , ϕ ( θ ) ) ω Open image in new window, θ [ r , 0 ) Open image in new window, with ϕ ( 0 ) = δ ( t ) = φ 2 ( t ) = C + φ ( t ) Open image in new window, i.e., for
φ 1 ( t + θ ) < ϕ ( θ ) < φ 2 ( t + θ ) , θ [ r , 0 ) , ϕ ( 0 ) = φ 2 ( t ) = C + φ ( t ) , Open image in new window
we have
F ( t , ϕ ) δ ( t ) = c ( t ) ϕ ( r ) φ 2 ( t ) = c ( t ) ϕ ( r ) + e c ( t ) exp ( e t 0 r t c ( s ) d s ) > c ( t ) φ 2 ( t r ) + e c ( t ) exp ( e t 0 r t c ( s ) d s ) = c ( t ) [ C + φ ( t r ) ] + e c ( t ) exp ( e t 0 r t c ( s ) d s ) = c ( t ) exp ( e t 0 r t r c ( s ) d s ) + e c ( t ) exp ( e t 0 r t c ( s ) d s ) = c ( t ) exp ( e t 0 r t c ( s ) d s ) [ e exp ( e t r t c ( s ) d s ) ] [ we use (8) ] c ( t ) exp ( e t 0 r t c ( s ) d s ) [ e exp ( e 1 e ) ] = 0 . Open image in new window
Therefore, F ( t , ϕ ) > δ ( t ) Open image in new window and (6) holds. Inequality (7) holds as well because for t t 0 Open image in new window, ϕ C r 1 Open image in new window and ( t + θ , ϕ ( θ ) ) ω Open image in new window, θ [ r , 0 ) Open image in new window, with ϕ ( 0 ) = ρ ( t ) = φ 1 ( t ) = C φ ( t ) Open image in new window, i.e., for
φ 1 ( t + θ ) < ϕ ( θ ) < φ 2 ( t + θ ) , θ [ r , 0 ) , ϕ ( 0 ) = φ 1 ( t ) = C φ ( t ) Open image in new window
we have
F ( t , ϕ ) ρ ( t ) = c ( t ) ϕ ( r ) φ 1 ( t ) = c ( t ) ϕ ( r ) + [ e c ( t ) exp ( e t 0 r t c ( s ) d s ) ] < c ( t ) φ 1 ( t r ) e c ( t ) exp ( e t 0 r t c ( s ) d s ) = c ( t ) [ C φ ( t r ) ] e c ( t ) exp ( e t 0 r t c ( s ) d s ) = c ( t ) [ 2 C exp ( e t 0 r t r c ( s ) d s ) ] e c ( t ) exp ( e t 0 r t c ( s ) d s ) = c ( t ) [ 2 C e exp ( e t 0 r t c ( s ) d s ) + exp ( e t 0 r t r c ( s ) d s ) ] = c ( t ) [ 2 C + exp ( e t 0 r t c ( s ) d s ) ( e + exp ( e t r t c ( s ) d s ) ) ] [ we use (8) ] c ( t ) [ 2 C + exp ( e t 0 r t c ( s ) d s ) ( e + exp ( e 1 e ) ) ] = 2 c ( t ) C < 0 . Open image in new window
Now, we will specify the system of initial functions p Z , ω Open image in new window mentioned in Theorem 4. For
z = ( t 0 , y ) Z , Open image in new window
( y Open image in new window varies within the interval [ φ 1 ( t 0 ) , φ 2 ( t 0 ) ] Open image in new window), we define
p ( z ) ( θ ) : = y + μ θ , θ [ r , 0 ] , | μ | m , Open image in new window
i.e., every initial function is a linear function described by formula (10). Since φ ( t ) < 0 Open image in new window, t [ t 0 r , ) Open image in new window, for the system of functions p Z , ω Open image in new window, both assumptions (i), (ii) in Definition 3 are valid. Indeed, this property implies
φ 2 ( t ) = φ ( t ) < 0 and φ 1 ( t ) = φ ( t ) > 0 Open image in new window
if t [ t 0 r , ) Open image in new window,
min [ t 0 r , t 0 ] { | φ 2 ( t ) | } = min [ t 0 r , t 0 ] { | φ 1 ( t ) | } = m Open image in new window
and
m φ i ( t ) m , t [ t 0 r , t 0 ] , i = 1 , 2 . Open image in new window
Therefore, every segment
y ( t ) = y + μ t , | μ | m , t [ t 0 r , t 0 ] Open image in new window
satisfies inequalities
φ 2 ( t ) < y ( t ) < φ 1 ( t ) Open image in new window
(12)

if y int Z Open image in new window, t [ t 0 r , t 0 ] Open image in new window. Consequently, (i) in Definition 3 holds.

If y Z Open image in new window, then inequalities (12) hold if t [ t 0 r , t 0 ) Open image in new window and (ii) is also valid.

Theorem 4 is also valid for this system. Consequently, there exists a point
z 0 = ( t 0 , y 0 ) Z ω Open image in new window
such that
( t , y ( t 0 , p ( z 0 ) ) ( t ) ) ω , t [ t 0 r , ) , Open image in new window
i.e.,
φ 1 ( t ) < y ( t 0 , p ( z 0 ) ) ( t ) < φ 2 ( t ) , t [ t 0 r , ) . Open image in new window
(13)
From inequalities (13) we conclude
lim t + y ( t 0 , p ( z 0 ) ) ( t ) = C Open image in new window
because of (11). This solution is positive, i.e.,
y ( t 0 , p ( z 0 ) ) ( t ) > 0 , t [ t 0 r , ) Open image in new window

due to positivity of φ 1 ( t ) Open image in new window.

Since the statement of the theorem holds for initial functions with μ = 0 Open image in new window, we can also conclude that due to linearity of equation (1), every constant positive initial function defines a positive solution.

If the solution y ( t 0 , p ( z 0 ) ) ( t ) Open image in new window does not coincide with the solution y ( t 0 , φ K , μ ) ( t ) Open image in new window, i.e., if y 0 K Open image in new window, then due to linearity, the sum or the difference of y ( t 0 , p ( z 0 ) ) ( t ) Open image in new window and a suitable positive solution generated by a positive constant initial function gives the solution y ( t 0 , φ K , μ ) ( t ) Open image in new window. It is only necessary to show that the solution y ( t 0 , φ K , μ ) ( t ) Open image in new window will be again positive. The condition for positivity is
φ 1 ( t 0 r ) > φ 2 ( t 0 ) φ 1 ( t 0 ) Open image in new window
or, after some computations,
4 C 1 + 2 exp ( e t 0 r t 0 c ( s ) d s ) . Open image in new window
The last inequality holds since
4 C 3 > 1 + 2 exp ( e t 0 r t 0 c ( s ) d s ) . Open image in new window

We finish the proof with the conclusion that the existence of positive limit K ( φ K , μ ) Open image in new window is proved. □

Theorem 6 Let all assumptions of Theorem  5 be valid. Then the solution y ( t 0 , φ K , μ ) Open image in new window of equation (1) is a positive dominant solution.

Proof Every positive solution y = y ( t ) Open image in new window of equation (1) on [ t 0 r , + ) Open image in new window is decreasing and therefore its limit lim t y ( t ) Open image in new window exists and is finite. The value of the limit can be either positive or zero. In the case of solution y ( t 0 , φ K , μ ) Open image in new window of equation (1), we have
lim t y ( t 0 , φ K , μ ) ( t ) = K ( φ K , μ ) > 0 . Open image in new window
By Theorem 1 there must exist another positive solution y = Y ( t ) Open image in new window of equation (1) on [ t 0 r , + ) Open image in new window such that either
lim t y ( t 0 , φ K , μ ) ( t ) Y ( t ) = 0 Open image in new window
(14)
or
lim t Y ( t ) y ( t 0 , φ K , μ ) ( t ) = 0 . Open image in new window
(15)
The first possibility (14) is impossible since in such a case there should exist a positive solution Y ( t ) Open image in new window of equation (1) on [ t 0 r , + ) Open image in new window with the property
lim t Y ( t ) = , Open image in new window

which is obviously false. The possibility (15) remains. Then, by Definition 1, a solution y ( t 0 , φ K , μ ) Open image in new window of equation (1) is a dominant solution on [ t 0 r , + ) Open image in new window. □

Remark 2 It is well known [[8], Theorem 3.3.1] that every continuous initial function φ, defined on the interval [ t 0 r , t 0 ] Open image in new window, such that φ ( t 0 ) > 0 Open image in new window, φ ( t 0 ) φ ( s ) Open image in new window, s [ t 0 r , t 0 ) Open image in new window, defines a positive solution on [ t 0 r , + ) Open image in new window 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 t Open image in new window 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.

The topic considered in this paper is also connected with problems on the existence of bounded solutions. We refer, e.g., to recent papers [28, 29, 30] and to the references therein.

Notes

Acknowledgements

This research was supported by the Grant No 1/0090/09 of the Grant Agency of Slovak Republic (VEGA).

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© Diblík and Kúdelčíková; licensee Springer 2012

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ŽilinaŽilinaSlovak Republic

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